A Clockwork Kidney - University of Melbourne

A C L O C KW O R K K I D N E YRO B E RT M O S SUsing hierarchical dynamical networks to modelemergent dynamics in the kidneySeptember 2008Department of Computer Science and Software EngineeringSchool of Engineering, The University of MelbourneSubmitted in total fulfilment of the requirementsof the degree of Doctor of Philosophy

Robert Moss. A Clockwork Kidney: Using hierarchical dynamicalnetworks to model emergent dynamics in the kidney© September 2008S U P E RV I S O R S:Edmund KazmierczakMichael KirleyPeter HarrisL O C AT I O N:MelbourneT I M E F R A M E:September 2008

The pure and simple truth is rarely pureand never simple.— Oscar WildeSee everything; overlook a great deal;correct a little.— Pope John XXIIIAll models are wrong. Some models are useful.— George E.P. BoxDedicated to my parents, for so many reasons.

simulation of whole-kidney function from the dynamics of individualnephrons is tractable. Furthermore, the work provides a basis forpredicting emergent effects of localised renal disease. With the continueddevelopment of this model, we hope that significant insightwill be gained into the onset, progression and treatment of renaldiseases.vi

D E C L A R AT I O NThis is to certify that(i) the thesis comprises only my original work towards the PhDexcept where indicated in the Preface;(ii) due acknowledgement has been made in the text to all othermaterial used;(iii) the thesis is less than 100,000 words in length, exclusive oftables, maps, bibliographies and appendices.Melbourne, September 2008Robert Moss

It is true that we live in a complex worldand strive to solve inherently complex problems,which often do require complex mechanisms.However, this should not diminish our desirefor elegant solutions, which convinceby their clarity and effectiveness.— Niklaus Wirth, 1984 Turing Award LectureA C K N O W L E D G E M E N T SI thank my supervisors Ed Kazmierczak and Mike Kirley, for theirboundless enthusiasm and support, their ideas and suggestions, andfor being so generous with their time. Making the difficult seem easyand the impossible merely difficult, I learnt everything I know of theworld of research under their tutelage.I owe a massive debt to Peter Harris, for his exceptional generosityand his readiness to share his expertise in a field to which I was anewcomer. My naive questions were always answered quickly andthoroughly, and his guidance was essential in ensuring that my workwas kept relevant to the world of renal modelling.Don Marsh and Randy Thomas have been very supportive of myefforts, and have provided both great assistance and invaluable feedbackon a frequent basis. I sincerely thank them, especially for all oftheir time that was spent patiently correcting my many beginners’mistakes.I discussed many of my ideas and problems with Gavin Baker andTariq Mahmood, receiving countless gems of advice and many helpfulsuggestions. Paul Fraser was always on hand to commiserate andempathise about the rigours of being a postgraduate student, and tocongratulate the hard-won successes.While conducting my research, I was financially supported by anAustralian Postgraduate Award (APA), funded by the Department ofEducation, Science and Training. I was also awarded the StawellScholarship by the School of Engineering to prepare a paper forpublication after the submission of this thesis.Finally, I thank my parents, who have been supportive not justthroughout my candidature, but who have also encouraged me froman early age to develop an interest in research. They are, almostcertainly, the only people who will read the published thesis in itsentirety.ix

C O N T E N T SI I N T RO D U C T I O N 11 I N T RO D U C T I O N 31.1 Aim 41.2 Overview 5II BAC K G RO U N D 72 T H E K I D N E Y 92.1 Physiology of the kidney 92.2 Renal modelling 122.3 Disease in the kidney 142.3.1 Symptoms of renal disease 142.3.2 The economic impact of renal disease 162.4 Modelling renal disease 162.5 Modelling the renal physiology as a network 173 N E T W O R K M O D E L S 193.1 Graph automata models 193.2 Complex network models 213.3 Hierarchy in real-world networks 233.4 Disturbed networks and disease 25III M O D E L S A N D E X P E R I M E N T S 274 M O D E L D E S I G N 294.1 Modelling approach 294.1.1 The effects of a discrete time model 304.1.2 Analysis of discrete network automata 324.2 Model structure 334.2.1 The single-nephron model 334.2.2 The multi-nephron model 344.2.3 The multi-column model 364.3 Network properties 384.4 State space equations and update rules 404.4.1 Choosing an appropriate time-step size 404.4.2 Discrete fluid flow and solute transport 414.4.3 Model input 434.4.4 State variables 434.4.5 State update rules 444.5 Summary 535 M O D E L DY NA M I C S 555.1 The single-nephron model 565.2 Two-nephron systems 625.3 An inhomogeneous 8-nephron system 675.4 A 72-nephron multi-column model 735.5 Summary of experiments 746 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T 796.1 Incorporating impairments into the model 796.1.1 Stochastic impairments 806.1.2 Stochastic spread of impairments 806.1.3 Deterministic impairments 81xi

xiiC O N T E N T S6.2 Choice of impairment 816.2.1 Implementation of the impairment 826.3 Measuring the effects of impairments 826.4 Impairments in an 8-nephron system 836.4.1 A single impaired nephron 846.4.2 Two impaired nephrons 886.4.3 Summary of experiments 926.5 Impairments on a larger scale 936.5.1 Impaired nephron pairs 946.5.2 Impaired columns 996.5.3 Summary of experiments 1066.6 Conclusions about system stability 1077 L A R G E-SCALE S I M U L AT I O N 1097.1 Introduction 1097.2 Performance of the implementation 1107.2.1 The single-threaded algorithm 1107.2.2 Performance analysis 1117.3 Improving performance with concurrency 1147.3.1 The concurrent algorithm 1147.3.2 Communication in the concurrent algorithm 1177.3.3 Performance of the implementation 1197.4 Automatic generation of large models 1227.4.1 Stochastic generation of large models 1227.4.2 Model growth as a rewriting system 123IV S U M M A RY O F R E S E A R C H 1278 D I S C U S S I O N A N D C O N C L U S I O N 1298.1 Contributions 1298.2 Limitations and further work 1308.2.1 Improved understanding of whole-kidney function 1308.2.2 Modelling of renal disease 1318.2.3 Pluggable virtual kidney 1318.2.4 Modelling kidney failure with reliability factors 1328.3 Conclusion 132B I B L I O G R A P H Y 133

L I S T O F F I G U R E SFigure 1 Physiology of the kidney. 9Figure 2 The functional segments of the nephron. 10Figure 3 The inverse-sigmoid signal generated by the TGFmechanism. 11Figure 4 An example of a 1D binary cellular automaton. 19Figure 5 The beehive rule. 20Figure 6 A logic gate implemented with the beehive rule. 21Figure 7 The iterative construction of a hierarchical network.24Figure 8 Choosing an appropriate sampling rate. 31Figure 9 The single-nephron model. 34Figure 10 The single-nephron tubule as a 1D network. 34Figure 11 Solute transport in the loop of Henle. 35Figure 12 The two-nephron network. 35Figure 13 A single column of interstitial fluid. 36Figure 14 The arterial tree for a multi-nephron model. 37Figure 15 The arterial tree for multiple columns of interstitialfluid. 37Figure 16 Diffusion between columns of interstitial fluid. 38Figure 17 Evolution of small-world network properties I. 39Figure 18 Evolution of small-world network properties II. 40Figure 19 Discrete fluid flow along tubule segments. 41Figure 20 Fluid flow into a sub-divided tubule segment. 42Figure 21 Solute transfer in the loop of Henle. 48Figure 22 Oscillations in medullary sodium concentration. 50Figure 23 Regular oscillations in single-nephron glomerularfiltration rate (SNGFR). 57Figure 24 Construction and maintenance of a medullarysodium gradient. 58Figure 25 The single-nephron dynamics in response to decreasedblood sodium concentration (k = 0.3 andP A = 110). 60Figure 26 The change in sodium concentration in the top(outer medulla) and bottom (inner medulla) layersof medullary fluid, in response to decreasedblood sodium concentration. 60Figure 27 The single-nephron dynamics in response to increasedblood sodium concentration (k = 0.3 andP A = 110). 61Figure 28 The change in sodium concentration in the top(outer medulla) and bottom (inner medulla) layersof medullary fluid, in response to increasedblood sodium concentration. 61Figure 29 Mean difference in filtration rates in a two-nephronsystem, over the parameter space (P A , k). 62Figure 30 Lyapunov stability over the parameter space (P A ,k). 63xiii

xivList of FiguresFigure 31 The change in SNGFR in response to changes inP A . 64Figure 32 The effect of varying the vascular coupling betweentwo identical nephrons I. 65Figure 33 The effect of varying the vascular coupling betweentwo identical nephrons II. 65Figure 34 The effect of varying the vascular coupling betweentwo inhomogeneous nephrons. 66Figure 35 The medullary sodium gradient in a two-nephronsystem when blood sodium concentration remainsat 135–145 mmol/L. 68Figure 36 The medullary sodium gradient in a two-nephronsystem when blood sodium concentration progressivelydrops to 110–120 mmol/L. 68Figure 37 The medullary sodium gradient in a two-nephronsystem when blood sodium concentration progressivelyrises to 160–170 mmol/L. 68Figure 38 Structure of the 8-nephron model. 69Figure 39 The medullary salt gradient in an eight-nephronsystem when blood sodium concentration is altered.70Figure 40 The salt concentration in the inner and outermedulla of an eight-nephron system when bloodsodium concentration is altered. 70Figure 41 The whole-system filtration rate (WSFR) of aneight-nephron system when blood sodium concentrationis altered. 70Figure 42 The WSFR of eight-nephron systems with differentlevels of vascular signalling. 71Figure 43 The WSFR power spectral density of eight-nephronsystems with different levels of vascular signalling. 72Figure 44 The high-level structure of the 72-nephron model. 73Figure 45 The WSFR of the 72-nephron system when γ =0. 75Figure 46 The individual whole-column filtration rates (WCFRs)in the 72-nephron system when γ = 0. 75Figure 47 Power spectral densities of the WSFR of the 72-nephron system for different values of γ. 76Figure 48 Power spectral densities of the WCFRs of the 72-nephron system for different values of γ. 77Figure 49 Average WCFRs and WSFRs of the 72-nephron systemfor different values of γ. 77Figure 50 The loops of Henle in the eight-nephron system.Each nephron is identified by a number from 1to 8, and the length of the loops are rated on ascale of 0 to 3. 83Figure 51 Variations in the WSFR of an eight-nephron system,in response to single impaired nephrons ofdifferent loop lengths. 85

List of FiguresxvFigure 52 Comparisons of WSFR between the original (unimpaired)eight-nephron system and the eight-nephronsystem where nephron 7 is impaired and D P =0.5. 86Figure 53 The power spectral densities of SNGFR in theeight-nephron system with a single impaired nephron. 87Figure 54 The difference in power spectral densities ofSNGFR between the original eight-nephron systemand the system with one impaired nephron. 87Figure 55 Variations in the WSFR of an eight-nephron system,in response to pairs of impaired nephronsof different loop lengths. 89Figure 56 Comparisons of WSFR between the original eightnephronsystem and the eight-nephron systemwhere both longest-looped nephrons are impairedand D P = 0.5. 90Figure 57 The power spectral densities of SNGFR in theeight-nephron system where both longest-loopnephrons are impaired. 91Figure 58 The difference in power spectral densities ofSNGFR between the original eight-nephron systemand the system with two impaired nephrons. 91Figure 59 The high-level structure of the 72-nephron model. 93Figure 60 Variations in the WSFR in the 72-nephron systemwhere all nephrons of a specific loop length areimpaired. 95Figure 61 Comparison of the WSFR in the original 72-nephronsystem to that of the 72-nephron system whereall nephrons of a specific loop length are impaired.96Figure 62 A comparison of the power spectral densities ofWCFR between the original 72-nephron systemand the system where all nephrons of a specificloop length are impaired. 97Figure 63 The difference in power spectral densities ofWCFR between the original 72-nephron systemand the system where all nephrons of a specificloop length are impaired. 98Figure 64 Variations in WCFR in the 72-nephron systemwhere all nephrons of loop length 3 are impaired. 99Figure 65 Variations in the WSFR in the 72-nephron simulationswhere all of the nephrons in a particularcolumn (x, y) are impaired. 100Figure 66 Comparison of the WSFR in the original 72-nephronsystem and the system where all of the nephronsin column 9 are impaired. 101Figure 67 A comparison of the spectral densities of WCFRbetween the original 72-nephron system and thesystem where all of the nephrons in column 9are impaired. 102

Figure 68 The difference in power spectral densities ofWCFR between the original 72-nephron systemand the system where all of the nephrons in column9 are impaired. 103Figure 69 The difference in the power spectral density ofcolumn 9 in the original 72-nephron system andthe system where all of the nephrons in column9 are impaired. 104Figure 70 Variations in WCFR in the system where all of thenephrons in column 9 are impaired. 105Figure 71 Comparing 1-column, 2-column and 4-columnsimulations. 112Figure 72 Execution time is O(n 3 ) for n nephrons per column.113Figure 73 The master/slave relationship between columns. 115Figure 74 Communication between the controller and columns. 116Figure 75 Communication between columns when performingthe inter-column diffusion. 117Figure 76 Reordering the communication in Fig. 74, in orderto reduce the delay between time-steps. 117Figure 77 Amdahl’s law. 121Figure 78 The improvement in performance as the simulationis distributed over multiple computers. 121Figure 79 A Lindenmayer system that resembles the arterialtree of the kidney. 125Figure 80 A simple model of compliant vessels. 131L I S T O F TA B L E STable 1 Some of the solutes regulated by the nephron. 10Table 2 Properties of the n-nephron single-column network.39Table 3 Properties of the n-nephron c 2 -column network. 40Table 4 The solutes support by the model. 43Table 5 State variables in the nephron model. 45Table 6 Model parameters. 45Table 7 The behaviour of each tubule segment in thenephron model. 46Table 8 Notation for model equations. 47Table 9 Distal tubule responses to hormone concentrationsin the filtrate. 52Table 10 Steady-state concentration levels. 56Table 11 Systematic decrease blood sodium concentration.60Table 12 Systematic increase in blood sodium concentration.61Table 13 The difference in the sodium gradient at standardand altered blood sodium concentrations. 70xvi

Table 14Table 15Table 16Table 17Increases in spectral density peaks in the SNGFRof nephrons in the system with one impairednephron. 86Increases in spectral density peaks in the SNGFRof nephrons in the system with two impairednephrons. 92Performance comparison of two models of identicalsize. 114Message content in inter-process communication.118A C R O N Y M SAARADHARFBOINCCKDCPUCRFDCTECFEPOESRFGFRGNIgARASRRTSISSNGFRTGFTHPOWCFRWSFRafferent arteriole resistanceantidiuretic hormoneacute renal failureBerkeley Open Infrastructure for Network Computingchronic kidney diseasecentral processing unitchronic renal failuredistal convoluted tubuleextracellular fluiderythropoietinend-stage renal failureglomerular filtration rateglomerular nephritisImmunoglobulin Arenin-angiotensin systemrenal replacement therapysusceptible-infected-susceptiblesingle-nephron glomerular filtration ratetubuloglomerular feedbackthrombopoietinwhole-column filtration ratewhole-system filtration ratexvii

Part II N T R O D U C T I O N

I N T R O D U C T I O N1A complex system is a system comprised of many interconnectedparts, whose behaviours are both variable and dependent on thebehaviour of the other parts [161, 146]. Such systems exhibit emergentbehaviour—behaviour that may be deterministic, but which hasproperties that can only be examined at a higher level than the individualparts [3]. The difficulty in understanding such systems isthat the behaviour of the whole is highly dependent on the individualparts, and each part is highly dependent on other parts [17]. Moresimply, a complex system is holistic—it is more than just the sum ofits parts—and we must understand how the interactions between theparts affect the whole.The kidney is one of the major organs responsible for homeostasis ofthe body. It regulates acid-base balance, extracellular fluid, electrolyteconcentrations, blood volume and blood pressure, as well as excretingmetabolic waste products. The kidney is also a complex system [133,173, 164], consisting of approximately 800,000 to 1,000,000 nephronsin the human [149]. The nephron is the basic filtration unit of thekidney, and is a convoluted tubule that adjusts the solute levels of thebody’s blood plasma. Nephrons are surrounded by interstitial fluid,into which solutes are secreted and reabsorbed, and by a complicatednetwork of capillaries that also exchange solutes with the surroundinginterstitial fluid. While the behaviour of individual nephrons canfluctuate widely and can even behave in a chaotic manner [112],the overall behaviour of the kidney remains stable. In particular,interactions occur between neighbouring nephrons that give rise tosynchronisations in pressure and flow oscillations—indicating thatgroups of nephrons are capable of producing emergent behaviour[166, 81, 112, 145, 167].In fact, the kidney is able to maintain its homeostatic functions overan astonishing range of bodily conditions, such as wide variations inblood pressure, extracellular fluid volume and solute concentrations.The kidney is also resilient to the effects of renal diseases and theloss of renal tissue (e.g. surgical removal)—a human can survive withonly one half of a single kidney. Although some of the regulatorymechanisms in the kidney are well understood, much less is knownabout how the functional stability of the entire kidney arises from theindividual components and their interactions.To the best of our knowledge, there has been relatively little researchinto how the many regulatory mechanisms affect higher-levelbehaviour in the kidney. Researchers have focused on understandingthe chemical processes and fluid mechanics at the cellular level, andmodelling the kidney’s vascular system. Many low-level propertiesof nephron tubule segments have been modelled [189, 177] and anumber of multi-nephron models have been proposed (mostly twonephronmodels [78, 81, 145], although a 22-nephron model alsoexists [112]). However, the standard methods and techniques in thisfield are not amenable to modelling and analysing the kidney as a3

4 I N T RO D U C T I O Ncomplex system [17, 201], due to the coupling between equations, thecomplexity of calculating solutions, and the inability to decouple thestructure and behaviour of the model. This thesis proposes a differentapproach that minimises the coupling between equations, reducesthe computational cost of solving the model equations, and decouplesthe model’s structure from its behaviour.1.1 A I MA knowledge gap exists between the understanding of the cellularprocesses and transporters that give rise to nephron function, andthe understanding of whole-kidney function. There has been littleresearch into how the dynamics of individual nephrons—and thecouplings, interactions and regulator mechanisms that arise betweennephrons—give rise to the dynamics observed at the whole-kidneylevel.The aim of this thesis is to address this gap by modelling the kidneyas a complex system and analysing it accordingly. The ultimate intentof this research is to provide an approach to renal modelling that iscapable of predicting whole-kidney function from the dynamics ofindividual nephrons, and can therefore be of practical use to clinicians.Specifically, this thesis makes the following contributions:• A modelling approach—hierarchical dynamical networks—whichcombines complex networks and graph automata into a singlemodel. This approach explicitly captures the structure and interactionsin multi-nephron systems, and decouples the structureand behaviour of the model. This approach allows emergentdynamics to be easily explored and analysed.• The development of a multi-nephron model that produces validbehaviour and renders the simulation of whole-kidney functioncomputationally tractable. Using this model, the emergent effectsof the couplings and interactions between nephrons canbe investigated.• An investigation into the dynamics of multi-nephron systemsthat focuses on whole-system and hierarchical properties ratherthan the dynamics of individual nephrons. As part of this investigation,the dynamics of a 72-nephron system are analysed—asystem significantly larger than existing multi-nephron models.• A study of whole-system stability in response to localised impairmentsin nephron function. This is the first study of theemergent dynamics of impaired nephron function, and serves asan illustration of how the emergent dynamics produced by renaldiseases may be predicted and analysed. The impaired multinephronsystems are shown to exhibit very stable behaviour,which we contend is a feature of both the model and the kidneyproper.• The computational cost of the model is shown to be low enoughthat the simulation of whole-kidney function is feasible for thefirst time. It is also demonstrated that simulations can be easily

1.2 OV E RV I E W 5distributed across multiple computers, resulting in a significantgain in performance. An implementation of the model that supportsparallel and distributed execution is presented, based onthe Join Calculus [59, 60].• In order to predict whole-kidney function, a whole-kidney modelmust be constructed. This thesis proposes two approaches forautomatically generating such models.1.2 OV E RV I E WRenal physiology is introduced in Chapter 2, followed by a surveyof existing tubule, whole-nephron and multi-nephron models. Thesemodels, which are able to capture precise low-level dynamics ofthe nephron tubule, are shown to be unsuitable for modelling andanalysing the kidney as a complex system. This chapter also presentsa brief discussion of renal disease and a review of renal diseasestudies.Graph automata and complex networks are introduced in Chapter3, and are shown to be better suited to modelling the kidneyas a complex system than the renal models presented in Chapter 2.Existing graph automata and complex network models are reviewed,followed by a discussion of how hierarchical properties of complexsystems can be captured with such models. The chapter concludes byreviewing the application of these models to studying the effects ofdisease.The modelling approach—hierarchical dynamical networks—is presentedin Chapter 4 and contrasted to continuous non-hierarchicalmodels. The structure of the multi-nephron model is presented andcompared to the structure of existing networks, and the update rulesof the model are then given.Chapter 5 presents the experimental design used to validate themodel and to explore the emergent dynamics produced by the model.This is followed by the results of each experiment in turn: the validationof the single-nephron dynamics; the validation of the two-nephrondynamics; a further exploration of the interactions between competingcoupling mechanisms in the two-nephron system; and finally ananalysis of the interactions between competing coupling mechanismsin 8-nephron and 72-nephron systems.A study of whole-system stability in response to localised impairmentsin nephron function is presented in Chapter 6. The chapterbegins by discussing how localised impairments can be incorporatedinto the model, and how the emergent effects of such impairmentscan be studied. A simple impairment is introduced into 8-nephronand 72-nephron systems and the stability of these systems is analysed.The impaired multi-nephron systems are shown to exhibit stablebehaviour, which we contend is a feature of both the model and thekidney proper.The implementation of the model simulation is presented in Chapter7 and the performance of this implementation is analysed, demonstratingthat the simulation of whole-kidney function from the dynamicsof individual nephrons is computationally tractable. It is alsodemonstrated that the simulations can easily be distributed across

6 I N T RO D U C T I O Nmultiple computers—a distributed implementation based on the JoinCalculus [59, 60] is presented and a performance analysis revealsthat the result in a significant gain in simulation performance. awhole-kidney model is required to predict whole-kidney function, andthe chapter concludes by proposing two methods for automaticallygenerating such models.The contributions of this thesis are summarised in Chapter 8. Thelimitations of this work are then discussed, and potential avenuesof further research to build on the work are proposed. Finally, theresearch outcomes are evaluated against the research aim.

Part IIB A C K G R O U N D

T H E K I D N E Y2This chapter begins by introducing the reader to the physiology of thekidney—the structure and function of the kidney is briefly presented—and a review of existing nephron tubule models is presented. This isfollowed by an overview of the major renal diseases, their significance,and their effects on the kidney, and a review of models of kidneydisease. The chapter is concluded with a short description of themodelling approach that will be used to develop the nephron modelpresented in this thesis.2.1 P H Y S I O L O G Y O F T H E K I D N E YThe kidney is one of the major organs involved in whole-body homeostasis.It is responsible for regulating solute concentrations in theblood, the volume, composition and avid-base balance of extra-cellularfluid, and blood pressure in the body. The kidney also plays a partin the body’s endocrine system, secreting hormones such as reninas part of the renin-angiotensin system (RAS), erythropoietin (EPO)to regulate red blood cell production, and thrombopoietin (THPO) toregulate platelet production.The functional unit of the kidney is the nephron, a long, segmentedtubule into which blood plasma is filtered (see Fig. 1a). As the filtrateflows along the nephron tubule, water is reabsorbed into thebloodstream and solutes are both reabsorbed into the bloodstreamand secreted into the tubule (see Table 1). The filtrate flows from thenephron into the collecting ducts (labelled “A” in Fig. 1a), which drainthe fluid into the bladder via the urethra. The human kidney containsaround one million nephrons [149], which are arranged in renal lobes(see Fig. 1b).(a) A single nephron andthe peritubular capillaries.(b) Nephrons (shown as glomeruli connected to arcuatearteries) are grouped in renal lobes.Figure 1: Physiology of the kidney.9

10 T H E K I D N E YSolute Formula Reabsorbed SecretedSodium Na + Yes YesPotassium K + Yes YesProton H + No YesChloride Cl − Yes YesCalcium Ca 2+ Yes NoBicarbonate HCO − 3 Yes YesPhosphate PO 3−4 Yes NoUrea (NH 2 ) 2 CO Yes YesGlucose C 6 H 12 O 6 Yes NoAmino acids various Yes NoTable 1: Some of the solutes regulated by the nephron.1Cortex29738Outer Medulla46Inner Medulla5Figure 2: A diagram showing the functional segments of the nephron.1: the afferent arteriole, which provides the nephron with blood; 2:the glomerulus, a network of capillaries that filters a fraction of theblood plasma into the nephron tubule (the remaining blood flows intothe peritubular capillaries that surround the tubule—not shown here,but shown in Fig. 1a); 3: the proximal tubule; 4: the descending limbof Henle; 5: the thin ascending limb of Henle; 6: the thick ascendinglimb of Henle; 7: the distal tubule; 8: the collecting ducts, whichdrains the filtrate into the bladder; 9: the TGF control mechanism (aninteraction between the macula densa and the filtration process).The structure of a single nephron is shown in Fig. 2. The nephronis surrounded by connective tissue, which holds the nephron tubulesand renal blood vessels, and the interstitial fluid, which can be dividedinto compartments within the cortex, the outer medulla andthe inner medulla. Blood flows through the afferent arteriole (1) andinto the glomerulus (2), where a fraction of the blood plasma is filteredinto the nephron tubule, and the remaining blood flows intothe peritubular capillaries (not shown). The filtrate—the plasma thathas been filtered into the tubule—flows through the proximal tubule(3), the descending and ascending limbs of the loop of Henle (4–6),

2.1 P H Y S I O L O G Y O F T H E K I D N E Y 11the distal tubule (7) and finally the collecting ducts (8). The loopof Henle acts as a counter-current exchange mechanism for NaCl,increasing the NaCl concentration in the interstitial fluid as the loopdescends. In addition to this counter-current mechanism, the productionof lactate in the tubule may contribute significantly to theaccumulation of sodium [76], and solutes such as urea also play arole in the concentrating mechanism [97, 98, 176]. Existing modelshave incorporated gradients—measured between the top and bottomof the loop of Henle—such as 150mmol/L [78], although the gradientcan be as large as 1200mmol/L in human kidneys [169] and over2000mmol/L in other species [158, 76].A major regulation mechanism (9) in the kidney is tubuloglomerularfeedback (TGF), a negative-feedback mechanism that regulates thefiltration rate of an individual nephron. A signal is sent to the afferentarteriole that controls its diameter (and thus the filtration rate),based on the delivery of solutes, mainly sodium chloride, to a regioncalled the macula densa (the end of the ascending limb of Henle,6–7 in Fig. 2). The filtration rate affects distal sodium delivery—highsodium levels indicate that the filtration rate is too high for sodium tobe reabsorbed effectively, while low sodium levels indicate that thefiltration rate is so low that too much sodium is being reabsorbed—and the TGF mechanism keeps the filtration rate within appropriatebounds (see Fig. 3). There is evidence that the TGF mechanism mayalso be mediated through the resistance of the efferent arteriole[151, 68, 23]—the arteriole that carries blood out of the glomerulus—and the permeability of the glomerular capillaries [130, 28, 181].Solute DeliveryFiltration RateFigure 3: The inverse-sigmoid signal generated by the TGF mechanism.Changes in the solute delivery to the macula densa result in aninverse response in the filtration rate. The system typically oscillatesin the highlighted region, about the equilibrium point indicated onthe graph.Specialised cells in the tubule wall generate a signal that controlsthe diameter of the afferent arteriole (which supplies blood to theglomerulus). The signal is not instantaneous, but has a delay of around

12 T H E K I D N E Y3–5 seconds [166, 145, 79]), and there is a further delay for thefiltrate to flow from the glomerulus to the macula densa (around10–15 seconds [166, 78, 79]). This negative-feedback mechanism hasbeen the focus of many multi-nephron models, and is known to havean inverse-sigmoid relationship [160, 166] with sodium delivery [159]at the macula densa (see Fig. 3).Two different types of interactions are known to occur betweennephrons in multi-nephron systems [166, 81, 112]: vascular signaling,where the TGF signal from one nephron also reaches nearbynephrons; and hemodynamic coupling, where the resistance of anafferent arteriole affects the hydrostatic pressure at nearby nephrons.Vascular signaling favours in-phase synchronisation [112], since anincrease or decrease in the TGF signal of one nephron is propagated toneighbouring nephrons. Conversely, hemodynamic coupling favoursanti-phase synchronisation [112], since the contraction of an afferentarteriole reduces the filtration rate of that nephron and increasesthe flow to—and therefore the filtration rate of—the neighbouringnephrons. One model has predicted that vascular signalling overwhelmsthe hemodynamic coupling, leading to in-phase synchronisationsin adjacent nephrons [112].The kidney has a hierarchical physical composition—the nephronconsists of a number of distinct segments; nephrons can be groupedinto interacting neighbourhoods (such as all nephrons joined to asingle cortical radial artery [112]); large numbers of these neighbourhoodsform renal pyramids; and the kidney consists of multiplesuch pyramids (e. g., from 8–18 in humans [55] to a single pyramidin smaller animals such as rodents and monotremes [211]). Thefunctional structure of the kidney is closely related to its physicalstructure—witness how the arrangement of the nephron tubule reflectsthe functional structure of the TGF mechanism—and thereforeexhibits a similar hierarchy. Hierarchy is a property of many complexsystems [164, 150, 47] and it would be desirable to capture this hierarchyexplicitly in our model, as opposed to existing nephron models,which only capture hierarchy implicitly (if at all). Hierarchy in modelsis discussed further in Sect. 3.3.2.2 R E NA L M O D E L L I N GThe study of renal physiology takes three forms: clinical studies thatfocus on diagnosing and treating kidney-related diseases; experimentalstudies that examine the behaviour and function of the kidney(in vivo or in vitro); and theoretical studies that propose models tocapture the behaviours and functions that have been observed experimentally.These fields encompass the study of more than just thebehaviour of the nephron tubule segments, and can be divided intothree general areas: cellular, tubular and vascular physiology.For the purpose of this thesis it is sufficient to concentrate ontubular physiology, since the models presented herein focus on thebehaviour of and interactions between nephrons. To this end, therange of theoretical models that have been proposed are now surveyed,as they not only provide a starting point for the design ofnew models, but also compare the models to the relevant portion

2.2 R E NA L M O D E L L I N G 13of the available experimental data. Several clinical studies are alsoreviewed in Sect. 2.4.The majority of nephron modelling has focused on replicating lowlevelbehaviour in individual tubule segments (e.g., the proximaltubule [190, 193], distal convoluted tubule [192, 192, 38, 37] andcollecting ducts [191, 191]). Experimental data shows that in thenephron, the flow rate and tubule pressure both oscillate, and modelshave shown that such oscillations can arise due to the TGF mechanism[78, 80]. Existing models are able to reproduce a wide range ofother experimental data, but uncertainty still remains about the rolesof transporters along the nephron tubule, due to the limitations ofexisting experimental techniques and the inability of current modelsto capture the essence of all observed phenomena [189].There are some published models that study the behaviour of multiplenephrons, such as their ability to regulate concentration gradientsin the medullary fluid [76, 108], or synchronisation phenomena thatcan arise due to pressure oscillations in blood flow [81, 112]. Thesemodels do not include the entire nephron, although at least onewhole-nephron model exists [170].These tubule models are generally formulated as systems of differentialequations that model the mechanics of fluid flow along tubulesegments and the transport of solutes and water across the tubulewalls. This approach has yielded models that are able to replicatespecific, detailed behaviours.For example, Weinstein [192] models the rat early distal convolutedtubule (DCT) as a tubule lined by DCT cells and intercellular space(both of which admit solute transport). Solute transport is governedby electrochemical potential gradients (via cotransporters) or coupledto metabolic energy (via an ATPase). This model updates previouswork, in drawing on new experimental data to refine the formulation(and inclusion) of several cotransporters, as well as including a morerealistic model of the tubule lining than previous models, allowingcell volume to vary and not requiring the intercellular and peritubularfluids to be at equilibrium. This model provides new observationsregarding cell volume homeostasis and new predictions concerningthe effects of alkalosis on the DCT.In short, this paper presents a model of the rat early DCT that isa significant extension of existing DCT models, by virtue of havingfewer underlying assumptions and more detailed cotransporter formulations.Despite these features, this model is a development ofthe published tubule models, and does not break new ground withrespect to the approach taken.Unlike discrete equations, which explicitly calculate the currentstate of a system based on the previous state of the system, systemsof continuous equations only place constraints upon the system. Solvingsuch constraints can be computationally very expensive (oftenrequiring numerical solving methods which discretise the equations).A unique solution may not exist, in which case linear combination ofthe solutions must be considered.The accuracy of these kinds of models is not without drawbacks, asthe models can be computationally very expensive. Due to the couplednature of the equations, the addition of another variable or a change

14 T H E K I D N E Yto the equation governing an existing variable can have significanteffects on the solutions of other equations in the model. Another drawbackis that the tubule models cannot necessarily be linked togetherto form models of an entire nephron tubule, as they may have differentunderlying assumptions or incompatible formulations (althoughsome tubule models have been successfully combined). Our approachovercomes these drawbacks by using difference equations, whichdecouple the continuous equations and provide explicit solutions foreach variable.2.3 D I S E A S E I N T H E K I D N E YRenal diseases can prevent the kidney from functioning correctly, resultingin renal failure. Renal failure is divided into acute and chronicforms. acute renal failure (ARF) is a rapid loss of renal function dueto kidney damage, which may be reversible if it is treated promptly.chronic kidney disease (CKD) is a progressive loss of renal functionover a period of months and years, which is divided into five stages,based on the decrease in the kidney filtration rate. The final stageis referred to as end-stage renal failure (ESRF), and requires someform of renal replacement therapy (RRT)—either a kidney transplantor dialysis. ARF may also result in ESRF. Regardless of the cause, ESRFmanifests as an abnormally low glomerular filtration rate (GFR) suchthat the kidney is unable to perform its function.This review focuses on diseases that cause CKD. The most commoncauses of CKD and ESRF in western countries are:• Diabetic nephropathy—thickening of the glomeruli.• Hypertension—chronically elevated blood pressure.• Glomerulonephritis—inflammation of the glomeruli.Together, these three conditions account for around 75% of all adultcases of chronic renal failure (CRF) [41]. In particular, diabetic nephropathyis the most common cause of ESRF in the USA [178, 53].2.3.1 Symptoms of renal diseaseWith the aim of studying how the onset of renal disease in individualnephrons can affect the whole-system dynamics of multi-nephronsystems, it is the symptoms of the renal diseases that are of greatestinterest, as the symptoms indicate the changes in dynamics causedby the diseases. The most common symptoms of renal disease arenow introduced.ProteinuriaProteinuria is the presence of abnormally large amounts of proteinin the urine—which can cause the urine to become foamy—and isa sign of potential renal damage. It was understood that a chargebarrier in the glomerulus prevents protein from entering the nephrontubule [180, 185, 39, 135] and that the presence of protein in theurine is due to deterioration in this charge barrier, allowing proteins

2.3 D I S E A S E I N T H E K I D N E Y 15such as albumin to enter the nephron tubule and ultimately the urine[69, 25]. However, the influence of this electrostatic interaction hasbeen questioned [45, 138, 73, 155] and there is evidence that proteinenters the nephron tubule under normal conditions, and that thepresence of protein in the urine is due to the failure of a reabsorptionmechanism in the proximal tubule cells [116, 32, 21, 44]. Proteinuriawas first described by William Charles Wells [63, 33], who suggestedthat the serum in the urine was derived from the blood [194] and gavethe first definition of the limits of normal albuminuria [195] (similarfigures are still used today [33]).HematuriaHematuria is the presence of red blood cells in the urine. Aside fromrenal diseases, hematuria can indicate the presence of kidney stonesor a tumour. Hematuria is a symptom of glomerulonephritic diseasessuch as Immunoglobulin A (IgA) disease (also known as Berger’snephropathy), which is the most common type of glomerulonephritisin adults world-wide [65, 20, 128].HypoalbuminemiaHypoalbuminemia is a condition where the level of albumin in theblood is abnormally low. It can be caused as a result of proteinuria,and may also indicate liver failure, cirrhosis or chronic hepatitis. Asalbumin is the major protein in the human body (contributing around60% of the total plasma protein by mass), a significant reduction inalbumin levels significantly lowers the oncotic pressure of the blood.Such a reduction leads to excessive amounts of water diffusing fromthe blood into the extra-cellular fluid, causing general swelling of thebody (edema).HyperlipidemiaHyperlipidemia is the presence of abnormally large amounts of lipidsin the blood, which can arise when the liver responds to albumin lossby secreting more protein into the bloodstream.HypertensionHypertension is a condition where the blood pressure is chronicallyelevated, and is one of the risk factors for cardiovascular diseases, aswell as a major cause of CRF. One of the major causes of hypertensionis a diet that is high in sodium [156, 92].Nephritic SyndromeNephritic syndrome is a collection of signs associated with glomerulardisorders, characterised by proteinuria, hematuria and hypertension.It is not a specific diagnosis and so the underlying causes can vary, butthe common feature is an inflammation of the glomeruli, leading tosalt and water retention and ultimately a reduction in kidney function.

16 T H E K I D N E YNephrotic SyndromeNephrotic syndrome is a disorder where large amounts of proteinleak from the blood into the urine. It is characterised by proteinuria,hypoalbuminemia, hyperlipidemia and edema (and occasionallyhypertension). Nephrotic syndrome has many specific causes (suchas diabetes and glomerulonephritis), but the common mechanismis inflammation causing damage to the glomeruli, allowing proteinssuch as albumin to pass through the kidneys and into the urine.2.3.2 The economic impact of renal diseaseIncidence of renal disease is becoming more prevalent in both westernand developing nations, placing a rapidly growing burden onglobal health care [36, 13, 183, 184]. Long-term kidney disease alsoincreases the risk of developing further complications such as cardiovasculardiseases, respiratory infections, bone and muscle problems,and anaemia [13]. Once ESRF occurs, the only option is to pursuepalliative treatments known as RRT. Dialysis is the most common formof treatment—approximately 77% of global RRT patients undergo dialysis,with the remaining 23% having functional transplants [125]—butit is also the most costly form of treatment [62] and is not withoutdrawbacks [154].The burden of premature death from diabetes is similar to that ofHIV/AIDS, with one in 20 deaths attributable to diabetes, and thenumber of people with diabetes is expected to more than double by2030 [136].These predictions of increased incidence of renal disease and theassociated increase in health-care costs highlight the importanceof improving our understanding of how renal diseases affect renalfunction. Studying the effects of renal diseases through the use ofmodels may lead to improvements in the diagnosis and treatment ofrenal diseases, helping to reduce the financial and social impacts ofthese diseases.2.4 M O D E L L I N G R E NA L D I S E A S EThe causes and mechanisms of renal disease have been investigated atthe cellular level [21, 32] and many clinical experiments [10, 86, 114,58, 91], laboratory experiments [95, 24, 84, 144] and case studies[172, 14, 148] have been performed. Rashidi and Khodarahmi presenta simple model of nephron loss in response to CRF [149], but there arefew other models that attempt to capture the effects of renal diseaseon the kidney or on multi-nephron systems. In particular, it does notappear that any existing tubule or nephron models have attempted tomodel the impact of renal disease on the nephron in the context ofmulti-nephron systems, precluding the study of emergent dynamicsproduced in the presence of, or in response to, renal disease.Thomson et al. present “the tubular hypothesis of glomerular filtration”[178], which uses a control theory model to explain severalnuances of kidney function in early diabetes. Early diabetes is knownto have strange effects on GFR, including diabetic hyperfiltration, a

2.5 M O D E L L I N G T H E R E NA L P H Y S I O L O G Y A S A N E T W O R K 17paradoxical effect of dietary sodium, and the renal response to dietaryprotein and amino acid infusion. The tubular hypothesis presumesthat these effects stem from primary effects on the proximal tubuleand loop of Henle, which impact the single-nephron glomerular filtrationrate (SNGFR) by feedback through the macula densa and theTGF mechanism. Such primary effects are categorised as “tubular”in comparison to all other possible effectors, which are classified as“vascular”. The difference between tubular and vascular effectorsis that vascular effectors have the same effect on GFR and sodiumdelivery to the macula densa—if the GFR is increased, so is the sodiumdelivery—while tubular effectors have contrary effects—if the GFR isincreased, the sodium delivery is decreased.The published data on the content of the early distal tubule indiabetes consistently lists sodium values substantially below normal,and experimental data shows that diabetic nephrons respond to thiswith hyperfiltration [72]. However, there is also a change in the TGFresponse which indicates that a vascular effect may have almost asmuch control over GFR.The diabetic kidney also exhibits a “salt paradox” where GFR variesinversely with dietary sodium intake, resulting in a lower GFR whensodium intake is increased, rather than the normal response of anelevated GFR. This cannot be explained by any vascular effectors, butcan be explained by the tubular hypothesis as long as the proximaltubule is the dominant mechanism linking GFR to extracellular fluid(ECF).Glycine infusion normally causes GFR to increase without alteringsodium levels at the macula densa, indicating that both vascular andtubular effects contribute to the increase in GFR, while in the diabetickidney the vascular response is minimal. Protein feeding alsoincreases GFR, by increasing tubular reabsorption to levels that decreasethe sodium delivery to the macula densa, despite the elevatedGFR. This tubular effect remains present in the diabetic kidney.In summary, Thomson et al. demonstrate that this tubular hypothesisand the associated control theory model can explain severalfactors that influence the behaviour of the kidney in early diabetes.Their paper does not focus on specific cellular mechanisms, preferringto treat the kidney as an integrated system of parts—an approachthat is reminiscent of our own. The major difference is that this modelstarts from experimental observations of the whole kidney to suggestthe effects that early diabetes has on the nephron tubule, while thefocus of our disease modelling is to start from known effects in thenephron tubule and to use our models to determine the resultingbehaviour at a higher level, such as the whole kidney.2.5 M O D E L L I N G T H E R E NA L P H Y S I O L O G Y A S A N E T W O R KAs will be discussed in the following chapter (Chapter 3), networkmodels are well suited to capturing both implicit and explicit hierarchiespresent in the system being modelled, and are also well suitedto studying the changes in system dynamics that arise in responseto alterations in the model structure and low-level behaviours. Inparticular, network models provide an advantage when exploring the

18 T H E K I D N E Yeffects of different couplings and interactions between parts of themodel, as these mechanisms are explicitly captured by the modelstructure.Furthermore, due to the use of discrete time equations, such modelsare able to scale to large systems without incurring the increase incomputational cost that is typical of continuous systems. This point isdiscussed further in Sect. 4.1.1 and the model presented in this thesisis demonstrated to scale to extremely large models in Chapter 7.

N E T W O R K M O D E L S3As mentioned in the introduction, there are two common classes ofnetwork model—networks with regular, fixed topologies and nodeswhose states evolve due to homogeneous update rules [201]; andnetworks with irregular topologies, where nodes have constant stateand dynamics are realised as changes in the network topology [139].However, the kidney has both an irregular static functional topology(closely tied to its physical structure) and localised states that evolveinhomogeneously over time. The model presented in this thesis willtherefore have features in common with both networks, and so bothclasses of network models are examined in this chapter. Firstly, definitionsof cellular and graph automata are presented, followed by areview of existing automata models. This is followed by an introductionto common network properties and a review of existing complexnetworks models.3.1 G R A P H AU T O M ATA M O D E L SA cellular automata A = {S, C, N, f} consists of a finite state spaceS, a configuration of the cells C ∈ S Z for a finite of infinite discreteregular grid of cells Z, a neighbourhood function N : Z → S n andan update rule f : S n → S. Time is discrete and at any time t, theconfiguration of an automaton is C(t) ∈ S Z . For each cell c ∈ Z, thestate at time t + 1 is calculated from the state of the neighbourhoodN(c) at time t, as shown in Eq. 3.1.c(t + 1) = f( N(c) ) (3.1)(a) An update rule—cells change statewhen their neighbours’ states differ.(b) The update rule appliedto an automaton.Figure 4: An example of a 1D binary cellular automaton that has aring topology—the cells at both ends are connected together.An example of a 1D automaton with a binary state space is shownin Fig. 4. The update rule f for this automaton (Fig. 4a) changes thestate of a cell when the states of the two neighbouring cells differ. Anapplication of this rule to a configuration is shown in Fig. 4b.Graph (or network) automata are an extension of cellular automata,where the topology of the cells Z is generalised to a network, ratherthan being restricted to a regular grid.19

20 N E T W O R K M O D E L SA graph (or network) G = {V, E} consists of a set of nodes (V),and edges that connect pairs of nodes (E ⊆ V × V). A graph is bydefault undirected, meaning that the edges (v 1 , v 2 ) and (v 2 , v 1 ) areequivalent, and for any (v 1 , v 2 ) ∈ E it is also true that (v 2 , v 1 ) ∈ E.Graphs may also be directed, in which case the edges (v 1 , v 2 ) and(v 2 , v 1 ) are distinct, and an edge (v 1 , v 2 ) is said to be directed fromv 1 to v 2 . The neighbourhood of a cell (node) c in a graph automata isthe set of cells (nodes) that are directly connected to c in the graph.Graph automata have been applied to the study of biological phenomena,such as computational operations in reaction-diffusion systems[2] and genetic regulatory networks [203].Adamatzky et al. [2] show how basic computational operationscan be implemented in a three-state hexagonal automaton. This automatonmodels a reaction-diffusion system using the “beehive rule”[206], which is presented in Fig. 5. The beehive rule exhibits gliderdynamics—that is, starting from a random configuration, it producesregular patterns that move across the cells whilst retaining theirshape (i. e., gliders). Adamatzky et al. show that by interpreting the directionin which the gliders travel as truth values, the beehive rule iscapable of producing logic gates (as depicted in Fig. 6), and performingactions such as multiplication. The result is that the automatonis logically universal—any sequential machine can be constructed inthe system.S2 Neighbours6543210S0S2 S0S0 S0 S2S0 S2 S2 S0S0 S0 S2 S2 S0S0 S2 S2 S2 S1 S1S0 S1 S2 S1 S2 S0 S00 1 2 3 4 5 6S1 Neighbours(a) The beehive update rule in matrix formS0 S0S1 S1 S2S1 S2Beehive ruleS2(b) An example of the beehive rule, for a cell that hastwo S1 neighbours and two S2 neighbours.Figure 5: The beehive rule operates on a hexagonal grid whose cellshave a state-space S = {S0, S1, S2} [2].While this is a significant achievement—it could potentially be a theoreticalprototype of physically-realised reaction-diffusion processors—the analysis and interpretation of the automaton’s behaviour is toodeeply rooted in visual observations of patterns in the cells. While

3.2 C O M P L E X N E T W O R K M O D E L S 21(a) t = 1 (b) t = 2 (c) t = 3 (d) t = 4¬x∧yxx∧¬yy(e) The logic gateFigure 6: A logic gate implemented with the beehive rule [2].such an analysis might be automated once the behaviour and theirinterpretations are known, it can not be readily applied to findingand analysing patterns of behaviour in larger, more complex models.Indeed, the state-space of a single nephron is so complicated that isnot obvious how to visualise more than a few rudimentary properties(such as changes in concentration of a single solute), which precludesthe possibility of discerning behavioural patterns visually.This failure of analysis techniques to scale to large models is notisolated to this single paper, rather it is typical of the graph automatamodels. Indeed, Wolfram acknowledges both the necessity of visualanalysis and the difficulties it entails [201, 3§12 pp. 111–3], remarkingthat as a result he ignored the possibility of cellular automataproducing complex behaviour for several years.One other shortcoming of standard graph automata is that all nodesare homogeneous, sharing the same state-space and update rules. Asdiscussed in Sect. 2.1, nephrons and nephron tubule segments arenot homogeneous, and so a graph automata model of a nephron mustallow for nodes with different state-spaces and update rules.Although nephrons are inhomogeneous, they do share a commonbasic structure and set of transport mechanisms. Each tubule segmentis typically lined by the same types of cells regardless of whichnephron it belongs to—although some tubules may not contain allpossible cell types—and so all instances of each segment that sharethe same cell types can be modelled homogeneously. By taking thisapproach, one does not obtain a graph automata with homogeneousnodes and update rules, but rather an automata with classes of nodesand update rules.However, when attention is turned to modelling large groups ofnephrons, it becomes clear that the analysis will necessarily be largelystatistical in nature, and that the models must have more complex,less regular topologies than the typical graph automata. Such networksare known as Complex Networks.3.2 C O M P L E X N E T W O R K M O D E L SLarge networks with complex topologies have been studied in thecontext of many biological phenomena, such as neural networks

22 N E T W O R K M O D E L S[54], protein interactions [90], metabolic reactions [4], and geneticregulation [132]. Some basic network properties are:D E G R E E The number of edges that connect to a node. For directedgraphs this can be extended to discern between edges thatlink to a node (INDEGREE) and edges that link from a node(OUTDEGREE).D E G R E E D I S T R I B U T I O N The relationship between the degree k andthe probability P(k) of a given node having that degree.D I A M E T E R The maximum length of the shortest paths between eachpair of nodes.C O N N E C T I O N P RO BA B I L I T Y The probability that two nodes selectedat random are connected by an edge.C L U S T E R I N G C O E FFI C I E N T For a specific node n, the fraction of thepossible edges between all neighbours of n that are present; fora network N, the average of the clustering coefficients for eachnode in the network.Based on these properties, complex networks can be classified intoa number of categories:R E G U L A R Each node has the same degree k and the network is saidto be K-REGULAR.R A N D O M Each pair of nodes is connected by an edge with uniformprobability p. The degree distribution of such networks is approximatelyGaussian.S M A L L W O R L D The shortest path between two nodes is likely tobe small as the shortest path length scales logarithmically orslower with network size n [188]. In contrast, the shortest pathlength scales linearly with n in a regular network.S C A L E-FREE The network has a power-law degree distribution. Many,perhaps even most, of the real-world networks that have beenstudied are scale-free [124]. The mechanism proposed as theexplanation for this observation is “growth by preferential attachment”[18]—when a node is added to the network, it isconnected to an existing node i with a probability proportionalto the degree of i.Wang and Chen [186], Newman [132] and Albert and Barabási [4]all review the progress and significant results in the current studiesof various complex networks, including the World Wide Web, humanlanguage, and paper authorship. The dynamics of these models arederived from their topology, either by adding or removing nodes fromthe network, or by percolating node states across the network (e. g.,to model the spread of disease). One of the main areas of researchusing these models is the analysis of network robustness 1 in responseto changes in the network (e. g.the removal of vertices or a change inpropagation parameters).1 Also called resilience or stability.

3.3 H I E R A R C H Y I N R E A L-WORLD N E T W O R K S 23The analysis methods used to determine network resilience typicallyfocus on how statistical properties such as reachability areaffected by changes to the network. For example, percolation theoryis concerned with the propagation of state across a network (e. g.,states such as “working” and “failed”). The parameters governing thestate propagation are altered and changes in vertex patterns are thenanalysed for properties such as reachability and connected sub-sets.This approach is widely used for modelling phenomena such as thespread of diseases in societies and the spread of computer virusesover computer networks.While this approach might initially seem well-suited to our aim ofmeasuring nephron resilience to the onset of renal disease, this is notthe case. Renal diseases affect factors such as filtration rates and thestrength of transport mechanisms, and it is not at all clear how theseeffects can be understood or analysed in terms of the usual kinds ofstatistical network properties.Also, the manner in which these network models capture the dynamicsof a system is not ideally suited to modelling groups of nephronsfor the following reasons:• We require classes of nodes and update rules, so we cannotmodel state updates as a simple propagation or percolationacross the network.• The topology of a network of nephrons remains fixed (ignoringmajor events such as surgical removal), so we cannot modelstate updates as changes to the network topology.However, multiple graph automata can be connected together by acomplex network, which can be used to capture high-level structureand hierarchy in a system. This hybrid approach allows us to separatethe modelling into two distinct concerns: modelling nephrons as graphautomata; and modelling groups of nephrons as complex networks ofthese automata.3.3 H I E R A R C H Y I N R E A L-WORLD N E T W O R K SHierarchy is important in complex systems, and in complex networksthe hierarchy emerges from the network model. Explicit hierarchyis typically used when designing systems [74, 89, 202], while thehierarchy in real-world systems is not often evident and must beextracted from the model.Most real-world complex networks exhibit four main hierarchicalproperties: a scale-free degree distribution [132]; small-worldbehaviour [186, 27]; a hierarchical structure in clustering features[150, 132]; and degree-degree correlations [131].To extract physical knowledge from a complex system, one mustfocus on the right level of description for that system [66]. As a result,there is great interest in the hierarchical properties of real-worldnetworks [115, 157] and how networks can be generated so that theyexhibit these hierarchical properties [174, 26, 67]. An example ofgenerating a network with hierarchical properties is illustrated inFig. 7.

24 N E T W O R K M O D E L S(a)n=0,N=5(b) n=1,N=25(c) n=2,N=125Figure 7: The iterative construction of a hierarchical network from afully-connected cluster of five nodes, as demonstrated by Ravasz andBarabási in [150].Existing studies of hierarchy in networks have focused on identifyinghighly clustered neighbourhoods in networks [150] and treatingthese neighbourhoods as networks in themselves [9]. In other words,the original system is divided into a number of modules. The definitionof a module in a complex dynamical system, and the importanceof inter-module dependencies, is not always unanimous nor wellquantified[187].Song et al. demonstrate that real-world networks consist of selfrepeatingpatterns on all scales [165]. This is unexpected, as it iswidely believed that such networks are not self-similar, due to thesmall-world property, which implies that the relationship between thenumber of nodes and the diameter of the network is exponential, andnot the power-law relation that is expected of self-similar networks.The self-similarity of these networks is demonstrated by calculatingtheir fractal dimension (using a “box-counting” method). They state:The box-counting method provides a powerful tool forfurther investigations of network properties because itenables a renormalization procedure, revealing that theself-similar properties and the scale-free degree distributionpersist irrespectively of the amount of coarse-grainingof the network.These properties help explain why real-world networks have ascale-free distribution, and the presence of self-similarity in differentnetworks—the World Wide Web, protein-protein interaction networks,cellular networks and social networks—suggests that this diverserange of networks may share common self-organisation dynamics.Graph automata can also demonstrate hierarchy in behaviour. Indeed,Wolfram classifies such automata into four classes, dependingon the hierarchy present in the time-state diagram [201, 6§2 p231],but little else has been done to investigate their hierarchical dynamics.Hierarchical automata do exist [74, 75, 113, 119], but are typicallyused to design modular systems [89] rather than to study the dynamicsof existing systems. In these applications, the automata are used

3.4 D I S T U R B E D N E T W O R K S A N D D I S E A S E 25to capture the intended dynamics of the system being designed, andare not used to study any emergent dynamics that were not explicitlydesigned into the system. In conclusion, although hierarchicalautomata have been used to design modular systems, there has beenlittle research into the emergent hierarchical properties of theseautomata, which is in stark contrast to the studies of hierarchy incomplex networks.3.4 D I S T U R B E D N E T W O R K S A N D D I S E A S EMany aspects of diseases have been studied through the use of networkmodels. For example, the development of breast cancer hasbeen modelled as an implicit graph automata [56] and the spread ofsexually-transmitted diseases has been studied using social networkmodels [104].One interesting study concerning the spread of epidemics across biological,social and communication networks is that of Pastor-Satorrasand Vespignani [142, 143], which examines the susceptible-infectedsusceptible(SIS) model [15, 127] on complex networks. Each node ofa network represents an individual system and each edge is a connectionalong which the infection can spread to other systems. Nodeshave two available states—“healthy” (i. e., susceptible to infection)and “infected”—and at each time-step healthy nodes are infected withrate ν if they are connected to one or more infected nodes. At thesame time, infected nodes are cured and again become susceptiblewith rate δ, which defines an effective spreading rate of λ = ν/δ.In networks such as random graphs, the most significant epidemiologicalresult is the existence of a non-zero epidemic threshold λ C[15]. In the case λ < λ C , the infection dies out exponentially fast;in the case λ > λ C , the infection spreads and becomes persistent.However, over a wide range of scale-free networks Pastor-Satorrasand Vespignani observed an absence of the epidemic threshold, whichimplies that scale-free networks are susceptible to persistent infectionregardless of the spreading rate—an observation that fits wellwith real-world data concerning the spread of computer viruses overthe Internet. Given that scale-free networks appear in a wide range ofareas [4], the results obtained here may help improve our understandingof computer virus epidemics and could well have applications inother social and biological systems.However, regardless of the scale of the models (e.g., cellular, social),the focus is on how diseases or infections spread. Our aim is to usedisease modelling to investigate how the onset of renal disease inindividual nephrons can affect the stability of an entire system ofnephrons. To achieve this aim, the ability of diseased nephrons toaffect neighbouring nephrons and the whole-system dynamics will bestudied, rather than studying how the disease is propagated. Withthis focus on system stability, not the spread of disease, the majorityof the existing disease modelling literature is not directly relevant.However, at least one study [12] has investigated the system-wideeffects of localised failure, in the context of power grids. Ash andNewth start with a simple model for representing cascading failureson complex networks (as given in e. g., [126, 50]). Each edge (i, j)

26 N E T W O R K M O D E L Sis assigned a weight g ij —the efficiency of moving between i and jon this edge—in the range [0,1]. Initially, g ij = 1 if the edge (i, j) ispresent in the network, and g ij = 0 if it is not.Flow between two nodes takes the most efficient (shortest) pathand the load L i (t) on node i at time t is equal to the number of suchpaths that pass through i at time t. Each node has a finite capacityψ i = L i (0) × α, where α is a measure of the stress on the systemat time t = 0. The network is operating at maximum capacity whenα = 1, and the load on the network is light when α ≫ 1. Once thecapacity of a node i is exceeded, its performance is diminished asper Eq. 3.2. Initially, the capacity of some node i is reduced, and ateach time-step the load is redistributed over other nodes accordingto Eq. 3.2.⎧⎨g ij (t) ×g ij (t + 1) =⎩g ij (0)ψ iL i (t)if L i (t) > ψ iotherwise(3.2)The efficiency of an Erdös–Rényi random graph [57] is comparedto that of a network that has been optimised using an evolutionaryalgorithm. For α < 1.35 the optimised network considerably outperformsthe random graph, while for α 1.35 the networks displaysimilar behaviour. This clearly demonstrates that the optimisation algorithmhas worked—network efficiency is greatly improved when thenetwork is operating near maximum capacity and a node is impaired—and the study goes on to analyse optimised networks for topologicalregularities that might explain the source of the increase in resilience.This study, unlike other network models, began with a functioningsystem in which the function of a single node was impaired and theresulting effect on the entire system was measured. While it mayappear that this approach is identical to the study of system stabilityoutlined in Sect. 1.1, this is not the case. This study incorporates achange in state—the reduction in capacity ψ i of some node i—andthis change is then propagated across the network, due to the loadredistribution captured in Eq. 3.2. Accordingly, the efficiency of thenetwork is simply a weighted measure of the number of nodes whosecapacities have been reduced. As such, this study is an applicationof percolation theory, where nodes are initially in one state (i. e.,full capacity) and the percolation of another state (i. e., reducedcapacity) is measured. In contrast, the approach outlined in Sect. 1.1introduces a localised impairment to a system and observes the effecton the system function, where the impairment is contained and doesnot spread across the system. The contribution of an evolutionaryalgorithm to optimise the efficiency of a network is also unrelated tothis approach, as it relies on the ability to change the topology of thesystem, which is not feasible in the context of renal disease.

Part IIIM O D E L S A N D E X P E R I M E N T S

M O D E L D E S I G N4This chapter presents the design of the model. To begin, Sect. 4.1 introducesthe modelling approach—hierarchical dynamical networks—and contrasts it with other approaches. The benefits of using a discretetime model are described and a brief overview of the analysistechniques for this model is presented. The structure of the model isthen described in Sect. 4.2, followed by an analysis of the structuralproperties of the model in Sect. 4.3. Finally, the underlying equationsand update rules that capture the model dynamics are presented inSect. 4.4.4.1 M O D E L L I N G A P P ROAC HAs we have chosen to model the kidney as a complex system, thecouplings between the individual parts (i. e., the tubule segments) areof more interest than the dynamics of the individual parts in isolation.To elaborate, it is of greater importance to model the major couplingand interaction mechanisms that arise in multi-nephron systems thanto develop highly accurate formulations for the individual tubulesegments. The kidney is known to be a remarkably resilient organ [71],capable of maintaining its function across a wide range of conditions;operating on the assumption that the kidney is indeed a complexsystem, our intuition is that this resilience is due to the regulatorymechanisms and other interactions that arise between the lower-levelcomponents of the kidney, rather than being primarily due to thebehaviour of the individual nephron tubule segments in isolation.This approach, with less emphasis on the behaviour of individualparts than on their interactions, is similar to the approaches usedin fluid dynamics and gas models, where the individual particles areessentially ignored, in favour of higher-level emergent propertiessuch as mean flow rate and pressure. In the words of Nobel laureateRichard Feynman [77]:We have noticed in nature that the behavior of a fluiddepends very little on the nature of the individual particlesin that fluid. For example, the flow of sand is very similarto the flow of water or the flow of a pile of ball bearings.We have therefore taken advantage of this fact to invent atype of imaginary particle that is especially simple for usto simulate. This particle is a perfect ball bearing that canmove at a single speed in one of six directions. The flow ofthese particles on a large enough scale is very similar tothe flow of natural fluids.We treat the nephron tubule segments analogously—the tubulesegments are modelled as highly idealised “imaginary particles” andare the building blocks of our model—and as such, they will be modelledas individual nodes in the network automata (see Sect. 3.1).29

30 M O D E L D E S I G NExisting network automata have typically used 2-state [205, 204],3-state [206, 2] and even n-state [61, 118, 129] state spaces. Ourstate spaces are significantly more complex than [61, 118, 129] andthe visual analysis methods used by existing models (and reviewed inSect. 3.1) do not scale to the study of such complex models.Complex systems exhibit hierarchy in their organisation [150, 187,47, 11, 164] and the kidney is no different. Systems of nephrons exhibithierarchy in their organisation and interact with each other ina variety of ways. Multi-nephron systems are modelled as multiplesingle-nephron network automata, connected to each other via an arterialtree. This tree of nephron models forms a complex network (seeSect. 3.2) but unlike the majority of existing complex network models,the hierarchy of this model will be represented explicitly in the modelstructure. We define the name hierarchical dynamical networks torefer to this approach of modelling a system as a hierarchical networkof graph automata.4.1.1 The effects of a discrete time modelContinuous models (in the form of systems of continuous equations)are the standard approach used in many fields, such as physics, chemistry,various engineering disciplines and commerce. The equationsin these models impose constraints upon the variables in the model,which are not always directly solvable. Various numerical methodsfor solving such systems exist, often by converting the equations intoa discrete form (and thus arriving at a discrete model). The systembeing modelled may in fact be discrete and the equations reflectinga more abstract view of the system—for example, although fluid dynamicsare modelled using continuous equations, the fluids beingmodelled consist of finite numbers of molecules.Unlike existing nephron models, the model presented here usesa discrete time model and the model dynamics are captured by systemsof difference equations. When continuous systems are solvednumerically—as is the case for a number of nephron models (e. g.,[78])—the system is translated into discrete equations. Using thisapproach, solutions are calculated iteratively until a fixed point isfound, which differs from the discrete time approach used here. Thischoice of a discrete time model has several advantages for modellinglarge systems, such as the kidney:• The equations explicitly calculate the state evolution of themodel, rather than capturing constraints on the state of themodel.• Coupling between equations does not affect the computationalcost of solving the equations—the computational cost of themodel is the sum of the computational costs of each equation.• The system dynamics are analysed by simulating the state evolutionof the model, not by obtaining a steady-state solution. Modelparameters and couplings can be changed during a simulation,and the dynamic response of the system can be observed.

4.1 M O D E L L I N G A P P ROAC H 31These advantages of the discrete-time approach are importantin order to meet our aims. Recall that our focus is on the natureof the interactions and couplings in multi-nephron systems. Basedon the assumption that the kidney is a complex system, it is theseinteractions that regulate the system dynamics. By taking a networkapproach to our model, we can explicitly represent these interactionsas edges in the network, and the use of discrete-time equations meansthat the equation coupling is equivalent to the explicit coupling in thenetwork. To illustrate this point, consider three state variables x(t),y(t) and z(t), such that:x(t + ∆) = F(x(t), y(t)) (4.1)y(t + ∆) = G(y(t), z(t)) (4.2)z(t + ∆) = H(z(t)) (4.3)Although x(t) is indirectly coupled to z(t) through y(t), the valueof z(t) and the behaviour of H(z(t)) do not affect the calculation ofx(t). At any given time t i in a model simulation, the value of all statevariables x(t i ), y(t i ) and z(t i ) are known. Calculating x(t i + ∆) issimple a matter of calculating F(x(t i ), y(t i )), where all the necessaryvalues are already known. The indirect coupling between x(t) andz(t) is realised, as the value of z(t i − ∆) affects y(t i ), which in turnaffects x(t i + ∆), but the computational cost of the whole system issimply the sum of the computational costs of the individual equations.The choice of the time-step size ∆ is important, as it determineswhich signals the model will be capable of reproducing [162] (illustratedin Fig. 8). It also affects the model equations, as the time-stepsize determines what activities occur in a single time-step. Finally,the time-step size may affect the number of time-steps that need tobe simulated for the system to reach its steady-state behaviour. Thechoice of an appropriate value for ∆ is discussed in Sect. 4.4.1.2SignalSamples (∆=2)1Value0-11 2 3 4 5Time (s)Figure 8: An inappropriate sampling rate can render a model incapableof reconstructing signals produced by the original system.

32 M O D E L D E S I G NWhen multiple parts of the model must reach an equilibrium in asingle time-step, systems of coupled equations can be used to capturethis explicit coupling (such an approach is used in Sect. 4.4.5). Forexample:x(t + ∆), y(t + ∆), z(t + ∆) = F(x(t), y(t), z(t)) (4.4)Simultaneous equations are generally more complex to solve thanindividual equations, but if the equations can be reduced to linearequations the system can be solved by Gaussian elimination. Theclassic algorithm uses O(n 3 ) arithmetic operations to solve for nequations in n variables. Improved algorithms have been developed,reaching O(n 2.807 ) [171] and O(n 2.376 ) [46], but these algorithmstend to have a larger constant and are only better suited to solvingsufficiently large systems of equations. So while the use of simultaneousequations can increase the computational complexity of themodel, the increase in complexity is bounded and can be determinedeasily. For small systems of simultaneous equations, the increasein complexity will have negligible impact on the complexity of themodel, which is not a guarantee that can be made for small systemsof coupled differential equations.4.1.2 Analysis of discrete network automataPerhaps the single biggest drawback of using discrete-time and differenceequations is that the model does not have a “solution” ofthe kind generated by continuous models—instead, the dynamics ofthe model is determined by analysing how the model state evolvesover time. In order to understand the dynamics of such a model, itmust be determined when the model has reached its steady-state(which is likely to be a cycle of states rather than a single state). Ingeneral, there is no algorithm to determine how quickly a discretemodel will reach a steady-state behaviour (a generalised algorithmis not possible, as it would amount to solving the Halting Problem[182]).The standard approach is to simulate the model from an initial stateand to search for patterns in the model dynamics. Typical analysistechniques for graph automata are inherently visual in nature anddo not scale to large systems (see Sect. 3.1). For complex modelssuch as the nephron model proposed here, alternative analysis techniqueshave to be applied, which are typically statistical in nature.Since many of the phenomena exhibited by multi-nephron systemsare oscillatory in nature (e. g., filtration rates and the tubuloglomerularfeedback (TGF) mechanism), signals analysis techniques such asFourier analysis and power spectral densities will be used. Lyapunovstability measures will also be used, to determine how sensitive themodel dynamics are to changes in the system. This notion of stabilityor sensitivity to conditions will prove particularly useful for determininghow stable the model’s dynamics are when we incorporate theeffect of diseases into the model (Chapter 6).It is possible that the choice of initial state will affect the steadystatebehaviour of the model (much like the choice of initial conditions

4.2 M O D E L S T RU C T U R E 33for systems of continuous equations) and the relationship betweenthe steady-state dynamics and the choice of initial conditions mustbe considered in order to come to any conclusions concerning thedynamics of the model. Similarly, the stability of the model in responseto changes in parameters may be harder to determine thanfor continuous models, as different parameter values can be usedin simulations, but there is no general approach to determine thehow dynamics of the model vary as a function of a parameter value(for exceedingly simple models, such an approach may be possible).Again, statistical methods will be used to measure such properties.4.2 M O D E L S T RU C T U R E4.2.1 The single-nephron modelThe nephron tubule consists of a number of functionally distinctsegments—typical tubule models focus on the behaviour of a singlesuch segment—and the single nephron model reflects this structureby treating the nephron tubule as a sequence of nodes, one for eachsuch segment.The nephron does not exist in isolation—the tubule sits in interstitialfluid with which it exchanges solutes (shown in Fig. 2); it isalso surrounded by a network of peritubular capillaries that alsoexchange solutes with the interstitial fluid (shown in Fig. 1a). Thesingle-nephron model must therefore take into account both of thesestructures. Note that while both the tubule and peritubular capillariesexchange solutes with the interstitial fluid, the tubule usesmany active and secondary transport mechanisms to do so, whilethe peritubular capillaries only undergo diffusion with the interstitialfluid.The interstitial fluid is divided into horizontal layers (as shown inFig. 2) whose compositions differ, and so the single-nephron modelreflects this layered structure. The peritubular capillaries surroundthe tubule, and our model divides these capillaries into segmentsthat correspond to the tubule segments. Rather than using additionalnodes to model these capillaries, each tubule segment node alsomodels the corresponding peritubular capillaries. That is, the stateof each tubule segment node encapsulates the state of the tubulesegment and the state of the surrounding peritubular capillaries.The model structure—showing fluid flow along the tubule and theTGF mechanism—is depicted in Fig. 9 and the complete network—including interstitial fluid nodes and edges for solute transport andTGF—is shown in Fig. 10.This network captures the functional topology of a nephron, butit is not detailed enough to permit behaviours such as the countercurrentexchange mechanisms in the loop of Henle. The model isrefined by dividing the limbs of Henle and the medullary fluid into anumber of layers (see Fig. 11) in order to model the counter-currentexchange mechanism—a good example of hierarchy in a complexsystem. Without dividing the medullary fluid into layers a salt gradientcould not exist, and without dividing the limbs into layers there wouldbe no counter-current exchange mechanism.

34 M O D E L D E S I G NCortexGlomerulusDistalTubuleProximalTubuleTGFOuter MedullaDescendingLimbAscendingThick LimbCollectingDuctsInner MedullaAscendingThin LimbFigure 9: The single-nephron model, showing tubule flow (solid arrows)and the TGF mechanism (dashed arrow). Solute transportsbetween the tubule and the surrounding fluid are not shown.GlomerulusProximalTubuleDescendingLimbAscendingThin LimbAscendingThick LimbDistalTubuleCollectingDuctsCortexOuterMedullaInnerMedullaFluid flow TG feedback Solute transportFigure 10: The single-nephron tubule as a 1D network automata,showing the edges that capture fluid flow, solute transport and TGF.Rather than dividing the nodes into sub-nodes, it would have beenpossible to model the layers as a number of state variables in theoriginal nodes. However, this approach would have hidden the hierarchicalstructure of the model, preventing the model from explicitlycapturing the layers and transports as nodes and edges. The numberof layers into which the loop of Henle and medullary fluid aredivided is a parameter of the model, allowing for control over the balancebetween approximating a continuous counter-current exchangemechanism and having a simple model structure.4.2.2 The multi-nephron modelExisting models of two-nephron systems have focused on the interactionsthat arise due to the oscillations in filtration rate and tubulepressure [112, 166, 19, 81, 99, 19], taking into account the regulatoryrole played by the TGF mechanism. These models have reproduced arange of experimentally observed phenomena, such as in-phase synchronisationof both single-nephron glomerular filtration rate (SNGFR)and pressure oscillations in the proximal tubule. However, thesemodels fail to take into account any interactions that may arise viaother channels, such as changes in the interstitial fluid due to so-

4.2 M O D E L S T RU C T U R E 35Outer MedullaDescending LimbH 2OH 2OH 2OH 2OH 2OH 2ONaNaNaNaNaNaThick AscendingLimbThin AscendingLimb150 mmol/LNa300 mmol/LInner MedullaFigure 11: Solute transport in the loop of Henle, demonstrating howthe limbs and the interstitial fluid are divided into layers. Diffusionof H 2 O occurs between the descending limb and the interstitial fluid,while Na is actively transported from the ascending limbs into theinterstitial fluid. The division into layers is necessary for the loop ofHenle to act as a counter-current exchange mechanism. An exampleof typical figures for the sodium gradient in the interstitial fluid [78]are shown on the right.lute secretion or reabsorption by a segment of the nephron tubule.The extension of the single-nephron network to a two-nephron networkis designed to allow for these kinds of interactions to arise andbe analysed—the network automata allows for parameters (such asinteractions and couplings) to be changed during a simulation.A two-nephron model was obtained by connecting two singlenephronmodels together, as shown in Fig. 12. The afferent arteriolesare treated as variable resistors placed in parallel across a fixed potential(the difference between arterial and venous pressure). Thus,by changing the resistance of its afferent arteriole, a nephron caninfluence the SNGFR of neighbouring nephrons—this interaction isreferred to as hemodynamic coupling. The afferent arterioles areassumed to be non-compliant, to simplify the arterial model.GlomerulusProximalTubuleDescendingLimbAscendingThin LimbAscendingThick LimbDistalTubuleCollectingDuctsCortexOuterMedullaInnerMedullaGlomerulusProximalTubuleDescendingLimbAscendingThin LimbAscendingThick LimbDistalTubuleCollectingDuctsFluid flow TG feedback Solute transportFigure 12: The two-nephron network, showing the edges that capturefluid flow, solute transport and TGF.The multi-nephron models are a straight-forward generalisation ofthe two-nephron model shown in Fig. 12, where multiple nephron

36 M O D E L D E S I G Nautomata share the same column of interstitial fluid (such a columnis shown in Fig. 13). Again, the afferent arterioles are assumed to benon-compliant and are treated as variable resistors.CortexOuter MedullaInner MedullaFigure 13: A single column of interstitial fluid, showing the divisionof the fluid into layers. Multiple nephron automata can be placed inthis column, sharing the interstitial fluid.For a small number of nephrons, it is sufficient to model thenephrons as parallel resistors placed across a potential equal tothe difference between the arterial and vascular pressures. However,larger multi-nephron models must take into account the structure ofthe arterial tree that supplies the nephrons with blood [112].The arterial tree for a column of interstitial fluid is modelled asa number of branches connected to a main vessel (see Fig. 14).Nephrons connected to the same branch are treated as analogous toresistors connected in parallel (with resistance equal to the resistanceof the afferent arterioles) over a potential (the difference betweenarterial and venous blood pressure). Nephron resistance is modelledsolely as the afferent arteriole resistance because the tubule and capillariesare non-compliant and therefore have a constant resistance,which we ignore for simplicity. Between each branch is a resistancewhich accounts for the extra distance the blood must travel to reachthe lower branches.4.2.3 The multi-column modelFor sufficiently large multi-nephron models it is not reasonable for allof the nephrons to share the same interstitial fluid, as this would allowall of the nephrons in the model to directly interact with each other—something that does not occur in sufficiently large systems (suchas the kidney as a whole)—and it enforces a globally-homogeneousinterstitial fluid, which is not physiologically reasonable.To separate nephrons into local neighbourhoods, and to allow theinterstitial fluid to evolve inhomogeneously, the multi-column modeldivides the fluid into a number of columns, as shown in Fig. 15.Each column is supplied with blood from an arterial tree that hasan identical structure to the arterial tree inside each column. Thecolumns are arranged on a 2D grid, and may thus neighbour othercolumns.

4.2 M O D E L S T RU C T U R E 37P AP VR BR BNNNNFigure 14: The arterial tree for a multi-nephron model, shown assolid lines. Each node “N” represents a single nephron network automatain the column of fluid, whose resistance is equal to that ofthe nephron’s afferent arteriole R A . P A and P V are the arterial andvenous pressures, respectively, and the resistance of the arterial vesselsis modelled as additional resistances R B . The dotted lines showthe venous tree, which is not included in the model.P AP VR CR CFigure 15: The arterial tree for multiple columns of interstitial fluid.Each column contains an arterial tree (as shown in Fig. 14). P Aand P V are the arterial and venous pressures, respectively, and theresistance of the arterial vessels is modelled as additional resistancesR C . The dotted lines show the venous tree, which is not included inthe model.To simulate a single time-step for this model, we simulate the timestepfor each column of interstitial fluid, and then permit diffusionbetween neighbouring columns. We make two assumptions concern-

38 M O D E L D E S I G Ning the nature of this diffusion. Firstly, we only permit horizontaldiffusion (i.e., there is no diffusion between different layers of theinterstitial fluid). Secondly, we only permit the diffusion to act on onequarter of the fluid in each column (see Fig. 16)—this prevents anyportion of the interstitial fluid from undergoing inter-column diffusionwith more than one column, as each quarter of the fluid diffuses withthe neighbour on the nearest side (if any).diffusionFigure 16: At each time-step, diffusion is performed between the fluidlayers in neighbouring columns, where one quarter of the fluid ineach layer is allowed to diffuse with a neighbouring layer.Note that the arterial tree and the 2D column arrangement areentirely separate and can be structured independently of each other—e. g., columns attached to the same branch of the arterial tree neednot be located near each other. As hemodynamic coupling is capturedby the arterial tree and the inter-column diffusion is captured by the2D arrangement, these two mechanisms can be controlled independently,allowing for the interactions between these mechanisms to beinvestigated.4.3 N E T W O R K P RO P E RT I E SNow that the structure of the model has been defined, it is possible tocalculate several standard network properties (as defined in Sect. 3.2)that will allow the connectivity properties of models of differentsizes to be compared, as well as providing a reference point forcomparing the model structure to other network models and realworldnetworks. Note that these properties only take into accountdirect connections between pairs of nodes, and can not be usedto determine the likelihood of interactions between nodes that arenot directly connected. All nodes are connected via paths across thenetwork, but emergent behaviour cannot be inferred from the networktopology—it must be analysed using simulations. Similarly, althoughnephrons can be coupled (e. g., hemodynamic coupling occurs overarterial tree), these network properties cannot infer any propertiesof the resulting dynamics.The network properties of the single-column model are shownin Table 2. Not much can be inferred from the values for a singlenephronnetwork, due to the small size of the network (14 nodes).Only when the model is extended to larger numbers of nephrons willit be possible to draw some conclusions about the network propertiesof the model. If we consider a two-nephron network (Fig. 12), wesee that in comparison to the single-nephron network, the averagedegree has increased, the connection probability and the clusteringcoefficient have dropped, and the network diameter is unchanged.

4.3 N E T W O R K P RO P E RT I E S 39Note that the clustering coefficient decreases for n < 4 and increasesfor n 4. For n > 5, we have C > p (as shown in Fig. 17), which isan indication that the network is a small-world network.Property n = 1 n = 2 value lim n→∞Mean Degree 3.83 4.8446n7n+56.57Diameter 6 6 6 6Clustering (C) 0.097 0.065n2 + 9n−3 1 4n−21Connection Pr (p) 0.174 0.1357n+50.07123n49n 2 +63n+20Table 2: Properties of the n-nephron single-column network.00.18Cp0.120.0601 3 5 7 9nFigure 17: As n increases, the clustering coefficient (C) becomeslarger than the connection probability (p).The same network properties for multi-column models—with c 2columns of interstitial fluid arranged in a c × c grid, each containingn nephrons—are shown in Table 3. Note that although the networkdiameter tends to ∞, it grows linearly with the number of columnsin the model. There is essentially no difference in the mean degree,clustering coefficient or connection probability between the singlecolumnand c 2 -column networks. In both cases the mean degree tendsto 46/7 ≈ 6.57; the clustering coefficient tends to 1/14 ≈ 0.071; and theconnection probability tends to 0.Similar to the single-column network, we find that as the networksize grows, we have C > p, which again indicates that the model isa small-world network. In fact, for c = 1 we find that C > p whenn > 5 (as per the single-column network) and that for c > 1 we find∀n : C > p. Both values quickly approach their asymptotic values, sothat even for small values of n and c the difference between C and pis close to maximal (as shown in Fig. 18).

40 M O D E L D E S I G NProperty Value lim c→∞Mean Degree46nc 2 +12c(c−1)46n+12(7n+5)c 2 7n+5Diameter 6 + (c − 1) 2 ∞Clustering (C)Connection Pr (p)nc 22 + n3n(3n−1)+4−4/c + n2n(2n−1)+4−4/c(7n+5)c 223nc 2 +6c(c−1)(7n+5)c 2 ((7n+5)c 2 −1)n14n+10Table 3: Properties of the n-nephron c 2 -column network.00.10.050-0.05-0.113n57 1 23c4Figure 18: As either n or c increases, the difference between theclustering coefficient (C) and the connection probability (p) quicklyapproaches the asymptotic value of 1/14 ≈ 0.071, as shown in this plotof C − p.4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S4.4.1 Choosing an appropriate time-step sizeAs mentioned previously (Sect. 4.1.1), the model uses discrete time.Each time-step maps the current state of the model, S(t) to the stateof the model at the next time-step S(t + ∆) (the time-step size ∆is aparameter of the simulation). A simulation of the model is achievedby repeatedly applying this mapping, beginning with the initial modelstate. This mapping is realised as a number of equations, whichsimulate the fluid flow and solute transports in the model, and updatethe state variables in the model.More importantly, the kidney exhibits multi-rate dynamics—nephronresponses can range from a matter of seconds (e. g., the TGF mechanism)to the order of hours (e. g., the secretion of hormones). Specifically,the SNGFR tends to oscillate with a period of 30–40s [78, 166,111] and the TGF signal has a delay of 3–5s [166, 145]. The modelaims to recreate these signals, and therefore needs a sampling rate

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 41capable of reproducing these signals [162]. For example, a time-stepsize of no greater than 1.5s is required to reconstruct a 3s delay inthe TGF signal. A time-step size of one second was ultimately chosen,as this is sufficient to reconstruct signals with frequencies less than1/2Hz [162] and therefore meets our needs.4.4.2 Discrete fluid flow and solute transportDue to the use of discrete time, bodies of fluid are modelled asdiscrete volumes, and flow along a vessel is treated as a sequenceof these discrete volumes. A vessel is able to hold a fixed numberof discrete volumes, and at each time-step an entering volume offluid forces another volume of fluid to exit the vessel (depicted inFig. 19)—each vessel acts as a queue. In this model, all vessels (i. e.,the arterial tree, the nephron tubule and the peritubular capillaries)are non-compliant, meaning their radius is fixed regardless of thehydraulic pressure of the fluid in vessel.v 0v 1v 2v 3v 0v 1v 2v 3Figure 19: Discrete fluid flow along a segment of tubule that containsthree discrete volumes of fluid (v 1 , v 2 and v 3 ). In one time-step avolume of fluid v 0 enters the tubule, causing the volume v 3 to exit thetubule.Existing models have modelled the tubule as compliant [111], modelledtubule segments as having constant resistance [19], and modelledthe tubule and afferent arteriole as non-compliant [99]. Treatingthe tubule and capillaries as non-compliant allows the model to ignorethe effects of hydrostatic pressure on resistance to fluid flow.Tubular compliance affects the speed of pulse propagation in thetubule, and the TGF time-delay can be increased to compensate fornon-compliance [99].The model does not take into hydrostatic pressure anywhere inthe tubule (but hydrostatic pressure is used to determine blood flowthrough the arterial tree and to calculate SNGFRs). As the tubule isassumed to have no compliance, the pressure will vary in proportionto the flow rate along the tubule. The filtrate enters the tubule indiscrete volumes, each of which travels along the tubule at the samerate, rather than mixing with the neighbouring fluid volumes.The tubule segment nodes control solute transport—each nodecontains update rules that govern the transports between the tubulefiltrate, the interstitial fluid and the peritubular capillaries. Tubulesegments that have been divided into sub-nodes (i. e., the loops ofHenle and the collecting ducts) perform the solute transport andfluid flow in several stages, in order to capture the dynamics of thecounter-current exchange mechanism (shown in detail in Fig. 20).The stages progress as follows:

42 M O D E L D E S I G N1. The fluid entering the tubule segment in the time-step is dividedinto sub-volumes, (the specific number is a parameter of themodel).2. A sub-volume enters the tubule segment.3. Solute transports are simulated for each sub-node.4. Steps 2 and 3 are repeated until all of the sub-volumes haveentered the tubule segment.V aV 0V 1V 1V aV b1V bV aVv bVv ba1Vv abV 2V 2V 1V 11V b1V b2V 3V 3V 2V 21V 11V 12V 3V 3V 3V 21V 21V 3Figure 20: A volume of fluid V 0 enters the tubule in the time-step.According to step 1, it is divided into two volumes, V a and V b . V b entersthe tubule, forcing V 3 to flow out of the tubule. Solute transportsare then performed for each sub-node (represented by the superscript“1”). V a then enters the tubule, forcing V2 1 to flow out of thetubule, and solute transports are performed again (represented bya superscript “2” for volumes that have undergone solute transporttwice, and by a superscript “1” for V a , which has undergone solutetransport once).For the sake of simplicity, the model ignores a number of factorsthat can affect solute transport and the composition of the tubulefiltrate:• Physiological factors such as conductivity, reflection and Michaelis-Menten kinetics.• Chemical processes in the tubule, such as glucose consumption,lactate production and hormone secretion.• Low-level structural features such as the various pathways betweenthe filtrate and the interstitial fluid.• Permeability is only taken into account for solute transports thatare dependant on a hormone concentration (e. g., sodium andwater reabsorption in the distal tubule and collecting ducts).

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 434.4.3 Model inputThe input for the model consists of the blood entering the arterial treeat each time-step. The input is characterised by the concentrationsof a number of solutes—the solutes used in the model are listed inTable 4. Except for water and red blood cells, all solutes are assumedto have a molarity of 0—they are not considered to contribute tofluid volume. The solute AvgProtein represents a typical protein inthe bloodstream and takes its properties from albumin, the mostabundant plasma protein in humans and other mammals.The input solute concentrations are specified as ranges, having aminimum and maximum value. At each time-step, the input concentrationsolute is chosen at random from the range, where each value inthe range has an equal likelihood of being chosen. Model parameterscan also be changed at any time-step in a simulation, making themodel interactive.Solute Molar Mass Has volumeWater 18.01 YesSodium 58.44 NoChlorine 35.453 NoPotassium 39.0983 Noantidiuretic hormone (ADH) 1.5e5 NoAldosterone 360.49 NoRBC 54.05e12 YesAvgProtein 6.5e4 NoTable 4: The solutes support by the model.4.4.4 State variablesThe glomerulus and ascending thick limb nodes have internal statevariables, which are used by the update rules to implement the dynamicsof these tubule segments. These state variables are listed inTable 5: R A (t), C A (t), C E (t), P G (t), Π G (t), ξ(t) and φ(t) are the statevariables for the glomerulus, and Na M (t) is the state variable for thethick ascending limb. The variables C A (t) and C E (t) are calculatedfrom the composition of the plasma at the afferent and efferent arterioles,respectively. Likewise, Na M (t) is the history of sodium deliveryto the macula densa and is calculated from the composition of thefiltrate leaving the thick ascending limb. The update rules for theremaining state variables are:

44 M O D E L D E S I G NR A (t) = ξ(t) × exp(ΛH A ) (4.5)P G (t) = P V + P A × R E /R A (t) × C A (t)/C E (t)(4.6)(1 + R E /R A (t)) × C A (t)/C E (t)ψφ(t) = ξ max −(4.7)1 + exp(k[Na M (t − D T GF ) − Na 1/2 ])( 2ζφ(t) + ξ(t − ∆)(ω∆ + 2 )ξ(t − 2∆)(ω∆)ξ(t) =2 −(ω∆) 21 + 2ζω∆ + 1(4.8)(ω∆)( )2CA + CΠ G = ΠE, Π(C) = aC + bC 2 (4.9)2The glomerulus state variables, and their associated update rules,are discrete forms of the equations for the same variables in [78],while the formula for calculating oncotic pressure Π(C) (and the associatedconstants a, and b) come from Deen et al. [52]. Higher-orderequations for ξ(t) have been used to simulate higher-frequency phenomenon[112, 166], but we have chosen to use the simpler secondorderequation. The model parameters used in these update rules arelisted in Table 6. The value of D T GF controls the time delay in theTGF mechanism, and need not be a multiple of ∆—the value can becalculated by linear interpolation between time-steps.Hemodynamic coupling arises naturally from the arterial tree network.Vascular signalling, however, does not arise naturally from thearterial tree or from the single-nephron model. To incorporate thiscoupling mechanism into the model, the update rule for the statevariable R A (t) must be adjusted. The strength of the TGF signal isproportional to ξ i (t) for each adjacent nephron i on the same branchof the arterial tree, and each such signal is assumed to have decayedto a fraction γ of the original signal. Thus, the update rule is now:R A (t) =(ξ(t) + ∑ i∈Nγξ i (t))× exp(ΛH A ) (4.10)Vascular signalling has been modelled in existing models by anidentical approach [112, 166].4.4.5 State update rulesThe update rules for each tubule segment node control the flow offluid and the transport of solutes along the tubule. A summary of thefunction of each node is given in Table 7. Solutes are transportedbetween the tubule filtrate and the interstitial fluid, and diffusionoccurs between the interstitial fluid and the peritubular capillaries.The model does not take into account hydrostatic pressure in theinterstitial fluid, so water is reabsorbed directly into the peritubularcapillaries to preserve the interstitial fluid volume.The state space for each node in the model covers the compositionof the fluid in the node—R N for N solutes, including water—and thenode’s state variables—R S V for S V state variables—resulting in a

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 45Variable Definition UnitR A (t) Afferent arteriole resistance. mmHg . min / nLC A (t)C E (t)Afferent arteriole protein concentration.Efferent arteriole protein concentration.P G (t) Hydrostatic pressure in theglomerulus.Π G (t) Oncotic pressure in theglomerulus.ξ(t)Afferent arteriole response toTGF.g/Lg/LmmHgmmHgmmHg . min / nLφ(t) The TGF signal. mmHg . min / nLNa M (t) Macula densa sodium delivery. mol/sTable 5: State variables in the nephron model.Parameter Definition UnitP A Arterial pressure. mmHgP V Venous pressure. mmHgP BCK fNa 1/2kD TGFωζH APressure at the start of thetubule.Permeability constant for theglomerulus.Sodium delivery for which theTGF response is half-maximal.Range parameter for the TGFsignal.Time delay in sending the TGFsignal.Natural frequency for the TGFmechanism.Damping coefficient for the TGFmechanism.mmHgnonemol/sL/mmolsrad/snoneΛ Viscous resistance coefficient. noneThe average afferent hematocrit.noneTable 6: Model parameters.

46 M O D E L D E S I G NSegmentBehaviourGlomerulusFilters a percentage of the plasma into thetubule.Proximal Tubule Reabsorbs 2/3 of the tubular Na and H 2 O.Descending LimbAscending LimbDistal TubuleCollecting DuctsDiffuses H 2 O to balance the Na concentrationsin the tubule and the medullary fluid.Actively transports sodium into the medullaryfluid to maintain a gradient between themedullary and tubular Na concentrations.Reabsorbs Na proportional to aldosterone concentrationand reabsorbs H 2 O proportional toADH concentration.Reabsorbs Na proportional to aldosterone concentrationand reabsorbs H 2 O proportional toADH concentration.Table 7: The behaviour of each tubule segment in the nephron model.state space of R N+S V . This state space is far more complex than thetypical n-state spaces of the cellular automata reviewed in Sect. 3.1.For simplicity, the update rules in the model describe only thetranslocation of sodium and water as essential components of thefiltrate. Equations for other solutes (such as chloride and potassium)can easily be incorporated into future, more complex models. Suchmodels will also need to account for the effects of changes in filtrationand reabsorption at the level of whole kidney function andthe consequent modulation of extracellular fluid volume and bloodpressure. In addition, the update rules take into account the effects ofthe hormones ADH and aldosterone, but neither hormone is secreted,reabsorbed or metabolised in the model at present.The notation used to present the update rules is described in Table8. The update rules are now presented.GlomerulusThe role of the glomerulus node is to filter a portion of the plasmafrom the blood entering the nephron via the afferent arteriole into thenephron tubule. As such, this node must calculate the SNGFR for thetime-step and divide the incoming blood accordingly. The equationused to calculate SNGFR is a standard equation [71, p. 317]:SNGFR = K f × (P G − P BC − Π G ) (4.11)Proximal tubuleThe proximal tubule reabsorbs 67% of the sodium and water in thetubule fluid. The water is transported directly to the capillaries (topreserve interstitial fluid volume), while the sodium is transported

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 47SymbolNa(t)H 2 O(t)X(t) IX(t) IFX(t) I+IFX(t) RX(t) DX(t) AX(t) MX(t) D+MX(t) M+ANa GDefinitionQuantity of sodium at time t (mol).Volume of water at time t (L).Quantity of X entering the tubule segment.Quantity of X in the interstitial fluid that surroundsthe tubule segment.Quantity of X entering the tubule segmentand in the interstitial fluid that surrounds thetubule.Quantity of X reabsorbed by the tubule segment.Quantity of X in the descending limb.Quantity of X in the ascending limb.Quantity of X in the medullary fluid.Quantity of X in the medullary fluid and descendinglimb.Quantity of X in the medullary fluid and ascendinglimb.The sodium gradient maintained between theascending limb and medullary fluid (mol/L).Table 8: Notation for model equations.

48 M O D E L D E S I G Ninto the interstitial fluid and diffused into the capillaries. The updaterules are:Na(t) R = 0.67 × Na(t) I (4.12)H 2 O(t) R = 0.67 × H 2 O(t) I (4.13)This constant behaviour is clearly far simpler than existing proximaltubule models (such as [193] and [190]), but it is able to capture someof the essential dynamics of the tubule segment:• The proximal reabsorption is essentially iso-osmotic [177], sosodium and water are reabsorbed in equal proportion.• The proximal tubule normally reabsorbs about 2/3 of the filteredload of sodium and water [30, 31, 71, p. 333].Loop of HenleThe major difference between the model presented here and existingmodels of the loop of Henle [76, 175, 108] is that in this model,the reabsorption processes are assumed to maintain an equilibriumbetween the filtrate and the interstitial fluid. The descending limb onlypermits the diffusion of water and the ascending limb only permitsthe active transport of sodium, as shown in Fig. 21. There are a largenumber of other transport mechanisms that are known to occur inthe limbs (e. g., salt secretion in the descending limb [108], lactateproduction [76] and the recycling of urea [176]), which are currentlynot modelled.DiMAiNat DiH 2Ot DiH 2ONaNat M H 2 Ot MNat AiH 2Ot AiDescending LimbInterstitial FluidAscending LimbFigure 21: Solute transfer in the loop of Henle, showing only thequantities of sodium and water present in the nodes.Consider a horizontal slice of the loop of Henle at time t, as shownin Fig. 21. At each time-step, the model simulates the diffusion ofwater from the descending limb and the active transport of sodiumfrom the ascending limb. The model assumes that both the diffusionand active transport reach equilibrium during the time-step. Thismeans that:• Enough water is diffused to balance the ratio of sodium to waterin the interstitial fluid and the descending limb.• Enough sodium is actively transported to maintain a gradientof Na G mol/L between the interstitial fluid and the ascendinglimb.Given these assumptions and the state of slice at time t, we areable to calculate the state at time t + ∆ (before the fluid flow along

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 49the tubule is simulated). Firstly, the model conserves the total waterand sodium in the system:H 2 O(t) D+M = H 2 O(t + ∆t) D+M (4.14)Na(t) M+A = Na(t + ∆t) M+A (4.15)The descending limb reaches an equilibrium with the medullaryfluid, maintaining a sodium concentration that is identical to thesodium concentration in the medullary fluid:Na(t) M Na(t)=DH 2 O(t + ∆t) M H 2 O(t + ∆t) D(4.16)H 2 O(t + ∆t) D =Na(t) D× H 2 O(t) D+MNa(t) D+M(4.17)H 2 O(t + ∆t) M =Na(t) MNa(t) D+M× (H 2 O(t) D+M ) (4.18)The ascending limb reaches an equilibrium with the medullary fluid,maintaining a sodium concentration that is Na G less than the sodiumconcentration in the medullary fluid:Na(t + ∆t) MH 2 O(t) MNa(t + ∆t) A == Na(t + ∆t) A+ Na G (4.19)H 2 O(t) A H 2O(t) A×H 2 O(t) M+A (4.20)(Na(t) M+A − Na G H 2 O(t) M )Na(t + ∆t) M =H 2O(t) MH 2 O(t) M+A×(Na(t) M+A − Na G H 2 O(t) A )(4.21)Simulations of the two-nephron model revealed a limitation of theabove equations—this system of equations assume that the loop ofHenle reaches an equilibrium with the medullary fluid in isolation–theequations ignore the possibility that other loops of Henle are alsointeracting with the medullary fluid. When two nephrons share thesame interstitial fluid, the above equations cause large oscillationsin the medullary sodium concentration (see Fig. 22), as the loops ofHenle both compensate for any imbalance in the equilibrium.The solution was for the equations to ensure that each layer ofinterstitial fluid reached equilibrium with all of the descending andascending limbs of Henle present in the fluid—a change that is onlya generalisation of the original equations. To achieve this new equilibrium,it was necessary for the transport equations to be solvedsimultaneously.Consider again a horizontal slice of the loop of Henle at time t, asshown in Fig. 21. At each time-step, the model simulates the diffusionof water from the descending limb and the active transport of sodiumfrom the ascending limb. The model assumes that both the diffusionand active transport reach equilibrium. This now means that:• Enough water is diffused to balance the ratio of sodium to waterin the interstitial fluid and all of the descending limbs.

50 M O D E L D E S I G N320310Sodium (mmol/L)300290280270260250100 110 120 130 140 150TimestepFigure 22: Oscillations in medullary sodium concentration.• Enough sodium is actively transported to maintain a gradient ofX mol/L between the interstitial fluid and all of the ascendinglimbs.For a system of n nephrons in a single column of interstitial fluid,the equations for the loop of Henle are based on the assumption thatthe n nephrons cooperatively maintain the sodium gradient. Eachnephron in the system reabsorbs some volume w i of water from thedescending limb Di and some amount s i of salt from the ascendinglimb Ai for each i ɛ {1..n}. The state of a single layer of a loop of Henleat time t is shown in Fig. 21. At equilibrium, we obtain three equationsfor the water transport in the descending limbs: the conservationequations (4.22) and (4.23), and the equilibrium equation (4.24).H 2 O(t + ∆) Di = H 2 O(t) Di − w i (4.22)H 2 O(t + ∆) M = H 2 O(t) M + ∑ iw i (4.23)Na(t) Di Na(t)=M(4.24)H 2 O(t + ∆) Di H 2 O(t + ∆) MLikewise, we obtain three equations for the sodium transport in theascending limbs: the conservation equations (4.25) and (4.26), andthe equilibrium equation (4.27).Na(t + ∆) Ai = Na(t) Ai − s i (4.25)Na(t + ∆) M = Na(t) M + ∑ is i (4.26)Na(t + ∆) MH 2 O(t) M= Na(t + ∆) AiH 2 O(t) Ai+ Na G (4.27)Both systems of equations (water transport and sodium transport)can be transformed into the form Ax = b, where:

4.4 S TAT E S PAC E E Q UAT I O N S A N D U P DAT E RU L E S 51⎛⎞1 α 1 α 1 · · · α 1α 2 1 α 2 · · · α 2A =⎜.⎝ .... .. ⎟⎠α N α N α N · · · 1⎡ ⎤ ⎡ ⎤ ⎡ ⎤w 1 s 1β 1w 2s 2...β 2x =⎢ . or, b =⎣ .⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ ⎦ ⎣ .⎥⎦w N s N β N(4.28)(4.29)The values of α i and β i depend on which system of equations weare solving. For the equations governing water reabsorption in thedescending limbs, we find:α i =Na(t) DiNa(t) M + Na(t) Di(4.30)β i = Na(t) MH 2 O(t) Di − Na(t) Di H 2 O(t) MNa(t) M + Na(t) Di(4.31)For the equations governing the active transport of sodium in theascending limbs, we find:α i =H 2 O(t) AiH 2 O(t) Ai + H 2 O(t) M(4.32)β i = Na(t) AiH 2 O(t) M − Na(t) M H 2 O(t) AiH 2 O(t) Ai + H 2 O(t) M+ Na (4.33)GH 2 O(t) Ai H 2 O(t) MH 2 O(t) Ai + H 2 O(t) MThe refined model solves both systems of linear equations usingGaussian elimination (with partial pivoting, to improve the stabilityof the solutions [179]), allowing equations 4.22, 4.23, 4.25 and 4.26to be solved.These equations enforce the sodium molarity at each macula densato be equal, assuming that the macula densas are located in thesame layer of interstitial fluid. This has the effect of making the TGFresponse of each nephron identical at every time-step, enforcingin-phase synchronisation. Experimental data, however, contradictsthis [167]. A more realistic response is obtained by altering the TGFmechanism to respond to sodium delivery (mmol/s) instead of sodiumconcentration, an approach used by Layton et al. [99]. As a result,the values of Na 1/2 used by the model are not directly comparable totypical values of 30–60 mmol/L [78].Distal tubuleThe first portion of the distal tubule—the early distal tubule—is similarin function to the thick ascending limb [71, p. 336], reabsorbingsodium but remaining impermeable to water. This portion of the

52 M O D E L D E S I G NADHAldosteroneMinimal 1 pM 0.01 nMHalf-maximal 10 pM 0.3 nMMaximal 100 pM 10 nMTable 9: Distal tubule responses to hormone concentrations in thefiltrate. The ADH response values are from Star et al. [168] and thealdosterone response values are from Christ et al. [40]. Note that theinhibitory effects of much higher concentrations [87] are ignored bythis model.tubule also forms part of the juxtaglomerular complex, generatingthe TGF signal that provides feedback control over the SNGFR. As illustratedin Fig. 9 and Fig. 10, the thick ascending limb node capturesthis behaviour.The distal tubule node focuses on capturing the dynamics of thelate distal tubule. This portion of the tubule has the following characteristics[71, p. 337]:• Sodium is reabsorbed at a rate that is regulated by hormones.In particular, the reabsorption rate is proportional to the aldosteroneconcentration in the filtrate.• The permeability of the tubule to water is controlled by the concentrationof ADH. When ADH concentration is high, the tubuleis permeable to water, but in the absence of ADH the tubule isessentially impermeable to water.Accordingly, the update rules for the distal tubule reabsorb sodiumand water in amounts that are proportional to the filtrate concentrationsof aldosterone and ADH, respectively. The minimal, half-maximaland maximal tubule responses are shown in Table 9.The update rules calculate the tubule response to ADH (Eq. 4.34)and aldosterone (Eq. 4.35)—the responses are kept within the range[0 . . . 1] (not shown in the equations)—and water and sodium arereabsorbed accordingly (Eq. 4.36 and Eq. 4.37). Note that theseequations do not account for any time delay between the hormoneconcentrations and the according tubule response [168, 40].R ADH (t) = log 10(ADH(t) I × 10 12 )2R Ald (t) = log 10(Aldosterone(t) I × 10 11 )(3)HH 2 O(t) R = R ADH (t) × H 2 O(t) I − Na(t) 2 O(t) I+IFINa(t) I+IF(4.34)(4.35)(4.36)Na(t) R = R Ald (t) × Na max × Na(t) I (4.37)The volume of reabsorbed water H 2 O R (t) is the volume of waterrequired to reach equilibrium with the interstitial fluid, multipliedby the tubule response R ADH (t). The quantity of reabsorbed sodium

4.5 S U M M A RY 53Na R (t) is the volume of filtrate sodium Na I (t) multiplied by the tubuleresponse R Ald (t) and a parameter Na max . This parameter representsthe maximal amount of sodium reabsorption that can be achieved bythe tubule, and was kept fixed at 0.9. The effects of this parameter onthe model dynamics were not investigated, as the model analysis didnot focus on the urine formation and concentration mechanisms.Collecting ductsThe cortical collecting ducts have functional characteristics similarto those of the late distal tubule [71, p. 336]. In particular, theyexhibit the same reabsorption characteristics for sodium and water.The medullary collecting ducts, while differing in function from thecortical ducts in several ways, also exhibit the same reabsorptioncharacteristics for sodium and water [71, pp. 337–8].Accordingly, the update rules for the collecting ducts are the sameas the update rules for the distal tubule—the collecting duct responsesto ADH and aldosterone are calculated (Eq. 4.38 and Eq. 4.39), andwater and sodium are reabsorbed accordingly (Eq. 4.40 and Eq. 4.41).R ADH (t) = log 10(ADH(t) I × 10 12 )2R Ald (t) = log 10(Aldosterone(t) I × 10 11 )(3)HH 2 O(t) R = R ADH (t) × H 2 O(t) I − Na(t) 2 O(t) I+IFINa(t) I+IF(4.38)(4.39)(4.40)Na(t) R = R Ald (t) × Na max × Na(t) I (4.41)4.5 S U M M A RYThis chapter has introduced the model design. A new modellingapproach—hierarchical dynamical networks—was introduced, followedby an evaluation of the benefits of this approach and anoverview of appropriate analysis techniques. The model structurewas then presented, and the structural properties of the model wereanalysed. Finally, the state variables and update rules of the modelwere documented. An experimental analysis of the model dynamics isnow presented in the following chapter.

M O D E L D Y N A M I C S5This chapter presents an analysis of the dynamics of the model describedin Chapter 4. The experimental design used to investigate thebehaviour of the model was as follows:• A single-nephron system was demonstrated to reproduce experimentallyobserved behaviour. Without performing this validation,the results of any experiments conducted on multi-nephronsystems would be invalid. The single-nephron system was shownto exhibit regular oscillations in single-nephron glomerular filtrationrate (SNGFR), due to the tubuloglomerular feedback (TGF)mechanism. It was also demonstrated that the system was capableof constructing and maintaining a stable sodium gradient inthe medullary fluid, over a wide range of sodium concentrations.• A two-nephron system was demonstrated to produce experimentallyobserved behaviour, consistent with existing investigationsof two-nephron models. Specifically, the synchronisations thatarise between the filtration rates of the two nephrons wereanalysed, and the effects of hemodynamic coupling and vascularsignalling were shown to produce results similar to thoseobserved in existing models. The two-nephron system also exhibiteda larger, more stable sodium gradient in the medullaryfluid than the single-nephron system. This is consistent with theadditional concentrating power associated with multiple loopsof Henle that is produced by existing models.• The strength of the hemodynamic coupling in the two-nephronsystem was explored for range of values for P A and k. By measuringthe Lyapunov stability of the two-nephron system fordifferent values of these parameters, it was confirmed that bothparameters had equivalent effects on the hemodynamic couplingin the system.• The analysis was then extended to an 8-nephron system, whichproduced a larger, more stable gradient than the two-nephronsystem, again consistent with results obtained from existingmodels.• The effects of hemodynamic coupling and vascular signallingon the 8-nephron system were then investigated. Rather thananalysing the dynamics of the individual nephrons, the analysisfocused on whole-system properties such as the whole-systemfiltration rate (WSFR). Power spectral densities were used todetermine the dominant frequencies present in the WSFR. Theresults showed that the degree of vascular signalling present inthe system not only affected the average WSFR, but also alteredthe dominant frequencies.• The analysis was finally extended to determine the effects ofvascular signalling on a 72-nephron multi-column system. Again,55

56 M O D E L DY NA M I C Sthe analysis did not focus on single-nephron dynamics, butrather on whole-column filtration rate (WCFR) and WSFR. Thepower spectral densities of the WCFR and WSFR again demonstratedthat the degree of vascular signalling did not only affectthe average filtration rates, but also affected the dominant frequencies.For example, in the absence of vascular signalling theWSFR exhibited a dominant low frequency. However, when vascularsignalling was introduced, the WSFR exhibited significanthigher-frequency signals.This model is also available via the KidneyGrid portal, which letsresearchers access a number of kidney models from around the world[42, 43]. The purpose of the KidneyGrid is to support collaborativeresearch and shared access to knowledge, both within the renal sciencecommunity and for external audiences (e. g., for educationalpurposes), without depending on single laboratories or specific programmingenvironments [42]. Currently, KidneyGrid users are ableto run simulations over ranges of parameters for two-nephron andeight-nephron systems. Visualisation is provided in the form of graphsof time-series data such as SNGFR and WSFR.The experimental results obtained from the experiments outlinedabove are now presented.5.1 T H E S I N G L E-NEPHRO N M O D E LTo validate the dynamics of the single-nephron model, the analysisfocused on the regulation of SNGFR mediated by the TGF mechanism,and the ability of the loop of Henle to construct and maintain a sodiumgradient in the medullary fluid. Throughout this analysis, the soluteconcentrations in the blood are—unless stated otherwise—as shownin Table 10.Solute Min (mmol/L) Max (mmol/L) SourceSODIUM 135 145 [96]CHLORIDE 95.0 103.0 [96]POTASSIUM 3.5 5.0 [96]Molecule Min (g/L) Max (g/L) SourceADH 0.0 4.7 × 10 −9 [1]ALDOSTERONE 5.0 × 10 −8 2.5 × 10 −7 [1]PROTEIN 54.0 54.0 [19]Table 10: Steady-state concentration levels.Experimental studies of SNGFR regulation (e. g., [103, 102, 78]) haveobserved regular oscillations in SNGFR, and tubule models support thehypothesis that these oscillations are caused by the TGF mechanism([78, 19]).The single-nephron model produced regular oscillations in SNGFRover a wide range of parameter values. These oscillations are driven

5.1 T H E S I N G L E-NEPHRO N M O D E L 57by the TGF mechanism, so a decrease in the strength of the feedbackshould result in smaller oscillations [78]. The single-nephron modelproduced this behaviour—plots of SNGFR for different values of theparameter k (the range parameter for the TGF signal) are shown inFig. 23—indicating that the model has successfully captured thisfeature of the TGF mechanism. The dynamics of the SNGFR were notinvestigated further for the single-nephron model, as the purposeof the analysis was to validate the behaviour of the single-nephronmodel, not to exhaustively explore the range of dynamics that it couldproduce.28282727SNGFR (nL / min)262524232221SNGFR (nL / min)26252423222120201950 100 150 200 250 300 350 400 450 5001950 100 150 200 250 300 350 400 450 500TimestepTimestep(a) k = 0.1 and P A = 110(b) k = 0.2 and P A = 11028282727SNGFR (nL / min)262524232221SNGFR (nL / min)26252423222120201950 100 150 200 250 300 350 400 450 5001950 100 150 200 250 300 350 400 450 500TimestepTimestep(c) k = 0.3 and P A = 110(d) k = 0.4 and P A = 110Figure 23: Regular oscillations in SNGFR.The construction and maintenance of a solute gradient in themedullary fluid is of great importance to the regulation of waterand sodium levels in the body. For example, this gradient allows thekidney to reabsorb a maximal amount of water when the body isdehydrated, producing highly concentrated urine [71, pp. 353]. Thisexistence of this gradient is due to the ability of the loop of Henleto act as a counter-current exchange mechanism [71, pp. 352]. Accordingly,the ability of the single-nephron model to construct andmaintain a gradient in the medulla was investigated.Figure 24 shows the medullary solute gradients for the systemswhose SNGFRs are shown in Fig. 23. The larger oscillations in SNGFRresult in larger variations in the magnitude of the salt gradient, but allfour systems demonstrate an ability to construct and maintain a stablesalt gradient in the medullary fluid. The magnitude of the gradientis at least an order of magnitude smaller than the salt gradientsobserved experimentally—the gradient between the top and bottomof the loop can be as large as 1200mmol/L in human kidneys [169]and over 2000mmol/L in other species [158, 76]. This difference inmagnitude is caused by three factors:

58 M O D E L DY NA M I C S• It does not take into account other solutes such as chloride andurea (e. g., if chloride were reabsorbed in quantities similar tosodium, the magnitude of the gradient would be doubled);• It does not have the concentrating power of multiple loops ofHenle [97, 98, 76, 108]—this increase in concentrating poweris demonstrated in Sect. 5.2; and• It does not take into account other processes such as the productionof lactate in the tubule, which may contribute significantlyto accumulation of sodium [76].Given these limitations, the ability of the model to maintain sucha gradient demonstrates that the equations for the loop of Henle(Sect. 4.4) are able to capture the basic behaviour of the countercurrentexchange mechanism.7070Sodium Gradient (mmol/L)605040302010Sodium Gradient (mmol/L)605040302010050 100 150 200 250 300 350 400 450 500050 100 150 200 250 300 350 400 450 500TimestepTimestep(a) k = 0.1 and P A = 110(b) k = 0.2 and P A = 1107070Sodium Gradient (mmol/L)605040302010Sodium Gradient (mmol/L)605040302010050 100 150 200 250 300 350 400 450 500050 100 150 200 250 300 350 400 450 500TimestepTimestep(c) k = 0.3 and P A = 110(d) k = 0.4 and P A = 110Figure 24: Construction and maintenance of a medullary sodiumgradient. The graphs show the difference in sodium concentrationbetween the outermost and innermost layers of medullary fluid.The stability of the sodium gradient across the medullary fluid wasfurther examined by changing the sodium delivery to the maculadensa, to provoke minimal and maximal TGF responses. In orderto change the sodium delivery, blood sodium levels were decreasedto 110 mmol/L and increased to 170 mmol/L. These levels exceedphysiologically normal levels—the body rigorously regulates the bloodsodium concentration—and only serve to illustrate the range of theTGF response.In all of the experiments shown in Fig. 24, the blood sodium concentrationwas 135–145 mmol/L (as listed in Table 10), and so therange of sodium delivery to the macula densa did not explore thelimits of the TGF response in the model. The stability of the sodium

5.1 T H E S I N G L E-NEPHRO N M O D E L 59gradient across the medullary fluid was further examined by changingthe sodium delivery to the macula densa, to provoke minimal andmaximal TGF responses. In order to change the sodium delivery, twoexperiments were conducted. In the first experiment, blood sodiumconcentration was decreased to 110 mmol/L; in the second experiment,blood sodium concentration was increased to 170 mmol/L. Theresults of these two experiments are now presented.For the first experiment, the blood sodium concentration was decreasedby 5 mmol/L every 500 time-steps (the decreases in bloodsodium concentration over time are listed in Table 11). The resultingSNGFR and medullary salt gradient are shown in Fig. 25. As the bloodsodium concentration decreased, so too did the sodium delivery tothe macula densa, and the TGF mechanism responded by dilating theafferent arteriole and increasing the SNGFR. As the SNGFR approachedits maximal value, the size of the oscillations decreased.Despite the decrease in blood sodium concentration, the medullarysalt gradient increased by only 11% when the blood sodium concentrationreached the minimal range of 110–120 mmol/L. This gradientincrease arose because, although the sodium concentration in eachlayer of medullary fluid decreased, the sodium concentration in thetop layers (the outer medulla) decreased more than the sodium concentrationin the bottom layers (the inner medulla). The difference insodium concentrations for these layers of medullary fluid are shownin Fig. 26.In the second experiment, the blood sodium concentration wasincreased by 5 mmol/L every 500 time-steps—the blood sodium concentrationranges are listed in Table 12. The resulting SNGFR andmedullary salt gradient are shown in Fig. 27. As the blood sodiumconcentration increased, so too did the sodium delivery to the maculadensa, and the TGF mechanism responded by constricting the afferentarteriole and decreasing the SNGFR. As the SNGFR approached itsminimal value, the size of the oscillations decreased.Despite the increase in blood sodium concentration, the medullarysalt gradient decreased by 11% when the blood sodium concentrationreached the maximal range of 160–170 mmol/L. This gradientdecrease arose because, although the sodium concentration in eachlayer of medullary fluid increased, the sodium concentration in thetop layers (the outer medulla) increased less than the sodium concentrationin the bottom layers (the inner medulla). The difference insodium concentrations for these layers of medullary fluid are shownin Fig. 28.In conclusion, this analysis of the single-nephron model dynamicsdemonstrate that the model is able to reproduce several patternsof behaviour that have been experimentally established. The singlenephronmodel exhibited regular oscillations in SNGFR whose amplitudeand frequency were comparable to both experimental data andexisting models. The model was also able to create and maintain astable salt gradient in the medullary fluid, over a wider range of bloodsodium concentrations than is physiologically reasonable.

60 M O D E L DY NA M I C STime-step Min (mmol/L) Max (mmol/L)0. . . 499 135 145500. . . 999 130 1401000. . . 1499 125 1351500. . . 1999 120 1302000. . . 2499 115 1252500. . . 3000 110 120Table 11: Systematic decrease blood sodium concentration.2970SNGFR (nL / min)2827262524232221Sodium Gradient (mmol/L)605040302010201000 2000 3000Timestep00 1000 2000 3000Timestep(a) The SNGFR(b) The medullary salt gradientFigure 25: The single-nephron dynamics in response to decreasedblood sodium concentration (k = 0.3 and P A = 110).190180Outer medullaInner medullaSodium (mmol/L)170160150140130120110500 1000 1500 2000 2500 3000TimestepFigure 26: The change in sodium concentration in the top (outermedulla) and bottom (inner medulla) layers of medullary fluid, inresponse to decreased blood sodium concentration.

5.1 T H E S I N G L E-NEPHRO N M O D E L 61Time-step Min (mmol/L) Max (mmol/L)0. . . 499 135 145500. . . 999 140 1501000. . . 1499 145 1551500. . . 1999 150 1602000. . . 2499 155 1652500. . . 3000 160 170Table 12: Systematic increase in blood sodium concentration.2870SNGFR (nL / min)2726252423222120Sodium Gradient (mmol/L)605040302010191000 2000 3000Timestep00 1000 2000 3000Timestep(a) The SNGFR(b) The medullary salt gradientFigure 27: The single-nephron dynamics in response to increasedblood sodium concentration (k = 0.3 and P A = 110).210200Outer medullaInner medullaSodium (mmol/L)190180170160150140500 1000 1500 2000 2500 3000TimestepFigure 28: The change in sodium concentration in the top (outermedulla) and bottom (inner medulla) layers of medullary fluid, inresponse to increased blood sodium concentration.

62 M O D E L DY NA M I C S5.2 T W O-NEPHRO N S Y S T E M SWith the single-nephron model dynamics validated, the focus of theanalysis turns to multi-nephron systems, which allow the interactionsand couplings between nephrons to be investigated. The multinephronanalysis begins with the simplest multi-nephron system—-atwo-nephron model, where each nephron has an identical structureand location in the renal fluid layers, and vascular signalling is absent(γ = 0).The first experiment investigated the effects that changing thearterial pressure (P A ) and the steepness of the TGF response (k) haveon the SNGFRs of the two nephrons. The state-space of P A × k wasexplored for values of P A ranging from a normal level of 108.5 mmHg[78] to a maximum of 114 mmHg in 0.5 mmHg increments, and forvalues of k ranging from 0.25 to 0.4 [78] in increments of 0.01. AsP A or k was increased, the two nephrons were observed to changefrom in-phase synchronisation to anti-phase synchronisation, with aregion in the middle where the nephrons were neither in-phase oranti-phase. This trend is summarised in Fig. 29, which shows themean absolute difference in filtration rates of the two nephrons overthe parameter space. This demonstrates the increasing the arterialpressure (P A ) and increasing the steepness of the TGF response (k)have similar effects on the two-nephron system, as suggested in [19].∆ (nL/min)20151050108110112P A (mmHg)0.400.360.320.28114 0.24 k (L/mmol)Figure 29: Mean difference in filtration rates in a two-nephron system,over the parameter space (P A , k).To measure the stability of the model in response to these parameterchanges, Lyapunov exponents were calculated about severalpoints in the phase space. An exponent L y for point y around pointx indicates that the behaviour of the system at y approaches thebehaviour of the system at x no slower than exp L y. The negativeLyapunov exponents (i. e., those where the behaviours of the two systemsconverge) about each of the chosen points are shown in Fig. 30.These graphs show that the system exhibits stable behaviour alongthe lines P A − k = C, for some fixed C. This further reinforces the

5.2 T W O-NEPHRO N S Y S T E M S 63hypothesis that P A and k have equivalent effects on the system [19],as an increase of one can be countered by a decrease in the other,and vice versa.-0.04-0.02L y-0.000.40L y(a) Lyapunov exponents at P A = 110.0, k = 0.33108110112P A (mmHg)114 0.24 0.32k (L/mmol)-0.04-0.02L y-0.000.40L y(b) Lyapunov exponents at P A = 113.0, k = 0.35108110112P A (mmHg)114 0.24 0.32k (L/mmol)Figure 30: Lyapunov stability over the parameter space (P A , k).The behaviour of the model was also examined a wider range ofarterial pressures P A (80–130 mmHg). At high pressures (115–130mmHg) the model exhibited anti-phase synchronisations, and dueto the non-compliant nature of the model, the difference betweenminimal and maximal filtration rates rose to 23 nL/min at P A = 130—shown in Fig. 31a. At low pressures (80–100 mmHg) the model exhibitedin-phase synchronisations, and the difference between minimaland maximal filtration rates decreased to 2.5 nL/min at P A = 80mmHg, again due to the non-compliant nature of the model—shownin Fig. 31b.

64 M O D E L DY NA M I C S552850SNGFR (nL / min)454035SNGFR (nL / min)272630500 600 700 800 900 100025500 600 700 800 900 1000TimestepTimestep(a) P A = 130 mmHg.(b) P A = 80 mmHg.Figure 31: The change in SNGFR in response to changes in P A .Anti-phase synchronisations have not been observed experimentally[167], as the vascular coupling between nephrons in the kidney overwhelmsthe hemodynamic coupling [112]. The two-nephron model hasbeen shown to produce anti-phase synchronisations in the absenceof vascular coupling, and it will now be shown that incorporatingvascular coupling into the model is sufficient to prevent anti-phasesynchronisations from arising, which is consistent with the resultspresented by Marsh et al. [112] and further validates the model.As hemodynamic and vascular coupling reinforce opposing behaviours,the two-nephron model is a suitable candidate for analysinghow the combination of these opposing mechanisms affect nephronbehaviour. The analysis so far has been conducted with hemodynamiccoupling present in the model (via the vascular network) and withoutany vascular coupling (γ = 0). As expected, the hemodynamiccoupling enforced anti-phase synchronisation between the two filtrationrates when the coupling was sufficiently strong (see Fig. 29).Accordingly, the two-nephron system exhibited regular anti-phaseoscillations for a range of P A and k values.To investigate the ability of vascular coupling to prevent anti-phasesynchronisation, the values P A = 110.5 mmHg and k = 0.3 were chosen,and the SNGFRs of the two nephrons were analysed over a rangeof γ values. The results are shown in Fig. 32, which demonstratesthat as γ is increased, the anti-phase nature of the SNGFR oscillationsis decreased until the system exhibits in-phase synchronisation. Themagnitude of the oscillations also decrease as γ is increased, sincevascular signalling causes the TGF signals to each afferent arterioleto approach the same value (i. e., when γ = 1 the signals will beidentical).If the strength of the hemodynamic coupling is increased—that is,if one or both of P A and k are increased—in the presence of vascularsignalling, then the anti-phase synchronisation becomes more prevalent.This is illustrated in Fig. 33, where k has been increased from0.2 to 0.3. Under these conditions, the effect of γ = 0.010 is similar tothe effect of γ = 0.005 when k = 0.2 (compare Fig. 32a and Fig. 33a).Similarly, for the case of γ = 0.11 the dynamics are similar to theeffect of γ = 0.008 when k = 0.2 (compare Fig. 32b and Fig. 33b).With two identical nephrons, the coupling mechanisms competeto force in-phase and anti-phase synchronisations on the system. Ifsome asymmetry is introduced into the model (e. g., if the time delay

5.2 T W O-NEPHRO N S Y S T E M S 6534343232SNGFR (nL / min)302826SNGFR (nL / min)302826242422500 600 700 800 900 100022500 600 700 800 900 1000TimestepTimestep(a) γ = 0.005: regular anti-phase oscillationsare present.(b) γ = 0.008: anti-phase oscillationsare less regular than for γ = 0.005.34323432SNGFR (nL / min)302826SNGFR (nL / min)302826242422500 600 700 800 900 100022500 600 700 800 900 1000TimestepTimestep(c) γ = 0.010: little evidence of in-phaseor anti-phase oscillations.(d) γ = 0.012: in-phase oscillations arepresent.Figure 32: The effect of varying the strength of vascular couplingbetween two identical nephrons, where P A = 110.5 mmHg and k =0.2.34343232SNGFR (nL / min)302826SNGFR (nL / min)302826242422500 600 700 800 900 100022500 600 700 800 900 1000TimestepTimestep(a) γ = 0.010: anti-phase oscillationsare present, similar to the behaviourwhen k = 0.2 and 0.005 < γ < 0.008.(b) γ = 0.011: anti-phase oscillationsare present, similar to the behaviourwhen k = 0.2 and γ = 0.008.Figure 33: The effect of varying the strength of vascular couplingbetween two identical nephrons, where P A = 110.5 mmHg and k =0.3.in one of the TGF mechanisms is altered), then the TGF responses ofthe two nephrons will be driven to different frequencies. The effectof such an asymmetry was investigated and the results are shown inFig. 34, where an additional two second delay was added to the TGFmechanism of one of the nephrons.As was the case for two identical nephrons, without any vascularcoupling the hemodynamic coupling forces an anti-phase synchronisationbetween the two SNGFRs, despite the different time delays inthe two TGF mechanisms. As the vascular coupling is increased, themagnitude of the oscillations decrease and become less regular—as

66 M O D E L DY NA M I C SSNGFR (nL / min)34333231302928272625242322500 600 700 800 900 1000SNGFR (nL / min)34333231302928272625242322500 600 700 800 900 1000TimestepTimestep(a) γ = 0.005 and identical TGF delays.(b) γ = 0.005 and different TGF delays.31313030SNGFR (nL / min)292827SNGFR (nL / min)29282726500 600 700 800 900 100026500 600 700 800 900 1000TimestepTimestep(c) γ = 0.008 and identical TGF delays.(d) γ = 0.008 and different TGF delays.3030SNGFR (nL / min)2928SNGFR (nL / min)292827500 600 700 800 900 100027500 600 700 800 900 1000TimestepTimestep(e) γ = 0.010 and identical TGF delays.(f) γ = 0.010 and different TGF delays.3030SNGFR (nL / min)2928SNGFR (nL / min)292827500 600 700 800 900 100027500 600 700 800 900 1000TimestepTimestep(g) γ = 0.012 and identical TGF delays.(h) γ = 0.012 and different TGF delays.2929SNGFR (nL / min)28SNGFR (nL / min)2827500 600 700 800 900 100027500 600 700 800 900 1000TimestepTimestep(i) γ = 0.020 and identical TGF delays.(j) γ = 0.020 and different TGF delays.Figure 34: The effect of varying the strength of vascular couplingbetween two nephrons, whose TGF delays differ by two seconds.

5.3 A N I N H O M O G E N E O U S 8-NEPHRO N S Y S T E M 67shown in Fig. 34d for γ = 0.08—in the same manner as for the systemof two identical nephrons. The two SNGFRs approach in-phase synchronisationas the vascular coupling is further increased. However, unlikethe system of two identical nephrons, some differences between thetwo SNGFRs remain even when γ = 0.20 (shown in Fig. 34j). Thus, thedifferent delays in the TGF mechanism prevent the two nephrons fromexhibiting identical SNGFRs.Now that the interactions between the hemodynamic and vascularcoupling mechanisms have been investigated, the analysis turns tothe medullary salt gradient produced by the two-nephron model.When the blood sodium concentration was within the normal rangeof 135–145 mmol/L, the two-nephron model was able to maintain astable salt gradient 46% larger than the gradient produced by thesingle-nephron model, as shown in Fig. 35. This increase in the saltgradient is due to the increased concentrating power of multipleloops of Henle.The two-nephron system also produced a more stable salt gradientthan the single-nephron system—Fig. 36 shows the salt gradientwhen blood sodium concentration was decreased as per Table 11, andFig. 37 shows the salt gradient when the blood sodium concentrationwas increased as per Table 12. In response to these changes inblood sodium concentration, the salt gradient in the two-nephronsystem exhibited smaller variations than the single-nephron system.These variations are summarised in Table 13. This result suggeststhat adding more nephrons to the model results in a larger, morestable gradient, which is consistent with existing medullary models[97, 98, 76, 108].5.3 A N I N H O M O G E N E O U S 8 - N E P H RO N S Y S T E MThe two-nephron system was used to analyse the interaction betweenthe competing coupling mechanisms of hemodynamic couplingand vascular signalling—and also to investigate the stability of themedullary salt gradient maintained by multiple loops of Henle—in thesimplest possible system. The analysis is now extended to an eightnephronsystem that consists of four nephron pairs, each pair havinglonger loops of Henle than the last. The structure of this system isshown in Fig. 38.The medullary salt gradient of the system was investigated over thesame changes in blood sodium concentration as the single-nephronand two-nephron systems (the changes are listed in Table 11 andTable 12). In comparison to these systems, the 8-nephron systemmaintained a larger, more stable medullary gradient, as illustratedin Fig. 39. The sodium concentrations at the top (outer medulla) andbottom (inner medulla) of the medullary fluid are shown in Fig. 40,and the oscillations in these concentrations are much smaller thanthose in the previous systems (e. g., compare to Fig. 36 and Fig. 37).This is further evidence that the combined dynamics of multipleloops of Henle result in a larger medullary gradient, which is againconsistent with existing medullary models [97, 98, 76, 108]. As shownin Fig. 41, in response to these changes in blood sodium concentration

68 M O D E L DY NA M I C SSodium Gradient (mmol/L)70605040302010Sodium (mmol/L)220210200190180170160150140130Outer medullaInner medulla00 1000 2000 3000120500 1000 1500 2000 2500 3000TimestepTimestep(a) The magnitude of the salt gradient.(b) The salt concentration in the innerand outer medulla.Figure 35: The medullary sodium gradient in a two-nephron systemwhen blood sodium concentration remains at 135–145 mmol/L.Sodium Gradient (mmol/L)70605040302010Sodium (mmol/L)220210200190180170160150140130Outer medullaInner medulla00 1000 2000 3000120500 1000 1500 2000 2500 3000TimestepTimestep(a) The magnitude of the salt gradient.(b) The salt concentration in the innerand outer medulla.Figure 36: The medullary sodium gradient in a two-nephron systemwhen blood sodium concentration progressively drops to 110–120mmol/L.70220Sodium Gradient (mmol/L)60504030201000 1000 2000 3000Sodium (mmol/L)210200190180170160150140130120Outer medullaInner medulla500 1000 1500 2000 2500 3000TimestepTimestep(a) The magnitude of the salt gradient.(b) The salt concentration in the innerand outer medulla.Figure 37: The medullary sodium gradient in a two-nephron systemwhen blood sodium concentration progressively rises to 160–170mmol/L.

5.3 A N I N H O M O G E N E O U S 8-NEPHRO N S Y S T E M 69CortexOuterMedullaInnerMedullaFigure 38: Structure of the 8-nephron model.the WSFR increases and decreases in the same manner as the SNGFRin the single-nephron and two-nephron systems.Note that while this 8-nephron system has exhibited a largermedullary salt gradient than the previous single-nephron and twonephronsystems, the increase is relatively small—21%—when comparedto the increase of 46% that was observed between the singlenephronand two-nephron systems. The likely causes for this are:the 8-nephron system may have approached the maximal medullarygradient that can be produced by the equations in Sect. 4.4.5; andthe value of Na 1/2 (see Table 6) is the same for each nephron, whichmay be unreasonable as the sodium delivery to the macula densa willdiffer depending on the length of the loop of Henle.While the two-nephron system was used to investigate the effectsof vascular signalling on the synchronisations between individualnephrons, the 8-nephron system was used to investigate the effectsof vascular signalling on the WSFR of a system of inhomogeneousnephrons. Figure 42 shows the WSFR of the 8-nephron system forseveral values of γ, the percentage of the TGF signal that leaks toneighbouring nephrons. These simulations demonstrated that thepresence of vascular signalling decreased the average WSFR and alsoled to smaller oscillations in WSFR.These trends are not surprising, since vascular signalling increasesthe TGF signal sent to each nephron—the nephron receives a fractionof neighbouring TGF signals in addition to its own TGF signal—whichleads to a decrease in the SNGFR of each nephron. And as shown inSect. 5.1, the size of the oscillations in SNGFR are decreased whenthe sodium delivery to the macula densa differs greatly from thehalf-maximal value Na 1/2 (see Fig. 3). Accordingly, increasing thedegree of vascular signalling in the 8-nephron system decreases theaverage SNGFR and the size of the oscillations in SNGFR, which resultsin similar effects on the WSFR.However, the power spectral densities of the different WSFRs (shownin Fig. 43) demonstrate that the degree of vascular signalling has amore complex effect on WSFR than simply reducing the average WSFRand the magnitude of WSFR oscillations. The different peaks exhibitedby the power spectral densities indicate that different dominant frequenciesare present in the WSFR, depending on the value of γ. Thismeans that the vascular signalling has affected the SNGFR frequenciesof the individual nephrons in the system, due to vascular signalling

70 M O D E L DY NA M I C SSodium (mmol/L) Single nephron Two nephrons110–120 +9.8% +0.6%160–170 −13.0% −7.5%Table 13: The difference in the sodium gradient at standard (135–145mmol/L) and altered blood sodium concentrations, which demonstratesthat the two-nephron system maintains a more stable gradientthan the single-nephron system.7070Sodium Gradient (mmol/L)605040302010Sodium Gradient (mmol/L)60504030201000 1000 2000 300000 1000 2000 3000TimestepTimestep(a) Blood sodium concentration progressivelydropped to 110–120 mmol/L.(b) Blood sodium concentration progressivelyraised to 160–170 mmol/L.Figure 39: The medullary salt gradient in an eight-nephron systemwhen blood sodium concentration is altered.Sodium (mmol/L)230220210200190180170160150140130120110Outer medullaInner medulla500 1000 1500 2000 2500 3000Sodium (mmol/L)230220210200190180170160150140130120110Outer medullaInner medulla500 1000 1500 2000 2500 3000TimestepTimestep(a) Blood sodium concentration progressivelydropped to 110–120 mmol/L.(b) Blood sodium concentration progressivelyraised to 160–170 mmol/L.Figure 40: The salt concentration in the inner and outer medulla ofan eight-nephron system when blood sodium concentration is altered.210210200200WSFR (nL/min)190180170WSFR (nL/min)190180170160160150500 1000 1500 2000 2500 3000150500 1000 1500 2000 2500 3000TimestepTimestep(a) Blood sodium concentration progressivelydropped to 110–120 mmol/L.(b) Blood sodium concentration progressivelyraised to 160–170 mmol/L.Figure 41: The WSFR of an eight-nephron system when blood sodiumconcentration is altered.

5.3 A N I N H O M O G E N E O U S 8-NEPHRO N S Y S T E M 71190185185180WSFR (nL/min)180175WSFR (nL/min)175170170165165500 1000 1500 2000 2500 3000160500 1000 1500 2000 2500 3000TimestepTimestep(a) γ = 0.000(b) γ = 0.001175164162170WSFR (nL/min)165160WSFR (nL/min)160158156154155500 1000 1500 2000 2500 3000152500 1000 1500 2000 2500 3000TimestepTimestep(c) γ = 0.002(d) γ = 0.003156144WSFR (nL/min)154152150148WSFR (nL/min)142140138146500 1000 1500 2000 2500 3000136500 1000 1500 2000 2500 3000TimestepTimestep(e) γ = 0.004(f) γ = 0.005Figure 42: The WSFR of eight-nephron systems with different levels ofvascular signalling.

72 M O D E L DY NA M I C S2e-206e-201.8e-201.6e-205e-20Power1.4e-201.2e-201e-208e-216e-21Power4e-203e-202e-204e-212e-211e-2000 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(a) γ = 0.000(b) γ = 0.0012.5e-197e-202e-196e-205e-20Power1.5e-191e-19Power4e-203e-202e-205e-201e-2000 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(c) γ = 0.002(d) γ = 0.0031e-208e-21Power9e-218e-217e-216e-215e-214e-213e-212e-211e-21Power7e-216e-215e-214e-213e-212e-211e-2100 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(e) γ = 0.004(f) γ = 0.005Figure 43: The WSFR power spectral density of eight-nephron systemswith different levels of vascular signalling.

5.4 A 7 2-NEPHRO N M U LT I-COLUMN M O D E L 73between nephrons whose loops of Henle have different lengths. Moreinterestingly, the spectral densities for γ = 0.000 (Fig. 43a) andγ = 0.005 (Fig. 43f) are very similar, despite the spectral densities for0.000 < γ < 0.005 showing no such similarity. This suggests that theWSFRs shown in Fig. 42a and Fig. 42f have similar properties—despitethe obvious differences in average and oscillation size—which is asurprising observation.5.4 A 7 2 - N E P H RO N M U LT I-COLUMN M O D E LThe eight-nephron system was used to analyse the interaction betweenthe competing coupling mechanisms of hemodynamic couplingand vascular signalling—and also to investigate the stability of themedullary salt gradient maintained by multiple loops of Henle—in asystem of inhomogeneous nephrons. The analysis is now extended toa 72-nephron system consisting of nine columns arranged in a 3 × 3grid, with each column having a structure identical to that of theeight-nephron system presented in Sect. 5.3. The 2D arrangementof the columns and the structure of the arterial tree are shown inFig. 44.P AP VR Cy23 6 9R C1 2 312 5 8R C4 5 601 4 70 1 2x7 8 9(a) The 2D arrangement ofinterstitial fluid columns.(b) The arterial tree connecting thecolumns.Figure 44: The high-level structure of the 72-nephron model. Eachcolumn is assigned a number 1–9. The internal structure of eachcolumn is identical, and is shown in Fig. 38.In the absence of vascular signalling (i. e., γ = 0), the WSFR exhibitsoscillations of 100–200 nL/min (see Fig. 45a). The power spectraldensity shows that these oscillations predominantly occur at a singlefrequency—the frequency at which the longest-looped nephronsnaturally oscillate. This is different to the power spectral density ofthe 8-nephron system, where higher-frequency peaks are present(see Fig. 43a), and indicates that the majority of higher-frequencysignals (such as the oscillations of the shorter-looped nephrons) canceleach other out in this system. Figure 46 shows the individualWCFRs, and the power spectral densities clearly indicate the presence

74 M O D E L DY NA M I C Sof higher-frequency oscillations in the WCFRs. Note that althoughthe structure of the individual columns are homogeneous, the WCFRplots and power spectral density plots differ, which indicates thatthe higher-level structure of the system (i. e., the 2D column arrangementand the arterial tree connecting the columns) is affecting thedynamics of each column.Once vascular signalling is introduced into the 72-nephron system,the patters in the WSFR become less clear. As shown in Fig. 47, theaddition of vascular signalling results in higher-frequency signalsbecoming more dominant than when gamma = 0, although this trendis not entirely consistent—witness Fig. 47c. Te individual WCFRs, however,show a consistent increase in the presence of higher-frequencysignals when compared to their behaviour when γ = 0—compare thepower spectral densities in Fig. 48 with those in Fig. 46.Finally, it was observed that as the degree of vascular signalling wasincreased, the average WSFR and WCFRs decreased—shown in Fig. 49—which is consistent with the results produced by the 8-nephron systemin Sect. 5.3.5.5 S U M M A RY O F E X P E R I M E N T SThe experiments in this chapter have demonstrated that the modelpresented in Chapter 4 produces valid dynamics across a range ofmulti-nephron systems.Firstly, a single-nephron system was shown to produce oscillationsin SNGFR mediated by the TGF mechanism, and was also shown tomaintain a stable salt gradient in the medullary fluid across thephysiological range of blood sodium concentrations.A two-nephron system was then used to investigate the interactionsthat arise between nephrons. Over the state space k × P A , the systemwas shown to transition between in-phase and anti-phase oscillationsin the absence of vascular signalling, and the parameters k and P Awere shown to have equivalent effects on these oscillations. Whenvascular signalling was introduced to the system, it was shown tooverwhelm the hemodynamic coupling present in the system and enforcein-phase oscillations, even when the nephrons featured differentdelays in the TGF mechanism. This system also maintained a larger,more stable medullary salt gradient than the single-nephron system,suggesting that the presence of multiple loops of Henle is responsiblefor the stability and magnitude of the gradient.An eight nephron system was then demonstrated to maintain amedullary salt gradient that was both larger and more stable than thetwo-nephron system, providing further evidence that additional loopsof Henle lead to larger, more stable medullary salt gradients. Theeffects of vascular signalling on the WSFR were then investigated, andit was shown that increased levels of vascular signalling decreasedthe average WSFR. Moreover, the spectral densities of the WSFR demonstratedthat the degree of vascular signalling affected the dominantfrequencies in the WSFR.Finally, a 72-nephron system of 9 columns was analysed. Althoughthe columns were homogeneous, the WCFRs differed, depending onwhich branch of the arterial tree the column was connected to. An

5.5 S U M M A RY O F E X P E R I M E N T S 759e-178e-177e-176e-17Power5e-174e-173e-172e-171e-1700 0.02 0.04Normalised Frequency(a) The WSFR.(b) The WSFR power spectral density.Figure 45: The WSFR of the 72-nephron system when γ = 0.WCFR (nL/min)280260240220200180160Power4e-173.5e-173e-172.5e-172e-171.5e-171e-175e-18140500 1000 1500 2000 2500 3000Timestep00 0.02 0.04Normalised Frequency(a) The WCFR of columns 1–3.(b) The spectral density of columns 1–3.2602.5e-17WCFR (nL/min)240220200180160140Power2e-171.5e-171e-175e-18120500 1000 1500 2000 2500 3000Timestep00 0.02 0.04Normalised Frequency(c) The WCFR of columns 4–6.(d) The spectral density of columns 4–6.2402e-172201.8e-171.6e-17WCFR (nL/min)200180160Power1.4e-171.2e-171e-178e-186e-181404e-182e-18120500 1000 1500 2000 2500 300000 0.02 0.04TimestepNormalised Frequency(e) The WCFR of columns 7–9.(f) The spectral density of columns 7–9.Figure 46: The individual WCFRs in the 72-nephron system when γ = 0.

76 M O D E L DY NA M I C S9e-171.8e-178e-171.6e-177e-171.4e-176e-171.2e-17Power5e-174e-17Power1e-178e-183e-176e-182e-174e-181e-172e-1800 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(a) γ = 0.000.(b) γ = 0.001.3e-173e-172.5e-172.5e-172e-172e-17Power1.5e-17Power1.5e-171e-171e-175e-185e-1800 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(c) γ = 0.002.(d) γ = 0.003.2.5e-179e-188e-182e-177e-18Power1.5e-171e-17Power6e-185e-184e-183e-185e-182e-181e-1800 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(e) γ = 0.004.(f) γ = 0.005.Figure 47: Power spectral densities of the WSFR of the 72-nephronsystem for different values of γ.

5.5 S U M M A RY O F E X P E R I M E N T S 771e-174e-189e-183.5e-188e-187e-183e-18Power6e-185e-184e-18Power2.5e-182e-181.5e-183e-182e-181e-181e-185e-1900 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(a) Columns 1–3 when γ = 0.003.(b) Columns 1–3 when γ = 0.005.9e-183.5e-188e-183e-187e-186e-182.5e-18Power5e-184e-18Power2e-181.5e-183e-182e-181e-181e-185e-1900 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(c) Columns 4–6 when γ = 0.003.(d) Columns 4–6 when γ = 0.005.3.5e-181.8e-183e-181.6e-182.5e-181.4e-181.2e-18Power2e-181.5e-18Power1e-188e-191e-186e-194e-195e-192e-1900 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(e) Columns 7–9 when γ = 0.003.(f) Columns 7–9 when γ = 0.005.Figure 48: Power spectral densities of the WCFRs of the 72-nephronsystem for different values of γ.18001700WSFR (nL/min)160015001400WCFR (nL/min)2202001801601404-67-913000 0.001 0.002 0.003 0.004 0.0050.001γ0.0031-30.005Columnγ(a) The change in average WSFR.(b) The change in average WCFR.Figure 49: Average WCFRs and WSFRs of the 72-nephron system fordifferent values of γ.

78 M O D E L DY NA M I C Sanalysis of the power spectral densities showed the arterial tree alsoinfluenced the dominant frequencies in the WCFRs. Increasing thedegree of vascular signalling in this system was shown to result inthe presence of higher-frequency signals in the WCFRs and WSFR, andalso to reduce the average WCFRs and WSFR.

E M E R G E N T E F F E C T S O F L O C A L I S E D I M PA I R M E N T6The dynamics of several multi-nephron systems have been analysedand presented in Chapter 5, providing a benchmark against whichthe behaviour of similar systems can be compared. In particular, suchbenchmarks allow the system-wide effects of localised changes indynamics (hereafter referred to as impairments) to be determined.The magnitude of the effect of an impairment on the whole-systemdynamics forms a measure of the stability or resilience of the system.The most obvious application of this approach—particularly usefulwhen applied to a whole-kidney model—is to model renal diseases asimpairments and to study how the diseases affect the whole-systemdynamics. Such studies could investigate which symptoms of a renaldisease have the most significant impact on renal function, or determinewhat fraction of the nephron population must become impairedto reach a particular stage of chronic kidney disease (CKD). The mostcommon symptoms of renal disease have been introduced in Sect. 2.3,and the application of network models and automata to studyingdisease has been reviewed in Sect. 3.4.This chapter presents an example of how to incorporate an impairmentinto the model and how to determine the impact of theimpairment on the dynamics of multiple systems and at differentlevels of hierarchy. This example introduces a simple impairment thatattempts to capture a single effect of proteinuria—the presence ofabnormally large amounts of protein in the urine (see Sect. 2.3 forfurther details). This impairment only serves to illustrate the generalapproach, and not to predict the effects that proteinuria might haveon kidney behaviour.To begin, the method by which impairments are incorporated intothe model is described, and the control that the modeller has over theincidence and spread of an impairment is discussed. Secondly, thechoice of the specific impairment is justified, and the design of theimpairment is presented. Thirdly, several techniques for measuringthe effects of the impairment on the system dynamics are outlined,focusing on the dynamics at several levels of hierarchy.The dynamics of several impaired systems are then presented andcompared to the dynamics of the original (unimpaired) systems. Twoexperiments investigate the effects that a single impaired nephronand pairs of impaired nephrons have on an 8-nephron system; twofurther experiments investigate the effect of impairments in the contextof a 72-nephron system. The chapter concludes by drawing someconclusions about the stability of the model in response to the chosenimpairment.6.1 I N C O R P O R AT I N G I M PA I R M E N T S I N T O T H E M O D E LFor an impairment to change the model dynamics, it must changeeither the model structure (Sect. 4.2) or the underlying equations79

80 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T(Sect. 4.4). The high-level structure of the model (Fig. 14 and Fig. 15)captures the physical structure of the kidney, which should not bealtered by an impairment. The lower-level structure (Fig. 12) capturesthe functional structure of the nephrons, which is closely tied to thephysical structure of the nephrons and should also not be alteredby an impairment. Instead, it is clear that impairments should onlyintroduce changes to the underlying model equations.To allow for both impaired and unimpaired nephrons to exist in amulti-nephron system, it is necessary for each nephron to includean impair state that indicates whether the nephron is impaired ornot. When the model equations are applied to a particular nephron,the impair state is examined and the appropriate modifications tothe model equations are applied. It is straightforward to extend thisapproach to support multiple impairments.Once the impairments have been defined, the evolution of thenephron impair states over time must be specified. All nephrons ina model are initially unimpaired and thus exhibit normal behaviour.The different ways in which a nephron may become impaired are nowdescribed.6.1.1 Stochastic impairmentsEach impairment is assigned a failure probability, which defines thelikelihood that a nephron will become impaired over a single timestep.This probability can be defined with respect to time (i. e., a fixedprobability for each time-step) or with respect to blood flow to thenephron at that time-step (i. e., a probability that can vary with eachtime-step).Similarly, each impaired is assigned a recovery probability, whichdefines the likelihood that an impaired nephron will become unimpaired.This probability can also be defined with respect to time or toblood flow to the nephron.Note that this approach models the transition between impair statesas a Markov Chain.6.1.2 Stochastic spread of impairmentsImpairments are also assigned a spreading probability that definesthe likelihood of that impairment spreading from an impaired nephronto a neighbouring nephron over a single time-step. For this to be welldefined,a definition of what constitutes a neighbouring nephron mustbe given.Nephrons in a single column of interstitial fluid are arranged in anumber of branches (see Sect. 4.2.2 and Fig. 14). If two nephronsin the same branch are next to each other, it is natural to considerthem as neighbours. The first nephron in a branch is also consideredto be a neighbour of the first nephron in the adjacent branchesabove and below it (again, see Fig. 14). More concisely, nephrons areneighbours if they are adjacent in either a depth-first or breadth-firsttraversal of the column’s nephron tree. Finally, the first nephron in anentire column is considered to be a neighbour of the first nephrons inadjacent columns (see Fig. 15 and Fig. 16).

6.2 C H O I C E O F I M PA I R M E N T 816.1.3 Deterministic impairmentsBy allowing impairments to arise, spread and recover in a stochasticmanner, it can be observed how they spread across the model, ina similar manner to the network models used in epidemiology (seeSect. 3.4). However, studying the effects that impaired nephrons haveon a system of nephrons—as opposed to studying the manner in whichthe impairments spread—requires explicit control over the impairstate of the nephrons.To this end the model also supports deterministic impairments,where any number of impairments can be introduced to particularnephrons in a system at specific time-steps. These impairments occurin parallel with the stochastic methods described previously, andfor absolute control over the impair states it is necessary to set thefailure, spreading and recovery probabilities of the impairment tozero.6.2 C H O I C E O F I M PA I R M E N TAs outlined in Sect. 2.3, there are many diseases that affect thekidney. For the purposes of disease modelling, however, only theactual symptoms of the disease are of interest—the specific causes(e.g., buildup of scar tissue, calcium deposits) are not directly relevant.As such, the modelling concern does not focus on a specific renaldisease, but rather on the symptoms of that disease. By designingimpairments for these symptoms, the effects of renal diseases can bemodelled as specific combinations of these impairments.As was stated in Sect. 2.3, diabetic nephropathy, hypertensionand glomerulonephritis account for around 75% of all adult cases ofchronic renal failure (CRF) [41], with diabetic nephropathy the mostcommon cause of CRF and end-stage renal failure (ESRF) in the USA[53]. The focus of the impairment example to be presented here istherefore on the symptoms presented by these diseases:E D E M A Excess fluid in the body.H E M AT U R I A Blood present in the urine.H Y P E R L I P I D E M I A Raised levels of lipids in the blood.H Y P E RT E N S I O N Chronically elevated blood pressure.H Y P OA L B U M I N E M I A Abnormally low albumin levels in the blood.P ROT E I N U R I A Protein present in the urine.Some of these symptoms are caused by the presence of othersymptoms—e.g., proteinuria can result in hypoalbuminemia, whichreduces the oncotic pressure in the blood and can lead to edema—and are not relevant in isolation. Other symptoms can be caused bychanging model parameters—e.g., hypertension can be simulatedby increasing the arterial pressure parameter—having a systemwideeffect as opposed to the per-nephron effects of impairmentsin our model, and must be controlled separately. Of the symptoms

82 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N Tdescribed here, the impairment that will now be presented only considersproteinuria—this impairment serves only as an example ofthis approach, and not as a complete, detailed model of one or moresymptoms.6.2.1 Implementation of the impairmentProteinuria is the presence of abnormally large amounts of protein inthe urine. As proteins are not secreted by the tubule, this conditionarises due to the presence of proteins in the tubule filtrate. Whilethis can have a number of effects on the function of the tubule, theonly effect that will be captured by the impairment is that on thereabsorption of water in the descending limb of Henle.The presence of protein in the filtrate increases the oncotic pressureof the filtrate and thus retards the diffusion of water from thefiltrate into the surrounding interstitial fluid. Accordingly, the impairmentwill capture this reduction in reabsorption by modifying theequations that govern the reabsorption of water in the limbs of Henle(as given in Sect. 4.4.5).Specifically, for each descending limb that is affected by the impairment,the amount of water reabsorbed into the interstitial fluidis some fraction D i of the amount that would be reabsorbed werethe tubule not affected by proteinuria. This is modelled by changingthe w i coefficients in Eq. 4.29 to take into account D i , resulting inEq. 6.1. For impaired nephrons, the value of D i is a model parameter,while for unimpaired nephrons D i = 1.⎡w 1′ w2′ x =⎢ . ⎣ .wN′⎤⎥⎦, w ′ i = w iD i(6.1)6.3 M E A S U R I N G T H E E FF E C T S O F I M PA I R M E N T SThe goal of this example impairment is to determine how systemdynamics are affected by impaired nephrons, and so the focus of theanalysis will be on whole-system dynamics rather than the dynamicsof individual tubule segments.Existing multi-nephron models have focused on single-nephronglomerular filtration rate (SNGFR) [99] and tubular pressure [112, 81];due to the design of the model, pressure is not calculated anywherein the tubule. Pressure can be treated as varying in proportion toflow rate, if the tubule is assumed to be non-compliant or only slightlycompliant. This point is discussed further in Sect. 4.4.2.In the experiments performed previously (Chapter 5), the analysisof the system dynamics focused on SNGFR—in particular, the frequencyof the oscillations and the phase (relative to other nephrons)—andso SNGFR will form the starting point of the analysis here. With theemphasis on whole-system dynamics, the analysis will also extend to

6.4 I M PA I R M E N T S I N A N 8-NEPHRO N S Y S T E M 83comparing the whole-system filtration rate (WSFR) of unimpaired andimpaired systems.Based on the assumption that the filtration rate at any level of hierarchy(e. g., single nephrons, multiple columns) can be characterisedby a small number of measurements (e. g., frequency, amplitude,phase and mean value), the low-level dynamics can be ignored. Instead,the analysis can determine the effects of the impairment—andthe degree to which the effects permeate across the system—bycomparing these characteristic measurements in the impaired andunimpaired systems at multiple levels of hierarchy.Once these effects have been analysed and quantified, it may transpirethat the impairment can be simplified—i. e., some of the changesto the model equations may be removed without affecting the systemdynamics—allowing the modeller to determine which aspects of theimpairment have the largest effects. To draw an analogy with flocks ofbirds [152, 153]—where three exceedingly simple rules for individualnavigation are sufficient to generate realistic aggregate motion foran entire flock—a thorough, detailed set of impairments may prove tobe more complex than necessary to model a chosen phenomena, suchas a particular renal disease.6.4 I M PA I R M E N T S I N A N 8 - N E P H RO N S Y S T E MThe first investigation was to analyse the effect that impaired nephronshave on the dynamics of the eight-nephron system presented inSect. 5.3. This system consists of four nephron pairs, each pair havinglonger loops of Henle than the last–this arrangement is shown inFig. 50.01 2 3 4 5 6 7 8Length123Figure 50: The loops of Henle in the eight-nephron system. Eachnephron is identified by a number from 1 to 8, and the length of theloops are rated on a scale of 0 to 3.Two separate experiments were conducted, to investigate the effectsof a single impaired nephron and the effects of two impairednephrons. These experiments are now presented and compared, andthe effects of impaired nephrons on the eight-nephron system aresummarised.In the following discussion, nephrons are frequently referred to bythe length of their loop of Henle (0 . . . 3) and their nephron number(1 . . . 8). The SNGFR oscillation frequency of the nephron pair with loopof Henle length L is also referred to as the natural frequency of looplength L, or the loop length L frequency.

84 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T6.4.1 A single impaired nephronThe first experiment was to investigate how a single impaired nephroncould affect the eight-nephron system. For each nephron in the system,simulations were run where the nephron was deterministicallyinfected at t = 1, 200 and the simulation ran until t = 3, 000. A simulationwas run for each value of D P from 0.5 to 0.95—in steps of0.05—and the model parameters and conditions were otherwise identicalto the experiments performed upon the system in Sect. 5.3.A comparison of the WSFR for each of these simulations is presentedin Fig. 51—Fig. 51a shows the average WSFRs and Fig. 51b shows themagnitude of the oscillations in WSFRs. From Fig. 51a it is clear thatthe impairment decreased the average WSFR, and that the magnitudeof this decrease is proportional to both D 1 and the length of thePimpaired nephron’s loop of Henle. However, Fig. 51b shows that theoscillations in WSFR were not affected by the impairment unless oneof the longest-loop nephrons was impaired and D P was small ( 0.6).A more detailed comparison between the original eight-nephronsystem and one of the impaired systems will now be given. Fig. 51shows that the impaired system that differed the most from the originalsystem was when one of the longest-loop nephron was infectedand D P = 0.5, and so it is with this impaired system that the originalsystem will be compared. Specifically, in this impaired system theimpaired nephron is nephron 7.The WSFR of the two systems are shown in Fig. 52a, where thedecrease in average WSFR due to the impairment can clearly be seen.The power spectral densities of the two WSFRs are shown in Fig. 52b,and the impaired system can be seen to have a more pronounced peakat a low frequency. This suggests that the SNGFRs of the longer-loopednephrons are experiencing larger oscillations in the impaired systemthan in the original system.The SNGFRs of the individual nephrons in the impaired system willnow be examined. The power spectral densities of the SNGFRs arepresented in Fig. 53. This figure shows that while nephrons 3–8 areessentially oscillating at a single frequency, nephrons 1 and 2 (thenephrons with the shortest loops) are oscillating at a combinationof three different frequencies. From the other power spectral densities,it is apparent that the spikes in the power spectral densities ofnephrons 1 and 2 correspond to the natural frequencies of the looplengths 0, 1 and 3—there does not appear to be a spike at the naturalfrequency of loop length 2. The frequency of the oscillations in SNGFRis clearly proportional to the length of the loops of Henle, as expected(see Sect. 5.3). Note that the impaired nephron (nephron 7) has alarger spike than nephron 8, while the other nephron pairs exhibitpower spectral densities that are identical.To compare these spectral densities to those of the same nephronsin the original system, plots of the differences in spectral densitiesbetween the original and impaired systems are given in Fig. 54.It is evident that nephrons 7 and 8 have larger oscillations in theimpaired system than in the original system, and as previously notedthe impaired nephron 7 is exhibiting larger oscillations than nephron8. Conversely, nephrons 3–6 exhibit a reduction in the oscillation

6.4 I M PA I R M E N T S I N A N 8-NEPHRO N S Y S T E M 85Average WSFR (nL/min)178177.6177.2176.8176.401Loop Length 23 0.4 0.6ImpairedOriginal10.8D P(a) The average WSFR over the simulations.Range in WSFR (nL/min)4.443.63.201Loop Length 23 0.4 0.6ImpairedOriginal10.8D P(b) The size of oscillations in WSFR over the simulations.Figure 51: Variations in the WSFR of an eight-nephron system, inresponse to single impaired nephrons of different loop lengths.magnitude. Nephrons 1 and 2 have increased oscillations at thenatural frequencies of loop lengths 1 and 3, but have decreasedoscillations at their own natural frequency (loop length 0).From the spikes at the natural frequency of loop length 3 thatare present in the power spectral densities of other nephron pairs,it is evident that the infection of nephron 7 has affected the SNGFRof nephrons 1 and 2. Also, although nephrons 3–6 exhibit reducedoscillations in SNGFR, their spectral densities show no spikes at thenatural frequency of the impaired nephron, which would appear to indicatethat these nephrons are somehow less affected by the impairednephron. The changes in spectral density peaks are summarised inTable 14.

86 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T180WSFR (nL/min)178176174Impaired172Original1200 1800 2400 3000Timestep(a) The WSFR in both the original and impaired eight-nephron systems.3e-202.5e-20ImpairedOriginal2e-20Power1.5e-201e-205e-2100 0.02 0.04Normalised Frequency(b) The power spectral density of WSFR in both the original and impaired eightnephronsystems.Figure 52: Comparisons of WSFR between the original (unimpaired)eight-nephron system and the eight-nephron system where nephron7 is impaired and D P = 0.5.Nephron Peak Increase Nephron Peak Increase1 0.25% 2 0.25%3 -77% 4 -77%5 -4.7% 6 -4.7%7 183% 8 153%Table 14: Increases in spectral density peaks in the SNGFR of nephronsin the system with one impaired nephron.

6.4 I M PA I R M E N T S I N A N 8-NEPHRO N S Y S T E M 872.52Nephron 1Nephron 23025Nephron 3Nephron 4Power1.51Power2015100.5500 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(a) Nephrons 1 and 2 (loop length 0).(b) Nephrons 3 and 4 (loop length 1).80007000Nephron 5Nephron 690008000Nephron 7Nephron 860007000Power5000400030002000Power600050004000300020001000100000 0.02 0.04Normalised Frequency00 0.02 0.04Normalised Frequency(c) Nephrons 5 and 6 (loop length 2).(d) Nephrons 7 and 8 (loop length 3).Figure 53: The power spectral densities of SNGFR in the eight-nephronsystem with a single impaired nephron.Power0.40.30.20.10-0.1-0.2-0.3Nephron 1Nephron 20 0.02 0.04Normalised FrequencyPower100-10-20-30-40-50-60-70-80-90-100Nephron 3Nephron 40 0.02 0.04Normalised Frequency(a) Nephrons 1 and 2 (loop length 0).(b) Nephrons 3 and 4 (loop length 1).Power500-50-100-150-200-250-300-350-400Nephron 5Nephron 60 0.02 0.04Normalised FrequencyPower40003500300025002000150010005000Nephron 7Nephron 80 0.02 0.04Normalised Frequency(c) Nephron 5 and 6 (loop length 2).(d) Nephrons 7 and 8 (loop length 3).Figure 54: The difference in power spectral densities of SNGFR betweenthe original eight-nephron system and the system with oneimpaired nephron.Returning attention to Fig. 52, despite changes in SNGFR spectraldensities at several frequencies, the WSFR spectral density only exhibitsan increase at the natural frequency of the impaired nephron,and a decrease in the average WSFR of 0.7%. Despite the spectral

88 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N Tdensities of individual nephrons varying from -77% to 183%, theaverage WSFR is not significantly affected, demonstrating that thiseight-nephron system can exhibit stable whole-system behaviourwhen the function of an individual nephron is impaired.6.4.2 Two impaired nephronsThe second experiment was to determine the effects of two impairednephrons on the same system. In this experiment, D P was kept fixedat 0.5 and for each pair of nephrons in the system, a simulationwas run where both nephrons were impaired at t = 1, 200 and thesimulation ran until t = 3, 000.A comparison of the WSFR for each of these simulations is presentedin Fig. 55—Fig. 55a shows the average WSFRs and Fig. 55b showsthe magnitude of the oscillations in WSFRs. From Fig. 55a it is clearthat the impairment decreased the average WSFR in proportion tothe lengths of the infected nephrons’ loops of Henle, as was the casefor a single impaired nephron. The magnitude of WSFR oscillations(Fig. 55b) also shows a similarity to the previous experiment, as it wasnot affected by the impairment unless at least one of the longest-loopnephrons was impaired.A comparison between the original system and the impaired systemwhere both longest-loop nephrons are impaired will now be given.This particular impaired system was chosen as it exhibits the lowestaverage WSFR of the systems under consideration (see Fig. 55a).The WSFR of the two systems are shown in Fig. 56a, where thedecrease in average WSFR due to the impairment is apparent. Thepower spectral densities of the two WSFRs are shown in Fig. 56b, andthe impaired system can be seen to have peak seven times larger atthe natural frequency of loop length 3. This peak is approximatelyfive times larger than the equivalent peak in the system with a singleimpaired nephron (Fig. 56b), indicating that the two impairednephrons have a much larger influence on the system dynamics thanthe single impaired nephron in Sect. 6.4.1.The analysis now turns to the SNGFRs of the individual nephrons.The power spectral densities of the SNGFRs are presented in Fig. 57.This figure shows that while nephrons 3–8 are essentially oscillating ata single frequency, nephrons 1 and 2 (the nephrons with the shortestloops) are oscillating at a combination of different frequencies. Thesespectral densities exhibit three main differences from the previoussystem that contained only a single impaired nephron:• Nephrons 1 and 2 feature a significantly larger peak at the naturalfrequency of loop length 3 (compare Fig. 53a and Fig. 57a).• Nephrons 3 and 4 exhibit smaller peaks, with differing magnitudes,which accentuate the small peaks at the natural frequenciesof loop lengths 0 and 3 (compare Fig. 53b and Fig. 57b).These additional peaks are present in the system with one impairednephron (Fig. 53b), but were minuscule in comparisonto the dominant peak.

6.4 I M PA I R M E N T S I N A N 8-NEPHRO N S Y S T E M 89Average WSFR (nL/min)17817717617501Loop Length 2ImpairedOriginal321 Loop Length3 0(a) The average WSFR over the simulations.Range in WSFR (nL/min)7654301Loop Length 2ImpairedOriginal321 Loop Length3 0(b) The size of oscillations in WSFR over the simulations.Figure 55: Variations in the WSFR of an eight-nephron system, inresponse to pairs of impaired nephrons of different loop lengths.• Nephrons 7 and 8 again exhibit peaks of different magnitude(compare Fig. 53d and Fig. 57d), but in this system both nephronshave identical impair states.The difference in spectral densities between nephrons 3 and 4, andbetween nephrons 7 and 8 is unexpected—both pairs of nephronshave identical impair states—but one possible explanation is that thedifferent peak sizes may be due to the relationship between the phaseof the individual nephron and the phases of the other nephrons in thesystem. This phenomena will not be investigated in any further detailhere—the changes in spectral densities will now be examined.Plots of the differences in spectral densities between the originaland impaired systems are given in Fig. 58. The plot for nephrons 1and 2 exhibit the large peak at the natural frequency of loop length 3

90 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T180178WSFR (nL/min)176174172Impaired170Original1200 1800 2400 3000Timestep(a) The WSFR in both the original and impaired eight-nephron systems.1.6e-191.4e-19ImpairedOriginal1.2e-19Power1e-198e-206e-204e-202e-2000 0.02 0.04Normalised Frequency(b) The power spectral density of WSFR in both the original and impaired eightnephronsystems.Figure 56: Comparisons of WSFR between the original eight-nephronsystem and the eight-nephron system where both longest-loopednephrons are impaired and D P = 0.5.(see Fig. 58a) and also show that the increase at their own naturalfrequency is approximately half of the increase shown when a singlenephron is impaired (compare Fig. 54a and Fig. 58a). Nephrons 3–6 again exhibit a decrease at their natural frequencies (Fig. 58b,Fig. 58c) and there is a difference in the magnitude of the decreasesin nephrons 3 and 4, but it is small enough to be almost invisible(Fig. 58b). The plot for nephrons 7 and 8 clearly shows the differencein peak amplitudes (Fig. 58d) that was observed in Fig. 57d.It is again evident that nephrons 7 and 8 have larger oscillations inthe impaired system than in the original system, and the impairmentof nephrons 7 and 8 has affected the SNGFR of nephrons 1 and 2.Although nephrons 3 and 4 do not exhibit an increase at the natural

6.4 I M PA I R M E N T S I N A N 8-NEPHRO N S Y S T E M 912.5Nephron 1Nephron 298Nephron 3Nephron 427Power1.51Power65430.52100 0.02 0.0400 0.02 0.04Normalised FrequencyNormalised Frequency(a) Nephrons 1 and 2 (loop length 0).(b) Nephrons 3 and 4 (loop length 1).80007000Nephron 5Nephron 670006000Nephron 7Nephron 860005000Power500040003000Power40003000200020001000100000 0.02 0.04Normalised Frequency00 0.02 0.04Normalised Frequency(c) Nephrons 5 and 6 (loop length 2).(d) Nephrons 7 and 8 (loop length 3).Figure 57: The power spectral densities of SNGFR in the eight-nephronsystem where both longest-loop nephrons are impaired.frequency of loop length 3 (Fig. 58b), the large decrease in the peak attheir own natural frequency has accentuated the peak at the naturalfrequency of loop length 3 (Fig. 57b). The changes in spectral densitypeaks are summarised in Table 15.Power1.41.210.80.60.40.20-0.2Nephron 1Nephron 20 0.02 0.04Normalised FrequencyPower200-20-40-60-80-100-120Nephron 3Nephron 40 0.02 0.04Normalised Frequency(a) Nephrons 1 and 2 (loop length 0).(b) Nephrons 3 and 4 (loop length 1).Power1000-100-200-300-400-500-600-700-800-900Nephron 5Nephron 60 0.02 0.04Normalised FrequencyPower2000180016001400120010008006004002000-200Nephron 7Nephron 80 0.02 0.04Normalised Frequency(c) Nephrons 5 and 6 (loop length 2).(d) Nephrons 7 and 8 (loop length 3).Figure 58: The difference in power spectral densities of SNGFR betweenthe original eight-nephron system and the system with twoimpaired nephrons.

92 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N TNephron Peak Increase Nephron Peak Increase1 -3% 2 -3%3 -95% 4 -95%5 -10% 6 -10%7 140% 8 111%Table 15: Increases in spectral density peaks in the SNGFR of nephronsin the system with two impaired nephrons.Returning attention to Fig. 56, despite changes in SNGFR spectraldensities at several frequencies, the WSFR spectral density only exhibitsan increase at the natural frequency of the impaired nephrons,and a decrease in the average WSFR of 1.4%—twice the decrease thatarose with a single impaired nephron, even though the WSFR spectraldensity showed a peak five times larger than in the system with asingle impaired nephron (compare Fig. 52b and Fig. 56b). Despite thespectral densities of individual nephrons varying from -95% to 140%,the average WSFR is not significantly affected, demonstrating that thiseight-nephron system exhibits stable whole-system behaviour whenthe function of the two longest-loop nephrons are impaired.6.4.3 Summary of experimentsThe results from both experiments indicate that the dynamics of thiseight-nephron system are more sensitive to changes in the longerloopednephrons than the shorter-looped nephrons. A possible explanationfor this observation is that the longer-looped nephrons playa more dominant role in the maintenance of the salt gradient in themedullary fluid, and that any change to their behaviour will thereforehave a larger impact on the system dynamics. The SNGFR of thelonger-looped nephrons also oscillates more slowly than the shorterloopednephrons—due to the longer time delay between the filtrateentering the tubule and reaching the macula densa—and accordinglyan impaired nephron might only affect neighbouring nephrons withshorter loops of Henle, as the shorter-looped nephrons can respondto imbalances more quickly than the longer-looped nephrons.Similarly, although the average WSFR only decreased by 0.7–1.4%,the amplitude of the oscillations in WSFR increased, which may indicatethat the system as a whole was producing a larger dynamic responseto the changed conditions. This might mean that the impairedsystems are less able to respond to further changes in conditions, asthey are oscillating closer to their maximal range. Such an effect ofimpairments on multi-nephron systems—that the system may be lessable to respond to changes in conditions, rather than just exhibitingchanged steady-state dynamics—will not be investigated further here.

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 936.5 I M PA I R M E N T S O N A L A R G E R S C A L EThe second investigation was to analyse the effect that impairednephrons have at a larger scale—the model used for this experimentwas a 72-nephron system, consisting of nine columns arranged in a 3 ×3 grid, the structure of each column being identical to that of the eightnephronsystem used in Sect. 6.4 (see Fig. 50). The 2D arrangementof the interstitial fluid columns is shown in Fig. 59a and the arterialtree structure is shown in Fig. 59b. In each simulation, nephronswere impaired at t = 1, 200 with D P = 0.5, and the simulations wererun until t = 3, 000.P AP VR CR C1 2 3y213 6 92 5 8R C4 5 601 4 70 1 2x7 8 9(a) The 2D arrangement ofinterstitial fluid columns.(b) The arterial tree connecting the columns.Figure 59: The high-level structure of the 72-nephron model. Eachcolumn is assigned a number 1–9.Two separate experiments were conducted. The first investigatedthe effects of impairing all nephrons in the model whose loops ofHenle were the same length, to determine how significantly the lengthof the loop affects the system dynamics. The difference between thisexperiment and the one conducted on the eight-nephron system(Sect. 6.4.1) is that the 72-nephron model incorporates inter-columndiffusion and a more complex arterial tree, which may reduce theeffect of the impairment.The second experiment investigated the effects of impairing thenephrons in an entire column, to determine how the impairmentaffects neighbouring columns. This experiment allowed the effectsof the 2D column arrangement and the arterial tree structure to becompared. These experiments are now presented and compared, andthe effects of impaired nephrons on the 72-nephron system are thensummarised.

94 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T6.5.1 Impaired nephron pairsIn this experiment, for each loop length L (see Fig. 50) all of thenephrons whose loops were length L were impaired at t = 1, 200. Thisexperiment is essentially a larger-scale equivalent of the experimentdetailed in Sect. 6.4.2.A comparison of the WSFR for each of these simulations is presentedin Fig. 60. The average WSFR decreased in proportion to the looplength of the impaired nephrons (Fig. 60a), as per the experimentsin Sect. 6.4. When the longest-loop nephrons in the model wereimpaired, the decrease in WSFR was 1.3%, which is almost identicalto the decrease observed in Sect. 6.4.2. As shown in Fig. 60b, thesize of the oscillations in WSFR varied with no discernible regularity,in contrast to the eight-nephron system (Fig. 55b).The remainder of this analysis contrasts the dynamics of the original72-nephron system with the dynamics of the system where allnephrons of loop length 3 are impaired, as this system demonstratedthe largest decrease in WSFR (see Fig. 60).The WSFR of both the original and impaired systems are presentedin Fig. 61, where the decrease in average WSFR is not immediatelyapparent (Fig. 61a). The power spectral densities of the two WSFRs(Fig. 61b) show that the impaired nephrons have caused larger peaksat the natural frequencies of loop lengths 1 and 3. There is also anadditional minor spike at a normalised frequency of approximately0.033, which is a higher frequency than even that of the shortestloopnephrons. The increases shown here are much smaller than theincreased spikes that arose when the two longest-loop nephrons wereimpaired in the eight-nephron model (compare Fig. 56b and Fig. 61b).The analysis now turns to the WCFRs of the individual columns inthe model, which is one level of hierarchy lower than the WSFR but isalso a level of hierarchy above the SNGFR. At this level of hierarchythe WCFR is directly comparable to the WSFR of the eight-nephronmodels that were investigated in Sect. 6.4, and thus any observationsmade in these previous experiments may be relevant here.The spectral densities of the WCFR for each column in the model areshown in Fig. 62. Columns 1–3 exhibited two dominant peaks, both ofwhich have increased in size in the impaired system—the increase inthe peak at the natural frequency of the nephrons with loop length 3has increased more than the secondary peak at the natural frequencyof nephrons with loop length 1 (Fig. 62a). Columns 4–9 all exhibitedthree dominant peaks—at the natural frequencies of nephrons withloop length 0, 1 and 2. There was a difference in relative sizes of thepeaks, with columns 4–6 having a larger peak at the frequency ofloop length 1 (Fig. 62b), and columns 7–9 having a larger peak at thefrequency of loop length 0 (Fig. 62c).All of the columns exhibited a larger spike at the natural frequencyof the longest-loop nephrons (Fig. 63), which is consistent with theexperiments conducted on the eight-nephron system. As was shown inSect. 6.4, this does not mean that nephrons with different natural frequenciesare unaffected. Apart from this trend, there is no observablecommon pattern in the changes in power spectral densities.

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 95Average WSFR (nL/min)1730172017101700ImpairedOriginal16900 1 2 3Loop Length(a) The average WSFR over the simulations.Range in WSFR (nL/min)250240230220210200ImpairedOriginal1900 1 2 3Loop Length(b) The size of oscillations in WSFR over the simulations.Figure 60: Variations in the WSFR in the 72-nephron system where allnephrons of a specific loop length are impaired.It is also important to note that the lowest-frequency peak in allthree graphs covers the natural frequencies of nephrons with looplengths 2 and 3. For columns 1–3, the loop length 3 frequency dominatedand the loop length 2 frequency formed a shoulder (Fig. 62a),while for columns 4–9 it was the loop length 2 frequency that dominatedand the loop length 3 frequency that formed a shoulder(Fig. 62b and Fig. 62c). This indicates that the impaired nephronsare possibly having a larger effect on columns 1–3, as these were theonly columns to show a clear increase in the peak size for the looplength 3 frequency.However, as was demonstrated in earlier experiments (Sect. 6.4),an increased peak at the loop length 3 frequency does not mean thatthe impaired nephrons are having a significant effect—in Fig. 56b

96 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T18501800ImpairedOriginalWSFR (nL/min)17501700165016001200 1800 2400 3000Timestep(a) The WSFR in both the original and impaired 72-nephron systems.1.2e-161e-16ImpairedOriginal8e-17Power6e-174e-172e-1700 0.02 0.04Normalised Frequency(b) The power spectral densities of WSFR in both the original and impaired72-nephron systems.Figure 61: Comparison of the WSFR in the original 72-nephron systemto that of the 72-nephron system where all nephrons of a specific looplength are impaired.this peak is greatly exaggerated in the impaired system, and yet theimpaired system exhibits a decrease in average WSFR of only 1.4%.In this experiment, columns 1–3 exhibited the smallest decrease inaverage WCFR and columns 7–9 exhibited the largest decrease, asdemonstrated in Fig. 64a, despite columns 1–3 also having the largestincrease in the loop length 3 peak (the changes in power spectraldensity of the columns are shown in Fig. 63).In summary, the WSFR and WCFR power spectral densities showedsmaller differences between the original and impaired 72-nephronsystems than those observed in the equivalent experiment on theeight-nephron system (Sect. 6.4.2), and the decrease in average WSFR

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 974.5e-174e-17ImpairedOriginal3.5e-173e-17Power2.5e-172e-171.5e-171e-175e-1800 0.02 0.04Normalised Frequency(a) Columns 1–3.3e-172.5e-17ImpairedOriginal2e-17Power1.5e-171e-175e-1800 0.02 0.04Normalised Frequency(b) Columns 4–6.Power2e-171.8e-171.6e-171.4e-171.2e-171e-178e-186e-184e-182e-18ImpairedOriginal00 0.02 0.04Normalised Frequency(c) Columns 7–9.Figure 62: A comparison of the power spectral densities of WCFRbetween the original 72-nephron system and the system where allnephrons of a specific loop length are impaired.

98 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N TPower1.2e-171e-178e-186e-184e-182e-180-2e-18-4e-18-6e-18-8e-180 0.02 0.04Normalised Frequency(a) Columns 1–3.Power7e-186e-185e-184e-183e-182e-181e-180-1e-18-2e-18-3e-180 0.02 0.04Normalised Frequency(b) Columns 4–6.1.5e-181e-185e-19Power0-5e-19-1e-18-1.5e-18-2e-180 0.02 0.04Normalised Frequency(c) Columns 7–9.Figure 63: The difference in power spectral densities of WCFR betweenthe original 72-nephron system and the system where all nephrons ofa specific loop length are impaired.

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 99Average WCFR (nL/min)2102001901801700x1ImpairedOriginal2 0 1y2(a) The average WCFR for each column in the original and impaired systems.Range in WCFR (nL/min)1301201101000x1ImpairedOriginal2 0 1y2(b) The range in WCFR for each column in the original and impaired systems.Figure 64: Variations in WCFR in the 72-nephron system where allnephrons of loop length 3 are impaired.of 1.3% is almost identical to the decrease of 1.4% observed in theeight-nephron experiment.6.5.2 Impaired columnsIn this experiment, for each column in the model a simulation wasperformed where all of the nephrons in that column were impairedat t = 1, 200. A comparison of the WSFR for each of these simulationsis presented in Fig. 65. Fig. 65a shows that the average WSFR decreasedthe most when the bottom-most columns in the arterial tree(columns 7–9) were impaired—a maximal decrease of 0.3%. All ofthe impaired simulations exhibited a smaller range in WSFR than theoriginal system (Fig. 65b), with the smallest range being produced

100 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N Twhen one of columns 4–6 was impaired. A potential explanation forthis observation is that columns 4–6 undergo inter-column diffusionwith columns at both top and bottom branches of the arterial tree,possibly resulting in a more wide-spread influence over the dynamicsof the other columns.Average WSFR (nL/min)1730172017100x1ImpairedOriginal2 0 1y2(a) The average WSFR over the simulations.Range in WSFR (nL/min)2302202102001900x1ImpairedOriginal2 0 1y2(b) The size of oscillations in WSFR over the simulations.Figure 65: Variations in the WSFR in the 72-nephron simulations whereall of the nephrons in a particular column (x, y) are impaired.The remainder of the analysis will contrast the dynamics of theoriginal 72-nephron system with the dynamics of the system whereall nephrons in column 9 are impaired, as this is one of the threesystems that demonstrates the largest decrease in WSFR (shown inFig. 65a).The WSFRs of the original and impaired systems are plotted inFig. 66a—the decrease in average WSFR is not discernible, as thedifference is 0.3%. The power spectral densities of the two WSFRs are

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 101shown in Fig. 66b, where the impaired system is again shown to havean enlarged peak at the natural frequency of loop length 3, and alsoa decreased peak at the natural frequency of loop length 1—a trendnot observed in the previous experiments.18501800ImpairedOriginalWSFR (nL/min)175017001650160015501200 1800 2400 3000Timestep(a) The WSFR in both the original and impaired 72-nephron systems.Power1e-169e-178e-177e-176e-175e-174e-173e-172e-171e-17ImpairedOriginal00 0.02 0.04Normalised Frequency(b) The power spectral densities of WSFR in both the original and impaired72-nephron systems.Figure 66: Comparison of the WSFR in the original 72-nephron systemand the system where all of the nephrons in column 9 are impaired.The analysis now turns to the WCFRs of the individual columns in themodel. The power spectral densities of the WCFRs—shown in Fig. 67—were very similar to those in the previous experiment (Fig. 62), exceptthat the mountain at the natural frequencies of loop lengths 2 and3 exhibited a small valley between the two peaks. Apart from thesechanges, there appears to be little difference between the spectraldensities of the WCFRs in the original and impaired systems.Fig. 68 shows the changes in WCFR spectral densities as a result ofthe impairment in column 9. In comparison to the changes in WCFR

102 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T4e-173.5e-17ImpairedOriginal3e-17Power2.5e-172e-171.5e-171e-175e-1800 0.02 0.04Normalised Frequency(a) Columns 1–3.2.5e-172e-17ImpairedOriginalPower1.5e-171e-175e-1800 0.02 0.04Normalised Frequency(b) Columns 4–6.Power2e-171.8e-171.6e-171.4e-171.2e-171e-178e-186e-184e-182e-18ImpairedOriginal00 0.02 0.04Normalised Frequency(c) Columns 7–8.Figure 67: A comparison of the spectral densities of WCFR between theoriginal 72-nephron system and the system where all of the nephronsin column 9 are impaired.

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 1031e-185e-19Power0-5e-19-1e-18-1.5e-180 0.02 0.04Normalised Frequency(a) Columns 1–3.3e-182.5e-182e-18Power1.5e-181e-185e-190-5e-19-1e-180 0.02 0.04Normalised Frequency(b) Columns 4–6.3e-182e-181e-18Power0-1e-18-2e-18-3e-180 0.02 0.04Normalised Frequency(c) Columns 7–8.Figure 68: The difference in power spectral densities of WCFR betweenthe original 72-nephron system and the system where all of thenephrons in column 9 are impaired.

104 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N T3e-172.5e-17ImpairedOriginal2e-17Power1.5e-171e-175e-1800 0.02 0.04Normalised Frequency(a) A direct comparison of the spectral densities.Power9e-188e-187e-186e-185e-184e-183e-182e-181e-180-1e-18-2e-180 0.02 0.04Normalised Frequency(b) The difference in spectral densities.Figure 69: The difference in the power spectral density of column 9in the original 72-nephron system and the system where all of thenephrons in column 9 are impaired.spectral densities from the previous experiment (Fig. 63), severalobservations can be made.• Columns 1–3 exhibit similar trends in both experiments (compareFig. 68a and Fig. 63a), but the magnitude of the increasesand decreases in peak size are approximately 5–10 times smaller.• Columns 4–6 also exhibit similar trends in both experiments, butagain the magnitude of the changes is reduced—this time 2–3times smaller— and there is no peak at the natural frequency ofloop length 0 in this experiment.• Columns 7–8 do not exhibit trends similar to those in the previousexperiment, and are also the only columns that exhibit

6.5 I M PA I R M E N T S O N A L A R G E R S C A L E 105Average WCFR (nL/min)2102001901801700x1ImpairedOriginal2 0 1y2(a) The average WCFR for each column in the impaired and original systems.Range in WCFR (nL/min)1301201100x1ImpairedOriginal2 0 1y2(b) The range in WCFR for each column in the impaired and original systems.Figure 70: Variations in WCFR in the system where all of the nephronsin column 9 are impaired.larger changes in these spectral densities than in the previousexperiment. This is most likely due to the impaired state of column9 exerting a larger influence on these two columns—via thearterial tree—than the effects of the impairment in the previousexperiment.The spectral density of the WCFR of column 9 is shown in Fig. 69a,and while it shares a similar shape with the WCFRs of the othercolumns (Fig. 67), it is the only column that exhibited such markedincreases in the natural frequencies of loop lengths 2 and 3. Thechanges in the spectral density are shown in Fig. 68, which is uniquein that all peaks except one are positive and that the peak size isinversely proportional to the normalised frequency.

106 E M E R G E N T E FF E C T S O F L O C A L I S E D I M PA I R M E N TA direct comparison of the individual WCFRs in the original andimpaired systems is given in Fig. 70. Only column 9 shows evidenceof a change in WCFR (Fig. 70a), which would indicate that the othercolumns are unaffected by the impairment of column 9. However,examining the size of the oscillations in WCFR reveals that column9 has influenced other columns (Fig. 70b)—columns 1–3 show adecrease of 0.5%, columns 4–6 show a decrease of 2.6%, column 7shows an increase of 0.6%, column 8 an increase of 0.8% and column9 an increase of 12%.Ignoring column 9, the changes in WCFR oscillation size are essentiallyuniform for all of the columns sharing the same branch of thearterial tree—there is only a 0.03% difference in the oscillation sizesof columns 5 and 6, even though column 6 undergoes inter-columndiffusion with the impaired column and column 5 does not. This wouldappear to indicate that the inter-column diffusion mechanism is ableto dampen out local changes, and that the arterial tree has a moredominant role in propagating the effects of the impairment acrossthe system.6.5.3 Summary of experimentsThe first experiment conducted on the 72-nephron system—the longestloopednephrons in each column were impaired—demonstrated a1.3% decrease in average WSFR. The same experiment, when conductedon the eight-nephron system (Sect. 6.4.2), demonstrated a1.4% decrease in average WSFR. These values were expected to besimilar, as the individual columns in this model share a structureidentical to that of the eight-nephron model, and the impairments inboth experiments were identical. The inter-column diffusion couldnot have caused any difference in average WSFR, since the interstitialfluid in each column evolved identically over time, due to the identicalstructure and impairment shared by all of the columns. It is possiblethat the difference in average WSFR between these experiments isdue to the arterial tree structure connecting the columns, but thedifference is so small that a more detailed investigation is required todetermine this for certain.In contrast, when the impairment was localised to a single columnin the 72-nephron system, the impairment did not appreciably affectthe dynamics of the system—average WSFR was decreased by only0.3% and neighbouring columns were not significantly affected (seeFig. 70). This is due to the inter-column diffusion (depicted in Fig. 16),which acts as an averaging mechanism for solute concentration gradientsin the interstitial fluid. The arterial tree was able to affectthe magnitude of the oscillations in WSFR (shown in Fig. 65b), butthe inter-column diffusion was able to keep the average WSFR almostconstant (shown in Fig. 65a), regardless of the choice of impairedcolumn. As the choice of column affects the number of neighbouringcolumns with which inter-column diffusion occurs, this indicates thatthe inter-column diffusion is a relatively strong agent for overcomingthe localised effects of this impairment.

6.6 C O N C L U S I O N S A B O U T S Y S T E M S TA B I L I T Y 1076.6 C O N C L U S I O N S A B O U T S Y S T E M S TA B I L I T YAs mentioned at the start of the chapter, the impairment presentedhere and the accompanying experiments serve only to illustrate theapproach of modelling and analysing the effects of a localised impairmenton multi-nephron systems. However, the experiments conductedhere demonstrated that a system consisting of a single column of interstitialis strongly resistant to the effects of the impairment. Furthermore,for a system consisting of multiple columns, the inter-columndiffusion mechanism further reduced the effects of the impairment, tothe point that there were little discernible effects on the whole-systemdynamics.Without making any predictions or observations concerning thereal-world effects of renal diseases on the kidney, these experimentsdemonstrated that the multi-nephron systems maintain remarkablystable dynamics when the function of individual nephrons are impaired.Despite the simplicity of the impairment model that was used,we contend that this stability is a feature of the model—and of thekidney proper—rather than a deficiency in the impairment model.

L A R G E - S C A L E S I M U L AT I O N77.1 I N T RO D U C T I O NSince one of the motivations for this model is to make whole-kidneysimulations computationally tractable, our attention now turns to howthe performance of the simulations scale as the size of the modelincreases. The experiments conducted in Chapter 5 and Chapter 6have only examined the behaviour of the model in the context of multinephronsystems whose sizes pale in comparison to that of an entirekidney. To discuss the simulation performance, several distinctionsmust be made:M O D E L The network model, as presented in Chapter 4.A L G O R I T H M A computational method for calculating the dynamicsof a given model.I M P L E M E N TAT I O N A computer program that executes an algorithm.P RO C E S S A sequence of computations, performed in serial.S I M U L AT I O N The act of using an implementation to calculate thedynamics of a given model.This chapter begins by presenting the algorithm that was used forthe experiments in Chapter 5 and Chapter 6, followed by a performanceanalysis of the resulting implementation. Despite the fact thatthis implementation is single-threaded, and is therefore unable tomake use of more than one central processing unit (CPU), it is shownthat using this implementation to simulate a whole-kidney model iscomputationally tractable.Secondly, the simulation performance can be greatly improved byutilising more than one CPU, and a new algorithm is presented thatperforms each time-step as a number of concurrent processes, eachof which can be performed on a different CPU. This algorithm wasimplemented using the Join Calculus [60, 59], which allows a singlesimulation to be distributed over multiple computers. A performanceanalysis demonstrates that this results in a significant performanceincrease when more than one CPU is available.Finally, to calculate the dynamics of an entire kidney, it is notsufficient for the simulation to be computationally tractable—themodel must first be constructed. The structure of the models used inChapter 5 and Chapter 6 were manually specified—the spatial parametersof each tubule segment, each nephron and the arterial tree—anapproach which clearly will not scale to models of 10 6 nephrons. Althoughno solution has currently been implemented, two techniquesfor automatically generating such large models are proposed.109

110 L A R G E-SCALE S I M U L AT I O N7.2 P E R F O R M A N C E O F T H E I M P L E M E N TAT I O N7.2.1 The single-threaded algorithmA pseudocode description of the single-threaded algorithm is given inListing 7.1.1 # determine the resistance of the arterial tree2 for column in model.columns:3 for nephron in column.nephrons:4 # calculated by an update rule (Eq. 4.5)5 nephron.update_resistance()6 # calculated from the nephron resistances (Fig. 14)7 # Ohm’s law and Kirchoff’s laws are used8 column.update_resistance()9 # calculated from the column resistances (Fig. 15)10 # Ohm’s law and Kirchoff’s laws are used11 R T = model.update_resistance()1213 # update the model parameters (Sect. 4.4.3)14 model.update_params()1516 # generate the input for the next time-step17 # volume = P A−P VR T(Sect. 4.2.2)18 input = model.input_blood(R T )1920 # distribute the blood to the nephrons21 for column in model.columns:22 # depends on the tree of column resistances (Fig. 15)23 column_input = model.distribute(input, column)24 for nephron in column.nephrons:25 # depends on the tree of nephron resistances (Fig. 14)26 nephron_input = column.distribute(column_input, nephron)27 nephron.set_input(nephron_input)2829 # simulate the time-step for each tubule segment (Sect. 4.4.5)30 # also update the state variables (Sect. 4.4.4)31 for column in model.columns:32 column.simulate_timestep()3334 # perform diffusion between neighbouring columns (Fig. 16)35 for column1, column2 in model.column_neighbours:36 column1.diffuse_with(column2)3738 # update the nephron disease states (Sect. 6.1)39 for column in model.columns:40 for nephron in column.nephrons:41 nephron.update_disease_state()Listing 7.1: The single-threaded algorithm in pseudocode.✆

7.2 P E R F O R M A N C E O F T H E I M P L E M E N TAT I O N 1117.2.2 Performance analysisThe single-threaded algorithm was implemented in OCaml [101] andwas compiled with the high-performance native-code compiler OCAM-LOPT. The simulations were performed on an Intel Core 2 Duo E6600with 2Gb of RAM, running Ubuntu Linux. The parameter values donot affect the simulation performance, as they do not affect the calculationsused to solve the model equations.The memory usage of the simulations (shown in Fig. 71a) growslinearly with the number of nephrons in the model, and the number ofcolumns into which the model is divided appears to have little effect.However, the execution times show significant differences, dependingon the number of columns in the model (Fig. 71b). The execution timegrows at O(n 3 ) for n nephrons per column, as demonstrated by thecurves fitted to the execution time plots (see Fig. 72). This is to beexpected, as solving the n × n matrix governing the loops of Henle(described in Sect. 4.4.5) is solved by Gaussian elimination using thestandard algorithm, which has a complexity of O(n 3 ).As demonstrated by Fig. 71c, when the number of nephrons ineach column is held constant, the execution time grows linearly withthe number of columns in the model. Thus, for a model containingc columns and n nephrons per column, the execution time for thesimulation is O(cn 3 ). This explains why the n 3 coefficients in Fig. 72decrease by a factor of four when c is doubled—n is halved to conservethe number of nephrons in the model, and so:c ′ = 2 × c (7.1)n ′ = n 2((O c ′ n ′3) ( n) ) ( 3 cn3)= O 2c × = O2 4(7.2)(7.3)For example, consider two models of 100,000 nephrons (Table 16).The first consists of 100 columns arranged in a 10 × 10 grid with1,000 nephrons per column; the second consists of 10,000 columnsarranged in a 100 × 100 grid with 10 nephrons per column. As theperformance of a model is O(cn 3 ), the performance of the first modelis O(10 11 ), compared to O(10 7 ) for the second model. This is not onlya huge improvement in performance, as the structure of the secondmodel more closely approximates a continuous body of interstitialfluid—the performance of the model improves the more the interstitialfluid structure approximates a continuous model.Using this implementation, a simulation of a 512-nephron model(8 nephrons per column) progressed at 444 nephron time-steps persecond and used 28Mb of RAM (56kb per nephron). Assuming thatthe linear relationships in Fig. 71a and Fig. 71c hold for a model of10 6 nephrons, this indicates that the simulation of a single time-stepfor a whole-kidney model would require 37.5 minutes and 53.4Gbof RAM. These orders of magnitude demonstrate that the implementationscales up to a whole-kidney model. We are not aware of anyexisting nephron models for which a whole-kidney simulation is computationallyfeasible.

112 L A R G E-SCALE S I M U L AT I O N981 column2 columns4 columns7RAM (MB)6543210 20 30 40 50 60 70Model size (nephrons)(a) Memory usage9008007001 column2 columns4 columns600Time (s)500400300200100010 20 30 40 50 60 70Model size (nephrons)(b) Execution time1401208 nephrons/column12 nephrons/column16 nephrons/column100Time (s)8060402000 10 20 30 40 50 60 70Number of nephrons(c) Execution timeFigure 71: Comparing 1-column, 2-column and 4-column simulations.

7.2 P E R F O R M A N C E O F T H E I M P L E M E N TAT I O N 113900800Execution TimeBest FitTime (s)7006005004003002001005.7×10 -2 n 2 + 2.5×10 -3 n 3010 20 30 40 50 60 70Number of nephrons (n)(a) Single-column simulations300250Execution TimeBest FitTime (s)2001501002.1 + 0.5n + 2.2×10 -2 n 2 + 6.1×10 -4 n 350010 20 30 40 50 60 70Number of nephrons (n)(b) Two-column simulationsTime (s)14012010080604020Execution TimeBest Fit1.2 + 0.7n + 8.4×10 -3 n 2 + 1.6×10 -4 n 3010 20 30 40 50 60 70Number of nephrons (n)(c) Four-column simulationsFigure 72: Execution time is O(n 3 ) for n nephrons per column.

114 L A R G E-SCALE S I M U L AT I O NModel 1 Model 2Nephrons 10 5 10 5c 10 2 10 4n 10 3 10 1Time O(10 11 ) O(10 7 )Table 16: Performance comparison of two models of identical size.7.3 I M P ROV I N G P E R F O R M A N C E W I T H C O N C U R R E N C YAlthough whole-kidney simulations are achievable with the monolithicalgorithm, the performance of such simulations can readily beimproved. While reductions in execution time may be achieved byoptimising the monolithic implementation, we show that much largerimprovements can be achieved by dividing the simulation into multipleprocesses that can be executed concurrently. If the processescan be distributed over multiple computers—employing significantlymore resources than available on a single computer—the resultingperformance gain of the distributed system can be remarkable [94].Typical distributed systems such as Berkeley Open Infrastructurefor Network Computing (BOINC) [35, 7], SETI@home [94, 8] and KidneyGrid[42, 43], and distributed algorithms such as MapReduce [51]operate by dividing the computational work into a number of distinctprocesses, executing each process independently, and performingsome analysis or calculation on the results.In contrast, the model presented in Chapter 4 requires the modelstate at time t to be calculated before the state at time t + ∆ canbegin to be calculated. As a result, if the simulation were broken intomultiple processes, the processes would need to communicate witheach other after every time-step. This is in contrast to the distributedsystems described previously, which perform multiple jobs concurrentlyand then combine the results in some manner. Accordingly, aconcurrent algorithm for the model simulation will necessarily differfrom such systems and cannot be implemented using an identicalapproach.7.3.1 The concurrent algorithmThe single-threaded algorithm Listing 7.1 can be divided into a numberof separate processes, based on the hierarchical structure ofthe model. Once the input has been generated and distributed tothe columns, the time-step can be simulated for each column independently,with the exception of the inter-column diffusion. Theinter-column diffusions can also be performed independently: foreach pair of neighbouring columns, assign one the “master”, which isresponsible for simulating the diffusion with the “slave” (see Fig. 73).This algorithm consists of a process for each column in the model(referred to as column processes and abbreviated to “Col1”, “Col2”,etc., in diagrams), and a process for the generation and distribution

7.3 I M P ROV I N G P E R F O R M A N C E W I T H C O N C U R R E N C Y 115of the model input (referred to as the “controller” and abbreviatedto “Ctrl” in diagrams). The algorithm is presented in pseudocodeform in Listing 7.2 and Listing 7.3, and the communication betweenthe processes is shown in Fig. 74 and Fig. 75. To reduce the delaybetween time-steps, the column processes can be changed so thatonce they receive input, they calculate the resistance of the arterialtree and immediately send this value to the controller (demonstratedin Fig. 76).1 # receive the resistances of each column in the model2 for column in model.columns:3 resistance = receive_message(column, send_resistance)4 R T = update_resistance(column, resistance)56 # generate the input for the next time-step7 # volume = P A−P VR T(Sect. 4.2.2)8 input = model.input_blood(R T )910 # distribute the blood to the columns11 for column in model.columns:12 # depends on the tree of column resistances (Fig. 15)13 column_input = model.distribute(input, column)14 send_message(column, column_input)Listing 7.2: The controller process in pseudocode.✆MasterMaster(a)Slave(b)SlaveFigure 73: The master/slave relationship between columns, for simulatingthe inter-column diffusion (see Fig. 16). The arrow (a) indicatesthat the column labelled “Master” performs the diffusion with thehighlighted column; the arrow (b) indicates that the highlighted columnperforms the diffusion with the column labelled “Slave”.The presentation of the column algorithm (Listing 7.2) has ignoredthe complexities of performing the inter-column diffusion. Each columnmust not only perform the diffusion between itself and neighbouringslaves, it must also communicate with neighbouring masters(see Fig. 75). This more detailed algorithm is presented in Listing 7.4.A significant feature of this concurrent algorithm is the total absenceof shared state between processes—the only shared data ispassed between processes using messages. This greatly simplifies

116 L A R G E-SCALE S I M U L AT I O N1 # determine the resistance of the arterial tree2 for nephron in column.nephrons:3 # calculated by an update rule (Eq. 4.5)4 nephron.update_resistance()5 # calculated from the nephron resistances (Fig. 14)6 R C = column.update_resistance()78 # send the column resistance to the controller9 send_message(controller, R C )1011 # receive the input blood for the next time-step12 receive_message(column_input)1314 # distribute the blood to the nephrons15 for nephron in column.nephrons:16 # depends on the tree of nephron resistances (Fig. 14)17 nephron_input = column.distribute(column_input, nephron)18 nephron.set_input(nephron_input)1920 # simulate the time-step for each tubule segment (Sect. 4.4.5)21 # also update the state variables (Sect. 4.4.4)22 column.simulate_timestep()2324 # perform diffusion between neighbouring columns (Fig. 16)25 for neighbour in column.slaves:26 column.diffuse_with(slave)2728 # update the nephron disease states (Sect. 6.1)29 for nephron in column.nephrons:30 nephron.update_disease_state()Listing 7.3: The column process in pseudocode.✆Ctrl Col1 Col2 ... ColNMessagesSend resistanceSend inputActionsGenerate inputSimulate time-stepFigure 74: Communication between the controller and columns. Ateach time-step, the controller sends input (blood) to each column; thecolumns then simulate the time-step; finally the columns send theirupdated arterial resistances to the controller.the implementation and excludes the possibility of problems suchas deadlock, livelock and race conditions, which are commonly as-

7.3 I M P ROV I N G P E R F O R M A N C E W I T H C O N C U R R E N C Y 117MasterSlaveMessagesRequest fluidSend fluidSend changesActionsCalculate changesUpdate fluidFigure 75: Communication between columns when performing theinter-column diffusion.Ctrl Col1 Col2 ... ColNMessagesSend resistanceSend inputActionsGenerate inputSimulate time-stepFigure 76: Reordering the communication in Fig. 74, in order toreduce the delay between time-steps.sociated with concurrent programs. Perhaps the biggest advantageis that the implementation can easily be distributed across multiplecomputers, since the lack of shared memory between computers hasno impact.7.3.2 Communication in the concurrent algorithmThe concurrent algorithm that has been described involves regularcommunication between the individual processes—for each time-step,two messages are sent between each column and the controller(Fig. 74) and three messages are sent between each pair of neighbouringcolumns (Fig. 75). If the columns are distributed across anetwork of computers, the limitations of the network hardware canimpact upon the performance of this communication, and thus reducethe simulation performance.

118 L A R G E-SCALE S I M U L AT I O N1 # save the current state of the interstitial fluid2 F C = copy(column.interstitial_fluid)34 # ask neighbouring slaves for their interstitial fluid5 for neighbour in column.slaves:6 send_message(neighbour, request_interstitial_fluid)78 # count how many slaves and masters have not completed the diffusion9 slaves = column.slave_count10 masters = column.master_count1112 while true:13 # receive messages from neighbouring columns14 message = receive_message()1516 # a master has requested the column’s interstitial fluid17 if message == request_interstitial_fluid:18 send_message(message.sender, F C )19 # a master has adjusted the interstitial fluid20 elif message == adjust_interstitial_fluid:21 column.adjust_interstitial_fluid(message.changes)22 master_count -= 123 # a slave has sent its interstitial fluid24 elif message == send_renal_fluid:25 # calculate the changes to the fluid in both columns26 local, remote = diffusion_changes(F C , message.fluid)27 # inform the slave to adjust its interstitial fluid28 send_message(message.sender, adjust_renal_fluid(remote))29 # adjust this column’s interstitial fluid30 column.adjust_interstitial_fluid(local)31 slave_count -= 132 # stop when all diffusions have been completed33 if slaves == 0 and masters = 0:34 breakListing 7.4: The algorithm for inter-column diffusion in pseudocode.✆Message Data Size (bytes)Send resistance double 8Send input double list (fluid) 8 × SRequest interstitial fluid none 0Send interstitial fluid fluid list 8 × S × LAdjust interstitial fluid fluid list 8 × S × LTable 17: Message content in inter-process communication. Themodel supports S solutes and each column has L layers of interstitialfluid.The content of each message sent between the individual processesis documented in Table 17. Based on this data, the maximum volumeof data sent by a column in a single time-step is given by Eq. 7.4.32 × S × L + 8 × S + 8 (7.4)

7.3 I M P ROV I N G P E R F O R M A N C E W I T H C O N C U R R E N C Y 119The model currently supports eight solutes (Sect. 4.4.3) and themodels used in Chapter 5 and Chapter 6 used up to sixteen layers ofinterstitial fluid, resulting in a maximum volume of 4168 bytes (4.07kb).To examine the worst-case volume of communication, consider awhole-kidney model (i. e., 10 6 nephrons) that consists of 10 6 columns,each containing a single nephron. Using Eq. 7.4, the volume of datasent at each time-step would be 3.88Gb—such a large volume ofcommunication would certainly reduce the simulation performance.This volume can be decreased by reducing the number of columnsin the model, but there are two further options that do not affectthe model structure: locating multiple columns on the same computer(i. e., sharing a single CPU or using multiple CPUs); and allowingcolumns to simulate the behaviour of multiple columns. Both of theseoptions were explored with the implementation.7.3.3 Performance of the implementationAlthough the concurrent algorithm differs significantly from the originalalgorithm (Listing 7.1), the concurrent implementation sharesmost of its code with the original implementation (Sect. 7.2.2). Theconcurrent implementation was written in JoCaml [110, 147, 100],an extension of OCaml that incorporates the Join Calculus [60, 59] toallow for concurrent and distributed programming.The Join Calculus—like other process calculi such as CCS [120],the π Calculus [121, 122], the Fusion Calculus [140, 141] and theAmbient Calculus [34]—is used to model concurrent systems. TheJoin Calculus is equivalent to the π Calculus [59], but is easier toimplement in a distributed setting [59].The Join Calculus uses a message-passing concurrency model verysimilar to the one used to present the concurrent algorithm (Listing7.2 and Listing 7.3), which greatly simplified the implementationprocess. To illustrate how well the JoCaml semantics match the concurrentalgorithm, the implementation of the inter-column diffusion(Listing 7.4) is presented in Listing 7.5. The only difference is thatthe loop and the variables slaves and masters are captured by themessage diff_state(column, F C , slaves, masters).Amdahl’s law [6] establishes the theoretical maximum speedup thatcan be gained by performing a computation with multiple processors,where a fraction p of the computation can be executed in paralleland the computation is performed on N identical processors (Eq. 7.5).According to this law, we expect to see a significant speedup asthe simulation is distributed over a greater number of processors,although the return diminishes and approaches a limit (see Eq. 7.6and Fig. 77).1S(N) =(1 − p) + p/Nlim S(N) = 1N→∞ 1 − p(7.5)(7.6)This law is based on the implicit assumption that p is independentof N, which is only the case if the size of the problem is fixed. If thisis not the case and the problem size can scale with the number of

120 L A R G E-SCALE S I M U L AT I O N101 # start the inter-column diffusion2 diff_start(column) =3 # save the current state of the interstitial fluid4 F C = copy(column.interstitial_fluid)5 # ask neighbouring slaves for their interstitial fluid6 for neighbour in column.slaves:7 spawn request_fluid(column, neighbour)8 # wait to receive messages from neighbouring columns9 diff_state(column, F C , column.slaves, column.masters)11 # no more masters or slaves to wait for12 diff_state(column, F C , 0, 0) =13 # the inter-column diffusion is complete14 diff_complete(column)1516 # a master requests the column’s interstitial fluid17 diff_state(column, F C , slaves, masters)18 & request_fluid(master, column) =19 # send the interstitial fluid to the master20 send_fluid(master, column, F C )21 # still have to wait for all of the masters and slaves22 diff_state(column, F C , slaves, masters)2324 # a master changes the column’s interstitial fluid25 diff_state(column, F C , slaves, masters)26 & adjust_fluid(master, column, changes) =27 # adjust the interstitial fluid28 column.adjust_fluid(changes)29 # there is now one less master to wait for30 diff_state(column, F C , slaves, masters - 1)3132 # a slave sends its interstitial fluid33 diff_state(column, F C , slaves, masters)34 & send_fluid(column, slave, fluid) =35 # calculate the changes to the fluid in both columns36 local, remote = diffusion_changes(F C , message.fluid)37 # inform the slave to adjust its interstitial fluid38 adjust_fluid(column, slave, remote)39 # adjust this column’s interstitial fluid40 column.adjust_interstitial_fluid(local)41 # there is now one less slave to wait for42 diff_state(column, F C , slaves - 1, masters)Listing 7.5: The JoCaml implementation of inter-column diffusion inpseudocode form.✆processors (e. g., the complexity of the model may be increased byrefining the equations, incorporating additional equations, or augmentingthe structure), this will usually increase the relative amountof the computation that can be executed in parallel. As a result, thespeedup gained by using additional processors can significantly exceedthe original limit proposed by Amdahl’s law (e. g., by an order ofmagnitude [70]).For the model presented here, the problem size can be expanded byincreasing the complexity of the model as described in the previous

7.3 I M P ROV I N G P E R F O R M A N C E W I T H C O N C U R R E N C Y 121Speedup2018161412108642Parallel %95%90%75%50%01 8 64 512 4096 32768N (processors)Figure 77: Amdahl’s law for the theoretical maximum speedup whenusing multiple processors.876Measurementsp = 95%p = 90%p = 85%Speedup543212 4 6 8 10 12N (processors)Figure 78: The improvement in performance as the simulation isdistributed over multiple computers.paragraph, and also by increasing the number of nephrons in themodel, although it is not sensible to have a model that containsmore nephrons than are present in a kidney. Since the portion ofthe implementation that must be executed in serial is very small,and the complexity of the model will only increase as it is improvedand extended, it is evident that the distributed implementation willsignificantly benefit from being executed over many processors.To determine the maximal speedup obtainable by distributing thesimulation over multiple processors, a system of 384 nephrons wassimulated for 1, 000 time-steps using 1, 2, 3, 4, 6 and 12 computers(not including the computer running the controller process). Thetiming measurements are plotted in Fig. 78, which clearly shows that

122 L A R G E-SCALE S I M U L AT I O Nthe distributed implementation corresponds to p ≈ 90%. Accordingly,the maximal speedup that can be obtained is given by Eq. 7.7. Thenetwork capacity may have reduced the performance somewhat, andit is possible that a larger model would have reduced the frequencyof communication and resulted in a larger performance increase. Alarger model was not used to test this hypothesis, due to the limitationsof model construction—for a description of this issue and aproposed solution, see Sect. 7.4.lim S(N) = 1N→∞ 1 − p ≈ 1= 10 (7.7)1 − 0.9Therefore, given sufficient hardware, the distributed implementationshould be capable of providing a 10-fold performance increaseover the original single-threaded implementation. Based on the performanceestimates for a whole-kidney simulation given in Sect. 7.2.2,this would result in the simulation of a single time-step for a wholekidneymodel in 3.75 minutes (i. e., 3:45).7.4 AU T O M AT I C G E N E R AT I O N O F L A R G E M O D E L STo investigate the dynamics of sufficiently large multi-nephron systems(e. g.whole-kidney models), it is necessary to first constructthe model. The structure of the multi-nephron systems presented inChapter 5 and Chapter 6 were specified manually, but this approachis clearly infeasible for larger systems, so there is a need for automatedmodel growth. The kidney has a regular, but inhomogeneous,structure that cannot be easily replicated using well-known networkgrowth techniques [4] (e. g., preferential attachment).The automatic growth of models has been explored in the context ofbone growth [29], and the dynamics of complex networks in responseto the addition of nodes has been studied across many fields [4]. Suchapproaches are not directly applicable to the growth of multi-nephronmodels because the addition of a nephron to an existing model is adiscrete action that cannot be performed at random, and the resultingmodel must have a physically-realistic structure.Instead, two alternative approaches to generate large multi-nephronsystems are proposed: a stochastic approach that uses statistical constraintson the model structure to generate a model that satisfiesthe constraints, and a rewriting system approach that generates thearterial tree and attaches nephrons to the tree in a stochastic manner.These two approaches are now presented in turn.7.4.1 Stochastic generation of large modelsThe stochastic approach allows the modeller to specify statisticalconstraints on various aspects of the model structure, from which amodel that satisfies these constraints is generated. Constraints canbe applied at three different levels of hierarchy: nephrons, interstitialfluid columns, and the entire model. Nephrons and interstitial fluidcolumns can also be divided into classes, each of which has its ownset of constraints.

7.4 AU T O M AT I C G E N E R AT I O N O F L A R G E M O D E L S 123For example, nephrons are often classified as cortical or juxtamedullary—thisapproach allows the modeller to define the constraintson these two classes independently. Similarly, columns ofinterstitial fluid can be divided into classes (e. g., based on how farinto the renal pyramid they descend) and have independent sets ofconstraints placed on them.The constraints on each nephron class are:• The length and location of each tubule segment in the interstitialfluid.• The value of nephron-specific model parameters such as k (seeTable 6).The constraints on each column class are:• The number of nephrons in the column.• The number of branches in the column’s arterial tree.• The distribution of nephron classes in the column.The constraints on the entire model structure are:• The number of interstitial fluid columns in the model.• The number of branches in the model’s arterial tree.• The distribution of column classes in the model.The stochastic nature of this approach is provided by acceptingthree different types of value for the constraints detailed above:• A single value (i. e., a constant);• A range from which a value will be chosen at random (e. g.,0 . . . 1); and• A list of values that associates a likelihood with each givenvalue.Neighbouring columns on a branch of the arterial tree are alsodefined to be neighbours in the 2D arrangement of columns, andthe only other neighbouring columns are those from neighbouringbranches. These limits on the model structure may reduce the abilityof the generated models to accurately mimic a real kidney, but theyare easily solved. By allowing more general constraints (such asassigning likelihoods to columns of the same class being neighbours),it can not be guaranteed that the constraints will be solvable.7.4.2 Model growth as a rewriting systemAn alternative, more complex, approach to growing such large modelsis to generate the arterial tree for the entire model and then attachnephrons to this tree. This requires an understanding of the renalarterial tree structure and a method for using this information togenerate an arterial tree for the model. Based on experimental studiesof vascular systems and the application of modelling techniques to

124 L A R G E-SCALE S I M U L AT I O Ncapture their structural properties, this approach should yield anaccurate model of the renal arterial tree, to which individual nephronscan then be attached.Vascular systems have been demonstrated to exhibit self-similarity,allowing their structure to be described by fractal models [22, 197,163, 207]. Studies of these fractal models suggest that there is an evolutionaryadvantage to branching systems having a fractal dimensions[196]—the fractal models are significantly less sensitive to structuralvariations than the classical scaling models [163, 196, 197].Lindenmayer systems [105, 115, 106, 107]—hereafter referredto as L-systems—are a class of rewriting system that are able torepresent fractal objects obtained by use of recursive transformations[85] (see Fig. 79 for an example), such as the fractal models that havebeen applied to vascular systems. Indeed, such systems have alreadybeen used to generate biological models such as virtual animals[82, 83], cellular processes [64], swarms [88] and vascular systems[207] including the human bronchial tree [22] and the human retina[93].In particular, the branching properties of the renal arterial tree[210] and the cardiovascular system [208, 209] have been shown toexhibit variations in branching parameters, resulting in multiple fractaldimensions. Radial, length and connectivity data for the arterialand venous trees of the rat kidney are also available [134].The fractal dimension of the renal arterial tree has been estimatedto be in the range 1.61–1.64 [49], which is consistent with valuesfound in artificial fractal structures generated by diffusion-limitedaggregation processes [200, 199, 198, 117] and similar to values proposedfor the retinal vasculature [123, 109]. However, the processesresponsible for the morphogenesis of the renal vasculature are stillnot fully understood [48].A straightforward approach to building the renal arterial tree is togrammatically evolve a rewriting system so that it builds a tree withthe desired fractal dimension [137] without analysing any graphicalrepresentations of the system [5]. This approach fails to take intoaccount the multifracticality of the renal arterial tree, but it is notimmediately clear what an impact (if any) this would have on themodel. It also uses the fractal dimension of the resulting tree as themeasure of fitness, although this is not necessarily a suitable metric,as noted by Zamir in [207]:Two such structures may have the same fractal dimensionby virtue of their space-filling properties, but havewidely different structures in their fluid dynamic designand function. A way of discussing fractal properties thatis particularly suited to open tree structures . . . is thatof considering L-system models by which the tree can begenerated.The alternative approach suggested by Zamir allows the renal arterialtree to exhibit multiple fractal dimensions, by using a parametricL-system [207] that allows the branching parameters to change asthe tree descends. To determine the changes in the parameters wouldrequire dividing the renal arterial tree into a finite number of layers—each of which would ideally demonstrate little variation in its branch

7.4 AU T O M AT I C G E N E R AT I O N O F L A R G E M O D E L S 125L × 0.7940ᴼL × 0.7940ᴼ40ᴼL × 0.79²LL40ᴼ(a) The rule takes a line of length Land attaches two lines to one end, withlength L × 0.79, at angles of ±40 ◦ tothe original line.(b) The third stage of the system.(c) The sixth stage of the system.(d) The tenth stage of the system.Figure 79: An example of a Lindenmayer system that superficiallyresembles the arterial tree of the kidney, as presented in [207].

126 L A R G E-SCALE S I M U L AT I O Nstructure—and derive independent parameter values for each layer.The data provided by Nordsletten et al. would be a useful referencein determining how to divide the renal arterial tree into appropriatelayers [134].Both of the approaches outlined above rely on measurements of thebranching characteristics of the renal arterial tree. For the purposesof growing a model of the renal arterial tree, it is enough to measurethe length, diameter and branching angle of each branch in the tree[207]. Notwithstanding the techniques used in [134], an automatedprocess built to capture these parameters would be extremely useful.Baker has demonstrated that Model-Image Registration can be usedto fit a parametric 3D model to the curvature of the human cochlea,based on medical images [16]. This suggests that a similar approachmay be able to determine the branching parameters based on medicalimages of the renal arterial tree.If such a rewriting system were used to generate a representationof the vascular network for the model, this representation wouldcontain sufficient information to not only generate the arterial treeof the model, but to also give each arterial branch a spatial location.Given a desired resolution for the interstitial fluid columns, their 2Darrangement would then be specified by this spatial information. Asa result, this hypothetical arterial network generated by a rewritingsystem would specify the entire structure of the model, except for thatof the individual nephrons. To specify the structure of each nephronin the model, a stochastic approach similar to the one outlined inSect. 7.4.1 would be sufficient.

Part IVS U M M A RY O F R E S E A R C H

D I S C U S S I O N A N D C O N C LU S I O N8The ultimate intent of this research was to provide an approach torenal modelling that is capable of predicting whole-kidney functionfrom the dynamics of individual nephrons, and can therefore beof practical use to clinicians. In this chapter, the contributions ofthis thesis are summarised. The limitations of the work are thendiscussed and several areas of further research to build on this workare nominated. Finally, the research outcomes are evaluated againstthe research aim.8.1 C O N T R I B U T I O N SThe standard methods and techniques used to model nephron behaviourare not well suited to studying interactions, couplings andemergent dynamics in large multi-nephron systems. This is due to severallimitations, chiefly the complexity of solving systems of coupledequations and the inability to decouple the structure and behaviourof the model. This thesis has proposed a new approach for modellingmulti-nephron systems, which minimises the coupling betweenequations, reduces the computational cost of solving the model equations,and decouples the structure and behaviour of the model. Thecontributions of this thesis are:• A modelling approach—hierarchical dynamical networks—whichcombines complex networks and graph automata into a singlemodelling framework. This approach explicitly captures thestructure and interactions in multi-nephron systems, and decouplesthe structure and behaviour of the model. This approachallows emergent dynamics to be easily explored and analysed,as discussed in Chapter 4.• The development of a multi-nephron model that produces validbehaviour in single-nephron and multi-nephron systems—asdemonstrated in Chapter 5—and renders the simulation ofwhole-kidney function from the dynamics of individual nephronscomputationally tractable. Using this model, the emergent effectsof the couplings and interactions between nephrons canbe investigated.• An investigation into the dynamics of multi-nephron systemsthat focused on whole-system and hierarchical properties ratherthan the dynamics of individual nephrons, conducted in Chapter5. As part of this investigation, the dynamics of a 72-nephronsystem were analysed—a system significantly larger than existingmulti-nephron models.• A study of whole-system stability in response to localised impairmentsin nephron function, conducted in Chapter 6. This wasthe first study of the emergent dynamics of impaired nephron129

130 D I S C U S S I O N A N D C O N C L U S I O Nfunction, and served as an illustration of how the emergentdynamics produced by renal diseases may be predicted andanalysed. The impaired multi-nephron systems exhibited verystable behaviour, which we contend is a feature of both themodel and the kidney proper.• In Chapter 7, the computational cost of the model was shown tobe low enough that the simulation of whole-kidney function isfeasible for the first time. It was also demonstrated that simulationscould be easily distributed across multiple computers, resultingin a significant gain in performance. An implementationof the model that supports parallel and distributed executionwas presented, based on the Join Calculus [59, 60].• In order to predict whole-kidney function, a whole-kidney modelmust be constructed. Two approaches for automatically generatingsuch models were proposed (but not implemented) inChapter 7.8.2 L I M I TAT I O N S A N D F U RT H E R W O R KThe model presented in this thesis is built on a number of simplifyingassumptions, described in Sect. 4.4.2. Most notably, all vessels arenon-compliant and the function of each tubule segment was modelledat a much higher level than in existing tubule models, which attemptto capture precise low-level dynamics. However, both of these limitationscan be removed by refining the update rules in the model.The update rules for each tubule node can be replaced by discreteversions of the continuous equations used in existing tubule modelswithout incurring unexpected increases in computational cost (seeSect. 4.1.1). Similarly, compliance can be modelled by changing howthe flow through a vessel is related to the pressure difference acrossthe vessel, as illustrated in Fig. 80.The other significant limitation on this work is the lack of a methodfor automatically generating models of large multi-nephron systems—without such a method, the structure of such models much be specifiedmanually, which is infeasible for sufficiently large systems. Anapproach to automatically generate multi-nephron models from statisticalconstraints on the model structure was presented in Sect. 7.4.1,but did not provide control over the structure of the arterial tree andthe 2D arrangement of the interstitial fluid columns. An improvedapproach was presented in Sect. 7.4.2, using a rewriting system togenerate the renal arterial tree and attaching nephrons to this tree.By analysing branching characteristics in medical images of renalarterial trees, this approach would generate models with realisticarterial trees and model structure.Several areas of further research that build on this thesis are nowproposed.8.2.1 Improved understanding of whole-kidney functionThe simulation of whole-kidney function combined with the interactivityof the model would allow for an improved understanding how the

8.2 L I M I TAT I O N S A N D F U RT H E R W O R K 131CompliantNon-compliantFlowPressureFigure 80: A simple model of compliant vessels, using a cubic equationto capture the relationship between pressure and flow, rather thana linear equation. The highlighted region shows the pressure rangeacross which the cubic equation approximates the dynamics of acompliant vessel.whole-kidney function adapts to changes in bodily conditions. Oneapplication for this knowledge would be to design dialysis machinesthat can better replicate whole-kidney function.8.2.2 Modelling of renal diseaseBased on the simple example presented in Chapter 6, detailed modelsof individual renal diseases can be developed and incorporatedinto whole-kidney models, in order to perform clinically-relevant experimentssuch as investigating how the presence of a particularrenal disease affects whole-kidney function. The model also allowsfor impairments to spread across the system, which would allowfor experiments into the progressive loss of kidney function to beconducted.8.2.3 Pluggable virtual kidneySince the modelling approach presented in Sect. 4.1 decouples theindividual equations, arbitrary differential equations can be convertedinto difference equations and incorporated into the model. Due to thelow computational cost of the model and the distributed simulationframework, a virtual kidney model could be produced that would allowresearchers to experiment with incorporating systems of equationsfrom different tubule models and to analyse the resulting dynamics atmultiple levels of hierarchy, from the individual tubule to the wholekidneyscale.

132 D I S C U S S I O N A N D C O N C L U S I O N8.2.4 Modelling kidney failure with reliability factorsA simple failure model could also be incorporated into a whole-kidneymodel, whereby each nephron n would be assigned a reliability factorR n = f(t, v) that is a function of the age of the nephron t and thevolume of fluid v that has flown through that nephron. Simulations ofa particular model would be used to determine the specific R n valuesfor some t ′ (e. g., 60 years). These R n values could then be appliedat the start of further simulations, allowing the dynamics of an agedkidney to be analysed, since the simulation would effectively havestarted at time t = t ′ . The feasibility of this study relies on choosinga reasonable function f(t, v)—given any such f(t, v), analysis of themodel dynamics and disease modelling experiments should proveto be useful benchmarks to compare the resulting dynamics againstthose observed in patients of the particular age t ′ .8.3 C O N C L U S I O NThe ultimate intent of this research was to provide the basis foran approach to renal modelling that is capable of predicting wholekidneyfunction from the dynamics of individual nephrons, and cantherefore be of practical use to clinicians.A unique modelling approach was proposed in Chapter 4, and theresulting models were shown to produce valid behaviour in Chapter 5.Chapter 5 also demonstrated that the models could be used to studyemergent dynamics in multi-nephron systems, and the applicationof this analysis to studying the effects of localised impairments onwhole-system dynamics was illustrated in Chapter 6. Finally, the computationalcost of the model was investigated in Chapter 7, whichdemonstrated that the simulation of whole-kidney function was computationallytractable, and that the performance of the simulationscould be improved by utilising multiple computers.In conclusion, the modelling and analysis techniques presentedin this thesis allow for emergent dynamics to be studied in largemulti-nephron systems. This work demonstrates that, for the firsttime, simulation of whole-kidney function from the dynamics of individualnephrons is tractable. Furthermore, the work provides a basisfor predicting emergent effects of localised renal disease. With thecontinued development of this model, we hope that significant insightwill be gained into the onset, progression and treatment of renaldiseases.

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C O L O P H O NThis thesis was typset with LAT E X 2ε using the Bera, Euler and Timestype faces. The typographic style is available for LAT E X via CTAN as“classicthesis”, except that this thesis uses Bera as the main fontand Times for smallcaps.Final Version as of 1st September 2008 at 21:00.