Factors influencing the accuracy of biomechanical breast models

This is an author produced version of an article that appears in:MEDICAL PHYSICSThe internet address for this paper is:https://publications.icr.ac.uk/2637/Copyright information:http://www.aip.org/pubservs/web_posting_guidelines.htmlPublished text:C Tanner, J A Schnabel, D L G Hill, D J Hawkes, M O Leach,D R Hose (2006) Factors influencing the accuracy ofbiomechanical breast models, Medical Physics, Vol. 33(6),1758-1769Institute of Cancer Research Repositoryhttps://publications.icr.ac.ukPlease direct all emails to:publications@icr.ac.uk

Factors influencing the accuracy of biomechanical breast modelsChristine Tanner, Julia A. Schnabel, Derek L. G. Hill, and David J. HawkesCentre of Medical Image Computing at University College London, Gower Street, London,WC1E 6BT, United KingdomMartin O. LeachCancer Research UK Clinical MR Research Group, The Institute of Cancer Research and the RoyalMarsden NHS Foundation Trust, Sutton, Surrey SM2 5PT, United KingdomD. Rodney HoseClinical Sciences Division, Medical Physics and Clinical Engineering, Royal Hallamshire Hospital,University of Sheffield, Sheffield S10 2JF, United KingdomReceived 24 August 2005; revised 29 March 2006; accepted for publication 30 March 2006;published 19 May 2006Recently it has been suggested that finite element methods could be used to predict breast deformationsin a number of applications, including comparison of multimodality images, validation ofimage registration and image guided interventions. Unfortunately knowledge of the mechanicalproperties of breast tissues is limited. This study evaluated the accuracy with which biomechanicalbreast models based on finite element methods can predict the displacements of tissue within thebreast in the practical clinical situation where the boundaries of the organ might be known reasonablyaccurately but there is some uncertainty on the mechanical properties of the tissue. For twodatasets, we investigate the influence of tissue elasticity values, Poisson’s ratios, boundary conditions,finite element solvers and mesh resolutions. Magnetic resonance images were acquired beforeand after compressing each volunteer’s breast by about 20%. Surface displacement boundary conditionswere derived from a three-dimensional nonrigid image registration. Six linear and threenonlinear elastic material models with and without skin were tested. These were compared tohyperelastic models. The accuracy of the models was evaluated by assessing the ability of themodel to predict the location of 12 corresponding anatomical landmarks. The accuracy was mostsensitive to the Poisson’s ratio and the boundary condition. Best results were achieved for accurateboundary conditions, appropriate Poisson’s ratios and models where fibroglandular tissue was atmost four times stiffer than fatty tissue. These configurations reduced the mean maximum distanceof the landmarks from 6.6 mm 12.4 mm to 2.1 mm 3.4 mm averaged over allexperiments. © 2006 American Association of Physicists in Medicine. DOI: 10.1118/1.2198315Key words: Finite element analysis, breast imaging, imaged guided interventions, deformationI. INTRODUCTIONBiomechanical breast models employing finite elementmethods FEMs have been explored for predicting mechanicaldeformations during biopsy procedures, 1,2 for modelingcompressions similar to x-ray mammography, 3–5 for the registrationof magnetic resonance MR and x-raymammograms, 6,7 for validating the nonrigid registration ofdynamic contrast-enhanced DCE MR mammograms, 8 fortesting reconstruction algorithms in elastography, 9–11 and as aforward model for elastography. 12 The use of mechanicalmodels in these applications can be seen as providing priorinformation about the expected deformations. This prior informationhelps to constrain possible solutions and thus toprovide a physical basis for interpolation, reconstruction andprediction for those cases in which there is insufficient informationin the image data. Previously described biomechanicalbreast models have been assessed based on the predictedlocation of anatomical landmarks selected in breast imagesacquired before and after in-vivo compression, 1,2,6,7,13 by visualcomparison of the simulated compressed breast imagewith the uncompressed breast, 3 or not at all. 4,5 Evaluationsbased on landmarks were limited due to the use of only veryfew landmarks 1,2 or a single dataset. 6,7,13The first aim of this study was to determine a suitablemodel configuration, that can be employed for simulatingplausible breast deformations during DCE MR mammographyfor the validation of nonrigid registration. 8 The secondobjective was to determine the most important modellingaspects. Biomechanical breast models are commonly criticizedfor modelling the mechanical breast properties in a toosimplistic or unrealistic manner by using rather poor estimatesof the biomechanical properties. Yet, the influence ofthe mechanical breast properties on the model accuracy incomparison to other modelling factors has not been systematicallyinvestigated for the practical clinical situation wherethe boundary conditions may be known rather well from imagingsystems yet the mechanical properties are rather poorlycharacterized.Biomechanical breast models mainly vary with respect tothe mesh generation, the boundary conditions employed, thetissue properties assumed and the solution strategies. This1758 Med. Phys. 33 „6…, June 2006 0094-2405/2006/33„6…/1758/12/$23.00 © 2006 Am. Assoc. Phys. Med. 1758

1759 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1759TABLE I. Preprocessing parameters and geometric FEM model configurations. All triangulations were smoothedby 120 iterations. Elements that exceeded an aspect ratio of 20 or an angle of 165° were classified as “badlyshaped” by ANSYS.Volunteer oneVolunteer twoGM1 GM2 GM1 GM2Preprocessing Isotropic voxel size mm 8 8 10 7.5Number of decimation steps 120 2 1 1Mesh configuration Number of nodes 51 072 102 102 58 614 117 347Number of elements 34 873 72 756 40 636 81 633Badly shaped elements % 0.057 0.003 0.000 0.004study assessed the influence of all of these aspects on theaccuracy with which the displacement of internal breaststructures can be predicted. Results are presented for twodatasets. Experiments included a comparison of Mooney-Rivlin hyperelastic models and finite deformation formulationswith simpler approaches.The evaluation was based on obtaining 3D MR imagesbefore and after compressing the breast by approximately20% in thickness. Breast surface displacements were calculatedby applying a full 3D nonrigid registration 14 to theseMR images. A subset of these surface displacements servedas boundary conditions for the FEM model. The results wereevaluated on the basis of the residual displacement error of12 manually identified point landmarks.II. MATERIALS AND METHODSA. DatasetsTwo volunteers were recruited for this study, a 37 year oldnulliparous pre-menopausal woman and a 65 year old multiparouspost-menopausal woman on hormone replacementtherapy. Tl weighted images of the volunteers were acquiredon a Philips 1.5T Intera using a 3D fast gradient echo sequencewith TR=16.9, TE=6.0, flip angle 35°, axial slicedirection, and a spatial resolution of 0.820.822 mm 3and 0.820.822.5 mm 3 , respectively. Each volunteer waspositioned in a fixation device for breast biopsies providedby Philips Medical Systems. The right breast was placedbetween two sagittal plates without any compression for thefirst image. For the second image, the plate on the breast’smedial side was kept immobile, while the plate on the lateralside was moved manually towards the immobile plate by asmuch as the volunteer could comfortably tolerate. The imagesshowed that this resulted in a compression of about20% in breast thickness for each volunteer from 75 mm to60 mm for volunteer one and from 78 mm to 63 mm forvolunteer two. Volunteers were asked not to move betweenthese two scans. Consent of volunteers was obtained in accordancewith Guy’s Hospital, London, UK, local researchethical approval 00/11/99. Figures 6a and 7a show exampleslices of these images.B. Nonrigid registrationSurface displacements were derived from a full 3D nonrigidregistration, rather than a 3D surface registration, 15 toimprove accuracy by avoiding segmentation and decimationerrors. The nonrigid registration algorithm described inRueckert et al. 14 was employed to register the image of thecompressed breast to the image of the uncompressed breast.The registration was not constrained to conserve volume becauseblood volume may have been reduced due to the compression.The images were nonrigidly registered using aregular grid of control points approximated by B-splines andnormalized mutual information 16 as a voxel based similaritymeasure. The registration algorithm was applied in a multiresolutionfashion, with an initial control point spacing of40 mm. At every resolution level this distance was halved.Based on a visual assessment of the registration accuracy, theprocess was terminated at a final control point spacing of10 mm for volunteer one and 2.5 mm for volunteer two. Acquisitionof the images within 15 minutes, in the same setting,with a high spatial resolution and without the administrationof contrast agent simplified the task of registration.Example slices of the registered images are shown in Figures6b and 7b.C. Biomechanical breast models1. Geometric FEM modelThe 3D MR images were first corrected for intensity nonuniformityusing the N3 program. 17 The images of the uncompressedbreast were then manually segmented into breastand background, using an interactive tool in ANALYZE. 18The breast regions were further segmented into fatty andfibroglandular tissue using thresholding followed by manualediting if necessary. The breast segmentations were blurredusing an isotropic 3D Gaussian kernel with a standard deviationof 1 mm. The blurred segmentations were then downsampledto an isotropic voxel size of 7.5 mm to 10 mmTable I such that different mesh resolutions could be testedand the limit of available elements was not exceeded duringthe meshing stage. A 3D triangulation of the outer surface ofthe breast was obtained using marching cubes, smoothingand decimation techniques from The Visualization Toolkit. 19Medical Physics, Vol. 33, No. 6, June 2006

1760 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1760FIG. 1. Geometric models for volunteer one. a–d 2Daxial example slice of the 3D MR image or biomechanicalbreast models at the same position showing aMR image of the uncompressed breast, b segmentationof a, c and d material assignment to the elementsof c GM1 and d GM2. e and f surfacerendering of the undeformed 3D mesh of e GM1 andf GM2.The triangulated volumes were meshed into 10-noded tetrahedralelements with the ANSYS FEM package. 20 These elementshave an additional node on each edge, which allowsfor quadratic interpolation of the element properties. Theskin was modelled by adding 1 mm thick triangular prismswith additional nodes in the middle of each edge on thesurface of the breast. Material properties were assigned to theelements according to the image segmentation.Two geometrical models, called GM1 and GM2, werecreated for each volunteer to investigate the combined influenceof the element size and shape. The GM1 models hadabout one-half as many elements as the GM2 models, seeTable I. This was achieved either by decimating the triangularsurface more volunteer one or by down-sampling theblurred segmentations more volunteer two. Figure 1 showsa 2D example slice of the breast image, the segmentation andthe corresponding meshes for volunteer one.2. Material modelsPublished values of the stress to strain relationshipYoung’s modulus of breast tissue types varyconsiderably. 21–23 Models were constructed that covered therange of values and complexity of published stress-strain relationshipsin order to determine their influence on predicteddeformations in a systematic and quantitative way.a Linear and piecewise-linear models: The individualtissue types were modelled as isotropic and homogeneous.Only displacement boundary conditions were applied, andhence only the ratios of the Young’s modulus of the differenttissue types were required. Six linear models were constructedwhere fibroglandular tissue was 1, 1.5, 5, 10, 15 or20 times stiffer than fatty tissue. These values cover andincrease the range of reported ex-vivo ratios 1.5 to 5.0 forup to 5% prestraining 21–23 and in-vivo ratios 1.3 toTABLE II. Young’s modulus E of fatty and fibroglandular tissue for linear and piecewise-linear material models.The same Young’s modulus was used for compression as for tension, i.e., Ee=E−e. The last column listsmeanE g /E f for strain e−0.25,0.Young’s modulus E in kPa for strain e0Name Fatty tissue E f FibroglandulartissueE g E gE fLinear MM1 1 1 1.0MM2 1 1.5 1.5MM3 1 5 5.0MM4 1 10 10.0MM5 1 15 15.0MM6 1 20 20.0Nonlinear MM7 718.16e+4.46 if 0e0.2515.1 exp10.0e 0.8184 if e0.25MM8 18.5 1001.3e 2 −272.7e+86.1 4.0MM9 519.7e 2 +2.4e+4.9 123 888.9e 3 −11 766.7e 2 +696.9e+12.1 15.1Medical Physics, Vol. 33, No. 6, June 2006

1761 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1761FIG. 2. Relative stress-strain relationshipsof fibroglandular to fatty tissuefor MM1–MM9. The right-hand figureshows details of the relationships withlow stress ratios.5.7. 24–27 Three nonlinear models were created in accordanceto previously reported values. 1,3,22 These nine models weretested without skin MM1-MM9 and with skin that was 10times stiffer than the fatty tissue in the linear modelsMM1S-MM9S. The nonlinear stress-strain functions wereapproximated by piecewise-linear functions. The materialproperties of these models are summarized in Table II. Theirrelative stress-strain functions are depicted in Figure 2.b Hyperelastic models: Hyperelastic materials can experiencelarge elastic strain that is recoverable and have generallyvery small compressibility. 28 The stress-strain relation ofhyperelastic materials is, by definition, completely definedby a strain-energy function . We assessed the neo-Hookeanmodel nH = 10 I 1 −3 and the five-parameter Mooney-Rivlin model MR = 10 I 1 −3+ 11 I 1 −3I 2 −3+ 01 I 2−3+ 20 I 1 −3 2 + 02 I 2 −3 2 , where ik are the model constants,I 1 = 1 2 + 2 2 + 3 2 , I 2 = 1 2 2 2 + 1 2 3 2 + 2 2 3 2 , and i are theeigenvalues of the stretch tensor. 28 These models were fittedin the least squares sense to the strain-stress relationship ofMM1-MM9 for strains within ±25%. Parameter fitting withANSYS led to unstable materials for the Mooney-Rivlinmodels. A constrained parameter fitting was developed instead.The constraints ensured that the work required for anarbitrary change in deformation is always positive. No convergenceproblems were experienced for models derived inthis way. The parameters of these models are listed in TableIII. Figure 3 shows their relative stress-strain relationships incomparison to the original functions. Neo-Hookean modelsapproximated linear models well. Mooney-Rivlin modelsachieved better fits for nonlinear models due to their greaternumber of free parameters.c Poisson’s ratios: The influence of the Poisson’s ratiowas assessed for values 0.499, 0.495, 0.45, 0.4, 0.3, 0.2, and0.1 for the linear and piecewise-linear material models.3. Boundary conditionsThe influence of boundary conditions was assessed forthree different setups, called BC1, BC2, and BC3. Displacementswere prescribed for the surface nodes on the lateral,medial and posterior side of the breast as illustrated in TableIV. BC1 and BC2 constrained 51% 54% of the surfacenodes for volunteer one two, while BC3 constrained 24%46%. BC3 constrained no posterior, but more lateral andTABLE III. Parameters of strain energy function for hyperelastic material models. The last column states meanE g /E f for strain e−0.25,0.Parameters of strain energy function in kPa forName Fatty tissue Fibroglandular tissueE gE fNeo-Hookean 10 10MM1nH 0.1289 0.1289 1.0MM2nH 0.1289 0.1933 1.5MM3nH 0.1289 0.6443 5.0MM4nH 0.1289 1.2887 10.0MM5nH 0.1289 1.9330 15.0MM6nH 0.1289 2.5774 20.0MM7nH 19.8360 17.3002 0.9MM8nH 2.4134 11.2613 4.6MM9nH 3.5714 105.0993 29.4Mooney–Rivlin 10 01 20 11 02 10 01 20 11 02MM7MR 46.42 −31.77 37.07 1.96 1.51 42.83 −36.54 51.83 7.33 0.52 0.6MM8MR 5.83 −3.14 0.90 0.64 0.08 26.07 −15.56 8.10 1.71 0.31 3.9MM9MR 10.00 −7.14 3.12 1.82 2.07 263.15 −231.32 387.50 55.81 −0.02 19.9Medical Physics, Vol. 33, No. 6, June 2006

1762 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1762FIG. 3. Relative stress-strain relationshipsof fibroglandular to fatty tissuefor hyperelastic models. MMinH denotesthe approximation of MMi by aneo-Hookean model and MMiMR representsthe five-parameter Mooney-Rivlin model of MMi.medial nodes than BC1 and BC2. The mean displacementmagnitude of the posterior medial nodes for BC1 was5.7 mm 1.5 mm for volunteer one and 5.9 mm 2.3 mmfor volunteer two. BC2 assumed zero displacements of thesenodes.The volume change introduced by a boundary conditionwas estimated from the volume of the original and thewarped surface triangulation. For BC1, the triangulation waswarped by the 3D nonrigid image registration, because alldisplacements prescribed by BC1 were defined by the outcomeof the image registration. This resulted in a volumechange of 0.1% −5.9% for volunteer one two and hencewe concluded that the deformation of the breast of volunteerone did not change volume. For BC2 and BC3, the triangulationwas warped by the FEM model which provided thebest mean displacement error for the boundary condition,resulting in an estimated volume change of −6.4%−6.5% for BC2 and −1.0% −9.45% for BC3.4. FEM model solutionAll FEM model solutions were obtained using a static orsteady state analysis, i.e., the models were assumed to haverelaxed to their lowest energy solution. The general approachwas to use the preconditioned conjugate gradient PCGsolver; to employ ANSYS’s automatic load step adjustmentwith initially one step linear models or four steps nonlinearmodels and maximally 500 steps; and to use an infinitesimaldeformation formulation. This strategy was testedagainst the frontal and the sparse direct solver 32 and a finitedeformation formulation, i.e., taking account of the geometricnonlinearity.For pure displacement approaches, nearly incompressiblematerial can lead to predicted displacements that are muchsmaller than they should be, i.e., locking, or to no convergence.The mixed u-p formulation, where pressure is anothersolution variable, and hyperelastic models were employed toovercome these problems for the finite deformationformulation. 28A continuous displacement field was derived from thenode displacements by interpolation with the quadratic shapefunction for 10-noded tetrahedral elements. 29D. Thin plate spline interpolationThe FEM solutions were compared to the result of interpolatingthe prescribed surface displacements by 3D thinplate spline basis functions 30 using The VisualizationToolkit. 19 The 3D thin plate spline basis functions are definedby h p x=x−x p , where x p denotes the 3D coordinateof the pth constrained node.TABLE IV. Configurations of three displacement boundary conditions BC1, BC2, and BC3. The surface nodesof the FE model were classified as posterior, medial and lateral as illustrated by the black areas on the leftfigures. Surface nodes were either free to relax free, displaced using the result of the 3D nonrigid registrationnreg or fixed to remain at their initial position fixed. BC3 contained additional nodes on the lateral andmedial side towards the posterior side to compensate for the completely unconstrained posterior side.Medical Physics, Vol. 33, No. 6, June 2006

1763 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1763FIG. 4. Examples of corresponding landmarks of thedeformed and undeformed breast. The images show orthogonal2D slices of the undeformed breast left-handside and the deformed breast right-hand side for volunteerand top and volunteer two bottom. The landmarksare located at the center of each circle.E. Landmark selectionThe accuracy of each model was assessed by manuallyidentifying for each breast 12 corresponding point landmarksin the set of images.Reference landmarks were selected in the image of thedeformed breast. Landmarks were generally placed at bifurcations,centers of small tissue regions or tips of spiculatedbranches, as illustrated in Figure 4. Two landmarks wereusually picked in every tenth slice for volunteer one and inevery seventh slice for volunteer two. Figure 5 shows thedistribution of the landmarks.The corresponding landmarks were then identified in theimage showing the undeformed breast. This was repeatedtwo more times by a single observer on different days. Theaverage position of these three measurements was assignedto each landmark.Figure 5 shows the displacement pattern of the landmarksfor both volunteers. These differ mainly in the inferior tosuperior direction. This was caused by the inferior part volunteerone or the superior part volunteer two of the undeformedbreast being closer to the lateral plate.F. Error measuresThe intraobserver variability of selecting a landmark wasmeasured by the mean Euclidean distance of the three measurementsto their average position. The intraobserver variabilityfor the 24 landmarks ranged from 0.05 mm to0.77 mm with a mean of 0.24 mm and a 90th percentile of0.50 mm.The performance of a model was quantified by the Euclideandistance between the position of the reference landmarkand the predicted position of the corresponding landmark byFIG. 5. Distribution and displacementof landmarks in the deformed breast.The images show the orthogonal meanintensity projections of the breast segmentation.The projected landmark positionsare marked by . Lines showthe projected displacements of thelandmarks. The minimum, mean andmaximum displacement of the 12landmarks of each volunteer are listedin mm for each direction u 1 , lateral tomedial; u 2 , posterior to anterior; u 3 ,inferiorto superior. u denotes thelength of the displacements.Medical Physics, Vol. 33, No. 6, June 2006

1764 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1764the model. This distance is called the displacement error. Thedistribution of the displacement errors from the 12 landmarksof each volunteer was summarized by its mean and itsmaximum.Statistical significance was not assessed because of thesmall number of datasets. By definition the best modelshould be that which minimizes the displacement error measure.Any model that produced a mean maximum displacementerror that exceeded the overall best results by less thanthe mean 90th percentile intraobserver variability of0.24 mm 0.50 mm was also classified as best.The sensitivity of the displacement errors was assessedwith respect to four factors, namely the mean Young’smoduli ratio of fibroglandular and fatty tissue, the Poisson’sratio, the boundary condition, and the addition of skin. Theanalysis was based on the model-free sensitivity index S i ,which measures the average reduction in variance of the displacementerror for sample quantity Y when fixing individualfactors F i . S i is given by 1−1/N Nj=1 varY F i = f j /varYwhere N is the number of possible values for factor F i andvarY F i = f j is the variance of Y after fixing factor F i tovalue f j . 31G. Overview of testsTest 1 — Direct solvers: Results obtained with the PCGsolver were compared to results obtained by two direct solvers,namely the frontal and the sparse direct solver. Testswere conducted for the coarser mesh GM1, nine modelswithout skin MM1–MM9, a Poisson’s ratio of 0.495 0.2for volunteer one two, the accurate boundary conditionBC1 and an infinitesimal deformation formulation.Test 2 — Mesh density: This test assessed whether doublingthe number of mesh elements provided different resultsGM1 versus GM2. The test configurations were otherwiseas in Test 1 when employing the PCG solver.Test 3 — Finite deformation formulation: An infinitesimaland a finite deformation formulation were compared in Test3. MM1–MM9, a Poisson’s ratio of 0.499 and the PCGsolver were employed for the infinitesimal deformation formulation.Incompressible hyperelastic models fitted toMM1–MM9, a mixed u-p approach and the sparse directsolver were used for the finite deformation formulation. Testswere only conducted for volunteer one and BC1, since thiswas the only deformation that preserved volume.Test 4 — Linear and piecewise-linear models, skin, Poisson’sratios, boundary conditions: Six linear and threepiecewise-linear models with and without skin MM1–MM9,MM1S–MM9S were compared for three boundary conditionsBC1, BC2, BC3 and Poisson’s ratios 0.499, 0.495,0.45, 0.4, 0.3, 0.2, and 0.1. All models were solved using thecoarser mesh GM1, the PCG solver and an infinitesimaldeformation formulation.III. RESULTSThis study assessed the influence of different tissue propertiesand Poisson’s ratios, boundary conditions, finite elementsolvers and mesh resolutions on the accuracy withwhich biomechanical breast models can predict the displacementsof internal breast structures.A. Visualization of resultsFigures 6a and 7a show 2D orthogonal example slicesof the MR breast images before and after compressing thebreast between two plates for volunteers one and two, respectively.The misalignment due to the compression isclearly visible on the difference images. Motion artifacts inthe difference images were greatly reduced after 3D nonrigidregistration Figures 6b and 7b with the mean maximumdisplacement error dropping from 6.41 mm12.98 mm to 0.92 mm 1.67 mm for volunteer one andfrom 6.80 mm 11.85 mm to 1.03 mm 2.74 mm for volunteertwo. Less artifact reduction was achieved by the biomechanicalbreast models, with the mean displacement errorranging from 1.88 mm to 4.58 mm for volunteer one Figures6c–6h and from 2.12 mm to 4.01 mm for volunteertwo Figures 7c–7h.B. Test 1: Direct solversEmploying the direct solvers frontal or sparse direct insteadof an iterative solver changed the mean maximumdisplacement errors of the material models from MM1 toMM9 by less than 0.0007 mm 0.0012 mm for both volunteers.In conclusion, employing a direct solver did not influencethe accuracy of the models at the precision of the landmarkselection.C. Test 2: Mesh densityA denser mesh GM2 changed the mean maximum displacementerror on average by 0.003 mm 0.004 mm forvolunteer one and by −0.088 mm −0.059 mm for volunteertwo. The mean maximum displacement error of the individualmodels changed by less than 0.097 mm 0.196 mmfor both volunteers. The accuracy of the models were thereforenot substantially changed by a finer mesh.D. Test 3: Finite deformation formulation „volunteerone, BC1 only…Finite deformation solutions were obtained for incompressiblehyperelastic models using a mixed u-p solutionstrategy for the volume preserving deformation, i.e., volunteerone and boundary condition BC1. These were comparedto linear and piecewise-linear models that used an infinitesimaldeformation formulation and a Poisson’s ratio of 0.499.None of the finite deformation formulations improved theresult of their infinitesimal counterpart. Neo-Hookean modelsincreased, the mean maximum displacement errors onaverage by 0.07 mm 0.15 mm when compared to the linearmodels and by 0.28 mm 0.34 mm for MM7–MM9. Approximatingthe nonlinear material models by Mooney-Rivlin models increased the mean maximum displacementerror by 0.14 mm 0.25 mm on average. The volumeMedical Physics, Vol. 33, No. 6, June 2006

1765 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1765FIG. 6. Axial, sagittal and coronal 2D example slices of volunteer one showing a original images with top compressed breast, middle uncompressedbreast, bottom difference image of uncompressed and compressed breast; b difference image after 3D nonrigid registration; c–h difference image of theFEM deformed image and the original image of the compressed breast for results with c, e, g smallest mean displacement error and d, f, h largestmean displacement error for BC1-3. The mean displacement error DE are listed for each case.changes introduced by the hyperelastic models 0.0% to0.1% resembled the volume change of the 3D image registration.The linear and piecewise-linear models reduced themesh volume by 0.7%.The mean maximum displacement error changed bymore than 0.24 mm 0.50 mm only when approximatingMM9 by a neo-Hookean model. This worse performance islikely to have been caused by the poor approximation of thestress-strain relationship as is apparent in Figure 3. The neo-Hookean models matched the linear models very well, butstill did not improve the results. It must therefore be concludedthat incompressible hyperelastic models and the useof a finite deformation formulation results in a similar accuracyto linear and piecewise-linear models with a Poisson’sratio of 0.499 and an infinitesimal deformation formulationfor a volume preserving deformation.FIG. 7. Axial, sagittal and coronal 2D example slices of volunteer two a of the original image with top compressed breast, middle uncompressed breast,bottom difference image of uncompressed and compressed breast; b difference image after 3D nonrigid registration; c–h difference image of the FEMdeformed image and the original image of the compressed breast for results with c, e, g smallest mean displacement error and d, f, h largest meandisplacement error for BC1-3. The mean displacement error DE are listed for each case.Medical Physics, Vol. 33, No. 6, June 2006

1766 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1766FIG. 8. Results for volunteer one, boundary condition BC1 and material models without skin. The mean and maximum displacement errors are displayed ascontour plots with respect to the meanE g /E f of nine material models x axis, see Table II and seven Poisson’s ratios y axis, 0.499, 0.495, 0.45, 0.4, 0.3,0.2, 0.1. The contour interval is 0.1 mm. Only the five lowest contours of each graph are labelled. The largest value of each graph is shown by the boxednumber. Best results, i.e., result within the mean 90th percentile intraobserver variability of 0.24 mm 0.50 mm from the smallest mean maximumdisplacement error of 1.88 mm 3.33 mm, are marked by stars. Dots represent the remaining experiments.E. Test 4: Linear and piecewise-linear models, skin,Poisson’s ratios, boundary conditionsThe effects of the average elastic ratio of fibroglandularand fatty tissue, and the Poisson’s ratio on the displacementerror are illustrated for BC1 and models without skin in Figures8 and 9. Table V provides an overview of the mainresults:iiiFor volunteer one and boundary condition BC1, thebest results were achieved by material models wherefibroglandular tissue was at most 4 times stiffer thanfatty tissue and by MM3S while keeping Poisson’sratios high v0.499,0.495, see Figure 8. Thesemodels reduced the mean maximum displacementerror to 1.96 mm 3.43 mm. Assuming that no motionhad occurred at the posterior and medial sideBC2 caused substantially higher errors, see TableV. Best results for BC2 required low Poisson’s ratiosv0.1,0.2,0.3, due to the introduction ofvolume changes. Keeping the posterior side unconstrainedBC3 was better than assuming no motionat the posterior side BC2 but was worse than BC1.Best models for BC3 had low Young’s moduli ratiosand high Poisson’s ratios. Very similar contour plotswere achieved when adding skin with a Young’smodulus of 10 kPa, with the mean maximum displacementerror of any model changing by less than0.21 mm 0.36 mm. The mean and maximum displacementerror was most sensitive to the boundarycondition, see Table VI. The most important factorafter fixing the boundary condition was either thePoisson’s ratio for BC1 apart from the mean errorof MM1–MM6, for BC2 and for the maximum errorof MM7–MM9 for BC3 or the Young’s moduli ratio.Thin plate spline interpolation of the prescribedsurface displacements provided only for BC1 amean and a maximum displacement error that wassimilar to the best FE models, see Table V.For volunteer two, the nonrigid registration introduceda volume change of −5.9%. Substantiallyworse results were therefore obtained at high Poisson’sratios, see Figure 9 and Table V. Best resultsfor BC1 were achieved by all models but MM6,MM9, MM9S while keeping Poisson’s ratios lowv0.1,0.2,0.3. Assuming no motion at the posteriorand medial side BC2 introduced lower errorsthan for volunteer one Table V. Not constrainingthe posterior side BC3 provided similar results asBC2. Best results for BC3 were achieved by linearmodels with skin and a low Poisson’s ratio. Addingskin had a bigger impact on the results than for volunteerone, with the mean maximum displacementerror changing on average, by −0.09 mm−0.42 mm and up to 0.31 mm 2.21 mm. Themean and maximum displacement error was mostMedical Physics, Vol. 33, No. 6, June 2006

1767 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1767FIG. 9. Results for volunteer two, boundary condition BC1 and material models without skin. The smallest mean maximum displacement error for BC1 was2.12 mm 3.15 mm.sensitive to the Poisson’s ratio, see Table VI. Themost important factor after fixing the Poisson’s ratiowas the boundary condition apart from one casemean error of MM7–MM9 for v=0.1 where theresults were more sensitive to the Young’s moduliratio. Thin plate spline interpolation of the boundaryconstraints Table V provided worse results thanthe best FE models.In conclusion, nine material models achieved for BC1 andappropriate Poisson’s ratios best results for both volunteers,namely all material models where fibroglandular tissue wasat most 4 times stiffer than fatty tissue and MM3S. Theirmean maximum displacement error was on average2.11 mm 3.37 mm. Less accurate or less constrainedboundary conditions BC2, BC3 led to higher errors. Keepingthe posterior side unconstrained BC3 was better forvolunteer one than assuming no motion at this side BC2.The boundary condition and the Poisson’s ratio had the highestinfluence on the displacement error.IV. CONCLUSIONSThis study evaluated the accuracy with which biomechanicalbreast models were able to predict the in-vivo displacementsof internal breast structures for a pre- and a postmenopausalvolunteer. The first objective of this study was tofind a suitable FEM configuration that can produce plausibledeformations during DCE MR mammography acquisition.For this purpose, any model which achieved, for accurateboundary conditions, a mean maximum displacement errorthat exceeded the best result by less than 0.24 mm0.50 mm for both volunteers was regarded as suitable. Thesecond aim was to investigate what modelling aspects aremost important.Biomechanical breast models were able to predict the displacementsof internal breast structures i.e., 12 correspondinganatomical landmarks for accurate boundary conditionsBC1, appropriate Poisson’s ratios and suitable elastic propertiesfibroglandular tissue is at most 4 times stiffer thanfatty tissue to a mean accuracy of 2.0 mm volunteer oneand 2.2 mm volunteer two for a deformation that introduceda mean displacement of 6.4 mm and 6.8 mm, respectively.Maximum errors reduced from 13 mm volunteer oneand 11.8 mm volunteer two to 3.4 mm and 3.3 mm, respectively.These suitable elastic properties are within therange of values reported by in-vivo elastography studies. 24–27The models were generally more sensitive to the choice ofPoisson’s ratio or boundary condition than to the elasticityproperties. Lower Poisson’s ratios improved the predictionwhen volume changes needed to be modelled. Assuming nomotion at the posterior and medial nodes, as often done whenmodelling breast compressions, 1–7 led to worse results forvolunteer one than keeping the posterior nodes unconstrained.Employing direct solvers or a finer mesh of improvedquality did not influence the accuracy. Using hyperelasticmaterial models and a finite deformation formulation did notimprove the accuracy.The robustness of the results to the elastic properties wereprobably caused by modelling only a few tissue types and byMedical Physics, Vol. 33, No. 6, June 2006

1768 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1768TABLE V. Configuration and mean results of the best FE models for Test 4, in comparison to the smallest displacement errors of the FE models and the resultsfrom a thin plate spline TPS interpolation. Interpolation results within 0.24 mm 0.50 mm from the smallest mean maximum displacement error aremarked by stars.Smallesterror mmBest FE modelsError mmTPS interp.Error mmMeanMaxMaterialmodelPoisson’sratio Mean Max Mean MaxVolunteer oneBC1 1.88 3.33 meanE g /E f 5, MM3S 0.495, 0.499 1.96 3.43* 1.69BC2 2.75 7.12 meanE g /E f 4, MM9S 0.1–0.3 2.85 7.33 3.32BC3 2.25 4.76 meanE g /E f 4, MM8S 0.495, 0.499 2.38 4.90 2.54* 3.46* 7.21* 4.61Volunteer twoBC1 2.12 3.15 all but MM6, MM9S 0.1–0.3 2.25 3.34* 2.20 4.29BC2 2.36 3.85 all but MM5, MM6 0.1–0.4 2.47 3.97 3.57 4.98BC3 2.52 3.80 MM1S–MM6S 0.1–0.3 2.66 3.90 2.83* 4.30the FEMs being greatly constrained by surface displacementsand hence acting as interpolants. A similar robustness wasreported by Ruiter et al.. 6,7 The results of this study comparefavorably with the performance reported by Azar et al. 1 andRuiter et al. 6,7 most likely because of improved boundaryconditions and much finer meshes. The 3D image registrationachieved mean errors of about 1.0 mm using just theinformation provided by the images and a mechanical unconstrainedtransformation model. Whether such an accuracycould be obtained prospectively with a patient specific, heterogeneousbiomechanical model and matching boundaryconditions, remains to be seen.Predicting breast deformations with the reported accuracywould improve the surgical precision for nonpalpable lesions,support the registration of x-ray and DCE MR mammogramsand provide plausible breast deformations for validationand teaching purposes. The main challenge in theseapplications is, therefore, to establish accurate boundary conditions.When registering, for example, a preoperative MRmammogram to the patient for surgical planning, this couldrequire image-visible surface markers, surface extraction ofthe patient’s breast using a stereo camera system, surfaceregistration and ultrasound based registration of the posteriorside and we are developing a system to do just that.The applied forces in these experiments have not beenmeasured and hence it is not known whether these wouldjustify the 6% volume change observed for volunteer two onthe basis that venous blood pressure values were exceeded.During DCE MR mammography acquisition, however, nosignificant change in external force is applied.Our investigation of the most important modelling aspectsis limited because only two volunteers were assessed, onlyone deformation scenario was evaluated and the appliedcompressions were much lower than during x-ray mammography.Nevertheless, we have no evidence to suggest that theproperties of the breast tissue examined are anything buttypical and the results of this study provide already valuableinformation for the practical clinical situation where theboundaries of the breast are known rather accurately.In conclusion, first all material models where fibroglandulartissue was no greater than 4 times stiffer than fattytissue provided the best results for both volunteers and canbe recommended in conjunction with a high Poisson’s ratiofor the simulation of plausible breast deformations duringDCE MR mammography. Second, the accuracy of the modelwas more affected by the choice of boundary condition orPoisson’s ratio than by the choice of elastic material proper-TABLE VI. Sensitivity analysis of the displacement error. The sensitivity index S i measures the average variance reduction of the mean or maximumdisplacement error when fixing the listed factor. Influential factors are indicated by large S i values.Sensitivity index S i in %Volunteer oneVolunteer twoMM1–MM6 MM7–MM9 MM1–MM6 MM7–MM9Fixed factor Mean Max Mean Max Mean Max Mean MaxYoung’s moduli ratio E g /E f 15 3 3 0 0 1 16 3Poisson’s ratio v 5 1 5 0 51 77 39 56Boundary condition 52 92 59 93 35 10 31 31Skin 0 0 0 0 3 4 0 0Medical Physics, Vol. 33, No. 6, June 2006

1769 Tanner et al.: Factors influencing the accuracy of biomechanical breast models 1769ties. Further studies are required to reconfirm the importanceof these modelling factors for other applications.ACKNOWLEDGMENTSThe authors would like to thank Kate McLeish, DavidAtkinson, Philipp Batchelor, and Andy D. Castellano-Smithfor their help in image acquisition or model construction.One of the authors C.T. was supported by EPSRC GR/M52779 and MIAS-IRC EPSRC GR/N14248/01, UK MRCD2025/31. One of the authors J.A.S. has received fundingfrom Easy Vision Medical IT, Philips Medical Systems, Best,The Netherlands.1 F. S. Azar, D. N. Metaxas, and M. D. Schnall, “A finite model of thebreast of predicting mechanical deformations during biopsy procedure,”IEEE Workshop on Mathematical Methods in Biomedical Image Analysis,South Carolina, USA, 2000, pp. 38–45.2 F. S. Azar, D. N. Metaxas, and M. D. 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Iterative solvers e.g., PCG solver start from an initial estimateof the solution and then improve it until convergence. Direct solversare generally more robust, but require more memory and are slower.Medical Physics, Vol. 33, No. 6, June 2006