FOURTEENTH ANNUAL EUROPEAN PRESSURE ULCER ...

Thursday September 1st
Proceedings of the 14th Annual European Pressure Ulcer Meeting
Oporto, Portugal
Mathematical models to describe wound healing and wound contraction
Fred Vermolen 1 , Etelvina Javierre 2 , Amit Gefen 3
1* Delft Institute of Applied Mathematics, Delft University of Technology, Delft, the Netherlands
F.J.Vermolen@tudelft.nl
2 University of Zaragoza, Zaragoza, Spain
3 University of Tel-Aviv, Tel-Aviv, Israel
Introduction
Wound healing phenomena, either occuring in
cutaneous wounds, or in trauma on organs or bone,
are crucial biological processes for the viability of a
living organism. These healing processes in general
proceed by signaling processes from, for instance
platelets, that trigger the cells in surrounding
undamaged tissues to come into action. This action
can be mobility, cell movement in the direction of a
signaling or growth factor concentration gradient or by
(biased) random motion, or the proliferation of cells
(division and growth). The simulations were described
in more detail in [1].
Methods
In this talk, we will review some of the mathematical
models we are working on. These models are
predominantly based on systems of reaction-diffusionconvection
equations and on the equations of viscoelasticity.
Here, we show some results from finiteelement
simulations and discuss some of the
implications. This model is applied to simulation of
wound contraction, angiogenesis and wound closure.
The processes of wound contraction and angiogenesis
take place in the dermis and are contain the ingress
and proliferation of (myo-)fibroblasts, endothelial cells,
and the regeneration of the capillary network. Wound
closure takes place in the epidermis as a result of
motility and proliferation of keratinocytes. Between the
two mechanisms, there is interaction via the oxygen
tension and secretion of signaling agents. This
interaction is dealt with in the model.
Besides the partial differential equations approach that
we use to model wound healing and wound
contraction, we will also discuss a brand new
formulation of a semi-stochastic cellular based model
on wound closure and growth of cell cultures. This
model is based on the mechanical forces that are
exerted and sensed by cells. Furthermore, the cellular
motion contains a somewhat random component. Cell
death and cell division are incorporated into this model
as stochastic processes as well. It is possible to
extend this formalism to chemical signaling. This semistochastic
approach has been described in [2].
If time allows, we will also show the newest
applications to modeling angiogenesis on fibrosis
impaired areas on the surface of a heart.
45
Results
We will show the results from simulations with input
date that are as realistic as possible. The results are
presented in terms of profiles of the various cell
densities over the dermal and epidermal region.
Furthermore, we show the contraction dynamics of a
wound over time.
Discussion
There is a huge variety of mathematical models for
wound healing. Most models are based on partial
differential equations and therewith, they are
continuum-based models. As soon as one arrives at a
dimension that is comparable to a cell size, then these
models will fail to predict the right behavior of the
phenomenon. Therefore, we are also interested at the
processes on a cellular level. It is one of our aims to
link the cellular models with the continuum models.
Furthermore, the value of the parameters is uncertain,
as well as most of the data is patient-specific. These
uncertainties introduce a certain randomness. We
think that we will be able to deal with this degree of
randomness by the use of stochastic finite-element
methods. This will be done in future studies.
Clinical relevance
The mathematical models will contribute in
understanding the fundamental relations between the
various subprocesses occurring in wound healing. This
understanding can give guidelines to improve
treatments such as preventing the formation of
hypertrophic scars, which is a mayor undesired sideeffect
during the healing of burns.
Acknowledgements
We are grateful for the financial support from
Agenschap.nl for funding the collaboration between
the Delft University and University of Zaragoza.
There is no Conflict of Interest
References
[1] F.J.Vermolen. et al., J. Tissue Viability. 19(2):43-
53, 2010
[2] F.J.Vermolen. et al., Biomechanics and modeling in
mechanobiology, to appear, 2011
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