Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 317

(myo-)fibroblasts, as well as wound closure by the keratinocytes that form the
basis of the epidermis (epithelium).
Many in-vitro experimental and clinical in-vivo studies have been carried out to
scrutinize the biological mechanisms that take place during the very complex
process of wound healing. Unfortunately, still many of the underlying biology is
still unclear despite the long lasting research in wound healing. In order to improve
and to prevent wounds, such as pressure ulcers or diabetic ulcers, it is important to
quantify the influence of the related partial processes taking place during the
healing of wounds. This quantification can be done using statistical analyses on
raw data using for instance genetic algorithms or other forms of artificial intelli-
gence such as neural networks. Since much data lack detailed quantitative aspects,
this holds for in-vivo data in particular, mathematical modeling is also a very
helpful tool for the quest of the interrelations between the parameters involved.
The challenge is either to build a complicated mathematical model that contains as
many of the biological parameters as possible, or to construct simple models that
contain a minimum number of parameters such that only those parameters and
processes that have the largest impact on the healing kinetics are taken into
account. The first class of models will involve many biological parameters that
need to be determined using complicated inverse modeling or any other type of
regression analysis, in which the valid question arises whether the set of param-
eters determined is the actual solution or that one should take another combination
of the parameters involved which reproduces the experiments (almost) equally
well. In other words, the question of uniqueness arises in a natural setting. This
concern is overcome by the construction of a simplified formalism of a certain
(partial) biological process occurring in wound healing. In this paper, we will
highlight the latter class of mathematical models: simplified models for partial
processes occurring during wound healing. We will look at models designed for
various scales and attempt to describe the relations between these models in terms
of the underlying biology and mathematics.
Since wound healing involves basic biological processes like cell migration as a
result of chemico-mechanical stimuli and random walk, cell proliferation and
growth, cell differentiation, cell death, secretion and signaling of growth factors,
we will incorporate many of these processes in a different way into the models at
the various scales considered. To apply these processes, one basically considers
the following mathematical approaches:
• Continuum-based partial differential equations involving transport (random
walk, chemo-tenso taxis) and mechanical balances (visco-elasticity) on a tissue
scale;
• Cellular scale involving discrete lattice models like the cellular Potts model,
cellular automata models (involving a minimization of a virtual energy with a
Monte-Carlo like scheme), or the continuous semi-stochastic approach by
Vermolen and Gefen [ 1 ] and Byrne and Drasdo [ 2 ];