Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 333

considered in the model. Next to these biological quantities, the local stress-strain
pattern as a result of the contractile forces exerted by fibroblasts are dealt with by
the solution of visco-elastic equations (Maxwell-model). This class of models is
useful to model processes like angiogenesis, wound contraction and wound closure
and this class has also be extended to model processes like tumor growth. An
advantage of this model class is its applicability to in-vivo cases. Unfortunately,
this class of models that contains a system of complicated nonlinear PDEs suffers
from the incorporation of huge number of parameters which often are hard to
measure. In this manuscript, we will not present a detailed model for wound
healing, however, we will give a very simplified flavor of this model class by the
consideration of a single partial differential equation that can be used to model
wound closure in an in-vitro setting. To this extent, we consider the relatively
simple Fisher-Kolmogorov equation, which reads as

o u

ot DDu ¼ ku ð 1 u

Þ; t [ 0 ; ð x ; y ; z Þ2 X ; ð 40 Þ

u 0

subject to some initial condition, that reads as

u ð 0 ; ð x ; y ; z ÞÞ ¼ u 0 ð x ; y ; z Þ¼ u 0 ;

ð x ; y ; z Þ 62 X w ;

ð 41 Þ

0 ; ð x ; y ; z Þ2 X n X w :

Here u denotes the cell density, u 0 denotes the undamaged equilibrium cell density
and X w denotes the area of the initial wound. Furthermore, in the PDE cell motion
is determined through random walk only and the right-hand side models growth of
the cell population towards an equilibrium. A snapshot at 12 : 5 h of the cell density
(number of cells per unit volume or area, normalized to unity) is shown in Fig. 7 .
We used an initial circular 'wound' of radius 1 mm, D ¼ 10 4 mm 2 = h ; and
k ¼ 0 : 7 h 1 . If one also encounters chemotaxis, then the cells move according to
the concentration gradient of a generic chemical. Each cell moves according to the
aforementioned concentration gradient, hence in the case of u cells per unit area or
volume, the amount and direction of cell movement are determined by the con-
centration gradient of the chemical multiplied by the cell density u, which gives

o u

ot DDu þrð ub r c Þ¼ ku ð 1 u

Þ; t [ 0 ; ð x ; y ; z Þ2 X ; ð 42 Þ

u 0

where b and c, respectively, denote the sensitivity and motility of the cells as a
result of chemotaxis, and the concentration of the chemical that gives rise to
chemotaxis. Here also a profile of the chemoattractant needs to be determined,
which already increases the parameter space considerably. Note that the above
equations corresponds to a drift term that is given by b r c in the stochastic
counterpart. This equation models mobility of cells towards the concentration
gradient of c, whereas reversing the sign would model mobility of cells away from
the concentration gradient. In the case of a bounded domain of computation, then