Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 322

It is easy to check the validity of the exact solution using Ito's calculus. The drift
term could possibly result from chemotaxis or fluid flow and induces a temporarily
shifting mean in the probability density, hence Eq. ( 5 ) is altered into

!

ð 2pr 2 t Þ 2 exp ð x lt Þ 2

1

f ð x ; y ; z Þ¼ : ð 11 Þ

2r 2 t

It can be shown by the use of some elementary algebra that this function solves the
Fokker-Planck equation

ot þrð lf Þ r 2

of

2 Df ¼ 0 ;

f ð 0 ; ð x ; y ; z ÞÞ ¼ d ð x X 0 Þ: ð 12 Þ

The above concepts are very standard and were, for the case of unbiased
random walk, originally derived by Einstein to study Brownian motion of a par-
ticle. Note that we modeled the bacteria as point-sources so far. The extension to
multiple bacteria, say n ; is somewhat straightforward upon approximating the
bacterial motion of each bacterium as independent stochastic processes. The
probability follows from the binomial distribution that is used to compute the
probability of k successes out of n trials where the probability of success is given
by p. Since then the probability that a certain region, say X possesses k n
bacteria is determined through

ð P ð t ; X ÞÞ k ð 1 P ð t ; X ÞÞ n k :

p ð t ; X; k Þ¼ n
k

ð 13 Þ

Hence the probability that this region X contains at least one bacteria is given by

p ð t ; X; k 1 Þ¼ 1 ð 1 P ð t ; X ÞÞ n nP ð t ; X Þ;

ð 14 Þ

where the last approximation is only accurate for P ð t ; X Þ 1 : This approximation
enables us to approximate the probability density function for n particles by
nf ð t ; ð x ; y ; z ÞÞ at those positions away from the initial bacterial positions. Note also
that for t [ 0 the probability density function(s) becomes finite at each position
and that we can take the limit meas ð X Þ! 0 ; to get an arbitrarily small probability
as the volume considered tends to zero. Hence the approach can be extended to
solving f in the case of a multi-bacterial environment under the application of the
superposition principle for linear diffusion equations. These concepts can be used
to model the bacterial density using the same partial differential equations. One
can also evaluate a convolution over the domain of computation to get the bacterial
density in case of a (piecewise) continuous initial bacterial distribution.

2.1.2 A Cell Deformation Model

In the literature, many models for cell deformation exist [ 26 , 17 ], to mention a few
of them. As far as we know, one of the major issues is that most of these models