Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 319

2 Mathematical Models: From Cell Scale to Tissue Scale

In this section, we consider some mathematical models at various scales where we
introduce the models first to make the present manuscript complete. In the next
chapter, we will describe the link between modeling at various scales in terms of
the underlying biology and mathematics. In this chapter, we will mainly focus on
cell migration, proliferation and death.

2.1 The Cell Scale

In this class of model, we consider the deformation during migration of individual
cells. The cells are assumed to migrate as a result of a chemical signal. We bear in
mind that the mathematical description of the influencing signal is generic and can
easily be adapted and used to model cell deformation and growth as a result of a
mechanical signal.

2.1.1 Random Walk: From Bacteria or Cells to Probability

The model can be applied to bacterial sources where individual bacteria make the
surrounding tissue more acid by the effective production of biotic lactates as a
result of the competition between the bacteria and cells for the available nutrients
and oxygen, which make white blood cells move towards the infectious bacteria,
or it can be applied, for instance, to the migration of fibroblasts or keratinocytes,
among others, towards the wound region due to the signaling agents released by
platelets that are in the coagulated area of the wound. In the case of modeling
individual randomly moving bacteria, we use a random walk model with a sto-
chastic differential equation based on Ito-processes. The model may also incor-
porate the bacteria in an upscaled way so that only bacterial densities are
considered. First, we consider the individual random walk of bacteria. Then, in
three dimensions, the equation of motion does not contain any deterministic drift,
hence for the motion of the bacterium, we obtain

dX ð t Þ¼ rdW ð t Þ; dY ð t Þ¼ rdW ð t Þ; dZ ð t Þ¼ rdW ð t Þ; for t [ 0 ;

ð 1 Þ

subject to the prescribed initial bacterial condition ð X ð 0 Þ; Y ð 0 Þ; Z ð 0 ÞÞ ¼
ð X 0 ; Y 0 ; Z 0 Þ; where the co-ordinate positions are independent. Here W ð t Þ is a
Wiener process, or Brownian Motion such that the position of the bacterium is
distributed normally with mean coordinates ð X 0 ; Y 0 ; Z 0 Þ and variance of r 2 t for
each coordinate direction. Formally, the Wiener process satisfies the following
requirements: