Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 324

agent that attracts cells, we assume all sources to be very small compared to cell
areas and therefore, we approximate these sources as point-sources. Here, we will
assume that a point source is able to move, for instance via Brownian motion, but
that the diffusion-field that surrounds it, sets in either instantaneously or gradually
builds up in time. These point sources can correspond to either bacteria or to points
on the cell boundary of other cells. First we consider the instantaneous diffusion
field. For this purpose, we consider a point-source in 2D that moves according to a
trajectory ð X ð t Þ; Y ð t ÞÞ moving under (biased) Brownian motion for instance [see
Eq. ( 1 )or( 9 )], for which we have for ð x ; y Þ2

2

R and t [ 0

DDc ¼ c ð t Þ d ð x X ð t Þ; y Y ð t ÞÞ: ð 15 Þ

Here c denotes the concentration of the chemical, which diffuses with a diffusion
coefficient D. Further, c denotes the strength of the point-source, which may
change in time as a result of being present or not being present, and d ð:Þ denotes
the Dirac Delta Function. The solution to this differential equation is given by

c ð t ; ð x ; y ÞÞ ¼ c ð t Þ

2pD ln ðð x X ð t ÞÞ 2 þð y Y ð t ÞÞ 2 Þ;

ð 16 Þ

2 ; which can be found in textbooks like for instance [ 27 ]. For the 3-D case, we
report that the Green's Function is given by

in R

1
4pD jj x X ð t Þjj ;

c ð t ; ð x ; y ; z ÞÞ ¼ ð 17 Þ

In the case of multiple, say n, sources, with intensities c j and positions
ð X j ð t Þ; Y j ð t ÞÞ; linearity of the diffusion equation allows us to use the superposition
principle, to obtain

c ð t ; ð x ; y ÞÞ ¼ X

n

c j ð t Þ

2pD ln ðjj x X j ð t Þjj 2 Þ:

ð 18 Þ

j ¼ 1

For a continuously distributed source-function Q ð t ; ð x ; y ÞÞ that is non-zero in X
R

2 ; we get the following convolution-based solution

2pD R
X

Q ð t ; ð x ; y ÞÞ ln ðjj x x jj 2 Þ dX ; c ð t ; ð x ; y ÞÞ ¼ 1

ð 19 Þ

where the above integral is evaluated over ð x ; y Þ: The 3D case can be treated
analogously.
For the case of a transient diffusion field, we proceed analogously to the steady-
state case with the application of delta-functions to deal with the point sources,
then we arrive at

o c 2 ; t [ 0 ;

ot DDc ¼ c ð t Þ d ð x X ð t Þ; y Y ð t ÞÞ;

for ð x ; y Þ2 R ð 20 Þ