Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 325

in which we assume that ð X ð t Þ; Y ð t ÞÞ represents a certain trajectory. Initially, the
concentration is assumed to be zero, and then the following solution for the 2-
dimensional case, as a Green's Function, is derived according to the principles
outlined in Evans [ 27 ]:

ds ;

c ð t ; ð x ; y ÞÞ ¼ R

t

4pD ð t s Þ exp jj x X ð s Þjj 2

c ð s Þ

ð 21 Þ

4D ð t s Þ

0

for the two-dimensional case and

ds ;

c ð t ; ð x ; y ; z ÞÞ ¼ R

t

ð 4pD ð t s ÞÞ 3 = 2 exp jj x X ð s Þjj 2

c ð s Þ

ð 22 Þ

4D ð t s Þ

0

for the 3-D case, see also Eq. ( 5 ). Using this Green's Function, any solution with
sources having a compact support, but non-zero measure can be constructed, or
any initial condition, by the application of superposition arguments that result into
a convolution. For completeness, we give the result for n discrete point sources at
the points ð X j ð t Þ; Y j ð t ÞÞ and strength c j ð t Þ for j 2f 1 ; ... ; n g :

ds ;

c ð t ; ð x ; y ÞÞ ¼ P

R

n t

4pD ð t s Þ exp jj x X j ð s Þjj 2

c j ð s Þ

ð 23 Þ

4D ð t s Þ

j ¼ 1

0

2 ; we get the following

as well as for a 'continuous' source that lives in X R

solution by the use of convolution

dXds :

c ð t ; ð x ; y ÞÞ ¼ R

R

t

4pD ð t s Þ exp jj x x jj 2

Q ð s ;ð x ; y ÞÞ

ð 24 Þ

4D ð t s Þ

0 X

3 is fully analogous.
Next we consider the dynamics of the points on the cell boundary. Inertia is
neglected in the present formalism. The computational domain may be given by a
flat two-dimensional substrate, where we consider projections of cells or by a
three-dimensional domain where cells move through extracellular matrix or a gel-
like medium. We divide the circumference of the cell into N points. On each point,
the cell detects a chemical signal and each point moves according to the con-
centration gradient that is constructed by a (sequence of) fundamental solutions.
Further the direction of motion, as well as the velocity of the points are determined
by the degree of deformation. To this extent, we use the following phenomeno-
logical law for the motion of the gridpoints on the cell boundary

The treatment in R

v i ¼ b r c ð t ; x i Þþ a x c ð t Þþ x i x i ð t Þ ð Þ; for i 2f 1 ; ; N g; ð 25 Þ

where b denotes a mobility parameter of the cell boundary. This parameter is a
measure for the deformation rate of the cell and also represents a measure of the
sensitivity of the cell boundary to the concentration gradient. This b-term models
chemotaxis. Further, the a-term models the 'desire' of the cell to attain its