Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 330

z i
jj z i jj :

z i ¼ ð 34 Þ

The velocity vector is constructed by the multiplication of this direction vector by
the signal strength that is sensed by the considered cell to obtain:

r i ð t Þ¼ a i M ð r i Þ z i :

ð 35 Þ

Here the cell velocity is modeled as instantaneous. In [ 18 ], we present a modifi-
cation to incorporate the inertia effects into this formulation. The above equation
can also be enriched with a stochastic contribution as is done in Vermolen and
Gefen [ 1 ] using a uniform distribution or using Brownian Motion as in Eq. ( 9 ).
Further, a i is a velocity parameter with a dimension

hi ¼ hi ; determined by

m 2 s
kg

m 3
Ns

2
b i

R 3
f

F i
F

a i ¼ : ð 36 Þ

Note that F is a property of the specific phenotype of the cell. The coefficient b i ;
with unit s 1 ; accounts for the mobility of the viable cell over the substrate surface.
In [ 18 ], we incorporate the concentration of an infectious agent that typically
results from bacteria. The cell-substrate friction effectively represents the averaged
contribution of focal adhesions along the entire base of the cell without considering
each localized connections of integrins. In [ 1 ], it is shown that a i is determined by

a i ¼ b i R 3
lF 2

F i ; ð 37 Þ

where l ( ¼ 0 : 2 following [ 28 ]) denotes the cell friction coefficient. In [ 18 ], inertia
is taken into account. Since this effect is known to be small, this effect is omitted in
the present manuscript.
The cells will push each other away if they are too close together. This will give
rise to repelling contact forces once these cells impinge elastically. The contact
forces are due to the linear deformation of the cell bodies. In the present manu-
script, the principles outlined in Gefen [ 28 ] are used. In [ 1 ], the derivation of the
invagination force based on the work in Gefen [ 28 ] is given. In this manuscript, the
final result for the strain energy density is given, which reads as

p

R E h 2

pR 3

M ij ¼ 6
15

; ð 38 Þ

where h ¼ max ð 2R jj r ij jj; 0 Þ is the indentation of cell i into cell j and vice versa.
The final result for the total strain energy density function becomes

M i ð r Þ¼ M i ð r Þ M ij ;

ð 39 Þ