Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 329

the substrate E s ; the strain energy density as a result of cell i pulling on the
substrate is computed by

F i

M i ¼ 1

2 r ¼ 1 2 E s ð r i Þ 2 ¼

2E s ð r i Þ p 2 R 4 ;

ð 28 Þ

where the last step can be evaluated using some standard Hookean relations from
mechanics, see [ 1 ]. In [ 1 ], we show by the use of finite-element simulations, that
the strain energy density away from the cell can be approximated by

jj r r i jj
R

M i ð r Þ¼ M i

ð 29 Þ

exp f k i g; for r 2 X ; i 2f 1 ; ... ; n g;

where r i represents the location of cell i, projected on X ; further k i is a measure of
how much the signal is attenuated, where we have

k i ¼ E s ð r i Þ
E i

: ð 30 Þ

Here E i represents the Young's Modulus of the cell. Using the additivity-property
of the strain energy density, the strain energy density for a superposition of cells is
given by

M ð r Þ¼ P M j ð r Þ¼ P

n n

jj r r j jj
R

M j exp f k j g:

ð 31 Þ

j ¼ 1 j ¼ 1

Let r i ð t Þ denote the position of the cell center i at time t, then using this expression,
the mechanical stimulus sensed by the ith cell is computed via

M ð r i Þ¼ X

n

M j ð r i Þ¼ X

n

jj r i r j jj
R

M j exp f k j g

j ¼ 1 j ¼ 1

ð 32 Þ

¼ M i þ X

n

jj r i r j jj
R

M j exp f k j g; for all i 2f 1 ; ... ; n g:

j ¼ 1 j 6¼ i

Note that the cell's own contribution is also incorporated in this formula. Next, we
repeat some of the formulas in Vermolen and Gefen [ 1 ] for the determination of
the direction in which the cell is moving. The direction is determined by all the
vectors connecting the other cells felt by the considered cell. The weight factors
are given by the strength of the signal, in this case the strain energy density,
experienced by the cell. This implies the following (deterministic) direction:

z i ¼ X

n

M j ð r i ð t ÞÞ r j r i

jj r j r i jj ; for all i 2f 1 ; ... ; n g;

ð 33 Þ

j ¼ 1 j 6¼ i

where a contributing term is mapped onto zero if jj r i r j jj ¼ 0 : The unit vector
follows from the normalization: