Biomedical Engineering Reference
In-Depth Information
π
g
ρ
c
=−
ρ
c
1
.
14
1
(ρ
H
+
)
1
.
2
.
(21.11)
10
−
17
ρ
o
ρ
g
3
.
65
·
10
−
10
·
+
ρ
g
+
7
.
21
·
10
−
3
The dependence of
π
o
on
ρ
o
, and
π
g
on
ρ
g
is 'Michaelis-Menten-like', giving rates
that vary monotonically from zero to a maximum asymptotic value as the respective
concentrations increase from
ρ
o
,
ρ
g
0.
Finally, the ECM production is assumed to be linearly proportional to the total
living cell population and is given by
π
e
ρ
c
=
=
10
−
8
ρ
c
.
8
.
27
·
(21.12)
21.2.3 Influence of Mechanics: Growth, Cell Death and Enhanced
Motility
Tumors start as single cells or multi cell colonies and grow over time resulting in
deformation of the tumor mass and the surrounding matrix due to growth. Bio-
logical growth is modeled by introducing a multiplicative split in the deformation
gradient
F
F
e
F
g
, where
F
e
and
F
g
are the deformation gradients induced due
to elastic strain and growth, respectively (Garikipati et al.,
2004
). Assuming only
dilatational growth and resorption,
F
g
is an isotropic tensor. It represents the kine-
matics of growth caused by cell proliferation, and is given by
=
=
G
ρ
c
I
,
F
g
(21.13)
where
I
is the second-order unit tensor, and
G
is any nonlinear function characteriz-
ing the swelling caused due to growth as a function of cell population. In this work,
the following dependence is assumed:
1
,
G
ρ
c
=
ρ
c
≤
ρ
h
.
(21.14)
ρ
c
1
ρ
h
), ρ
c
>ρ
h
.
2
(
1
+
Here,
ρ
h
is the critical cell population density required to occupy all available free
volume without causing any mechanical stress. This growth causes mechanical in-
teractions with the surrounding growth matrix and results in significant compressive
stresses, which, when beyond a certain threshold, can inhibit cell proliferation. This
stress inhibited proliferation is modeled by Eq. (
21.9
), where the second term con-
taining the trace of the Cauchy stress is only considered when it is compressive and
above a threshold value. Further, as a possible explanation of symmetry-breaking
in tumor shapes leading to ellipsoidal tumors, it is assumed that cancer cells, being
highly motile, can migrate away from regions of high compressive stresses within
the tumor mass. This is modeled by the advection terms in Eqs. (
21.1
) and (
21.2
),
and given by
V
c
(σ)
=
∇
M
σ
ii
,
(21.15)
