Biomedical Engineering Reference
In-Depth Information
21.2.2 Species Sources and Chemomechanical Coupling
The source terms are expressed as mass rate per unit volume with units of fg
/
µm
3
s
(fg
femtogram). Many of the empirical relations listed here are from Casciari
et al. (
1992
) or their modified relations from Narayanan et al. (
2010
). For brevity of
the expressions, only the primary cancer cell population is considered in the source
terms listed here. For the cell populations, the source terms model the initial stages
of the tumor growth, characterized by exponential growth, i.e.
=
ρ
0
J
t
D
π
c
ρ
c
,τ
c
=
1
τ
inv
e
tτ
inv
,
τ
inv
=
0
.
693
/τ
c
,
τ
c
=
log 2
.
(21.7)
Here,
ρ
0
is the initial cell concentration and
t
D
is the doubling time of the cells,
while
J
is the determinant of the deformation gradient, which is driven by volumet-
ric growth of the tumor, in addition to its elastic deformation. The cell doubling time
is dependent on the oxygen and glucose concentrations (Casciari et al.,
1992
), and
the pH (
ρ
H
+
) of the medium (Casciari et al.,
1992
; Bourrat-Floeck et al.,
1991
). We
have used the equation proposed by Casciari et al. (
1992
) for the doubling times:
ρ
g
ρ
o
ρ
H
+
0
.
46
.
t
c
opt
0
.
014
10
−
2
10
3
+
1
.
8
·
+
7
.
3
·
t
D
=
(21.8)
ρ
g
ρ
o
This exponential growth is balanced by the sink
π
c
n
, representing the concentration
of dead cells produced primarily due to aging, depletion of nutrients or environmen-
tal factors like surrounding pressure. These terms are modeled as follows:
π
c
n
ρ
c
,σ,ρ
o
,ρ
g
=
ρ
c
κ
1
.
0
]
(ρ
0
/ρ
g
)
2
e
2
.
0
−[
(ρ
0
/ρ
o
)
2
+
−
+
ρ
c
κ
1
.
0
−
e
−
(σ
ii
/P
c
)
2
.
(21.9)
Here, the first term on the right hand side denotes cell death due to aging and the
second term represents stress-driven cell death. Although it is known that cells have
different tolerances to different stress states, here, stress-driven cell death is assumed
to only depend on the hydrostatic component of the stress, represented by the trace
σ
ii
of the Cauchy stress,
κ
is the cell death rate constant and
P
c
is the threshold
value of pressure at which stress-driven cell death is significant. Equation (
21.9
)
represents a possible chemomechanical coupling which accounts for the effect of
mechanics on cell proliferation.
The oxygen and glucose consumption rates were also adapted from Casciari et al.
(
1992
) to be consistent with the different units used here, and scaled by the local
cell concentration to be expressed as mass rates per unit volume. The resulting rate
functions take on field values that vary over time and space, and have the forms
ρ
c
7
.
68
,
10
−
15
ρ
g
(ρ
H
+
)
0
.
92
ρ
o
π
o
ρ
c
=−
3
.
84
·
10
−
7
·
+
(21.10)
ρ
o
10
−
4
+
1
.
47
·
