Biomedical Engineering Reference
In-Depth Information
Fig. 21.2
Different
in vitro
shapes of LS174T tumors. In all cases, the initial tumor seed was
spheroidal
dead cells as a separate concentration field. Also, keeping in view recent sugges-
tions in the literature (Valastyan and Weinberg,
2011
), a secondary cell population
of more motile and proliferating cancer stem cells in considered in certain simula-
tions. Now the reaction-transport equations and various source and sink terms are
presented.
21.2.1 Species Transport
The evolution of these biological and chemical species is modeled using the follow-
ing set of reaction-transport partial differential equations. The primary transport of
these species is due to diffusion. However, to account for active transport of cells
in response to mechanical/chemical signaling, advection terms have also been in-
cluded. The concentrations of ECM and dead cells have no associated transport,
π
c
ρ
c
,τ
c
−
π
c
n
ρ
c
,σ,ρ
o
,ρ
g
−∇·
−
ρ
c
−
ρ
c
D
c
V
c
(σ)
ρ
c
,
(21.1)
˙
=
∇
∇
π
s
ρ
s
,τ
s
−
π
s
n
ρ
s
,σ,ρ
o
,ρ
g
−∇·
−
ρ
s
−
ρ
s
D
s
V
s
(σ)
ρ
s
,
(21.2)
˙
=
∇
∇
π
e
ρ
c
,ρ
s
,
ρ
e
˙
=
(21.3)
π
c
n
ρ
c
,σ,ρ
o
,ρ
g
+
π
s
n
ρ
s
,σ,ρ
o
,ρ
g
,
ρ
n
˙
=
(21.4)
π
o
ρ
c
,ρ
s
,ρ
o
,ρ
g
−∇·
−
ρ
o
,
ρ
o
D
o
˙
=−
∇
(21.5)
π
g
ρ
c
,ρ
s
,ρ
o
,ρ
g
−∇·
−
ρ
g
.
ρ
g
D
g
˙
=−
∇
(21.6)
Here, the species are identified by the superscripts:
c
-cancer cells (primary cell pop-
ulation),
s
-cancerous stem cells (secondary cell population),
e
-ECM,
n
-dead cells,
o
-oxygen and
g
-glucose. The subpopulations of dead cells are separately identified
as
c
n
-dead cancer cells and
s
n
-dead stem cells.
π
x
are the species source terms,
D
x
and
V
x
the diffusivity and advective velocity, respectively. In addition,
τ
x
rep-
resents the cell doubling time which controls the proliferation rate and
σ
is the
Cauchy stress induced due to growth and external interactions. As can be seen from
the equations, the source terms control the growth dynamics via cell proliferation
and nutrient consumption, hence a detailed discussion of these terms follows.
