Biomedical Engineering Reference
In-Depth Information
We postulate that such
energy expenditures
are determined by both an intrinsic
factor and an external effect:
S =
r
S
p
j
,
[4]
where the intrinsic factor
r
S
corresponds to the inverse capability of tumor cells
for spatial movement, while the external effect is attributable to mechanical con-
finements
p
j
. Note that higher values of
r
S
imply greater costs of spatial move-
ment in terms of cells' energy expenditures; it is thus an inverse capability term.
3.2.4. Cell Death
As shown in Figure 1, cancer cells that experience dwindling supplies
of nutrients or escalating levels of toxic metabolites can either turn quiescent,
a reversible state, or undergo cell apoptosis/death. In our model, cell death is
a stochastic event such that if either of two conditions occur—(i) nutrient re-
serves fall below the lower threshold G
j
< G
L
, or (ii) levels of toxic metabolites
rise above the upper threshold U
j
> U
U
, then for tumor cells residing in such loca-
tions, the likelihood of death becomes positive and is proportional to the toxic-
ity levels:
Pr
death,
j
= U
j
/(
k
U
+ U
j
),
[5]
where
k
U
is the parameter representing the inverse sensitivity of cell death to
toxicity level. Note that higher values of
k
U
correspond to lower probability of
death; hence the "inverse" term. Other than quiescence, cell death is a nonre-
versible event. In fact, cell death is particularly imminent for the non-
proliferating and non-migrating quiescent cells trapped inside a cluster due to
their inability to escape to more favorable locations. Eventually, these dead cells
start to form an emerging central necrotic region, a hallmark of highly malignant
brain tumors (25).
3.2.5. Nutrient Sources
The supplies of nutrients in our model can be either
replenished
(e.g.,
through neighboring blood vessels) or
non-replenished
(representing scattering
traces of nutrient in the intercellular space). In every period
t
, the change in the
levels of nutrients in every location is due to the current (new) production of
nutrients,
g
G
G, diffusion from the surrounding lattice sites, /#(
D
G
/G), and nutri-
ent depletion,
r
G
I:
s
G
=+
g
G
¸
(
D
G I
)
r
.
[6]
G
G
G
s
t
where the parameter
g
G
represents the fixed rate of nutrient production,
D
G
is the
diffusion coefficient of nutrients, and
r
G
controls how fast a tumor cell metabo-
