Biomedical Engineering Reference
In-Depth Information
lizes the on-site nutrient sources. For replenished sources of nutrients we set
g
G
> 0, while for non-replenished sources
g
G
= 0. Nonetheless, this setup allows for
a feedback between the two distinct types of nutrient sources: the initially non-
replenished intercellular nutrients may get recharged through diffusion from the
replenished source.
3.2.6. Mechanical Confinements
As a first approximation, in (6), we assumed a static distribution of me-
chanical confinements. It is a "static" distribution in the sense that the levels of
tissue resistance remain constant over time regardless of cell behavior. Subse-
quently, in (7), we adopted the more realistic notion of an
adaptive grid lattice
such that locations that have been traversed by migrating tumor cells experience
a reduction in mechanical confinements:
s
p
=
r
I
.
[7]
p
s
t
Equation [7] specifies that mechanical confinements "decay" at the rate of
r
p
per
viabl
e
cell regardless of its phenotype, so that for a given c
o
nstant cell popula-
tion
prI
¸ time steps. Over
the course of a simulation, however, it is rarely the case that the cell population
remains constant, and thus the time when mechanical confinements disappears
at a particular lattice site is a stochastic variable. Biologically, the fall in me-
chanical confinements represents the degeneration of extracellular matrix due to
cell invasion and secretion of proteases, i.e., matrix-degrading enzymes (41).
According to Eq. [7], the presence of any viable tumor cell (regardless of pheno-
type) reduces tissue consistency. However, we argue that only invasive cells are
capable of taking advantage of the deformed grid lattice by following the path of
declining resistance. Assuming an underlying tendency of invasive cells to limit
their energy expenditure as captured by Eq. [4], this process would in turn en-
courage even more tumor cells to invade the host tissue further following the
paths that traversed by their peers.
I
, mechanical confinements go to zero after
,
/(
)
tj
p
j
3.2.7. Toxic Metabolites
In our model, the levels of toxic metabolites (which here can represent a
combination of specific inhibitory soluble factors released by tumor cells, ly-
sosomal content from dying tumor cells or tissue hypoxia) evolve according to
the following function:
s
U
=
¸
(
D
U I
)
+
r
.
[8]
U
U
s
t
