Biomedical Engineering Reference
In-Depth Information
Table 1
. Fixed parameters in the model along with their corresponding
denotation and specific values
Parameter Equation Value
The positive impact of nutrients,
q
G
2
10
The negative impact of toxicity,
q
U
2
-10
The negative impact of resistance,
q
p
2
-10
The negative effect of overcrowding,
c
3
0.1
Inverse sensitivity of cell death to toxicity level,
k
U
5
10
The rate of nutrient production,
g
G
6
0.1
The rate of nutrient depletion,
r
G
6
0.01
Diffusion coefficient of nutrients,
D
G
6
0.001
The rate of mechanical resistance reduction,
r
p
7
0.001
The rate of toxicity accumulation,
r
U
8
0.02
Diffusion coefficient of toxic metabolites,
D
U
8
0.001
r
G
. As discussed in ยง4.1, the parameter
r
S
captures the extent of local search rela-
tive to global search. In particular, as
r
S
, global search is completely elimi-
nated, leaving only local search as the sole probing mechanism. At the other
extreme, as
r
S
0, global search becomes the dominant
cell receptor mechanism
since tumor cells can invade any location with no spatial constraint. Therefore,
with lower values of
r
S
we expect to see acceleration of the tumor's average ve-
locity since it takes less time for the first tumor cell to reach the second nutrient
source. Interestingly, we found that the performance of the tumor system exhib-
its a
phase transition
at a critical value
r
S
*. At
r
S
>
r
S
*, where local search is the
dominant probing mechanism, raising
r
S
results in slower average velocity yet
longer lifetime of the tumor system. In contrast, at lower values
r
S
<
r
S
*, where
global search dominates, incrementing
r
S
by a small amount (and thus introduc-
ing a modicum of local search) actually increases the average velocity (i.e.,
shortens the time it takes for the first tumor cell to invade the peak of the second
nutrient source) yet decreases the lifetime of the tumor system. The term "phase
transition" here thus corresponds to the nonlinear behavior of the tumor system:
starting at low values of
r
S
, increasing that value will accelerate the tumor's spa-
tial expansion, yet there is a threshold level
r
S
* beyond which the velocity starts
to decline as
r
S
increases further toward maximum local search. The choice of
r
S
also reflects the potential tradeoff between the velocity of spatial expansion (i.e.,
a clinically relevant measure for the aggression of the tumor system) and the
