Biomedical Engineering Reference
In-Depth Information
Sufficient conditions (
15
) of the Main Stability Theorem
4.2
imply large ranges
for the four
treatment parameters c
, which are directly related
to the four treatment strategies in Fig.
1
. The values of initial conditions have
been varied between 1 and 10
5
(
t
)
,
s
1
(
t
)
,
a
(
t
)
,
r
2
(
t
)
for the populations of effector and tumor cells,
E
(
0
)
and
T
(
0
)
. Global stability conditions (
15
) hold for any initial conditions
10
3
for our baseline run). Sufficient conditions (
15
)
are tested numerically by solving system (
5
) in Matlab using
ode15s
(a solver
for stiff systems). Since conditions (
15
) are sufficient, we combine techniques
from uncertainty and sensitivity analysis (see [
33
] for a review) to efficiently and
comprehensively investigate treatment combinations and how they might affect
cancer progression.
Regions of the parameter space where cancer is cleared are searched by sampling
the parameter space in the ranges defined in Table
1
. We only vary the four
treatment
parameters
, while all others are kept constant at their baseline values (see
Baseline
column in Table
1
). Samples are generated from uniform distributions and the
sampling scheme used is known as Latin hypercube sampling (LHS) [
37
]. LHS
scheme comprises three main steps: (1) definition of probability density functions
to use as a priori distributions for the parameters under analysis, (2) number N of
samples to perform and (3) independent sampling of each parameter. The last step
assumes that each parameter distribution is divided into N subintervals of equal
probability and that the sampling is preformed without replacement. The accuracy of
LHS is comparable to simple random sampling schemes but more efficient (i.e., with
a significant reduction in the number of samples needed). In our study we use a
sample size of 10,000 and tested numerically the impact of combining only constant
treatment strategies, although conditions (
15
) are also valid for time-varying inputs.
(we use
E
(
0
)=
C
(
0
)=
5.1
Sensitivity Analysis as a Way to Determine
Optimal Parameters for Treatment
In conjunction with uncertainty analysis, we use a generalized correlation coefficient
(partial rank correlation coefficient, PRCC) to guide our understanding of which
treatment parameter(s) contribute most to drive cancer proliferation or clearance
(our model outcomes). PRCC is one of the most popular sensitivity indexes used
for the analysis of deterministic models [
33
]. PRCCs results can be interpreted as
a degree of correlation between input and output variability: PRCCs vary between
−
1 and 1 and can be applied to any nonlinear monotonic relationship. A test of
significance is also available: only PRCCs that are significantly different from zero
are shown in this study. In order to select an optimal combination of treatments, a
pairwise comparison between PRCCs has been performed by a generalized z-test
(see p. 183 in [
33
]) and a ranking of the treatments is generated. Uncertainty and
sensitivity analysis results are shown in Table
2
. We review these techniques and
Search WWH ::
Custom Search