Biomedical Engineering Reference
In-Depth Information
Tabl e 2 Uncertainty and
sensitivity analysis results
Name
Definition
PRCC
Ranking
a
(
t
)
Cancer clearance term
0
.
1993
1st
s 1
(
t
)
Immunotherapy term
0
.
1061
2nd
c
(
t
)
Cancer antigenicity
0
.
0814
3rd
r 2
0791 3rd
PRCC values significantly different from 0 (with p
(
t
)
Cancer growth rate
0
.
<
0
.
01) PRCCs ranking based on generalized z-test ( p
<
0
.
05)
others in [ 33 ]; our Matlab scripts to perform LHS, PRCC as well as other uncertainty
and sensitivity analysis techniques are available online at http://malthus.micro.med.
umich.edu/lab/usanalysis.html .
Tab le 2 shows how all four parameters have PRCCs that are statistically different
from zero (with p
01): not surprisingly they are all negatively correlated
with cancer cell count (i.e., increasing their values from the baseline, decreases
cancer cell count). Two treatment parameters, a
<
0
.
(cancer clearance and
immunotherapy terms, respectively), have the highest impact on reducing cancer
cells. The other two parameters (i.e., c
(
t
)
and s 1 (
t
)
) have similar PRCCs (they share
the same ranking since they are not statistically different from each other), so they
are equally effective in reducing cancer cell count. Figure 4 shows scatter plots
of parameters versus cancer-cell counts, resulting from our extensive uncertainty
analysis with an LHS scheme of 10,000 samples. We classify the outputs in four
groups: complete clearance (green dots, no cancer cells), partial clearance (blue
dots, cancer cell count below the initial condition T
(
t
)
and r 2 (
t
)
10 3 ), small growth (red
dots, cancer cell count between 10 3 and 10 6 ), and large growth (black dots, cancer
cell count above 10 6 ). Cases where conditions ( 15 ) are satisfied are included in the
“green” region of Fig. 4 . There is clearly a synergy between immunotherapy and
cancer clearance terms ( s 1 (
(
0
)=
t
)
and a
(
t
)
): both must be large to achieve complete
clearance (green). High values for s 1 (
are always associated with lower
cancer cell counts, but no correlation can be inferred between either of these two
parameters and antigenicity ( c
t
)
or a
(
t
)
). Below we show two
examples of numerical simulations leading to complete clearance, when conditions
( 15 ) are either satisfied or not (Fig. 5 ). Clearance is usually achieved fast when
conditions ( 15 ) are satisfied.
(
t
)
) or cancer growth rate ( r 2 (
t
)
6
Conclusion and Discussion
Using mathematical models to explore important problems in biology is an ever-
increasing tool towards shedding light on these complex systems. Cancer modeling
has had a recent and successful history of making predictions that can assist in
hypothesis generation leading to experimental and perhaps clinical verification.
For example, gene therapy is a relatively young idea in treatment of diseases,
 
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