Biomedical Engineering Reference
In-Depth Information
Fig. 3
The quadratic
function
K
K
(
E
)
K(E)
K(h(t))
β
K(0)
A
0
B
E
h(t)
with
)=
ε
+
θ
r
2
(
t
)
h
(
t
.
(20)
a
(
t
)
Checking the discriminant of quadratic polynomial
K
(
E
)
one proves that
K
(
E
)=
0
has always two real roots
A
<
0,
B
>
0 of opposite signs such that ˜
γ
>
0. Since the
leading coefficient ˜
α
<
0 the quadratic polynomial
K
(
E
)
is positive in the interval
(
A
,
B
)
and negative outside. The first inequality of Eq. (
15
), in the case
T
>
T
0
,is
equivalent to condition
K
(
h
(
t
))
>
β
⇒
h
(
t
)
∈
[
0
,
B
)
(see Fig.
3
).
At the same time
K
(
0
)=
γ
=
˜
fc
(
t
)
T
(
t
)+
s
1
(
t
)
f
>
σ
. This implies that one will
have
E
(
t
)
>
β
0
/
(
E
+
f
)
with
β
0
=
min
(
σ
,
β
)
(21)
once the inequality (
19
) is violated, forcing
E
(
t
)
to increase. Since
h
(
t
)
is non-
t
∗
and will
increasing function, the inequality (
19
) will be satisfied for certain
t
=
t
∗
. As follows from the proof of Theorem
4.1
and Eq. (
13
), for
hold then for all
t
≥
t
∗
the function
T
t
≥
(
t
)
will decrease till it takes the value
T
=
T
0
.If
T
0
=
0(Case
II) then one uses Theorem
4.1
again to derive Eq. (
16
).
One can interpret the significance of the inequality
fc
(
t
)
T
0
+
s
1
(
t
)
>
σ
>
0in
Eq. (
15
) as follows. Once antigenicity is switched off, i.e.
c
(
t
)=
0, treatment,
s
1
(
t
)
,
should be nonzero and vice versa. In the case of complete tumor clearance (
T
0
=
0),
the treatment term should be always positive above a certain level. Indeed, partial
clearance does not exclude future regressing via the “escape” effect.
5
Numerical Simulations
Because the analytical results hold only for very small regions of parameter
space,we would like to explore the gene therapy model more fully. To this end,
we will apply statistical sampling techniques and numerical analysis to the system.
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