Biomedical Engineering Reference
In-Depth Information
(
)
If
v
is the concentration of the agent in blood, we assume that its effectiveness is
modulated by an unimodal non-negative function
t
γ
(
ρ
)
γ
(
)=
with
0
0. This leads to
the new equation
pF
q
p
q
p
v
p
=
−
γ
(
t
)
p
,
(28)
while the equation for
q
is unchanged. The above change deeply impacts the
behavior of the dynamical system. It is not difficult to show that, under constant
continuous infusion therapy, the model allows that the tumor-vessels system is
multi-stable. This multi-stability may have both beneficial and detrimental “side
effects” [
45
,
46
]. On one hand, multi-stability allowed [
45
] to explain the pruning
effect as a change of attractor of a tumor under chemotherapy. For example, the tem-
porary delivery of anti-angiogenic therapy before an uninterrupted chemotherapy
may move an orbit in the
space from the basin of attraction of a locally stable
equilibrium with a large tumor size to the basin of attraction of another equilibrium
with a small tumor size. On the other hand, the actual drug concentration profiles
are affected by large bounded stochastic fluctuations, so that—as shown in [
46
]—
the tumor volume may undergo detrimental noise-induced transitions from small to
large equilibrium sizes. These stochastic transition phenomena might thus explain
some cases of resistance to chemotherapy as due to non-genetic mechanisms.
(
p
,
q
)
8
Optimal Protocols for Combined Anti-angiogenic
and Chemotherapies
In view of the high cost and limited availability of anti-angiogenic agents and
because of harmful side effects of cytotoxic drugs, it is not feasible to give
indefinite administrations of agents. The practically relevant question is how an
a priori given, limited amount of anti-angiogenic and chemotherapeutic agents is
best administered. Clearly, while anti-angiogenic agents are mostly limited because
of their cost, chemotherapeutic agents must be limited because of their toxic side
effects. For optimization problems, it is no longer possible to keep the model
general, but a specific choice needs to be made for the growth function
F
and all
the other functional terms that define the model. Here, for sake of definiteness, we
consider the following model for tumor anti-angiogenesis that is based on [
16
]:
[AC]
for a free terminal time
T
, minimize the objective
J
(
u
)=
p
(
T
)
subject to
the dynamics
p
ln
q
p
p
=
ξ
−
ϕ
pv
,
p
(
0
)=
p
0
,
(29)
dp
3
q
q
=
bp
−
−
μ
q
−
γ
qu
−
η
qv
,
q
(
0
)=
q
0
,
(30)
y
=
u
,
y
(
0
)=
0
,
(31)
z
=
v
,
z
(
0
)=
0
,
(32)
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