Biomedical Engineering Reference
In-Depth Information
ln
p
q
b
p
2
3
ξ
d
b
q
p
1
/
3
−
(
μ
+
p
2
/
3
γ
u
sin
(
p
,
q
)=
Ψ
(
p
,
q
)=
ξ
+
q
+
)
(34)
or, equivalently, using Eq. (
33
), in terms of x by
1
3
ξ
+
bx
ln
x
1
2
3
ξ
−
bx
γ
u
sin
(
x
)=
+
.
(35)
S
There exists exactly one connected arc on the singular curve
along which the
≤
(
)
≤
singular control is admissible, i.e., satisfies the bounds
0
u
sin
x
u
max
.This
x
∗
,
x
u
]
where x
∗
and x
u
are the unique solutions to
[
arc is defined over an interval
x
∗
)=
x
u
)=
(
(
the equations u
sin
0
and u
sin
u
max
. At these points the singular control
=
=
saturates at the control limits u
u
max
.
An important feature of this solution is that it becomes the basis for the optimal
solution of the combination therapy problem
[AC]
. Indeed, for a typical initial
condition with
p
0
and u
q
, optimal controls for the combination therapy problem have
the following structure:
optimal controls for the anti-angiogenic agent follow the
optimal angio-monotherapy and then, at a specific time, chemotherapy becomes
active and is given in one full dose session
. The formulas for the singular control
and singular arc need to be adjusted to the presence of chemotherapy, but in this case
it is not possible that both controls are singular simultaneously. More specifically,
we have the following result:
<
Proposition 8.2 ([
48
]).
If the optimal anti-angiogenic dose rate u is singular on
an open interval I, then the chemotherapeutic agent v is bang-bang on I with at
most one switching on I from v
=
0
to v
=
v
max
and we have the following relation
between the controls u and v:
γ
u
sin
(
t
)+(
η
−
ϕ
)
v
(
t
)=
Ψ
(
p
(
t
)
,
q
(
t
))
(36)
with
defined by Eq. (
34
). Given v, this determines the anti-angiogenic dose rate
with a jump-discontinuity where chemotherapy becomes active.
Ψ
This structure allows to set up a minimization problem over a 1-dimensional
parameter
that denotes the time when chemotherapy becomes active. We illustrate
this for an initial condition
τ
q
0
where the anti-angiogenic inhibitor
will immediately be applied at full dose. In principle, the time
(
p
0
,
q
0
)
with
p
0
<
when chemotherapy
is activated can lie anywhere in the interval of definition. For example, if the amount
z
max
of chemotherapeutic agents is high, then it is possible that chemotherapy
already becomes active along the interval when the anti-angiogenic dose is at
maximum. Analogously, if this amount is very low, it is possible that this activation
will only occur after all anti-angiogenic inhibitors have been used up. Typically,
however, this time
τ
will lie somewhere in the interval where the anti-angiogenic
dosage follows the singular monotherapy structure and this is illustrated in Fig.
2
a.
τ
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