Biomedical Engineering Reference
In-Depth Information
over all piecewise continuous (respectively, Lebesgue measurable) functions
u
:
[
0
,
T
]
→
[
0
,
u
max
]
and
v
:
[
0
,
T
]
→
[
0
,
v
max
]
for which the corresponding trajectory
satisfies the endpoint constraints
y
(
T
)
≤
y
max
and
z
(
T
)
≤
z
max
.
The constants
u
max
and
v
max
denote the maximum dose rates of the anti-
angiogenic and cytotoxic agents, respectively, and Eqs. (
31
)and(
32
) limit the
overall amounts of each drug given to
y
max
and
z
max
. In this formulation, for the
moment, the dosages
u
and
v
are identified with their concentrations. We comment
on the changes that occur to the optimal solutions if a standard linear model for the
pharmacokinetics of the drugs is included in Sect.
10
. Variations of this model are
considered, for example, in [
30
,
56
,
62
,
63
].
It follows from the dynamics (in fact, in greater generality) that for any
admissible control pair
, solutions to this dynamical system
exist for all times and remain positive. Some properties of optimal controls have
been established for a more general formulation in [
48
], but complete solutions
require a specification of the dynamics. We also denote by
[A]
the special case of
problem
[AC]
that corresponds to an anti-angiogenic monotherapy. This problem is
obtained from the formulation above by simply setting
z
max
=
(
u
,
v
)
defined on
[
0
,
∞
)
0 which eliminates
the chemotherapeutic agent. For the monotherapy problem
[A]
, a complete solution
in form of a regular synthesis of optimal controlled trajectories has been given in
[
26
] and the significance of this solution lies in the fact that the optimal solution for
the combination therapy problem indeed does build upon this synthesis. This is a
nontrivial feature which does not hold for solutions to optimal control problems for
nonlinear systems in general, but seems to be prevalent for the models combining
anti-angiogenic treatments with chemotherapy and also radiotherapy. We therefore
start with a brief review of the optimal synthesis for the monotherapy problem.
An optimal
synthesis
of controlled trajectories acts like a GPS system. It provides
a full “road map” of how optimal protocols look like depending on the current
conditions of the state variables in the problem, both qualitatively and quantita-
tively
. Given any particular point
that represents the tumor volume and the
current value of the carrying capacity, and any value
y
that represents the amount
of inhibitors that have already been used up, equivalently, the amount remaining
to be used, it tells how to choose the control
u
. Figure
1
, for specific parameter
values that have been taken from [
16
], gives a two-dimensional rendering of such a
synthesis for the monotherapy problem when the variable
y
has been omitted. The
actual numerical values of the parameters are not important for our presentation
here since, indeed, the optimal synthesis looks qualitatively identical regardless
of these numerical values [
26
]. In this chapter, we do not pursue quantitative
results, but our aim is to describe
robust qualitative results
about optimal controls
that are transferable to other models and give insights about the structure of
optimal solutions that can be useful in more general situations as well. In fact,
given our theoretically optimal analytical solution, for a typical initial condition, a
straightforward Matlab code just takes seconds to compute the optimal control and
corresponding trajectory. Since we have a full understanding of the theoretically
(
p
,
q
)
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