Biomedical Engineering Reference
In-Depth Information
Under the conditions of the Pontryagin's maximum principle given in Eqs. (
17
)
and (
18
), we have
d
d
t
H
)) =
∂
H
∂
p
0
p
0
(
t
,
x
(
t
)
,
p
(
t
)
,
,
u
(
t
t
(
t
,
x
(
t
)
,
p
(
t
)
,
,
u
(
t
))
and thus if
f
,
f
0
and
U
t
do not depend on
t
,then
H
does not depend on
t
and
p
0
(
,
(
)
,
(
)
,
,
)
max
v
∈
U
t
H
is constant.
Thanks to Pontryagin's maximum principle one is often able to determine the
optimal control as a function of the adjoint vector. Nevertheless, the adjoint vector
is not easy to compute. It is defined through its value at the terminal point,
p
t
x
t
p
t
v
(
)
,
and solutions to the associated boundary value problem are difficult to compute, nor
need they be unique.
For Pontryagin's maximum principle to be applicable, the cell population model
must be a set of ODEs controlled by drug infusions, as presented in Sect.
3.1
.The
authors generally minimise the number of cancer cells at final time with a bound on
the instantaneous drug flow. The total dose is either constrained to be bounded or is
part of the objective, a smaller dose improving the objective. When the information
provided by Pontryagin's maximum principle is enough to know the optimal control,
as in [
49
,
56
,
75
,
85
], it gives the control, i.e., the solution to the problem, in an
explicit formula, without any discretisation. This is then a very valuable information.
Unfortunately, for most optimal control problems, and optimal control arising from
chemotherapy problems are not an exception, we do not have enough information
and we have to use a numerical algorithm to solve the problem.
T
6.3
Numerical Methods for Optimal Control Problems
Two classes of numerical methods exist for optimal control problems, namely
indirect methods, also called shooting methods, and direct methods.
Shooting Method
The shooting method is based on the observation that, if ever we knew the value
p
0
=(
p
0
,
p
0
,...,
p
0
)
of the adjoint state at the initial point, we could get the
optimal controls time by time. Thus we define the shooting function
G
(
p
0
)
such
that
G
satisfies the final conditions (recall that
T
is the
final time). The shooting method simply consists in solving the equation
G
(
p
0
)=
0 if and only if
p
(
T
)
(
p
0
)=
0,
with variable
p
0
, for instance by a Newton method.
A variant of the shooting method was used in [
85
] for chemotherapy optimisa-
tion. Ledzewicz et al. considered two drugs that act on a Gompertzian model: one is
an anti-angiogenic, which controls the carrying capacity of the tumour and the other
is a cytotoxic drug, which controls a death term. The pharmacodynamics of the
Search WWH ::
Custom Search