Biomedical Engineering Reference
In-Depth Information
6
Therapeutic Optimisation Procedures
In the previous sections we have described the drugs used in chemotherapy, various
cell population dynamic models and the objectives and constraints considered in
chemotherapy optimisation problems that have been published. Those three topics
can be seen as components of an optimisation model. In order to get quantitative
results, the parameters of this model should be estimated by using the techniques
presented in Sect.
5
. Then, one has to choose an optimisation procedure to solve the
optimisation problem considered.
When choosing an optimisation procedure, one first needs to identify what are
the optimisation variables. For chemotherapy optimisation, there are two main
situations: either the optimisation variables are some parameters of a predefined
infusion scheme or they are the infusion scheme itself, represented by a time-
dependent control function
u
(
t
)
(cf. Sect.
4.1
).
6.1
Graphic Optimisation
“Graphic optimisation” simply consists in plotting the value of the objective for all
admissible points. It is a very simple scheme and the only requirement for its success
is that the admissible set must have a nonempty interior. It also provides graphics to
present the result.
Graphic optimisation suits particularly the case of a predefined infusion scheme
with only a few parameters. For instance, this technique was used by Webb [
138
]
and Panetta and Adam [
115
]. These authors considered models of the McKendrick
type (see Sect.
3
) and they searched for the best period of periodic drug infusions,
i.e., the period of predefined drug infusion schemes that minimises the growth rate
of cancer cells. Altinok et al. [
5
] proposed a cellular automaton model controlled
by two predefined infusion schedules of drugs where the parameter is the phase
difference between a circadian clock and the drug infusion.
The drawback of this method is that when the number of parameters grows, the
time necessary for the resolution of the problem grows exponentially. Moreover,
graphics are less practical when the dimension exceeds 3. The classical solution is,
rather, to make use of a more evolved optimisation algorithm. However, if one needs
an evolved optimisation algorithm anyway, one might as well consider an optimal
control problem, in which the whole infusion schedule is the optimisation variable.
6.2
Pontryagin's Maximum Principle
An optimal control problem is an optimisation problem where the objective is a
function of the state variables
x
n
(
·
)
:
R
+
→
R
of a dynamical system and of the
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