Biomedical Engineering Reference
In-Depth Information
their control functions. Various examples of such drug effects have been given
in Sect.
2
. Introducing pharmacokinetics (i.e., evolution of concentrations) for the
drugs chosen produces additional equations to the cell population dynamic model,
and their pharmacodynamics (i.e., actual drug actions) modify this cell dynamics
according to the target and to the effect of the drugs. Then, optimisation of cancer
treatments can be represented as an optimal control problem on this controlled
dynamic system. In this section, we first discuss how the drug infusions are taken
into account in the model, then we give examples of objective functions and
constraints considered in the literature on the treatment of cancers.
4.1
Classes of Control Functions: What Is Fixed and What
May Be Optimised
We introduce a vector space
X
, called the state space. At each time
t
∈
R
+
,the
state of the system is
x
X
. This variable lists all the data necessary to represent
the system. It should at least contain the number (or density) of cells for each type
considered. The state may have coordinates for healthy and cancer cells and for
each phase considered. In a PDE model, the state may also distinguish between
ages or between locations of cells. In the PDE case,
X
has infinite dimension.
The state should also contain concentrations of drugs in each compartment of the
pharmacokinetic model.
We denote by
u
the control function,
u
:
(
t
)
∈
U
. It represents the (multi)drug
infusion schedule time by time, one coordinate per drug. The dynamics of the
biological system can thus be written as
R
+
→
x
(
t
)=
f
(
t
,
x
(
t
)
,
u
(
t
))
,
where
f
:
, under standard hypothesis on
the dynamics
f
, the state is uniquely defined and we will denote the state variable
associated with the control
u
R
+
×
X
×
U
→
X
. Given a control
u
(
·
)
. Examples of such functions are given in
Sect.
3
, for instance, in Eqs. (
1
)-(
4
), (
11
)-(
12
)and(
15
).
Alternatively, instead of a control function, one may consider simpler predefined
infusion schemes with only a small number of control parameters. Such infusion
schemes may represent either a simple model for an early study or a consequence
of technical constraints such as the fact that oral drugs can only be administered
at fixed hours (at meal time for instance). Then
u
(
·
)
by
x
u
(
·
)
m
∈
R
is a set of parameters and
the dynamics is
x
. Examples of such parameters are the period of a
periodic scheme [
114
,
138
] or the phase difference between a circadian clock and
the time of drug infusion initiation [
5
] (see also Sect.
3
).
(
t
)=
f
u
(
t
,
x
(
t
))
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