Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 272

Malthusian growth model, where growth exponents are the targets of control, such
a constraint becomes a lower bound on the asymptotic growth rate of the healthy
cell population [ 138 ].
In the same way, a drug used in a treatment must reach a minimal concentration
at the level of its target (which blood levels reflect only very indirectly) to
produce therapeutic effects. Classically, clinical pharmacologists are accustomed to
appreciating such efficacy levels by lower threshold blood levels, that are themselves
estimated as functions of pharmacokinetic parameters such as first and second half-
life times and distribution volume of the drug, with confidence interval estimates for
a general population of patients. As in the case of toxicity, a more dynamic view is
possible, by considering drug levels that decrease the number of cancer cells, that
is, which yield a negative growth rate in the cancer cell population.
This leads to the definition of admissible sets for drug infusion flows, the union

of and of a therapeutic range containing the infusion levels that are at the same

time efficient and not too toxic (such a constraint is considered in [ 136 ]). Those
admissible sets are rather difficult to take into account, however, as they lead to
complex combinatorial problems.
An approach that is consequently often chosen (see [ 85 ] for instance) is to forget
this constraint in the model and to a posteriori check that the optimal drug infusion
schedules found are high enough to be efficient when they are nonzero.
That may be an elementary reason why so-called bang-bang controls (i.e., all-
or-none) are of major interest in chemotherapy optimisation: they are defined as
controls such that at each time, either the drug infusion flow is the smallest possible
(i.e., 0), or it is the highest possible. Even though it is now easy to use in the clinic
(and also in ambulatory conditions) programmable pumps that may deliver drug
flows according to any predefined schedule with long-lasting autonomy, solutions
to optimisation problems often turn out to be bang-bang (tap open-tap closed).
But solutions to optimisation problems in cancer chemotherapy are not always
bang-bang, when considerations other than on simple parallel growth of the two
populations are taken into account, and this includes competition, when the two
populations are in contact, e.g. in the bone marrow normal haematopoietic and
leukaemic cells, or when both populations are submitted to a common—but
differently exerted—physiological control, such as by circadian clocks [ 15 ].
Another interesting approach, relying on two models, one of them including the
cell division cycle [ 115 ], and putting the optimal control problem with toxicity
constraints, is developed in [ 54 ]. The optimal control problem is solved by using
the industrial software gPROMS R

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.

Drug Resistance

Whereas therapeutic efficacy and limitation of toxic adverse effects are the first
concern when dealing with chemotherapy, the frequent development of drug
resistances in the target cancer cell populations is certainly the second bigger issue
in the clinic. The development of such resistances may come from overexpression