Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 259

2.4 Representation of Drug Targets

It is appropriate to consider anticancer drugs, cytotoxic or cytostatic, and their
targets, through their effects on the cell cycle in cell populations. This of course
assumes that a model of the cell cycle in a proliferating cell population is given.
Nevertheless we also firstly mention other models, which either do not include the
cell cycle or do not describe events at the level of a cell population.
In a review article [ 75 ], Kimmel and Swierniak considered two possibilities to
represent in a mathematical model the action of cytotoxic drugs on their targets
in a proliferating cell population: either by a possible direct effect on cell death,
enhancing it, presumably by launching or accelerating the apoptotic cascade in one
or more phases of the cell cycle, or by a blockade of one or more transitions between
two phases, arresting the cycle at some checkpoint, most often with the involvement
of protein p53, and only secondarily launching cell death.
This is indeed a general alternative in the representation of the effects of
cytotoxics. If no cell cycle phase structure has been put in the population dynamic
model used to represent the evolution of the cell populations at stake, i.e., when
no account is taken of cell cycle phases in these populations, then only the first
possibility exists: modulation of a death term.
As regards cytostatics (which by definition are not supposed to kill cells, at least
not directly), the representation of their action in physiological models with age
structure for the cell cycle should be somewhat different. It can be done either by
a slowdown of the progression speed in the G 1 phase (or in the proliferating phase
in a one-phase model) or by an action on the exchanges between non-proliferating
( G 0 ) and proliferating phases when a G 0 phase is represented in the model.
It is also possible to combine cytostatic and cytotoxic effects in the same
model. In [ 69 ], for instance, the authors use an age-structured model with a 1-
phase proliferative subpopulation exchanging cells with a non-proliferative cell
compartment to combine a slowdown effect on proliferation for the cytostatic effect
with an increase in the cell death term for the cytotoxic effect—of the same drug,
lapatinib, a tyrosine kinase inhibitor, in their case, the variation between these effects
depending on the dose. Acting on two different targets in a cell cycle model by two
different drugs, a cytotoxic and a cytostatic one, in the same cell population is thus
possible, and such models are thus amenable to study and optimise combination
therapies, such as cetuximab+irinotecan advocated in [ 44 ].

3 Overview of Cell Population Dynamics Designed

for Cancer Treatment

In this section, we present an overview of several kinds of cell population models
that have been used in modelling anticancer treatments. Some works we refer to do
not include cancer treatment optimisation, but it is important to mention them as