Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 283

7 Focus: Cancer Therapeutics to Control Long-Term Cell

Population Behaviour in Structured Cell Population
Models

7.1 Linear and Nonlinear Models

We have presented in [ 26 , 27 ] a method based on the control of eigenvalues in
an age-structured model, yielding the numerical solution of an optimal control
problem in the context of cancer chronotherapeutics, and we sum up below some
of its results as regards system modelling, identification, and theoretical therapeutic
optimisation. To this goal, we used an age-structured cell population model, since
our aim was to represent the action of cytotoxic anticancer drugs, which always
act onto the cell division cycle in a proliferating cell population. The model
chosen, of the McKendrick type [ 102 ], is linear. This may be considered as a harsh
simplification to describe biological reality, which involves nonlinear feedbacks
to represent actual growth conditions such as population size limitation due to
space scarcity. Nonetheless, having in mind that linear models in biology are just
linearisations of more complex models (for instance considering the fact a first
course of chemotherapy will most often kill enough cells to make room for a non
space-limited cell population to thrive in the beginning) we think that it is worth
studying population growth and its asymptotic behaviour in linear conditions and
thus analyse it using its growth (or Malthus) exponent. This first eigenvalue of the
linear system may be considered as governing the asymptotic behaviour, at each
point where it has been linearised, of a more complex nonlinear system, as described
in [ 21 , 22 ].

7.2 Age-Structured Models for Tissue Proliferation

and Its Control

We know that circadian clocks [ 87 - 92 ] normally control cell proliferation, by gating
at checkpoints between cell cycle phases (i.e., by letting cells pass to the next phase
only conditionally). We also know that circadian clock disruption has been reported
to be a possible cause of lack of physiologically control on tissue proliferation in
cancer [ 91 ], a fact that we will represent in our model to distinguish between cancer
and healthy cell populations.
The representation of the dynamics of the division cycle in proliferating cell
proliferations by physiologically structured PDEs is thus a natural frame to model
proliferation in cell populations, healthy or tumour. The inclusion in such prolifera-
tion models of targets for its control, physiological (circadian) and pharmacological
(by drugs supposed to act directly on checkpoints), allows to develop mathematical