Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 274

Ideally, the optimal solution of a therapeutic control problem should take into
account both the drug resistance (using evolutionary cell population dynamics) and
the toxicity constraints, but these constraints have usually been treated separately so
far. Whereas the difficult problem of drug resistance control is certainly one of our
concerns in a cell Darwinian perspective, in the sequel we shall present only results
for the (easier) toxicity control problem.

5 Identification of Parameters: The Target Model

and Drug Effects

5.1 Methods of Parameter Identification

In the many works dedicated to modelling pharmacological control of tumour
growth and its optimisation that have been published in the last 40 years, when
the issue of confronting a theoretical optimisation method with actual data has been
tackled, quite different attitudes have been displayed. When identifying parameters
of a biological model, one may use different methods, according to the nature of
considered experimental data, their precision and reliability, and, also of course
according to the scientific background of people in charge of identification. One
may distinguish between at least three types of methods, all of which, to yield the
best estimation of the parameters at stake usually rely on least squares minimisation,
otherwise said minimisation of a L 2 distance between experimental quantitative
observations and numerical features of the model, either direct outputs, such as cell
numbers, or computed statistical parameters, such as mean cell cycle times.

Probabilistic Methods

The first method is based on the theory of parameter estimation in statistical models,
and supposes that a probability measure, depending on a set of statistical parameters,
e.g., mean and variance of a probability density function (p.d.f.), is a priori given in a
space of constitutive parameters of the model, e.g., coefficients in a set of differential
equations. In its simplest form, estimation will result from the minimisation of the L 2
distance between a model p.d.f. and a corresponding observed histogram, yielding
with precision a best set of parameters for the p.d.f. It may also result from more
elaborated principles, such as maximum likelihood estimation (including the use
of computational algorithms of the expectation-maximisation (EM) type, with or
without the assumption of an underlying Markov chain), see the statistical literature
on the subject, e.g., [ 86 , 137 ] for a general presentation (To this class of methods
may also be related attempts to characterise by its statistical properties a chaotic
deterministic system, as studied for instance in [ 82 ], when no actual model is given
of the system, which is only supposed to have trajectories converging towards a
chaotic attractor—on which they are dense—an attractor which by definition is
endowed with an invariant ergodic measure.).