Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 260

they could be used as a first step towards cancer therapy optimisation. As so much
has been published in the last 50 years, we do not claim to be exhaustive, only
recollecting the main models used to describe the fate of cell populations submitted
to cancer treatments.

3.1 ODE Models for Growing Cell Populations

with Drug Control

The first models of tumour growth were developed to reproduce and explain
experimentally observed tumour growth curves. The most common ones are
the exponential model ( d d t = λ

K ,where K is
the maximum tumour size, or “carrying capacity” of the environment), and the
Gompertz ( d d t = λ

N 1

N ), the logistic ( d d t = λ

N

−

N ln N , where again K is the carrying capacity). Contrary to
the exponential model, the logistic and the Gompertz models take into account
the possible limitation of growth due, for instance, to a lack of space or resources,
assuming that the instantaneous growth rate d N d t depends on the carrying capacity
of the environment. The Gompertz model was initially developed in the context
of insurance [ 62 ] and was first used in the nineties to fit experimental data of
tumour growth [ 81 ]. A lot of studies on drug control are based on these models
[ 13 , 15 , 35 , 96 - 98 , 106 - 108 ]. For instance, Murray [ 106 , 107 ] considered a two-
population Gompertz growth model with a loss term to model the effect of the
cytotoxic drug. He considered both tumour and normal cells in order to take into
account possible side-effects of the treatment on the population of normal cells.
Murray's purpose was to minimise the size of the population of tumour cells at the
end of the treatment while keeping the population of normal cells above a given
threshold. In [ 108 ], Murray took into account cell resistance to chemotherapy and
applied the problem of optimising drug schedules to a two-drug chemotherapy. In
[ 96 ], Martin developed a model to determine chemotherapy schedules that would
minimize the size of the tumour at the end of the treatment, under constraints
of maximal drug doses (individual doses and cumulative dose), ensuring that the
tumour decrease might be faster than a given threshold. In further works, Martin
et al. also introduced tumour cell resistance to chemotherapy [ 97 , 98 ].
More recently, one of us and his co-workers [ 15 , 35 ] investigated the effects
of oxaliplatin on tumour cells and healthy cells. To model tumour growth, they
used a Gompertz model modified by a “therapeutic efficacy term” as a death term
depending on the drug concentration.
In [ 35 ], the work presented in [ 15 ] is extended by coupling this model of
tumour growth to a model of healthy cell growth and to a three-compartment
model describing the time evolution of the concentration of oxaliplatin in plasma,
healthy tissue and tumour tissue. This work was done on the basis of experimental
data related to oxaliplatin PK-PD and tumour growth curves with or without drug
injection to determine the model parameters and compared time-scheduled infusion
schemes with constant ones. In the same work, the drug infusion schedule was