Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 262

proliferative and quiescent, by representing time variations of their densities, and
allow exchanges of cells from one population to the other. They are based on the
fact that only proliferative cells are sensitive to chemotherapies and they allow to
study the effect on tumour growth of several treatment schedules and to determine
the optimal ones. For instance, in [ 114 , 115 ], Panetta et al. proposed the following
model

x 1
x 2

α − μ − ηβ
μ

x 1
x 2

d
d t

= ,

(5)

− β − γ

where x 1 and x 2 represent the cycling and non-cycling tumour cell mass, re-
spectively,

α the cycling growth rate, μ the rate at which cycling cells become

non-cycling, η the natural decay of cycling cells, β the rate at which non-cycling

cells become cycling, and the natural decay of non-cycling cells. All these

parameters are supposed to be constant and positive. By adding a drug-induced
death term in the equation on cycling cells, the authors investigated the effects
on tumour growth of two kinds of periodic chemotherapies: a pulsed one and a
piecewise continuous one. They also considered the effect on a population of non-
tumour cells (or normal cells) in order to determine optimal drug schedules. Some
authors later based themselves on this model to determine optimal chemotherapy
schedules [ 56 , 83 ]. Using experimental data, Ribba et al. [ 124 ] introduced a third
kind of cells, the necrotic cells, and the carrying capacity in order to investigate the
effect of an antiangiogenic treatment on tumour growth dynamics and on hypoxic
and necrotic tissues within the tumour.
Kozusko et al. [ 80 ] deepened the work of Panetta et al. [ 115 ] by developing a
model of tumour growth integrating two compartments within the cell cycle: one for
cells in phases G 1 and S and another for cells in G 2 and M . They based their model
on experimental data to represent the effect on tumour growth of an antimitotic agent
(curacin A), that prevents cells from dividing. They modeled the blockade of cells in
the G 2 /

γ

M phase of the cell cycle according to the treatment dose, and distinguished
resistant cells from sensitive ones. This model was able to predict a minimum
dose of treatment able to stop growth of both kinds of cells. To analyse the effect
on tumour growth of another anticancer treatment (mercaptopurine) according to
varying degrees of cell resistance, Panetta et al. [ 117 ] modified the model introduced
by Kozusko et al. [ 80 ] by distinguishing phases G 0 / G 1 , S and G 2 / M .

3.2 PDE Models with Spatial Dynamics for Tumour Growth

and Drug Effects

ODE models presented above do not integrate any spatial dimension. They were
historically developed to explain in vitro tumour growth curves. Obviously in vivo
tumour growth depends on its environment. For instance, it depends on the