Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 268

tumour cell dynamics under periodic age-specific chemotherapy. Hinow et al. [ 69 ]
developed an age-structured PDE model to investigate the cytotoxic and cytostatic
effects on tumour growth of a cancer drug, lapatinib, on the basis of biological
experiments. The authors distinguished between proliferative and non-proliferative
cells and assumed that only proliferative cells were ageing.
Oneofuswithhisco-workers[ 27 , 40 , 41 ] considered a multiphase age-structured
PDE model in which they introduced time dependency of the parameters (death rate,
transition rate from one phase of the cell cycle to the next one). They investigated the
effects of a circadian control on the tumour growth rate with and without a periodic
cell cycle phase specific chronotherapy. More details about [ 27 ] can be found in
Sect. 7 .
Basse et al. [ 16 , 18 , 19 ] developed a phase- and size-structured model of a cell
population submitted to paclitaxel, a cancer agent that induces mitotic arrest of
the cell cycle and cell death. The size of the cell was considered as determined
by its DNA content and some of the model parameters were determined by fitting
experimental flow cytometry data. On the basis of the work of Spinelli et al. [ 130 ],
Basse et al. [ 20 ] developed a phase- and age-structured model of a cell population
submitted to a chemotherapy. They considered several cancer agents and assumed
that these chemotherapies affected tumour cell population dynamics by modifying
cell cycle phase transition functions or by killing cells in the mitotic phase.
We b b [ 138 ] proposed a both age- and size-structured model for normal and
tumour cell dynamics under chemotherapy, on the basis of the McKendrick model
with an additional transport term. He supposed that the two population differed by
their mean cell cycle duration, which was longer for tumour cells, and modelled the
effects of the chemotherapy by a time periodic death term. His aim was to take
the resonance phenomenon into account to determine optimal period of the cancer
treatment in order to induce the lowest tumour growth rate and the highest normal
cell population growth rate.
Finally, we present some examples of models that are not actually physiologically
structured PDE models but that derive from them.
Ubezio and co-workers [ 94 , 95 , 104 , 135 ] based themselves on an age- and phase-
structured PDE model to develop a discrete age-structured model of cell cycle
describing the time evolution of the number of cells of age a at time t in the phases
G 1 , S and G 2 /

M of the cell cycle. This model also takes into account the inter-cell
variability in phase duration. The potential effects of drug (blocking cells in G 1 or
in G 2 , etc) were modelled through separate parameters. Thanks to this model and
experimental data, Montalenti et al. [ 104 ] investigated the effects of several doses
of cisplatin on ovarian carcinoma cells. Although cisplatin is known to block cells
mostly (but not only, since as an alkylating agent inducing double strand breaks
throughout the cell cycle, it is not phase-specific) in G 2 , they also analysed the
effects of cisplatin on cells in G 1 .In[ 94 , 95 ], based on experimental data, Lupi et al.
investigated the effects of topotecan and melphalan, respectively, on ovarian cells.
Later, Ubezio et al. [ 135 ] deepened the previous works of their team by examining
the effects of five drugs (doxorubicin, cisplatin, topotecan, paclitaxel and melphalan)
on ovarian cancer cells using several doses.