Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 267

5-fluorouracil (5-FU), an anticancer drug known to block cells in the S phase. They
modelled the effects of this drug by increasing the probability that cells submitted
to 5-FU while in S phase exit from the cell cycle at the next G 2 / M transition.

They compared the cytotoxic efficacy of continuous administration of 5-FU and of
several chronomodulated therapies that differed from their administration peak time.
Later, in [ 5 ], Altinok et al. analysed the cytotoxic effects of 5-FU chronotherapies
according to their administration peak time and to the cell cycle mean duration.
As they did for 5-FU, Altinok et al. also investigated the effects of oxaliplatin
chronomodulated therapies on tumour cells. Contrary to 5-FU, oxaliplatin is an
anticancer agent that is not phase-specific. Therefore the authors modelled the
effects of oxaliplatin in a non phase-specific way, by increasing the probability
for exposed cells of exiting the cell cycle at the next checkpoint ( G 1 /

S or G 2 / M

transitions).

3.4 Physiologically Structured PDE Models for the Cell Cycle

and Drug Effects

Time and space are not the only two variables on which tumour growth depends.
In fact, tumour growth also depends on the physiological properties of cancer cells,
that can be for instance age of the cells (i.e., time since the last cell division), mass
or volume of the cells, or their DNA content. To take this phenomenon into account,
the McKendrick PDE framework is the best suited
⎧
⎨

∂ n

)+ ∂
∂ t (

a , t

a [

g ( a ) n ( a , t )] + d ( a ) n ( a , t )= 0 ( t > 0 , a > a min ) ,

∂

∞
a min β (

(15)

n ( a min , t )= s ) n ( s , t ) d s ( t > 0 ) ,

⎩

( , )= ( ) ( > ) ,

n a 0 n 0 a a a min

( , )

where n is the density of tumour cells with the characteristic a (age, mass,

volume, DNA content, etc.) at time t , g is the tumour growth rate, d is the death
rate,

a t

≥

β is the tumour cell birth rate, a min 0 is the minimum value of a . Note that

g , d , depend on a .

Physiologically structured cell population dynamics models have been exten-
sively studied in the last 25 years, see e.g. [ 7 - 9 , 14 , 17 , 20 , 27 , 40 , 41 , 65 , 69 , 71 , 74 ,
103 , 123 , 138 ]. For instance, Iwata et al. [ 71 ] developed a model of the dynamics of
the colony size distribution of metastatic tumours, assuming that both primary and
metastatic tumour growth depended on the size of the tumour. The authors proposed
a Gompertz equation to model the primary tumour growth and a McKendrick
type equation to model the evolution of the colony size distribution of metastases.
Kheifetz et al. [ 74 ] proposed a model for tumour cell age distribution to investigate

β