Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 288

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0 5 10 15 20 25 0 5 10 15 20 25

time (h) time (h)

Fig. 3 Drug and circadian controls, healthy cell population case ( a , left ) and cancer cell population
case ( b , right ). Cosine-like functions modelling the drug and circadian controls for transition from
G 1 to S / G 2 / M ( dash-dotted line ) and for transition from S / G 2 / M to G 1 in healthy cells. The

“natural” (drug-free) control for S / G 2 / M to G 1 transition corresponds to the solid line , the drug-

induced one to the dashed line

control on these transitions by circadian clocks, together with free-running drug
infusion regimens. The drug infusion regimens were optimised using a Uzawa
method with an augmented Lagrangian (see Sect. 6.3 or [ 29 ] for algorithmical
details), aiming at decreasing the growth rate in a cancer cell population (objective)
while preserving the same in a healthy cell population (constraint) by maintaining
it over a prescribed threshold. The idea is the same as in [ 15 ], except that we deal
here with cell population growth exponents instead of cell numbers.
We considered two cell populations, that we called cancer cells and healthy cells.
In these simulations, we took into account cell death via a constant death rate,
the same for both populations. We made the two cell populations only differ by
their circadian control function and we assumed that there was no interaction

between the two populations, healthy and cancer. We took for this circadian control
a continuous piecewise cosine-like function for each phase (Fig. 3 a). We assumed
that cancer cell populations still obey circadian control at these main checkpoints but
less faithfully, and we modelled their behaviour by a looser answer to the circadian
control signal (Fig. 3 b).
Transitions from one phase to the other are described by the transition rates
K i → i + 1 (

ψ

t , x ) . We took them with the form

K i → i + 1 ( t , x )= κ ( x ) ψ ( t )( 1 − g i ( t )) ,

i i

where is the transition rate of the cell without circadian control identified

from FUCCI data,

κ ( x )

ψ ( t ) is the natural circadian control modelled by a cosine-like i

function and g i is the effect at the cell level of the drug infusion at time t on

the transition rate from phase i to phase i

( t )

+ 1. No drug corresponds to g i ( t )= 0,

a transition-blocking infusion corresponds to g i 1. We assumed that the drug

has the same effect on both populations, which couples their behaviours through the
drug infusions.

( t )=