Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 289

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0 5 10 15 20 25

time (h)

Fig. 4 Locally optimal periodic drug infusion strategy (function g 2 , see text for details) found by
the optimisation algorithm

We solved the constrained optimisation problem [ 26 , 27 ] with Uzawa's algorithm.
After convergence of the algorithm, we get the locally optimal strategy, shown in
our case on Fig. 4 , defining on

[ ]

0; 24 the 24 h-periodic function g 2 (assuming that

=

g 1 0, that is here mimicking the action of an anticancer drug—as 5-fluorouracil—

active in S phase only). It can be seen that it mainly consists in forbidding transitions
when healthy cells do no change phase, thus harming cancer cells only.
The result of the optimised drug infusion regimen rates is shown in Fig. 5 ,where
it can be seen that the asymptotic growth rate of cancer cells, initially positive and
higher than the one of healthy cells, has been rendered negative by the periodic
treatment exerted on transition rates while the new growth rate of healthy cells,
though moderately affected by the treatment, remains positive.
Note that the FUCCI technology only enables us to distinguish between cells
in G 1 and S

/ /

M , without distinction between S , G 2 and M . However, we may
note that the method used in [ 27 ] to identify phase transitions relies in fact on the
probability distribution of durations of phases. Since the duration of the phase M
is known to be most of the time very short, with almost zero variability within cell
populations, it would be legitimate to consider it as fixed, as 1 h, say, and that the
recorded variability of S

G 2

/ / /

G 2 . Thus, we could
have considered that we were dealing in this identification process with a transition
function from S

G 2 M is in fact the variability of S

/ / /

M to G 1 . In this case, the
transition function from M to G 1 could have been modelled by an indicator function,
representing the fact that mitotic cells divide and enter the G 1 phase once they
are old enough. Under these assumptions, we could have applied our optimisation

G 2 to M instead of the one from S G 2