Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 271

4.2 Objective Functions: Measuring the Output

An optimisation problem consists in maximising or minimising a given real-valued
objective function that models the objective we want to reach.
The main purpose of a cancer treatment is to minimise the number of cancer
cells. When the model takes into account the number of cancer cells directly [ 5 , 15 ,
49 , 56 , 75 , 85 , 118 , 136 ], the objective function is simply the value of the coordinate
of the state variable corresponding to the number of cancer cells at a time T , T
being either fixed or controlled. To take into account the drug effects, Swierniak et
al. [ 75 , 134 ] defined a performance index to minimise the number of tumour cells
at the end of the treatment while minimising the cumulated drug dose (viewed as a
measure of the cumulated drug effects on healthy cells).
The optimisation problem can also be formalised as the minimisation of the
asymptotic growth rate of the cancer cell population [ 27 , 115 , 138 ]. Hence, the
number of cancer cells will increase more slowly, or even eventually decrease.
We will present this approach in a linear frame (hence controlling eigenvalues) in
Sect. 7 .
Alternatively, in [ 134 ], Swierniak et al. discussed the problem of maximising
both the final number of normal cells and the cumulated drug effects on tumour
cells. They concluded that this approach led to optimisation principles similar to
those developed to solve the problem of minimising both the final number of tumour
cells and the cumulated drug effects on healthy cells.

4.3 Constraints, Technological and Biological,

Static or Dynamic

Toxicity Constraints

A critical issue in cancer treatment is due to the fact that drugs usually exert their
effects not only on cancer cells but also on healthy cells. A simple way to minimise
the number of cancer cells is to deliver a huge quantity of drug to the patient, who
is however then certainly exposed at high lethal risk. In order to avoid such “toxic
solutions”, one may set constraints in the optimisation problem, which thus becomes
an optimisation problem under constraints.
Putting an upper bound on the drug instantaneous flow [ 56 ] and/or on the total
drug dose is a simple way to prevent too high a toxicity for a given treatment.
A bound on total dose may also represent a budget limit for expensive drugs [ 85 ].
However, fixed bounds on drug doses are not dynamic, i.e., they do not take
into account specificities of the patient's metabolism and response to the treatment,
other than by adapting daily doses to fixed coarse parameters such as body surface
or weight (as is most often the case in the clinic so far). In order to get closer to
actual toxicity limits, and hoping for a better result, it is possible to consider instead
a lower bound on the number of healthy cells, as in [ 15 ]. In the same way, using a