Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 282

Other differentiable optimisation algorithms may also be used, depending on
the properties of the problem at stake. In general, all these algorithms give a local
optimal solution quickly but they do not give any guarantee that the control solution
produced is a global optimum. In order to overcome this drawback, some authors
chose to use stochastic algorithms instead.
Stochastic algorithms
Stochastic algorithms are algorithms that use a random number generator to find the
optimal solutions of a given problem. These random numbers are used to explore
the admissible control set with the hope that the optimal controls will eventually be
hit. Each stochastic optimisation algorithm is a compromise between focusing on
good solutions and letting enough freedom to exploration in order not to miss the
global optimum. See [ 68 , 129 ] for more details on this subject.
In [ 1 ], Agur et al. considered an age-structured cell cycle model with determin-
istic cycle phase lengths. The drug under consideration is toxic for cells in one
of the phases only. They considered a composite objective function that takes into
account the number of cancer and healthy cells in the end and a measure of the
survival of the patient. They assumed that a patient survives if at no time the number
of healthy cells falls below a threshold. The authors compared three versions of
simulated annealing. They first defined the neighbourhood of every point of the
admissible set, that is, at every point, they defined the possible ways to go to
another point. This neighbourhood should be large enough to give freedom to the
algorithm but not too large because otherwise the computational cost of searching
the neighbourhood would be dissuasive. Then simulated annealing gives the rule for
the acceptance or rejection of a neighbour, which gets stricter when a parameter, call
the temperature, decreases. In theory, if the temperature is decreased properly, the
iterates converge to an optimum of the problem. In practice, convergence may be
desperately slow. The other two heuristics presented in the paper do the same work
but with simplified rules, that do not guarantee convergence to an optimum but have
smaller computational costs.
Villasana et al. proposed in [ 136 ] an ODE model with three types of cells: cancer
cells in interphase (i.e., G 1 , S and G 2 ), cancer cells in mitosis phase ( M ) and healthy
cells. Each type of cells has a particular dynamic and there are interactions between
them. They considered a combination of a cytotoxic and of a cytostatic drug and
they wanted to minimise the number of cancer cells while keeping the number
of healthy cells above a threshold. They used the covariance matrix adaptation
evolutionary strategy (CMA-ES) to solve this problem. This is an algorithm based
on probabilistic mutations of the current iterates and on a selection of the best
ones [ 68 ]. The covariance matrix adaptation is a way to give the mutations directions
for them to be more effective.