TITLE:
Modeling and Numerical Solution of a Cancer Therapy Optimal Control Problem
AUTHORS:
Melina-Lorén Kienle Garrido, Tim Breitenbach, Kurt Chudej, Alfio Borzì
KEYWORDS:
Cancer, Radiotherapy, Anti-Angiogenesis, Sparse Controls, Optimal Control, Pontryagin’s Maximum Principle, SQH Method
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.8,
August
30,
2018
ABSTRACT: A mathematical optimal-control tumor therapy framework consisting of radio-
and anti-angiogenesis control strategies that are included in a tumor
growth model is investigated. The governing system, resulting from the combination
of two well established models, represents the differential constraint
of a non-smooth optimal control problem that aims at reducing the volume
of the tumor while keeping the radio- and anti-angiogenesis chemical dosage
to a minimum. Existence of optimal solutions is proved and necessary conditions
are formulated in terms of the Pontryagin maximum principle. Based
on this principle, a so-called sequential quadratic Hamiltonian (SQH) method
is discussed and benchmarked with an “interior point optimizer—a
mathematical programming language” (IPOPT-AMPL) algorithm. Results of
numerical experiments are presented that successfully validate the SQH solution
scheme. Further, it is shown how to choose the optimisation weights in
order to obtain treatment functions that successfully reduce the tumor volume
to zero.