Modeling and Numerical Solution of a Cancer Therapy Optimal Control Problem

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.

Set-valued analysis of anti-angiogenic therapy and radiotherapy

“…The following complementary coupled dynamics between the tumor volume p ∈ (0, ∞), and the time-varying carrying capacity q ∈ (0, ∞), are considered from [28].…”

“…• [39] Operates (PMP) to optimally control Hahnfeldt's sub-model, with the objective function of minimizing the size of cancer. • [28] Proposes of the model (1), a Sequential Quadratic Hamiltonian (SQH) method to choose the optimisation weights, in order to obtain treatment functions that successfully reduce the tumor volume to zero.…”

“…As in (1d) we have r 0 = 0 for the initial value of the tumor repair r, and we consider v 0 = 0 as the initial value of the parameter control v, for the parameter κ of the dynamics (10b) we propose κ = 0.5, as in [28] to combine Hahnfeldt and Ergun dynamics, while the following table 1 gives the numerical values of the model (10) parameters. 1, but kept on decreasing state all over time therapy in accordance with (2b) and (2c), caused by growth limitation of the carrying capacity q due to combined anti-angiogenic therapy and radiotherapy (u, w), and by direct effect of the radiotherapy w(p, q) on the tumor volume p, while the tumor repair r is augmented.…”

Comparison of metaheuristic optimization algorithms for numerical solutions of optimal control problems

“…These algorithms can be used in many different fields such as extracting features in image processing, [37][38][39][40] engineering problems, 13,[41][42][43][44] solving NP hard problems, [45][46][47] classification, clustering 48 and health. 36,49 Such algorithms provide successful results in cases where the objective function is discontinuous and in limited optimization operations. Tawhid et al developed the WOFPA algorithm by combining the whale optimization algorithm and the flower pollination algorithm.…”

“…This algorithm is applied in many different fields such as GA. These algorithms can be used in many different fields such as extracting features in image processing, 37–40 engineering problems, 13,41–44 solving NP hard problems, 45–47 classification, clustering 48 and health 36,49 . Such algorithms provide successful results in cases where the objective function is discontinuous and in limited optimization operations.…”