Optimal control for a bone metastasis with radiotherapy model using a linear objective functional

Radiation is known to cause genetic damage to highly proliferative cells such as cancer cells. However, the radiotherapy effects  to  bone cells is not completely known. In this work we present a mathematical modeling framework to test hypotheses related to the radiation-induced effects on bone metastasis. Thus, we pose an optimal control problem based  on a Komarova model describing the interactions between cancer cells and  bone cells   at a single site of bone remodeling.  The radiotherapy treatment  is included in the form of a functional which   minimizes the use of radiation using a penalty function. Moreover, we are interested to model the 'on' and the 'off' time states of the radiation schedules;  so we propose an optimal control problem with a L1-type objective functional. Bang-bang or singular arc solutions are the  obtained optimal control solutions. We characterize both solutions types   and explicitly give necessary optimality conditions for them. We present numerical simulations to analyze the different possible radiation effects on the bone and cancer cells. We also evaluate  the more significant parameters to shift from a bang-bang solution to a singular arc solution and vice versa. Additionally, we study a fractionated radiotherapy model.