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Metastatic disease is a lethal stage of cancer progression. It is characterized by the spread of aberrant cells from a primary tumor to distant tissues like the bone. Several treatments are used to deal with bone metastases formation, but they are palliative since the disease is considered incurable. Computational and mathematical models are used to understand the underlying mechanisms of how bone metastasis evolves. In this way, new therapies aiming to reduce or eliminate the metastatic burden in the bone tissue may be proposed. We present an optimal control approach to analyze some common treatments for bone metastasis. In particular, we focus on denosumab treatment, an anti-resorptive therapy, and radiotherapy treatment which has a cell killing action. We base our work in a variant of an existing model introduced by Komarova. The new model incorporates a logistic equation in order to describe the bone metastasis evolution. We provide proofs of existence and uniqueness of solutions to the corresponding optimal control problems for each treatment. Moreover, we present some numerical simulations to analyze the effectiveness of both treatments when different interactions between cancer and bone cells occur. A discussion of the obtained results is provided.

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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.