Bone metastasis treatment modeling via optimal control

Abstract

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.

When optimal is not the best: cost effectiveness analysis for HPV epidemic models

“…Theorem 4.1 in [14, Chapter III] ensures the existence of an optimal control and the corresponding solution for this problem. Proofs of such statements can be found in [8,23].…”

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

“…Osteoblastic lesions are characterized by CCs-promoted over-activation of OBs but this cross-talk is largely unknown with the CCs-OCs one (Ubellacker and McAllister, 2016;Ottewell, 2016). The mathematical model that we present next codies these interactions by simplifying the network of chemical reactions using power-law functions (Komarova type models) (Jerez and Camacho, 2018;Camacho and Jerez, 2019). Let C(t), B(t) and M (t) denote the density of OCs, OBs and CCs at time t, respectively.…”

“…Local stability conditions for such equilibrium points are discussed in Jerez and Camacho (2018); Camacho and Jerez (2019) to know how the cancer-invasion equilibrium changes locally by variations of certain parameters it is very useful for simulations.…”

“…In the context of bone metastatic diseases there are few works related to treatment optimization, for example: In Warman et al (2018) based on an evolutionary game model describing prostate cancer bone metastasis progression, they studied optimal chemotherapy scheduling. Lemos et al (2016), Camacho and Jerez (2019) and Jerez et al (2021) considered an optimal control problem of the Komarova type, see Komarova et al (2003); the goal of the rst one was to obtain a therapy to accelerate bone mass recovery in the case of the myeloma bone disease, and for the second and third work the goal was to study the eciency of the radiotherapy, denosumab and antigen receptor T-cell treatments by proposing continuous control solutions in metastatic bone disease.…”

Optimal Control against the Human Papillomavirus: Protection versus Eradication of the Infection

“…Furthermore, for such functionals, the optimal controls can be obtained as explicit functions of the state and adjoint variables via the Pontryagin maximum principle. Although the financial cost of public health interventions to prevent epidemics most likely grows linearly with the magnitude of the corresponding control [24], we choose objective functionals with a quadratic dependence on the controls following the current trend of the literature [17,18,[25][26][27].…”