Optimal Control Analysis of a Mathematical Model for Breast Cancer

Nonlinear dynamics of estrogen receptor-positive breast cancer integrating experimental data: A novel spatial modeling approach

Oncology research has focused extensively on estrogen hormones and their function in breast cancer proliferation. Mathematical modeling is essential for the analysis and simulation of breast cancers. This research presents a novel approach to examine the therapeutic and inhibitory effects of hormone and estrogen therapies on the onset of breast cancer. Our proposed mathematical model comprises a nonlinear coupled system of partial differential equations, capturing intricate interactions among estrogen, cytotoxic T lymphocytes, dormant cancer cells, and active cancer cells. The model's parameters are meticulously estimated through experimental studies, and we conduct a comprehensive global sensitivity analysis to assess the uncertainty of these parameter values. Remarkably, our findings underscore the pivotal role of hormone therapy in curtailing breast tumor growth by blocking estrogen's influence on cancer cells. Beyond this crucial insight, our proposed model offers an integrated framework to delve into the complexity of tumor progression and immune response under hormone therapy. We employ diverse experimental datasets encompassing gene expression profiles, spatial tumor morphology, and cellular interactions. Integrating multidimensional experimental data with mathematical models enhances our understanding of breast cancer dynamics and paves the way for personalized treatment strategies. Our study advances our comprehension of estrogen receptor-positive breast cancer and exemplifies a transformative approach that merges experimental data with cutting-edge mathematical modeling. This framework promises to illuminate the complexities of cancer progression and therapy, with broad implications for oncology.

Chemoimmunotherapy Administration Protocol Design for the Treatment of Leukemia through Mathematical Modeling and In Silico Experimentation

Cancer with all its more than 200 variants continues to be a major health problem around the world with nearly 10 million deaths recorded in 2020, and leukemia accounted for more than 300,000 cases according to the Global Cancer Observatory. Although new treatment strategies are currently being developed in several ongoing clinical trials, the high complexity of cancer evolution and its survival mechanisms remain as an open problem that needs to be addressed to further enhanced the application of therapies. In this work, we aim to explore cancer growth, particularly chronic lymphocytic leukemia, under the combined application of CAR-T cells and chlorambucil as a nonlinear dynamical system in the form of first-order Ordinary Differential Equations. Therefore, by means of nonlinear theories, sufficient conditions are established for the eradication of leukemia cells, as well as necessary conditions for the long-term persistence of both CAR-T and cancer cells. Persistence conditions are important in treatment protocol design as these provide a threshold below which the dose will not be enough to produce a cytotoxic effect in the tumour population. In silico experimentations allowed us to design therapy administration protocols to ensure the complete eradication of leukemia cells in the system under study when considering only the infusion of CAR-T cells and for the combined application of chemoimmunotherapy. All results are illustrated through numerical simulations. Further, equations to estimate cytotoxicity of chlorambucil and CAR-T cells to leukemia cancer cells were formulated and thoroughly discussed with a 95% confidence interval for the parameters involved in each formula.

Mathematical modeling and machine learning for public health decision-making: the case of breast cancer in Benin

Breast cancer is the most common type of cancer in women. Its mortality rate is high due to late detection and cardiotoxic effects of chemotherapy. In this work, we used the Support Vector Machine (SVM) method to classify tumors and proposed a new mathematical model of the patient dynamics of the breast cancer population. Numerical simulations were performed to study the behavior of the solutions around the equilibrium point. The findings revealed that the equilibrium point is stable regardless of the initial conditions. Moreover, this study will help public health decision-making as the results can be used to minimize the number of cardiotoxic patients and increase the number of recovered patients after chemotherapy.

Strategies for the treatment of breast cancer: from classical drugs to mathematical models

Breast cancer is one of the most common cancers and generally affects women. It is a heterogeneous disease that presents different entities, different biological characteristics, and differentiated clinical behaviors. With this in mind, this literature review had as its main objective to analyze the path taken from the simple use of classical drugs to the application of mathematical models, which through the many ongoing studies, have been considered as one of the reliable strategies, explaining the reasons why chemotherapy is not always successful. Besides, the most commonly mentioned strategies are immunotherapy, which includes techniques and therapies such as the use of antibodies, cytokines, antitumor vaccines, oncolytic and genomic viruses, among others, and nanoparticles, including metallic, magnetic, polymeric, liposome, dendrimer, micelle, and others, as well as drug reuse, which is a process by which new therapeutic indications are found for existing and approved drugs. The most commonly used pharmacological categories are cardiac, antiparasitic, anthelmintic, antiviral, antibiotic, and others. For the efficient development of reused drugs, there must be a process of exchange of purposes, methods, and information already available, and for their better understanding, computational mathematical models are then used, of which the methods of blind search or screening, based on the target, knowledge, signature, pathway or network and the mechanism to which it is directed, stand out. To conclude it should be noted that these different strategies can be applied alone or in combination with each other always to improve breast cancer treatment.

Personalized Immunotherapy Treatment Strategies for a Dynamical System of Chronic Myelogenous Leukemia

Simple Summary

As computer performance continues to grow at more affordable costs, mathematical modelling and in silico experimentation begin to play a larger role in understanding cancer evolution. The aim of our work is to formulate a control strategy for the Adaptive Cellular Therapy (ACT) that can fully eradicates the Chronic Myelogenous Leukemia (CML) cells population in a mathematical model describing interactions between naive T cells, effector T cells and CML cancer cells in the circulatory blood system. Mathematical analysis and numerical simulations allow us to conclude that it is possible to design a personalized administration protocol for the ACT in the form of a pulse train with asymmetrical waves and a fixed amplitude to achieve complete CML cancer cells eradication. The amplitude of the impulse on which the treatment is applied is given by an arithmetical combination of the parameters of the system with at least a duty cycle of 45 min/day.

Abstract

This paper is devoted to exploring personalized applications of cellular immunotherapy as a control strategy for the treatment of chronic myelogenous leukemia described by a dynamical system of three first-order ordinary differential equations. The latter was achieved by applying both the Localization of Compact Invariant Sets and Lyapunov’s stability theory. Combination of these two approaches allows us to establish sufficient conditions on the immunotherapy treatment parameter to ensure the complete eradication of the leukemia cancer cells. These conditions are given in terms of the system parameters and by performing several in silico experimentations, we formulated a protocol for the therapy application that completely eradicates the leukemia cancer cells population for different initial tumour concentrations. The formulated protocol does not dangerously increase the effector T cells population. Further, complete eradication is considered when solutions go below a finite critical value below which cancer cells cannot longer persist; i.e., one cancer cell. Numerical simulations are consistent with our analytical results.

Optimal Control Theory for Personalized Therapeutic Regimens in Oncology: Background, History, Challenges, and Opportunities

Optimal control theory is branch of mathematics that aims to optimize a solution to a dynamical system. While the concept of using optimal control theory to improve treatment regimens in oncology is not novel, many of the early applications of this mathematical technique were not designed to work with routinely available data or produce results that can eventually be translated to the clinical setting. The purpose of this review is to discuss clinically relevant considerations for formulating and solving optimal control problems for treating cancer patients. Our review focuses on two of the most widely used cancer treatments, radiation therapy and systemic therapy, as they naturally lend themselves to optimal control theory as a means to personalize therapeutic plans in a rigorous fashion. To provide context for optimal control theory to address either of these two modalities, we first discuss the major limitations and difficulties oncologists face when considering alternate regimens for their patients. We then provide a brief introduction to optimal control theory before formulating the optimal control problem in the context of radiation and systemic therapy. We also summarize examples from the literature that illustrate these concepts. Finally, we present both challenges and opportunities for dramatically improving patient outcomes via the integration of clinically relevant, patient-specific, mathematical models and optimal control theory.

Effect of Linearization in a WNT Signaling Model

A nonlinear model consisting of a system of coupled ordinary differential equations (ODE), describing a biological process linked with cancer development, is linearized using Taylor series and tested against different magnitudes of input perturbations, in order to investigate the extent to which the linearization is accurate. The canonical wingless/integrated (WNT) signaling pathway is considered. The linearization procedure is described, and special considerations for linearization validity are analyzed. The analytical properties of nonlinear and linearized systems are studied, including aspects such as existence of steady state and initial value sensitivity. Linearization is a useful tool for speeding up drug response computations or for providing analytical answers to problems such as required drug concentrations. A Monte Carlo-based error testing workflow is employed to study the errors introduced by the linearization for different input conditions and parameter vectors. The deviations between the nonlinear and the linearized system were found to increase in a polynomial fashion w.r.t. the magnitude of tested perturbations. The linearized system closely followed the original one for perturbations of magnitude within 10% of the base input vector which yielded the state-space fixed point used for the linearization.