An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors

Adam Glick,Antonio Mastroberardino

doi:10.3390/math5040049

Adam Glick, Antonio Mastroberardino

Open Access

https://doi.org/10.3390/math5040049

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Abstract

Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved.

Highlights

Cancer is a disease of unregulated cell growth
Current clinical treatments using angiogenesis inhibitors to prevent either of these from happening favor the results from the bang-bang control where a patient receives an injection of a drug over discrete time periods, generally 1–2 injections per week for several weeks, but our research shows that great tumor control can be achieved using the continuous administration of a drug, or daily injections over the course of treatment
Because of the unfavorable results with bang-bang control and singular arcs for tumors depending on initial conditions, we investigate the two compartment model investigated in [32] but subject it to a continuous optimal control protocol

Summary

Introduction

Cancer is a disease of unregulated cell growth. Unlike normal adult cells that divide only to replace dead or dying cells, cancer cells continue to grow, forming more abnormal cells. Cancerous cells arise when proto-oncogenes undergo a genetic change and become oncogenes, or tumorcoding genes. Cancer cells are often found to contain broken or morphed chromosomes that can cause irregular gene expression. Cancer invades healthy cells to obtain nutrients by parasitising its host organism. According to the National Institute of Health’s National Cancer Institute, 1.6 million new cases of cancer are predicted to be diagnosed in 2017 and over 600,000 people would die from the disease [1,2]

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