Abstract
In this chapter, we consider the second major feature of the tumor microenvironment: interactions between the tumor and the immune system. Fundamental principles that have already been outlined in the introduction (see Section 1.3.4) will be expanded upon in this chapter. As a vehicle for the analysis we use the classical model by Stepanova [303] and some of its modifications that have been developed in the literature. This model captures the main features that we want to discuss here—immune surveillance and tumor dormancy—and, at the same time, being low-dimensional and minimally parameterized, has the advantage of allowing us to easily visualize associated geometric features (regions of attractions, stability boundaries, etc.). We formulate an optimal control problem whose objective to be minimized is tailored to the inherent multi-stable structure that these systems have. These problems are considered under chemotherapy and under combinations of chemotherapy with a rudimentary form of an immune boost.
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Notes
- 1.
The reader may find the short fragment “On Exactitude in Science” by Jorge Luis Borges [33] of interest.
- 2.
The numerical computations were carried out by our former graduate students Mohammad Naghneian and Mozhdeh Moselman Faraji Sadat.
- 3.
The numerical calculations were carried out by Behrooz Amini.
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Schättler, H., Ledzewicz, U. (2015). Optimal Control for Mathematical Models of Tumor Immune System Interactions. In: Optimal Control for Mathematical Models of Cancer Therapies. Interdisciplinary Applied Mathematics, vol 42. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2972-6_8
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