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### Session S01 - Modeling and Computation for Control and Optimization of Biological and Physical Systems

Thursday, July 15, 15:30 ~ 15:55 UTC-3

## A two-strain model of tumor growth

### Fabio Milner

#### Arizona State University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak58e1bb68cf6ae41cb186cb9862c66a79').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy58e1bb68cf6ae41cb186cb9862c66a79 = 'fm&#105;ln&#101;r' + '&#64;'; addy58e1bb68cf6ae41cb186cb9862c66a79 = addy58e1bb68cf6ae41cb186cb9862c66a79 + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text58e1bb68cf6ae41cb186cb9862c66a79 = 'fm&#105;ln&#101;r' + '&#64;' + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak58e1bb68cf6ae41cb186cb9862c66a79').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy58e1bb68cf6ae41cb186cb9862c66a79 + '\'>'+addy_text58e1bb68cf6ae41cb186cb9862c66a79+'<\/a>';

Cancer in most organs develops in the form of solid tumors that begin a spheroids consisting entirely of cancerous (parenchyma) cells. When the tumor reaches the size of approximately 1 mm, it needs blood vessels (vascularization) to develop in it in order to be able to develop further. A model of vascularized tumor growth with competing parenchyma cells of two different strains will be presented. A related model for a single strain of parenchyma was described by Chen and Friedman in [1]. The model consists of a free-boundary value problem for a system of nonlinear parabolic PDEs describing the dynamics of the parenchyma cells and vascular endothelial cells (VECs), coupled with an ODE describing the dynamics of vascular density and an algebraic equation describing a variable used as proxy for resource availability locally. The tumor is assumed to have radial symmetry and the mathematical domain is then taken to be a sphere with moving boundary, with the boundary moving at the unique physically determined velocity that changes local tumor volume exactly to maintain density.

Some theoretical analytical properties of the model will be presented, and then the focus will shift to simulations to study the possible development of hypertumors [2]. Results from several simulations will be shown and biological interpretations provided, including how selection for increased proliferation or for increased angiogenesis may lead to tumors that are structurally like integrated tissues or like segregated ecological niches, as well as to hypertumors.

References

[1] D. Chen, and A. Friedman, A two-phase free boundary problem with discontinuous velocity: Application to tumor model", $J. Math. Anal. Applic.$ 399 (1), 378-393 (2013).

[2] J. Nagy, Competition and Natural Selection in a Mathematical Model of Cancer", $Bull. Math. Biol.$ 66, 663-687 (2004).

Joint work with R. L. Alvarez (Podium Education-Data Science and Curriculum-Austin, TX , USA) and J.D. Nagy (Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ, USA).