Existence of solutions of cancer invasion parabolic system with integrable data
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Existence of solutions of cancer invasion parabolic system with integrable data L. Shangerganesh1 · V. N. Deiva Mani2,3 · S. Karthikeyan2 Received: 17 January 2018 / Accepted: 12 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this work, we consider the model which describes the interactions of cancer cells with healthy cells and matrix degrading enzymes. This model consists of three unknown parameters namely cancer cell density, extra cellular matrix density and matrix degradation enzymes concentration. The main goal is to study the existence of weak-renormalized solutions for the considered chemotaxis-haptotaxis cancer invasion parabolic system under the assumptions of no growth conditions and integrable data. Keywords Cancer invasion system · Weak solution · Renormalized solution Mathematics Subject Classification 35K65 · 92D25
1 Introduction Extracellular matrix (ECM) degradation by matrix degrading enzymes (MDEs) (which are produced by cancer cells) is a primary reason for the tumor pathogenesis. Enzymes that degrade ECM are abundant in tumors. Further cancer metastasis is the spread of cancer cells to other healthy cells and organs of the body where the tumor originates. Moreover the formation of new tumors through blood vessels is the single event that results in the death of most patients with cancer. In general, chemotaxis is the movement of invasive cancer cells in a direction which produces the spatial gradients in response to an extracellular chemical gradient and haptotaxis is the cancer cell invasion along a gradient of ECM bound. From a mathematical point of view, many mathematical models have been developed for the description of the
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L. Shangerganesh [email protected] V. N. Deiva Mani [email protected] S. Karthikeyan [email protected]
1
Department of Applied Sciences, National Institute of Technology Goa, Ponda, Goa, India
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Department of Mathematics, Periyar University, Salem, India
3
Department of Mathematics, Anna University Regional Campus, Coimbatore, India
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progress of cancer in different stages, for example, see [10,20,21,27,28,32,35] and also the references therein. Among them, most of the frequently discussed models of cancer invasion belong to either one of the categories of haptotaxis models [20,35] or chemotaxis-haptotaxis models, see [22]. We consider a mathematical model of chemotaxis-haptotaxis cancer invasion parabolic system with nonlinear diffusive operator as in [45]. In this model, we assume that a diffusive chemical substance which is produced by tumor cells called MDEs degrades the non-diffusive static healthy tissues, that is, ECM. Further we have assumed that cancer cells are attracted by both enzymes and healthy tissues in the sense that the cancer cells bias their movement along the gradients of the concentrations of both ECM and MDE. In otherwords, the movement of tumor cells is assumed to be governed by random motion, chemotaxis a
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