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On the avascular ellipsoidal tumour growth model within a nutritive environment

Part of: Parabolic equations and systems Mathematical biology in general Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  18 September 2018

GEORGE FRAGOYIANNIS ,
FOTEINI KARIOTOU  and
PANAYIOTIS VAFEAS
GEORGE FRAGOYIANNIS
Affiliation:
Department of Chemical Engineering, University of Patras, Patras, Greece email: [email protected]; [email protected]
FOTEINI KARIOTOU
Affiliation:
School of Science and Technology, Hellenic Open University, Patras, Greece email: [email protected]
PANAYIOTIS VAFEAS*
Affiliation:
Department of Chemical Engineering, University of Patras, Patras, Greece email: [email protected]; [email protected]
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Abstract

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

Keywords

Mathematical modellingboundary value problemsavascular tumour growthellipsoidal geometry

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35C10: Series solutions 35K57: Reaction-diffusion equations 92B99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 111 - 142
Copyright
© Cambridge University Press 2018 

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