Eur. Phys. J. Plus (2018) 133: 85
https://doi.org/10.1140/epjp/i2018-11932-1

Regular Article

Tumour model with intrusive morphology, progressive phenotypical heterogeneity and memory

¹ Institute for Groundwater Studies, Faculty of Natural and Agricultural Science, University of Free State, 9300, Bloemfontein, South Africa
² Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), 11566, Riyadh, Saudi Arabia

^* e-mail: AtanganaA@ufs.ac.za

Received: 2 January 2018
Accepted: 24 January 2018
Published online: 1 March 2018

Abstract

The model of a tumour, taking into account invasive morphology, progressive phenotypical heterogeneity and also memory, is developed and analyzed in this paper. Three models are investigated: first we consider the model describing the proliferation concentrates in proximity of tumour boundaries, in which the oxygen levels are pronounced. Then we consider the model where the oxygen around the tumour is considered to be unchanged by the vascular system. Finally, we investigate the model of growth of tumours using the concept of non-local operators with the Mittag-Leffler kernel. We provide the numerical solution using the extended 3/8 Simpson method for the new trends of fractional integration for the proliferation concentrates in the proximity of the tumour model. Then we provide the exact solutions of the Gompertz model with three different fractional differentiations involving power law, exponential decay law and the Mittag-Leffler law.