Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation

It is well established that gliomas are heterogeneous (polyclonal), that the degree of heterogeneity always rises with grade. It is believed that the more cancerous cells have a greater propensity to mutate, increasing heterogeneity. Consequently, it is anticipated that a tumour will have a variety of cell types. In this work, a fractional diffusion model for tumour growth where two cell populations are assumed, which could have different diffusivity and proliferation rates, is studied and analysed. The coupled system is solved analytically via the zeroth order finite Hankel transform employing the fractional derivatives with exponential and Mittag-Leffler kernels, respectively and the obtained solutions simulated using MATLAB. Important highlights of the simulations include: (i.) using the Caputo fractional derivative, and considering the scenario when the rate of loss of cell population u(r,t) is α=0.5, keeping the other values of the parameters ν=1.2,β=0.1, it is observed that, near the centre of the tumour where biopsy is to be carried out, the tumour cell concentrations u(r,t) dominates v(r,t) when t=0.5 while tumour cell population v(r,t) dominates u(r,t) at time t=2.0; (ii.) with the Atangana–Baleanu derivative, and considering the same scenario it is observed that, near the centre of the tumour, the cell concentration v(r,t) dominates u(r,t) when t=0.5 while the dominance is much higher at time t=2.0. Thus, it is concluded that, the nature of kernel in the fractional operator could indeed alter the dominance between the tumour cell concentrations.