Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth

Abstract

In this paper we examine spatio-temporal pattern formation in reaction-diffusion systems on the surface of the unit sphere in 3D. We first generalise the usual linear stability analysis for a two-chemical system to this geometrical context. Noting the limitations of this approach (in terms of rigorous prediction of spatially heterogeneous steady-states) leads us to develop, as an alternative, a novel numerical method which can be applied to systems of any dimension with any reaction kinetics. This numerical method is based on the method of lines with spherical harmonics and uses fast Fourier transforms to expedite the computation of the reaction kinetics. Numerical experiments show that this method efficiently computes the evolution of spatial patterns and yields numerical results which coincide with those predicted by linear stability analysis when the latter is known. Using these tools, we then investigate the rĵle that pre-pattern (Turing) theory may play in the growth and development of solid tumours. The theoretical steady-state distributions of two chemicals (one a growth activating factor, the other a growth inhibitory factor) are compared with the experimentally and clinically observed spatial heterogeneity of cancer cells in small, solid spherical tumours such as multicell spheroids and carcinomas. Moreover, we suggest a number of chemicals which are known to be produced by tumour cells (autocrine growth factors), and are also known to interact with one another, as possible growth promoting and growth inhibiting factors respectively. In order to connect more concretely the numerical method to this application, we compute spatially heterogeneous patterns on the surface of a growing spherical tumour, modelled as a moving-boundary problem. The numerical results strongly support the theoretical expectations in this case. Finally in an appendix we give a brief analysis of the numerical method.

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

“…In contrast to the case of a fixed domain, numerous different stripe and spot patterns occur depending on perturbations of the initial conditions for the model. Further examples of studies of RDEs illustrating the role of domain growth can be found in papers by Varea et al (1999), Chaplain et al (2001), Liaw et al (2001), Painter et al (1999), Crampin et al ( , 1999, Oster and Bressloff (2006), Madzvamuse et al (2003Madzvamuse et al ( , 2005, Madzvamuse (2005) and for a review see Plaza et al (2004).…”

Nonlinear simulation of the effect of microenvironment on tumor growth

“…Greenspan (1976), Byrne and Chaplain (1996a), Byrne and Chaplain (1996b)) used ordinary differential equations (ODE) to model cancer as a homogeneous population, as well as partial differential equation (PDE) models restricted to spherical geometries. Linear and weakly nonlinear analyses have been performed to assess the stability of spherical tumors to asymmetric perturbations (e.g., Chaplain et al (2001), Byrne and Matthews (2002), Cristini et al (2003), and Li et al (2006), and discussed in the reviews by Araujo and McElwain (2004a) and Byrne et al (2006)) as a means to characterize the degree of aggression. Various interactions of the tumor with the microenvironment, such as stress-induced limitations of tumor growth, have also been studied in this context (e.g., Jones et al (2000), Mollica (2002, 2004), Roose et al (2003), McElwain (2004b, 2005), and Ambrosi and Guana (2006)).…”

Nonlinear Modeling and Simulation of Tumor Growth

“…Early work including [90,39,40] used ordinary differential equations to model cancer as a homogeneous population, as well as partial differential equation models restricted to spherical geometries. Linear and weakly nonlinear analyses have been performed to assess the stability of spherical tumors to asymmetric perturbations [49,42,59,127,16,37] in order to characterize the degree of aggression. Various interactions of a tumor with the microenvironment, such as stress-induced limitations to growth, have also been studied [108,6,7,176,17,18,5].…”