Multiphase modelling of vascular tumour growth in two spatial dimensions

Abstract

In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters is investigated.

A Mathematical Model of the Growth of Uterine Myomas

“…We proposed a simple linear stress-strain relationship, which, on differentiation with respect to time, was made applicable to an evolution problem with accumulated stress. Coupled to this we modelled fluid flow in the blood and extracellular space using standard approaches for modelling flow in a porous media; in particularly, we treated separately arterial and venous flow, which distinguishes this model to those studied in [4,16]. Since the growth of myomas requires the conversion of extracellular fluid into cellular material, the fluid transport within myomas plays a major role in their growth; the current model considers two fluid sources, which we termed "vascular-influx" and "passive-influx".…”

“…The separate treatment of arteries and veins distinguishes the current model to that of related vascular tumour models, which consider only a single vasculature phase [4,16]; the benefits of doing this will be made clear after equation (2.5). There being no other material in the regions means that the sum of these fractions is equal to unity, i.e.…”

“…see, for example, [1,16]. For the fluid phases, we adopt the usual assumptions for (Darcy) flow in a porous media, whereby the stress tensors reduce to 9) and the corresponding body forces account for the drag between the solid-fluid phases, hence…”

“…[6,21,22]. Byrne and coworkers [4,16] used multi-phase, continuum models to study vascular tumour growth, in which all phases behave as fluids, except for an additional pressure term for the cellular phase that forces cells to move in order to alleviate stress under high compression. However, these continuum models are aimed at describing soft tissue solid tumours and the assumption that all phases are fluid-like are not suited for the fibrous material in myomas.…”

A multi-scale analysis of drug transport and response for a multi-phase tumour model

“…This transport is most appropriately modelled on the length scale associated with the full extent of the tumour tissue; however, the transport properties of the drug and growth dynamics of the tumour are highly dependent on the evolving microstructural properties of the tumour and its micro-vasculature , Perfahl et al (2011)). In this work we extend the multiscale analyses of O'Dea et al (2014), , and by employing a multiphase model of tumour growth, of the type developed in Breward et al (2002) and , and exploited in the context of vascular tumour growth in Breward et al (2004) and Hubbard and Byrne (2013), in which we explicitly incorporate microscale dynamics into a system of homogenized partial differential equations (PDEs) for tumour growth, response, and transport.…”