Modelling solid tumour growth using the theory of mixtures

Abstract

In this paper the theory of mixtures is used to develop a two-phase model of an avascular tumour, which comprises a solid, cellular, phase and a liquid phase. Mass and momentum balances which are used to derive the governing equations are supplemented by constitutive laws that distinguish the two phases and enable the stresses within the tumour to be calculated. Novel features of the model include the dependence of the cell proliferation rate on the cellular stress and the incorporation of mass exchange between the two phases. A combination of numerical and analytical techniques is used to investigate the sensitivity of equilibrium tumour configurations to changes in the model parameters. Variation of parameters such as the maximum cell proliferation rate and the rate of natural cell death yield results which are consistent with analyses performed on simpler tumour growth models and indicate that the two-phase formulation is a natural extension of the earlier models. New predictions relate to the impact of mechanical effects on the tumour's equilibrium size which decreases under increasing stress and/or external loading. In particular, as a parameter which measures the reduction in cell proliferation due to cell stress is increased a critical value is reached, above which the tumour is eliminated.

Multiphase flow models of biogels from crawling cells to bacterial biofilms

“…A standard assumption is that there is a solvation stress that is the difference between the hydrostatic pressure and the interphase pressure (Lubkin and Jackson, 2002;Breward et al, 2002;Byrne and Preziosi, 2003;Jackson and Byrne, 2002). There is also a contractile stress that acts on the cell phase and is the difference between the hydrostatic pressure and the cell phase pressure.…”

Cell adhesion mechanisms and stress relaxation in the mechanics of tumours

“…In fact, some attempts were done in the literature to describe growing tumours alternatively as elastic solids McElwain 2004, 2005a,b;Jones et al 2000;Roose et al 2003), or fluids (Ambrosi and Preziosi 2002;Breward et al 2002Breward et al , 2003Byrne et al 2003;Byrne and Preziosi 2004;Chaplain et al 2006;Chen et al 2001;Franks et al 2003a,b;Franks and King 2003), including stress relaxation while avoiding the split of the deformation gradient into growth and deformation. Usual assumptions in the former case are incompressibility of the cell component and a linear elastic behaviour.…”

“…The mechanical stress-strain constitutive equation. Most models use fluid-like constitutive equations (Ambrosi and Preziosi 2002;Breward et al 2002Breward et al , 2003Byrne et al 2003;Byrne and Preziosi 2004;Chen et al 2001;Franks et al 2003a,b;Franks and King 2003), others adopt a linear elastic one McElwain 2004, 2005a,b;Jones et al 2000;Roose et al 2003). Tumours are therefore sometimes considered as an aggregate of soft balloons that roll on each other in a continuum limit (visco-elastic fluid) or take inspiration from biological soft tissues models (non-linear elasticity).…”

Reoxygenation and Split-Dose Response to Radiation in a Tumour Model with Krogh-Type Vascular Geometry

“…However, the hypothesis that total cell volume fraction is constant is an oversimplification in the transient that follows a treatment, when an increased volume fraction of extracellular fluid has been observed (Zhao et al, 1996). This hypothesis might be relaxed by adopting the two-fluid model of tumour tissue (Byrne and Preziosi, 2003).…”