Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications

Resorting to a multiphase modelling framework, tumours are described here as a mixture of tumour and host cells within a porous structure constituted by a remodelling extracellular matrix (ECM), which is wet by a physiological extracellular fluid. The model presented in this article focuses mainly on the description of mechanical interactions of the growing tumour with the host tissue, their influence on tumour growth, and the attachment/detachment mechanisms between cells and ECM. Starting from some recent experimental evidences, we propose to describe the interaction forces involving the extracellular matrix via some concepts coming from viscoplasticity. We then apply the model to the description of the growth of tumour cords and the formation of fibrosis

Initial/boundary-value problems of tumor growth within a host tissue

This paper concerns multiphase models of tumor growth in interaction with a surrounding tissue, taking into account also the interplay with diffusible nutrients feeding the cells. Models specialize in nonlinear systems of possibly degenerate parabolic equations, which include phenomenological terms related to specific cell functions. The paper discusses general modeling guidelines for such terms, as well as for initial and boundary conditions, aiming at both biological consistency and mathematical robustness of the resulting problems. Particularly, it addresses some qualitative properties such as a priori nonnegativity, boundedness, and uniqueness of the solutions. Existence of the solutions is studied in the one-dimensional time-independent case.Comment: 30 pages, 5 figure

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size & boundary condition

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour

Slip flow through channels with varying elliptic cross section

The low Reynolds number fluid flow through a channel with varying elliptic cross section is studied, with the inclusion of slip on the fluid–solid interface. An analytical expression for the flux through the channel is obtained, assuming that the slip length is small in comparison to the width of the channel, and a simplified expression for the flux in the limit of small eccentricity is also derived. Numerical results show that there is excellent agreement between the analytical expressions and the finite element solution for three examples presented in this paper. The expression for the flux is reformulated in terms of the semi-major and semi-minor radii, and the area and circumference, in order to estimate the flow through irregular channels. It is shown that the resulting expressions are extensions to existing estimation methods, the ‘Aissen approximation’ and ‘hydraulic radius’ methods, to include the effect of slip flow through channels with non-uniform cross section. Numerical results show that there is good agreement between the suggested flux estimation methods and the finite element solution for a pore throat extracted from a Berea sandstone. Furthermore, the suggested methods significantly outperform two common methods used to estimate the flux through an irregular channel, the ‘volume matching’ and ‘minimum radius’ methods

A dynamic network model for the action of low salinity on two-phase flow

Experimental evidence shows that decreasing the salinity of the injection water during the oil recovery process can lead to an increase in the amount of oil recovered. While the ion-exchange reactions which cause this effect are well understood in an industrial setting, there is a limited understanding of how to quantitatively describe the macroscale low salinity effect in terms of the microscale mechanisms. In this paper, we derive a dynamic network model for the salinity-dependent two-phase flow of oil and water through a porous medium in which the salinity of the water affects the thickness of the thin water layer separating the oil phase from the solid surface through the multicomponent ionic exchange mechanism, which results in a salinity-dependent slip condition on the effective oil-solid interface. We solve the network model numerically for a drainage stage followed by waterflood stage on a 30 × 30 network with random pore and throat radii distributions, and present results averaged over multiple simulations. Low-salinity waterflooding is compared with high-salinity waterflooding in both secondary and tertiary mode. Our model is able to reproduce the low salinity effect observed experimentally, in which the amount of oil produced increases as the salinity of the injection brine decreases

Surface-tension- and injection-driven spreading of a thin viscous film

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form t k , we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of k. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these

Slip flow through channels with varying elliptic cross section

The low Reynolds number fluid flow through a channel with varying elliptic cross section is studied, with the inclusion of slip on the fluid–solid interface. An analytical expression for the flux through the channel is obtained, assuming that the slip length is small in comparison to the width of the channel, and a simplified expression for the flux in the limit of small eccentricity is also derived. Numerical results show that there is excellent agreement between the analytical expressions and the finite element solution for three examples presented in this paper. The expression for the flux is reformulated in terms of the semi-major and semi-minor radii, and the area and circumference, in order to estimate the flow through irregular channels. It is shown that the resulting expressions are extensions to existing estimation methods, the ‘Aissen approximation’ and ‘hydraulic radius’ methods, to include the effect of slip flow through channels with non-uniform cross section. Numerical results show that there is good agreement between the suggested flux estimation methods and the finite element solution for a pore throat extracted from a Berea sandstone. Furthermore, the suggested methods significantly outperform two common methods used to estimate the flux through an irregular channel, the ‘volume matching’ and ‘minimum radius’ methods

The effect of ions on the motion of an oil slug through a charged capillary

Numerous experimental studies have documented that injecting low salinity water into an oil reservoir can increase the amount of oil recovered. However, due to the complexity of the chemical interactions involved in this process, there has been much debate over the dominant mechanism causing this effect. In order to further understand one proposed mechanism, multicomponent ionic exchange, we study the motion of an oil slug through a clay pore throat filled with saline water. The pore throat is modelled as a capillary tube connecting two bulk regions of water. We assume that the surfaces of the oil and the capillary are negatively charged and that, due to repulsion between these surfaces, the oil slug is separated from the capillary surface by a thin film of water. Ion interactions at the oil-water and clay-water interfaces are modelled using the law of mass action. By using lubrication theory to describe the thin-film ow in the water layer separating the oil from the clay surface, and the macroscopic ow through the capillary, we derive expressions for the thickness of the wetting film, and the velocity of the oil slug, given a pressure difference across the ends of the capillary. Numerical results show that the thickness of the water layer and the velocity of the oil slug increase as the salinity of the water is reduced, suggesting that this mechanism contributes to the low salinity effect. An analytical solution is presented in the limit in which the applied pressure is small

Surface-tension- and injection-driven spreading of a thin viscous film

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form t k , we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of k. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these