Mathematical biomedicine and modeling avascular tumor growth

In this chapter we review existing continuum models of avascular tumor growth, explaining howthey are inter related and the biophysical insight that they provide. The models range in complexity and include one-dimensional studies of radiallysymmetric growth, and two-dimensional models of tumor invasion in which the tumor is assumed to comprise a single population of cells. We also present more detailed, multiphase models that allow for tumor heterogeneity. The chapter concludes with a summary of the different continuum approaches and a discussion of the theoretical challenges that lie ahead

Turbulence damping as a measure of the flow dimensionality

The dimensionality of turbulence in fluid layers determines their properties. We study electromagnetically driven flows in finite depth fluid layers and show that eddy viscosity, which appears as a result of three-dimensional motions, leads to increased bottom damping. The anomaly coefficient, which characterizes the deviation of damping from the one derived using a quasi-two-dimensional model, can be used as a measure of the flow dimensionality. Experiments in turbulent layers show that when the anomaly coefficient becomes high, the turbulent inverse energy cascade is suppressed. In the opposite limit turbulence can self-organize into a coherent flow.Comment: 4 pages, 4 figure

Multiphase modelling of vascular tumour growth in two spatial dimensions

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 are investigated

Calculus from the past: multiple delay systems arising in cancer cell modelling

Non-local calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of non-local problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including the model of the cell cycle developed in Simms K, Bean N, & Koeber A. [10]. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents

Modeling Stem/Progenitor Cell-Induced Neovascularization and\ud Oxygenation around Solid Implants

Tissue engineering constructs and other solid implants with biomedical applications, such as drug delivery devices or bioartificial organs, need oxygen (O2) to function properly. To understand better the vascular integration of such devices, we recently developed a novel model sensor containing O2-sensitive crystals, consisting of a polymeric capsule limited by a nano-porous filter. The sensor was implanted in mice with hydrogel alone (control) or hydrogel embedded with mouse CD117/c-kit+ bone marrow progenitor cells (BMPC) in order to stimulate peri-implant neovascularization. The sensor provided local partial O2 pressure (pO2) using non-invasive electron paramagnetic resonance (EPR) signal measurements. A consistently higher level of per-implant oxygenation was observed in the cell-treatment case as compared to the control over a 10-week period. In order to provide a mechanistic explanation of these experimental observations, we present in this paper a mathematical model, formulated as a system of coupled partial differential equations, that simulates peri-implant vascularization. In the control case, vascularization is considered to be the result of a Foreign Body Reaction (FBR) while in the cell-treatment case, adipogenesis in response to paracrine stimuli produced by the stem cells is assumed to induce neovascularization. The model is validated by fitting numerical predictions of local pO2 to measurements from the implanted sensor. The model is then used to investigate further the potential for using stem cell treatment to enhance the vascular integration of biomedical implants. We thus demonstrate how mathematical modeling combined with experimentation can be used to infer how vasculature develops around biomedical implants in control and stem celltreated cases

Towards whole-organ modelling of tumour growth

Multiscale approaches to modelling biological phenomena are growing rapidly. We present here some recent results on the formulation of a theoretical framework which can be developed into a fully integrative model for cancer growth. The model takes account of vascular adaptation and cell-cycle dynamics. We explore the effects of spatial inhomogeneity induced by the blood flow through the vascular network and of the possible effects of p27 on the cell cycle. We show how the model may be used to investigate the efficiency of drug-delivery protocols

A design principle for vascular beds: the effects of complex blood rheology

We propose a design principle that extends Murray's original optimization principle for vascular architecture to account for complex blood rheology. Minimization of an energy dissipation function enables us to determine how rheology affects the morphology of simple branching networks. The behavior of various physical quantities associated with the networks, such as the wall shear stress and the flow velocity, is also determined. Our results are shown to be qualitatively and quantitatively compatible with independent experimental observations and simulations