H. M. Byrne

A Two-Phase Model of Solid Tumour Growth

Many solid tumour growth models are formulated as systems of parabolic and/or hyperbolic equations. Here an alternative, two-phase theory is developed to describe solid tumour growth. Versions of earlier models are recovered when suitable limits of the new model are taken. We contend that the multiphase approach represents a more general, and natural, modelling framework for studying solid tumour growth than existing theories.

A hybrid approach to multi-scale modelling of cancer

In this paper, we review multi-scale models of solid tumour growth and discuss a middle-out framework that tracks individual cells. By focusing on the cellular dynamics of a healthy colorectal crypt and its invasion by mutant, cancerous cells, we compare a cell-centre, a cell-vertex and a continuum model of cell proliferation and movement. All models reproduce the basic features of a healthy crypt: cells proliferate near the crypt base, they migrate upwards and are sloughed off near the top. The models are used to establish conditions under which mutant cells are able to colonize the crypt either by top-down or by bottom-up invasion. While the continuum model is quicker and easier to implement, it can be difficult to relate system parameters to measurable biophysical quantities. Conversely, the greater detail inherent in the multi-scale models means that experimentally derived parameters can be incorporated and, therefore, these models offer greater scope for understanding normal and diseased crypts, for testing and identifying new therapeutic targets and for predicting their impacts.

Crypt dynamics and colorectal cancer: advances in mathematical modelling

Abstract.   Mathematical modelling forms a key component of systems biology, offering insights that complement and stimulate experimental studies. In this review, we illustrate the role of theoretical models in elucidating the mechanisms involved in normal intestinal crypt dynamics and colorectal cancer. We discuss a range of modelling approaches, including models that describe cell proliferation, migration, differentiation, crypt fission, genetic instability, APC inactivation and tumour heterogeneity. We focus on the model assumptions, limitations and applications, rather than on the technical details. We also present a new stochastic model for stem‐cell dynamics, which predicts that, on average, APC inactivation occurs more quickly in the stem‐cell pool in the absence of symmetric cell division. This suggests that natural niche succession may protect stem cells against malignant transformation in the gut. Finally, we explain how we aim to gain further understanding of the crypt system and of colorectal carcinogenesis with the aid of multiscale models that cover all levels of organization from the molecular to the whole organ.

Topological chaos in inviscid and viscous mixers

Topological chaos may be used to generate highly effective laminar mixing in a simple batch stirring device. Boyland, Aref & Stremler (2000) have computed a material stretch rate that holds in a chaotic flow, provided it has appropriate topological properties, irrespective of the details of the flow. Their theoretical approach, while widely applicable, cannot predict the size of the region in which this stretch rate is achieved. Here, we present numerical simulations to support the observation of Boyland et al. that the region of high stretch is comparable with that through which the stirring elements move during operation of the device. We describe a fast technique for computing the velocity field for either inviscid, irrotational or highly viscous flow, which enables accurate numerical simulation of dye advection. We calculate material stretch rates, and find close agreement with those of Boyland et al. ,i rrespective of whether the fluid is modelled as inviscid or viscous, even though there are significant differences between the flow fields generated in the two cases.

Continuum approximations of individual-based models for epithelial monolayers

This work examines a 1D individual-based model (IBM) for a system of tightly adherent cells, such as an epithelial monolayer. Each cell occupies a bounded region, defined by the location of its endpoints, has both elastic and viscous mechanical properties and is subject to drag generated by adhesion to the substrate. Differential-algebraic equations governing the evolution of the system are obtained from energy considerations. This IBM is then approximated by continuum models (systems of partial differential equations) in the limit of a large number of cells, N, when the cell parameters vary slowly in space or are spatially periodic (and so may be heterogeneous, with substantial variation between adjacent cells). For spatially periodic cell properties with significant cell viscosity, the relationship between the mean cell pressure and length for the continuum model is found to be history dependent. Terms involving convective derivatives, not normally included in continuum tissue models, are identified. The specific problem of the expansion of an aggregate of cells through cell growth (but without division) is considered in detail, including the long-time and slow-growth-rate limits. When the parameters of neighbouring cells vary slowly in space, the O(1/N(2)) error in the continuum approximation enables this approach to be used even for modest values of N. In the spatially periodic case, the neglected terms are found to be O(1/N). The model is also used to examine the acceleration of a wound edge observed in wound-healing assays.

A Mathematical Model of Trophoblast Invasion

In this paper we present a simple mathematical model to describe the initial phase of placental development during which trophoblast cells invade the uterine tissue as a continuous mass of cells. The key physical variables involved in this crucial stage of mammalian development are assumed to be the invading trophoblast cells, the uterine tissue, trophoblast-derived proteases that degrade the uterine tissue, and protease inhibitors that neutralise the action of the proteases. Numerical simulations presented here are in good qualitative agreement with experimental observations and show how changes in the system parameters influence the rate and degree of trophoblast invasion. In particular we suggest that chemotactic migration is a key feature of trophoblast invasion and that the rate at which proteases are produced is crucial to the successful implantation of the embryo. For example, both insufficient and excess production of the proteases may result in premature halting of the trophoblasts. Such behaviour may represent the pathological condition of failed trophoblast implantation and subsequent spontaneous abortion.

Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour

In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.

A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism

Cell migration involves different mechanisms in different cell types and tissue environments. Changes in migratory behaviour have been observed experimentally and associated with phenotypic switching in various situations, such as the migration-proliferation dichotomy of glioma cells, the epithelial-mesenchymal transition or the mesenchymal-amoeboid transition of fibrosarcoma cells in the extracellular matrix (ECM). In the present study, we develop a modelling framework to account for changes in migratory behaviour associated with phenotypic switching. We take into account the influence of the ECM on cell motion and more particularly the alignment process along the fibers. We use a mesoscopic description to model two cell populations with different migratory properties. We derive the corresponding continuum (macroscopic) model by appropriate rescaling, which leads to a generic reaction-diffusion system for the two cell phenotypes. We investigate phenotypic adaptation to dense and sparse environments and propose two complementary transition mechanisms. We study these mechanisms by using a combination of linear stability analysis and numerical simulations. Our investigations reveal that when the cell migratory ability is reduced by a crowded environment, a diffusive instability may appear and lead to the formation of aggregates of cells of the same phenotype. Finally, we discuss the importance of the results from a biological perspective.

Mathematical Modelling of Angiogenesis in Wound Healing: Comparison of Theory and Experiment

In this paper we present a simple mathematical model for angiogenesis in wound healing and then compare the results of theoretical predictions from computer simulations with actual experimental data. Numerical simulations of the model equations exhibit many of the characteristic features of wound healing in soft tissue. For example, the steady propagation of the wound healing unit through the wound space, the development of a dense band of capillaries near the leading edge of the unit, and the elevated vessel density associated with newly healed wounds, prior to vascular remodelling, are all discernible from the simulations. The qualitative accuracy of the initial model is assessed by comparing the numerical results with independent clinical measurements that show how the surface area of a range of wounds changes over time. The model is subsequently modified to include the effect of vascular remodelling and its impact on the spatio-temporal structure of the vascular network investigated. Predictions are made concerning the effect that changes in physical parameters have on the healing process and also regarding the manner in which remodelling is initiated.

Continuum Modelling of In Vitro Tissue Engineering: A Review

By providing replacements for damaged tissues and organs, in vitro tissue engineering has the potential to become a viable alternative to donor-provided organ transplant, which is increasingly hampered by a shortage of available tissue. The complexity of the myriad biophysical and biochemical processes that together regulate tissue growth renders almost impossible understanding by experimental investigation alone. Mathematical modelling applied to tissue engineering represents a powerful tool with which to investigate how the different underlying processes interact to produce functional tissues for implantation. The aim of this review is to demonstrate how a combination of mathematical modelling, analysis and in silico computation, undertaken in collaboration with experimental studies, may lead to significant advances in our understanding of the fundamental processes that regulate biological tissue growth and the optimal design of in vitro methods for generating replacement tissues that are fully functional. With this in mind, we review the state-of-the-art in theoretical research in the field of in vitro tissue engineering, concentrating on continuum modelling of cell culture in bioreactor systems and with particular emphasis on the generation of new tissues from cells seeded on porous scaffolds. We highlight the advantages and limitations of different mathematical modelling approaches that can be used to study aspects of cell population growth. We also discuss future mathematical and computational challenges and interesting open questions.