Mark A. J. Chaplain

Growth of necrotic tumors in the presence and absence of inhibitors

In this article a model for the evolution of a spherically symmetric, nonnecrotic tumor is presented. The effects of nutrients and inhibitors on the existence and stability of time-independent solutions are studied. With a single nutrient and no inhibitors present, the trivial solution, which corresponds to a state in which no tumor is present, persists for all parameter values, whereas the nontrivial solution, which corresponds to a tumor of finite size, exists for only a prescribed range of parameters, which corresponds to a balance between cell proliferation and cell death. Stability analysis, based on a two-timing method, suggests that, where it exists, the nontrivial solution is stable and the trivial solution unstable. Otherwise, the trivial solution is stable. Modification to these characteristic states brought about by the presence of different types of inhibitors are also investigated and shown to have significant effect. Implications of the model for the treatment of cancer are also discussed.

Multiscale modelling and nonlinear simulation of vascular tumour growth.

In this article, we present a new multiscale mathematical model for solid tumour growth which couples an improved model of tumour invasion with a model of tumour-induced angiogenesis. We perform nonlinear simulations of the ulti-scale model that demonstrate the importance of the coupling between the development and remodeling of the vascular network, the blood flow through the network and the tumour progression. Consistent with clinical observations, the hydrostatic stress generated by tumour cell proliferation shuts down large portions of the vascular network dramatically affecting the flow, the subsequent network remodeling, the delivery of nutrients to the tumour and the subsequent tumour progression. In addition, extracellular matrix degradation by tumour cells is seen to have a dramatic affect on both the development of the vascular network and the growth response of the tumour. In particular, the newly developing vessels tend to encapsulate, rather than penetrate, the tumour and are thus less effective in delivering nutrients.

Mathematical Modelling of Tumour Invasion and Metastasis

In this paper we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix) and matrix-degradative enzymes associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of individual cells and hence it is possible to examine the implications of metastatic spread. Finally, the results of the models are compared with actual clinical observations and the implications of the model for improved surgical treatment of patients are considered.

Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development

The growth and development of solid tumours occurs in two distinct stages-the avascular growth phase and the vascular growth phase. During the former growth phase the tumour remains in a diffusion-limited, dormant state of a few millimetres in diameter (cf. multicell spheroids, carcinoma in situ) while during the latter growth phase, invasion and metastasis may take place. In order to accomplish the transition from avascular to vascular growth, solid tumours may secrete diffusible substances known as tumour angiogenesis factors (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events. Capillary sprouts are formed which migrate towards the tumour, eventually penetrating it and permitting vascular growth to take place. This paper will present several mathematical models which deal with the various stages of growth and development of solid tumours.

Modeling the Influence of the E-Cadherin-β-Catenin Pathway in Cancer Cell Invasion: A Multiscale Approach

In this article, we show, using a mathematical multiscale model, how cell adhesion may be regulated by interactions between E-cadherin and beta-catenin and how the control of cell adhesion may be related to cell migration, to the epithelial-mesenchymal transition and to invasion in populations of eukaryotic cells. E-cadherin mediates cell-cell adhesion and plays a critical role in the formation and maintenance of junctional contacts between cells. Loss of E-cadherin-mediated adhesion is a key feature of the epithelial-mesenchymal transition. beta-catenin is an intracellular protein associated with the actin cytoskeleton of a cell. E-cadherins bind to beta-catenin to form a complex which can interact both with neighboring cells to form bonds, and with the cytoskeleton of the cell. When cells detach from one another, beta-catenin is released into the cytoplasm, targeted for degradation, and downregulated. In this process there are multiple protein-complexes involved which interact with beta-catenin and E-cadherin. Within a mathematical individual-based multiscale model, we are able to explain experimentally observed patterns solely by a variation of cell-cell adhesive interactions. Implications for cell migration and cancer invasion are also discussed.

Modelling the role of cell-cell adhesion in the growth and development of carcinomas

In this paper, a mathematical model is presented to describe the evolution of an avascular solid tumour in response to an externally-supplied nutrient. The growth of the tumour depends on the balance between expansive forces caused by cell proliferation and cell-cell adhesion forces which exist to maintain the tumours compactness. Cell-cell adhesion is incorporated into the model using the Gibbs-Thomson relation which relates the change in nutrient concentration across the tumour boundary to the local curvature, this energy being used to preserve the cell-cell adhesion forces. Our analysis focuses on the existence and uniqueness of steady, radially-symmetric solutions to the model, and also their stability to time-dependent and asymmetric perturbations. In particular, our analysis suggests that if the energy needed to preserve the bonds of adhesion is large then the radially-symmetric configuration is stable with respect to all asymmetric perturbations, and the tumour maintains a radially-symmetric structure-this corresponds to the growth of a benign tumour. As the energy needed to maintain the tumours compactness diminishes so the number of modes to which the underlying radially-symmetric solution is unstable increases-this corresponds to the invasive growth of a carcinoma. The strength of the cell-cell bonds of adhesion may at some stage provide clinicians with a useful index of the invasive potential of a tumour.

MATHEMATICAL MODELLING OF CANCER CELL INVASION OF TISSUE: THE ROLE OF THE UROKINASE PLASMINOGEN ACTIVATION SYSTEM

The growth of solid tumours proceeds through two distinct phases: the avascular and the vascular phase. It is during the latter stage that the insidious process of cancer invasion of peritumoral tissue can and does take place. Vascular tumours grow rapidly allowing the cancer cells to establish a new colony in distant organs, a process that is known as metastasis. The progression from a single, primary tumour to multiple tumours in distant sites throughout the body is known as the metastatic cascade. This is a multistep process that first involves the over-expression by the cancer cells of proteolytic enzyme activity, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs). uPA itself initiates the activation of an enzymatic cascade that primarily involves the activation of plasminogen and subsequently its matrix degrading protein plasmin. Degradation of the matrix then enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body. In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix) which focuses on the role of the plasminogen activation system. The model consists of a system of reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, urokinase plasminogen activator (uPA), uPA inhibitors, plasmin and the host tissue. The focus of the modelling is on the spatio-temporal dynamics of the uPA system and how this influences the migratory properties of the cancer cells through random motility, chemotaxis and haptotaxis. The results obtained from numerical computations carried out on the model equations produce rich, dynamic heterogeneous spatio-temporal solutions and demonstrate the ability of rather simple models to produce complicated dynamics, all of which are associated with tumour heterogeneity and cancer cell progression and invasion.

Paradoxical Dependencies of Tumor Dormancy and Progression on Basic Cell Kinetics

Even after a tumor is established, it can early on enter a state of dormancy marked by balanced cell proliferation and cell death. Disturbances to this equilibrium may affect cancer risk, as they may cause the eventual lifetime clinical presentation of a tumor that might otherwise have remained asymptomatic. Previously, we showed that cell death, proliferation, and migration can play a role in shifting this dynamic, making the understanding of their combined influence on tumor development essential. We developed an individual cell-based computer model of the interaction of cancer stem cells and their nonstem progeny to study early tumor dynamics. Simulations of tumor growth show that three basic components of tumor growth--cell proliferation, migration, and death--combine in unexpected ways to control tumor progression and, thus, clinical cancer risk. We show that increased proliferation capacity in nonstem tumor cells and limited cell migration overall lead to space constraints that inhibit proliferation and tumor growth. By contrast, increasing the rate of cell death produces the expected tumor size reduction in the short term, but results ultimately in paradoxical accelerated long-term growth owing to the liberation of cancer stem cells and formation of self-metastases.

Free boundary value problems associated with the growth and development of multicellular spheroids

In this paper a generalized model for the growth of avascular tumours is presented. The formulation leads naturally to the incorporation of free boundaries which define the outer tumour surface explicitly and various inner surfaces implicitly. A combination of numerical simulations, asymptotic analysis and perturbation techniques is used to study the model and yields results, which agree well with experimentally-observed phenomena.

Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity

Solid tumours grow through two distinct phases: the avascular and the vascular phase. During the avascular growth phase, the size of the solid tumour is restricted largely by a diffusion-limited nutrient supply and the solid tumour remains localised and grows to a maximum of a few millimetres in diameter. However, during the vascular growth stage the process of cancer invasion of peritumoral tissue can and does take place. A crucial component of tissue invasion is the over-expression by the cancer cells of proteolytic enzyme activity, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs). uPA itself initiates the activation of an enzymatic cascade that primarily involves the activation of plasminogen and subsequently its matrix degrading protein plasmin. Degradation of the matrix then enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body. In this paper we consider a relatively simple mathematical model of cancer cell invasion of tissue (extracellular matrix) which focuses on the role of a generic matrix degrading enzyme such as uPA. The model consists of a system of reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the host tissue. The results obtained from numerical computations carried out on the model equations produce dynamic, heterogeneous spatio-temporal solutions and demonstrate the ability of a rather simple model to produce complicated dynamics, all of which are associated with tumour heterogeneity and cancer cell progression and invasion.