A mathematical analysis of a model for tumour angiogenesis

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.     

Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions

To ensure its sustained growth, a tumour may secrete chemical compounds which cause neighbouring capillaries to form sprouts which then migrate towards it, furnishing the tumour with an increased supply of nutrients. In this paper a mathematical model is presented which describes the migration of capillary sprouts in response to a chemoattractant field set up by a tumour-released angiogenic factor, sometimes termed a tumour angiogenesis factor (TAF). The resulting model admits travelling wave solutions which correspond either to successful neovascularization of the tumour or failure of the tumour to secure a vascular network, and which exhibit many of the characteristic features of angiogenesis. For example, the increasing speed of the vascular front, and the evolution of an increasingly developed vascular network behind the leading capillary tip front (the brush-border effect) are both discernible from the numerical simulations. Through the development and analysis of a simplified caricature model, valuable insight is gained into how the balance between chemotaxis, tip proliferation and tip death affects the tumour's ability to induce a vascular response from neighbouring blood vessels. In particular, it is possible to define the success of angiogenesis in terms of known parameters, thereby providing a potential framework for assessing the viability of tumour neovascularization in terms of measurable quantities.     

Angiogenesis and vascular remodelling in normal and cancerous tissues

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.     

A cellular automaton model for tumour growth in inhomogeneous environment

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.     

Modelling the Role of Angiogenesis and Vasculogenesis in Solid Tumour Growth

Recent experimental evidence suggests that vasculogenesis may play an important role in tumour vascularisation. While angiogenesis involves the proliferation and migration of endothelial cells (ECs) in pre-existing vessels, vasculogenesis involves the mobilisation of bone-marrow-derived endothelial progenitor cells (EPCs) into the bloodstream. Once blood-borne, EPCs home in on the tumour site, where subsequently they may differentiate into ECs and form vascular structures. In this paper, we develop a mathematical model, formulated as a system of nonlinear ordinary differential equations (ODEs), which describes vascular tumour growth with both angiogenesis and vasculogenesis contributing to vessel formation. Submodels describing exclusively angiogenic and exclusively vasculogenic tumours are shown to exhibit similar growth dynamics. In each case, there are three possible scenarios: the tumour remains in an avascular steady state, the tumour evolves to a vascular equilibrium, or unbounded vascular growth occurs. Analysis of the full model reveals that these three behaviours persist when angiogenesis and vasculogenesis act simultaneously. However, when both vascularisation mechanisms are active, the tumour growth rate may increase, causing the tumour to evolve to a larger equilibrium size or to expand uncontrollably. Alternatively, the growth rate may be left unaffected, which occurs if either vascularisation process alone is able to keep pace with the demands of the growing tumour. To clarify further the effects of vasculogenesis, the full model is also used to compare possible treatment strategies, including chemotherapy and antiangiogenic therapies aimed at suppressing vascularisation. This investigation highlights how, dependent on model parameter values, targeting both ECs and EPCs may be necessary in order to effectively reduce tumour vasculature and inhibit tumour growth.     

The mechanism of regression of solid SL2 lymphosarcoma after local IL-2 therapy

Treatment of cancer by the administration of interleukin-2 (IL-2) at the tumour site is a very effective approach. The mechanism of this tumour regression is not clear, although it is generally assumed that it involves an IL-2-stimulated immune reaction. There are, however, no immune parameters that consistently correlate with the therapeutic effect. We have studied the histopathological events in a subcutaneously (s.c.) growing SL2 lymphosarcoma (transplantation of tumour cells at day 0) treated with peritumoural IL-2 injections at days 10-14. Most IL-2-treated tumours had already begun to regress from day 12 onwards, showing that local IL-2 therapy was also effective in the present study. The immediate reaction after local IL-2 administration is vascular leakage from the surrounding circulation, causing oedema within the tumour and in a broad zone surrounding it. The presence of oedema is always accompanied by markedly increased tumour necrosis. After a few days extensive angiogenesis occurs at the border between the oedematous area and the healthy connective tissue. Leucocytes, mainly macrophages, migrate via the newly formed blood vessels to gain access to the necrotising tumour site, where they form a granuloma. These macrophages phagocytose the dead tumour material. During growth, the SL-2 tumours infiltrate the surrounding tissue. The infiltrating tumour strands are apparently attacked by macrophages, as the tumour cells in close proximity to the latter are progressively destroyed. Therefore, the body of the tumour and the infiltrating tumour strands are attacked in different ways. The primary effect of IL-2 administration at the tumour site is vascular leakage that causes oedema in and around the tumour. This is followed by extensive angiogenesis, with the resulting migration of white cells from the circulation, which form a granuloma around the tumour. Both the oedema and the granuloma cause tumour regression.     

Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth,Pre-Existing Vessel Co-Option,Angiogenesis and Blood Perfusion

We propose a coupled mathematical modelling system to investigate glioblastoma growth in response to dynamic changes in chemical and haemodynamic microenvironments caused by pre-existing vessel co-option, remodelling, collapse and angiogenesis. A typical tree-like architecture network with different orders for vessel diameter is designed to model pre-existing vasculature in host tissue. The chemical substances including oxygen, vascular endothelial growth factor, extra-cellular matrix and matrix degradation enzymes are calculated based on the haemodynamic environment which is obtained by coupled modelling of intravascular blood flow with interstitial fluid flow. The haemodynamic changes, including vessel diameter and permeability, are introduced to reflect a series of pathological characteristics of abnormal tumour vessels including vessel dilation, leakage, angiogenesis, regression and collapse. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. The simulation focuses on the avascular phase of tumour development and stops at an early phase of angiogenesis. The model is able to demonstrate the main features of glioblastoma growth in this phase such as the formation of pseudopalisades, cell migration along the host vessels, the pre-existing vasculature co-option, angiogenesis and remodelling. The model also enables us to examine the influence of initial conditions and local environment on the early phase of glioblastoma growth.     

Modelling the Cell Cycle and Cell Movement in Multicellular Tumour Spheroids

This paper analyses a recent mathematical model of avascular tumour spheroid growth which accounts for both cell cycle dynamics and chemotactic driven cell movement. The model considers cells to exist in one of two compartments: proliferating and quiescent, as well as accounting for necrosis and apoptosis. One particular focus of this paper is the behaviour created when proliferating and quiescent cells have different chemotactic responses to an extracellular nutrient supply. Two very different steady-state behaviours are identified corresponding to those cases where proliferating cells move either more quickly or more slowly than quiescent cells in response to a gradient in the extracellular nutrient supply. The case where proliferating cells move more rapidly leads to the commonly accepted spheroid structure of a thin layer of proliferating cells surrounding an inner quiescent core. In the case where proliferating cells move more slowly than quiescent cells the model predicts an interesting structure of a thin layer of quiescent cells surrounding an inner core of proliferating and quiescent cells. The sensitivity of this tumour structure to the cell cycle model parameters is also discussed. In particular variations in the steady-state size of the tumour and the types of transient behaviour are explored. The model reveals interesting transient behaviour with sharply delineated regions of proliferating and quiescent cells.     

Tumor-associated fibroblasts (part II): Functional impact on tumor tissue

The article focuses on the functional impact of tumor-associated fibroblasts (TAF) on its surrounding cells. It intends to cover the recent knowledge on TAF, the phenotype, and expression profile of which have been described in the first part of the review series (Kunz-Schughart and Knuechel, 2002). The present review is subdivided into two main chapters: (1) functional impact of TAF on tumor cells and (2) fibroblast-host cell interactions in tumor tissue. In the first paragraph of chapter (1) about the role of fibroblasts in tumor cell growth and differentiation it is revealed, how strongly cellular interaction is dependent on fibroblast and tumor cell type as well as the spatial ratio between the cells. The variation of cellular behavior depending on quantity of molecules holds also true for the group of ECM molecules, e.g. the balance between MMPs and TIMPs, which provide an interesting therapeutic target in tumor tissue. This is one of the topics addressed in the second paragraph which focuses on tumor cell dissemination. Chapter (2) addresses the relation of TAF to other intra- or peritumoral host cells. The hypoxia-related angiogenesis induction of fibroblasts via growth factor secretion (e.g. VEGF) is considered as important as the immune modulatory properties of fibroblasts on immune cells, such as monocytes/macrophages. These cellular properties can be tested under controlled conditions in three-dimensional heterologous cultures of human cells, providing the chance for systematic modification to assess therapeutic effects in an in vivo like environment.     

Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours

Objectives: Mathematical models are useful for studying vascular and avascular tumours, because these allow for more logical experimental design and provide valuable insights into the underlying mechanisms of their growth and development. The processes of avascular tumour growth and the development of capillary networks through tumour‐induced angiogenesis have already been extensively investigated, albeit separately. Despite the clinical significance of vascular tumours, few studies have combined these approaches to develop a single comprehensive growth and development model. Materials and methods: We develop a continuum‐based mathematical model of vascular tumour growth. In the model, angiogenesis is initiated through the release of angiogenic growth factors (AGFs) by cells in the hypoxic regions of the tumour. The nutrient concentration within the tumour reflects the influence of capillary growth and invasion induced by AGF. Results and conclusions: Parametric and sensitivity studies were performed to evaluate the influence of different model parameters on tumour growth and to identify the parameters with the most influence, which include the rates of proliferation, apoptosis and necrosis, as well as the diffusion of sprout tips and the size of the region affected by angiogenesis. An optimization was performed for values of the model parameters that resulted in the best agreement with published experimental data. The resulting model solution matched the experimental data with a high degree of correlation (r = 0.85).     

Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis.

The growth of new capillary blood vessels, or angiogenesis, is a prominent component of numerous physiological and pathological conditions. An understanding of the co-ordination of underlying cellular behaviors would be helpful for therapeutic manipulation of the process. A probabilistic mathematical model of angiogenesis is developed based upon specific microvessel endothelial cell (MEC) functions involved in vessel growth. The model focuses on the roles of MEC random motility and chemotaxis, to test the hypothesis that these MEC behaviors are of critical importance in determining capillary growth rate and network structure. Model predictions are computer simulations of microvessel networks, from which questions of interest are examined both qualitatively and quantitatively. Results indicate that a moderate MEC chemotactic response toward an angiogenic stimulus, similar to that measured in vitro in response to acidic fibroblast growth factor, is necessary to provide directed vascular network growth. Persistent random motility alone, with initial budding biased toward the stimulus, does not adequately provide directed network growth. A significant degree of randomness in cell migration direction, however, is required for vessel anastomosis and capillary loop formation, as simulations with an overly strong chemotactic response produce network structures largely absent of these features. The predicted vessel extension rate and network structure in the simulations are quantitatively consistent with experimental observations of angiogenesis in vivo. This suggests that the rate of vessel outgrowth is primarily determined by MEC migration rate, and consequently that quantitative in vitro migration assays might be useful tools for the prescreening of possible angiogenesis activators and inhibitors. Finally, reduction of MEC speed results in substantial inhibition of simulated angiogenesis. Together, these results predict that both random motility and chemotaxis are MEC functions critically involved in determining the rate and morphology of new microvessel network growth.     

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 multi-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.     

PDGFRbeta+ perivascular progenitor cells in tumours regulate pericyte differentiation and vascular survival

The microvasculature consists of endothelial cells and their surrounding pericytes. Few studies on the regulatory mechanisms of tumour angiogenesis have focused on pericytes. Here we report the identification of tumour-derived PDGFRbeta (+) (platelet-derived growth factor receptor beta) progenitor perivascular cells (PPCs) that have the ability to differentiate into pericytes and regulate vessel stability and vascular survival in tumours. A subset of PDGFRbeta (+) PPCs is recruited from bone marrow to perivascular sites in tumours. Specific inhibition of PDGFRbeta signalling eliminates PDGFRbeta (+) PPCs and mature pericytes around tumour vessels, leading to vascular hyperdilation and endothelial cell apoptosis in pancreatic islet tumours of transgenic Rip1Tag2 mice.     

A model of vascular compression in solid tumours

Often when a vascularized solid tumour reaches a diameter of 1–3 cm, central vascular compression occurs and the blood flow is confined to the peripheral outer layer. Here a model is presented which develops the concept that this vascular compression is due to the combined effect of the proliferation of cells inside the tumour and the pressure exerted on the tumour by the surrounding tissue. It is hypothesized that this occlusion eventually causes central necrosis.     

Role of copper in tumour angiogenesis--clinical implications.

The formation of new blood vessels is the initial step in progressive tumour development and metastasis. The first stage in tumour angiogenesis is the activation of endothelial cells. Copper ions stimulate proliferation and migration of endothelial cells. It has been shown that serum copper concentration increases as the cancer disease progresses and correlates with tumour incidence and burden. Copper ions also activate several proangiogenic factors, e.g., vascular endothelial growth factor, basic fibroblast growth factor, tumour necrosis factor alpha and interleukin 1. This review concerns a brief introduction into the basics of tumour blood vessel development as well as the regulatory mechanisms of this process. The role of copper ions in tumour angiogenesis is discussed. The new antiangiogenic therapies based on a reduction of copper levels in tumour microenvironment are reviewed.     

Increased apoptosis and angiogenesis in gastric low-grade mucosa-associated lymphoid tissue-type lymphoma by Helicobacter heilmannii infection in C57/BL6 mice

Helicobacter heilmannii has been reported to cause gastric low-grade mucosa-associated lymphoid tissue-type (MALT) lymphoma, but its precise pathophysiological mechanism remains to be clarified. We recently established a model of gastric B-cell MALT lymphoma in C57BL/6 mice by means of peroral infection of H. heilmannii primarily obtained from cynomolgus monkeys. Using this model, macroscopic, immunohistochemical, and electron microscopic observations of MALT lymphomas were carried out in order to examine the development of apoptosis and angiogenesis. Enhancement of the microvascular network and an increase in vascular endothelial growth factor-A were detected in the central region of the MALT lymphoma tissue in the infected mouse stomach, while vascular endothelial growth factor-C was detected at the margins of the MALT lymphomas. In addition, many H. heilmannii-invaded parietal cells showed caspase-3 immunoreactivity in the fundic mucosal tissue surrounding the MALT lymphoma. In conclusion, in H. heilmannii-induced MALT lymphoma, enhanced immunoreactivity of vascular endothelial growth factor-A and factor-C was observed in areas encircled by increased parietal cell apoptosis, which indicates the pathophysiological relevance of both angiogenesis and apoptosis in MALT lymphoma formation.