Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumour growth, linking the relatively harmless avascular growth phase and the potentially fatal vascular growth phase. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration, proliferation; degradation of tissue; new capillary vessel (sprout) formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is blood flow associated with the nascent network, the subsequent growth of the network evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix and perfusion-related haemodynamic forces in a manner that may be described as both adaptive and dynamic. In this paper we present a mathematical model which simultaneously couples vessel growth with blood flow through the vessels--dynamic adaptive tumour-induced angiogenesis (DATIA). This new mathematical model presents a theoretical and computational investigation of the process and highlights a number of important new targets for therapeutic intervention. In contrast to earlier flow models, where the effects of perfusion (blood flow) were essentially evaluated a posteriori, i.e. after generating a hollow network, blood flow in the model described in this paper has a direct impact during capillary growth, with radial adaptations and network remodelling occurring as immediate consequences of primary anastomoses. Capillary network architectures resulting from the dynamically adaptive model are found to differ radically from those obtained using earlier models. The DATIA model is used to examine the effects of changing various physical and biological model parameters on the developing vascular architecture and the delivery of chemotherapeutic drugs to the tumour. Subsequent simulations of chemotherapeutic treatments under different parameter regimes lead to the identification of a number of new therapeutic targets for tumour management.     

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.     

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions

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.     

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.     

Vascularization is a general requirement for growth of plant and animal tumours

Solid-tumour growth in animals as in humans depends on angiogenesis. Tumours that fail to induce the formation of new blood vessels do not enlarge beyond a few millimetres in diameter. Plant tumours induced by Agrobacterium tumefaciens can reach diameters of more than 100 mm, thus raising the question of how they are sufficiently supplied with nutrients and water. Until recently, these rapidly growing tumours were considered unorganized or partly organized masses. However, in analogy to animal and human tumours, growth of leaf and stem tumours depends on neovascularization. Plant tumour cells induce the formation of a sophisticated vascular network consisting of water-conducting vessels and assimilate-transporting sieve elements. Similar to animal and human tumours that overexpress angiogenic growth factors, plant tumours overexpress the T-DNA-encoded vascularization-promoting growth factors auxin and cytokinin upon AGROBACTERIUM: infection. High auxin levels induce ethylene emission from the tumours, which has a strong impact on tumour and host stem, as well as on root structure and function. Ethylene apparently stimulates abscisic acid synthesis in the leaves above the tumour, which reduces transpiration and thus protects the host plant from rapid wilting. Hence, for the elucidation of phytohormone-dependent vascular development in plants, such tumours are regarded as an excellent model system. The comparison of analogous requirement of neovascularization for tumour growth in plants, as in animals and humans, is discussed in terms of interdisciplinary strategies of possible prevention and therapy.     

Jumping the barrier: VE-cadherin, VEGF and other angiogenic modifiers in cancer

The endothelial barrier controls the passage of fluids, nutrients and cells through the vascular wall. This physiological function is closely related to developmental and adult angiogenesis, blood pressure control, as well as immune responses. Moreover, cancer progression is frequently characterized by disorganized and leaky blood vessels. In this context, vascular permeability drives tumour-induced angiogenesis, blood flow disturbances, inflammatory cell infiltration and tumour cell extravasation. Although various molecules have been implicated, the vascular endothelial adhesion molecule, VE-cadherin (vascular endothelial cadherin), has emerged as a critical player involved in maintaining endothelial barrier integrity and homoeostasis. Indeed, VE-cadherin coordinates the endothelial cell-cell junctions through its adhesive and signalling properties. Of note, many angiogenic and inflammatory mediators released into the tumour microenvironment influence VE-cadherin behaviour. Therefore restoring VE-cadherin function could be one very promising target for vascular normalization in cancer therapies. In this review, we will mainly focus on recent discoveries concerning the molecular mechanisms involved in modulating VE-cadherin plasticity in cancer.     

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.     

Cell elongation is key to in silico replication of in vitro vasculogenesis and subsequent remodeling

Vasculogenesis, the de novo growth of the primary vascular network from initially dispersed endothelial cells, is the first step in the development of the circulatory system in vertebrates. In the first stages of vasculogenesis, endothelial cells elongate and form a network-like structure, called the primary capillary plexus, which subsequently remodels, with the size of the vacancies between ribbons of endothelial cells coarsening over time. To isolate such intrinsic morphogenetic ability of endothelial cells from its regulation by long-range guidance cues and additional cell types, we use an in vitro model of human umbilical vein endothelial cells (HUVEC) in Matrigel. This quasi-two-dimensional endothelial cell culture model would most closely correspond to vasculogenesis in flat areas of the embryo like the yolk sac. Several studies have used continuum mathematical models to explore in vitro vasculogenesis: such models describe cell ensembles but ignore the endothelial cells' shapes and active surface fluctuations. While these models initially reproduce vascular-like morphologies, they eventually stabilize into a disconnected pattern of vascular "islands." Also, they fail to reproduce temporally correct network coarsening. Using a cell-centered computational model, we show that the endothelial cells' elongated shape is key to correct spatiotemporal in silico replication of stable vascular network growth. We validate our simulation results against HUVEC cultures using time-resolved image analysis and find that our simulations quantitatively reproduce in vitro vasculogenesis and subsequent in vitro remodeling.     

VEGF and imaging of vessels in rheumatoid arthritis

Angiogenesis is a prominent feature of rheumatoid synovitis. Formation of new blood vessels permits a supply of nutrients and oxygen to the augmented inflammatory cell mass and so contributes to perpetuation of joint disease. Vascular endothelial growth factor (VEGF) is a potent endothelial cell-specific growth factor that is upregulated by proinflammatory cytokines and by hypoxia. Serum VEGF concentrations are elevated in rheumatoid arthritis (RA) and correlate with disease activity. Furthermore, serum VEGF measured at first presentation in RA is highly significantly correlated with radiographic progression of disease over the subsequent year. Power Doppler ultrasonography is a sensitive method for demonstrating the presence of blood flow in small vessels and there is a very close relation between the presence or absence of vascular flow signal on power Doppler imaging and the rate of early synovial enhancement on dynamic gadolinium-enhanced magnetic resonance imaging (MRI) of joints with RA. Images obtained by both dynamic enhanced MRI and power Doppler ultrasonography correlate with vascularity of synovial tissue as assessed histologically. In early RA, there is a striking association between joint erosions assessed on high-resolution ultrasonography and vascular signal in power Doppler mode. Collectively, these findings implicate vascular pannus in the erosive phase of disease and strongly suggest that proangiogenic molecules such as VEGF are targets for novel therapies in RA. Animal model data supports this concept. It seems likely that serological and imaging measures of vascularity in RA will become useful tools in the assessment of disease activity and response to therapy.     

Dll4-Notch Signalling Blockade Synergizes Combined Ultrasound-Stimulated Microbubble and Radiation Therapy in Human Colon Cancer Xenografts

Tumour vasculature acts as an essential lifeline for tumour progression and facilitates metastatic spread. Novel vascular targeting strategies aiming to sustain vascular shutdown could potentially induce substantial damage, resulting in a significant tumour growth delay. We investigated the combination of two novel complementary vascular targeting agents with radiation therapy in a strategy aiming to sustain vascular disruption. Experiments were carried out with delta-like ligand 4 (Dll4) blockade (angiogenesis deregulator) treatment administered in combination with a radiation-based vascular destruction treatment in a highly aggressive well-perfused colon cancer tumour line implanted in female athymic nude mice. Tumours were treated with permutations of radiation, ultrasound-stimulated microbubbles (USMB) and Dll4 monoclonal antibody (mAb). Tumour vascular response was assessed with three-dimensional power Doppler ultrasound to measure active flow and immunohistochemistry. Tumour response was assessed with histochemical assays and longitudinal measurements of tumour volume. Our results suggest a significant tumour response in animals treated with USMB combined with radiation, and Dll4 mAb, leading to a synergistic tumour growth delay of up to 24 days. This is likely linked to rapid cell death within the tumour and a sustained tumour vascular shutdown. We conclude that the triple combination treatments cause a vascular shutdown followed by a sustained inhibition of angiogenesis and tumour cell death, leading to a rapid tumour vascular-based ‘collapse’ and a significant tumour growth delay.     

Visualisation and stereological assessment of blood and lymphatic vessels

The physiological processes involved in tissue development and regeneration also include the parallel formation of blood and lymphatic vessel circulations which involves their growth, maturation and remodelling. Both vascular systems are also frequently involved in the development and progression of pathological conditions in tissues and organs. The blood vascular system circulates oxygenated blood and nutrients at appropriate physiological levels for tissue survival, and efficiently removes all waste products including carbon dioxide. This continuous network consists of the heart, aorta, arteries, arterioles, capillaries, post-capillary venules, venules, veins and vena cava. This system exists in an interstitial environment together with the lymphatic vascular system, including lymph nodes, which aids maintenance of body fluid balance and immune surveillance. To understand the process of vascular development, vascular network stability, remodelling and/or regression in any research model under any experimental conditions, it is necessary to clearly and unequivocally identify and quantify all elements of the vascular network. By utilising stereological methods in combination with cellular markers for different vascular cell components, it is possible to estimate parameters such as surface density and surface area of blood vessels, length density and length of blood vessels as well as absolute vascular volume. This review examines the current strategies used to visualise blood vessels and lymphatic vessels in two- and three-dimensions and the basic principles of vascular stereology used to quantify vascular network parameters.     

Critical Transitions in Early Embryonic Aortic Arch Patterning and Hemodynamics

Transformation from the bilaterally symmetric embryonic aortic arches to the mature great vessels is a complex morphogenetic process, requiring both vasculogenic and angiogenic mechanisms. Early aortic arch development occurs simultaneously with rapid changes in pulsatile blood flow, ventricular function, and downstream impedance in both invertebrate and vertebrate species. These dynamic biomechanical environmental landscapes provide critical epigenetic cues for vascular growth and remodeling. In our previous work, we examined hemodynamic loading and aortic arch growth in the chick embryo at Hamburger-Hamilton stages 18 and 24. We provided the first quantitative correlation between wall shear stress (WSS) and aortic arch diameter in the developing embryo, and observed that these two stages contained different aortic arch patterns with no inter-embryo variation. In the present study, we investigate these biomechanical events in the intermediate stage 21 to determine insights into this critical transition. We performed fluorescent dye microinjections to identify aortic arch patterns and measured diameters using both injection recordings and high-resolution optical coherence tomography. Flow and WSS were quantified with 3D computational fluid dynamics (CFD). Dye injections revealed that the transition in aortic arch pattern is not a uniform process and multiple configurations were documented at stage 21. CFD analysis showed that WSS is substantially elevated compared to both the previous (stage 18) and subsequent (stage 24) developmental time-points. These results demonstrate that acute increases in WSS are followed by a period of vascular remodeling to restore normative hemodynamic loading. Fluctuations in blood flow are one possible mechanism that impacts the timing of events such as aortic arch regression and generation, leading to the variable configurations at stage 21. Aortic arch variations noted during normal rapid vascular remodeling at stage 21 identify a temporal window of increased vulnerability to aberrant aortic arch morphogenesis with the potential for profound effects on subsequent cardiovascular morphogenesis.     

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).     

The role of acidity in solid tumour growth and invasion

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this paper, we develop a simple model of three-dimensional tumour growth to examine the role of acidosis in the interaction between normal and tumour cell populations. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Analysis of the model allows us to assess novel therapies directed towards changing the level of acidity within the tumour.     

The spatial-temporal redistribution of pulmonary blood flow with postnatal growth.

The pulmonary vascular tree undergoes remarkable postnatal development and remodeling. While a number of studies have characterized longitudinal changes in vascular function with growth, none have explored regional patterns of vascular remodeling. We therefore studied six neonatal pigs to see how regional blood flow changes with growth. We selected pigs because of their rapid growth and their similarities to human development with respect to the pulmonary vascular tree. Fluorescent microspheres of varying colors were injected into the pulmonary circulation to mark regional blood on days 3, 12, 27, 43, and 71 after birth. The animals were awake and in the prone posture for all injections. The lungs were subsequently removed, air dried, and sectioned into approximately 2-cm(3) pieces. Flow on each injection day was determined for each piece. Despite the increase in the hydrostatic gradient in the lung with growth, there was a strong correlation between blood flow to the same lung piece when compared on days 3 and 71 (0.73 +/- 0.12). Although a dorsal-ventral gradient of perfusion did not exist on day 3, blood flow increased more in the dorsal region by day 12 and then gradually became more uniform by day 71. Although most of the lung pieces did not show any discernable pattern of blood flow redistribution, there were spatial patterns of blood flow redistribution that were similar across animals. Our findings suggest that local mechanisms, shared across animals, guide regional changes in vascular resistance or vasoregulation during postnatal development. In the pig, these mechanisms act to produce more uniform flow in the normal posture for an ambulating quadruped. The stimuli for these changes have not yet been identified.     

The role of blood flow and microRNAs in blood vessel development

The circulatory system is the first organ system that develops during embryogenesis, and is essential for embryo viability and survival. Crucial for developing a functional vasculature are the specification of arterial-venous identity in vessels and the formation of a hierarchical branched vascular network. Sprouting angiogenesis, intussusception, and flow driven remodeling events collectively contribute to establishing the vascular architecture. At the molecular level, arterial-venous identity and branching are regulated by genetically hardwired mechanisms involving Notch, vascular endothelial growth factor and neural guidance molecule signaling pathways, modulated by hemodynamic factors. MicroRNAs are small, non-coding RNAs that act as silencers to fine-tune the gene expression profile. MicroRNAs are known to influence cell fate decisions, and microRNA expression can be controlled by blood flow, thus placing microRNAs potentially at the center of the genetic cascades regulating vascular differentiation. In the present review, we summarize current progress regarding microRNA functions in blood vessel development with an emphasis on studies performed in zebrafish and mouse models.     

The mathematical modelling of tumour angiogenesis and invasion

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. Endothelial cells which form the lining of neighbouring blood vessels 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. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model.