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.     

Continuous and discrete mathematical models of tumor-induced angiogenesis

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.     

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.     

Vascularization of normal human thyroid tissue transplanted to nude mice

The vascularization of normal human thyroid tissue transplanted to nude, athymic mice was examined by light, electron microscopy and autoradiography after continuous infusion of 3H-thymidine during 2, 4 and 6 days after transplantation. Labelled vascular sprouts were found in the surrounding host connective tissue after 2 days, in between peripheral follicles after 4 days and in central parts of the transplants after 6 days. The autoradiographic observations indicate that the sprouts originated from the surrounding host tissue. The amount of sprouts increased up to a maximum after 2 weeks of transplantation. At this time large interfollicular areas were occupied by sprouts. At later observations (3-5 weeks) sprouts occurred together with typical fenestrated capillaries. After 7 weeks all sprouts had differentiated into mature vessels. Our observations suggest that the transplanted thyroid tissue induces the formation of vascular sprouts in the surrounding host connective tissue. The sprouts then penetrate and vascularize the thyroid tissue.     

Changes in the vascular extracellular matrix during embryonic vasculogenesis and angiogenesis

We have previously characterized monoclonal antibodies against chick brain cells. One of them (14-2B2) brightly stained all capillaries in frozen sections of chick brain. Here we show that this antibody is directed against chick fibronectin. Using this antibody and polyclonal antibodies against laminin, we have studied the development of the vascular extracellular matrix. Vasculogenesis, the development of capillaries from in situ differentiating endothelial cells, was studied in yolk sac blood islands and intraembryonic dorsal aorta. Blood islands produced high levels of fibronectin but not laminin. Early intraembryonic capillaries all expressed fibronectin but little if any laminin. The dorsal aorta of a 6-day-old chick embryo has several layers of fibronectin-producing cells, but is devoid of laminin. Laminin expression commenced at Day 8 and by Day 10 an adult-like distribution was found in the aortic vascular wall. Angiogenesis, the formation of capillaries from preexisting vessels, was studied during brain development. Capillary sprouts invading the neuroectoderm at Embryonic Day 4 migrated in a fibronectin-rich matrix devoid of laminin. Ultrastructural immunolocalization demonstrated the presence of fibronectin exclusively on the abluminal site of the endothelial cells. Beginning on Day 6, laminin codistributed with fibronectin in brain capillaries. We conclude that immature capillaries migrate and proliferate in a fibronectin-rich extracellular matrix, which is subsequently remodeled acquiring basement membrane-like characteristics. We suggest that laminin expression is an early indication of vascular maturation.     

Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial cell migration, and organize themselves into a branched, connected network structure. Subsequent cell proliferation near the sprout-tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we initially generate theoretical capillary networks (which are morphologically similar to those networks observed in vivo) using the discrete mathematical model of Anderson and Chaplain. This discrete model describes the formation of a capillary sprout network via endothelial cell migratory and proliferative responses to external chemical stimuli (tumour angiogenic factors, TAF) supplied by a nearby solid tumour, and also the endothelial cell interactions with the extracellular matrix. The main aim of this paper is to extend this work to examine fluid flow through these theoretical network structures. In order to achieve this we make use of flow modelling tools and techniques (specifically, flow through interconnected networks) from the field of petroleum engineering. Having modelled the flow of a basic fluid through our network, we then examine the effects of fluid viscosity, blood vessel size (i.e., diameter of the capillaries), and network structure/geometry, upon: (i) the rate of flow through the network; (ii) the amount of fluid present in the complete network at any one time; and (iii) the amount of fluid reaching the tumour. The incorporation of fluid flow through the generated vascular networks has highlighted issues that may have major implications for the study of nutrient supply to the tumour (blood/oxygen supply) and, more importantly, for the delivery of chemotherapeutic drugs to the tumour. Indeed, there are also implications for the delivery of anti-angiogenesis drugs to the network itself. Results clearly highlight the important roles played by the structure and morphology of the network, which is, in turn, linked to the size and geometry of the nearby tumour. The connectedness of the network, as measured by the number of loops formed in the network (the anastomosis density), is also found to be of primary significance. Moreover, under certain conditions, the results of our flow simulations show that an injected chemotherapy drug may bypass the tumour altogether.     

TUMOR ANGIOGENESIS : Rapid Induction of Endothelial Mitoses Demonstrated by Autoradiography

Walker ascites tumor cells and an extract derived from such cells (tumor angiogenesis factor, TAF) were injected into the subcutaneous tissue of rats by using a dorsal air sac technique. At intervals thereafter, thymidine-³H was injected into the air sac and the tissues were examined by autoradiography and electron microscopy. Autoradiographs of 1µ thick Epon sections showed thymidine-³H labeling in endothelial cells of small vessels 1–3 mm from the site of implantation, as early as 6–8 hr after exposure to live tumor cells At this time interval endothelial cells appeared histologically normal. DNA synthesis by endothelium subsequently increased and within 48 hr new blood vessel formation was detected. The presence of thymidine-³H-labeled endothelial nuclei, endothelial mitoses, and regenerating-type endothelium was confirmed by electron microscopy. TAF also induced neovascularization and endothelial cell DNA synthesis after 48 hr. A similar response was not evoked in saline controls. Formic acid, which elicited a more intense inflammatory response, was associated with less endothelial labeling and neovascularization at the times studied. Pericytes and other connective tissue cells were also stimulated by live tumor cells and TAF. The mechanism of new blood vessel formation induced by tumors is still unknown but our findings argue against cytoplasmic contact or nonspecific inflammation as prerequisites for tumor angiogenesis.     

Whole Mount Immunofluorescent Staining of the Neonatal Mouse Retina to Investigate Angiogenesis In vivo

Angiogenesis is the complex process of new blood vessel formation defined by the sprouting of new blood vessels from a pre-existing vessel network. Angiogenesis plays a key role not only in normal development of organs and tissues, but also in many diseases in which blood vessel formation is dysregulated, such as cancer, blindness and ischemic diseases. In adult life, blood vessels are generally quiescent so angiogenesis is an important target for novel drug development to try and regulate new vessel formation specifically in disease. In order to better understand angiogenesis and to develop appropriate strategies to regulate it, models are required that accurately reflect the different biological steps that are involved. The mouse neonatal retina provides an excellent model of angiogenesis because arteries, veins and capillaries develop to form a vascular plexus during the first week after birth. This model also has the advantage of having a two-dimensional (2D) structure making analysis straightforward compared with the complex 3D anatomy of other vascular networks. By analyzing the retinal vascular plexus at different times after birth, it is possible to observe the various stages of angiogenesis under the microscope. This article demonstrates a straightforward procedure for analyzing the vasculature of a mouse retina using fluorescent staining with isolectin and vascular specific antibodies.     

Distinct signalling pathways regulate sprouting angiogenesis from the dorsal aorta and the axial vein

Angiogenesis, the formation of new blood vessels from pre-existing vessels, is critical to most physiological processes and many pathological conditions. During zebrafish development, angiogenesis expands the axial vessels into a complex vascular network that is necessary for efficient oxygen delivery. Although the dorsal aorta and the axial vein are spatially juxtaposed, the initial angiogenic sprouts from these vessels extend in opposite directions, indicating that distinct cues may regulate angiogenesis of the axial vessels. We found that angiogenic sprouts from the dorsal aorta are dependent on vascular endothelial growth factor A (Vegf-A) signalling, and do not respond to bone morphogenetic protein (Bmp) signals. In contrast, sprouts from the axial vein are regulated by Bmp signalling independently of Vegf-A signals, indicating that Bmp is a vein-specific angiogenic cue during early vascular development. Our results support a paradigm whereby different signals regulate distinct programmes of sprouting angiogenesis from the axial vein and dorsal aorta, and indicate that signalling heterogeneity contributes to the complexity of vascular networks.     

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.     

Formation of the branching pattern of blood vessels in the wall of the avian yolk sac studied by a computer simulation

The present study was performed to provide data to support the notion previously believed but not proved experimentally or theoretically, that blood vessels are formed by the selection of capillaries in the network. In an attempt to understand the mechanism of formation of blood vessel branching structures, the transformation of a capillary network to a branching system in the wall of quail yolk sac was successively recorded by a series of photographs, and a computer simulation was carried out for the process of in vivo vascularization based on the photographs. The simulation demonstrated that a positive feedback system participated in the formation of a branching structure. That is, vessels which had been much used were enlarged, whereas less used vessels were reduced in their size and finally extinguished. The enlarged vessels became major components of the branching system. As the body of an embryo grew, it was observed that polygonal capillary networks enlarged, which led each polygon of the network to divide into a few finer polygons. Then, some of the capillary vessels were again selected and formed a branching system. This process repeated during the body growth, indicating that the vascular system developed adaptively to the body growth. A region where the growth was fast, received much blood flow and produced finer networks of capillaries. Thus, it was experimentally demonstrated for the first time that capillaries in the network are successively selected by a positive feedback mechanism and form blood vessels.     

A heterogeneous in vitro three dimensional model of tumour-stroma interactions regulating sprouting angiogenesis

Angiogenesis, the formation of new blood vessels, is an essential process for tumour progression and is an area of significant therapeutic interest. Different in vitro systems and more complex in vivo systems have been described for the study of tumour angiogenesis. However, there are few human 3D in vitro systems described to date which mimic the cellular heterogeneity and complexity of angiogenesis within the tumour microenvironment. In this study we describe the Minitumour model--a 3 dimensional human spheroid-based system consisting of endothelial cells and fibroblasts in co-culture with the breast cancer cell line MDA-MB-231, for the study of tumour angiogenesis in vitro. After implantation in collagen-I gels, Minitumour spheroids form quantifiable endothelial capillary-like structures. The endothelial cell pre-capillary sprouts are supported by the fibroblasts, which act as mural cells, and their growth is increased by the presence of cancer cells. Characterisation of the Minitumour model using small molecule inhibitors and inhibitory antibodies show that endothelial sprout formation is dependent on growth factors and cytokines known to be important for tumour angiogenesis. The model also shows a response to anti-angiogenic agents similar to previously described in vivo data. We demonstrate that independent manipulation of the different cell types is possible, using common molecular techniques, before incorporation into the model. This aspect of Minitumour spheroid analysis makes this model ideal for high content studies of gene function in individual cell types, allowing for the dissection of their roles in cell-cell interactions. Finally, using this technique, we were able to show the requirement of the metalloproteinase MT1-MMP in endothelial cells and fibroblasts, but not cancer cells, for sprouting angiogenesis.     

The formation of the insulo-acinar portal system in the pancreas of human fetuses]

In human fetuses 12-20-week-old peculiarities in creation and development of the pancreatic hemo-microcirculatory bed have been studied in connection of its incretory part formation. The vascular glomerulus begins to form on the 12th-14th week as transformation of capillaries of the exocrinic parenchyma into the glomerular capillaries, which, on their getting out of the glomeruli, turn into vessels. The latter participate in formation of the insulo-acinar portal system. Certain structures have been revealed for adaptation of an elevated volume of the blood stream in the glomeruli place, where the glomerular microvessels pass into the acinar anastomoses. The latter perform shunting of the glomerular capillaries with the arteriole, which brings blood, enriched in insular hormones, to distantly situating acinar cells.     

Vasculogenesis and angiogenesis: two distinct morphogenetic mechanisms establish embryonic vascular pattern

We are using a monoclonal antibody, QH-1, as a label for angioblasts in quail embryos to study vascular development. Our previous experiments showed that major embryonic blood vessels, such as the dorsal aortae and posterior cardinal veins, develop from angioblasts of mesodermal origin that appear in the body of the embryo proper (Coffin and Poole: Development, 102:735-748, '88). We theorized that there are two separate processes for blood vessel development that occur in quail embryos. One mechanism termed "vasculogenesis" forms blood vessels in place by the aggregation of angioblasts into a cord. The other mechanism, termed "angiogenesis," is the formation of new vessels by sprouting of capillaries from existing vessels. Here we report the results of microsurgical transplantation experiments designed to determine the extent of cell migration taking place during blood vessel formation. Comparison of the chimeras to normal embryos suggests that the vascular pattern develops, in part, from the normally restricted points of entry of angioblasts into the head from the ventral and dorsal aortae. Transplantations of quail mesoderm (1-15 somite stage) into the head of 5-15 somite chick hosts resulted in extensive sprouting and in migration of single and small groups of angioblasts away from the graft sites. Transplantations into the trunk resulted in incorporation of the graft into the normal vascular pattern of the host. Lateral plate mesoderm was incorporated into the dorsal aortae and individual sprouts grew between somites and along the neural tube to contribute to the intersomitic and vertebral arteries, respectively.     

A Novel Xenograft Model in Zebrafish for High-Resolution Investigating Dynamics of Neovascularization in Tumors

Tumor neovascularization is a highly complex process including multiple steps. Understanding this process, especially the initial stage, has been limited by the difficulties of real-time visualizing the neovascularization embedded in tumor tissues in living animal models. In the present study, we have established a xenograft model in zebrafish by implanting mammalian tumor cells into the perivitelline space of 48 hours old Tg(Flk1:EGFP) transgenic zebrafish embryos. With this model, we dynamically visualized the process of tumor neovascularization, with unprecedented high-resolution, including new sprouts from the host vessels and the origination from VEGFR2⁺ individual endothelial cells. Moreover, we quantified their contributions during the formation of vascular network in tumor. Real-time observations revealed that angiogenic sprouts in tumors preferred to connect each other to form endothelial loops, and more and more endothelial loops accumulated into the irregular and chaotic vascular network. The over-expression of VEGF165 in tumor cells significantly affected the vascularization in xenografts, not only the number and size of neo-vessels but the abnormalities of tumor vascular architecture. The specific inhibitor of VEGFR2, SU5416, significantly inhibited the vascularization and the growth of melanoma xenografts, but had little affects to normal vessels in zebrafish. Thus, this zebrafish/tumor xenograft model not only provides a unique window to investigate the earliest events of tumoral neoangiogenesis, but is sensitive to be used as an experimental platform to rapidly and visually evaluate functions of angiogenic-related genes. Finally, it also offers an efficient and cost-effective means for the rapid evaluation of anti-angiogenic chemicals.     

Angiogenic network formation in the developing vertebrate trunk

We have used time-lapse multiphoton microscopy of living Tg(fli1:EGFP)y1 zebrafish embryos to examine how a patterned, functional network of angiogenic blood vessels is generated in the early vertebrate trunk. Angiogenic vascular sprouts emerge from the longitudinal trunk axial vessels (the dorsal aorta and posterior cardinal vein) in two spatially and temporally distinct steps. Dorsal aorta-derived sprouts form an initial primary network of vascular segments, followed by emergence of vein-derived secondary vascular sprouts that interact and interconnect dynamically with the primary network to initiate vascular flow. Using transgenic silent heart mutant embryos, we show that the gross anatomical patterning of this network of vessels does not require blood circulation. However, our results suggest that circulatory flow dynamics play an important role in helping to determine the pattern of interconnections between the primary network and secondary sprouts, and thus the final arterial or venous identity of the vessels in the functional network. We discuss a model to explain our results combining genetic programming of overall vascular architecture with hemodynamic determination of circulatory flow patterns.     

Ultrastructural localization of fibronectin and laminin in human granulation tissue in relation to capillary development

The distribution of fibronectin (FN) and laminin (LM) at developing capillaries during various developmental stages, from capillary sprouts to relatively developed capillaries, was studied by light- and electron-microscopy immunocytochemistry. By light-microscope, FN immunoreactivity was diffusely distributed throughout the stroma of the granulation tissues, while for LM it was preferentially distributed at the perivascular region with the various developmental stages of the immature capillaries. Ultrastructural study revealed that capillary sprouts were closely surrounded by plentiful deposits of immunoreaction with the FN, but only faintly for LM. Relatively developed capillaries with large and tall endothelium were surrounded by plentiful immunoreactive products with both FN and LM, and immunoreactivities in the cisternae of rER of the endothelium and/or the pericytes were also shown. Cytoplasmic interdigitations between the endothelium and the pericyte of developing capillaries were recognized without an immunoreaction for FN and LM. These results mean that the capillary sprouts are associated with a prepatterned FN-rich and LM-poor perivascular matrix, whereas relatively developed capillaries are associated with a FN- and LM-rich perivascular matrix which would have been produced by the capillary endothelium and/or pericytes.     

Morphology and tyrosine-hydroxylase immunohistochemistry of the systemic secondary vessel system of the blue catfish,Arius graeffei

Fish have a secondary vessel system which emerges from the primary vasculature via large numbers of coiled origins. The precise role of this vessel system is unknown. Vascular casting techniques and scanning electron microscopy reveal that the secondary vessels of the blue catfish, Arius graeffei, originate from dorsal, lateral, and ventral segmental primary arteries and from the caudal dorsal aorta. These vessels anastomose with each other to form larger secondary arteries which parallel the primary vessels for their entire length. Secondary vessels do not appear to form a capillary bed in the skin in A. graeffei as they do in some fish species. Coiled secondary vessel origins are abundant within the tunica media and adventitia of the primary vessels from which they emerge. The origins of the secondary vessels are surrounded by the extensive cytoplasmic processes of specialized endothelial cells. These processes extend for up to 6 μm into the lumen of the primary vessel. Ultrastructurally the coiled secondary capillaries consist of an endothelial cell tube which is surrounded by a single layer of pericytes. These endothelial cells extend large numbers of microvilli into the lumen of the coiled secondary capillary. Nerve terminals are commonly associated with the coiled secondary capillaries. Immunohistochemistry has revealed the presence of tyrosine-hydroxylase, an enzyme involved in catecholamine synthesis in nerve varicosities close to secondary vessels in A. graeffei. This vessel system could therefore be regulated by adrenergic nerves. © 1996 Wiley-Liss, Inc.     

A Cell-Based Model of Extracellular-Matrix-Guided Endothelial Cell Migration During Angiogenesis

Angiogenesis, the formation of new blood vessels sprouting from existing ones, occurs in several situations like wound healing, tissue remodeling, and near growing tumors. Under hypoxic conditions, tumor cells secrete growth factors, including VEGF. VEGF activates endothelial cells (ECs) in nearby vessels, leading to the migration of ECs out of the vessel and the formation of growing sprouts. A key process in angiogenesis is cellular self-organization, and previous modeling studies have identified mechanisms for producing networks and sprouts. Most theoretical studies of cellular self-organization during angiogenesis have ignored the interactions of ECs with the extra-cellular matrix (ECM), the jelly or hard materials that cells live in. Apart from providing structural support to cells, the ECM may play a key role in the coordination of cellular motility during angiogenesis. For example, by modifying the ECM, ECs can affect the motility of other ECs, long after they have left. Here, we present an explorative study of the cellular self-organization resulting from such ECM-coordinated cell migration. We show that a set of biologically-motivated, cell behavioral rules, including chemotaxis, haptotaxis, haptokinesis, and ECM-guided proliferation suffice for forming sprouts and branching vascular trees.