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.     

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.     

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.     

A novel angiogenesis model using interstitial cells derived from skeletal muscle

A novel angiogenesis model using co-culture of endothelial and interstitial cells was developed, in which bovine capillary endothelial cells (BCEC) formed capillary-like structure on the monolayer of interstitial cells (like smooth muscle cells) isolated from rat skeletal muscle. The capillary formation of BCEC occurred even under serum-free conditions. Insulin stimulated the capillary growth under serum-free condition, but not BCEC growth in the single culture system. These results suggested that the insulin effect on the capillary growth was brought about indirectly through the interstitial cells, and that this co-culture system may be useful for the study of angiogenesis (especially in skeletal muscle).     

In-silico fragment-based identification of novel angiogenesis inhibitors

Inhibition of vascular endothelial growth factor receptor-2 (VEGFR2) kinase blocks angiogenesis, the process of generating new capillary blood vessels that are important in tumor growth. To identify small molecule inhibitors of the VEGFR2 kinase, we undertook a computer assisted fragment-based screening that used 3-D structural models of the VEGFR2 kinase, and hinge region as a fragment anchor point. Seven novel non-cytotoxic compounds were identified which limited the induction of capillary networks by human umbilical vein endothelial cells in the low micromolar range.     

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.     

Thrombospondin modules and angiogenesis

Angiogenesis is a complex, multifactorial process that involves signals from endothelial cells and from the stoma. Extracellular matrix proteins participate in the modulation of growth factor response, contribute to the architecture of the vasculature and provide signals for the stabilization of mature capillary beds. The identification of the relevant extracellular matrix molecules and the characterization of their effects has been a central focus of research in vascular biology. Thrombospondin-1 is an extracellular glycoprotein first to be recognized as an inhibitor of angiogenesis more than a decade ago. Since then, much has been learned about its ability to regulate vascular growth in several angiogenesis models, functional domains have been identified, and mechanisms of action determined. This review summarizes current understanding on the effects of thrombospondin-1 and -2 during the process of angiogenesis. We will also extend our comments to ADAMTS1, a member of a relatively novel group of matrix metalloproteinases with thrombospondin repeats and shown to affect endothelial cell function and angiogenesis.     

Extracellular matrix and cell shape: potential control points for inhibition of angiogenesis.

Capillary endothelial (CE) cells require two extracellular signals in order to switch from quiescence to growth and back to differentiation during angiogenesis: soluble angiogenic factors and insoluble extracellular matrix (ECM) molecules. Soluble endothelial mitogens, such as basic fibroblast growth factor (FGF), act over large distances to trigger capillary growth, whereas ECM molecules act locally to modulate cell responsiveness to these soluble cues. Recent studies reveal that ECM molecules regulate CE cell growth and differentiation by modulating cell shape and by activating intracellular chemical signaling pathways inside the cell. Recognition of the importance of ECM and cell shape during capillary morphogenesis has led to the identification of a series of new angiogenesis inhibitors. Elucidation of the molecular mechanism of capillary regulation may result in development of even more potent angiogenesis modulators in the future.     

Comparison of expression of cathepsins B and L and MMP2 in endothelial cells and in capillary sprouting in collagen gel

The lysosomal cysteine proteinases cathepsins B and L are known to play an important role in the invasive growth of tumor cells, but their association with angiogenesis has been less well studied. The aim of this study was to determine the possible role of endothelial cell-associated cathepsins B and L in induced capillary growth in the aorta ring model of angiogenesis. Specific inhibitors of cysteine proteinases did not inhibit capillary growth in aorta ring culture and only slightly inhibited the degradation of surrounding collagen. In contrast, strong inhibition of both processes by the matrix metalloproteinase inhibitor BB-94 was observed, indicating the importance of endogenous MMP production in angiogenesis. In support of this finding, we demonstrated a significant increase in endogenous endothelial mRNA of MMP2, but not of cathepsins B and L, in proliferating primary human dermal microvascular endothelial cells (HMVEC-d) in culture. However, MMP2 mRNA expression was increased only when the cells were embedded in collagen but not when they were grown on plastic, regardless of the addition of the growth factors VEGF or bFGF. Moreover, on plastic the impairment of MMP2 induction by growth factors was observed. The differential effect of growth factors implies the crosstalk with integrin signaling as a consequence of binding to the different matrix. This study suggests that endothelial cell-associated cathepsins B and L are not involved in the invasive growth of capillaries from existing blood vessels and that the presence of collagen is necessary for MMP2 expression in endothelial cells.     

Vasculogenesis and angiogenesis in embryonic-stem-cell-derived embryoid bodies

Embryonic stem cells (ESC) have been established previously from the inner cell mass cells of mouse blastocysts. In suspension culture, they spontaneously differentiate to blood-island-containing cystic embryoid bodies (CEB). The development of blood vessels from in situ differentiating endothelial cells of blood islands, a process which we call vasculogenesis, was induced by injecting ESC into the peritoneal cavity of syngeneic mice. In the peritoneum, fusion of blood islands and formation of an in vivo-like primary capillary plexus occurred. Transplantation of ESC and ESC-derived complex and cystic embryoid bodies (ESC-CEB) onto the quail chorioallantoic membrane (CAM) induced an angiogenic response, which was directed by nonyolk sac endoderm structures. Neither yolk sac endoderm from ESC-CEB nor normal mouse yolk sac tissue induced angiogenesis on the quail CAM. Extracts from ESC-CEB stimulated the proliferation of capillary endothelial cells in vitro. Mitogenic activity increase during in vitro culture and differentiation of ESC. Almost all growth factor activity was associated with the cells. The ESC-CEB derived endothelial cell growth factor bound to heparin-sepharose. The identification of acidic fibroblast growth factor (FGF)in heparin-sepharose-purified material was accomplished by immunoblot experiments involving antibodies against acidic and basic FGF. We conclude that vasculogenesis, the development of blood vessels from in situ differentiating endothelial cells, and angiogenesis, the sprouting of capillaries from preexisting vessels are very early events during embryogenesis which can be studied using ESC differentiating in vitro. Our results suggest that vasculogenesis and angiogenesis are differently regulated.     

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.     

Ischemia-induced angiogenesis is impaired in aminopeptidase A deficient mice via down-regulation of HIF-1α

Aminopeptidase A (APA; EC 3.4.11.7) is a transmembrane metalloprotease with several functions in tumor angiogenesis. To investigate the role of APA in the process of ischemia-induced angiogenesis, we evaluated the cellular angiogenic responses under hypoxic conditions and the process of perfusion recovery in the hindlimb ischemia model of APA-deficient (APA-KO; C57Bl6/J strain) mice.Western blotting of endothelial cells (ECs) isolated from the aorta of APA-KO mice revealed that the accumulation of hypoxia-inducible factor-1α (HIF-1α) protein in response to hypoxic challenge was blunted. Regarding the proteasomal ubiquitination, a proteasome inhibitor MG-132 restored the reduced accumulation of HIF-1α in ECs from APA-KO mice similar to control mice under hypoxic conditions. These were associated with decreased growth factor secretion and capillary formation in APA-KO mice. In the hindlimb ischemia model, perfusion recovery in APA-KO mice was decreased in accordance with a significantly lower capillary density at 2 weeks. Regarding vasculogenesis, no differences were observed in cell populations and distribution patterns between wild type and APA-KO mice in relation to endothelial progenitor cells.Our results suggested that Ischemia-induced angiogenesis is impaired in APA-KO mice partly through decreased HIF-1α stability by proteasomal degradation and subsequent suppression of HIF-1α-driven target protein expression such as growth factors. APA is a functional target for ischemia-induced angiogenesis.     

Purified monocyte-derived angiogenic substance (angiotropin) stimulates migration, phenotypic changes, and "tube formation" but not proliferation of capillary endothelial cells in vitro

Activated monocytes (macrophages, histiocytes) induce the formation of new blood vessels by secretion product(s). From conditioned serum-free media of porcine peripheral monocytes treated with concanavalin A, a substance with very strong angiogenic activity in vivo, designated as angiotropin, has been isolated and purified to homogeneity. We investigated the biological action of the monocyte-derived angiogenic substance on cultured capillary and large vessel (aorta) endothelial cells and on 3T3 fibroblasts, mimicking steps of the angiogenic pathway in vitro. We found that angiotropin does not stimulate the proliferation of capillary endothelial and 3T3 cells; however, in concentrations less than 1 ng/ml, it enhances random migration of capillary endothelial cells but not of 3T3 cells. On confluent monolayers of capillary and aortic endothelial cells angiotropin leads to defined changes of cell morphology that are dose dependent and reversible. In the presence of angiotropin, capillary endothelial cells rapidly form tubelike structures on gelatinized plates. This organizational state is not reached with aortic endothelial cells. The results indicate that the biological action of monocytic angiotropin is different from that of the angiogenic growth factors that stimulate the proliferation of endothelial cells and nonlymphoid mesenchymal cells and keep endothelial cells in the contact-inhibited epitheloid cell phenotype. We propose that angiotropin is directly involved in monocyte-induced angiogenesis.     

血管内皮细胞和心脏组织块的立体培养

本文旨在对比研究二维平面与三维立体培养模式下,内皮细胞和心脏组织形态学的差异。采用胶内、胶上、三明治模式、玻片培养小室模型等多种I型胶原立体培养模型,通过免疫荧光技术及显微形态学观察组织和细胞的生长情况。在二维平面培养中,原代心脏血管内皮细胞呈铺路石样排列;而在三维胶原培养模式中,内皮细胞呈长梭状形态,并迁入胶原培养介质中,和体内血管新生及血管生成过程中的内皮细胞活化表型相似。加入血管内皮生长因子(vascular endo- thelial growth factor VEGF)能增强内皮细胞管状结构的形成。在三维胶原中,心脏组织块生长良好,迁出的细胞将相邻组织块连接起来,组织块有自发的搏动。本工作表明,改进的薄层胶原培养、玻片培养小室模型和动脉条模型是较好的研究血管生成和血管新生的工具。在三维培养的情况下,内皮细胞通过空间增殖、迁移和锚定,可形成管状结构,比二维平面培养更适合用于血管新生的研究。不同的立体培养模型可用于不同目的的研究。     

Albendazole inhibits endothelial cell migration, tube formation, vasopermeability, VEGF receptor-2 expression and suppresses retinal neovascularization in ROP model of angiogenesis

The angiogenic process begins with the cell proliferation and migration into the primary vascular network, and leads to vascularization of previously avascular tissues and organs as well to growth and remodeling of the initially homogeneous capillary plexus to form a new microcirculation. Additionally, an increase in microvascular permeability is a crucial step in angiogenesis. Vascular endothelial growth factor (VEGF) plays a central role in angiogenesis. We have previously reported that albendazole suppresses VEGF levels and inhibits malignant ascites formation, suggesting a possible effect on angiogenesis. This study was therefore designed to investigate the antiangiogenic effect of albendazole in non-cancerous models of angiogenesis. In vitro, treatment of human umbilical vein endothelial cells (HUVECs) with albendazole led to inhibition of tube formation, migration, permeability and down-regulation of the VEGF type 2 receptor (VEGFR-2). In vivo albendazole profoundly inhibited hyperoxia-induced retinal angiogenesis in mice. These results provide new insights into the antiangiogenic effects of albendazole.     

Retinal capillary endothelial cells prefer different substrates for growth and migration

Recent studies have shown that the extracellular matrix modifies the behaviour of endothelial cells. We have studied the effects of extracellular matrix components on retinal capillary endothelial cell migration and proliferation. Bovine retinal capillary endothelial cells were selectively cultured from collagenase-digested microvessel fragments. In a filter system for the assessment of migration, endothelial cells responded to substrate-bound fibronectin but not to soluble fibronectin. Cell migration on collagen- or gelatin-coated filters was minimal, and these cells failed to adopt a spread morphology, remaining instead as round cells. Cell replication was quantified using a protein dye binding assay for adherent cells in 96 well plates. Serum was essential for growth irrespective of the substrate. Cells harvested from microvessel cultures proliferated more rapidly on collagen- and gelatin-coated plastic than on fibronectin and were unaffected by additions to the medium such as endothelial cell conditioned medium, whereas cells proliferating directly from the microvessels grew at a faster rate on fibronectin and also responded to conditioned medium supplement. When cultured on collagen gels, initial microvessel cells and harvested cells required surface fibronectin in order to adopt a cobblestone morphology. These results show that fibronectin is a requirement for bovine retinal capillary endothelial cell migration, but proliferation of these cells can be supported, with slight differences, by both fibronectin and collagen provided serum growth factors are present. These findings are relevant to the early phase of angiogenesis in which migration and proliferation of endothelial cells occurs.     

A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation

This work studies a fundamental problem in blood capillary growth: how the cell proliferation or death induces the stress response and the capillary extension or regression. We develop a one-dimensional viscoelastic model of blood capillary extension/regression under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting a viscoelastic response from the cells, which again induces extension or regression of the capillary. Nonlinear analysis captures two cases when the biologically meaningful solution exists: (1) the cell density decreases from root to tip, which may occur in vessel regression; (2) the cell density is time-independent and is of small variation along the capillary, which may occur in capillary extension without proliferation. The linear analysis with perturbation in cell density due to proliferation or death predicts the global biological solution exists provided the change in cell density is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes, which validate the linear analysis theory. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.     

Neprilysin inhibits angiogenesis via proteolysis of fibroblast growth factor-2

Neprilysin is a cell surface peptidase that catalytically inactivates neuropeptide substrates and functions as a tumor suppressor via its enzymatic function and multiple protein-protein interactions. We investigated whether neutral endopeptidase could inhibit angiogenesis in vivo utilizing a murine corneal pocket angiogenesis model and found that it reduced fibroblast growth factor-2-induced angiogenesis by 85% (p < 0.01) but had no effect on that of vascular endothelial growth factor. Treatment with recombinant neprilysin, but not enzymatically inactive neprilysin, resulted in a slight increase in basic fibroblast growth factor electrophoretic mobility from proteolytic cleavage between amino acids Leu-135 and Gly-136, which was inhibited by the neutral endopeptidase inhibitor CGS24592 and heparin. Cleavage kinetics were rapid, comparable with that of other known neprilysin substrates. Functional studies involving neprilysin-expressing vascular endothelial cells demonstrated that neutral endopeptidase inhibition significantly enhanced fibroblast growth factor-mediated endothelial cell growth, capillary array formation, and signaling, whereas exogenous recombinant neprilysin inhibited signaling. Recombinant constructs confirmed that cleavage products neither promoted capillary array formation nor induced signaling. Moreover, mutation of the cleavage site resulted in concomitant loss of cleavage and increased the potency of fibroblast growth factor-2 to induce capillary array formation. These data indicate that neprilysin proteolytically inactivates fibroblast growth factor-2, resulting in negative regulation of angiogenesis.     

Plasmalemmal vacuolar H+-ATPase is decreased in microvascular endothelial cells from a diabetic model

Angiogenesis requires invasion of extracellular matrix (ECM) proteins by endothelial cells and occurs in hypoxic and acidic environments that are not conducive for cell growth and survival. We hypothesize that angiogenic cells must exhibit a unique system to regulate their cytosolic pH in order to cope with these harsh conditions. The plasmalemmal vacuolar type H(+)-ATPase (pmV-ATPase) is used by cells exhibiting an invasive phenotype. Because angiogenesis is impaired in diabetes, we hypothesized that pmV-ATPase is decreased in microvascular endothelial cells from diabetic rats. The in vitro angiogenesis assays demonstrated that endothelial cells were unable to form capillary-like structures in diabetes. The proton fluxes were slower in cells from diabetic than normal model, regardless of the presence or absence of Na(+) and HCO(3) (-) and were suppressed by V-H(+)-ATPase inhibitors. Immunocytochemical data revealed that pmV-ATPases were inconspicuous at the plasma membrane of cells from diabetic whereas in normal cells were prominent. The pmV-ATPase activity was lower in cells from diabetic than normal models. Inhibition of V-H(+)-ATPase suppresses invasion/migration of normal cells, but have minor effects in cells from diabetic models. These novel observations suggest that the angiogenic abnormalities in diabetes involve a decrease in pmV-ATPase in microvascular endothelial cells.