A Coupled Mathematical Model of Cell Migration,Vessel Cooption and Tumour Microenvironment during the Initiation of Micrometastases

We propose a coupled mathematical model for the detailed quantitative analyses of initial microtumour and micrometastases formation by including cancer cell migration, host vessel cooption and changes in microenvironment. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. Migration probability of a migrating cell is assumed to be influenced by local chemical microenvironment. Pre-existing vessel cooption and remodelling are introduced according to the local haemodynamical microenvironment, such as interstitial pressure and vessel wall permeability. After the tumour cells and tumour vessels distribution are updated, the chemical substances are coupled calculated with the haemodynamical environment. The simulation results clearly reproduce the tumour cells migrate and proliferate along the pre-existing vessels at the very early stage of growth, which are consistent with many published experimental observations. In addition, the model demonstrates the interactions of tumour cells with the pre-existing vessels, which are believed to be essential for initial adhesion, proliferation, invasion, and micrometastases establishment. Quantitative analysis of tumour expansion in longitudinal and transverse directions shows that the cooption and migration along host vessels will be inhibited once angiogenesis phase occurs. The influences of the ability of cell migration and the inclusion of vessel cooption on the formation of micrometastases are discussed.     

Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion

We propose a mathematical modelling system to investigate the dynamic process of tumour cell proliferation, death and tumour angiogenesis by fully coupling the vessel growth, tumour growth and blood perfusion. Tumour growth and angiogenesis are coupled by the chemical microenvironment and the cell-matrix interaction. The haemodynamic calculation is carried out on the updated vasculature. The domains of intravascular, transcapillary and interstitial fluid flow were coupled in the model to provide a comprehensive solution of blood perfusion variables. An estimation of vessel collapse is made according to the wall shear stress criterion to provide feedback on vasculature remodelling. The simulation can show the process of tumour angiogenesis and the spatial distribution of tumour cells for periods of up to 24 days. It can show the major features of tumour and tumour microvasculature during the period such as the formation of a large necrotic core in the tumour centre with few functional vessels passing through, and a well circulated tumour periphery regions in which the microvascular density is high and associated with more aggressive proliferating cells of the growing tumour which are all consistent with physiological observations. The study also demonstrated that the simulation results are not dependent on the initial tumour and networks, which further confirms the application of the coupled model feedback mechanisms. The model enables us to examine the interactions between angiogenesis and tumour growth, and to study the dynamic response of a solid tumour to the changes in the microenvironment. This simulation framework can be a foundation for further applications such as drug delivery and anti-angiogenic therapies.     

A Continuum Mathematical Model of the Developing Murine Retinal Vasculature

Angiogenesis, the process of new vessel growth from pre-existing vasculature, is crucial in many biological situations such as wound healing and embryogenesis. Angiogenesis is also a key regulator of pathogenesis in many clinically important disease processes, for instance, solid tumour progression and ocular diseases. Over the past 10–20 years, tumour-induced angiogenesis has received a lot of attention in the mathematical modelling community and there have also been some attempts to model angiogenesis during wound healing. However, there has been little modelling work of vascular growth during normal development. In this paper, we describe an in silico representation of the developing retinal vasculature in the mouse, using continuum mathematical models consisting of systems of partial differential equations. The equations describe the migratory response of cells to growth factor gradients, the evolution of the capillary blood vessel density, and of the growth factor concentration. Our approach is closely coupled to an associated experimental programme to parameterise our model effectively and the simulations provide an excellent correlation with in vivo experimental data. Future work and development of this model will enable us to elucidate the impact of molecular cues upon vasculature development and the implications for eye diseases such as diabetic retinopathy and neonatal retinopathy of prematurity.     

Signal transduction in vasculogenesis and developmental angiogenesis

The vasculature is a highly specialized organ that functions in a number of key physiological tasks including the transport of oxygen and nutrients to tissues. Formation of the vascular system is an essential and rate-limiting step in development and occurs primarily through two main mechanisms, vasculogenesis and angiogenesis. Both vasculogenesis, the de novo formation of vessels, and angiogenesis, the growth of new vessels from pre-existing vessels by sprouting, are complex processes that are mediated by the precise coordination of multiple cell types to form and remodel the vascular system. A host of signaling molecules and their interaction with specific receptors are central to activating and modulating vessel formation. This review article summarizes the current state of research involving signaling molecules that have been demonstrated to function in the regulation of vasculogenesis and angiogenesis, as well as molecules known to play a role in vessel maturation, hypoxia-driven angiogenesis and arterial-venous specification.     

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.     

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.     

3D Multi-Cell Simulation of Tumor Growth and Angiogenesis

We present a 3D multi-cell simulation of a generic simplification of vascular tumor growth which can be easily extended and adapted to describe more specific vascular tumor types and host tissues. Initially, tumor cells proliferate as they take up the oxygen which the pre-existing vasculature supplies. The tumor grows exponentially. When the oxygen level drops below a threshold, the tumor cells become hypoxic and start secreting pro-angiogenic factors. At this stage, the tumor reaches a maximum diameter characteristic of an avascular tumor spheroid. The endothelial cells in the pre-existing vasculature respond to the pro-angiogenic factors both by chemotaxing towards higher concentrations of pro-angiogenic factors and by forming new blood vessels via angiogenesis. The tumor-induced vasculature increases the growth rate of the resulting vascularized solid tumor compared to an avascular tumor, allowing the tumor to grow beyond the spheroid in these linear-growth phases. First, in the linear-spherical phase of growth, the tumor remains spherical while its volume increases. Second, in the linear-cylindrical phase of growth the tumor elongates into a cylinder. Finally, in the linear-sheet phase of growth, tumor growth accelerates as the tumor changes from cylindrical to paddle-shaped. Substantial periods during which the tumor grows slowly or not at all separate the exponential from the linear-spherical and the linear-spherical from the linear-cylindrical growth phases. In contrast to other simulations in which avascular tumors remain spherical, our simulated avascular tumors form cylinders following the blood vessels, leading to a different distribution of hypoxic cells within the tumor. Our simulations cover time periods which are long enough to produce a range of biologically reasonable complex morphologies, allowing us to study how tumor-induced angiogenesis affects the growth rate, size and morphology of simulated tumors.     

Emergent vascular network inhomogeneities and resulting blood flow patterns in a growing tumor

Tumors acquire sufficient oxygen and nutrient supply by coopting host vessels and neovasculature created via angiogenesis, thereby transforming a highly ordered network into chaotic heterogeneous tumor specific vasculature. Vessel regression inside the tumor leads to large regions of necrotic tissue interspersed with isolated surviving vessels. We extend our recently introduced model to incorporate Fahraeus-Lindqvist- and phase separation effects, refined tissue oxygen level computation and drug flow computations. We find, unexpectedly, that collapse and regression accelerates rather than diminishes the perfusion and that a tracer substance flowing through the remodeled network reaches all parts of the tumor vasculature very well. The reason for decreased drug delivery well known in tumors should therefore be different from collapse and vessel regression. Implications for drug delivery in real tumors are discussed.     

Effects of Dual Targeting of Tumor Cells and Stroma in Human Glioblastoma Xenografts with a Tyrosine Kinase Inhibitor against c-MET and VEGFR2

Anti-angiogenic treatment of glioblastoma with Vascular Endothelial Growth Factor (VEGF)- or VEGF Receptor 2 (VEGFR2) inhibitors normalizes tumor vessels, resulting in a profound radiologic response and improved quality of life. This approach however does not halt tumor progression by diffuse infiltration, as this phenotype is less angiogenesis dependent. Combined inhibition of angiogenesis and diffuse infiltrative growth would therefore be a more effective treatment approach in these tumors. The HGF/c-MET axis is important in both angiogenesis and cell migration in several tumor types including glioma. We therefore analyzed the effects of the c-MET- and VEGFR2 tyrosine kinase inhibitor cabozantinib (XL184, Exelixis) on c-MET positive orthotopic E98 glioblastoma xenografts, which routinely present with angiogenesis-dependent areas of tumor growth, as well as diffuse infiltrative growth. In in vitro cultures of E98 cells, cabozantinib effectively inhibited c-MET phosphorylation, concomitant with inhibitory effects on AKT and ERK1/2 phosphorylation, and cell proliferation and migration. VEGFR2 activation in endothelial cells was also effectively inhibited in vitro. Treatment of BALB/c nu/nu mice carrying orthotopic E98 xenografts resulted in a significant increase in overall survival. Cabozantinib effectively inhibited angiogenesis, resulting in increased hypoxia in angiogenesis-dependent tumor areas, and induced vessel normalization. Yet, tumors ultimately escaped cabozantinib therapy by diffuse infiltrative outgrowth via vessel co-option. Of importance, in contrast to the results from in vitro experiments, in vivo blockade of c-MET activation was incomplete, possibly due to multiple factors including restoration of the blood-brain barrier resulting from cabozantinib-induced VEGFR2 inhibition. In conclusion, cabozantinib is a promising therapy for c-MET positive glioma, but improving delivery of the drug to the tumor and/or the surrounding tissue may be needed for full activity.     

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.     

On the preservation of vessel bifurcations during flow-mediated angiogenic remodelling

During developmental angiogenesis, endothelial cells respond to shear stress by migrating and remodelling the initially hyperbranched plexus, removing certain vessels whilst maintaining others. In this study, we argue that the key regulator of vessel preservation is cell decision behaviour at bifurcations. At flow-convergent bifurcations where migration paths diverge, cells must finely tune migration along both possible paths if the bifurcation is to persist. Experiments have demonstrated that disrupting the cells’ ability to sense shear or the junction forces transmitted between cells impacts the preservation of bifurcations during the remodelling process. However, how these migratory cues integrate during cell decision making remains poorly understood. Therefore, we present the first agent-based model of endothelial cell flow-mediated migration suitable for interrogating the mechanisms behind bifurcation stability. The model simulates flow in a bifurcated vessel network composed of agents representing endothelial cells arranged into a lumen which migrate against flow. Upon approaching a bifurcation where more than one migration path exists, agents refer to a stochastic bifurcation rule which models the decision cells make as a combination of flow-based and collective-based migratory cues. With this rule, cells favour branches with relatively larger shear stress or cell number. We found that cells must integrate both cues nearly equally to maximise bifurcation stability. In simulations with stable bifurcations, we found competitive oscillations between flow and collective cues, and simulations that lost the bifurcation were unable to maintain these oscillations. The competition between these two cues is haemodynamic in origin, and demonstrates that a natural defence against bifurcation loss during remodelling exists: as vessel lumens narrow due to cell efflux, resistance to flow and shear stress increases, attracting new cells to enter and rescue the vessel from regression. Our work provides theoretical insight into the role of junction force transmission has in stabilising vasculature during remodelling and as an emergent mechanism to avoid functional shunting.     

Glioblastoma: A Pathogenic Crosstalk between Tumor Cells and Pericytes

Cancers likely originate in progenitor zones containing stem cells and perivascular stromal cells. Much evidence suggests stromal cells play a central role in tumor initiation and progression. Brain perivascular cells (pericytes) are contractile and function normally to regulate vessel tone and morphology, have stem cell properties, are interconvertible with macrophages and are involved in new vessel formation during angiogenesis. Nevertheless, how pericytes contribute to brain tumor infiltration is not known. In this study we have investigated the underlying mechanism by which the most lethal brain cancer, Glioblastoma Multiforme (GBM) interacts with pre-existing blood vessels (co-option) to promote tumor initiation and progression. Here, using mouse xenografts and laminin-coated silicone substrates, we show that GBM malignancy proceeds via specific and previously unknown interactions of tumor cells with brain pericytes. Two-photon and confocal live imaging revealed that GBM cells employ novel, Cdc42-dependent and actin-based cytoplasmic extensions, that we call flectopodia, to modify the normal contractile activity of pericytes. This results in the co-option of modified pre-existing blood vessels that support the expansion of the tumor margin. Furthermore, our data provide evidence for GBM cell/pericyte fusion-hybrids, some of which are located on abnormally constricted vessels ahead of the tumor and linked to tumor-promoting hypoxia. Remarkably, inhibiting Cdc42 function impairs vessel co-option and converts pericytes to a phagocytic/macrophage-like phenotype, thus favoring an innate immune response against the tumor. Our work, therefore, identifies for the first time a key GBM contact-dependent interaction that switches pericyte function from tumor-suppressor to tumor-promoter, indicating that GBM may harbor the seeds of its own destruction. These data support the development of therapeutic strategies directed against co-option (preventing incorporation and modification of pre-existing blood vessels), possibly in combination with anti-angiogenesis (blocking new vessel formation), which could lead to improved vascular targeting not only in Glioblastoma but also for other cancers.     

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

We formulate a theoretical model to analyze the vascular remodelling process of an arterio-venous vessel network during solid tumour growth. The model incorporates a hierarchically organized initial vasculature comprising arteries, veins and capillaries, and involves sprouting angiogenesis, vessel cooption, dilation and regression as well as tumour cell proliferation and death. The emerging tumour vasculature is non-hierarchical, compartmentalized into well-characterized zones and transports efficiently an injected drug-bolus. It displays a complex geometry with necrotic zones and “hot spots” of increased vascular density and blood flow of varying size. The corresponding cluster size distribution is algebraic, reminiscent of a self-organized critical state. The intra-tumour vascular-density fluctuations correlate with pressure drops in the initial vasculature suggesting a physical mechanism underlying hot spot formation.     

Vasculature in pre-blastema and nerve-dependent blastema stages of regenerating forelimbs of the adult newt, Notophthalmus viridescens

Immunocytochemistry utilizing a monoclonal antibody (BV1; blood vessel 1) highly reactive to the vasculature of the adult newt showed that a developing vasculature was present during early, pre-blastema, and early-bud blastema stages of forelimb regeneration in this species. Infusion of Prussian Blue and DiI into the brachial artery further delineated the intactness of this early vasculature. Finally, macroscopic observations of vascular flow underneath the apical epithelial cap (AEC) and microsurgical removal of the AEC and observation of subsequent bleeding buttressed the conclusion that an intact vasculature exists during early nerve-dependent stages of newt forelimb regeneration. The results suggest that this process of neovascular formation is angiogenesis, i.e., the formation of new vessels from pre-existing vessels in the stump. Furthermore, angiogenesis is an ongoing process initiated early after amputation. Blastema cells and the AEC are likely sourcesof factors that stimulate neovascularization.     

Numerical modelling of the angiogenesis process in wound contraction

Angiogenesis consists of the growth of new blood vessels from the pre-existing vasculature. This phenomenon takes place in several biological processes, including wound healing. In this work, we present a mathematical model of angiogenesis applied to skin wound healing. The developed model includes biological (capillaries and fibroblasts), chemical (oxygen and angiogenic growth factor concentrations) and mechanical factors (cell traction forces and extracellular matrix deformation) that influence the evolution of the healing process. A novelty from previous works, apart from the coupling of angiogenesis and wound contraction, is the more realistic modelling of skin as a hyperelastic material. Large deformations are addressed using an updated Lagrangian approach. The coupled non-linear model is solved with the finite element method, and the process is studied over two wound geometries (circular and elliptical) of the same area. The results indicate that the elliptical wound vascularizes two days earlier than the circular wound but that they experience a similar contraction level, reducing its size by 25 %.     

The involvement of endothelial progenitor cells in tumor angiogenesis

Endothelial progenitor cells (EPCs) have been isolated from peripheral blood CD34, VEGFR-2, or AC 133 (CD133) antigen-positive cells, which may home to site of neovascularization and differentiate into endothelial cells in situ. Endothelial cells contribute to tumor angiogenesis, and can originate from sprouting or co-option of neighbouring pre-existing vessels. Emerging evidence indicate that bone marrow-derived circulating EPCs can contribute to tumor angiogenesis and growth of certain tumors. This review article will summarize the literature data concerning this new role played by EPCs in tumor angiogenesis.     

Vessel abnormalization: another hallmark of cancer? Molecular mechanisms and therapeutic implications

As a result of excessive production of angiogenic molecules, tumor vessels become abnormal in structure and function. By impairing oxygen delivery, abnormal vessels fuel a vicious cycle of non-productive angiogenesis, which creates a hostile microenvironment from where tumor cells escape through leaky vessels and which renders tumors less responsive to chemoradiation. While anti-angiogenic strategies focused on inhibiting new vessel growth and destroying pre-existing vessels, clinical studies showed modest anti-tumor effects. For many solid tumors, anti-VEGF treatment offers greater clinical benefit when combined with chemotherapy. This is partly due to a normalization of the tumor vasculature, which improves cytotoxic drug delivery and efficacy and offers unprecedented opportunities for anti-cancer treatment. Here, we overview key novel molecular players that induce vessel normalization.     

The angiopoietins and Tie2/Tek: adding to the complexity of cardiovascular development

The expansion or remodelling of pre-existing blood vessels, known as angiogenesis, by either nascent sprouting, intercalated or intussusceptive growth is a highly regulated process. Angiogenesis is critical not only during normal embryonic vascular development, but also in the progression of several diseases, including cancer, psoriasis, and diabetes. Mouse molecular genetic experiments have shown that the angiopoietins and their receptor Tie2/Tek are indispensable for embryonic vessel development. The importance of the angiopoietin-signalling pathway has also been shown to extend beyond development, into in vitro and in vivo experimental models of angiogenic growth. Currently the precise role of the angiopoietins remains unclear. However, what is emerging from genetic, xenograft transplant, histochemical and cell culture experiments are that the response of endothelial cells to angiopoietins appears to be context and endothelial cell type specific.