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.     

Neovascularization and Hematopoietic Stem Cells

Vasculogenesis and angiogenesis are the major forms of blood vessel formation. Angiogenesis is the process where new vessels grow from pre-existing blood vessels, and is very important in the functional recovery of pathological conditions, such as wound healing and ischemic heart diseases. The development of better animal model and imaging technologies in past decades has greatly enriched our understanding on vasculogenesis and angiogenesis processes. Hypoxia turned out to be an important driving force for angiogenesis in various ischemic conditions. It stimulates expression of many growth factors like vascular endothelial growth factor, platelet-derived growth factor, insulin-like growth factor, and fibroblast growth factor, which play critical role in induction of angiogenesis. Other cellular components like monocytes, T cells, neutrophils, and platelets also play significant role in induction and regulation of angiogenesis. Various stem/progenitor cells also being recruited to the ischemic sites play crucial role in the angiogenesis process. Pre-clinical studies showed that stem/progenitor cells with/without combination of growth factors induce neovascularization in the ischemic tissues in various animal models. In this review, we will discuss about the fundamental factors that regulate the angiogenesis process and the use of stem cells as therapeutic regime for the treatment of ischemic diseases.     

A Computational Model for Predicting Nanoparticle Accumulation in Tumor Vasculature

Vascular targeting of malignant tissues with systemically injected nanoparticles (NPs) holds promise in molecular imaging and anti-angiogenic therapies. Here, a computational model is presented to predict the development of tumor neovasculature over time and the specific, vascular accumulation of blood-borne NPs. A multidimensional tumor-growth model is integrated with a mesoscale formulation for the NP adhesion to blood vessel walls. The fraction of injected NPs depositing within the diseased vasculature and their spatial distribution is computed as a function of tumor stage, from 0 to day 24 post-tumor inception. As the malignant mass grows in size, average blood flow and shear rates increase within the tumor neovasculature, reaching values comparable with those measured in healthy, pre-existing vessels already at 10 days. The NP vascular affinity, interpreted as the likelihood for a blood-borne NP to firmly adhere to the vessel walls, is a fundamental parameter in this analysis and depends on NP size and ligand density, and vascular receptor expression. For high vascular affinities, NPs tend to accumulate mostly at the inlet tumor vessels leaving the inner and outer vasculature depleted of NPs. For low vascular affinities, NPs distribute quite uniformly intra-tumorally but exhibit low accumulation doses. It is shown that an optimal vascular affinity can be identified providing the proper balance between accumulation dose and uniform spatial distribution of the NPs. This balance depends on the stage of tumor development (vascularity and endothelial receptor expression) and the NP properties (size, ligand density and ligand-receptor molecular affinity). Also, it is demonstrated that for insufficiently developed vascular networks, NPs are transported preferentially through the healthy, pre-existing vessels, thus bypassing the tumor mass. The computational tool described here can effectively select an optimal NP formulation presenting high accumulation doses and uniform spatial intra-tumor distributions as a function of the development stage of the malignancy.     

Proteome analysis of the rat cornea during angiogenesis

Angiogenesis is the formation of new blood vessels from the pre-existing vasculature. However, the study of this complex process is often hampered by the lack of a suitable cell-based model and the tools to study the biochemical events that lead to new blood vessel growth. The most widely accepted model for angiogenesis is the in vivo rat corneal model. In this model, the cornea, which is normally an avascular tissue, is stimulated to undergo angiogenesis in response to silver nitrate cauterization or to the implantation of an exogenous angiogenic agent. The physical changes associated with the new vessel growth can be readily monitored visually, but the regulated biochemical events that result in the growth and remodeling of the new vessels are much more challenging to decipher. In this report, a proteomics approach is evaluated for its utility in deciphering the biochemical events during a time course of angiogenic stimulation. At various time points post-silver nitrate cautery, corneas were harvested, solubilized, and analyzed by two-dimensional gel electrophoresis. Protein expression profiles at the various stages of angiogenesis were compared to those of control corneas. One hundred and eleven differentially-expressed proteins were identified by either matrix-assisted laser desorption/ionization-time of flight mass spectrometry or liquid chromatography-coupled electrospray ionization tandem mass spectrometry. Many of the proteins up-regulated during the angiogenesis process were identified as blood-related proteins, thus validating the development of functional blood vessels in the normally avascular tissue of the cornea. Furthermore, detection of differentially-regulated proteins in cauterized versus control tissue clearly validated the utility of a proteomics approach to study this model of angiogenesis. However, in order to get at the key regulatory proteins in the angiogenesis process, it is clear that additional scale-up and enrichment approaches will be required.     

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.     

Branching exponent heterogeneity and wall shear stress distribution in vascular trees

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D(1))(z) = (D(2))(z) + (D(3))(z); D(1) is the parent vessel diameter and D(2) and D(3) are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance of uniform shear stress. A mean z of 3 has been considered confirmation of such a control mechanism. The objective of the present study was to evaluate the consequences of a heterogeneous distribution of z values about the mean with regard to this uniform shear stress hypothesis. Simulations were carried out on model structures otherwise conforming to the criteria consistent with uniform shear stress when z = 3 but with varying distributions of z. The result was that when there was significant heterogeneity in z approaching that found in a real arterial tree, the coefficient of variation in shear stress was comparable to the coefficient of variation in z and nearly independent of the mean value of z. A systematic increase in mean shear stress with decreasing vessel diameter was one component of the variation in shear stress even when the mean z = 3. The conclusion is that the influence of shear stress in determining vessel diameters is not, per se, manifested in a mean value of z. In a vascular tree having a heterogeneous distribution in z values, a particular mean value of z (e.g., z = 3) apparently has little bearing on the uniform shear stress hypothesis.     

Crystal structure of an angiogenesis inhibitor bound to the FGF receptor tyrosine kinase domain.

Angiogenesis, the sprouting of new blood vessels from pre-existing ones, is an essential physiological process in development, yet also plays a major role in the progression of human diseases such as diabetic retinopathy, atherosclerosis and cancer. The effects of the most potent angiogenic factors, vascular endothelial growth factor (VEGF), angiopoietin and fibroblast growth factor (FGF) are mediated through cell surface receptors that possess intrinsic protein tyrosine kinase activity. In this report, we describe a synthetic compound of the pyrido[2,3-d]pyrimidine class, designated PD 173074, that selectively inhibits the tyrosine kinase activities of the FGF and VEGF receptors. We show that systemic administration of PD 173074 in mice can effectively block angiogenesis induced by either FGF or VEGF with no apparent toxicity. To elucidate the determinants of selectivity, we have determined the crystal structure of PD 173074 in complex with the tyrosine kinase domain of FGF receptor 1 at 2.5 A resolution. A high degree of surface complementarity between PD 173074 and the hydrophobic, ATP-binding pocket of FGF receptor 1 underlies the potency and selectivity of this inhibitor. PD 173074 is thus a promising candidate for a therapeutic angiogenesis inhibitor to be used in the treatment of cancer and other diseases whose progression is dependent upon new blood vessel formation.     

Arteriogenesis versus angiogenesis: similarities and differences

Cardiovascular diseases account for more than half of total mortality before the age of 75 in industrialized countries. To develop therapies promoting the compensatory growth of blood vessels could be superior to palliative surgical interventions. Therefore, much effort has been put into investigating underlying mechanisms. Depending on the initial trigger, growth of blood vessels in adult organisms proceeds via two major processes, angiogenesis and arteriogenesis. While angiogenesis is induced by hypoxia and results in new capillaries, arteriogenesis is induced by physical forces, most importantly fluid shear stress. Consequently, chronically elevated fluid shear stress was found to be the strongest trigger under experimental conditions. Arteriogenesis describes the remodelling of pre-existing arterio-arteriolar anastomoses to completely developed and functional arteries. In both growth processes, enlargement of vascular wall structures was proposed to be covered by proliferation of existing wall cells. Recently, increasing evidence emerges, implicating a pivotal role for circulating cells, above all blood monocytes, in vascular growth processes. Since it has been shown that monocytes/ macrophage release a cocktail of chemokines, growth factors and proteases involved in vascular growth, their contribution seems to be of a paracrine fashion. A similar role is currently discussed for various populations of bone-marrow derived stem cells and endothelial progenitors. In contrast, the initial hypothesis that these cells -after undergoing a (trans-)differentiation- contribute by a structural integration into the growing vessel wall, is increasingly challenged.     

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.     

Impaired flow-dependent control of vascular tone and remodeling in P2X4-deficient mice

The structure and function of blood vessels adapt to environmental changes such as physical development and exercise. This phenomenon is based on the ability of the endothelial cells to sense and respond to blood flow; however, the underlying mechanisms remain unclear. Here we show that the ATP-gated P2X4 ion channel, expressed on endothelial cells and encoded by P2rx4 in mice, has a key role in the response of endothelial cells to changes in blood flow. P2rx4(-/-) mice do not have normal endothelial cell responses to flow, such as influx of Ca(2+) and subsequent production of the potent vasodilator nitric oxide (NO). Additionally, vessel dilation induced by acute increases in blood flow is markedly suppressed in P2rx4(-/-) mice. Furthermore, P2rx4(-/-) mice have higher blood pressure and excrete smaller amounts of NO products in their urine than do wild-type mice. Moreover, no adaptive vascular remodeling, that is, a decrease in vessel size in response to a chronic decrease in blood flow, was observed in P2rx4(-/-) mice. Thus, endothelial P2X4 channels are crucial to flow-sensitive mechanisms that regulate blood pressure and vascular remodeling.     

Angioarrestin: a unique angiopoietin-related protein with anti-angiogenic properties

The process of angiogenesis plays a pivotal role in embryogenesis, wound healing, and tumorigenesis through the growth of new blood vessels from pre-existing vasculature. Among the angiogenic factors recently identified as specific for vascular endothelium are the angiopoietins. In depth characterization of the angiopoietins has allowed investigators to better understand the molecular basis of blood vessel formation and vascular endothelial cell function. In this review, we describe angiopoietins and related family members, with particular emphasis on a recently identified protein known as angioarrestin. Our investigations clearly demonstrate that angioarrestin is an anti-angiogenic molecule. The effects of angioarrestin on tumor cell progression and specific aspects of the angiogenic cascade in in vitro models are further discussed.     

Blood flow patterns and pressure loss in the ascending aorta: A comparative study on physiological and aneurysmal conditions

An aortic aneurysm is defined as a balloon-shaped bulging of all three histologic components of the aortic vessel walls (intima, media and adventitia). This dilation results from vessel weakening owing to remodeling, i.e. due to cystic degeneration of the Tunica media (Marfan), progression of atherosclerosis or presence of a bicuspid aortic valve. The growth rate of the aortic diameter varies from patient to patient and may progress until the aneurysm ultimately ruptures. The role of hemodynamics, i.e. blood flow patterns, and shear stresses that are supposed to intensify during aneurysm growth are not yet fully understood, but thought to play a key role in the enlargement process. The aim of this study is to characterize the aortic blood flow in a silicone model of a pathological aorta with ascending aneurysm, to analyze the differences in the blood flow pattern compared to a healthy aortic model, and to single out possible blood flow characteristics measurable using phase contrast magnetic resonance imaging (MRI) that could serve as indicators for aneurysm severity. MRI simulations were performed under physiological, pulsatile flow conditions using data obtained from optical three dimensional particle tracking measurements. In comparison to the healthy geometry, elevated turbulence intensity and pressure loss are measured in the diseased aorta, which we propose as a complimentary indicator for assessing the aneurysmal severity. Our results shed a light on the interplay between the blood flow dynamics and their contribution to the pathophysiology and possible role for future risk assessment of ascending aortic aneurysms.     

The Corneal Micropocket Assay: A Model of Angiogenesis in the Mouse Eye

The mouse corneal micropocket assay is a robust and quantitative in vivo assay for evaluating angiogenesis. By using standardized slow-release pellets containing specific growth factors that trigger blood vessel growth throughout the naturally avascular cornea, angiogenesis can be measured and quantified. In this assay the angiogenic response is generated over the course of several days, depending on the type and dose of growth factor used. The induction of neovascularization is commonly triggered by either basic fibroblast growth factor (bFGF) or vascular endothelial growth factor (VEGF). By combining these growth factors with sucralfate and hydron (poly-HEMA (poly(2-hydroxyethyl methacrylate))) and casting the mixture into pellets, they can be surgically implanted in the mouse eye. These uniform pellets slowly-release the growth factors over five or six days (bFGF or VEGF respectively) enabling sufficient angiogenic response required for vessel area quantification using a slit lamp. This assay can be used for different applications, including the evaluation of angiogenic modulator drugs or treatments as well as comparison between different genetic backgrounds affecting angiogenesis. A skilled investigator after practicing this assay can implant a pellet in less than 5 min per eye.     

Changes in the rheological properties of blood vessel tissue remodeling in the course of development of diabetes.

Rheological properties of blood vessels are expected to change in disease process if the structure of the vessel wall changes. This is illustrated in diabetes, which can be induced in rat by a single injection of Streptozocin. One of the rheological properties of the blood vessel is the stress-strain relationship. The nonlinear stress-strain relationship of arteries is best expressed as derivations of a strain-energy function. In this paper, the stress-strain relations are measured and the coefficients in the strain energy function of arteries are determined for diabetic and control rats. The meaning of these coefficients are explained. The influence of diabetes on the elastic property of the arteries is expressed by the changes of these coefficients. A point of departure of the present paper from all other blood vessel papers published so far is that all strains used here are referred to the zero-stress state of the arteries, whereas all other papers refer strains to the no-load state. The existence of a large difference between the zero-stress state and no-load state of arteries is one of our recent findings. We have explained that the use of zero-stress state as a basis of strain measurements reveals that the in vivo circumferential stress distribution is quite uniform in the vessel wall at the homeostatic condition. It also makes the strain energy function much more accurate than those in which the residual stress is ignored. Using these new results, the stress and strain distribution in normal and diabetic arteries are presented.     

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.     

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.     

VEGF receptor protein-tyrosine kinases: structure and regulation

The human VEGF family consists of VEGF (VEGF-A), VEGF-B, VEGF-C, VEGF-D, and placental growth factor (PlGF). The VEGF family of receptors consists of three protein-tyrosine kinases (VEGFR1, VEGFR2, and VEGFR3) and two non-protein kinase co-receptors (neuropilin-1 and neuropilin-2). These components participate in new blood vessel formation from angioblasts (vasculogenesis) and new blood vessel formation from pre-existing vasculature (angiogenesis). Interaction between VEGFR1 and VEGFR2 or VEGFR2 and VEGFR3 alters receptor tyrosine phosphorylation.     

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.     

A multiphase model describing vascular tumour growth

In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.     

Fluid shear stress and the vascular endothelium: for better and for worse

As blood flows, the vascular wall is constantly subjected to physical forces, which regulate important physiological blood vessel responses, as well as being implicated in the development of arterial wall pathologies. Changes in blood flow, thus generating altered hemodynamic forces are responsible for acute vessel tone regulation, the development of blood vessel structure during embryogenesis and early growth, as well as chronic remodeling and generation of adult blood vessels. The complex interaction of biomechanical forces, and more specifically shear stress, derived by the flow of blood and the vascular endothelium raise many yet to be answered questions:How are mechanical forces transduced by endothelial cells into a biological response, and is there a "shear stress receptor"?Are "mechanical receptors" and the final signaling pathways they evoke similar to other stimulus-response transduction systems?How do vascular endothelial cells differ in their response to physiological or pathological shear stresses?Can shear stress receptors or shear stress responsive genes serve as novel targets for the design of diagnostic and therapeutic modalities for cardiovascular pathologies?The current review attempts to bring together recent findings on the in vivo and in vitro responses of the vascular endothelium to shear stress and to address some of the questions raised above.