A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins

Mathematical Modeling and Analysis of CD200-CD200R in Cancer Treatment

CD200 is a cell membrane protein that binds to its receptor, CD200 receptor (CD200R). The CD200 positive tumor cells inhibit the cellular functions of M1 and M2 macrophages and dendritic cells (DCs) through the CD200-CD200R complex, resulting in downregulation of Interleukin-10 and Interleukin-12 productions and affecting the activation of cytotoxic T lymphocytes. In this work, we provide two ordinary differential equation models, one complete model and one simplified model, to investigate how the binding affinities of CD200R and the populations of M1 and M2 macrophages affect the functions of the CD200-CD200R complex in tumor growth. Our simulations demonstrate that (i) the impact of the CD200-CD200R complex on tumor promotion or inhibition highly depends on the binding affinity of the CD200R on M2 macrophages and DCs to the CD200 on tumor cells, and (ii) a stronger binding affinity of the CD200R on M1 macrophages or DCs to the CD200 on tumor cells induces a higher tumor cell density in the CD200 positive tumor. Thus, the CD200 blockade would be an efficient treatment method in this case. Moreover, the simplified model shows that the binding affinity of CD200R on macrophages is the major factor to determine the treatment efficacy of CD200 blockade when the binding affinities of CD200R on M1 and M2 macrophages are significantly different to each other. On the other hand, both the binding affinity of CD200R and the population of macrophages are the major factors to determine the treatment efficacy of CD200 blockade when the binding affinities of CD200R on M1 and M2 macrophages are close to each other. We also analyze the simplified model to investigate the dynamics of the positive and trivial equilibria of the CD200 positive tumor case and the CD200 deficient tumor case. The bifurcation diagrams show that when M1 macrophages dominate the population, the tumor cell density of the CD200 positive tumor is higher than the one of CD200 deficient tumor. Moreover, the dynamics of tumor cell density change from tumor elimination to tumor persistence to oscillation, as the maximal proliferation rate of tumor cells increases.

Building Mathematical Models for Vascular Growth and Inhibition

Microvascular channel growth and inhibition, such as what occurs in vasculogenic mimicry, are generally represented in tables or shown in bar graphs. Although informative, those representations lack accurate predictions on dosage or the opportunity to report an unbiased metric when one wants to compare different signal dependence, for instance, the concentration of different drugs or enzymes or expression levels of particular genes.Mathematical model building is an exercise that makes you think of which are the key variables of a particular phenomenon and how they affect the targeting experimental output.Starting from early blood vessel formation and regression (number of vessels) due to an inducer/inhibitor effect, we show how a conceptual mathematical model may be built. As an example, the model was used to parameterize aloin bioactivity on a chick yolk sac membrane (YSM) assay with respect to its vasculogenic and vessel regression properties. A separable functional form where vessel formation and cell death occur as mutually exclusive concentration or signal-dependent functions showed that there was a good correlation with experimental data. Although an analytical solution for that simple case is presented, parameter determination and parametric analysis may be carried out numerically by solving the system of ordinary differential equations that represents the model using nonlinear regression for parameter determinations. Such model formulation thus allows for a more objective evaluation concentration dependence and is suggested as a novel method to evaluate blood vessel formation and inhibition as well as a general model for quantitative balance between chemical stimulation and toxicity.

Capillary network formation and structure in a modified discrete mathematical model of angiogenesis

Angiogenesis, as part of cancer development, involves hierarchical complicated events and processes. Multiple studies have revealed the significance of the formation and structure of tumor-induced capillary networks. In this study, a discrete mathematical model of angiogenesis is studied and modified to capture the realistic physics of capillary network formation. Modifications are performed on the mathematical foundations of an existing discrete model of angiogenesis. The main modifications are the imposition of the matrix density effect, implementation of realistic boundary and initial conditions, and improvement of the method of governing equations based on physical observation. Results show that endothelial cells accelerate angiogenesis and capillary formation as they migrate toward the tumor and clearly exhibit the physical concept of haptotactic movement. On the other hand, consideration of blood flow-induced stress leads to a dynamic adaptive vascular network of capillaries which intelligibly reflects the brush border effect . The present modified model of capillary network formation is based on the physical rationale that defines a clear mathematical and physical interpretation of angiogenesis, which is likely to be used in cancer development modeling and anti-angiogenic therapies.

A NUMERICAL PROOF THAT CERTAIN CELLS FOLLOW THE TRAILS OF THE DIFFUSIONS OF SOME CHEMICALS IN THE EXTRACELLULAR MATRIX

In this work, we obtain the numerical solutions of a 2D mathematical model of tumor angiogenesis originally presented in [Pamuk S, ÇAY İ, Sazci A, A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix, Math Biosci 306:32–48, 2018] to numerically prove that the certain cells, the endothelials (EC), pericytes (PC) and macrophages (MC) follow the trails of the diffusions of some chemicals in the extracellular matrix (ECM) which is, in fact, inhomogeneous. This leads to branching, the sprouting of a new neovessel from an existing vessel. Therefore, anastomosis occurs between these sprouts. In our figures we do see these branching and anastomosis, which show the fact that the cells diffuse according to the structure of the ECM. As a result, one sees that our results are in good agreement with the biological facts about the movements of certain cells in the Matrix.

Mathematical simulation of tumour angiogenesis: angiopoietin balance is a key factor in vessel growth and regression

Excessive tumour growth results in a hypoxic environment around cancer cells, thus inducing tumour angiogenesis, which refers to the generation of new blood vessels from pre-existing vessels. This mechanism is biologically and physically complex, with various mathematical simulation models proposing to reproduce its formation. However, although temporary vessel regression is clinically known, few models succeed in reproducing this phenomenon. Here, we developed a three-dimensional simulation model encompassing both angiogenesis and tumour growth, specifically including angiopoietin. Angiopoietin regulates both adhesion and migration between vascular endothelial cells and wall cells, thus inhibiting the cell-to-cell adhesion required for angiogenesis initiation. Simulation results showed a regression, i.e. transient decrease, in the overall length of new vessels during vascular network formation. Using our model, we also evaluated the efficacy of administering the drug bevacizumab. The results highlighted differences in treatment efficacy: (1) earlier administration showed higher efficacy in inhibiting tumour growth, and (2) efficacy depended on the treatment interval even with the administration of the same dose. After thorough validation in the future, these results will contribute to the design of angiogenesis treatment protocols.

A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.

Extracellular matrix compression temporally regulates microvascular angiogenesis

Mechanical cues influence tissue regeneration, and although vasculature is known to be mechanically sensitive, little is known about the effects of bulk extracellular matrix deformation on the nascent vessel networks found in healing tissues. Previously, we found that dynamic matrix compression in vivo potently regulated revascularization during bone tissue regeneration; however, whether matrix deformations directly regulate angiogenesis remained unknown. Here, we demonstrated that load initiation time, magnitude, and mode all regulate microvascular growth, as well as upstream angiogenic and mechanotransduction signaling pathways. Immediate load initiation inhibited angiogenesis and expression of early sprout tip cell selection genes, while delayed loading enhanced microvascular network formation and upstream signaling pathways. This research provides foundational understanding of how extracellular matrix mechanics regulate angiogenesis and has critical implications for clinical translation of new regenerative medicine therapies and physical rehabilitation strategies designed to enhance revascularization during tissue regeneration.

Modelling skin wound healing angiogenesis: A review

The occurrence of wounds is a main health concern in Western society due to their high frequency and treatment cost. During wound healing, the formation of a functional blood vessel network through angiogenesis is an essential process. Angiogenesis allows the reestablishment of the normal blood flow, the sufficient exchange of oxygen and nutrients and the removal of metabolic waste, necessary for cell proliferation and viability. Mathematical and computational models provide new tools to improve the healing process. In fact, over the last thirty years, in silico models have been continuously formulated to describe the effect of several biological and mechanical factors in angiogenesis during wound healing. Additionally, with different levels of complexity, these models allow coupling the human skin structure, to distinct cell types and growth factors, to study extracellular matrix composition and to understand its deformation. This paper discusses how in silico models, which are more economical and less time-consuming comparatively to laboratory methodologies, can help test new strategies to promote/optimize angiogenesis. The continuum, cell-based and hybrid mathematical models of wound healing angiogenesis are reviewed in the present paper, in order to identify possible improvements. Accordingly, the development of higher dimension models incorporating multiscale analysis at molecular, cellular and tissue level remains a challenge that future models should consider.

Angiogenic Factors produced by Hypoxic Cells are a leading driver of Anastomoses in Sprouting Angiogenesis-a computational study

Angiogenesis - the growth of new blood vessels from a pre-existing vasculature - is key in both physiological processes and on several pathological scenarios such as cancer progression or diabetic retinopathy. For the new vascular networks to be functional, it is required that the growing sprouts merge either with an existing functional mature vessel or with another growing sprout. This process is called anastomosis. We present a systematic 2D and 3D computational study of vessel growth in a tissue to address the capability of angiogenic factor gradients to drive anastomosis formation. We consider that these growth factors are produced only by tissue cells in hypoxia, i.e. until nearby vessels merge and become capable of carrying blood and irrigating their vicinity. We demonstrate that this increased production of angiogenic factors by hypoxic cells is able to promote vessel anastomoses events in both 2D and 3D. The simulations also verify that the morphology of these networks has an increased resilience toward variations in the endothelial cell's proliferation and chemotactic response. The distribution of tissue cells and the concentration of the growth factors they produce are the major factors in determining the final morphology of the network.

The proportion of cancer-related entries in PubMed has increased considerably; is cancer truly "The Emperor of All Maladies"?

In this work, the public database of biomedical literature PubMed was mined using queries with combinations of keywords and year restrictions. It was found that the proportion of Cancer-related entries per year in PubMed has risen from around 6% in 1950 to more than 16% in 2016. This increase is not shared by other conditions such as AIDS, Malaria, Tuberculosis, Diabetes, Cardiovascular, Stroke and Infection some of which have, on the contrary, decreased as a proportion of the total entries per year. Organ-related queries were performed to analyse the variation of some specific cancers. A series of queries related to incidence, funding, and relationship with DNA, Computing and Mathematics, were performed to test correlation between the keywords, with the hope of elucidating the cause behind the rise of Cancer in PubMed. Interestingly, the proportion of Cancer-related entries that contain “DNA”, “Computational” or “Mathematical” have increased, which suggests that the impact of these scientific advances on Cancer has been stronger than in other conditions. It is important to highlight that the results obtained with the data mining approach here presented are limited to the presence or absence of the keywords on a single, yet extensive, database. Therefore, results should be observed with caution. All the data used for this work is publicly available through PubMed and the UK’s Office for National Statistics. All queries and figures were generated with the software platform Matlab and the files are available as supplementary material.

Modeling Longitudinal Preclinical Tumor Size Data to Identify Transient Dynamics in Tumor Response to Antiangiogenic Drugs

Experimental evidence suggests that antiangiogenic therapy gives rise to a transient window of vessel normalization, within which the efficacy of radiotherapy and chemotherapy may be enhanced. Preclinical experiments that measure components of vessel normalization are invasive and expensive. We have developed a mathematical model of vascular tumor growth from preclinical time‐course data in a breast cancer xenograft model. We used a mixed‐effects approach for model parameterization, leveraging tumor size data to identify a period of enhanced tumor growth that could potentially correspond to the transient window of vessel normalization. We estimated the characteristics of the window for mice treated with an anti‐VEGF antibody (bevacizumab) or with a bispecific anti‐VEGF/anti‐angiopoietin‐2 antibody (vanucizumab). We show how the mathematical model could theoretically be used to predict how to coordinate antiangiogenic therapy with radiotherapy or chemotherapy to maximize therapeutic effect, reducing the need for preclinical experiments that directly measure vessel normalization parameters.

Vascular phenotype identification and anti-angiogenic treatment recommendation: A pseudo-multiscale mathematical model of angiogenesis

The development of anti-angiogenic drugs for cancer therapy has yielded some promising candidates, but novel approaches for interventions to angiogenesis have led to disappointing results. In addition, there is a shortage of biomarkers that are predictive of response to anti-angiogenic treatments. Consequently, the complex biochemical and physiological basis for tumour angiogenesis remains incompletely understood. We have adopted a mathematical approach to address these issues, formulating a spatially averaged multiscale model that couples the dynamics of VEGF, Ang1, Ang2 and PDGF, with those of mature and immature endothelial cells and pericyte cells. The model reproduces qualitative experimental results regarding pericyte coverage of vessels after treatment by anti-Ang2, anti-VEGF and combination anti-VEGF/anti-Ang2 antibodies. We used the steady state behaviours of the model to characterise angiogenic and non-angiogenic vascular phenotypes, and used mechanistic perturbations representing hypothetical anti-angiogenic treatments to generate testable hypotheses regarding transitions to non-angiogenic phenotypes that depend on the pre-treatment vascular phenotype. Additionally, we predicted a synergistic effect between anti-VEGF and anti-Ang2 treatments when applied to an immature pre-treatment vascular phenotype, but not when applied to a normalised angiogenic pre-treatment phenotype. Based on these findings, we conclude that changes in vascular phenotype are predicted to be useful as an experimental biomarker of response to treatment. Further, our analysis illustrates the potential value of non-spatial mathematical models for generating tractable predictions regarding the action of anti-angiogenic therapies.

Agent-based model of angiogenesis simulates capillary sprout initiation in multicellular networks

Many biological processes are controlled by both deterministic and stochastic influences. However, efforts to model these systems often rely on either purely stochastic or purely rule-based methods. To better understand the balance between stochasticity and determinism in biological processes a computational approach that incorporates both influences may afford additional insight into underlying biological mechanisms that give rise to emergent system properties. We apply a combined approach to the simulation and study of angiogenesis, the growth of new blood vessels from existing networks. This complex multicellular process begins with selection of an initiating endothelial cell, or tip cell, which sprouts from the parent vessels in response to stimulation by exogenous cues. We have constructed an agent-based model of sprouting angiogenesis to evaluate endothelial cell sprout initiation frequency and location, and we have experimentally validated it using high-resolution time-lapse confocal microscopy. ABM simulations were then compared to a Monte Carlo model, revealing that purely stochastic simulations could not generate sprout locations as accurately as the rule-informed agent-based model. These findings support the use of rule-based approaches for modeling the complex mechanisms underlying sprouting angiogenesis over purely stochastic methods.

A multiphase model for chemically- and mechanically- induced cell differentiation in a hollow fibre membrane bioreactor: minimising growth factor consumption

We present a simplified two-dimensional model of fluid flow, solute transport, and cell distribution in a hollow fibre membrane bioreactor. We consider two cell populations, one undifferentiated and one differentiated, with differentiation stimulated either by growth factor alone, or by both growth factor and fluid shear stress. Two experimental configurations are considered, a 3-layer model in which the cells are seeded in a scaffold throughout the extracapillary space (ECS), and a 4-layer model in which the cell-scaffold construct occupies a layer surrounding the outside of the hollow fibre, only partially filling the ECS. Above this is a region of free-flowing fluid, referred to as the upper fluid layer. Following previous models by the authors (Pearson et al. in Math Med Biol, 2013, Biomech Model Mechanbiol 1-16, 2014a, we employ porous mixture theory to model the dynamics of, and interactions between, the cells, scaffold, and fluid in the cell-scaffold construct. We use this model to determine operating conditions (experiment end time, growth factor inlet concentration, and inlet fluid fluxes) which result in a required percentage of differentiated cells, as well as maximising the differentiated cell yield and minimising the consumption of expensive growth factor.

The Force at the Tip--Modelling Tension and Proliferation in Sprouting Angiogenesis

Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art.

Sprouting angiogenesis—a process by which new blood vessels grow from existing ones—is an ubiquitous phenomenon in health and disease of higher organisms, playing a crucial role in organogenesis, wound healing, inflammation, as well as on the onset and progression of over 50 different diseases such as cancer, rheumatoid arthritis and diabetes. Mathematical models have the ability to suggest relevant hypotheses with respect to the mechanisms of cell movement and rearrangement within growing vessel sprouts. The inclusion of both biochemical and mechanical processes in a mathematical model of sprouting angiogenesis permits to describe sprout extension as a function of the forces exerted by the cells in the tissue. It also allows to question the regulation of biochemical processes by mechanical forces and vice-versa. In this work we present a compact model of sprouting angiogenesis that includes the mechanical characteristics of the vessel and the tissue. We use this model to suggest the mechanism for the regulation of proliferation within sprout formation. We conclude that the tip cell has the role of creating a tension in the cells that follow its lead. On those first cells of the stalk, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. The modelling strategy used, deemed phase-field, permits to describe the evolution of the shape of different domains in complex systems. It is focused on the movement of the interfaces between the domains, and not on an exhaustive description of the transport properties within each domain. For this reason, it requires a reduced number of parameters, and has been used extensively in modelling other biological phenomena such as tumor growth. The coupling of mechanical and biochemical processes in a compact mathematical model of angiogenesis will enable the study of lumen formation and aneurisms in the near future. Also, this framework will allow the study of the action of flow in vessel remodelling, since local forces can readily be coupled with cell movement to obtain the final vessel morphology.

A mathematical model for lymphangiogenesis in normal and diabetic wounds

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis) very few have been proposed for the regeneration of the lymphatic network. Moreover, lymphangiogenesis is markedly distinct from angiogenesis, occurring at different times and in a different manner. Here a model of five ordinary differential equations is presented to describe the formation of lymphatic capillaries following a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system is then solved numerically and the results are compared with the available biological literature. Finally, a parameter sensitivity analysis of the model is taken as a starting point for suggesting new therapeutic approaches targeting the enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper understanding of the phenomenon in question, clarifying the main factors involved. In particular, the balance between TGF-β and VEGF levels, rather than their absolute values, is identified as crucial to effective lymphangiogenesis. In addition, the results indicate lowering the macrophage-mediated activation of TGF-β and increasing the basal lymphatic endothelial cell growth rate, inter alia, as potential treatments. It is hoped the findings of this paper may be considered in the development of future experiments investigating novel lymphangiogenic therapies.

Systems biology of the microvasculature

The vascular network carries blood throughout the body, delivering oxygen to tissues and providing a pathway for communication between distant organs. The network is hierarchical and structured, but also dynamic, especially at the smaller scales. Remodeling of the microvasculature occurs in response to local changes in oxygen, gene expression, cell-cell communication, and chemical and mechanical stimuli from the microenvironment. These local changes occur as a result of physiological processes such as growth and exercise, as well as acute and chronic diseases including stroke, cancer, and diabetes, and pharmacological intervention. While the vasculature is an important therapeutic target in many diseases, drugs designed to inhibit vascular growth have achieved only limited success, and no drug has yet been approved to promote therapeutic vascular remodeling. This highlights the challenges involved in identifying appropriate therapeutic targets in a system as complex as the vasculature. Systems biology approaches provide a means to bridge current understanding of the vascular system, from detailed signaling dynamics measured in vitro and pre-clinical animal models of vascular disease, to a more complete picture of vascular regulation in vivo. This will translate to an improved ability to identify multi-component biomarkers for diagnosis, prognosis, and monitoring of therapy that are easy to measure in vivo, as well as better drug targets for specific disease states. In this review, we summarize systems biology approaches that have advanced our understanding of vascular function and dysfunction in vivo, with a focus on computational modeling.

Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis

Interleukin-35 (IL-35), a cytokine from the Interleukin-12 cytokine family, has been considered as an anti-inflammatory cytokine which promotes tumor progression and tumor immune evasion. It has also been demonstrated that IL-35 is secreted by regulatory T cells. Recent mouse experiments have shown that IL-35 produced by cancer cells promotes tumor growth via enhancing myeloid cell accumulation and angiogenesis, and reducing the infiltration of activated CD8[Formula: see text] T cells into tumor microenvironment. In the present paper we develop a mathematical model based on these experimental results. We include in the model an anti-IL-35 drug as treatment. The extended model (with drug) is used to design protocols of anti-IL-35 injections for treatment of cancer. We find that with a fixed total amount of drug, continuous injection has better efficacy than intermittent injections in reducing the tumor load while the treatment is ongoing. We also find that the percentage of tumor reduction under anti-IL-35 treatment improves when the production of IL-35 by cancer is increased.

Involvement of tumor macrophage HIFs in chemotherapy effectiveness: mathematical modeling of oxygen, pH, and glutathione

The four variables, hypoxia, acidity, high glutathione (GSH) concentration and fast reducing rate (redox) are distinct and varied characteristics of solid tumors compared to normal tissue. These parameters are among the most significant factors underlying the metabolism and physiology of solid tumors, regardless of their type or origin. Low oxygen tension contributes to both inhibition of cancer cell proliferation and therapeutic resistance of tumors; low extracellular pH, the reverse of normal cells, mainly enhances tumor invasion; and dysregulated GSH and redox potential within cancer cells favor their proliferation. In fact, cancer cells under these microenvironmental conditions appreciably alter tumor response to cytotoxic anti-cancer treatments. Recent experiments measured the in vivo longitudinal data of these four parameters with tumor development and the corresponding presence and absence of tumor macrophage HIF-1α or HIF-2α in a mouse model of breast cancer. In the current paper, we present a mathematical model-based system of (ordinary and partial) differential equations to monitor tumor growth and susceptibility to standard chemotherapy with oxygen level, pH, and intracellular GSH concentration. We first show that our model simulations agree with the corresponding experiments, and then we use our model to suggest treatments of tumors by altering these four parameters in tumor microenvironment. For example, the model qualitatively predicts that GSH depletion can raise the level of reactive oxygen species (ROS) above a toxic threshold and result in inhibition of tumor growth.

A systems biology view of blood vessel growth and remodelling

Blood travels throughout the body in an extensive network of vessels – arteries, veins and capillaries. This vascular network is not static, but instead dynamically remodels in response to stimuli from cells in the nearby tissue. In particular, the smallest vessels – arterioles, venules and capillaries – can be extended, expanded or pruned, in response to exercise, ischaemic events, pharmacological interventions, or other physiological and pathophysiological events. In this review, we describe the multi-step morphogenic process of angiogenesis – the sprouting of new blood vessels – and the stability of vascular networks in vivo. In particular, we review the known interactions between endothelial cells and the various blood cells and plasma components they convey. We describe progress that has been made in applying computational modelling, quantitative biology and high-throughput experimentation to the angiogenesis process.

A hybrid model of the role of VEGF binding in endothelial cell migration and capillary formation

Vascular endothelial growth factor (VEGF) is the most studied family of soluble, secreted mediators of endothelial cell migration, survival, and proliferation. VEGF exerts its function by binding to specific tyrosine kinase receptors on the cell surface and transducing the effect through downstream signaling. In order to study the influence of VEGF binding on endothelial cell motion, we develop a hybrid model of VEGF-induced angiogenesis, based on the theory of reinforced random walks. The model includes the chemotactic response of endothelial cells to angiogenic factors bound to cell-surface receptors, rather than approximating this as a function of extracellular chemical concentrations. This allows us to capture biologically observed phenomena such as activation and polarization of endothelial cells in response to VEGF gradients across their lengths, as opposed to extracellular gradients throughout the tissue. We also propose a novel and more biologically reasonable functional form for the chemotactic sensitivity of endothelial cells, which is also governed by activated cell-surface receptors. This model is able to predict the threshold level of VEGF required to activate a cell to move in a directed fashion as well as an optimal VEGF concentration for motion. Model validation is achieved by comparison of simulation results directly with experimental data.

The role of CD200-CD200R in tumor immune evasion

CD200 is a cell membrane protein that interacts with CD200 receptor (CD200R) of myeloid lineage cells. During tumor initiation and progression, CD200-positive tumor cells can interact with M1 and M2 macrophages through CD200-CD200R-compex, and downregulate IL-10 and IL-12 productions secreted primarily by M2 and M1 macrophages, respectively. In the tumor microenvironment, IL-10 inhibits the activation of cytotoxic T lymphocytes (CTL), while IL-12 enhances CTL activation. In this paper, we used a system approach to determine the combined effect of CD200-CD200R interaction on tumor proliferation by developing a mathematical model. We demonstrate that blocking CD200 on tumor cells may have opposite effects on tumor proliferation depending on the "affinity" of the macrophages to form the CD200-CD200R-complex with tumor cells. Our results help understanding the complexities of tumor microenvironment.

Hypoxia inducible factors-mediated inhibition of cancer by GM-CSF: a mathematical model

Under hypoxia, tumor cells, and tumor-associated macrophages produce VEGF (vascular endothelial growth factor), a signaling molecule that induces angiogenesis. The same macrophages, when treated with GM-CSF (granulocyte/macrophage colony-stimulating factor), produce sVEGFR-1 (soluble VEGF receptor-1), a soluble protein that binds with VEGF and inactivates its function. The production of VEGF by macrophages is regulated by HIF-1α (hypoxia inducible factor-1α), and the production of sVEGFR-1 is mediated by HIF-2α. Recent experiments measured the effect of inhibiting tumor growth by GM-CSF treatment in mice with HIF-1α-deficient or HIF-2α-deficient macrophages. In the present paper, we represent these experiments by a mathematical model based on a system of partial differential equations. We show that the model simulations agree with the above experiments. The model can then be used to suggest strategies for inhibiting tumor growth. For example, the model qualitatively predicts the extent to which GM-CSF treatment in combination with a small molecule inhibitor that stabilizes HIF-2α will reduce tumor volume and angiogenesis.

Modeling the inhibition of breast cancer growth by GM-CSF

M-CSF is overexpressed in breast cancer and is known to stimulate macrophages to produce VEGF resulting in angiogenesis. It has recently been shown that the growth factor GM-CSF injected into murine breast tumors slowed tumor growth by secreting soluble VEGF receptor-1 (sVEGFR-1) that binds and inactivates VEGF. This study presents a mathematical model that includes all the components above, as well as MCP-1, tumor cells, and oxygen. The model simulations are representative of the in vivo data through predictions of tumor growth using different protocol strategies for GM-CSF for the purpose of predicting higher degrees of treatment success. For example, our model predicts that once a week dosing of GM-CSF would be less effective than daily, twice a week, or three times a week treatment because of the presence of essential factors required for the anti-tumor effect of GM-CSF.

Cancer stem cells in solid tumors: is 'evading apoptosis' a hallmark of cancer?

Conventional wisdom has long held that once a cancer cell has developed it will inevitably progress to clinical disease. Updating this paradigm, it has more recently become apparent that the tumor interacts with its microenvironment and that some environmental bottlenecks, such as the angiogenic switch, must be overcome for the tumor to progress. In parallel, attraction has been drawn to the concept that there is a minority population of cells - the cancer stem cells - bestowed with the exclusive ability to self-renew and regenerate the tumor. With therapeutic targeting issues at stake, much attention has shifted to the identification of cancer stem cells, the thinking being that the remaining non-stem population, already fated to die, will play a negligible role in tumor development. In fact, the newly appreciated importance of intercellular interactions in cancer development also extends in a unique and unexpected way to interactions between the stem and non-stem compartments of the tumor. Here we discuss recent findings drawn from a hybrid mathematical-cellular automaton model that simulates growth of a heterogeneous solid tumor comprised of cancer stem cells and non-stem cancer cells. The model shows how the introduction of cell fate heterogeneity paradoxically influences the tumor growth dynamic in response to apoptosis, to reveal yet another bottleneck to tumor progression potentially exploitable for disease control.

A cell-based model of endothelial cell migration, proliferation and maturation during corneal angiogenesis

The motivation of this work stems from two critical experimental observations associated with corneal angiogenesis: (1) angiogenesis will not succeed without endothelial cell proliferation, and (2) proliferation mainly occurs at the leading edge of developing sprouts (Sholley et al., Lab. Invest. 51:624-634, 1984). To discover the underlying mechanisms of these phenomena, we develop a cell-based mathematical model that integrates a mechanical model of elongation with a biochemical model of cell phenotype variation regulated by angiopoietins within a developing sprout. This model allows for a detailed study of the relative roles of endothelial cell migration, proliferation, and maturation. The model is validated by quantitatively comparing its predictions with data derived from corneal angiogenesis experiments. We conclude that cell elasticity and cell-to-cell adhesion allow only limited sprout extension in the absence of proliferation, and the maturation process combined with bioavailability of VEGF can explain the localization of proliferation to the leading edge. We also use this model to investigate the effects of X-ray irradiation, Ang-2 inhibition, and extracellular matrix anisotropy on sprout morphology and extension.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Random walk models in biology

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

A stochastic model of tumor angiogenesis

Angiogenesis is a prerequisite for the growth, invasion and metastasis of solid tumors. The aim of this study was to investigate the changes of microvessel densities in tumor with a method of probability. We showed that the stages of tumor angiogenesis may follow a probabilistic pattern compatible with Markov chain and this could be of prognostic importance. The system was simulated with 10,000 patients in order to show the applicability of this method. This approach may improve classification schemes and perhaps make diagnosis and treatment more individualized to patients.

A hybrid model for three-dimensional simulations of sprouting angiogenesis

Recent advances in cancer research have identified critical angiogenic signaling pathways and the influence of the extracellular matrix on endothelial cell migration. These findings provide us with insight into the process of angiogenesis that can facilitate the development of effective computational models of sprouting angiogenesis. In this work, we present the first three-dimensional model of sprouting angiogenesis that considers explicitly the effect of the extracellular matrix and of the soluble as well as matrix-bound growth factors on capillary growth. The computational model relies on a hybrid particle-mesh representation of the blood vessels and it introduces an implicit representation of the vasculature that can accommodate detailed descriptions of nutrient transport. Extensive parametric studies reveal the role of the extracellular matrix structure and the distribution of the different vascular endothelial growth factors isoforms on the dynamics and the morphology of the generated vascular networks.

Modeling the VEGF–Bcl-2–CXCL8 Pathway in Intratumoral Agiogenesis

Recent experiments show that vascular endothelial growth factor (VEGF) is the crucial mediator of downstream events that ultimately lead to enhanced endothelial cell survival and increased vascular density within many tumors. The newly discovered pathway involves up-regulation of the anti-apoptotic protein Bcl-2, which in turn leads to increased production of interleukin-8 (CXCL8). The VEGF-Bcl-2-CXCL8 pathway suggests new targets for the development of anti-angiogenic strategies including short interfering RNA (siRNA) that silence the CXCL8 gene and small molecule inhibitors of Bcl-2. In this paper, we present and validate a mathematical model designed to predict the effect of the therapeutic blockage of VEGF, CXCL8, and Bcl-2 at different stages of tumor progression. In agreement with experimental observations, the model predicts that curtailing the production of CXCL8 early in development can result in a delay in tumor growth and vascular development; however, it has little effect when applied at late stages of tumor progression. Numerical simulations also show that blocking Bcl-2 up-regulation, either at early stages or after the tumor has fully developed, ensures that both microvascular and tumor cell density stabilize at low values representing growth control. These results provide insight into those aspects of the VEGF-Bcl-2-CXCL8 pathway, which independently and in combination, are crucial mediators of tumor growth and vascular development. Continued quantitative modeling in this direction may have profound implications for the development of novel therapies directed against specific proteins and chemokines to alter tumor progression.

Modeling the effects of vasculature evolution on early brain tumor growth

Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.

Mathematical modeling of fracture healing in mice: comparison between experimental data and numerical simulation results

The combined use of experimental and mathematical models can lead to a better understanding of fracture healing. In this study, a mathematical model, which was originally established by Bailón-Plaza and van der Meulen (J Theor Biol 212:191-209, 2001), was applied to an experimental model of a semi-stabilized murine tibial fracture. The mathematical model was implemented in a custom finite volumes code, specialized in dealing with the model's requirements of mass conservation and non-negativity of the variables. A qualitative agreement between the experimentally measured and numerically simulated evolution in the cartilage and bone content was observed. Additionally, an extensive parametric study was conducted to assess the influence of the model parameters on the simulation outcome. Finally, a case of pathological fracture healing and its treatment by administration of growth factors was modeled to demonstrate the potential therapeutic value of this mathematical model.