Computational models of VEGF-associated angiogenic processes in cancer

Integrated cancer tissue engineering models for precision medicine

Tumors are not merely cancerous cells that undergo mindless proliferation. Rather, they are highly organized and interconnected organ systems. Tumor cells reside in complex microenvironments in which they are subjected to a variety of physical and chemical stimuli that influence cell behavior and ultimately the progression and maintenance of the tumor. As cancer bioengineers, it is our responsibility to create physiologic models that enable accurate understanding of the multi-dimensional structure, organization, and complex relationships in diverse tumor microenvironments. Such models can greatly expedite clinical discovery and translation by closely replicating the physiological conditions while maintaining high tunability and control of extrinsic factors. In this review, we discuss the current models that target key aspects of the tumor microenvironment and their role in cancer progression. In order to address sources of experimental variation and model limitations, we also make recommendations for methods to improve overall physiologic reproducibility, experimental repeatability, and rigor within the field. Improvements can be made through an enhanced emphasis on mathematical modeling, standardized in vitro model characterization, transparent reporting of methodologies, and designing experiments with physiological metrics. Taken together these considerations will enhance the relevance of in vitro tumor models, biological understanding, and accelerate treatment exploration ultimately leading to improved clinical outcomes. Moreover, the development of robust, user-friendly models that integrate important stimuli will allow for the in-depth study of tumors as they undergo progression from non-transformed primary cells to metastatic disease and facilitate translation to a wide variety of biological and clinical studies.

Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics

For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T + ), those producing testosterone ( T P ), and those independent of testosterone ( T - ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T + cells (highest frequency in zone 1) benefit from interacting with T P cells (highest frequency in zone 2). Zone 1 T + cells interacting with cells in zone 2 experience a higher likelihood of encountering a T P cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.

3D hybrid modelling of vascular network formation

We develop an off-lattice, agent-based model to describe vasculogenesis, the de novo formation of blood vessels from endothelial progenitor cells during development. The endothelial cells that comprise our vessel network are viewed as linearly elastic spheres that move in response to the forces they experience. We distinguish two types of endothelial cells: vessel elements are contained within the network and tip cells are located at the ends of vessels. Tip cells move in response to mechanical forces caused by interactions with neighbouring vessel elements and the local tissue environment, chemotactic forces and a persistence force which accounts for their tendency to continue moving in the same direction. Vessel elements are subject to similar mechanical forces but are insensitive to chemotaxis. An angular persistence force representing interactions with the local tissue is introduced to stabilise buckling instabilities caused by cell proliferation. Only vessel elements proliferate, at rates which depend on their degree of stretch: elongated elements have increased rates of proliferation, and compressed elements have reduced rates. Following division, the fate of the new cell depends on the local mechanical environment: the probability of forming a new sprout is increased if the parent vessel is highly compressed and the probability of being incorporated into the parent vessel increased if the parent is stretched. Simulation results reveal that our hybrid model can reproduce the key qualitative features of vasculogenesis. Extensive parameter sensitivity analyses show that significant changes in network size and morphology are induced by varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and the sprouting probability to mechanical stretch. Varying the chemotactic sensitivity directly influences the directionality of the networks. The degree of branching, and thereby the density of the networks, is influenced by the sprouting probability. Glyphs that simultaneously depict several network properties are introduced to show how these and other network quantities change over time and also as model parameters vary. We also show how equivalent glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, in the longer term, for model validation. We conclude that our biomechanical hybrid model can generate vascular networks that are qualitatively similar to those generated from in vitro and in vivo experiments.

Effects of endothelial cell proliferation and migration rates in a computational model of sprouting angiogenesis

Angiogenesis, the recruitment of new blood vessels, is a critical process for the growth, expansion, and metastatic dissemination of developing tumors. Three types of cells make up the new vasculature: tip cells, which migrate in response to gradients of vascular endothelial growth factor (VEGF), stalk cells, which proliferate and extend the vessels, and phalanx cells, which are quiescent and support the sprout. In this study we examine the contribution of tip cell migration rate and stalk cell proliferation rate on the formation of new vasculature. We calculate several vascular metrics, such as the number of vascular bifurcations per unit volume, vascular segment length per unit volume, and vascular tortuosity. These measurements predict that proliferation rate has a greater effect on the spread and extent of vascular growth compared to migration rate. Together, these findings provide strong implications for designing anti-angiogenic therapies that may differentially target endothelial cell proliferation and migration. Computational models can be used to predict optimal anti-angiogenic therapies in combination with other therapeutics to improve outcome.

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.

Optimal chemotherapy for leukemia: a model-based strategy for individualized treatment

Acute Lymphoblastic Leukemia, commonly known as ALL, is a predominant form of cancer during childhood. With the advent of modern healthcare support, the 5-year survival rate has been impressive in the recent past. However, long-term ALL survivors embattle several treatment-related medical and socio-economic complications due to excessive and inordinate chemotherapy doses received during treatment. In this work, we present a model-based approach to personalize 6-Mercaptopurine (6-MP) treatment for childhood ALL with a provision for incorporating the pharmacogenomic variations among patients. Semi-mechanistic mathematical models were developed and validated for i) 6-MP metabolism, ii) red blood cell mean corpuscular volume (MCV) dynamics, a surrogate marker for treatment efficacy, and iii) leukopenia, a major side-effect. With the constraint of getting limited data from clinics, a global sensitivity analysis based model reduction technique was employed to reduce the parameter space arising from semi-mechanistic models. The reduced, sensitive parameters were used to individualize the average patient model to a specific patient so as to minimize the model uncertainty. Models fit the data well and mimic diverse behavior observed among patients with minimum parameters. The model was validated with real patient data obtained from literature and Riley Hospital for Children in Indianapolis. Patient models were used to optimize the dose for an individual patient through nonlinear model predictive control. The implementation of our approach in clinical practice is realizable with routinely measured complete blood counts (CBC) and a few additional metabolite measurements. The proposed approach promises to achieve model-based individualized treatment to a specific patient, as opposed to a standard-dose-for-all, and to prescribe an optimal dose for a desired outcome with minimum side-effects.

Stimulus-responsive ultrasound contrast agents for clinical imaging: motivations, demonstrations, and future directions

Microbubble ultrasound contrast agents allow imaging of the vasculature with excellent resolution and signal-to-noise ratios. Contrast in microbubbles derives from their interaction with an ultrasound wave to generate signal at harmonic frequencies of the stimulating pulse; subtracting the elastic echo caused by the surrounding tissue can enhance the specificity of these harmonic signals significantly. The nonlinear acoustic emission is caused by pressure-driven microbubble size fluctuations, which in both theoretical descriptions and empirical measurements was found to depend on the mechanical properties of the shell that encapsulates the microbubble as well as stabilizes it against the surrounding aqueous environment. Thus biochemically induced switching between a rigid 'off' state and a flexible 'on' state provides a mechanism for sensing chemical markers for disease. In our research, we coupled DNA oligonucleotides to a stabilizing lipid monolayer to modulate stiffness of the shell and thereby induce stimulus-responsive behavior. In initial proof-of-principle studies, it was found that signal modulation came primarily from DNA crosslinks preventing the microbubble size oscillations rather than merely damping the signal. Next, these microbubbles were redesigned to include an aptamer sequence in the crosslinking strand, which not only allowed the sensing of the clotting enzyme thrombin but also provided a general strategy for sensing other soluble biomarkers in the bloodstream. Finally, the thrombin-sensitive microbubbles were validated in a rabbit model, presenting the first example of an ultrasound contrast agent that could differentiate between active and inactive clots for the diagnosis of deep venous thrombosis.

The Proteome of a Healthy Human during Physical Activity under Extreme Conditions

The review examines the new approaches in modern systems biology, in terms of their use for a deeper understanding of the physiological adaptation of a healthy human in extreme environments. Human physiology under extreme conditions of life, or environmental physiology, and systems biology are natural partners. The similarities and differences between the object and methods in systems biology, the OMICs (proteomics, transcriptomics, metabolomics) disciplines, and other related sciences have been studied. The latest data on environmental human physiology obtained using systems biology methods are discussed. The independent achievements of systems biology in studying the adaptation of a healthy human to physical activity, including human presence at high altitude, to the effects of hypoxia and oxidative stress have been noted. A reasonable conclusion is drawn that the application of the methods and approaches used in systems biology to study the molecular pattern of the adaptive mechanisms that develop in the human body during space flight can provide valuable fundamental knowledge and fill the picture of human metabolic pathways.

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.

Compartment model predicts VEGF secretion and investigates the effects of VEGF trap in tumor-bearing mice

Angiogenesis, the formation of new blood vessels from existing vasculature, is important in tumor growth and metastasis. A key regulator of angiogenesis is vascular endothelial growth factor (VEGF), which has been targeted in numerous anti-angiogenic therapies aimed at inhibiting tumor angiogenesis. Systems biology approaches, including computational modeling, are useful for understanding this complex biological process and can aid in the development of novel and effective therapeutics that target the VEGF family of proteins and receptors. We have developed a computational model of VEGF transport and kinetics in the tumor-bearing mouse, which includes three-compartments: normal tissue, blood, and tumor. The model simulates human tumor xenografts and includes human (VEGF121 and VEGF165) and mouse (VEGF120 and VEGF164) isoforms. The model incorporates molecular interactions between these VEGF isoforms and receptors (VEGFR1 and VEGFR2), as well as co-receptors (NRP1 and NRP2). We also include important soluble factors: soluble VEGFR1 (sFlt-1) and α-2-macroglobulin. The model accounts for transport via macromolecular transendothelial permeability, lymphatic flow, and plasma clearance. We have fit the model to available in vivo experimental data on the plasma concentration of free VEGF Trap and VEGF Trap bound to mouse and human VEGF in order to estimate the rates at which parenchymal cells (myocytes and tumor cells) and endothelial cells secrete VEGF. Interestingly, the predicted tumor VEGF secretion rates are significantly lower (0.007-0.023 molecules/cell/s, depending on the tumor microenvironment) than most reported in vitro measurements (0.03-2.65 molecules/cell/s). The optimized model is used to investigate the interstitial and plasma VEGF concentrations and the effect of the VEGF-neutralizing agent, VEGF Trap (aflibercept). This work complements experimental studies performed in mice and provides a framework with which to examine the effects of anti-VEGF agents, aiding in the optimization of such anti-angiogenic therapeutics as well as analysis of clinical data. The model predictions also have implications for biomarker discovery with anti-angiogenic therapies.

Applying systems biology methods to the study of human physiology in extreme environments

Systems biology is defined in this review as ‘an iterative process of computational model building and experimental model revision with the aim of understanding or simulating complex biological systems’. We propose that, in practice, systems biology rests on three pillars: computation, the omics disciplines and repeated experimental perturbation of the system of interest. The number of ethical and physiologically relevant perturbations that can be used in experiments on healthy humans is extremely limited and principally comprises exercise, nutrition, infusions (e.g. Intralipid), some drugs and altered environment. Thus, we argue that systems biology and environmental physiology are natural symbionts for those interested in a system-level understanding of human biology. However, despite excellent progress in high-altitude genetics and several proteomics studies, systems biology research into human adaptation to extreme environments is in its infancy. A brief description and overview of systems biology in its current guise is given, followed by a mini review of computational methods used for modelling biological systems. Special attention is given to high-altitude research, metabolic network reconstruction and constraint-based modelling.

Manipulating the microvasculature and its microenvironment

The microvasculature is a dynamic cellular system necessary for tissue health and function. Therapeutic strategies that target the microvasculature are expanding and evolving, including those promoting angiogenesis and microvascular expansion. When considering how to manipulate angiogenesis, either as part of a tissue construction approach or a therapy to improve tissue blood flow, it is important to know the microenvironmental factors that regulate and direct neovessel sprouting and growth. Much is known concerning both diffusible and matrix-bound angiogenic factors, which stimulate and guide angiogenic activity. How the other aspects of the extravascular microenvironment, including tissue biomechanics and structure, influence new vessel formation is less well known. Recent research, however, is providing new insights into these mechanisms and demonstrating that the extent and character of angiogenesis (and the resulting new microcirculation) is significantly affected. These observations and the resulting implications with respect to tissue construction and microvascular therapy are addressed.

Multiscale imaging and computational modeling of blood flow in the tumor vasculature

The evolution in our understanding of tumor angiogenesis has been the result of pioneering imaging and computational modeling studies spanning the endothelial cell, microvasculature and tissue levels. Many of these primary data on the tumor vasculature are in the form of images from pre-clinical tumor models that provide a wealth of qualitative and quantitative information in many dimensions and across different spatial scales. However, until recently, the visualization of changes in the tumor vasculature across spatial scales remained a challenge due to a lack of techniques for integrating micro- and macroscopic imaging data. Furthermore, the paucity of three-dimensional (3-D) tumor vascular data in conjunction with the challenges in obtaining such data from patients presents a serious hurdle for the development and validation of predictive, multiscale computational models of tumor angiogenesis. In this review, we discuss the development of multiscale models of tumor angiogenesis, new imaging techniques capable of reproducing the 3-D tumor vascular architecture with high fidelity, and the emergence of "image-based models" of tumor blood flow and molecular transport. Collectively, these developments are helping us gain a fundamental understanding of the cellular and molecular regulation of tumor angiogenesis that will benefit the development of new cancer therapies. Eventually, we expect this exciting integration of multiscale imaging and mathematical modeling to have widespread application beyond the tumor vasculature to other diseases involving a pathological vasculature, such as stroke and spinal cord injury.

Integration of experimental and computational approaches to sprouting angiogenesis

PURPOSE OF REVIEW

We summarize recent experimental and computational studies that investigate molecular and cellular mechanisms of sprouting angiogenesis. We discuss how experimental tools have unveiled new opportunities for computational modeling by providing detailed phenomenological descriptions and conceptual models of cell-level behaviors underpinned by high-quality molecular data. Using recent examples, we show how new understanding results from bridging computational and experimental approaches.

RECENT FINDINGS

Experimental data extends beyond the tip cell vs. stalk cell paradigm, and involves numerous molecular inputs such as vascular endothelial growth factor and Notch. This data is being used to generate and validate computational models, which can then be used to predict the results of hypothetical experiments that are difficult to perform in the laboratory, and to generate new hypotheses that account for system-wide interactions. As a result of this integration, descriptions of critical gradients of growth factor-receptor complexes have been generated, and new modulators of cell behavior have been described.

SUMMARY

We suggest that the recent emphasis on the different stages of sprouting angiogenesis, and integration of experimental and computational approaches, should provide a way to manage the complexity of this process and help identify new regulatory paradigms and therapeutic targets.

Pharmacokinetics and pharmacodynamics of VEGF-neutralizing antibodies

Background

Vascular endothelial growth factor (VEGF) is a potent regulator of angiogenesis, and its role in cancer biology has been widely studied. Many cancer therapies target angiogenesis, with a focus being on VEGF-mediated signaling such as antibodies to VEGF. However, it is difficult to predict the effects of VEGF-neutralizing agents. We have developed a whole-body model of VEGF kinetics and transport under pathological conditions (in the presence of breast tumor). The model includes two major VEGF isoforms VEGF₁₂₁ and VEGF₁₆₅, receptors VEGFR1, VEGFR2 and co-receptors Neuropilin-1 and Neuropilin-2. We have added receptors on parenchymal cells (muscle fibers and tumor cells), and incorporated experimental data for the cell surface density of receptors on the endothelial cells, myocytes, and tumor cells. The model is applied to investigate the action of VEGF-neutralizing agents (called "anti-VEGF") in the treatment of cancer.

Results

Through a sensitivity study, we examine how model parameters influence the level of free VEGF in the tumor, a measure of the response to VEGF-neutralizing drugs. We investigate the effects of systemic properties such as microvascular permeability and lymphatic flow, and of drug characteristics such as the clearance rate and binding affinity. We predict that increasing microvascular permeability in the tumor above 10^-5 cm/s elicits the undesired effect of increasing tumor interstitial VEGF concentration beyond even the baseline level. We also examine the impact of the tumor microenvironment, including receptor expression and internalization, as well as VEGF secretion. We find that following anti-VEGF treatment, the concentration of free VEGF in the tumor can vary between 7 and 233 pM, with a dependence on both the density of VEGF receptors and co-receptors and the rate of neuropilin internalization on tumor cells. Finally, we predict that free VEGF in the tumor is reduced following anti-VEGF treatment when VEGF₁₂₁ comprises at least 25% of the VEGF secreted by tumor cells.

Conclusions

This study explores the optimal drug characteristics required for an anti-VEGF agent to have a therapeutic effect and the tumor-specific properties that influence the response to therapy. Our model provides a framework for investigating the use of VEGF-neutralizing drugs for personalized medicine treatment strategies.

A two-compartment model of VEGF distribution in the mouse

Vascular endothelial growth factor (VEGF) is a key regulator of angiogenesis – the growth of new microvessels from existing microvasculature. Angiogenesis is a complex process involving numerous molecular species, and to better understand it, a systems biology approach is necessary. In vivo preclinical experiments in the area of angiogenesis are typically performed in mouse models; this includes drug development targeting VEGF. Thus, to quantitatively interpret such experimental results, a computational model of VEGF distribution in the mouse can be beneficial. In this paper, we present an in silico model of VEGF distribution in mice, determine model parameters from existing experimental data, conduct sensitivity analysis, and test the validity of the model.

The multiscale model is comprised of two compartments: blood and tissue. The model accounts for interactions between two major VEGF isoforms (VEGF₁₂₀ and VEGF₁₆₄) and their endothelial cell receptors VEGFR-1, VEGFR-2, and co-receptor neuropilin-1. Neuropilin-1 is also expressed on the surface of parenchymal cells. The model includes transcapillary macromolecular permeability, lymphatic transport, and macromolecular plasma clearance. Simulations predict that the concentration of unbound VEGF in the tissue is approximately 50-fold greater than in the blood. These concentrations are highly dependent on the VEGF secretion rate. Parameter estimation was performed to fit the simulation results to available experimental data, and permitted the estimation of VEGF secretion rate in healthy tissue, which is difficult to measure experimentally. The model can provide quantitative interpretation of preclinical animal data and may be used in conjunction with experimental studies in the development of pro- and anti-angiogenic agents. The model approximates the normal tissue as skeletal muscle and includes endothelial cells to represent the vasculature. As the VEGF system becomes better characterized in other tissues and cell types, the model can be expanded to include additional compartments and vascular elements.