A review of mathematical models for the formation of vascular networks

Formation of vascular-like structures using a chemotaxis-driven multiphase model

We propose a continuum model for pattern formation, based on the multiphase model framework, to explore in vitro cell patterning within an extracellular matrix (ECM). We demonstrate that, within this framework, chemotaxis-driven cell migration can lead to the formation of cell clusters and vascular-like structures in 1D and 2D respectively. The influence on pattern formation of additional mechanisms commonly included in multiphase tissue models, including cell-matrix traction, contact inhibition, and cell-cell aggregation, are also investigated. Using sensitivity analysis, the relative impact of each model parameter on the simulation outcomes is assessed to identify the key parameters involved. Chemoattractant-matrix binding is further included, motivated by previous experimental studies, and found to reduce the spatial scale of patterning to within a biologically plausible range for capillary structures. Key findings from the in-depth parameter analysis of the 1D models, both with and without chemoattractant-matrix binding, are demonstrated to translate well to the 2D model, obtaining vascular-like cell patterning for multiple parameter regimes. Overall, we demonstrate a biologically-motivated multiphase model capable of generating long-term pattern formation on a biologically plausible spatial scale both in 1D and 2D, with applications for modelling in vitro vascular network formation.

Radiomics Multifactorial in Silico Model for Spatial Prediction of Glioblastoma Progression and Recurrence: A Proof-of-Concept

BACKGROUND

Radiomics-based prediction of glioblastoma spatial progression and recurrence may improve personalized strategies. However, most prototypes are based on limited monofactorial Gompertzian models of tumor growth. The present study consists of a proof of concept on the accuracy of a radiomics multifactorial in silico model in predicting short-term spatial growth and recurrence of glioblastoma.

METHODS

A radiomics-based biomathematical multifactorial in silico model was developed using magnetic resonance imaging (MRI) data from a 53-year-old patient with newly diagnosed glioblastoma of the right supramarginal gyrus. Raw and optimized models were derived from the MRI at diagnosis and matched to the preoperative MRI obtained 28 days after diagnosis to test the accuracy in predicting the short-term spatial growth of the tumor. An additional optimized model was derived from the early postoperative MRI and matched to the MRI documenting tumor recurrence to test spatial accuracy in predicting the location of recurrence. The spatial prediction accuracy of the model was reported as an average Jaccard index.

RESULTS

Optimized models yielded an average Jaccard index of 0.69 and 0.26 for short-term tumor growth and long-term recurrence site, respectively.

CONCLUSIONS

The present radiomics-based multifactorial in silico model was feasible, reliable, and accurate for short-term spatial prediction of glioblastoma progression. The predictive value for the spatial location of recurrence was still low, and refinements in the description of tissue reorganization in the peritumoral and resected areas may be critical to optimize accuracy further.

Computational modeling of angiogenesis: The importance of cell rearrangements during vascular growth

Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre-existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the "snail-trail" hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low-oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail-trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role. This article is categorized under: Cardiovascular Diseases > Computational Models Cancer > Computational Models.

Long-term day-by-day tracking of microvascular networks sprouting in fibrin gels: From detailed morphological analyses to general growth rules

Understanding and controlling of the evolution of sprouting vascular networks remains one of the basic challenges in tissue engineering. Previous studies on the vascularization dynamics have typically focused only on the phase of intense growth and often lacked spatial control over the initial cell arrangement. Here, we perform long-term day-by-day analysis of tens of isolated microvasculatures sprouting from endothelial cell-coated spherical beads embedded in an external fibrin gel. We systematically study the topological evolution of the sprouting networks over their whole lifespan, i.e., for at least 14 days. We develop a custom image analysis toolkit and quantify (i) the overall length and area of the sprouts, (ii) the distributions of segment lengths and branching angles, and (iii) the average number of branch generations-a measure of network complexity. We show that higher concentrations of vascular endothelial growth factor (VEGF) lead to earlier sprouting and more branched networks, yet without significantly affecting the speed of growth of individual sprouts. We find that the mean branching angle is weakly dependent on VEGF and typically in the range of 60°-75°, suggesting that, by comparison with the available diffusion-limited growth models, the bifurcating tips tend to follow local VEGF gradients. At high VEGF concentrations, we observe exponential distributions of segment lengths, which signify purely stochastic branching. Our results-due to their high statistical relevance-may serve as a benchmark for predictive models, while our new image analysis toolkit, offering unique features and high speed of operation, could be exploited in future angiogenic drug tests.

Optimal mechanical interactions direct multicellular network formation on elastic substrates

Cells self-organize into functional, ordered structures during tissue morphogenesis, a process that is evocative of colloidal self-assembly into engineered soft materials. Understanding how intercellular mechanical interactions may drive the formation of ordered and functional multicellular structures is important in developmental biology and tissue engineering. Here, by combining an agent-based model for contractile cells on elastic substrates with endothelial cell culture experiments, we show that substrate deformation-mediated mechanical interactions between cells can cluster and align them into branched networks. Motivated by the structure and function of vasculogenic networks, we predict how measures of network connectivity like percolation probability and fractal dimension as well as local morphological features including junctions, branches, and rings depend on cell contractility and density and on substrate elastic properties including stiffness and compressibility. We predict and confirm with experiments that cell network formation is substrate stiffness dependent, being optimal at intermediate stiffness. We also show the agreement between experimental data and predicted cell cluster types by mapping a combined phase diagram in cell density substrate stiffness. Overall, we show that long-range, mechanical interactions provide an optimal and general strategy for multicellular self-organization, leading to more robust and efficient realizations of space-spanning networks than through just local intercellular interactions.

Soliton approximation in continuum models of leader-follower behavior

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations (PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equations at leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

Approaches to vascular network, blood flow, and metabolite distribution modeling in brain tissue

The cardiovascular system plays a key role in the transport of nutrients, ensuring a continuous supply of all cells of the body with the metabolites necessary for life. The blood supply to the brain is carried out by the large arteries located on its surface, which branch into smaller arterioles that penetrate the cerebral cortex and feed the capillary bed, thereby forming an extensive branching network. The formation of blood vessels is carried out via vasculogenesis and angiogenesis, which play an important role in both embryo and adult life. The review presents approaches to modeling various aspects of both the formation of vascular networks and the construction of the formed arterial tree. In addition, a brief description of models that allows one to study the blood flow in various parts of the circulatory system and the spatiotemporal metabolite distribution in brain tissues is given. Experimental study of these issues is not always possible due to both the complexity of the cardiovascular system and the mechanisms through which the perfusion of all body cells is carried out. In this regard, mathematical models are a good tool for studying hemodynamics and can be used in clinical practice to diagnose vascular diseases and assess the need for treatment.

Bio-inspired microfluidics: A review

Biomicrofluidics, a subdomain of microfluidics, has been inspired by several ideas from nature. However, while the basic inspiration for the same may be drawn from the living world, the translation of all relevant essential functionalities to an artificially engineered framework does not remain trivial. Here, we review the recent progress in bio-inspired microfluidic systems via harnessing the integration of experimental and simulation tools delving into the interface of engineering and biology. Development of "on-chip" technologies as well as their multifarious applications is subsequently discussed, accompanying the relevant advancements in materials and fabrication technology. Pointers toward new directions in research, including an amalgamated fusion of data-driven modeling (such as artificial intelligence and machine learning) and physics-based paradigm, to come up with a human physiological replica on a synthetic bio-chip with due accounting of personalized features, are suggested. These are likely to facilitate physiologically replicating disease modeling on an artificially engineered biochip as well as advance drug development and screening in an expedited route with the minimization of animal and human trials.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Mechanical regulation of the early stages of angiogenesis

Favouring or thwarting the development of a vascular network is essential in fields as diverse as oncology, cardiovascular disease or tissue engineering. As a result, understanding and controlling angiogenesis has become a major scientific challenge. Mechanical factors play a fundamental role in angiogenesis and can potentially be exploited for optimizing the architecture of the resulting vascular network. Largely focusing on in vitro systems but also supported by some in vivo evidence, the aim of this Highlight Review is dual. First, we describe the current knowledge with particular focus on the effects of fluid and solid mechanical stimuli on the early stages of the angiogenic process, most notably the destabilization of existing vessels and the initiation and elongation of new vessels. Second, we explore inherent difficulties in the field and propose future perspectives on the use of in vitro and physics-based modelling to overcome these difficulties.

On the role of mechanical signals on sprouting angiogenesis through computer modeling approaches

Sprouting angiogenesis, the formation of new vessels from preexisting vasculature, is an essential process in the regeneration of new tissues as well as in the development of some diseases like cancer. Although early studies identified chemical signaling as the main driver of this process, many recent studies have shown a strong role of mechanical signals in the formation of new capillaries. Different types of mechanical signals (e.g., external forces, cell traction forces, and blood flow-induced shear forces) have been shown to play distinct roles in the process; however, their interplay remains still largely unknown. During the last decades, mathematical and computational modeling approaches have been developed to investigate and better understand the mechanisms behind mechanically driven angiogenesis. In this manuscript, we review computational models of angiogenesis with a focus on models investigating the role of mechanics on the process. Our aim is not to provide a detailed review on model methodology but to describe what we have learnt from these models. We classify models according to the mechanical signals being investigated and describe how models have looked into their role on the angiogenic process. We show that a better understanding of the mechanobiology of the angiogenic process will require the development of computer models that incorporate the interactions between the multiple mechanical signals and their effect on cellular responses, since they all seem to play a key in sprout patterning. In the end, we describe some of the remaining challenges of computational modeling of angiogenesis and discuss potential avenues for future research.

A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein

Chemotaxis, the migration of cells in response to chemical stimulus, is an important concept in the angiogenesis model. In most angiogenesis models, chemotaxis is defined as the migration of a sprout tip in response to the upgradient of the VEGF (vascular endothelial growth factor). However, we found that angiogenesis induced by performing arterial patch grafting on rabbits occurred under the decreasing VEGFA gradient. Data show that the VEGFA concentration peaked at approximately 0.3 to 0.5 cm away from the arterial patch and decreased as the measurement approaches the patch. We also observed that the new blood vessels formed are twisted and congested in some areas, in a distinguishable manner from non-pathological blood vessels. To explain these observations, we developed a mathematical model and compared the results from numerical simulations with the experimental data. We introduced a new chemotactic velocity using the temporal change in the chemoattractant gradient to govern the sprout tip migration. We performed a hybrid simulation to illustrate the growth of new vessels. Results indicated the speed of growth of new vessels oscillated before reaching the periphery of the arterial patch. Crowded and congested blood vessel formation was observed during numerical simulations. Thus, our numerical simulation results agreed with the experimental data.

In Vitro, In Vivo, and In Silico Models of Lymphangiogenesis in Solid Malignancies

Lymphangiogenesis (LA) is the formation of new lymphatic vessels by lymphatic endothelial cells (LECs) sprouting from pre-existing lymphatic vessels. It is increasingly recognized as being involved in many diseases, such as in cancer and secondary lymphedema, which most often results from cancer treatments. For some cancers, excessive LA is associated with cancer progression and metastatic dissemination to the lymph nodes (LNs) through lymphatic vessels. The study of LA through in vitro, in vivo, and, more recently, in silico models is of paramount importance in providing novel insights and identifying the key molecular actors in the biological dysregulation of this process under pathological conditions. In this review, the different biological (in vitro and in vivo) models of LA, especially in a cancer context, are explained and discussed, highlighting their principal modeled features as well as their advantages and drawbacks. Imaging techniques of the lymphatics, complementary or even essential to in vivo models, are also clarified and allow the establishment of the link with computational approaches. In silico models are introduced, theoretically described, and illustrated with examples specific to the lymphatic system and the LA. Together, these models constitute a toolbox allowing the LA research to be brought to the next level.

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.

Simulation of the process of angiogenesis: Quantification and assessment of vascular patterning in the chicken chorioallantoic membrane

Angiogenesis, the formation of new blood vessels from pre-existing ones, begins during embryonic development and continues throughout life. Sprouting angiogenesis is a well-defined process, being mainly influenced by vascular endothelial growth factor (VEGF). In this study, we propose a meshless-based model capable of mimicking the angiogenic response to several VEGF concentrations. In this model, endothelial cells migrate according to a diffusion-reaction equation, following the VEGF gradient concentration. The chick chorioallantoic membrane (CAM) assay was used to model the branching process and to validate the obtained numerical results. To analyse the angiogenic response, the total vessel number and the total vessel length presented in the CAM images and in the simulations for all the VEGF concentrations tested were quantified. In both the CAM assay and simulation, the treatments with VEGF increased the total vessel number and the total vessel length. The obtained quantitative results were very similar between the two methodologies used. The proposed model accurately simulates the capillary network pattern concerning its structure and morphology, for the lowest VEGF concentration tested. For the highest VEGF concentration, the capillary network predicted by the model was less accurate when compared to the one presented in the CAM assay but this may be explained by changes in blood vessel width at higher VEGF concentrations. This remains to be tested.

Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data

Tumor-associated vasculature is responsible for the delivery of nutrients, removal of waste, and allowing growth beyond 2-3 mm3. Additionally, the vascular network, which is changing in both space and time, fundamentally influences tumor response to both systemic and radiation therapy. Thus, a robust understanding of vascular dynamics is necessary to accurately predict tumor growth, as well as establish optimal treatment protocols to achieve optimal tumor control. Such a goal requires the intimate integration of both theory and experiment. Quantitative and time-resolved imaging methods have emerged as technologies able to visualize and characterize tumor vascular properties before and during therapy at the tissue and cell scale. Parallel to, but separate from those developments, mathematical modeling techniques have been developed to enable in silico investigations into theoretical tumor and vascular dynamics. In particular, recent efforts have sought to integrate both theory and experiment to enable data-driven mathematical modeling. Such mathematical models are calibrated by data obtained from individual tumor-vascular systems to predict future vascular growth, delivery of systemic agents, and response to radiotherapy. In this review, we discuss experimental techniques for visualizing and quantifying vascular dynamics including magnetic resonance imaging, microfluidic devices, and confocal microscopy. We then focus on the integration of these experimental measures with biologically based mathematical models to generate testable predictions.

Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis

Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology.

Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress-Strain Constitutive Equations

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin-Voigt model of linear viscoelasticity to represent the stress-strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress-strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin-Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress-strain relation of the ECM, while no patterns are observed when the Kelvin-Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress-strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

Cerebrovascular development: mechanisms and experimental approaches

The cerebral vasculature plays a central role in human health and disease and possesses several unique anatomic, functional and molecular characteristics. Despite their importance, the mechanisms that determine cerebrovascular development are less well studied than other vascular territories. This is in part due to limitations of existing models and techniques for visualisation and manipulation of the cerebral vasculature. In this review we summarise the experimental approaches used to study the cerebral vessels and the mechanisms that contribute to their development.

Recent progress understanding pathophysiology and genesis of brain AVM-a narrative review

Considerable progress has been made over the past years to better understand the genetic nature and pathophysiology of brain AVM. For the actual review, a PubMed search was carried out regarding the embryology, inflammation, advanced imaging, and fluid dynamical modeling of brain AVM. Whole-genome sequencing clarified the genetic origin of sporadic and familial AVM to a large degree, although some open questions remain. Advanced MRI and DSA techniques allow for better segmentation of feeding arteries, nidus, and draining veins, as well as the deduction of hemodynamic parameters such as flow and pressure in the individual AVM compartments. Nonetheless, complete modeling of the intranidal flow structure by computed fluid dynamics (CFD) is not possible so far. Substantial progress has been made towards understanding the embryology of brain AVM. In contrast to arterial aneurysms, complete modeling of the intranidal flow and a thorough understanding of the mechanical properties of the AVM nidus are still lacking at the present time.

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.

Comparative analysis of continuum angiogenesis models

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological “snail-trail” model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

A New Mathematical Model for Controlling Tumor Growth Based on Microenvironment Acidity and Oxygen Concentration

Hypoxia and the pH level of the tumor microenvironment have a great impact on the treatment of tumors. Here, the tumor growth is controlled by regulating the oxygen concentration and the acidity of the tumor microenvironment by introducing a two-dimensional multiscale cellular automata model of avascular tumor growth. The spatiotemporal evolution of tumor growth and metabolic variations is modeled based on biological assumptions, physical structure, states of cells, and transition rules. Each cell is allocated to one of the following states: proliferating cancer, nonproliferating cancer, necrotic, and normal cells. According to the response of the microenvironmental conditions, each cell consumes/produces metabolic factors and updates its state based on some stochastic rules. The input parameters are compatible with cancer biology using experimental data. The effect of neighborhoods during mitosis and simulating spatial heterogeneity is studied by considering multicellular layer structure of tumor. A simple Darwinist mutation is considered by introducing a critical parameter (Nmm) that affects division probability of the proliferative tumor cells based on the microenvironmental conditions and cancer hallmarks. The results show that Nmm regulation has a significant influence on the dynamics of tumor growth, the growth fraction, necrotic fraction, and the concentration levels of the metabolic factors. The model not only is able to simulate the in vivo tumor growth quantitatively and qualitatively but also can simulate the concentration of metabolic factors, oxygen, and acidity graphically. The results show the spatial heterogeneity effects on the proliferation of cancer cells and the rest of the system. By increasing Nmm, tumor shrinkage and significant increasing in the oxygen concentration and the pH value of the tumor microenvironment are observed. The results demonstrate the model's ability, providing an essential tool for simulating different tumor evolution scenarios of a patient and reliable prediction of spatiotemporal progression of tumors for utilizing in personalized therapy.

A multiscale model of complex endothelial cell dynamics in early angiogenesis

We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

Angiogenesis, the process by which new blood vessels are formed by sprouting from the pre-existing vascular bed, plays a key role in both physiological and pathological processes, including tumour growth. The structure of a growing vascular network is determined by the coordinated behaviour of endothelial cells in response to various signalling cues. Recent experimental studies have highlighted the importance of cell rearrangements as a driver for sprout elongation. However, the functional role of this phenomenon remains unclear. We formulate a new multiscale model of angiogenesis which, by accounting explicitly for the complex dynamics of endothelial cells within growing angiogenic sprouts, is able to reproduce generic features of angiogenic structures (branching, chemotactic sensitivity, cell mixing, etc.) as emergent properties of its dynamics. We validate our model against experimental data and then use it to quantify the phenomenon of cell mixing in vascular networks generated by endothelial cells of different lineages. Our results show that there is a direct correlation between the time evolution of cell mixing in a growing vascular network and its branching structure, thus paving the way for understanding the functional role of cell rearrangements in angiogenesis.

Evaluating snail-trail frameworks for leader-follower behavior with agent-based modeling

Branched networks constitute a ubiquitous structure in biology, arising in plants, lungs, and the circulatory system; however, the mechanisms behind their creation are not well understood. A commonly used model for network morphogenesis proposes that sprouts develop through interactions between leader (tip) cells and follower (stalk) cells. In this description, tip cells emerge from existing structures, travel up chemoattractant gradients, and form new networks by guiding the movement of stalk cells. Such dynamics have been mathematically represented by continuum "snail-trail" models in which the tip cell flux contributes to the stalk cell proliferation rate. Although snail-trail models constitute a classical depiction of leader-follower behavior, their accuracy has yet to be evaluated in a rigorous quantitative setting. Here, we extend the snail-trail modeling framework to two spatial dimensions by introducing a novel multiplicative factor to the stalk cell rate equation, which corrects for neglected network creation in directions other than that of the migrating front. Our derivation of this factor demonstrates that snail-trail models are valid descriptions of cell dynamics when chemotaxis dominates cell movement. We confirm that our snail-trail model accurately predicts the dynamics of tip and stalk cells in an existing agent-based model (ABM) for network formation [Pillay et al., Phys. Rev. E 95, 012410 (2017)10.1103/PhysRevE.95.012410]. We also derive conditions for which it is appropriate to use a reduced, one-dimensional snail-trail model to analyze ABM results. Our analysis identifies key metrics for cell migration that may be used to anticipate when simple snail-trail models will accurately describe experimentally observed cell dynamics in network formation.

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

Computational feasibility of simulating whole-organ vascular networks

The human body contains approximately 20 billion blood vessels, which transport nutrients, oxygen, immune cells, and signals throughout the body. The brain's vasculature includes up to 9 billion of these vessels to support cognition, motor processes, and myriad other vital functions. To model blood flowing through a vasculature, a geometric description of the vessels is required. Previously reported attempts to model vascular geometries have produced highly-detailed models. These models, however, are limited to a small fraction of the human brain, and little was known about the feasibility of computationally modeling whole-organ-sized networks. We implemented a fractal-based algorithm to construct a vasculature the size of the human brain and evaluated the algorithm's speed and memory requirements. Using high-performance computing systems, the algorithm constructed a vasculature comprising 17 billion vessels in 1960 core-hours, or 49 minutes of wall-clock time, and required less than 32 GB of memory per node. We demonstrated strong scalability that was limited mainly by input/output operations. The results of this study demonstrated, for the first time, that it is feasible to computationally model the vasculature of the whole human brain. These findings provide key insights into the computational aspects of modeling whole-organ vasculature.

Computational models of melanoma

Genes, proteins, or cells influence each other and consequently create patterns, which can be increasingly better observed by experimental biology and medicine. Thereby, descriptive methods of statistics and bioinformatics sharpen and structure our perception. However, additionally considering the interconnectivity between biological elements promises a deeper and more coherent understanding of melanoma. For instance, integrative network-based tools and well-grounded inductive in silico research reveal disease mechanisms, stratify patients, and support treatment individualization. This review gives an overview of different modeling techniques beyond statistics, shows how different strategies align with the respective medical biology, and identifies possible areas of new computational melanoma research.

From tumour perfusion to drug delivery and clinical translation of in silico cancer models

In silico cancer models have demonstrated great potential as a tool to improve drug design, optimise the delivery of drugs to target sites in the host tissue and, hence, improve therapeutic efficacy and patient outcome. However, there are significant barriers to the successful translation of in silico technology from bench to bedside. More precisely, the specification of unknown model parameters, the necessity for models to adequately reflect in vivo conditions, and the limited amount of pertinent validation data to evaluate models' accuracy and assess their reliability, pose major obstacles in the path towards their clinical translation. This review aims to capture the state-of-the-art in in silico cancer modelling of vascularised solid tumour growth, and identify the important advances and barriers to success of these models in clinical oncology. Particular emphasis has been put on continuum-based models of cancer since they - amongst the class of mechanistic spatio-temporal modelling approaches - are well-established in simulating transport phenomena and the biomechanics of tissues, and have demonstrated potential for clinical translation. Three important avenues in in silico modelling are considered in this contribution: first, since systemic therapy is a major cancer treatment approach, we start with an overview of the tumour perfusion and angiogenesis in silico models. Next, we present the state-of-the-art in silico work encompassing the delivery of chemotherapeutic agents to cancer nanomedicines through the bloodstream, and then review continuum-based modelling approaches that demonstrate great promise for successful clinical translation. We conclude with a discussion of what we view to be the key challenges and opportunities for in silico modelling in personalised and precision medicine.

Gibbs point field model quantifies disorder in microvasculature of U87-glioblastoma

Microvascular proliferation in glioblastoma multiforme is a biological key mechanism to facilitate tumor growth and infiltration and a main target for treatment interventions. The vascular architecture can be obtained by Single Plane Illumination Microscopy (SPIM) to evaluate vascular heterogeneity in tumorous tissue. We make use of the Gibbs point field model to quantify the order of regularity in capillary distributions found in the U87 glioblastoma model in a murine model and to compare tumorous and healthy brain tissue. A single model parameter Γ was assigned that is linked to tissue-specific vascular topology through Monte-Carlo simulations. Distributions of the model parameter Γ differ significantly between glioblastoma tissue with mean 〈Γ_G〉=2.1±0.4, as compared to healthy brain tissue with mean 〈Γ_H〉=4.9±0.4, suggesting that the average Γ-value allows for tissue differentiation. These results may be used for diagnostic magnetic resonance imaging, where it has been shown recently that Γ is linked to tissue-inherent relaxation parameters.

Hybrid Automata Library: A flexible platform for hybrid modeling with real-time visualization

The Hybrid Automata Library (HAL) is a Java Library developed for use in mathematical oncology modeling. It is made of simple, efficient, generic components that can be used to model complex spatial systems. HAL's components can broadly be classified into: on- and off-lattice agent containers, finite difference diffusion fields, a GUI building system, and additional tools and utilities for computation and data collection. These components are designed to operate independently and are standardized to make them easy to interface with one another. As a demonstration of how modeling can be simplified using our approach, we have included a complete example of a hybrid model (a spatial model with interacting agent-based and PDE components). HAL is a useful asset for researchers who wish to build performant 1D, 2D and 3D hybrid models in Java, while not starting entirely from scratch. It is available on GitHub at https://github.com/MathOnco/HAL under the MIT License. HAL requires the Java JDK version 1.8 or later to compile and run the source code.

Phenotypes of Vascular Flow Networks

Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.

A particle-based model for endothelial cell migration under flow conditions

Endothelial cells (ECs) play a major role in the healing process following angioplasty to inhibit excessive neointima. This makes the process of EC healing after injury, in particular EC migration in a stented vessel, important for recovery of normal vessel function. In that context, we present a novel particle-based model of EC migration and validate it against in vitro experimental data. We have developed a particle-based model of EC migration under flow conditions in an in vitro vessel with obstacles. Cell movement in the model is a combination of random walks and directed movement along the local flow velocity vector. For model calibration, a set of experimental data for cell migration in a similarly shaped channel has been used. We have calibrated the model for a baseline case of a channel with no obstacles and then applied it to the case of a channel with ridges on the bottom surface, representative of stent strut geometry. We were able to closely reproduce the cell migration speed and angular distribution of their movement relative to the flow direction reported in vitro. The model also reproduces qualitative aspects of EC migration, such as entrapment of cells downstream from the flow-disturbing ridge. The model has the potential, after more extensive in vitro validation, to study the effect of variation in strut spacing and shape, through modification of the local flow, on EC migration. The results of this study support the hypothesis that EC migration is strongly affected by the direction and magnitude of local wall shear stress.

Lattice and continuum modelling of a bioactive porous tissue scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

Glioblastoma multiforme restructures the topological connectivity of cerebrovascular networks

Glioblastoma multiforme alters healthy tissue vasculature by inducing angiogenesis and vascular remodeling. To fully comprehend the structural and functional properties of the resulting vascular network, it needs to be studied collectively by considering both geometric and topological properties. Utilizing Single Plane Illumination Microscopy (SPIM), the detailed capillary structure in entire healthy and tumor-bearing mouse brains could be resolved in three dimensions. At the scale of the smallest capillaries, the entire vascular systems of bulk U87- and GL261-glioblastoma xenografts, their respective cores, and healthy brain hemispheres were modeled as complex networks and quantified with fundamental topological measures. All individual vessel segments were further quantified geometrically and modular clusters were uncovered and characterized as meta-networks, facilitating an analysis of large-scale connectivity. An inclusive comparison of large tissue sections revealed that geometric properties of individual vessels were altered in glioblastoma in a relatively subtle way, with high intra- and inter-tumor heterogeneity, compared to the impact on the vessel connectivity. A network topology analysis revealed a clear decomposition of large modular structures and hierarchical network organization, while preserving most fundamental topological classifications, in both tumor models with distinct growth patterns. These results augment our understanding of cerebrovascular networks and offer a topological assessment of glioma-induced vascular remodeling. The findings may help understand the emergence of hypoxia and necrosis, and prove valuable for therapeutic interventions such as radiation or antiangiogenic therapy.

On the Impact of Chemo-Mechanically Induced Phenotypic Transitions in Gliomas

Tumor microenvironment is a critical player in glioma progression, and novel therapies for its targeting have been recently proposed. In particular, stress-alleviation strategies act on the tumor by reducing its stiffness, decreasing solid stresses and improving blood perfusion. However, these microenvironmental changes trigger chemo-mechanically induced cellular phenotypic transitions whose impact on therapy outcomes is not completely understood. In this work we analyze the effects of mechanical compression on migration and proliferation of glioma cells. We derive a mathematical model of glioma progression focusing on cellular phenotypic plasticity. Our results reveal a trade-off between tumor infiltration and cellular content as a consequence of stress-alleviation approaches. We discuss how these novel findings increase the current understanding of glioma/microenvironment interactions and can contribute to new strategies for improved therapeutic outcomes.

Agent-based modeling of morphogenetic systems: Advantages and challenges

The complexity of morphogenesis poses a fundamental challenge to understanding the mechanisms governing the formation of biological patterns and structures. Over the past century, numerous processes have been identified as critically contributing to morphogenetic events, but the interplay between the various components and aspects of pattern formation have been much harder to grasp. The combination of traditional biology with mathematical and computational methods has had a profound effect on our current understanding of morphogenesis and led to significant insights and advancements in the field. In particular, the theoretical concepts of reaction–diffusion systems and positional information, proposed by Alan Turing and Lewis Wolpert, respectively, dramatically influenced our general view of morphogenesis, although typically in isolation from one another. In recent years, agent-based modeling has been emerging as a consolidation and implementation of the two theories within a single framework. Agent-based models (ABMs) are unique in their ability to integrate combinations of heterogeneous processes and investigate their respective dynamics, especially in the context of spatial phenomena. In this review, we highlight the benefits and technical challenges associated with ABMs as tools for examining morphogenetic events. These models display unparalleled flexibility for studying various morphogenetic phenomena at multiple levels and have the important advantage of informing future experimental work, including the targeted engineering of tissues and organs.

Modelling of endothelial cell migration and angiogenesis in microfluidic cell culture systems

Tumour-induced angiogenesis is a complex biological process that involves growth of new blood vessels within the tumour microenvironment and is an important target for cancer therapies. Significant efforts have been undertaken to develop theoretical models as well as in vitro experimental models to study angiogenesis in a highly controllable and accessible manner. Various mathematical models have been developed to study angiogenesis, but these have mostly been applied to in vivo assays. Recently, microfluidic cell culture systems have emerged as useful and powerful tools for studying cell migration and angiogenesis processes, but thus far, mathematical angiogenesis models have not yet been applied to microfluidic geometries. Integrating mathematical and computational modelling with microfluidic-based assays has potential to enable greater control over experimental parameters, provide new insights into fundamental angiogenesis processes and assist in accelerating design and optimization of operating conditions. Here, we describe the development and application of a combined mathematical and computational modelling approach tailored specifically for microfluidic cell culture systems. The objective was to allow optimization of the engineering design of microfluidic systems, where the model may be used to test the impact of various geometric parameters on cell migration and angiogenesis processes, and assist in identifying optimal device dimensions to achieve desired readouts. We employed two separate continuum mathematical models that treated cell density, vessel length density and vascular endothelial growth factor (VEGF) concentration as continuous average variables, and we implemented these models numerically using finite difference discretization and a Method of Lines approach. We examined the average response of cells to VEGF gradients inside our microfluidic device, including the time-dependent changes in cell density and vessel density, and how they were affected by changes in device geometries including the migration port width and length. Our study demonstrated that mathematical modelling can be integrated with microfluidics to offer new perspectives on emerging problems in biomicrofluidics and cancer biology.

Data driven mathematical modeling reveals the dynamic mechanism of MSC-induced neovascularization

Coculture of mesenchymal stem cells (MSCs) and vascular endothelial cells (ECs) in vitro leads to the formation of a capillary-like reticular structure by ECs, which has great potential as a better substitute for artificial blood vessels in terms of stability and functionality. To investigate the mechanisms of the early neovascularization induced by MSCs, we analyzed the kinematic features of the motion of ECs and concluded that the dynamic interaction between cells and the extracellular matrix would reveal the capillary-like structure formation. Based on this hypothesis, we proposed a mathematical model to simulate the vascular-like migration pattern of ECs in silico, which was confirmed by in vitro studies. These in vitro studies validated that the dynamic secretion and degradation of collagen I is the critical factor for capillary structure formation. The model proposed based on cell tracking, single cell sequencing, and mathematical simulation provides a better understanding of the neovascularization process induced by MSCs and a possible simple explanation guiding this important cellular behavior.-Yu, Y., Situ, Q., Jia, W., Li, J., Wu, Q., Lei, J. Data driven mathematical modeling reveals the dynamic mechanism of MSC-induced neovascularization.

Simulation of Multispecies Desmoplastic Cancer Growth via a Fully Adaptive Non-linear Full Multigrid Algorithm

A fully adaptive non-linear full multigrid (FMG) algorithm is implemented to computationally simulate a model of multispecies desmoplastic tumor growth in three spatial dimensions. The algorithm solves a thermodynamic mixture model employing a diffuse interface approach with Cahn-Hilliard-type fourth-order equations that are coupled, non-linear, and numerically stiff. The tumor model includes extracellular matrix (ECM) as a major component with elastic energy contribution in its chemical potential term. Blood and lymphatic vasculatures are simulated via continuum representations. The model employs advection-reaction-diffusion partial differential equations (PDEs) for the cell, ECM, and vascular components, and reaction-diffusion PDEs for the elements diffusing from the vessels. This study provides the details of the numerical solution obtained by applying the fully adaptive non-linear FMG algorithm with finite difference method to solve this complex system of PDEs. The results indicate that this type of computational model can simulate the extracellular matrix-rich desmoplastic tumor microenvironment typical of fibrotic tumors, such as pancreatic adenocarcinoma.

A local uPAR-plasmin-TGFβ1 positive feedback loop in a qualitative computational model of angiogenic sprouting explains the in vitro effect of fibrinogen variants

In experimental assays of angiogenesis in three-dimensional fibrin matrices, a temporary scaffold formed during wound healing, the type and composition of fibrin impacts the level of sprouting. More sprouts form on high molecular weight (HMW) than on low molecular weight (LMW) fibrin. It is unclear what mechanisms regulate the number and the positions of the vascular-like structures in cell cultures. To address this question, we propose a mechanistic simulation model of endothelial cell migration and fibrin proteolysis by the plasmin system. The model is a hybrid, cell-based and continuum, computational model based on the cellular Potts model and sets of partial-differential equations. Based on the model results, we propose that a positive feedback mechanism between uPAR, plasmin and transforming growth factor β1 (TGFβ1) selects cells in the monolayer for matrix invasion. Invading cells releases TGFβ1 from the extracellular matrix through plasmin-mediated fibrin degradation. The activated TGFβ1 further stimulates fibrin degradation and keeps proteolysis active as the sprout invades the fibrin matrix. The binding capacity for TGFβ1 of LMW is reduced relative to that of HMW. This leads to reduced activation of proteolysis and, consequently, reduced cell ingrowth in LMW fibrin compared to HMW fibrin. Thus our model predicts that endothelial cells in LMW fibrin matrices compared to HMW matrices show reduced sprouting due to a lower bio-availability of TGFβ1.

Therapies for a range of medical conditions, including cancer, wound healing and diabetic retinopathy can benefit from a better control over the growth of blood vessels. The chemical properties of fibrin, the material that forms scabs in wounds and can also occur in large concentrations in tumors, can regulate the degree of blood vessel growth (angiogenesis). Angiogenesis can be mimicked in cell cultures. These allow us to modulate the chemical properties of fibrin and study the effect on angiogenesis. Fibrin occurs in high molecular weight (HMW) and in low molecular weight (LMW) forms. Interestingly, there is more ingrowth of angiogenic-like structures into HMW than in LMW fibrin, but the mechanisms are poorly understood. To get more insight into these, we constructed a computational model. Using the model, we propose and analyse a hypothetical mechanism for sprouting that could explain the differences in endothelial cell sprouting in LMW and HMW fibrin matrices. Our model suggests that cells digest fibrin, thus creating space for ingrowth. At the same time, digestion frees growth factors bound to fibrin, that activates further secretion of digestive enzymes by the cells. We propose that the resulting positive feedback loop spontaneously selects cells in the endothelial monolayer for ingrowth and helps the blood vessel sprout move deeper into the fibrin. This could be a complementary mechanism to lateral-inhibition by Delta-Notch for the selection of leader cells, also called ‘tip cells’. Our model predicts that endothelial cells in LMW fibrin compared to HMW fibrin show reduced sprouting due to a lower bio-availability of TGFβ1.

Modeling triple-negative breast cancer heterogeneity: Effects of stromal macrophages, fibroblasts and tumor vasculature

A hallmark of breast tumors is its spatial heterogeneity that includes its distribution of cancer stem cells and progenitor cells, but also heterogeneity in the tumor microenvironment. In this study we focus on the contributions of stromal cells, specifically macrophages, fibroblasts, and endothelial cells on tumor progression. We develop a computational model of triple-negative breast cancer based on our previous work and expand it to include macrophage infiltration, fibroblasts, and angiogenesis. In vitro studies have shown that the secretomes of tumor-educated macrophages and fibroblasts increase both the migration and proliferation rates of triple-negative breast cancer cells. In vivo studies also demonstrated that blocking signaling of selected secreted factors inhibits tumor growth and metastasis in mouse xenograft models. We investigate the influences of increased migration and proliferation rates on tumor growth, the effect of the presence on fibroblasts or macrophages on growth and morphology, and the contributions of macrophage infiltration on tumor growth. We find that while the presence of macrophages increases overall tumor growth, the increase in macrophage infiltration does not substantially increase tumor growth and can even stifle tumor growth at excessive rates.

Modelling of retinal vasculature based on genetically tuned parametric L-system

Structures of retinal blood vessels are of great importance in diagnosis and treatment of diseases that affect the eyes. Parametric Lindenmayer system (L-system) is one of the powerful rule-based methods that has a great capability for generating tree-like structures using simple rewriting rules. In this study, a novel framework, which can be used to model the retinal vasculature based on available images, has been proposed. This framework presents a solution to special instance of a general open problem, the L-system inverse problem, in which L-system rules should be obtained based on images representing a particular tree-like structure. In this study, genetic algorithm with a novel objective function based on feature matching and an L-system grammar comparison has been used along with nonlinear regression to solve the parametric L-system inverse problem. The resulting L-system growth rules have been employed to predict inaccessible vascular branches. Graphical and quantitative comparison between model results and target structures of different case studies reveals that the proposed framework can be used to generate the structure of retinal blood vessels accurately. Even in the cases lacking sufficient image data, it can provide acceptable predictions.

The impact of exclusion processes on angiogenesis models

Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.

Translational models of tumor angiogenesis: A nexus of in silico and in vitro models

Emerging evidence shows that endothelial cells are not only the building blocks of vascular networks that enable oxygen and nutrient delivery throughout a tissue but also serve as a rich resource of angiocrine factors. Endothelial cells play key roles in determining cancer progression and response to anti-cancer drugs. Furthermore, the endothelium-specific deposition of extracellular matrix is a key modulator of the availability of angiocrine factors to both stromal and cancer cells. Considering tumor vascular network as a decisive factor in cancer pathogenesis and treatment response, these networks need to be an inseparable component of cancer models. Both computational and in vitro experimental models have been extensively developed to model tumor-endothelium interactions. While informative, they have been developed in different communities and do not yet represent a comprehensive platform. In this review, we overview the necessity of incorporating vascular networks for both in vitro and in silico cancer models and discuss recent progresses and challenges of in vitro experimental microfluidic cancer vasculature-on-chip systems and their in silico counterparts. We further highlight how these two approaches can merge together with the aim of presenting a predictive combinatorial platform for studying cancer pathogenesis and testing the efficacy of single or multi-drug therapeutics for cancer treatment.

Graph analysis of cell clusters forming vascular networks

This manuscript describes the experimental observation of vasculogenesis in chick embryos by means of network analysis. The formation of the vascular network was observed in the area opaca of embryos from 40 to 55 h of development. In the area opaca endothelial cell clusters self-organize as a primitive and approximately regular network of capillaries. The process was observed by bright-field microscopy in control embryos and in embryos treated with Bevacizumab (Avastin^®), an antibody that inhibits the signalling of the vascular endothelial growth factor (VEGF). The sequence of images of the vascular growth were thresholded, and used to quantify the forming network in control and Avastin-treated embryos. This characterization is made by measuring vessels density, number of cell clusters and the largest cluster density. From the original images, the topology of the vascular network was extracted and characterized by means of the usual network metrics such as: the degree distribution, average clustering coefficient, average short path length and assortativity, among others. This analysis allows to monitor how the largest connected cluster of the vascular network evolves in time and provides with quantitative evidence of the disruptive effects that Avastin has on the tree structure of vascular networks.