A review of mathematical models for the formation of vascular networks

A Network Model to Explore the Effect of the Micro-environment on Endothelial Cell Behavior during Angiogenesis

Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-function mutations. These properties make our model a suitable platform for the understanding of the molecular mechanisms underlying some pathologies. For example, it is possible to follow the changes in the activation patterns caused by mutations that promote Tip EC behavior and inhibit Phalanx EC behavior, that lead to the conditions associated with retinal vascular disorders and tumor vascularization. Moreover, the model describes how mutations that promote Phalanx EC behavior are associated with the development of arteriovenous and venous malformations. These results suggest that the network model that we propose has the potential to be used in the study of how the modulation of the EC extracellular micro-environment may improve the outcome of vascular disease treatments.

A mathematical model of tumour angiogenesis: growth, regression and regrowth

Cancerous tumours have the ability to recruit new blood vessels through a process called angiogenesis. By stimulating vascular growth, tumours get connected to the circulatory system, receive nutrients and open a way to colonize distant organs. Tumour-induced vascular networks become unstable in the absence of tumour angiogenic factors (TAFs). They may undergo alternating stages of growth, regression and regrowth. Following a phase-field methodology, we propose a model of tumour angiogenesis that reproduces the aforementioned features and highlights the importance of vascular regression and regrowth. In contrast with previous theories which focus on vessel remodelling due to the absence of flow, we model an alternative regression mechanism based on the dependency of tumour-induced vascular networks on TAFs. The model captures capillaries at full scale, the plastic dynamics of tumour-induced vessel networks at long time scales, and shows the key role played by filopodia during angiogenesis. The predictions of our model are in agreement with in vivo experiments and may prove useful for the design of antiangiogenic therapies.

Geometries of vasculature bifurcation can affect the level of trophic damage during formation of a brain ischemic lesion

Ischemic lesion is a common cause of various diseases in humans. Brain tissue is especially sensitive to this type of damage. A common reason for the appearance of an ischemic area is a stop in blood flow in some branch of the vasculature system. Then, a decreasing concentration gradient results in a low mean level of oxygen in surrounding tissues. After that, the biochemical ischemic cascade spreads. In this review, we examine these well-known events from a new angle. It is stressed that there is essential evidence to predict the formation of an ischemic micro-area at the base of vascular bifurcation geometries. Potential applications to improve neuroprotection are also discussed.

Angiogenesis model with Erlang distributed delays

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

A Validated Multiscale In-Silico Model for Mechano-sensitive Tumour Angiogenesis and Growth

Vascularisation is a key feature of cancer growth, invasion and metastasis. To better understand the governing biophysical processes and their relative importance, it is instructive to develop physiologically representative mathematical models with which to compare to experimental data. Previous studies have successfully applied this approach to test the effect of various biochemical factors on tumour growth and angiogenesis. However, these models do not account for the experimentally observed dependency of angiogenic network evolution on growth-induced solid stresses. This work introduces two novel features: the effects of hapto- and mechanotaxis on vessel sprouting, and mechano-sensitive dynamic vascular remodelling. The proposed three-dimensional, multiscale, in-silico model of dynamically coupled angiogenic tumour growth is specified to in-vivo and in-vitro data, chosen, where possible, to provide a physiologically consistent description. The model is then validated against in-vivo data from murine mammary carcinomas, with particular focus placed on identifying the influence of mechanical factors. Crucially, we find that it is necessary to include hapto- and mechanotaxis to recapitulate observed time-varying spatial distributions of angiogenic vasculature.

Modeling angiogenesis: A discrete to continuum description

Angiogenesis is the process by which new blood vessels develop from existing vasculature. During angiogenesis, endothelial tip cells migrate via diffusion and chemotaxis, loops form via tip-to-tip and tip-to-sprout anastomosis, new tip cells are produced via branching, and a vessel network forms as endothelial cells follow the paths of tip cells. The latter process is known as the snail trail. We use a mean-field approximation to systematically derive a continuum model from a two-dimensional lattice-based cellular automaton model of angiogenesis in the corneal assay, based on the snail-trail process. From the two-dimensional continuum model, we derive a one-dimensional model which represents angiogenesis in two dimensions. By comparing the discrete and one-dimensional continuum models, we determine how individual cell behavior manifests at the macroscale. In contrast to the phenomenological continuum models in the literature, we find that endothelial cell creation due to tip cell movement (vessel formation via the snail trail) manifests as a source term of tip cells on the macroscale. Further, we find that phenomenological continuum models, which assume that endothelial cell creation is proportional to the flux of tip cells in the direction of increasing chemoattractant concentration, qualitatively capture vessel formation in two dimensions, but must be modified to accurately represent vessel formation. Additionally, we find that anastomosis imposes restrictions on cell density, which, if violated, leads to ill-posedness in our continuum model. We also deduce that self-loops should be excluded when tip-to-sprout anastomosis is active in the discrete model to ensure propagation of the vascular front.

Solitonlike attractor for blood vessel tip density in angiogenesis

Recently, numerical simulations of a stochastic model have shown that the density of vessel tips in tumor-induced angiogenesis adopts a solitonlike profile [Sci. Rep. 6, 31296 (2016)2045-232210.1038/srep31296]. In this work, we derive and solve the equations for the soliton collective coordinates that indicate how the soliton adapts its shape and velocity to varying chemotaxis and diffusion. The vessel tip density can be reconstructed from the soliton formulas. While the stochastic model exhibits large fluctuations, we show that the location of the maximum vessel tip density for different replicas follows closely the soliton peak position calculated either by ensemble averages or by solving an alternative deterministic description of the density. The simple soliton collective coordinate equations may also be used to ascertain the response of the vessel network to changes in the parameters and thus to control it.

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.

Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights

Gliomas are highly invasive brain tumours characterised by poor prognosis and limited response to therapy. There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve tumour blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation induced by blood vessel occlusion/collapse. In contrast, the therapeutic intention of normalising the abnormal tumour vasculature is to improve the efficacy of conventional treatment modalities. Although these strategies have shown therapeutic potential, it remains unclear why they both often fail to control glioma growth. To shed some light on this issue, we propose a mathematical model based on the migration/proliferation dichotomy of glioma cells in order to investigate why vaso-modulatory interventions have shown limited success in terms of tumour clearance. We found the existence of a critical cell proliferation/diffusion ratio that separates glioma responses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the front speed and increase the infiltration width, for those in the other regime, the invasion speed increases and infiltration width decreases. We discuss how these in silico findings can be used to guide individualised vaso-modulatory approaches to improve treatment success rates.

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

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

Spatio-temporal Models of Lymphangiogenesis in Wound Healing

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: (1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case and (2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here, we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.

Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks

Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.

Image based modeling of tumor growth

Tumors are a main cause of morbidity and mortality worldwide. Despite the efforts of the clinical and research communities, little has been achieved in the past decades in terms of improving the treatment of aggressive tumors. Understanding the underlying mechanism of tumor growth and evaluating the effects of different therapies are valuable steps in predicting the survival time and improving the patients' quality of life. Several studies have been devoted to tumor growth modeling at different levels to improve the clinical outcome by predicting the results of specific treatments. Recent studies have proposed patient-specific models using clinical data usually obtained from clinical images and evaluating the effects of various therapies. The aim of this review is to highlight the imaging role in tumor growth modeling and provide a worthwhile reference for biomedical and mathematical researchers with respect to tumor modeling using the clinical data to develop personalized models of tumor growth and evaluating the effect of different therapies.

Spheroid Formation of Hepatocarcinoma Cells in Microwells: Experiments and Monte Carlo Simulations

The formation of spherical aggregates during the growth of cell population has long been observed under various conditions. We observed the formation of such aggregates during proliferation of Huh-7.5 cells, a human hepatocarcinoma cell line, in a microfabricated low-adhesion microwell system (SpheroFilm; formed of mass-producible silicone elastomer) on the length scales up to 500 μm. The cell proliferation was also tracked with immunofluorescence staining of F-actin and cell proliferation marker Ki-67. Meanwhile, our complementary 3D Monte Carlo simulations, taking cell diffusion and division, cell-cell and cell-scaffold adhesion, and gravity into account, illustrate the role of these factors in the formation of spheroids. Taken together, our experimental and simulation results provide an integrative view of the process of spheroid formation for Huh-7.5 cells.

Coherent modelling switch between pointwise and distributed representations of cell aggregates

Biological systems are typically formed by different cell phenotypes, characterized by specific biophysical properties and behaviors. Moreover, cells are able to undergo differentiation or phenotypic transitions upon internal or external stimuli. In order to take these phenomena into account, we here propose a modelling framework in which cells can be described either as pointwise/concentrated particles or as distributed masses, according to their biological determinants. A set of suitable rules then defines a coherent procedure to switch between the two mathematical representations. The theoretical environment describing cell transition is then enriched by including cell migratory dynamics and duplication/apoptotic processes, as well as the kinetics of selected diffusing chemicals influencing the system evolution. Finally, biologically relevant numerical realizations are presented: in particular, they deal with the growth of a tumor spheroid and with the initial differentiation stages of the formation of the zebrafish posterior lateral line. Both phenomena mainly rely on cell phenotypic transition and differentiated behaviour, thereby constituting biological systems particularly suitable to assess the advantages of the proposed model.

An imaging-based computational model for simulating angiogenesis and tumour oxygenation dynamics

Tumour growth, angiogenesis and oxygenation vary substantially among tumours and significantly impact their treatment outcome. Imaging provides a unique means of investigating these tumour-specific characteristics. Here we propose a computational model to simulate tumour-specific oxygenation changes based on the molecular imaging data. Tumour oxygenation in the model is reflected by the perfused vessel density. Tumour growth depends on its doubling time (T d) and the imaged proliferation. Perfused vessel density recruitment rate depends on the perfused vessel density around the tumour (sMVDtissue) and the maximum VEGF concentration for complete vessel dysfunctionality (VEGFmax). The model parameters were benchmarked to reproduce the dynamics of tumour oxygenation over its entire lifecycle, which is the most challenging test. Tumour oxygenation dynamics were quantified using the peak pO2 (pO2peak) and the time to peak pO2 (t peak). Sensitivity of tumour oxygenation to model parameters was assessed by changing each parameter by 20%. t peak was found to be more sensitive to tumour cell line related doubling time (~30%) as compared to tissue vasculature density (~10%). On the other hand, pO2peak was found to be similarly influenced by the above tumour- and vasculature-associated parameters (~30-40%). Interestingly, both pO2peak and t peak were only marginally affected by VEGFmax (~5%). The development of a poorly oxygenated (hypoxic) core with tumour growth increased VEGF accumulation, thus disrupting the vessel perfusion as well as further increasing hypoxia with time. The model with its benchmarked parameters, is applied to hypoxia imaging data obtained using a [(64)Cu]Cu-ATSM PET scan of a mouse tumour and the temporal development of the vasculature and hypoxia maps are shown. The work underscores the importance of using tumour-specific input for analysing tumour evolution. An extended model incorporating therapeutic effects can serve as a powerful tool for analysing tumour response to anti-angiogenic therapies.

An integrated approach to quantitative modelling in angiogenesis research

Angiogenesis, the process by which new vessels form from existing ones, plays an important role in many developmental processes and pathological conditions. We study angiogenesis in the context of a highly controllable experimental environment: the cornea micropocket assay. Using a multidisciplinary approach that combines experiments, image processing and analysis, and mathematical modelling, we aim to provide mechanistic insight into the action of two angiogenic factors, vascular endothelial growth factor A (VEGF-A) and basic fibroblast growth factor (bFGF). We use image analysis techniques to extract quantitative data, which are both spatially and temporally resolved, from experimental images, and we develop a mathematical model, in which the corneal vasculature evolves in response to both VEGF-A and bFGF. The experimental data are used for model parametrization, while the mathematical model is used to assess the utility of the cornea micropocket assay and to characterize proposed synergies between VEGF-A and bFGF.

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

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

Tumour angiogenesis as a chemo-mechanical surface instability

The hypoxic conditions within avascular solid tumours may trigger the secretion of chemical factors, which diffuse to the nearby vasculature and promote the formation of new vessels eventually joining the tumour. Mathematical models of this process, known as tumour angiogenesis, have mainly investigated the formation of the new capillary networks using reaction-diffusion equations. Since angiogenesis involves the growth dynamics of the endothelial cells sprouting, we propose in this work an alternative mechanistic approach, developing a surface growth model for studying capillary formation and network dynamics. The model takes into account the proliferation of endothelial cells on the pre-existing capillary surface, coupled with the bulk diffusion of the vascular endothelial growth factor (VEGF). The thermo-dynamical consistency is imposed by means of interfacial and bulk balance laws. Finite element simulations show that both the morphology and the dynamics of the sprouting vessels are controlled by the bulk diffusion of VEGF and the chemo-mechanical and geometric properties at the capillary interface. Similarly to dendritic growth processes, we suggest that the emergence of tree-like vessel structures during tumour angiogenesis may result from the free boundary instability driven by competition between chemical and mechanical phenomena occurring at different length-scales.

Computer Simulations of the Tumor Vasculature: Applications to Interstitial Fluid Flow, Drug Delivery, and Oxygen Supply

Tumor vasculature, the blood vessel network supplying a growing tumor with nutrients such as oxygen or glucose, is in many respects different from the hierarchically organized arterio-venous blood vessel network in normal tissues. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature. Integrative models, based on detailed experimental data and physical laws, implement, in silico, the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. This chapter provides an overview over the current status of computer simulations of vascular remodeling during tumor growth including interstitial fluid flow, drug delivery, and oxygen supply within the tumor. The model predictions are compared with experimental and clinical data and a number of longstanding physiological paradigms about tumor vasculature and intratumoral solute transport are critically scrutinized.

Primary Phenomenon in the Network Formation of Endothelial Cells: Effect of Charge

Blood vessels are essential organs that are involved in the supply of nutrients and oxygen and play an important role in regulating the body's internal environment, including pH, body temperature, and water homeostasis. Many studies have examined the formation of networks of endothelial cells. The results of these studies have revealed that vascular endothelial growth factor (VEGF) affects the interactions of these cells and modulates the network structure. Though almost all previous simulation studies have assumed that the chemoattractant VEGF is present before network formation, vascular endothelial cells secrete VEGF only after the cells bind to the substrate. This suggests VEGF is not essential for vasculogenesis especially at the early stage. Using a simple experiment, we find chain-like structures which last quite longer than it is expected, unless the energetically stable cluster should be compact. Using a purely physical model and simulation, we find that the hydrodynamic interaction retard the compaction of clusters and that the chains are stabilized through the effects of charge. The charge at the surface of the cells affect the interparticle potential, and the resulting repulsive forces prevent the chains from folding. The ions surrounding the cells may also be involved in this process.

Autonomy and Non-autonomy of Angiogenic Cell Movements Revealed by Experiment-Driven Mathematical Modeling

Angiogenesis is a multicellular phenomenon driven by morphogenetic cell movements. We recently reported morphogenetic vascular endothelial cell (EC) behaviors to be dynamic and complex. However, the principal mechanisms orchestrating individual EC movements in angiogenic morphogenesis remain largely unknown. Here we present an experiment-driven mathematical model that enables us to systematically dissect cellular mechanisms in branch elongation. We found that cell-autonomous and coordinated actions governed these multicellular behaviors, and a cell-autonomous process sufficiently illustrated essential features of the morphogenetic EC dynamics at both the single-cell and cell-population levels. Through refining our model and experimental verification, we further identified a coordinated mode of tip EC behaviors regulated via a spatial relationship between tip and follower ECs, which facilitates the forward motility of tip ECs. These findings provide insights that enhance our mechanistic understanding of not only angiogenic morphogenesis, but also other types of multicellular phenomenon.

The macrophage and its role in inflammation and tissue repair: mathematical and systems biology approaches

Macrophages are central to the inflammatory response and its ability to resolve effectively. They are complex cells that adopt a range of subtypes depending on the tissue type and stimulus that they find themselves under. This flexibility allows them to play multiple, sometimes opposing, roles in inflammation and tissue repair. Their central role in the inflammatory process is reflected in macrophage dysfunction being implicated in chronic inflammation and poorly healing wounds. In this study, we discuss recent attempts to model mathematically and computationally the macrophage and how it partakes in the complex processes of inflammation and tissue repair. There are increasing data describing the variety of macrophage phenotypes and their underlying transcriptional programs. Dynamic mathematical and computational models are an ideal way to test biological hypotheses against experimental data and could aid in understanding this multi-functional cell and its potential role as an attractive therapeutic target for inflammatory conditions and tissue repair.

A mathematical model for lymphangiogenesis in normal and diabetic wounds

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

Systems biology of the microvasculature

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

Integrative models of vascular remodeling during tumor growth

Malignant solid tumors recruit the blood vessel network of the host tissue for nutrient supply, continuous growth, and gain of metastatic potential. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature that is in many respects different from the hierarchically organized arterio-venous blood vessel network of the host tissues. Integrative models based on detailed experimental data and physical laws implement in silico the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. In this review, we give an overview over the current status of integrative models describing tumor growth, vascular remodeling, blood and interstitial fluid flow, drug delivery, and concomitant transformations of the microenvironment. WIREs Syst Biol Med 2015, 7:113-129. doi: 10.1002/wsbm.1295 For further resources related to this article, please visit the WIREs website.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

Structural changes in thestrial blood-labyrinth barrier of aged C57BL/6 mice

Tight control over cochlear blood flow (CoBF) and the blood-labyrinth barrier (BLB) in the striavascularis is critical for maintaining the ionic, fluid and energy balance necessary for hearing function. Inefficient CoBF and disruption of BLB integrity have long been considered major etiologic factors in a variety of hearing disorders. In this study, we investigate structural changes in the BLB of the striavascularis in age-graded C57BL/6 mice (1 to 21 months) with a focus on changes in two blood barrier accessory cells, namely pericytes (PCs) and perivascular-resident macrophage-like melanocytes (PVM/Ms). Decreased capillary density was detectable at 6 months, with significant capillary degeneration seen in 9- to 21-month-old mice. Reduced capillary density was highly correlated with lower numbers of PCs and PVM/Ms. "Drop-out" of PCs and "activation" of PVM/Ms were seen at 6 months, with drastic changes being observed by 21 months. With newly established in vitro three-dimensional cell-based co-culture models, we demonstrate that PCs and PVM/Ms are essential for maintaining cochlear vascular architecture and stability.

Cell-based computational modeling of vascular morphogenesis using Tissue Simulation Toolkit

Computational modeling has become a widely used tool for unraveling the mechanisms of higher level cooperative cell behavior during vascular morphogenesis. However, experimenting with published simulation models or adding new assumptions to those models can be daunting for novice and even for experienced computational scientists. Here, we present a step-by-step, practical tutorial for building cell-based simulations of vascular morphogenesis using the Tissue Simulation Toolkit (TST). The TST is a freely available, open-source C++ library for developing simulations with the two-dimensional cellular Potts model, a stochastic, agent-based framework to simulate collective cell behavior. We will show the basic use of the TST to simulate and experiment with published simulations of vascular network formation. Then, we will present step-by-step instructions and explanations for building a recent simulation model of tumor angiogenesis. Demonstrated mechanisms include cell-cell adhesion, chemotaxis, cell elongation, haptotaxis, and haptokinesis.

Algorithmically generated rodent hepatic vascular trees in arbitrary detail

Physiologically realistic geometric models of the vasculature in the liver are indispensable for modelling hepatic blood flow, the main connection between the liver and the organism. Current in vivo imaging techniques do not provide sufficiently detailed vascular trees for many simulation applications, so it is necessary to use algorithmic refinement methods. The method of Constrained Constructive Optimization (CCO) (Schreiner et al., 2006) is well suited for this purpose. Its results after calibration have been previously compared to experimentally acquired human vascular trees (Schwen and Preusser, 2012). The goal of this paper is to extend this calibration to the case of rodents (mice and rats), the most commonly used animal models in liver research. Based on in vivo and ex vivo micro-CT scans of rodent livers and their vasculature, we performed an analysis of various geometric features of the vascular trees. Starting from pruned versions of the original vascular trees, we applied the CCO procedure and compared these algorithmic results to the original vascular trees using a suitable similarity measure. The calibration of the postprocessing improved the algorithmic results compared to those obtained using standard CCO. In terms of angular features, the average similarity increased from 0.27 to 0.61, improving the total similarity from 0.28 to 0.40. Finally, we applied the calibrated algorithm to refine measured vascular trees to the (higher) level of detail desired for specific applications. Having successfully adapted the CCO algorithm to the rodent model organism, the resulting individual-specific refined hepatic vascular trees can now be used for advanced modeling involving, e.g., detailed blood flow simulations.

Effect of treatment on the global dynamics of delayed pathological angiogenesis models

For three different types of angiogenesis models with variable delays, we consider either continuous or impulse therapy that eradicates tumor cells and suppresses angiogenesis. For the cancer-free solution, explicit conditions of global stability for the continuous and impulsive systems are obtained, together with delay-dependent estimates for the rates of decay for the tumor volume and pathological angiogenesis.

Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro

In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to ECM mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the ECM, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.