Dynamics of angiogenesis during wound healing: a coupled in vivo and in silico study

Diabetic wound healing of aloe vera major phytoconstituents through TGF-β1 suppression via in-silico docking, molecular dynamic simulation and pharmacokinetic studies

To restore the integrity of the skin and subcutaneous tissue, the wound healing process involves a complex series of well-orchestrated biochemical and cellular events. Due to the existence of various active components, accessibility and few side effects, some plant extracts and their phytoconstituents are recognised as viable options for wound healing agents. To find possible inhibitors of diabetic wound healing, four main constituents of aloe vera were identified from the literature. TGF-β1 and the compounds were studied using molecular docking to see how they interacted with the active site of target protein (PDB ID: 6B8Y). The pharmacokinetics investigation of the aloe emodin with the highest dock score complied with all the Lipinski's rule of five and pharmacokinetics criteria. Conformational change in the docked complex of Aloe emodin was investigated with the Amber simulation software, via a molecular dynamic (MD) simulation. The MD simulations of aloe emodin bound to TGF-β1 showed the significant structural rotations and twists occurring from 0 to 200 ns. The estimate of the aloe emodin-TGF-β1 complex's binding free energy has also been done using MM-PBSA/GBSA techniques. Additionally, aloe emodin has a wide range of enzymatic activities since their probability active (Pa) values is >0.700. 'Aloe emodin', an active extract of aloe vera, has been identified as the key chemical in the current investigation that can inhibit diabetic wound healing. Both in-vitro and in-vivo experiments will be used in a wet lab to confirm the current computational findings.Communicated by Ramaswamy H. Sarma.

Matricellular Proteins in the Homeostasis, Regeneration, and Aging of Skin

Matricellular proteins are secreted extracellular proteins that bear no primary structural functions but play crucial roles in tissue remodeling during development, homeostasis, and aging. Despite their low expression after birth, matricellular proteins within skin compartments support the structural function of many extracellular matrix proteins, such as collagens. In this review, we summarize the function of matricellular proteins in skin stem cell niches that influence stem cells' fate and self-renewal ability. In the epidermal stem cell niche, fibulin 7 promotes epidermal stem cells' heterogeneity and fitness into old age, and the transforming growth factor-β-induced protein ig-h3 (TGFBI)-enhances epidermal stem cell growth and wound healing. In the hair follicle stem cell niche, matricellular proteins such as periostin, tenascin C, SPARC, fibulin 1, CCN2, and R-Spondin 2 and 3 modulate stem cell activity during the hair cycle and may stabilize arrector pili muscle attachment to the hair follicle during piloerections (goosebumps). In skin wound healing, matricellular proteins are upregulated, and their functions have been examined in various gain-and-loss-of-function studies. However, much remains unknown concerning whether these proteins modulate skin stem cell behavior, plasticity, or cell-cell communications during wound healing and aging, leaving a new avenue for future studies.

Chronic Wound Healing Models

In this paper, we review and analyze the commonly available wound healing models reported in the literature and discuss their advantages and issues, considering their relevance and translational potential to humans. Our analysis includes different in vitro and in silico as well as in vivo models and experimental techniques. We further explore the new technologies in the study of wound healing to provide an all encompassing review of the most efficient ways to proceed with wound healing experiments. We revealed that there is not one model of wound healing that is superior and can give translatable results to human research. Rather, there are many different models that have specific uses for studying certain processes or stages of wound healing. Our analysis suggests that when performing an experiment to assess stages of wound healing or different therapies to enhance healing, one must consider not only the species that will be used but also the type of model and how this can best replicate the physiology or pathophysiology in humans.

Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation

A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.

Pericytes and shear stress each alter the shape of a self-assembled vascular network

Blood vessel morphology is dictated by mechanical and biochemical cues. Flow-induced shear stress and pericytes both play important roles, and they have previously been studied using on-chip vascular networks to uncover their connection to angiogenic sprouting and network stabilization. However, it is unknown which shear stress values promote angiogenesis, how pericytes are directed to sprouts, and how shear stress and pericytes affect the overall vessel morphology. Here, we employed a microfluidic device to study these phenomena in three-dimensional (3D) self-assembled vasculature. Computational fluid dynamics solver (COMSOL) simulations indicated that sprouts form most frequently at locations of relatively low shear stresses (0.5-1.5 dyn cm^-2). Experimental results show that pericytes limit vascular diameter. Interestingly, when treated with imatinib or crenolanib, which are chemotherapeutic drugs and inhibitors of platelet-derived growth factor receptor β (PDGFRβ), the pericyte coverage of vessels decreased significantly but vessel diameter remained unchanged. This furthers our understanding of the mechanisms underlying vascular development and demonstrates the value of this microfluidic device in future studies on drug development and vascular biology.

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.

Endothelial Cell Metabolism in Vascular Functions

Simple Summary

Recent findings in the field of vascular biology are nourishing the idea that targeting the endothelial cell metabolism may be an alternative strategy to antiangiogenic therapy, as well as a novel therapeutic approach for cardiovascular disease. Deepening the molecular mechanisms regulating how ECs re-adapt their metabolic status in response to the changeable conditions of the tissue microenvironment may be beneficial to develop novel innovative treatments to counteract the aberrant growth of vasculature.

Abstract

The endothelium is the innermost layer of all blood and lymphatic vessels composed of a monolayer of specialized endothelial cells (ECs). It is regarded as a dynamic and multifunctional endocrine organ that takes part in essential processes, such as the control of blood fluidity, the modulation of vascular tone, the regulation of immune response and leukocyte trafficking into perivascular tissues, and angiogenesis. The inability of ECs to perform their normal biological functions, known as endothelial dysfunction, is multi-factorial; for instance, it implicates the failure of ECs to support the normal antithrombotic and anti-inflammatory status, resulting in the onset of unfavorable cardiovascular conditions such as atherosclerosis, coronary artery disease, hypertension, heart problems, and other vascular pathologies. Notably, it is emerging that the ability of ECs to adapt their metabolic status to persistent changes of the tissue microenvironment could be vital for the maintenance of vascular functions and to prevent adverse vascular events. The main purpose of the present article is to shed light on the unique metabolic plasticity of ECs as a prospective therapeutic target; this may lead to the development of novel strategies for cardiovascular diseases and cancer.

Label-free concurrent 5-modal microscopy (Co5M) resolves unknown spatio-temporal processes in wound healing

The non-invasive investigation of multiple biological processes remains a methodological challenge as it requires capturing different contrast mechanisms, usually not available with any single modality. Intravital microscopy has played a key role in dynamically studying biological morphology and function, but it is generally limited to resolving a small number of contrasts, typically generated by the use of transgenic labels, disturbing the biological system. We introduce concurrent 5-modal microscopy (Co5M), illustrating a new concept for label-free in vivo observations by simultaneously capturing optoacoustic, two-photon excitation fluorescence, second and third harmonic generation, and brightfield contrast. We apply Co5M to non-invasively visualize multiple wound healing biomarkers and quantitatively monitor a number of processes and features, including longitudinal changes in wound shape, microvascular and collagen density, vessel size and fractality, and the plasticity of sebaceous glands. Analysis of these parameters offers unique insights into the interplay of wound closure, vasodilation, angiogenesis, skin contracture, and epithelial reformation in space and time, inaccessible by other methods. Co5M challenges the conventional concept of biological observation by yielding multiple simultaneous parameters of pathophysiological processes in a label-free mode.

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.

Long-Term Imaging of Wound Angiogenesis with Large Scale Optoacoustic Microscopy

Wound healing is a well-coordinated process, necessitating efficient formation of new blood vessels. Vascularization defects are therefore a major risk factor for chronic, non-healing wounds. The dynamics of mammalian tissue revascularization, vessel maturation, and remodeling remain poorly understood due to lack of suitable in vivo imaging tools. A label-free large-scale optoacoustic microscopy (LSOM) approach is developed for rapid, non-invasive, volumetric imaging of tissue regeneration over large areas spanning up to 50 mm with a depth penetration of 1.5 mm. Vascular networks in dorsal mouse skin and full-thickness excisional wounds are imaged with capillary resolution during the course of healing, revealing previously undocumented views of the angiogenesis process in an unperturbed wound environment. Development of an automatic analysis framework enables the identification of key features of wound angiogenesis, including vessel length, diameter, tortuosity, and angular alignment. The approach offers a versatile tool for preclinical research in tissue engineering and regenerative medicine, empowering label-free, longitudinal, high-throughput, and quantitative studies of the microcirculation in processes associated with normal and impaired vascular remodeling, and analysis of vascular responses to pharmacological interventions in vivo.

Simulation of angiogenesis in three dimensions: Application to cerebral cortex

The vasculature is a dynamic structure, growing and regressing in response to embryonic development, growth, changing physiological demands, wound healing, tumor growth and other stimuli. At the microvascular level, network geometry is not predetermined, but emerges as a result of biological responses of each vessel to the stimuli that it receives. These responses may be summarized as angiogenesis, remodeling and pruning. Previous theoretical simulations have shown how two-dimensional vascular patterns generated by these processes in the mesentery are consistent with experimental observations. During early development of the brain, a mesh-like network of vessels is formed on the surface of the cerebral cortex. This network then forms branches into the cortex, forming a three-dimensional network throughout its thickness. Here, a theoretical model is presented for this process, based on known or hypothesized vascular response mechanisms together with experimentally obtained information on the structure and hemodynamics of the mouse cerebral cortex. According to this model, essential components of the system include sensing of oxygen levels in the midrange of partial pressures and conducted responses in vessel walls that propagate information about metabolic needs of the tissue to upstream segments of the network. The model provides insights into the effects of deficits in vascular response mechanisms, and can be used to generate physiologically realistic microvascular network structures.

Mathematical Modeling Can Advance Wound Healing Research

Significance: For over 30 years, there has been sustained interest in the development of mathematical models for investigating the complex mechanisms underlying each stage of the wound healing process. Despite the immense associated challenges, such models have helped usher in a paradigm shift in wound healing research. Recent Advances: In this article, we review contributions in the field that span epidermal, dermal, and corneal wound healing, and treatments of nonhealing wounds. The recent influence of mathematical models on biological experiments is detailed, with a focus on wound healing assays and fibroblast-populated collagen lattices. Critical Issues: We provide an overview of the field of mathematical modeling of wound healing, highlighting key advances made in recent decades, and discuss how such models have contributed to the development of improved treatment strategies and/or an enhanced understanding of the tightly regulated steps that comprise the healing process. Future Directions: We detail some of the open problems in the field that could be addressed through a combination of theoretical and/or experimental approaches. To move the field forward, we need to have a common language between scientists to facilitate cross-collaboration, which we hope this review can support by highlighting progress to date.

Optical Coherence Tomography Angiography Monitors Cutaneous Wound Healing under Angiogenesis-Promoting Treatment in Diabetic and Non-Diabetic Mice

During wound healing, the rapid re-establishment of a functional microcirculation in the wounded tissue is of utmost importance. We applied optical coherence tomography (OCT) angiography to evaluate vascular remodeling in an excisional wound model in the pinnae of C57BL/6 and db/db mice receiving different proangiogenic topical treatments. Analysis of the high-resolution OCT angiograms, including the four quantitative parameters vessel density, vessel length, number of bifurcations, and vessel tortuosity, revealed changes of the microvasculature and allowed identification of the overlapping wound healing phases hemostasis, inflammation, proliferation, and remodeling. Angiograms acquired in the inflammatory phase in the first days showed a dilation of vessels and recruitment of pre-existing capillaries. In the proliferative phase, angiogenesis with the sprouting of new capillaries into the wound tissue led to an increase of the OCT angiography parameters vessel density, normalized vessel length, number of bifurcations, and vessel tortuosity by 28–47%, 39–52%, 33–48%, and 3–8% versus baseline, respectively. After the peak observed on study days four to seven, the parameters slowly decreased but remained still elevated 18 days after wounding, indicating a continuing remodeling phase. Our study suggests that OCT angiography has the potential to serve as a valuable preclinical research tool in studies investigating impaired vascular remodeling during wound healing and potential new treatment strategies.

A preliminary study of endothelial cell migration during angiogenesis using a meshless method approach

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the "influence-domain" approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two-dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

Sprouting Angiogenesis: A Numerical Approach with Experimental Validation

A functional vascular network is essential to the correct wound healing. In sprouting angiogenesis, vascular endothelial growth factor (VEGF) regulates the formation of new capillaries from pre-existing vessels. This is a very complex process and mathematical formulation permits to study angiogenesis using less time-consuming, reproducible and cheaper methodologies. This study aimed to mimic the chemoattractant effect of VEGF in stimulating sprouting angiogenesis. We developed a numerical model in which endothelial cells migrate according to a diffusion-reaction equation for VEGF. A chick chorioallantoic membrane (CAM) bioassay was used to obtain some important parameters to implement in the model and also to validate the numerical results. We verified that endothelial cells migrate following the highest VEGF concentration. We compared the parameters-total branching number, total vessel length and branching angle-that were obtained in the in silico and the in vivo methodologies and similar results were achieved (p-value smaller than 0.5; n = 6). For the difference between the total capillary volume fractions assessed using both methodologies values smaller than 15% were obtained. In this study we simulated, for the first time, the capillary network obtained during the CAM assay with a realistic morphology and structure.

Eden Model Simulation of Re-Epithelialization and Angiogenesis of an Epidermal Wound

Among the vital processes of cutaneous wound healing are epithelialization and angiogenesis. The former leads to the successful closure of the wound while the latter ensures that nutrients are delivered to the wound region during and after healing is completed. These processes are regulated by various cytokines and growth factors that subtend their proliferation and migration into the wound region until full healing is attained. Wound epithelialization can be enhanced by the administration of epidermal stem cells (ESC) or impaired by the presence of an infection. This paper uses the Eden model of a growing cluster to independently simulate the processes of epithelialization and angiogenesis in a cutaneous wound for different geometries. Further, simulations illustrating bacterial infection are provided. Our simulation results demonstrate contraction and closure for any wound geometry due to a collective migration of epidermal cells from the wound edge in fractal form and the diffusion of capillary sprouts with the laying down of capillary blocks behind moving tips into the wound area.

Modelling skin wound healing angiogenesis: A review

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

A two-phase model of early fibrous cap formation in atherosclerosis

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) are recruited from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and dangerously vulnerable to rupture. We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using data from a range of in vitro and in vivo studies, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development.

The panniculus carnosus muscle: an evolutionary enigma at the intersection of distinct research fields

The panniculus carnosus is a thin striated muscular layer intimately attached to the skin and fascia of most mammals, where it provides skin twitching and contraction functions. In humans, the panniculus carnosus is conserved at sparse anatomical locations with high interindividual variability, and it is considered of no functional significance (most possibly being a remnant of evolution). Diverse research fields (such as anatomy, dermatology, myology, neuroscience, surgery, veterinary science) use this unique muscle as a model, but several unknowns and misconceptions remain in the literature. In this article, we review what is currently known about panniculus carnosus structure, development, anatomical location, response to environmental stimuli and potential function(s), with the aim of putting together the evidence arising from the different research communities and raising interest in this unique muscle, which we postulate as an ideal model for both vascular and muscular research.

Study on the tumor-induced angiogenesis using mathematical models

We studied angiogenesis using mathematical models describing the dynamics of tip cells. We reviewed the basic ideas of angiogenesis models and its numerical simulation technique to produce realistic computer graphics images of sprouting angiogenesis. We examined the classical model of Anderson-Chaplain using fundamental concepts of mass transport and chemical reaction with ECM degradation included. We then constructed two types of numerical schemes, model-faithful and model-driven ones, where new techniques of numerical simulation are introduced, such as transient probability, particle velocity, and Boolean variables.

A hyperspectral vessel image registration method for blood oxygenation mapping

Blood oxygenation mapping by the means of optical oximetry is of significant importance in clinical trials. This paper uses hyperspectral imaging technology to obtain in vivo images for blood oxygenation detection. The experiment involves dorsal skin fold window chamber preparation which was built on adult (8-10 weeks of age) female BALB/c nu/nu mice and in vivo image acquisition which was performed by hyperspectral imaging system. To get the accurate spatial and spectral information of targets, an automatic registration scheme is proposed. An adaptive feature detection method which combines the local threshold method and the level-set filter is presented to extract target vessels. A reliable feature matching algorithm with the correlative information inherent in hyperspectral images is used to kick out the outliers. Then, the registration images are used for blood oxygenation mapping. Registration evaluation results show that most of the false matches are removed and the smooth and concentrated spectra are obtained. This intensity invariant feature detection with outliers-removing feature matching proves to be effective in hyperspectral vessel image registration. Therefore, in vivo hyperspectral imaging system by the assistance of the proposed registration scheme provides a technique for blood oxygenation research.

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.

Multimodality imaging and mathematical modelling of drug delivery to glioblastomas

Patients diagnosed with glioblastoma, an aggressive brain tumour, have a poor prognosis, with a median overall survival of less than 15 months. Vasculature within these tumours is typically abnormal, with increased tortuosity, dilation and disorganization, and they typically exhibit a disrupted blood-brain barrier (BBB). Although it has been hypothesized that the 'normalization' of the vasculature resulting from anti-angiogenic therapies could improve drug delivery through improved blood flow, there is also evidence that suggests that the restoration of BBB integrity might limit the delivery of therapeutic agents and hence their effectiveness. In this paper, we apply mathematical models of blood flow, vascular permeability and diffusion within the tumour microenvironment to investigate the effect of these competing factors on drug delivery. Preliminary results from the modelling indicate that all three physiological parameters investigated-flow rate, vessel permeability and tissue diffusion coefficient-interact nonlinearly to produce the observed average drug concentration in the microenvironment.

On the mathematical modeling of wound healing angiogenesis in skin as a reaction-transport process

Over the last 30 years, numerous research groups have attempted to provide mathematical descriptions of the skin wound healing process. The development of theoretical models of the interlinked processes that underlie the healing mechanism has yielded considerable insight into aspects of this critical phenomenon that remain difficult to investigate empirically. In particular, the mathematical modeling of angiogenesis, i.e., capillary sprout growth, has offered new paradigms for the understanding of this highly complex and crucial step in the healing pathway. With the recent advances in imaging and cell tracking, the time is now ripe for an appraisal of the utility and importance of mathematical modeling in wound healing angiogenesis research. The purpose of this review is to pedagogically elucidate the conceptual principles that have underpinned the development of mathematical descriptions of wound healing angiogenesis, specifically those that have utilized a continuum reaction-transport framework, and highlight the contribution that such models have made toward the advancement of research in this field. We aim to draw attention to the common assumptions made when developing models of this nature, thereby bringing into focus the advantages and limitations of this approach. A deeper integration of mathematical modeling techniques into the practice of wound healing angiogenesis research promises new perspectives for advancing our knowledge in this area. To this end we detail several open problems related to the understanding of wound healing angiogenesis, and outline how these issues could be addressed through closer cross-disciplinary collaboration.

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.

Angiogenesis in diabetes and obesity

The prevalence of diabetes mellitus and obesity continues to increase globally. Diabetic vascular complications are the main chronic diabetic complications and associated with mortality and disability. Angiogenesis is a key pathological characteristic of diabetic microvascular complications. However, there are two tissue-specific paradoxical changes in the angiogenesis in diabetic microvascular complications: an excessive uncontrolled formation of premature blood vessels in some tissues, such as the retina, and a deficiency in the formation of small blood vessels in peripheral tissues, such as the skin. This review will discuss the paradoxical phenomena of angiogenesis and its underlying mechanism in obesity, diabetes and diabetic complications.

Arteries are formed by vein-derived endothelial tip cells

Tissue vascularization entails the formation of a blood vessel plexus, which remodels into arteries and veins. Here we show, by using time-lapse imaging of zebrafish fin regeneration and genetic lineage tracing of endothelial cells in the mouse retina, that vein-derived endothelial tip cells contribute to emerging arteries. Our movies uncover that arterial-fated tip cells change migration direction and migrate backwards within the expanding vascular plexus. This behaviour critically depends on chemokine receptor cxcr4a function. We show that the relevant Cxcr4a ligand Cxcl12a selectively accumulates in newly forming bone tissue even when ubiquitously overexpressed, pointing towards a tissue-intrinsic mode of chemokine gradient formation. Furthermore, we find that cxcr4a mutant cells can contribute to developing arteries when in association with wild-type cells, suggesting collective migration of endothelial cells. Together, our findings reveal specific cell migratory behaviours in the developing blood vessel plexus and uncover a conserved mode of artery formation.

Sprouting of new blood vessels depends on the migration of endothelial tip cells into surrounding tissue. Here the authors reveal the existence of a distinct migratory signalling circuit that guides endothelial cells from developing veins to the leading tip position in developing arteries.

Angiogenesis: an adaptive dynamic biological patterning problem

Formation of functionally adequate vascular networks by angiogenesis presents a problem in biological patterning. Generated without predetermined spatial patterns, networks must develop hierarchical tree-like structures for efficient convective transport over large distances, combined with dense space-filling meshes for short diffusion distances to every point in the tissue. Moreover, networks must be capable of restructuring in response to changing functional demands without interruption of blood flow. Here, theoretical simulations based on experimental data are used to demonstrate that this patterning problem can be solved through over-abundant stochastic generation of vessels in response to a growth factor generated in hypoxic tissue regions, in parallel with refinement by structural adaptation and pruning. Essential biological mechanisms for generation of adequate and efficient vascular patterns are identified and impairments in vascular properties resulting from defects in these mechanisms are predicted. The results provide a framework for understanding vascular network formation in normal or pathological conditions and for predicting effects of therapies targeting angiogenesis.

The blood vessels provide an efficient system for transport of substances to all parts of the body. They are capable of growing or regressing during development, in response to changing functional needs, and in disease states. This is achieved by structural adaptation, i.e. changes in the diameters and other characteristics of existing vessels, and by angiogenesis, i.e. growth of new blood vessels. Here, we address the question: How do the processes of structural adaptation and angiogenesis lead to the formation of organized vessel networks that can supply the changing needs of the tissue? We carried out theoretical simulations of network growth and adaptation, including vessel blood flows, oxygen transport to tissue, and the generation of a growth factor in low-oxygen regions, which stimulates angiogenesis by sprouting from existing vessels. We showed that the processes of over-abundant random angiogenesis together with structural adaptation including pruning of redundant vessels can generate adequate and efficient vessel networks that are capable of continuously adapting to changing tissue needs. Our work provides insight into the biological mechanisms that are essential for formation and maintenance of functional vessel networks, and may lead to new strategies for controlling blood vessel formation in diseases.

A hybrid discrete-continuum mathematical model of pattern prediction in the developing retinal vasculature

Pathological angiogenesis has been extensively explored by the mathematical modelling community over the past few decades, specifically in the contexts of tumour-induced vascularisation and wound healing. However, there have been relatively few attempts to model angiogenesis associated with normal development, despite the availability of animal models with experimentally accessible and highly ordered vascular topologies: for example, growth and development of the vascular plexus layers in the murine retina. The current study aims to address this issue through the development of a hybrid discrete-continuum mathematical model of the developing retinal vasculature in neonatal mice that is closely coupled with an ongoing experimental programme. The model of the functional vasculature is informed by a range of morphological and molecular data obtained over a period of several days, from 6 days prior to birth to approximately 8 days after birth. The spatio-temporal formation of the superficial retinal vascular plexus (RVP) in wild-type mice occurs in a well-defined sequence. Prior to birth, astrocytes migrate from the optic nerve over the surface of the inner retina in response to a chemotactic gradient of PDGF-A, formed at an earlier stage by migrating retinal ganglion cells (RGCs). Astrocytes express a variety of chemotactic and haptotactic proteins, including VEGF and fibronectin (respectively), which subsequently induce endothelial cell sprouting and modulate growth of the RVP. The developing RVP is not an inert structure; however, the vascular bed adapts and remodels in response to a wide variety of metabolic and biomolecular stimuli. The main focus of this investigation is to understand how these interacting cellular, molecular, and metabolic cues regulate RVP growth and formation. In an earlier one-dimensional continuum model of astrocyte and endothelial migration, we showed that the measured frontal velocities of the two cell types could be accurately reproduced by means of a system of five coupled partial differential equations (Aubert et al. in Bull. Math. Biol. 73:2430-2451, 2011). However, this approach was unable to generate spatial information and structural detail for the entire retinal surface. Building upon this earlier work, a more realistic two-dimensional hybrid PDE-discrete model is derived here that tracks the migration of individual astrocytes and endothelial tip cells towards the outer retinal boundary. Blood perfusion is included throughout plexus development and the emergent retinal architectures adapt and remodel in response to various biological factors. The resulting in silico RVP structures are compared with whole-mounted retinal vasculatures at various stages of development, and the agreement is found to be excellent. Having successfully benchmarked the model against wild-type data, the effect of transgenic over-expression of various genes is predicted, based on the ocular-specific expression of VEGF-A during murine development. These results can be used to help inform future experimental investigations of signalling pathways in ocular conditions characterised by aberrant angiogenesis.

The mouse dorsal skin fold chamber as a means for the analysis of tissue engineered skin

The therapy of extensive and deep burn wounds is still a challenging task for reconstructive plastic surgery. The outcome is generally not satisfactory, neither from the functional nor from the aesthetic aspect. Several available skin substitutes are used but there is need for optimization of new skin substitutes which have to be tested in vitro as well as in vivo. Here, we show that the dorsal skin fold chamber preparation of mice is well suited for the testing of skin substitutes in vivo. Dermal skin constructs consisting of matriderm(®) covered with a collagen type I gel were inserted into full thickness skin wounds in the skin fold chambers. The skin substitutes integrated well into the adjacent skin and got epithelialized from the wound edges within 11 days. The epithelialization by keratinocytes is the prerequisite that also cell-free dermal substitutes might be used in the case of the lack of sufficient areas to gain split thickness skin grafts. Further advantage of the chambers is the lack of wound contraction, which is common but undesired in rodent wound healing. Furthermore, this model allows a sophisticated histological as well as immunohistochemical analysis. As such, we conclude that this model is well suited for the analysis of tissue engineered skin constructs. Besides epithelialization the mode and extend of neovascularization and contraction of artificial grafts may be studied under standardized conditions.

Dynamics of angiogenesis during murine retinal development: a coupled in vivo and in silico study

The manner in which the superficial retinal vascular plexus (RVP) develops in neonatal wild-type mice is relatively well documented and poses an interesting challenge to the mathematical modelling community. Prior to birth, astrocyte sprouting and proliferation begin around the edge of the optic nerve head, and subsequent astrocyte migration in response to a chemotactic gradient of platelet-derived growth factor (PDGF)-A results in the formation of a dense scaffold on the surface of the inner retina. Astrocytes express a variety of chemotactic and haptotactic proteins that subsequently induce endothelial cell sprouting and modulate growth of the RVP. An experimentally informed, two-dimensional hybrid partial differential equation-discrete model is derived to track the outward migration of individual astrocyte and endothelial tip cells in response to the appropriate biochemical cues. Blood perfusion is included throughout the development of the plexus, and the evolving retinal trees are allowed to adapt and remodel by means of several biological stimuli. The resulting wild-type in silico RVP structures are compared with corresponding experimental whole mounts taken at various stages of development, and agreement between the respective vascular morphologies is found to be excellent. Subsequent numerical predictions help elucidate some of the key biological processes underlying retinal development and demonstrate the potential of the virtual retina for the investigation of various vascular-related diseases of the eye.

Wound healing angiogenesis: the clinical implications of a simple mathematical model

Nonhealing wounds are a major burden for health care systems worldwide. In addition, a patient who suffers from this type of wound usually has a reduced quality of life. While the wound healing process is undoubtedly complex, in this paper we develop a deterministic mathematical model, formulated as a system of partial differential equations, that focusses on an important aspect of successful healing: oxygen supply to the wound bed by a combination of diffusion from the surrounding unwounded tissue and delivery from newly formed blood vessels. While the model equations can be solved numerically, the emphasis here is on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated and wound-bed angiogenesis can progress. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. We use our model to discuss the clinical use of treatments such as hyperbaric oxygen therapy, wound bed debridement, and revascularisation therapy that have the potential to initiate healing in chronic, stalled wounds.

Synergistic Regulation of Angiogenic Sprouting by Biochemical Factors and Wall Shear Stress

The process of sprouting angiogenesis involves activating endothelial cells in a quiescent monolayer of an existing vessel to degrade and migrate into the underlying matrix to form new blood vessels. While the roles of biochemical factors in angiogenic sprouting have been well characterized, the roles of fluid forces have received much less attention. This review summarizes results that support a role for wall shear stress in post-capillary venules as a mechanical factor capable of synergizing with biochemical factors to stimulate pro-angiogenic signaling in endothelial cells and promote sprout formation.

Temporal changes in microvessel leakiness during wound healing discriminated by in vivo fluorescence recovery after photobleaching

Regeneration of injured tissue is a dynamic process, critically dependent on the formation of new blood vessels and restructuring of the nascent plexus. Endothelial barrier function, a functional correlate of vascular restructuring and maturation, was quantified via intravital microscopic analysis of 150 kDa FITC-dextran-perfused blood vessels within discrete wounds created in the panniculus carnosus (PC) muscle of dorsal skinfold chamber (DSC) preparations in mice. Time to recovery of half-peak fluorescence intensity (t(1/2)) within individual vessel segments in three functional regions of the wound (pre-existing vessels, angiogenic plexus and blind-ended vessels (BEVs)) was quantified using in vivo fluorescence recovery after photobleaching (FRAP) and linear regression analysis of recovery profiles. Plasma flux across the walls of new vessel segments, particularly BEVs, was greater than that of pre-existing vessels at days 5-7 after injury (P < 0.05). TNP-470 reduced the permeability of BEVs at the leading edge of the advancing vascular plexus as measured by the decrease in luminal t(1/2) (P < 0.05), confirming the utility of FRAP as a quantitative measure of endothelial barrier function. Furthermore, these data are suggestive of a role for TNP-470 in selection for less leaky vascular segments within healing wounds. Increased FITC-dextran leakage was observed from pre-existing vessels after treatment with TNP-470 (P < 0.05), consistent with induction of transient vascular damage, although the significance of this finding is unclear. Using in vivo FRAP this study demonstrates the relationship between temporal changes in microvascular macromolecular flux and the morphology of maturing vascular segments. This combination of techniques may be useful to assess the therapeutic potential of angiogenic agents in restoring pre-injury levels of endothelial barrier function, following the establishment of a functional vascular plexus such as in models of wounding or tumour development.