Wound angiogenesis as a function of tissue oxygen tension: a mathematical model

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.

Modeling the effects of systemic mediators on the inflammatory phase of wound healing

The normal wound healing response is characterized by a progression from clot formation, to an inflammatory phase, to a repair phase, and finally, to remodeling. In many chronic wounds there is an extended inflammatory phase that stops this progression. In order to understand the inflammatory phase in more detail, we developed an ordinary differential equation model that accounts for two systemic mediators that are known to modulate this phase, estrogen (a protective hormone during wound healing) and cortisol (a hormone elevated after trauma that slows healing). This model describes the interactions in the wound between wound debris, pathogens, neutrophils and macrophages and the modulation of these interactions by estrogen and cortisol. A collection of parameter sets, which qualitatively match published data on the dynamics of wound healing, was chosen using Latin Hypercube Sampling. This collection of parameter sets represents normal healing in the population as a whole better than one single parameter set. Including the effects of estrogen and cortisol is a necessary step to creating a patient specific model that accounts for gender and trauma. Utilization of math modeling techniques to better understand the wound healing inflammatory phase could lead to new therapeutic strategies for the treatment of chronic wounds. This inflammatory phase model will later become the inflammatory subsystem of our full wound healing model, which includes fibroblast activity, collagen accumulation and remodeling.

Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis

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

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

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

Synergistic effects of 3D ECM and chemogradients on neurite outgrowth and guidance: a simple modeling and microfluidic framework

During nervous system development, numerous cues within the extracellular matrix microenvironment (ECM) guide the growing neurites along specific pathways to reach their intended targets. Neurite motility is controlled by extracellular signal sensing through the growth cone at the neurite tip, including chemoattractive and repulsive cues. However, it is difficult to regenerate and restore neurite tracts, lost or degraded due to an injury or disease, in the adult central nervous system. Thus, it is important to evaluate the dynamic interplay between ECM and the concentration gradients of these cues, which would elicit robust neuritogenesis. Such information is critical in understanding the processes involved in developmental biology, and in developing high-fidelity neurite regenerative strategies post-injury, and in drug discovery and targeted therapeutics for neurodegenerative conditions. Here, we quantitatively investigated this relationship using a combination of mathematical modeling and in vitro experiments, and determined the synergistic role of guidance cues and ECM on neurite outgrowth and turning. Using a biomimetic microfluidic system, we have shown that cortical neurite outgrowth and turning under chemogradients (IGF-1 or BDNF) within 3D scaffolds is highly regulated by the source concentration of the guidance cue and the physical characteristics of the scaffold. A mechanistic-driven partial differential equation model of neurite outgrowth has been proposed, which could also be used prospectively as a predictive tool. The parameters for the chemotaxis term in the model are determined from the experimental data using our microfluidic assay. Resulting model simulations demonstrate how neurite outgrowth was critically influenced by the experimental variables, which was further supported by experimental data on cell-surface-receptor expressions. The model results are in excellent agreement with the experimental findings. This integrated approach represents a framework for further elucidation of biological mechanisms underlying neuronal responses of specialized cell types, during various stages of development, and under healthy or diseased conditions.

Nonlinear finite element simulations of injuries with free boundaries: application to surgical wounds

Wound healing is a process driven by biochemical and mechanical variables in which a new tissue is synthesised to recover original tissue functionality. Wound morphology plays a crucial role in this process, as the skin behaviour is not uniform along different directions. In this work, we simulate the contraction of surgical wounds, which can be characterised as elongated and deep wounds. Because of the regularity of this morphology, we approximate the evolution of the wound through its cross section, adopting a plane strain hypothesis. This simplification reduces the complexity of the computational problem; while allows for a thorough analysis of the role of wound depth in the healing process, an aspect of medical and computational relevance that has not yet been addressed. To reproduce wound contraction, we consider the role of fibroblasts, myofibroblasts, collagen and a generic growth factor. The contraction phenomenon is driven by cell-generated forces. We postulate that these forces are adjusted to the mechanical environment of the tissue where cells are embedded through a mechanosensing and mechanotransduction mechanism. To solve the nonlinear problem, we use the finite element method (FEM) and an updated Lagrangian approach to represent the change in the geometry. To elucidate the role of wound depth and width on the contraction pattern and evolution of the involved species, we analyse different wound geometries with the same wound area. We find that deeper wounds contract less and reach a maximum contraction rate earlier than superficial wounds.

A systems biology view of blood vessel growth and remodelling

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

[Uterine ishtmique transmural hernia: results of its repair on symptoms and fertility]

OBJECTIVE

To study the effect of the surgical repair (isthmorraphy) of the large scar dehiscence after cesarean on symptoms and fertility for women who desire pregnancy.

PATIENTS AND METHODS

In this retrospective study, 14 symptomatic patients, who desire a new pregnancy underwent a surgical repair by laparotomy, laparoscopic or vaginal technique. Five women experienced failure of Assistance Reproductive Technique (IVF or ICSI) for idiopathic secondary infertility. The dehiscent scars were evaluated by ultrasound, hysterography, hysteroscopy and magnetic resonance imaging.

OUTCOME

Symptoms improvement was found in 92% of case. Ten pregnancy (71%) was obtained after surgical repair, 6 spontaneous and 4 after Assistance Reproductive Technique. Among the 5 women initially followed in the reproductive unit, 4 became pregnant, 3 after IVF or ICSI and 1 spontaneous. No operative complication occurred. The subsequent pregnancy was unremarkable with no uterine rupture.

DISCUSSION

Large scar defect after cesarean can take shape of a complete absent of the anterior wall of the uterus. No incident has been proved in this condition. There is a lack of data concerning these isthmocele. The experience of hysteroscopic repair cannot be applied to these real large diverticule of the scar cesarean. The results of this study suggest a link between the isthmocele and reversible symptoms after surgery. The first results concerning the subsequent fertility after surgical repair seem interesting

CONCLUSION

When a large scare defect (isthmocele) is found in symptomatic woman (pelvic pain, bleeding uterine, infertility), a surgical repair can be proposed, especially for woman who desire a new pregnancy.

A computational model of fibroblast and macrophage spatial/temporal dynamics in foreign body reactions

The implantation of medical devices often triggers several immune responses, one kind of which is categorized as foreign body reactions. It is well established that macrophages and many other cells participate in the complex processes of foreign body reactions, and cause severe inflammations and fibrotic capsule formation in surrounding tissues. However, the detailed mechanisms of macrophage responses, recruitment and activation, in foreign body reactions are not totally understood. In the meantime, mathematical models have been proposed to systematically decipher the behavior of this complex system of multiple cells, proteins and biochemical processes in wound healing responses. Based on these early works, this study introduces a mathematical model in two spatial dimensions to investigate the transient behavior of macrophages, fibroblasts and their interactions during the formation of fibrotic tissue. We find that the simulation results are consistent with the experimental observations. These findings support that the model can reveal quantitative insights for studying foreign body reaction processes.

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

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

Changes in Cesarean section scar dimensions during pregnancy: a prospective longitudinal study

OBJECTIVES

To describe changes in Cesarean section (CS) scars longitudinally throughout pregnancy, and to relate initial scar measurements, demographic variables and obstetric variables to subsequent changes in scar features and to final pregnancy outcome.

METHODS

In this prospective observational study we used transvaginal sonography (TVS) to examine the CS scar of 320 consecutive pregnant women at 11-13, 19-21 and 32-34 weeks' gestation. For scars visible on TVS, the hypoechoic part was measured in three dimensions and the residual myometrial thickness (RMT) was also measured. Analyses were carried out using one-way repeated measures ANOVA and mixed modeling. The incidence of subsequent scar rupture was recorded.

RESULTS

The CS scar was visible in 284/320 cases (89%). Concerning length and depth of the hypoechoic part of the scar and RMT, the larger the initial scar measurement, the larger the decrease observed during pregnancy. For the hypoechoic part of the scar, the width increased on average by 1.8 mm per trimester, while the depth and length decreased by 1.8 and 1.9 mm, respectively (false discovery rate P < 0.0001). Mean RMT in the first trimester was 5.2 mm and on average decreased by 1.1 mm per trimester. Two cases (0.62%) of uterine scar rupture were confirmed following a trial of vaginal delivery; these had a mean RMT of 0.5 mm at second scan and an average decrease of 2.6 mm over the course of pregnancy.

CONCLUSION

This study establishes reference data and confirms that the dimensions of CS scars change throughout pregnancy. Scar rupture was associated with a smaller RMT and greater decrease in RMT during pregnancy. There is the potential to test absolute values and observed changes in CS scar measurements as predictors of uterine scar rupture and outcome in trials of vaginal birth after Cesarean section.

Systems-based approaches toward wound healing

Wound healing in the pediatric patient is of utmost clinical and social importance because hypertrophic scarring can have aesthetic and psychological sequelae, from early childhood to late adolescence. Wound healing is a well-orchestrated reparative response affecting the damaged tissue at the cellular, tissue, organ, and system scales. Although tremendous progress has been made toward understanding wound healing at the individual temporal and spatial scales, its effects across the scales remain severely understudied and poorly understood. Here, we discuss the critical need for systems-based computational modeling of wound healing across the scales, from short-term to long-term and from small to large. We illustrate the state of the art in systems modeling by means of three key signaling mechanisms: oxygen tension-regulating angiogenesis and revascularization; transforming growth factor-β (TGF-β) kinetics controlling collagen deposition; and mechanical stretch stimulating cellular mitosis and extracellular matrix (ECM) remodeling. The complex network of biochemical and biomechanical signaling mechanisms and the multiscale character of the healing process make systems modeling an integral tool in exploring personalized strategies for wound repair. A better mechanistic understanding of wound healing in the pediatric patient could open new avenues in treating children with skin disorders such as birth defects, skin cancer, wounds, and burn injuries.

The role of CD200-CD200R in tumor immune evasion

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

A non-local evolution equation model of cell-cell adhesion in higher dimensional space

A model for cell-cell adhesion, based on an equation originally proposed by Armstrong et al. [A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243 (2006), pp. 98-113], is considered. The model consists of a nonlinear partial differential equation for the cell density in an N-dimensional infinite domain. It has a non-local flux term which models the component of cell motion attributable to cells having formed bonds with other nearby cells. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.

Stability analysis of a model for foreign body fibrotic reactions

Implanted medical devices often trigger immunological and inflammatory reactions from surrounding tissues. The foreign body-mediated tissue responses may result in varying degrees of fibrotic tissue formation. There is an intensive research interest in the area of wound healing modeling, and quantitative methods are proposed to systematically study the behavior of this complex system of multiple cells, proteins, and enzymes. This paper introduces a kinetics-based model for analyzing reactions of various cells/proteins and biochemical processes as well as their transient behavior during the implant healing in 2-dimensional space. In particular, we provide a detailed modeling study of different roles of macrophages (MΦ) and their effects on fibrotic reactions. The main mathematical result indicates that the stability of the inflamed steady state depends primarily on the reaction dynamics of the system. However, if the said equilibrium is unstable by its reaction-only system, the spatial diffusion and chemotactic effects can help to stabilize when the model is dominated by classical and regulatory macrophages over the inflammatory macrophages. The mathematical proof and counter examples are given for these conclusions.

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

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

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.

Overcoming hypoxia to improve tissue-engineering approaches to regenerative medicine

The current clinical successes of tissue engineering are limited primarily to low-metabolism, acellular, pre-vascularized or thin tissues. Mass transport has been identified as the primary culprit, limiting the delivery of nutrients (such as oxygen and glucose) and removal of wastes, from tissues deep within a cellular scaffold. While strategies to develop sufficient vasculature to overcome hypoxia in vitro are promising, inconsistencies between the in vitro and the in vivo environments may still negate the effectiveness of large-volume tissue-engineered scaffolds. While a common theme in tissue engineering is to maximize oxygen supply, studies suggest that moderate oxygenation of cellular scaffolds during in vitro conditioning is preferable to high oxygen levels. Aiming for moderate oxygen values to prevent hypoxia while still promoting angiogenesis may be obtained by tailoring in vitro culture conditions to the oxygen environment the scaffold will experience upon implantation. This review discusses the causes and effects of tissue-engineering hypoxia and the optimization of oxygenation for the minimization of in vivo hypoxia.

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell-cell contacts

A phenomenological model for the evolution of shape transition of cells is considered. These transitions are determined by the emission of growth-factors, as well as mechanical interaction if cells are subjected to hard impingement. The originality of this model necessitates a formal treatment of the mathematical model, as well as the presentation of elementary cases in order to illustrate the consistence of the model. We will also show some small-scale relevant applications.

Modeling the inhibition of breast cancer growth by GM-CSF

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

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.

Modeling stem/progenitor cell-induced neovascularization and oxygenation around solid implants

Tissue engineering constructs and other solid implants with biomedical applications, such as drug delivery devices or bioartificial organs, need oxygen (O(2)) to function properly. To understand better the vascular integration of such devices, we recently developed a novel model sensor containing O(2)-sensitive crystals, consisting of a polymeric capsule limited by a nanoporous filter. The sensor was implanted in mice with hydrogel alone (control) or hydrogel embedded with mouse CD117/c-kit+ bone marrow progenitor cells in order to stimulate peri-implant neovascularization. The sensor provided local partial O(2) pressure (pO(2)) using noninvasive electron paramagnetic resonance signal measurements. A consistently higher level of peri-implant oxygenation was observed in the cell-treatment case than in the control over a 10-week period. To provide a mechanistic explanation of these experimental observations, we present in this article a mathematical model, formulated as a system of coupled partial differential equations, that simulates peri-implant vascularization. In the control case, vascularization is considered to be the result of a foreign body reaction, while in the cell-treatment case, adipogenesis in response to paracrine stimuli produced by the stem cells is assumed to induce neovascularization. The model is validated by fitting numerical predictions of local pO(2) to measurements from the implanted sensor. The model is then used to investigate further the potential for using stem cell treatment to enhance the vascular integration of biomedical implants. We thus demonstrate how mathematical modeling combined with experimentation can be used to infer how vasculature develops around biomedical implants in control and stem cell-treated cases.

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.

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

OBJECTIVE

The most critical determinant of restoration of tissue structure during wound healing is the re-establishment of a functional vasculature, which largely occurs via angiogenesis, specifically endothelial sprouting from the pre-existing vasculature.

MATERIALS AND METHODS

We used confocal microscopy to capture sequential images of perfused vascular segments within the injured panniculus carnosus muscle in the mouse dorsal skin-fold window chamber to quantify a range of microcirculatory parameters during the first nine days of healing. This data was used to inform a mathematical model of sequential growth of the vascular plexus. The modeling framework mirrored the experimental circular wound domain and incorporated capillary sprouting and endothelial cell (EC) sensing of vascular endothelial growth factor gradients.

RESULTS

Wound areas, vessel densities and vessel junction densities obtained from the corresponding virtual wound were in excellent agreement both temporally and spatially with data measured during the in vivo healing process. Moreover, by perturbing the proliferative ability of ECs in the mathematical model, this leads to a severe reduction in vascular growth and poor healing. Quantitative measures from this second set of simulations were found to correlate extremely well with experimental data obtained from animals treated with an agent that targets endothelial proliferation (TNP-470).

CONCLUSION

Our direct combination and comparison of in vivo longitudinal analysis (over time in the same animal) and mathematical modeling employed in this study establishes a useful new paradigm. The virtual wound created in this study can be used to investigate a wide range of experimental hypotheses associated with wound healing, including disorders characterized by aberrant angiogenesis (e.g., diabetic models) and the effects of vascular enhancing/disrupting agents or therapeutic interventions such as hyperbaric oxygen.

Tumor angiogenesis and vascular patterning: a mathematical model

Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

Module-based multiscale simulation of angiogenesis in skeletal muscle

Background

Mathematical modeling of angiogenesis has been gaining momentum as a means to shed new light on the biological complexity underlying blood vessel growth. A variety of computational models have been developed, each focusing on different aspects of the angiogenesis process and occurring at different biological scales, ranging from the molecular to the tissue levels. Integration of models at different scales is a challenging and currently unsolved problem.

Results

We present an object-oriented module-based computational integration strategy to build a multiscale model of angiogenesis that links currently available models. As an example case, we use this approach to integrate modules representing microvascular blood flow, oxygen transport, vascular endothelial growth factor transport and endothelial cell behavior (sensing, migration and proliferation). Modeling methodologies in these modules include algebraic equations, partial differential equations and agent-based models with complex logical rules. We apply this integrated model to simulate exercise-induced angiogenesis in skeletal muscle. The simulation results compare capillary growth patterns between different exercise conditions for a single bout of exercise. Results demonstrate how the computational infrastructure can effectively integrate multiple modules by coordinating their connectivity and data exchange. Model parameterization offers simulation flexibility and a platform for performing sensitivity analysis.

Conclusions

This systems biology strategy can be applied to larger scale integration of computational models of angiogenesis in skeletal muscle, or other complex processes in other tissues under physiological and pathological conditions.

A mathematical model for chronic wounds

Chronic wounds are often associated with ischemic conditions whereby the blood vascular system is damaged. A mathematical model which accounts for these conditions is developed and computational results are described in the two-dimensional radially symmetric case. Preliminary results for the three-dimensional axially symmetric case are also included.

A mathematical model for foreign body reactions in 2D

The foreign body reactions are commonly referred to the network of immune and inflammatory reactions of human or animals to foreign objects placed in tissues. They are basic biological processes, and are also highly relevant to bioengineering applications in implants, as fibrotic tissue formations surrounding medical implants have been found to substantially reduce the effectiveness of devices. Despite of intensive research on determining the mechanisms governing such complex responses, few mechanistic mathematical models have been developed to study such foreign body reactions. This study focuses on a kinetics-based predictive tool in order to analyze outcomes of multiple interactive complex reactions of various cells/proteins and biochemical processes and to understand transient behavior during the entire period (up to several months). A computational model in two spatial dimensions is constructed to investigate the time dynamics as well as spatial variation of foreign body reaction kinetics. The simulation results have been consistent with experimental data and the model can facilitate quantitative insights for study of foreign body reaction process in general.

A continuum mathematical model of the developing murine retinal vasculature

Angiogenesis, the process of new vessel growth from pre-existing vasculature, is crucial in many biological situations such as wound healing and embryogenesis. Angiogenesis is also a key regulator of pathogenesis in many clinically important disease processes, for instance, solid tumour progression and ocular diseases. Over the past 10-20 years, tumour-induced angiogenesis has received a lot of attention in the mathematical modelling community and there have also been some attempts to model angiogenesis during wound healing. However, there has been little modelling work of vascular growth during normal development. In this paper, we describe an in silico representation of the developing retinal vasculature in the mouse, using continuum mathematical models consisting of systems of partial differential equations. The equations describe the migratory response of cells to growth factor gradients, the evolution of the capillary blood vessel density, and of the growth factor concentration. Our approach is closely coupled to an associated experimental programme to parameterise our model effectively and the simulations provide an excellent correlation with in vivo experimental data. Future work and development of this model will enable us to elucidate the impact of molecular cues upon vasculature development and the implications for eye diseases such as diabetic retinopathy and neonatal retinopathy of prematurity.

2D luminescence imaging of pH in vivo

Luminescence imaging of biological parameters is an emerging field in biomedical sciences. Tools to study 2D pH distribution are needed to gain new insights into complex disease processes, such as wound healing and tumor metabolism. In recent years, luminescence-based methods for pH measurement have been developed. However, for in vivo applications, especially for studies on humans, biocompatibility and reliability under varying conditions have to be ensured. Here, we present a referenced luminescent sensor for 2D high-resolution imaging of pH in vivo. The ratiometric sensing scheme is based on time-domain luminescence imaging of FITC and ruthenium(II)tris-(4,7-diphenyl-1,10-phenanthroline). To create a biocompatible 2D sensor, these dyes were bound to or incorporated into microparticles (aminocellulose and polyacrylonitrile), and particles were immobilized in polyurethane hydrogel on transparent foils. We show sensor precision and validity by conducting in vitro and in vivo experiments, and we show the versatility in imaging pH during physiological and chronic cutaneous wound healing in humans. Implementation of this technique may open vistas in wound healing, tumor biology, and other biomedical fields.

A Predictive Tool for Foreign Body Fibrotic Reactions Using 2-Dimensional Computational Model

It is well established that implanted medical devices often trigger immunological and inflammatory reactions. Such foreign body-mediated tissue responses may result in fibrotic tissue formation surrounding the implants. Despite of intensive research in the area of wound healing, few methods are currently available to systematically predict the quantitative behavior of the complex system of multiple cells, proteins and enzymes during foreign body-associated fibrotic reactions. This study introduces a kinetics-based predictive tool to analyze outcomes of reactions of various cells/proteins and biochemical processes and to understand transient behavior during the entire implant healing period up to several months in time. A computational model in two spatial dimensions is constructed to investigate the time dynamics as well as spatial variation of fibrotic reaction kinetics. Our results have shown that this model can be used to predict many features in a systematic way and also complement the traditional immunological methodology by experiments or empirical data predictions.

Mathematical modeling in wound healing, bone regeneration and tissue engineering

The processes of wound healing and bone regeneration and problems in tissue engineering have been an active area for mathematical modeling in the last decade. Here we review a selection of recent models which aim at deriving strategies for improved healing. In wound healing, the models have particularly focused on the inflammatory response in order to improve the healing of chronic wound. For bone regeneration, the mathematical models have been applied to design optimal and new treatment strategies for normal and specific cases of impaired fracture healing. For the field of tissue engineering, we focus on mathematical models that analyze the interplay between cells and their biochemical cues within the scaffold to ensure optimal nutrient transport and maximal tissue production. Finally, we briefly comment on numerical issues arising from simulations of these mathematical models.