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

A mathematical model for wound healing in the reef-building coral Pocillopora damicornis

Coral reefs, among the most diverse ecosystems on Earth, currently face major threats from pollution, unsustainable fishing practices , and perturbations in environmental parameters brought on by climate change. Corals also sustain regular wounding from other sea life and human activity. Recent reef restoration practices have even involved intentional wounding by systematically breaking coral fragments and relocating them to revitalize damaged reefs, a practice known as microfragmentation. Despite its importance, very little research has explored the inner mechanisms of wound healing in corals. Some reef-building corals have been observed to initiate an immunological response to wounding similar to that observed in mammalian species. Utilizing prior models of wound healing in mammalian species as the mathematical basis, we formulated a mechanistic model of wound healing, including observations of the immune response and tissue repair in scleractinian corals for the species Pocillopora damicornis. The model consists of four differential equations which track changes in remaining wound debris, number of cells involved in inflammation, number of cells involved in proliferation, and amount of wound closure through re-epithelialization. The model is fit to experimental wound size data from linear and circular shaped wounds on a live coral fragment. Mathematical methods, including numerical simulations and local sensitivity analysis, were used to analyze the resulting model. The parameter space was also explored to investigate drivers of other possible wound outcomes. This model serves as a first step in generating mathematical models for wound healing in corals that will not only aid in the understanding of wound healing as a whole, but also help optimize reef restoration practices and predict recovery behavior after major wounding events.

Oxygen Therapy to Enhance Wound Healing After Revascularization

Oxygen is one of the important factors for wound healing and infection control. The revascularization procedure is amended to correct the tissue hypoxia problem by increasing the blood flow to obtain an adequate amount of oxygen. Hypoxic wounds are still the issue in the cases of unsuccessful or incomplete revascularization. The issue needs to be clarified and confirmed by proper methods for management to achieve wound healing and prevent limb loss. Oxygen therapy may benefit in the case of remaining hypoxia or wound infection in postrevascularization.

Mathematical and numerical analyses of cellular, molecular and angiogenic parameters of a rat skin wound healing model

The inflammatory phase is an important event in the skin wound healing process. The deposition of granulation tissue in the wound bed and the rebuilding of the vascular network occur as inflammation diminishes. An angiogenic component in the formation of granulation tissue is the secretion of vascular endothelial growth factor, which assists in the chemotaxis, proliferation, and replication of fibroblasts. In this paper, we develop a mathematical model of skin wound healing angiogenic factors based on inflammatory cells (macrophages and neutrophils) and mediators (interleukin 6 and interleukin 10). We highlight the importance of this process in vascular endothelial growth factor release and in the formation of new capillary tips. We used a mathematical model of partial differential equations based on the reaction-diffusion-advection equations. In order to calibrate the parameters, we considered an in vivo model composed by four treatments: hydroalcoholic extract and oil-resin of Copaifera langsdorffii at 10% concentration, collagenase, and Lanette cream. Using the laboratory data for the wound edge, our mathematical model estimated the values of vascular endothelial growth factor concentration, and tips density in the center of the wound with a maximum error of 2.9%, and predicted healing time required for each treatment. The region of viability for the parameters, in the proposed model, was found through numerical simulations from the Interleukin 6 and 10 dysregulation and we obtained that, among the parameters analyzed, the greatest influencer in the dynamics of the system is the one, which represents the production of Interleukin 10 during phagocytosis.

Self-organized and directed branching results in optimal coverage in developing dermal lymphatic networks

Branching morphogenesis is a ubiquitous process that gives rise to high exchange surfaces in the vasculature and epithelial organs. Lymphatic capillaries form branched networks, which play a key role in the circulation of tissue fluid and immune cells. Although mouse models and correlative patient data indicate that the lymphatic capillary density directly correlates with functional output, i.e., tissue fluid drainage and trafficking efficiency of dendritic cells, the mechanisms ensuring efficient tissue coverage remain poorly understood. Here, we use the mouse ear pinna lymphatic vessel network as a model system and combine lineage-tracing, genetic perturbations, whole-organ reconstructions and theoretical modeling to show that the dermal lymphatic capillaries tile space in an optimal, space-filling manner. This coverage is achieved by two complementary mechanisms: initial tissue invasion provides a non-optimal global scaffold via self-organized branching morphogenesis, while VEGF-C dependent side-branching from existing capillaries rapidly optimizes local coverage by directionally targeting low-density regions. With these two ingredients, we show that a minimal biophysical model can reproduce quantitatively whole-network reconstructions, across development and perturbations. Our results show that lymphatic capillary networks can exploit local self-organizing mechanisms to achieve tissue-scale optimization.

Mathematical investigation of normal and abnormal wound healing dynamics: local and non-local models

The movement of cells during (normal and abnormal) wound healing is the result of biomechanical interactions that combine cell responses with growth factors as well as cell-cell and cell-matrix interactions (adhesion and remodelling). It is known that cells can communicate and interact locally and non-locally with other cells inside the tissues through mechanical forces that act locally and at a distance, as well as through long non-conventional cell protrusions. In this study, we consider a non-local partial differential equation model for the interactions between fibroblasts, macrophages and the extracellular matrix (ECM) via a growth factor (TGF-$ \beta $) in the context of wound healing. For the non-local interactions, we consider two types of kernels (i.e., a Gaussian kernel and a cone-shaped kernel), two types of cell-ECM adhesion functions (i.e., adhesion only to higher-density ECM vs. adhesion to higher-/lower-density ECM) and two types of cell proliferation terms (i.e., with and without decay due to overcrowding). We investigate numerically the dynamics of this non-local model, as well as the dynamics of the localised versions of this model (i.e., those obtained when the cell perception radius decreases to 0). The results suggest the following: (ⅰ) local models explain normal wound healing and non-local models could also explain abnormal wound healing (although the results are parameter-dependent); (ⅱ) the models can explain two types of wound healing, i.e., by primary intention, when the wound margins come together from the side, and by secondary intention when the wound heals from the bottom up.

Modelling the Effect of Matrix Metalloproteinases in Dermal Wound Healing

With over 2 million people in the UK suffering from chronic wounds, understanding the biochemistry and pharmacology that underpins these wounds and wound healing is of high importance. Chronic wounds are characterised by high levels of matrix metalloproteinases (MMPs), which are necessary for the modification of healthy tissue in the healing process. Overexposure of MMPs, however, adversely affects healing of the wound by causing further destruction of the surrounding extracellular matrix. In this work, we propose a mathematical model that focuses on the interaction of MMPs with dermal cells using a system of partial differential equations. Using biologically realistic parameter values, this model gives rise to travelling waves corresponding to a front of healthy cells invading a wound. From the arising travelling wave analysis, we observe that deregulated apoptosis results in the emergence of chronic wounds, characterised by elevated MMP concentrations. We also observe hysteresis effects when both the apoptotic rate and MMP production rate are varied, providing further insight into the management (and potential reversal) of chronic wounds.

Angiogenic Microvascular Wall Shear Stress Patterns Revealed Through Three-dimensional Red Blood Cell Resolved Modeling

The wall shear stress (WSS) exerted by blood flowing through microvascular capillaries is an established driver of new blood vessel growth, or angiogenesis. Such adaptations are central to many physiological processes in both health and disease, yet three-dimensional (3D) WSS characteristics in real angiogenic microvascular networks are largely unknown. This marks a major knowledge gap because angiogenesis, naturally, is a 3D process. To advance current understanding, we model 3D red blood cells (RBCs) flowing through rat angiogenic microvascular networks using state-of-the-art simulation. The high-resolution fluid dynamics reveal 3D WSS patterns occurring at sub-endothelial cell (EC) scales that derive from distinct angiogenic morphologies, including microvascular loops and vessel tortuosity. We identify the existence of WSS hot and cold spots caused by angiogenic surface shapes and RBCs, and notably enhancement of low WSS regions by RBCs. Spatiotemporal characteristics further reveal how fluctuations follow timescales of RBC "footprints." Altogether, this work provides a new conceptual framework for understanding how shear stress might regulate EC dynamics in vivo.

Computer simulation of the wound process (review of literature)

Relevance. Computer simulation is a mathematical modeling process performed on a computer that is designed to predict the behavior or results of a real or physical system. Computer simulation has a number of advantages over classical models of animal experiments: the cheapness of the method (the need to acquire and maintain animals disappears by itself), the speed of obtaining results, the absence of bioethical problems, the ability to change the conditions of the experiment, etc.

he purpose of this study is to review the methods of computer simulation of the wound process, to identify the shortcomings of the models and propose ways to solve them, as well as to select the best existing model for describing wound regeneration.

Material and methods. In the course of this work, an analysis was made of foreign and domestic literature on the problem of computer modeling of the wound process.

Results. After analyzing the relevant literature on this topic, the problem is seen precisely in the insufficiently studied process of wound regeneration, since many different cells, cytokines, growth factors, enzymes, fibrillar proteins, etc. take part in it. The models that currently exist describe wound regeneration only in an extremely generalized way, which does not allow us to apply them in clinical situations. Analyzing literature sources, we came to the conclusion that both numerical approaches, both cellular-biochemical (the first type of models) and phenomenological (the second type) are applicable in the case of wound modeling and can be used very successfully. The problem is that on the basis of one approach it is impossible to display a complete picture of wound healing, in this way it is possible to predict only individual regeneration parameters necessary for certain purposes due to the complexity and versatility of this typical pathophysiological process.

Conclusion. Computer modeling of wounds is still a controversial and complex topic. Existing models are not intended to describe all the processes occurring in a healing wound. It is much more productive to describe the various phenomena during healing separately. This is due to the fact that many elements are involved in the regeneration of the skin, which are almost impossible to take into account in full. The available models are of exclusively scientific value, consisting in attempts to understand all complex processes and interactions. Practical application is difficult, since existing models require specific input data that require highly specialized equipment. If we abstract from all this, then the best existing model of the first type is the model of the authors Yangyang Wang, Christian F. Guerrero-Juarez, Yuchi Qiu and co-authors, in addition to it, any of the described phenomenological models will do. 

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.

Advances in non-invasive biosensing measures to monitor wound healing progression

Impaired wound healing is a significant financial and medical burden. The synthesis and deposition of extracellular matrix (ECM) in a new wound is a dynamic process that is constantly changing and adapting to the biochemical and biomechanical signaling from the extracellular microenvironments of the wound. This drives either a regenerative or fibrotic and scar-forming healing outcome. Disruptions in ECM deposition, structure, and composition lead to impaired healing in diseased states, such as in diabetes. Valid measures of the principal determinants of successful ECM deposition and wound healing include lack of bacterial contamination, good tissue perfusion, and reduced mechanical injury and strain. These measures are used by wound-care providers to intervene upon the healing wound to steer healing toward a more functional phenotype with improved structural integrity and healing outcomes and to prevent adverse wound developments. In this review, we discuss bioengineering advances in 1) non-invasive detection of biologic and physiologic factors of the healing wound, 2) visualizing and modeling the ECM, and 3) computational tools that efficiently evaluate the complex data acquired from the wounds based on basic science, preclinical, translational and clinical studies, that would allow us to prognosticate healing outcomes and intervene effectively. We focus on bioelectronics and biologic interfaces of the sensors and actuators for real time biosensing and actuation of the tissues. We also discuss high-resolution, advanced imaging techniques, which go beyond traditional confocal and fluorescence microscopy to visualize microscopic details of the composition of the wound matrix, linearity of collagen, and live tracking of components within the wound microenvironment. Computational modeling of the wound matrix, including partial differential equation datasets as well as machine learning models that can serve as powerful tools for physicians to guide their decision-making process are discussed.

On modeling the multiscale mechanobiology of soft tissues: Challenges and progress

Tissues grow and remodel in response to mechanical cues, extracellular and intracellular signals experienced through various biological events, from the developing embryo to disease and aging. The macroscale response of soft tissues is typically nonlinear, viscoelastic anisotropic, and often emerges from the hierarchical structure of tissues, primarily their biopolymer fiber networks at the microscale. The adaptation to mechanical cues is likewise a multiscale phenomenon. Cell mechanobiology, the ability of cells to transform mechanical inputs into chemical signaling inside the cell, and subsequent regulation of cellular behavior through intra- and inter-cellular signaling networks, is the key coupling at the microscale between the mechanical cues and the mechanical adaptation seen macroscopically. To fully understand mechanics of tissues in growth and remodeling as observed at the tissue level, multiscale models of tissue mechanobiology are essential. In this review, we summarize the state-of-the art modeling tools of soft tissues at both scales, the tissue level response, and the cell scale mechanobiology models. To help the interested reader become more familiar with these modeling frameworks, we also show representative examples. Our aim here is to bring together scientists from different disciplines and enable the future leap in multiscale modeling of tissue mechanobiology.

What good is maths in studies of wound healing?

Wound healing is an aspect of normal physiology that we all take for granted until it goes wrong, such as, for example, the scarring that results from a severe burn, or those patients who suffer from debilitating chronic wounds that fail to heal. Ever since wound repair research began as a discipline, clinicians and basic scientists have collaborated to try and understand the cell and molecular mechanisms that underpin healthy repair in the hope that this will reveal clues for the therapeutic treatment of pathological healing. In recent decades mathematicians and physicists have begun to join in with this important challenge. Here we describe examples of how mathematical modeling married to biological experimentation has provided insights that biology alone could not fathom. To date, these studies have largely focused on wound re-epithelialization and inflammation, but we also discuss other components of wound healing that might be ripe for similar interdisciplinary approaches.

Simulating the behaviour of glioblastoma multiforme based on patient MRI during treatments

Glioblastoma multiforme is a brain cancer that still shows poor prognosis for patients despite the active research for new treatments. In this work, the goal is to model and simulate the evolution of tumour associated angiogenesis and the therapeutic response to glioblastoma multiforme. Multiple phenomena are modelled in order to fit different biological pathways, such as the cellular cycle, apoptosis, hypoxia or angiogenesis. This leads to a nonlinear system with 4 equations and 4 unknowns: the density of tumour cells, the [Formula: see text] concentration, the density of endothelial cells and the vascular endothelial growth factor concentration. This system is solved numerically on a mesh fitting the geometry of the brain and the tumour of a patient based on a 2D slice of MRI. We show that our numerical scheme is positive, and we give the energy estimates on the discrete solution to ensure its existence. The numerical scheme uses nonlinear control volume finite elements in space and is implicit in time. Numerical simulations have been done using the different standard treatments: surgery, chemotherapy and radiotherapy, in order to conform to the behaviour of a tumour in response to treatments according to empirical clinical knowledge. We find that our theoretical model exhibits realistic behaviours.

Intratumoral VEGF nanotrapper reduces gliobastoma vascularization and tumor cell mass

Glioblastoma multiforme (GBM) is the most aggressive and invasive malignant brain cancer. GBM is characterized by a dramatic metabolic imbalance leading to increased secretion of the pro-angiogenic factor VEGF and subsequent abnormal tumor vascularization. In 2009, FDA approved the intravenous administration of bevacizumab, an anti-VEGF monoclonal antibody, as a therapeutic agent for patients with GBM. However, the number of systemic side effects and reduced accessibility of bevacizumab to the central nervous system and consequently to the GBM tumor mass limited its effectiveness in improving patient survival. In this study, we combined experimental and computational modelling to quantitatively characterize the dynamics of VEGF secretion and turnover in GBM and in normal brain cells and simultaneous monitoring of vessel growth. We showed that sequestration of VEGF inside GBM cells, can be used as a novel target for improved bevacizumab-based therapy. We have engineered the VEGF nanotrapper, a cargo system that allows cellular uptake of bevacizumab and inhibits VEGF secretion required for angiogenesis activation and development. Here, we show the therapeutic efficacy of this nanocargo in reducing vascularization and tumor cell mass of GBM in vitro and in vivo cancer models.

Intrinsic Antibacterial and Conductive Hydrogels Based on the Distinct Bactericidal Effect of Polyaniline for Infected Chronic Wound Healing

Most chronic wounds suffer from infections, and their treatment is challenging. The usage of antibiotics may lead to bacterial resistance and adverse side effects. Positively charged substances have shown promise, but their applications are usually limited by certain cytotoxicity or complex synthesis. Doped polyaniline that carries a high density of positive charges would be a promising candidate due to its good biocompatibility and easy availability, but its interaction with bacteria has not been elucidated. Herein, the distinct bactericidal effect of polyaniline against Gram-positive bacteria has been verified. The antibacterial activity may result from the specific interaction with lipoteichoic acid to destroy the Gram-positive bacterial cell wall. Polyaniline and a macromolecular dopant (sulfonated hyaluronic acid) are used to construct a flexible hydrogel with skin-mimic electrical conductivity. The in vivo results demonstrate that electrical stimulation (ES) through this hydrogel is superior to ES via separated electrodes (the ES strategy used clinically) for promoting infected chronic wound healing.

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.

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.

Use of Magnetic Resonance Imaging (MRI) in the Management of Diagnostic Uncertainty in Low-Resource Settings: A Case Report of Cesarean Ectopic Pregnancy in a Tertiary Hospital in Ghana

Patient: Female, 35-year-old

Final Diagnosis: Cesarean section ectopic pregnancy

Symptoms: Amenorrhea

Medication:—

Clinical Procedure: Exploratory laparotomy • MRI • ultrasonography

Specialty: Obstetrics and Gynecology

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.

Mathematical Model of Chronic Dermal Wounds in Diabetes and Obesity

Chronic dermal-wound patients frequently suffer from diabetes type 2 and obesity; without treatment or early intervention, these patients are at risk of amputation. In this paper, we identified four factors that impair wound healing in these populations: excessive production of glycation, excessive production of leukotrient, decreased production of stromal derived factor (SDF-1), and insulin resistance. We developed a mathematical model of wound healing that includes these factors. The model consists of a system of partial differential equations, and it demonstrates how these four factors impair the closure of the wound, by reducing the oxygen flow into the wound area and by blocking the transition from pro-inflammatory macrophages to anti-inflammatory macrophages. The model is used to assess treatment by insulin injection and by oxygen infusion.

Hysteroscopic treatment of cesarean scar defect

OBJECTIVE

To investigate the effect of hysteroscopic surgery on the outcomes of obstetrics and gynecology among patients with cesarean section diverticulum.

METHODS

Ninety-nine infertile patients with cesarean section diverticulum received hysteroscopic treatment and were retrospectively analyzed. Patients were followed for 1 year.

RESULTS

The study included ninety-nine symptomatic patients with cesarean section diverticulum. After surgery, the menstrual periods of patients were improved from 11.15 ± 4.44 to 7.69 ± 2.85 days. Forty-seven (47/99) women became pregnant after surgery. The number of patients who became pregnant with an anteflexion uterus after hysteroscopic surgery is 32 (32/57), and the number of women who became pregnant with a retroflexion uterus is 15 (15/42).

CONCLUSION

Hysteroscopic surgery could improve the PCSD-associated prolonged menstrual bleeding, and satisfactory obstetrical outcomes could be achieved by the surgery treatment in women with cesarean defect.

Three-dimensional numerical simulation of soft-tissue wound healing using constrained-mixture anisotropic hyperelasticity and gradient-enhanced damage mechanics

Healing of soft biological tissues is the process of self-recovery or self-repair after injury or damage to the extracellular matrix (ECM). In this work, we assume that healing is a stress-driven process, which works at recovering a homeostatic stress metric in the tissue by replacing the damaged ECM with a new undamaged one. For that, a gradient-enhanced continuum healing model is developed for three-dimensional anisotropic tissues using the modified anisotropic Holzapfel-Gasser-Ogden constitutive model. An adaptive stress-driven approach is proposed for the deposition of new collagen fibres during healing with orientations assigned depending on the principal stress direction. The intrinsic length scales of soft tissues are considered through the gradient-enhanced term, and growth and remodelling are simulated by a constrained-mixture model with temporal homogenization. The proposed model is implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three numerical examples have been achieved to illustrate the performance of the proposed model in simulating the healing process with various damage situations, converging towards stress homeostasis. The orientations of newly deposited collagen fibres and the sensitivity to intrinsic length scales are studied through these examples, showing that both have a significant impact on temporal evolutions of the stress distribution and on the size of the damage region. Applications of the approach to carry out in silico experiments of wound healing are promising and show good agreement with existing experiment results.

Identifying possible risk factors for cesarean scar pregnancy based on a retrospective study of 291 cases

Aim

Cesarean scar pregnancy (CSP) is a rare but life‐threatening type of ectopic pregnancy. This study's aim is to investigate the clinical characteristics and possible risk factors for cesarean scar pregnancy.

Methods

A clinically randomized, unpaired and retrospective case–control study was implemented. A study group of 291 CSP patients and a control group of 317 full‐term pregnant women with a history of cesarean section (CS) were recruited in our hospital from May 2013 to October 2018. Their demographic characteristics and medical and obstetric history were collected.

Results

Only symptoms suggestive of an impending abortion, such as vaginal bleeding with or without abdominal pain, were identified as the clinical characteristics of CSP. Maternal age older than 35 years, gravidity higher than 3 (especially gravidity higher than 5), more than two induced abortions (especially more than five abortions), an interval of less than 5 years (especially less than 2 years) between the current pregnancy and the last CS, history of CS performed in a rural hospital, history of induced abortions after CS and retroposition of the uterus were possible independent risk factors for CSP.

Conclusion

CSP is a result of a combination of multiple factors associated with CS. There are no unique early clinical features of CSP. As a unique type of ectopic pregnancy, early diagnosis, early termination and early clearance should be the treatment principles. Further research is needed to evaluate the relationship between the cesarean scar defect and CSP in the future.

Chemoreactive Nanotherapeutics by Metal Peroxide Based Nanomedicine

The advances of nanobiotechnology and nanomedicine enable the triggering of in situ chemical reactions in disease microenvironment for achieving disease-specific nanotherapeutics with both intriguing therapeutic efficacy and mitigated side effects. Metal peroxide based nanoparticles, as one of the important but generally ignored categories of metal-involved nanosystems, can function as the solid precursors to produce oxygen (O2) and hydrogen peroxide (H2O2) through simple chemical reactions, both of which are the important chemical species for enhancing the therapeutic outcome of versatile modalities, accompanied with the unique bioactivity of metal ion based components. This progress report summarizes and discusses the most representative paradigms of metal peroxides in chemoreactive nanomedicine, including copper peroxide (CuO2), calcium peroxide (CaO2), magnesium peroxide (MgO2), zinc peroxide (ZnO2), barium peroxide (BaO2), and titanium peroxide (TiOx) nanosystems. Their reactions and corresponding products have been broadly explored in versatile disease treatments, including catalytic nanotherapeutics, photodynamic therapy, radiation therapy, antibacterial infection, tissue regeneration, and some synergistically therapeutic applications. This progress report particularly focuses on the underlying reaction mechanisms on enhancing the therapeutic efficacy of these modalities, accompanied with the discussion on their biological effects and biosafety. The existing gap between fundamental research and clinical translation of these metal peroxide based nanotherapeutic technologies is finally discussed in depth.

Computational modelling suggests complex interactions between interstitial flow and tumour angiogenesis

Angiogenesis, the growth of capillaries from pre-existing ones, plays a key role in cancer progression. Tumours release tumour angiogenic factors (TAFs) into the extracellular matrix (ECM) that trigger angiogenesis once they reach the vasculature. The neovasculature provides nutrients and oxygen to the tumour. In the ECM, the interstitial fluid moves driven by pressure differences and may affect the distribution of tumour TAFs, and, in turn, tumour vascularization. In this work, we propose a hybrid mathematical model to investigate the influence of fluid flow in tumour angiogenesis. Our model shows the impact of interstitial flow in a time-evolving capillary network using a continuous approach. The flow model is coupled to a model of angiogenesis that includes tip endothelial cells, filopodia, capillaries and TAFs. The TAF transport equation considers not only diffusive mechanisms but also the convective transport produced by interstitial flow. Our simulations predict a significant alteration of the new vascular networks, which tend to grow more prominently against the flow. The model suggests that interstitial flow may produce increased tumour malignancies and hindered treatments.

Identifying important parameters in the inflammatory process with a mathematical model of immune cell influx and macrophage polarization

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage polarization, with the simplifying assumption that macrophages can be classified as M1 or M2. With this model, we were able to reproduce the expected timing of sequential influx of immune cells and mediators in a general inflammatory setting. We then fit this model to in vivo experimental data obtained from a mouse peritonitis model of inflammation, which is widely used to evaluate endogenous processes in response to an inflammatory stimulus. Model robustness is explored with local structural and practical identifiability of the proposed model a posteriori. Additionally, we perform sensitivity analysis that identifies the population of apoptotic neutrophils as a key driver of the inflammatory process. Finally, we simulate a selection of proposed therapies including points of intervention in the case of delayed neutrophil apoptosis, which our model predicts will result in a sustained inflammatory response. Our model can therefore provide hypothesis testing for therapeutic interventions that target macrophage phenotype and predict outcomes to be validated by subsequent experimentation.

Using experimental data and mathematical analysis, we develop a model for the inflammatory response that includes macrophage polarization between M1 and M2 phenotypes. Dysfunction of this phenotypic switch can disrupt the timely influx and egress of immune cells during the healing process and lead to chronic wounds or disease. The modulation of macrophage population has been suggested as a strategy to dampen inflammation in diseases that feature chronic inflammation, such as diabetes and atherosclerosis. It is therefore important that we learn more about which components of the system drive the population level switch in phenotype. Our model is able to reproduce the expected timing of sequential influx of neutrophils and macrophages in response to an inflammatory stimulus. Model parameters were estimated with weighted least squares fitting to in vivo experimental data from a mouse model of peritonitis while considering identifiability of parameter sets. We perform sensitivity analysis that identifies primary drivers of the system, and predict the effects of variations in these key parameters on immune cell populations.

Influences of sodium and glycosaminoglycans on skin oedema and the potential for ulceration: a finite-element approach

Venous ulcers are chronic transcutaneous wounds common in the lower legs. They are resistant to healing and have a 78% chance of recurrence within 2 years. It is commonly accepted that venous ulcers are caused by the insufficiency of the calf muscle pump, leading to blood pooling in the lower legs, resulting in inflammation, skin oedema, tissue necrosis and eventually skin ulceration. However, the detailed physiological events by which inflammation contributes to wound formation are poorly understood. We therefore sought to develop a model that simulated the inflammation, using it to determine the internal stresses and pressure on the skin that contribute to venous ulcer formation. A three-layer finite-element skin model (epidermis, dermis and hypodermis) was developed to explore the roles in wound formation of two inflammation identifiers: glycosaminoglycans (GAG) and sodium. A series of parametric studies showed that increased GAG and sodium content led to oedema and increased tissue stresses of 1.5 MPa, which was within the reported range of skin tissue ultimate tensile stress (0.1-40 MPa). These results suggested that both the oedema and increased fluid pressure could reach a threshold for tissue damage and eventual ulcer formation. The models presented here provide insights to the pathological events associated with venous insufficiency, including inflammation, oedema and skin ulceration.

Predictive Approach Identifies Molecular Targets and Interventions to Restore Angiogenesis in Wounds With Delayed Healing

Impaired angiogenesis is a hallmark of wounds with delayed healing, and currently used therapies to restore angiogenesis have limited efficacy. Here, we employ a computational simulation-based approach to identify influential molecular and cellular processes, as well as protein targets, whose modulation may stimulate angiogenesis in wounds. We developed a mathematical model that captures the time courses for platelets, 9 cell types, 29 proteins, and oxygen, which are involved in inflammation, proliferation, and angiogenesis during wound healing. We validated our model using previously published experimental data. By performing global sensitivity analysis on thousands of simulated wound-healing scenarios, we identified six processes (among the 133 modeled in total) whose modulation may improve angiogenesis in wounds. By simulating knockouts of 25 modeled proteins and by simulating different wound-oxygenation levels, we identified four proteins [namely, transforming growth factor (TGF)-β, vascular endothelial growth factor (VEGF), fibroblast growth factor-2 (FGF-2), and angiopoietin-2 (ANG-2)], as well as oxygen, as therapeutic targets for stimulating angiogenesis in wounds. Our modeling results indicated that simultaneous inhibition of TGF-β and supplementation of either FGF-2 or ANG-2 could be more effective in stimulating wound angiogenesis than the modulation of either protein alone. Our findings suggest experimentally testable intervention strategies to restore angiogenesis in wounds with delayed healing.

Gradient-enhanced continuum models of healing in damaged soft tissues

Healing of soft biological tissue is the process of self-recovering or self-repairing the injured or damaged extracellular matrix (ECM). Healing is assumed to be stress-driven, with the objective of returning to a homeostatic stress metrics in the tissue after replacing the damaged ECM with new undamaged one. However, based on the existence of intrinsic length scales in soft tissues, it is thought that computational models of healing should be non-local. In the present study, we introduce for the first time two gradient-enhanced constitutive healing models for soft tissues including non-local variables. The first model combines a continuum damage model with a temporally homogenized growth model, where the growth direction is determined according to local principal stress directions. The second one is based on a gradient-enhanced healing model with continuously recoverable damage variable. Both models are implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three two-dimensional situations simulating the healing process of soft tissues are modeled numerically with both models, and their application for simulation of balloon angioplasty is provided by illustrating the change of damage field and geometry in the media layer throughout the healing process.

A mathematical model of the use of supplemental oxygen to combat surgical site infection

Infections are a common complication of any surgery, often requiring a recovery period in hospital. Supplemental oxygen therapy administered during and immediately after surgery is thought to enhance the immune response to bacterial contamination. However, aerobic bacteria thrive in oxygen-rich environments, and so it is unclear whether oxygen has a net positive effect on recovery. Here, we develop a mathematical model of post-surgery infection to investigate the efficacy of supplemental oxygen therapy on surgical-site infections. A 4-species, coupled, set of non-linear partial differential equations that describes the space-time dependence of neutrophils, bacteria, chemoattractant and oxygen is developed and analysed to determine its underlying properties. Through numerical solutions, we quantify the efficacy of different supplemental oxygen regimes on the treatment of surgical site infections in wounds of different initial bacterial load. A sensitivity analysis is performed to investigate the robustness of the predictions to changes in the model parameters. The numerical results are in good agreement with analyses of the associated well-mixed model. Our model findings provide insight into how the nature of the contaminant and its initial density influence bacterial infection dynamics in the surgical wound.

Modelling optimal delivery of bFGF to chronic wounds using ODEs

In this paper, we present an ordinary differential equation model depicting the interactions of basic fibroblast growth factor (bFGF) and its binding agents in a chronic wound. The delivery of bFGF was treated as a control variable and is coupled to an objective functional. By optimising the objective functional with respect to the control, predictions for optimal delivery rates of bFGF are proposed. The optimal control is then validated by comparing the cost of the objective functional for the optimal delivery rate and several alternative delivery rates. This paper addresses two objectives of effective drug delivery to chronic wounds. The first is to provide insight for the priority of delivering bFGF: to minimise the quantity of bFGF, or to optimise the distribution of bound bFGF. For effective concentrations of bound bFGF, the optimisation of bound bFGF must be prioritised over the minimisation of bFGF delivered. The second objective is to comment on the effect of the proteolytic environment within the wound, with the concentration of bound bFGF starting to decrease late in the treatment period for highly proteolytic environments. This will lead to long term complications with wound closure after the treatment has been completed. Also, it was found that for highly proteolytic environments, the cost of delivering bFGF increased. The need for optimal drug delivery is made apparent by the burden of chronic wounds on the medical industry across the developed world.

Experimental and computational analyses reveal dynamics of tumor vessel cooption and optimal treatment strategies

Cooption of the host vasculature is a strategy that some cancers use to sustain tumor progression without-or before-angiogenesis or in response to antiangiogenic therapy. Facilitated by certain growth factors, cooption can mediate tumor infiltration and confer resistance to antiangiogenic drugs. Unfortunately, this mode of tumor progression is difficult to target because the underlying mechanisms are not fully understood. Here, we analyzed the dynamics of vessel cooption during tumor progression and in response to antiangiogenic treatment in gliomas and brain metastases. We followed tumor evolution during escape from antiangiogenic treatment as cancer cells coopted, and apparently mechanically compressed, host vessels. To gain deeper understanding, we developed a mathematical model, which incorporated compression of coopted vessels, resulting in hypoxia and formation of new vessels by angiogenesis. Even if antiangiogenic therapy can block such secondary angiogenesis, the tumor can sustain itself by coopting existing vessels. Hence, tumor progression can only be stopped by combination therapies that judiciously block both angiogenesis and cooption. Furthermore, the model suggests that sequential blockade is likely to be more beneficial than simultaneous blockade.

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.

The innate immune response to ischemic injury: a multiscale modeling perspective

Background

Cell death as a result of ischemic injury triggers powerful mechanisms regulated by germline-encoded Pattern Recognition Receptors (PRRs) with shared specificity that recognize invading pathogens and endogenous ligands released from dying cells, and as such are essential to human health. Alternatively, dysregulation of these mechanisms contributes to extreme inflammation, deleterious tissue damage and impaired healing in various diseases. The Toll-like receptors (TLRs) are a prototypical family of PRRs that may be powerful anti-inflammatory targets if agents can be designed that antagonize their harmful effects while preserving host defense functions. This requires an understanding of the complex interactions and consequences of targeting the TLR-mediated pathways as well as technologies to analyze and interpret these, which will then allow the simulation of perturbations targeting specific pathway components, predict potential outcomes and identify safe and effective therapeutic targets.

Results

We constructed a multiscale mathematical model that spans the tissue and intracellular scales, and captures the consequences of targeting various regulatory components of injury-induced TLR4 signal transduction on potential pro-inflammatory or pro-healing outcomes. We applied known interactions to simulate how inactivation of specific regulatory nodes affects dynamics in the context of injury and to predict phenotypes of potential therapeutic interventions. We propose rules to link model behavior to qualitative estimates of pro-inflammatory signal activation, macrophage infiltration, production of reactive oxygen species and resolution. We tested the validity of the model by assessing its ability to reproduce published data not used in its construction.

Conclusions

These studies will enable us to form a conceptual framework focusing on TLR4-mediated ischemic repair to assess potential molecular targets that can be utilized therapeutically to improve efficacy and safety in treating ischemic/inflammatory injury.

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

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

Reactive Oxygen Species and NOX Enzymes Are Emerging as Key Players in Cutaneous Wound Repair

Our understanding of the role of oxygen in cell physiology has evolved from its long-recognized importance as an essential factor in oxidative metabolism to its recognition as an important player in cell signaling. With regard to the latter, oxygen is needed for the generation of reactive oxygen species (ROS), which regulate a number of different cellular functions including differentiation, proliferation, apoptosis, migration, and contraction. Data specifically concerning the role of ROS-dependent signaling in cutaneous wound repair are very limited, especially regarding wound contraction. In this review we provide an overview of the current literature on the role of molecular and reactive oxygen in the physiology of wound repair as well as in the pathophysiology and therapy of chronic wounds, especially under ischemic and hyperglycemic conditions.

A homeostatic-driven turnover remodelling constitutive model for healing in soft tissues

Remodelling of soft biological tissue is characterized by interacting biochemical and biomechanical events, which change the tissue's microstructure, and, consequently, its macroscopic mechanical properties. Remodelling is a well-defined stage of the healing process, and aims at recovering or repairing the injured extracellular matrix. Like other physiological processes, remodelling is thought to be driven by homeostasis, i.e. it tends to re-establish the properties of the uninjured tissue. However, homeostasis may never be reached, such that remodelling may also appear as a continuous pathological transformation of diseased tissues during aneurysm expansion, for example. A simple constitutive model for soft biological tissues that regards remodelling as homeostatic-driven turnover is developed. Specifically, the recoverable effective tissue damage, whose rate is the sum of a mechanical damage rate and a healing rate, serves as a scalar internal thermodynamic variable. In order to integrate the biochemical and biomechanical aspects of remodelling, the healing rate is, on the one hand, driven by mechanical stimuli, but, on the other hand, subjected to simple metabolic constraints. The proposed model is formulated in accordance with continuum damage mechanics within an open-system thermodynamics framework. The numerical implementation in an in-house finite-element code is described, particularized for Ogden hyperelasticity. Numerical examples illustrate the basic constitutive characteristics of the model and demonstrate its potential in representing aspects of remodelling of soft tissues. Simulation results are verified for their plausibility, but also validated against reported experimental data.

Spatio-temporal Models of Lymphangiogenesis in Wound Healing

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

Standardized ultrasonographic approach for the assessment of risk factors of incomplete healing of the cesarean section scar in the uterus

OBJECTIVE

To identify factors related to the healing of a Cesarean uterine incision using the standardized ultrasonographic approach of scar assessment in the non-pregnant uterus.

STUDY DESIGN

Measurements of the uterine scar were taken from 409 women with a history of at least one low transverse cesarean section (CS) with a single layer uterine closure. Residual myometrial thickness (RMT), width (W) and depth (D) of the triangular hypoechoic scar niche, D/RMT ratio and clinical characteristics were analyzed. For statistical analysis, the Mann-Whitney U test, chi-square test, Spearman's rank correlation coefficient, ANOVA test, and logistic regression were used.

RESULTS

268 women presented with a scar defect. RMT values were significantly correlated with the number of CSs (R=-0.17) and uterus retroflection (R=-0.15). The presence of a scar defect was significantly associated with lower RMT values (R=-0.33), greater gestational age (R=0.10), and younger maternal age (R=-0.11). The mean RMT value was significantly smaller in women with CSs performed in the second stage of labor (0.62) when compared to women with CSs in the first stage of labor (0.97) or without cervical dilatation (0.91).

CONCLUSION

A standardized approach of CS scar assessment in the non-pregnant uterus helps to identify women at risk of long-term complications of CS.

Multiscale Mechano-Biological Finite Element Modelling of Oncoplastic Breast Surgery-Numerical Study towards Surgical Planning and Cosmetic Outcome Prediction

Surgical treatment for early-stage breast carcinoma primarily necessitates breast conserving therapy (BCT), where the tumour is removed while preserving the breast shape. To date, there have been very few attempts to develop accurate and efficient computational tools that could be used in the clinical environment for pre-operative planning and oncoplastic breast surgery assessment. Moreover, from the breast cancer research perspective, there has been very little effort to model complex mechano-biological processes involved in wound healing. We address this by providing an integrated numerical framework that can simulate the therapeutic effects of BCT over the extended period of treatment and recovery. A validated, three-dimensional, multiscale finite element procedure that simulates breast tissue deformations and physiological wound healing is presented. In the proposed methodology, a partitioned, continuum-based mathematical model for tissue recovery and angiogenesis, and breast tissue deformation is considered. The effectiveness and accuracy of the proposed numerical scheme is illustrated through patient-specific representative examples. Wound repair and contraction numerical analyses of real MRI-derived breast geometries are investigated, and the final predictions of the breast shape are validated against post-operative follow-up optical surface scans from four patients. Mean (standard deviation) breast surface distance errors in millimetres of 3.1 (±3.1), 3.2 (±2.4), 2.8 (±2.7) and 4.1 (±3.3) were obtained, demonstrating the ability of the surgical simulation tool to predict, pre-operatively, the outcome of BCT to clinically useful accuracy.

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.

Free boundary problems in biology

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.

The Force at the Tip--Modelling Tension and Proliferation in Sprouting Angiogenesis

Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art.

Sprouting angiogenesis—a process by which new blood vessels grow from existing ones—is an ubiquitous phenomenon in health and disease of higher organisms, playing a crucial role in organogenesis, wound healing, inflammation, as well as on the onset and progression of over 50 different diseases such as cancer, rheumatoid arthritis and diabetes. Mathematical models have the ability to suggest relevant hypotheses with respect to the mechanisms of cell movement and rearrangement within growing vessel sprouts. The inclusion of both biochemical and mechanical processes in a mathematical model of sprouting angiogenesis permits to describe sprout extension as a function of the forces exerted by the cells in the tissue. It also allows to question the regulation of biochemical processes by mechanical forces and vice-versa. In this work we present a compact model of sprouting angiogenesis that includes the mechanical characteristics of the vessel and the tissue. We use this model to suggest the mechanism for the regulation of proliferation within sprout formation. We conclude that the tip cell has the role of creating a tension in the cells that follow its lead. On those first cells of the stalk, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. The modelling strategy used, deemed phase-field, permits to describe the evolution of the shape of different domains in complex systems. It is focused on the movement of the interfaces between the domains, and not on an exhaustive description of the transport properties within each domain. For this reason, it requires a reduced number of parameters, and has been used extensively in modelling other biological phenomena such as tumor growth. The coupling of mechanical and biochemical processes in a compact mathematical model of angiogenesis will enable the study of lumen formation and aneurisms in the near future. Also, this framework will allow the study of the action of flow in vessel remodelling, since local forces can readily be coupled with cell movement to obtain the final vessel morphology.

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.