Modeling collagen remodeling

Extreme Behaviors in Fibrous Material Remodeling: Auxetic to Non-Auxetic Transition and Phase Segregation

Fibrous materials, prevalent in biological tissues and engineered composites, undergo remodeling in response to mechanical loads, leading to plastic changes in fiber orientation. A previously developed continuum model describes this remodeling process. Building on that framework, the present study examines the extreme behaviors of such materials. Analytical results for the homogeneous response under tensile loading reveal three distinct classes: in class (A), fibers asymptotically approach a specific angle; in class (B), fibers align perpendicularly to the load direction; and in class (C), fibers align either with the load direction or perpendicularly, depending on their initial orientation. Numerical simulations are employed to analyze the non-homogeneous material response in a standard tensile test, demonstrating how differences in behavior arise from the material class and the initial fiber orientation distribution. This investigation focuses on the extreme behaviors of material classes (A) and (C), emphasizing phase segregation and transitions between auxetic and non-auxetic behavior.

Three-dimensional anisotropic unified continuum model for simulating the healing of damaged soft biological tissues

The soft biological tissues have the ability to heal and self-repair after damage or injury. During the healing process, damaged tissues are replaced by newly produced undamaged tissue to restore homeostasis. Computational modeling serves as an effective tool for simulating the healing process and understanding the underlying mechanisms. In previous work, we developed the first unified continuum damage model for the healing of soft biological tissues. However, the initial theory lacked generalizability to more realistic scenarios and applicability to biomechanical problems due to the simplicity of the isotropic constitutive model and two-dimensional simulations. Therefore, we further improve our approach by developing a three-dimensional anisotropic unified healing model to address more realistic challenges. By using the Holzapfel-Gasser-Ogden model as the hyperelastic term, the influence of the collagen fibers is considered and the reorientation of fibers in healing is simulated. Three numerical examples related to hypertension, aneurysm, and restenosis of the atherosclerotic artery after balloon angioplasty are presented to demonstrate the effectiveness of the proposed model. By comparing numerical solutions and reference solutions, we demonstrate the ability of the proposed model in simulating long-term tissue healing process and analyze the impact of anisotropic terms.

Phosphorylation of collagen fibrils enhances intrafibrillar mineralization and dentin remineralization

The hierarchical assembly of nanoapatite within a type I collagen matrix was achieved through biomimetic mineralization in vitro, cooperatively regulated by non-collagenous proteins and small biomolecules. Here, we demonstrated that IP6 could significantly promote intrafibrillar mineralization in two- and three-dimensional collagen models through binding to collagen fibrils via hydrogen bonds (the interaction energy ∼10.21 kJ mol-1), as confirmed by the FTIR spectra and isothermal experimental results. In addition, we find that IP6 associated with dental collagen fibrils can also enhance the remineralization of calcium-depleted dentin and restore its mechanical properties similar to the natural dentin within 4 days. The promoting effect is mainly due to the chemical modification of IP6, which alters the interfacial physicochemical properties of collagen fibrils, strengthening the interaction of calcium phosphate minerals and mineral ions with collagen fibrils. This strategy of interfacial regulation to accelerate the mineralization of collagen fibrils is essential for dental repair and the development of a clinical product for the remineralization of hard tissue.

Strategic outline of interventions targeting extracellular matrix for promoting healthy longevity

The extracellular matrix (ECM), composed of interlinked proteins outside of cells, is an important component of the human body that helps maintain tissue architecture and cellular homeostasis. As people age, the ECM undergoes changes that can lead to age-related morbidity and mortality. Despite its importance, ECM aging remains understudied in the field of geroscience. In this review, we discuss the core concepts of ECM integrity, outline the age-related challenges and subsequent pathologies and diseases, summarize diagnostic methods detecting a faulty ECM, and provide strategies targeting ECM homeostasis. To conceptualize this, we built a technology research tree to hierarchically visualize possible research sequences for studying ECM aging. This strategic framework will hopefully facilitate the development of future research on interventions to restore ECM integrity, which could potentially lead to the development of new drugs or therapeutic interventions promoting health during aging.

Layer-specific fiber distribution in arterial tissue modeled as a constrained mixture

Collagen fibers and their orientation greatly influence an artery's mechanical characteristics, determining its transversely isotropic behavior. It is generally assumed that these fibers are deposited along a preferred direction to maximize the load bearing capacity of the vessel wall. This implies a large spatial variation in collagen orientation which can be reconstructed in numerical models using so-called reorientation algorithms. Until now, these algorithms have used the classical continuum mechanics modeling framework which requires knowledge of tissue-level parameters and the artery's stress-free reference state, which is inaccessible in a clinical context. We present an algorithm to compute the preferred fiber distribution compatible with the constrained mixture theory, which orients two collagen fiber families according to the loading experienced by the isotropic non-collagenous extracellular matrix, without requiring prior knowledge of the stress-free state. Because consensus is lacking whether stress or stretch is the determining factor behind the preferred fiber distribution, we implemented both approaches and compared the results with experimental microstructural data of an abdominal aorta. The stress-based algorithm was able to describe several experimentally observed transitions of the fiber distribution across the intima, media and adventitia.

Mechano-biological and bio-mechanical pathways in cutaneous wound healing

Injuries to the skin heal through coordinated action of fibroblast-mediated extracellular matrix (ECM) deposition, ECM remodeling, and wound contraction. Defects involving the dermis result in fibrotic scars featuring increased stiffness and altered collagen content and organization. Although computational models are crucial to unravel the underlying biochemical and biophysical mechanisms, simulations of the evolving wound biomechanics are seldom benchmarked against measurements. Here, we leverage recent quantifications of local tissue stiffness in murine wounds to refine a previously-proposed systems-mechanobiological finite-element model. Fibroblasts are considered as the main cell type involved in ECM remodeling and wound contraction. Tissue rebuilding is coordinated by the release and diffusion of a cytokine wave, e.g. TGF-β, itself developed in response to an earlier inflammatory signal triggered by platelet aggregation. We calibrate a model of the evolving wound biomechanics through a custom-developed hierarchical Bayesian inverse analysis procedure. Further calibration is based on published biochemical and morphological murine wound healing data over a 21-day healing period. The calibrated model recapitulates the temporal evolution of: inflammatory signal, fibroblast infiltration, collagen buildup, and wound contraction. Moreover, it enables in silico hypothesis testing, which we explore by: (i) quantifying the alteration of wound contraction profiles corresponding to the measured variability in local wound stiffness; (ii) proposing alternative constitutive links connecting the dynamics of the biochemical fields to the evolving mechanical properties; (iii) discussing the plausibility of a stretch- vs. stiffness-mediated mechanobiological coupling. Ultimately, our model challenges the current understanding of wound biomechanics and mechanobiology, beside offering a versatile tool to explore and eventually control scar fibrosis after injury.

Computer Model-Driven Design in Cardiovascular Regenerative Medicine

Continuing advances in genomics, molecular and cellular mechanobiology and immunobiology, including transcriptomics and proteomics, and biomechanics increasingly reveal the complexity underlying native tissue and organ structure and function. Identifying methods to repair, regenerate, or replace vital tissues and organs remains one of the greatest challenges of modern biomedical engineering, one that deserves our very best effort. Notwithstanding the continuing need for improving standard methods of investigation, including cell, organoid, and tissue culture, biomaterials development and fabrication, animal models, and clinical research, it is increasingly evident that modern computational methods should play increasingly greater roles in advancing the basic science, bioengineering, and clinical application of regenerative medicine. This brief review focuses on the development and application of computational models of tissue and organ mechanobiology and mechanics for purposes of designing tissue engineered constructs and understanding their development in vitro and in situ. Although the basic approaches are general, for illustrative purposes we describe two recent examples from cardiovascular medicine-tissue engineered heart valves (TEHVs) and tissue engineered vascular grafts (TEVGs)-to highlight current methods of approach as well as continuing needs.

Implementing computational modeling in tissue engineering: where disciplines meet

In recent years, the mathematical and computational sciences have developed novel methodologies and insights that can aid in designing advanced bioreactors, microfluidic set-ups or organ-on-chip devices, in optimizing culture conditions, or predicting long-term behavior of engineered tissues in vivo. In this review, we introduce the concept of computational models and how they can be integrated in an interdisciplinary workflow for Tissue Engineering and Regenerative Medicine (TERM). We specifically aim this review of general concepts and examples at experimental scientists with little or no computational modeling experience. We also describe the contribution of computational models in understanding TERM processes and in advancing the TERM field by providing novel insights.

Increased Collagen Crosslinking in Stiff Clubfoot Tissue: Implications for the Improvement of Therapeutic Strategies

Congenital clubfoot is a complex musculoskeletal deformity, in which a stiff, contracted tissue forms in the medial part of the foot. Fibrotic changes are associated with increased collagen deposition and lysyl oxidase (LOX)-mediated crosslinking, which impair collagen degradation and increase the tissue stiffness. First, we studied collagen deposition, as well as the expression of collagen and the amount of pyridinoline and deoxypyridinoline crosslinks in the tissue of relapsed clubfoot by immunohistochemistry, real-time PCR, and enzyme-linked immunosorbent assay (ELISA). We then isolated fibroblast-like cells from the contracted tissue to study the potential inhibition of these processes in vitro. We assessed the effects of a LOX inhibitor, β-aminopropionitrile (BAPN), on the cells by a hydroxyproline assay, ELISA, and Second Harmonic Generation imaging. We also evaluated the cell-mediated contraction of extracellular matrix in 3D cell-populated collagen gels. For the first time, we have confirmed significantly increased crosslinking and excessive collagen type I deposition in the clubfoot-contracted tissue. We successfully reduced these processes in vitro in a dose-dependent manner with 10–40 µg/mL of BAPN, and we observed an increasing trend in the inhibition of the cell-mediated contraction of collagen gels. The in vitro inhibitory effects indicate that BAPN has good potential for the treatment of relapsed and resistant clubfeet.

Development of a Collagen Fibre Remodelling Rupture Risk Metric for Potentially Vulnerable Carotid Artery Atherosclerotic Plaques

Atherosclerotic plaque rupture in carotid arteries can lead to stroke which is one of the leading causes of death or disability worldwide. The accumulation of atherosclerotic plaque in an artery changes the mechanical properties of the vessel. Whilst healthy arteries can continuously adapt to mechanical loads by remodelling their internal structure, particularly the load-bearing collagen fibres, diseased vessels may have limited remodelling capabilities. In this study, a local stress modulated remodelling algorithm is proposed to explore the mechanical response of arterial tissue to the remodelling of collagen fibres. This stress driven remodelling algorithm is used to predict the optimum distribution of fibres in healthy and diseased human carotid bifurcations obtained using Magnetic Resonance Imaging (MRI). In the models, healthy geometries were segmented into two layers: media and adventitia and diseased into four components: adventitia, media, plaque atheroma and lipid pool (when present in the MRI images). A novel meshing technique for hexahedral meshing of these geometries is also demonstrated. Using the remodelling algorithm, the optimum fibre patterns in various patient specific plaques are identified and the role that deviations from these fibre configurations in plaque vulnerability is shown. This study provides critical insights into the collagen fibre patterns required in carotid artery and plaque tissue to maintain plaque stability.

Numerical Simulation of Corneal Fibril Reorientation in Response to External Loading

Purpose: To simulate numerically the collagen fibril reorientation observed experimentally in the cornea. Methods: Fibril distribution in corneal strip specimens was monitored using X-ray scattering while under gradually increasing axial loading. The data were analysed at each strain level in order to quantify the changes in the angular distribution of fibrils with strain growth. The resulting relationship between stain and fibril reorientation was adopted in a constitutive model to control the mechanical anisotropy of the tissue material. The outcome of the model was validated against the experimental measurements before using the model in simplified representations of two surgical procedures. Results: The numerical model was able to reproduce the experimental measurements of specimen deformation and fibril reorientation under uniaxial loading with errors below 8.0%. With tissue removal simulated in a full eye numerical model, fibril reorientation could be predicted around the affected area, and this change both increased with larger tissue removal and reduced gradually away from that area. Conclusion: The presented method can successfully simulate fibril reorientation with changes in the strain regime affecting cornea tissue. Analyses based on this method showed that fibrils tend to align parallel to the tissue cut following keratoplasty operations. With the ability to simulate fibril reorientation, numerical modelling can have a greater potential in modelling the behaviour following surgery and injury to the cornea.

Growth and remodelling of living tissues: perspectives, challenges and opportunities

One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.

EWMA Document: Negative Pressure Wound Therapy

1. Introduction Since its introduction in clinical practice in the early 1990's negative pressure wounds therapy (NPWT) has become widely used in the management of complex wounds in both inpatient and outpatient care.¹ NPWT has been described as a effective treatment for wounds of many different aetiologies^2,3 and suggested as a gold standard for treatment of wounds such as open abdominal wounds,^4-6 dehisced sternal wounds following cardiac surgery^7,8 and as a valuable agent in complex non-healing wounds.^9,10 Increasingly, NPWT is being applied in the primary and home-care setting, where it is described as having the potential to improve the efficacy of wound management and help reduce the reliance on hospital-based care.¹¹ While the potential of NPWT is promising and the clinical use of the treatment is widespread, highlevel evidence of its effectiveness and economic benefits remain sparse.^12-14 The ongoing controversy regarding high-level evidence in wound care in general is well known. There is a consensus that clinical practice should be evidence-based, which can be difficult to achieve due to confusion about the value of the various approaches to wound management; however, we have to rely on the best available evidence. The need to review wound strategies and treatments in order to reduce the burden of care in an efficient way is urgent. If patients at risk of delayed wound healing are identified earlier and aggressive interventions are taken before the wound deteriorates and complications occur, both patient morbidity and health-care costs can be significantly reduced. There is further a fundamental confusion over the best way to evaluate the effectiveness of interventions in this complex patient population. This is illustrated by reviews of the value of various treatment strategies for non-healing wounds, which have highlighted methodological inconsistencies in primary research. This situation is confounded by differences in the advice given by regulatory and reimbursement bodies in various countries regarding both study design and the ways in which results are interpreted. In response to this confusion, the European Wound Management Association (EWMA) has been publishing a number of interdisciplinary documents^15-19 with the intention of highlighting: The nature and extent of the problem for wound management: from the clinical perspective as well as that of care givers and the patients Evidence-based practice as an integration of clinical expertise with the best available clinical evidence from systematic research The nature and extent of the problem for wound management: from the policy maker and healthcare system perspectives The controversy regarding the value of various approaches to wound management and care is illustrated by the case of NPWT, synonymous with topical negative pressure or vacuum therapy and cited as branded VAC (vacuum-assisted closure) therapy. This is a mode of therapy used to encourage wound healing. It is used as a primary treatment of chronic wounds, in complex acute wounds and as an adjunct for temporary closure and wound bed preparation preceding surgical procedures such as skin grafts and flap surgery. Aim An increasing number of papers on the effect of NPWT are being published. However, due to the low evidence level the treatment remains controversial from the policy maker and health-care system's points of view-particularly with regard to evidence-based medicine. In response EWMA has established an interdisciplinary working group to describe the present knowledge with regard to NPWT and provide overview of its implications for organisation of care, documentation, communication, patient safety, and health economic aspects. These goals will be achieved by the following: Present the rational and scientific support for each delivered statement Uncover controversies and issues related to the use of NPWT in wound management Implications of implementing NPWT as a treatment strategy in the health-care system Provide information and offer perspectives of NPWT from the viewpoints of health-care staff, policy makers, politicians, industry, patients and hospital administrators who are indirectly or directly involved in wound management.

Phosphorylated chitosan to promote biomimetic mineralization of type I collagen as a strategy for dentin repair and bone tissue engineering

This novel biomimetic mineralization technique provides an efficient method to produce an advanced mineralized matrix.

Collagen fibre orientation and dispersion govern ultimate tensile strength, stiffness and the fatigue performance of bovine pericardium

The durability of bovine pericardium leaflets employed in bioprosthetic heart valves (BHVs) can significantly limit the longevity of heart valve prostheses. Collagen fibres are the dominant load bearing component of bovine pericardium, however fibre architecture within leaflet geometries is not explicitly controlled in the manufacture of commercial devices. Thus, the purpose of this study was to ascertain the influence of pre-determined collagen fibre orientation and dispersion on the mechanical performance of bovine pericardium. Three tissue groups were tested in uniaxial tension: cross-fibre tissue (XD); highly dispersed fibre-orientations (HD); or preferred-fibre tissue (PD). Both the XD and PD tissue were tested under cyclic loading at 1.5 Hz and a stress range of 2.7 MPa. The results of the static tensile experiments illustrated that collagen fibre orientation and degree of alignment significantly influenced the material's response, whereby, there was a statistically significant decrease in material properties between the XD groups and both the PD and HD groups for ultimate tensile strength and stiffness (p < 0.01). Furthermore, HD tissue had a stiffness of approximately 58% of the PD group, and XD tissue had a stiffness of approximately 18% of the PD group. The dynamic behaviour of the XD and PD groups was extremely distinct; for example a Weibull analysis indicated that the 50% probability of failure in specimens with fibres orientated perpendicular (XD) to the loading direction occurred at 375 cycles. Due to this failure, XD specimens survived on average less than 20% of the cycles completed by those in which fibres were aligned along the loading direction (PD). The results from this study indicate that fibre architecture is a significant factor in determining static strength and fatigue life in bovine pericardium, and thus must be incorporated in the design process to improve future device durability.

A novel computational remodelling algorithm for the probabilistic evolution of collagen fibre dispersion in biaxially strained vascular tissue

In this work, we constructed a novel collagen fibre remodelling algorithm that incorporates the complex nature of random evolution acting on single fibres causing macroscopic fibre dispersion. The proposed framework is different from the existing remodelling algorithms, in that the microscopic random force on cellular scales causing a rotational-type Brownian motion alone is considered as an aspect of vascular tissue remodelling. A continuum mechanical framework for the evolution of local dispersion and how it could be used for modeling the evolution of internal radius of biaxially strained artery structures under constant internal blood pressure are presented. A linear evolution form for the statistical fibre dispersion is employed in the model. The random force component of the evolution, which depends on the mechanical stress stimuli, is described by a single parameter. Although the mathematical form of the proposed model is simple, there is a strong link between the microscopic evolution of collagen dispersion on the cellular level and its effects on the macroscopic visible world through mechanical variables. We believe that the proposed algorithm utilizes a better understanding of the relationship between the evolution rates of mean fibre direction and fibre dispersion. The predictive capability of the algorithm is presented using experimental data. The model has been simulated by solving a single-layered axisymmetric artery (adventitia) deformation problem. The algorithm performed well for estimating the quantitative features of experimental anisotropy, the mean fibre direction vector and the dispersion (κ) measurements under strain-dependent evolution assumptions.

Emergence of Collagen Orientation Heterogeneity in Healing Infarcts and an Agent-Based Model

Spatial heterogeneity of matrix structure can be an important determinant of tissue function. Although bulk properties of collagen structure in healing myocardial infarcts have been characterized previously, regional heterogeneity in infarct structure has received minimal attention. Herein, we quantified regional variations of collagen and nuclear orientations over the initial weeks of healing after infarction in rats, and employed a computational model of infarct remodeling to test potential explanations for the heterogeneity we observed in vivo. Fiber and cell orientation maps were generated from infarct samples acquired previously at 1, 2, 3, and 6 weeks postinfarction in a rat ligation model. We analyzed heterogeneity by calculating the dot product of each fiber or cell orientation vector with every other fiber or cell orientation vector, and plotting that dot product versus distance between the fibers or cells. This analysis revealed prominent regional heterogeneity, with alignment of both fibers and cell nuclei in local pockets far exceeding the global average. Using an agent-based model of fibroblast-mediated collagen remodeling, we found that similar levels of heterogeneity can spontaneously emerge from initially isotropic matrix via locally reinforcing cell-matrix interactions. Specifically, cells that sensed fiber orientation at a distance or remodeled fibers at a distance by traction-mediated reorientation or aligned deposition gave rise to regionally heterogeneous structures. However, only the simulations in which cells deposited collagen fibers aligned with their own orientation reproduced experimentally measured patterns of heterogeneity across all time points. These predictions warrant experimental follow-up to test the role of such mechanisms in vivo and identify opportunities to control heterogeneity for therapeutic benefit.

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.

Modulation of collagen fiber orientation by strain-controlled enzymatic degradation

Collagen fiber anisotropy has a significant influence on the function and mechanical properties of cardiovascular tissues. We investigated if strain-dependent collagen degradation can explain collagen orientation in response to uniaxial and biaxial mechanical loads. First, decellularized pericardial samples were stretched to a fixed uniaxial strain and after adding a collagen degrading enzyme (collagenase), force relaxation was measured to calculate the degradation rate. This data was used to identify the strain-dependent degradation rate. A minimum was observed in the degradation rate curve. It was then demonstrated, for the first time, that biaxial strain in combination with collagenase alters the collagen fiber alignment from an initially isotropic distribution to an anisotropic distribution with a mean alignment corresponding with the strain at the minimum degradation rate, which may be in between the principal strain directions. When both strains were smaller than the minimum degradation point, fibers tended to align in the direction of the larger strain and when both strains were larger than the minimum degradation, fibers mainly aligned in the direction of the smaller strain. However, when one strain was larger and one was smaller than the minimum degradation point, the observed fiber alignment was in between the principal strain directions. In the absence of collagenase, uniaxial and biaxial strains only had a slight effect on the collagen (re)orientation of the decellularized samples.

STATEMENT OF SIGNIFICANCE

Collagen fiber orientation is a significant determinant of the mechanical properties of native tissues. To mimic the native-like collagen alignment in vitro, we need to understand the underlying mechanisms that direct this alignment. In the current study, we aimed to control collagen fiber orientation by applying biaxial strains in the presence of collagenase. We hypothesized that strain-dependent collagen degradation can describe specific collagen orientation when biaxial mechanical strains are applied. Based on this hypothesis, collagen fibers align in the direction where the degradation is minimal. Pericardial tissues, as isotropic collagen matrices, were decellularized and subjected to a fixed uniaxial strain. Then, collagenase was added to initiate the collagen degradation and the relaxation of force was measured to indicate the degradation rate. The V-shaped relationship between degradation rate and strain was obtained to identify the minimum degradation rate point. It was then demonstrated, for the first time, that biaxial strain in combination with collagenase alters the collagen fiber alignment from almost isotropic to a direction corresponding with the strain at the minimum degradation rate.

Heart valve tissue engineering: how far is the bedside from the bench?

Heart disease, including valve pathologies, is the leading cause of death worldwide. Despite the progress made thanks to improving transplantation techniques, a perfect valve substitute has not yet been developed: once a diseased valve is replaced with current technologies, the newly implanted valve still needs to be changed some time in the future. This situation is particularly dramatic in the case of children and young adults, because of the necessity of valve growth during the patient's life. Our review focuses on the current status of heart valve (HV) therapy and the challenges that must be solved in the development of new approaches based on tissue engineering. Scientists and physicians have proposed tissue-engineered heart valves (TEHVs) as the most promising solution for HV replacement, especially given that they can help to avoid thrombosis, structural deterioration and xenoinfections. Lastly, TEHVs might also serve as a model for studying human valve development and pathologies.

The Evolution of Collagen Fiber Orientation in Engineered Cardiovascular Tissues Visualized by Diffusion Tensor Imaging

The collagen architecture is the major determinant of the function and mechanical behavior of cardiovascular tissues. In order to engineer a functional and load-bearing cardiovascular tissue with a structure that mimics the native tissue to meet in vivo mechanical demands, a complete understanding of the collagen orientation mechanism is required. Several methods have been used to visualize collagen architecture in tissue-engineered (TE) constructs, but they either have a limited imaging depth or have a complicated set up. In this study, Diffusion Tensor Imaging (DTI) is explored as a fast and reliable method to visualize collagen arrangement, and Confocal Laser Scanning Microscopy (CLSM) was used as a validation technique. Uniaxially constrained TE strips were cultured for 2 days, 10 days, 3 and 6 weeks to investigate the evolution of the collagen orientation with time. Moreover, a comparison of the collagen orientation in high and low aspect ratio (length/width) TE constructs was made with both methods. Both methods showed similar fiber orientation in TE constructs. Collagen fibers in the high aspect ratio samples were mostly aligned in the constrained direction, while the collagen fibers in low aspect ratio strips were mainly oriented in the oblique direction. The orientation changed to the oblique direction by extending culture time and could also be visualized. DTI captured the collagen orientation differences between low and high aspect ratio samples and with time. Therefore, it can be used as a fast, non-destructive and reliable tool to study the evolution of the collagen orientation in TE constructs.

A theoretical model of the endothelial cell morphology due to different waveforms

Endothelial cells are key units in the regulatory biological process of blood vessels. They represent an interface to transmit variations on the fluid dynamic changes. They are able to adapt its cytoskeleton, by means of microtubules reorientation and F-actin reorganization, due to new mechanical environments. Moreover, they are responsible for initiating a huge cascade of biological processes, such as the release of endothelins (ET-1), in charge of the constriction of the vessel and growth factors such as TGF-β and PDGF. Although a huge efforts have been made in the experimental characterization and description of these two issues the computational modeling has not gained such an attention. In this work we study the 3D remodeling of endothelial cells based on the main features of blood flow. In particular we study how different oscillatory shear index and the time average wall shear stresses modify the endothelial cell shape. We found our model fitted the experimental works presented before in in vitro studies. We also include our model within a computational fluid dynamics simulation of a carotid artery to evaluate endothelial cell shape index which is a key predictor of atheroma plaque formation. Moreover, our approach can be coupled with models of collagen and smooth muscle cell growth, where remodeling and the associated release of chemical substance are involved.

Review of collagen I hydrogels for bioengineered tissue microenvironments: characterization of mechanics, structure, and transport

Type I collagen hydrogels have been used successfully as three-dimensional substrates for cell culture and have shown promise as scaffolds for engineered tissues and tumors. A critical step in the development of collagen hydrogels as viable tissue mimics is quantitative characterization of hydrogel properties and their correlation with fabrication parameters, which enables hydrogels to be tuned to match specific tissues or fulfill engineering requirements. A significant body of work has been devoted to characterization of collagen I hydrogels; however, due to the breadth of materials and techniques used for characterization, published data are often disjoint and hence their utility to the community is reduced. This review aims to determine the parameter space covered by existing data and identify key gaps in the literature so that future characterization and use of collagen I hydrogels for research can be most efficiently conducted. This review is divided into three sections: (1) relevant fabrication parameters are introduced and several of the most popular methods of controlling and regulating them are described, (2) hydrogel properties most relevant for tissue engineering are presented and discussed along with their characterization techniques, (3) the state of collagen I hydrogel characterization is recapitulated and future directions are proposed. Ultimately, this review can serve as a resource for selection of fabrication parameters and material characterization methodologies in order to increase the usefulness of future collagen-hydrogel-based characterization studies and tissue engineering experiments.

A method for predicting collagen fiber realignment in non-planar tissue surfaces as applied to glenohumeral capsule during clinically relevant deformation

Previously developed experimental methods to characterize micro-structural tissue changes under planar mechanical loading may not be applicable for clinically relevant cases. Such limitation stems from the fact that soft tissues, represented by two-dimensional surfaces, generally do not undergo planar deformations in vivo. To address the problem, a method was developed to directly predict changes in the collagen fiber distribution of nonplanar tissue surfaces following 3D deformation. Assuming that the collagen fiber distribution was known in the un-deformed configuration via experimental methods, changes in the fiber distribution were predicted using 3D deformation. As this method was solely based on kinematics and did not require solving the stress balance equations, the computational efforts were much reduced. In other words, with the assumption of affine deformation, the deformed collagen fiber distribution was calculated using only the deformation gradient tensor (obtained via an in-plane convective curvilinear coordinate system) and the associated un-deformed collagen fiber distribution. The new method was then applied to the glenohumeral capsule during simulated clinical exams. To quantify deformation, positional markers were attached to the capsule and their 3D coordinates were recorded in the reference position and three clinically relevant joint positions. Our results showed that at 60deg of external rotation, the glenoid side of the posterior axillary pouch had significant changes in fiber distribution in comparison to the other sub-regions. The larger degree of collagen fiber alignment on the glenoid side suggests that this region is more prone to injury. It also compares well with previous experimental and clinical studies indicating maximum principle strains to be greater on the glenoid compared to the humeral side. An advantage of the new method is that it can also be easily applied to map experimentally measured collagen fiber distribution (obtained via methods that require flattening of tissue) to their in vivo nonplanar configuration. Thus, the new method could be applied to many other nonplanar fibrous tissues such as the ocular shell, heart valves, and blood vessels.

Computational modeling of chemical reactions and interstitial growth and remodeling involving charged solutes and solid-bound molecules

Mechanobiological processes are rooted in mechanics and chemistry, and such processes may be modeled in a framework that couples their governing equations starting from fundamental principles. In many biological applications, the reactants and products of chemical reactions may be electrically charged, and these charge effects may produce driving forces and constraints that significantly influence outcomes. In this study, a novel formulation and computational implementation are presented for modeling chemical reactions in biological tissues that involve charged solutes and solid-bound molecules within a deformable porous hydrated solid matrix, coupling mechanics with chemistry while accounting for electric charges. The deposition or removal of solid-bound molecules contributes to the growth and remodeling of the solid matrix; in particular, volumetric growth may be driven by Donnan osmotic swelling, resulting from charged molecular species fixed to the solid matrix. This formulation incorporates the state of strain as a state variable in the production rate of chemical reactions, explicitly tying chemistry with mechanics for the purpose of modeling mechanobiology. To achieve these objectives, this treatment identifies the specific theoretical and computational challenges faced in modeling complex systems of interacting neutral and charged constituents while accommodating any number of simultaneous reactions where reactants and products may be modeled explicitly or implicitly. Several finite element verification problems are shown to agree with closed-form analytical solutions. An illustrative tissue engineering analysis demonstrates tissue growth and swelling resulting from the deposition of chondroitin sulfate, a charged solid-bound molecular species. This implementation is released in the open-source program FEBio ( www.febio.org ). The availability of this framework may be particularly beneficial to optimizing tissue engineering culture systems by examining the influence of nutrient availability on the evolution of inhomogeneous tissue composition and mechanical properties, the evolution of construct dimensions with growth, the influence of solute and solid matrix electric charge on the transport of cytokines, the influence of binding kinetics on transport, the influence of loading on binding kinetics, and the differential growth response to dynamically loaded versus free-swelling culture conditions.

Modelling the influence of endothelial heterogeneity on the progression of arterial disease: application to abdominal aortic aneurysm evolution

We sophisticate a fluid-solid growth computational framework for modelling aneurysm evolution. A realistic structural model of the arterial wall is integrated into a patient-specific geometry of the vasculature. This enables physiologically representative distributions of haemodynamic stimuli, obtained from a rigid-wall computational fluid dynamics analysis, to be linked to growth and remodelling algorithms. Additionally, a quasistatic structural analysis quantifies the cyclic deformation of the arterial wall so that collagen growth and remodelling can be explicitly linked to the cyclic deformation of vascular cells. To simulate aneurysm evolution, degradation of elastin is driven by reductions in wall shear stress (WSS) below homeostatic thresholds. Given that the endothelium exhibits spatial and temporal heterogeneity, we propose a novel approach to define the homeostatic WSS thresholds: We allow them to be spatially and temporally heterogeneous. We illustrate the application of this novel fluid-solid growth framework to model abdominal aortic aneurysm (AAA) evolution and to examine how the influence of the definition of the WSS homeostatic threshold influences AAA progression. We conclude that improved understanding and modelling of the endothelial heterogeneity is important for modelling aneurysm evolution and, more generally, other vascular diseases where haemodynamic stimuli play an important role.

Efficacy of tendon stem cells in fibroblast-derived matrix for tendon tissue engineering

BACKGROUND AIMS

After injury, tendons often heal with poor tissue quality and inferior mechanical properties. Tissue engineering using tendon stem cells (TSCs) is a promising approach in the repair of injured tendon. Tenogenic differentiation of TSCs needs an appropriate environment. More recently, the acellular extracellular matrix (ECM) generated from fibroblasts has been used to construct various engineering tissues. In this study, we successfully developed an engineered tendon tissue formed by seeding TSCs in de-cellularized fibroblast-derived matrix (dFM).

METHODS

Patellar TSCs and dermal fibroblast were isolated and cultured. Using the method of osmotic shock, dFM was obtained from dermal fibroblast. ECM proteins in dFM were examined. TSCs at passage 3 were seeded in dFM for 1 week. Proliferative capacity and characterization of TSCs cultured in dFM were determined by population doubling time, immunofluorescence staining and quantitative reverse transcriptase polymerase chain reaction. Engineered tendon tissue was prepared with dFM and TSCs. Its potentials for neo-tendon formation and promoting tendon healing were investigated.

RESULTS

dFM is suitable for growth and tenogenic differentiation of TSCs in vitro. Neo-tendon tissue was formed with tendon-specific protein expression when TSCs were implanted together with dFM. In a patellar tendon injury model, implantation of engineered tendon tissue significantly improved the histologic and mechanical properties of injured tendon.

CONCLUSIONS

The findings obtained from our study provide a basis for potential use of engineered tendon tissue containing dFM and TSCs in tendon repair and regeneration.

Development of a computational biomechanical model of the human upper-airway soft-tissues toward simulating obstructive sleep apnea

Numerous challenges are faced in investigations aimed at developing a better understanding of the pathophysiology of obstructive sleep apnea (OSA). The anatomy of the tongue and other upper-airway tissues, and the ability to model their behavior, are central to such investigations. We present details of the construction and development of a soft-tissue model of the human upper airway, with the ultimate goal of simulating obstructive sleep apnea. The steps taken to produce a representative anatomical geometry, of which the associated muscle histology is also captured, are documented. An overview of the mathematical models used to describe tissue behavior, both at a macro- and microscopic level, is given. A neurological model, which mimics the proprioceptive capabilities of the body, is described as it is applies to control of the active dynamics of the tongue. A simplified scenario, which allows for the manipulation of several environmental influences, is presented. It is demonstrated that the response of the genioglossus is qualitatively similar to that determined through experimental techniques. Furthermore, insights into the stress distribution developed within the tongue are discussed. It is shown that changes in almost any aspect of the breathing or physiological conditions invoke a significant change in the response of the airway dilators. The results of this study provide further evidence of the importance of modeling and simulation techniques as an aid in understanding the complex behavior of the human body.

Failure mechanisms in fibrous scaffolds

Polymeric fibrous scaffolds have been considered as replacements for load-bearing soft tissues, because of their ability to mimic the microstructure of natural tissues. Poor toughness of fibrous materials results in failure, which is an issue of importance to both engineering and medical practice. The toughness of fibrous materials depends on the ability of the microstructure to develop toughening mechanisms. However, such toughening mechanisms are still not well understood, because the detailed evolution at the microscopic level is difficult to visualize. A novel and simple method was developed, namely, a sample-taping technique, to examine the detailed failure mechanisms of fibrous microstructures. This technique was compared with in situ fracture testing by scanning electron microscopy. Examination of three types of fibrous networks showed that two different failure modes occurred in fibrous scaffolds. For brittle cracking in gelatin electrospun scaffolds, the random network morphology around the crack tip remained during crack propagation. For ductile failure in polycaprolactone electrospun scaffolds and nonwoven fabrics, the random network deformed via fiber rearrangement, and a large number of fiber bundles formed across the region in front of the notch tip. These fiber bundles not only accommodated mechanical strain, but also resisted crack propagation and thus toughened the fibrous scaffolds. Such understanding provides insight for the production of fibrous materials with enhanced toughness.

Interstitial growth and remodeling of biological tissues: tissue composition as state variables

Growth and remodeling of biological tissues involves mass exchanges between soluble building blocks in the tissue's interstitial fluid and the various constituents of cells and the extracellular matrix. As the content of these various constituents evolves with growth, associated material properties, such as the elastic modulus of the extracellular matrix, may similarly evolve. Therefore, growth theories may be formulated by accounting for the evolution of tissue composition over time in response to various biological and mechanical triggers. This approach has been the foundation of classical bone remodeling theories that successfully describe Wolff's law by establishing a dependence between Young's modulus and bone apparent density and by formulating a constitutive relation between bone mass supply and the state of strain. The goal of this study is to demonstrate that adding tissue composition as state variables in the constitutive relations governing the stress-strain response and the mass supply represents a very general and straightforward method to model interstitial growth and remodeling in a wide variety of biological tissues. The foundation for this approach is rooted in the framework of mixture theory, which models the tissue as a mixture of multiple solid and fluid constituents. A further generalization is to allow each solid constituent in a constrained solid mixture to have its own reference (stress-free) configuration. Several illustrations are provided, ranging from bone remodeling to cartilage tissue engineering and cervical remodeling during pregnancy.

Force-induced fibronectin assembly and matrix remodeling in a 3D microtissue model of tissue morphogenesis

Encapsulations of cells in type-I collagen matrices are widely used three-dimensional (3D) in vitro models of wound healing and tissue morphogenesis and are common constructs for drug delivery and for in vivo implantation. As cells remodel the exogenous collagen scaffold, they also assemble a dense fibronectin (Fn) matrix that aids in tissue compaction; however, the spatio-temporal (re)organization of Fn and collagen in this setting has yet to be quantitatively investigated. Here, we utilized microfabricated tissue gauges (μTUGs) to guide the contraction of microscale encapsulations of fibroblasts within collagen gels. We combined this system with a Foerster Radius Energy Transfer (FRET) labeled biosensor of Fn conformation to probe the organization, conformation and remodeling of both the exogenous collagen and the cell-assembled Fn matrices. We show that within hours, compact Fn from culture media adsorbed to the collagen scaffold. Over the course of tissue remodeling, this Fn-coated collagen scaffold was compacted into a thin, sparsely populated core around which cells assembled a dense fibrillar Fn shell that was rich in both cell and plasma derived Fn. This resulted in two separate Fn populations with different conformations (compact/adsorbed and extended/fibrillar) in microtissues. Cell contractility and microtissue geometry cooperated to remodel these two populations, resulting in spatial gradients in Fn conformation. Together, these results highlight an important spatio-temporal interplay between two prominent extracellular matrix (ECM) molecules (Fn and collagen) and cellular traction forces, and will have implications for future studies of the force-mediated remodeling events that occur within collagen scaffolds either in 3D in vitro models or within surgical implants in vivo.

Continuum mixture models of biological growth and remodeling: past successes and future opportunities

Biological growth processes involve mass exchanges that increase, decrease, or replace material that constitutes cells, tissues, and organs. In most cases, such exchanges alter the structural makeup of the material and consequently affect associated mechanobiological responses to applied loads. Given that the type and extent of changes in structural integrity depend on the different constituents involved (e.g., particular cytoskeletal or extracellular matrix proteins), the continuum theory of mixtures is ideally suited to model the mechanics of growth and remodeling. The goal of this review is twofold: first, to highlight a few illustrative examples that show diverse applications of mixture theory to describe biological growth and/or remodeling; second, to identify some open problems in the fields of modeling soft-tissue growth and remodeling.

Laser microfabricated poly(glycerol sebacate) scaffolds for heart valve tissue engineering

Microfabricated poly(glycerol sebacate) (PGS) scaffolds may be applicable to tissue engineering heart valve leaflets by virtue of their controllable microstructure, stiffness, and elasticity. In this study, PGS scaffolds were computationally designed and microfabricated by laser ablation to match the anisotropy and peak tangent moduli of native bovine aortic heart valve leaflets. Finite element simulations predicted PGS curing conditions, scaffold pore shape, and strut width capable of matching the scaffold effective stiffnesses to the leaflet peak tangent moduli. On the basis of simulation predicted effective stiffnesses of 1.041 and 0.208 MPa for the scaffold preferred (PD) and orthogonal, cross-preferred (XD) material directions, scaffolds with diamond-shaped pores were microfabricated by laser ablation of PGS cured 12 h at 160°C. Effective stiffnesses measured for the scaffold PD (0.83 ± 0.13 MPa) and XD (0.21 ± 0.03 MPa) were similar to both predicted values and peak tangent moduli measured for bovine aortic valve leaflets in the circumferential (1.00 ± 0.16 MPa) and radial (0.26 ± 0.03 MPa) directions. Scaffolds cultivated with fibroblasts for 3 weeks accumulated collagen (736 ± 193 μg/g wet weight) and DNA (17 ± 4 μg/g wet weight). This study provides a basis for the computational design of biomimetic microfabricated PGS scaffolds for tissue-engineered heart valves.

A mathematical model of the process of ligament repair: effect of cold therapy and mechanical stress

This article proposes a mathematical model that predicts the wound healing process of the ligament after a sprain, grade II. The model describes the swelling, expression of the platelet-derived growth factor (PDGF), formation and migration of fibroblasts into the injury area and the expression of collagen fibers. Additionally, the model can predict the effect of ice treatment in reducing inflammation and the action of mechanical stress in the process of remodeling of collagen fibers. The results obtained from computer simulation show a high concordance with the clinical data previously reported by other authors.

A computational model to describe the collagen orientation in statically cultured engineered tissues

Collagen provides cardiovascular tissues with the ability to withstand haemodynamic loads. A similar network is essential to obtain in tissue-engineered (TE) samples of the same nature. Yet, the mechanism of collagen orientation is not fully understood. Typically collagen remodelling is linked to mechanical loading. However, TE constructs also show an oriented collagen network when developed under static culture. Experiments under these conditions also indicate that the tissue gradually compacts due to contractile stresses developed in the α-actin fibres of the cells. Therefore, it is hypothesised that cellular contractile stresses are responsible for collagen orientation. A model describing the cellular α-actin turnover and the stresses developed by them is integrated in a structural constitutive model describing the mechanical behaviour of collagen fibres. Results show that the model can successfully capture the sample compaction, tissue stress generation and its heterogeneous collagen arrangement.

Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms

Biomechanical factors play fundamental roles in the natural history of abdominal aortic aneurysms (AAAs) and their responses to treatment. Advances during the past two decades have increased our understanding of the mechanics and biology of the human abdominal aorta and AAAs, yet there remains a pressing need for considerable new data and resulting patient-specific computational models that can better describe the current status of a lesion and better predict the evolution of lesion geometry, composition, and material properties and thereby improve interventional planning. In this paper, we briefly review data on the structure and function of the human abdominal aorta and aneurysmal wall, past models of the mechanics, and recent growth and remodeling models. We conclude by identifying open problems that we hope will motivate studies to improve our computational modeling and thus general understanding of AAAs.

On a Class of Admissible Constitutive Behaviors in Free-Floating Engineered Tissues

A commonly used assay for studying cell - matrix interactions is the free-floating fibroblast populated collagen lattice, which was introduced in 1979. Briefly, fibroblasts are seeded within an initially thin, amorphous, untethered, circular gel consisting of reconstituted fibrillar collagen. Although the gel remains traction free and circular, the cells typically contract the gel to less than 50% of its original diameter within hours to days. Cellular mechanotransduction mechanisms are fundamental to this contraction, but there has not been a careful study of the associated mechanics. In this paper, we model the initial contraction of a circular gel by assuming a homogeneous, axisymmetric finite deformation while allowing possible radial variations in material properties, including material symmetry. We show that trivial solutions alone (i.e., no deformation, no contraction) are admitted by equilibrium and boundary conditions unless radial variations exist in the material behavior, including cell contraction. Although more complete data are needed to model better this initial-boundary value problem, the present results are consistent with both the salient features of the gel assay and recent observations reported in the literature that cells often introduce regional variations in tissue properties in vivo in an attempt to achieve, maintain, or restore mechanical homeostasis.

Computational model of soft tissues in the human upper airway

This paper presents a three-dimensional finite element model of the tongue and surrounding soft tissues with potential application to the study of sleep apnoea and of linguistics and speech therapy. The anatomical data was obtained from the Visible Human Project, and the underlying histological data was also extracted and incorporated into the model. Hyperelastic constitutive models were used to describe the material behaviour, and material incompressibility was accounted for. An active Hill three-element muscle model was used to represent the muscular tissue of the tongue. The neural stimulus for each muscle group was determined through the use of a genetic algorithm-based neural control model. The fundamental behaviour of the tongue under gravitational and breathing-induced loading is investigated. It is demonstrated that, when a time-dependent loading is applied to the tongue, the neural model is able to control the position of the tongue and produce a physiologically realistic response for the genioglossus.

Closed incision management with negative pressure wound therapy (CIM): biomechanics

A novel closed incision management with negative pressure wound therapy (CIM) has been developed for convenient use with closed incisions that has the potential to be beneficial for patients at risk for postoperative complications. Incisions are typically under lateral tension. This study explored the biomechanical mechanisms by which integrity of the incisional closure is enhanced by CIM. CIM was hypothesized to affect local stresses around closed incisions in a beneficial manner. Finite element analyses (FEA) indicated that application of CIM decreased the lateral stresses ~50% around the incision and changed the direction of the stresses to a distribution that is typical of intact tissue. Bench evaluations corroborated findings that CIM significantly increased the force required to disrupt the closed incision by ~50% as compared with closure alone. In conclusion, using 2 FEAs and bench modeling, CIM was shown to reduce and normalize tissue stresses and bolster appositional forces at the incision.

The fiber orientation in the coronary arterial wall at physiological loading evaluated with a two-fiber constitutive model

A patient-specific mechanical description of the coronary arterial wall is indispensable for individualized diagnosis and treatment of coronary artery disease. A way to determine the artery's mechanical properties is to fit the parameters of a constitutive model to patient-specific experimental data. Clinical data, however, essentially lack information about the stress-free geometry of an artery, which is necessary for constitutive modeling. In previous research, it has been shown that a way to circumvent this problem is to impose extra modeling constraints on the parameter estimation procedure. In this study, we propose a new modeling constraint concerning the in-situ fiber orientation (β (phys)). β (phys), which is a major contributor to the arterial stress-strain behavior, was determined for porcine and human coronary arteries using a mixed numerical-experimental method. The in-situ situation was mimicked using in-vitro experiments at a physiological axial pre-stretch, in which pressure-radius and pressure-axial force were measured. A single-layered, hyperelastic, thick-walled, two-fiber material model was accurately fitted to the experimental data, enabling the computation of stress, strain, and fiber orientation. β (phys) was found to be almost equal for all vessels measured (36.4 ± 0.3)°, which theoretically can be explained using netting analysis. In further research, this finding can be used as an extra modeling constraint in parameter estimation from clinical data.

Differential histomechanical response of carotid artery in relation to species and region: mathematical description accounting for elastin and collagen anisotropy

The selection of a mathematical descriptor for the passive arterial mechanical behavior has been long debated in the literature and customarily constrained by lack of pertinent data on the underlying microstructure. Our objective was to analyze the response of carotid artery subjected to inflation/extension with phenomenological and microstructure-based candidate strain-energy functions (SEFs), according to species (rabbit vs. pig) and region (proximal vs. distal). Histological variations among segments were examined, aiming to explicitly relate them with the differential material response. The Fung-type model could not capture the biphasic response alone. Combining a neo-Hookean with a two-fiber family term alleviated this restraint, but force data were poorly captured, while consideration of low-stress anisotropy via a quadratic term allowed improved simulation of both pressure and force data. The best fitting was achieved with the quadratic and Fung-type or four-fiber family SEF. The latter simulated more closely than the two-fiber family the high-stress response, being structurally justified for all artery types, whereas the quadratic term was justified for transitional and muscular arteries exhibiting notable elastin anisotropy. Diagonally arranged fibers were associated with pericellular medial collagen, and circumferentially and longitudinally arranged fibers with medial and adventitial collagen bundles, evidenced by the significant correlations of SEF parameters with quantitative histology.

A mathematical model of medial collateral ligament repair: migration, fibroblast proliferation and collagen formation

The partial rupture of ligament fibres leads to an injury known as grade 2 sprain. Wound healing after injury consists of four general stages: swelling, release of platelet-derived growth factor (PDGF), fibroblast migration and proliferation and collagen production. The aim of this paper is to present a mathematical model based on reaction-diffusion equations for describing the repair of the medial collateral ligament when it has suffered a grade 2 sprain. We have used the finite element method to solve the equations of this. The results have simulated the tissue swelling at the time of injury, predicted PDGF influence, the concentration of fibroblasts migrating towards the place of injury and reproduced the random orientation of immature collagen fibres. These results agree with experimental data reported by other authors. The model describes wound healing during the 9 days following such injury.