Evaluation of fundamental hypotheses underlying constrained mixture models of arterial growth and remodelling

Continuum Growth Mechanics: Reconciling Two Common Frameworks

The objective of this study was to investigate whether the two most common growth mechanics modeling frameworks, the constrained-mixture growth model and the kinematic growth model, could be reconciled mathematically. The purpose of this effort was to provide practical guidelines for potential users of these modeling frameworks. Results showed that the kinematic growth model is mathematically consistent with a special form of the constrained-mixture growth model, where only one generation of a growing solid exists at any given time, overturning its entire solid mass at each instant of growth in order to adopt the reference configuration dictated by the growth deformation. The thermodynamics of the kinematic growth model, along with the specialized constrained-mixture growth model, requires a cellular supply of chemical energy to allow deposition of solid mass under a stressed state. A back-of-the-envelope calculation shows that the amount of chemical energy required to sustain biological growth under these models is negligibly small, when compared to the amount of energy normally consumed daily by the human body. In conclusion, this study successfully reconciled the two most popular growth theories for biological growth and explained the special circumstances under which the constrained-mixture growth model reduces to the kinematic growth model.

Multiscale insights into postnatal aortic development

Despite its vital importance for establishing proper cardiovascular function, the process through which the vasculature develops and matures postnatally remains poorly understood. From a clinical perspective, an ability to mechanistically model the developmental time course in arteries and veins, as well as to predict how various pathologies and therapeutic interventions alter the affected vessels, promises to improve treatment strategies and long-term clinical outcomes, particularly in pediatric patients suffering from congenital heart defects. In the present study, we conducted a multiscale investigation into the postnatal development of the murine thoracic aorta, examining key allometric relations as well as relationships between in vivo mechanical stresses, collagen and elastin expression, and the gradual accumulation of load-bearing constituents within the aortic wall. Our findings suggest that the production of fibrillar collagens in the developing aorta associates strongly with the ratio of circumferential stresses between systole and diastole, hence emphasizing the importance of a pulsatile mechanobiological stimulus. Moreover, rates of collagen turnover and elastic fiber compaction can be inferred directly by synthesizing transcriptional data and quantitative histological measurements of evolving collagen and elastin content. Consistent with previous studies, we also observed that wall shear stresses acting on the aorta are similar at birth and in maturity, supporting the hypothesis that at least some stress targets are established early in development and maintained thereafter, thus providing a possible homeostatic basis to guide future experiments and inform future predictive modeling.

A multiscale computational model of arterial growth and remodeling including Notch signaling

Blood vessels grow and remodel in response to mechanical stimuli. Many computational models capture this process phenomenologically, by assuming stress homeostasis, but this approach cannot unravel the underlying cellular mechanisms. Mechano-sensitive Notch signaling is well-known to be key in vascular development and homeostasis. Here, we present a multiscale framework coupling a constrained mixture model, capturing the mechanics and turnover of arterial constituents, to a cell-cell signaling model, describing Notch signaling dynamics among vascular smooth muscle cells (SMCs) as influenced by mechanical stimuli. Tissue turnover was regulated by both Notch activity, informed by in vitro data, and a phenomenological contribution, accounting for mechanisms other than Notch. This novel framework predicted changes in wall thickness and arterial composition in response to hypertension similar to previous in vivo data. The simulations suggested that Notch contributes to arterial growth in hypertension mainly by promoting SMC proliferation, while other mechanisms are needed to fully capture remodeling. The results also indicated that interventions to Notch, such as external Jagged ligands, can alter both the geometry and composition of hypertensive vessels, especially in the short term. Overall, our model enables a deeper analysis of the role of Notch and Notch interventions in arterial growth and remodeling and could be adopted to investigate therapeutic strategies and optimize vascular regeneration protocols.

Biomechanics models predict increasing smooth muscle tone as a novel therapeutic target for central arterial dysfunction in hypertension

INTRODUCTION

Vasodilation can paradoxically increase arterial stiffness in older, hypertensive adults. This study modeled increasing smooth muscle tone as a therapeutic strategy to improve central arterial dysfunction in hypertension using participant-specific simulations.

METHODS

Participant-specific models of the carotid artery were parameterized from vascular ultrasound measures of nitroglycerin-induced vasodilation in 18 hypertensive veterans. The acute changes in carotid artery mechanics were simulated for changes of ±2, ±4, and ±6% in smooth muscle tone and ±5, ±10, and ±15 mmHg in mean arterial pressure (MAP). The chronic carotid artery adaptations were simulated based on the hypothesis that the carotid artery will remodel wall-cross sectional area to maintain mechanical homeostasis.

RESULTS

A 6% increase in smooth muscle tone acutely decreased carotid pulse wave velocity from 6.89 ± 1.24 m/s to 5.83 ± 1.73 m/s, and a 15 mmHg decrease in MAP decreased carotid pulse wave velocity to 6.17 ± 1.23 m/s. A 6% increase in smooth muscle tone acutely decreased wall stress from 76.2 ± 12.3 to 64.2 ± 10.4 kPa, and a 15 mmHg decrease in MAP decreased wall stress to 60.6 ± 10.7 kPa. A 6% increase in smooth muscle tone chronically decreased wall cross-sectional area from 18.3 ± 5.4 to 15.2 ± 4.9 mm 2, and a 15 mmHg decrease in MAP decreased wall cross-sectional area to 14.3 ± 4.6 mm 2 .

CONCLUSION

In participant-specific simulation, increasing smooth muscle tone can have a stronger or equivalent effect on carotid artery mechanics compared with decreasing blood pressure. Increasing central arterial smooth muscle tone may be a novel therapeutic target to improve central arterial dysfunction in older, hypertensive adults and should be a focus of future research.

Shear stress regulation in cylindrical arteries through medial growth and nitric oxide release

The mechanisms employed by blood vessels in order to adapt to their hemodynamic environment are important for our general understanding of disease and development. Changes in arterial geometry are generally induced by two effects: vasodilation and/or constriction; and growth and remodeling ("G &R"). The first can occur over short periods of a few minutes, while the second usually occurs over timescales of weeks or months. The free radical Nitric oxide (NO) is one of the few biological signaling molecules that is gaseous. When smooth muscle cells internalize NO, they lengthen and ultimately induce a relaxation of the artery. Platelet-Derived Growth Factor (PDGF) is a growth factor released by smooth muscle cells and platelets that regulates cell growth and division. In this paper we present a single-layered, axisymmetric hyperelastic model for a deforming, growing artery in which the opening angle is regulated by NO and growth is induced by PDGF. Our model describes vasodilation and G &R in a long cylindrical artery regulated by a steady-state Poiseuille flow. The transport of NO released by the endothelium is governed by a diffusion equation with a shear-stress dependent flux boundary condition. Arterial opening angle is assumed to be a Hill function of the wall-averaged NO concentration. We find that both growth and NO help to regulate shear stress with respect to the flow rate, but regulation through growth occurs only at large times. In contrast, regulation through NO is immediate but can only occur as long as the opening angle is able to continually decrease as a function of flow rate. Our model is calibrated using experimental data from ligated, control, and anastomosed carotid arteries of adult and weanling rabbits. Our results generate shear stress/flow rate and lumen radius/flow rate curves that agree with experimental data from control and NO-inhibited rabbit carotid arteries.

Intracellular signaling control of mechanical homeostasis in the aorta

Mature arteries exhibit a preferred biomechanical state in health evidenced by a narrow range of intramural and wall shear stresses. When stresses are perturbed by changes in blood pressure or flow, homeostatic mechanisms tend to restore target values via altered contractility and/or cell and matrix turnover. In contrast, vascular disease associates with compromised homeostasis, hence we must understand mechanisms underlying mechanical homeostasis and its robustness. Here, we use a multiscale computational model wherein mechanosensitive intracellular signaling pathways drive arterial growth and remodeling. First, we identify an ensemble of cell-level parameterizations where tissue-level responses are well-regulated and adaptive to hemodynamic perturbations. The responsible mechanism is persistent multiscale negative feedback whereby mechanosensitive signaling drives mass turnover until homeostatic target stresses are reached. This demonstrates how robustness emerges despite inevitable cell and individual heterogeneity. Second, we investigate tissue-level effects of signaling node knockdowns (ATIR, ROCK, TGF[Formula: see text]RII, PDGFR, ERK1/2) and find general agreement with experimental reports of fault tolerance. Robustness against structural changes manifests via low engagement of the node under baseline stresses or compensatory multiscale feedback via upregulation of additional pathways. Third, we show how knockdowns affect collagen and smooth muscle turnover at baseline and with perturbed stresses. In several cases, basal production is not remarkably affected, but sensitivities to stress deviations, which influence feedback strength, are reduced. Such reductions can impair adaptive responses, consistent with previously reported aortic vulnerability despite grossly normal appearances. Reduced stress sensitivities thus form a candidate mechanism for how robustness is lost, enabling transitions from health towards disease.

Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms

Thoracic aortic aneurysm (TAA) is a localized dilatation of the aorta that can lead to life-threatening dissection or rupture. In vivo assessments of TAA progression are largely limited to measurements of aneurysm size and growth rate. There is promise, however, that computational modelling of the evolving biomechanics of the aorta could predict future geometry and properties from initiating mechanobiological insults. We present an integrated framework to train a deep operator network (DeepONet)-based surrogate model to identify TAA contributing factors using synthetic finite-element-based datasets. For training, we employ a constrained mixture model of aortic growth and remodelling to generate maps of local aortic dilatation and distensibility for multiple TAA risk factors. We evaluate the performance of the surrogate model for insult distributions varying from fusiform (analytically defined) to complex (randomly generated). We propose two frameworks, one trained on sparse information and one on full-field greyscale images, to gain insight into a preferred neural operator-based approach. We show that this continuous learning approach can predict the patient-specific insult profile associated with any given dilatation and distensibility map with high accuracy, particularly when based on full-field images. Our findings demonstrate the feasibility of applying DeepONet to support transfer learning of patient-specific inputs to predict TAA progression.

The effects of viscoelasticity on residual strain in aortic soft tissues

Residual stress is thought to play a critical role in modulating stress distributions in soft biological tissues and in maintaining the mechanobiological stress environment of cells. Residual stresses in arteries and other tissues are classically assessed through opening angle experiments, which demonstrate the continuous release of residual stresses over hours. These results are then assessed through nonlinear biomechanical models to provide estimates of the residual stresses in the intact state. Although well studied, these analyses typically focus on hyperelastic material models despite significant evidence of viscoelastic phenomena over both short and long timescales. In this work, we extended the state-of-the-art structural tensor model for arterial tissues from Holzapfel and Ogden for fractional viscoelasticity. Models were tuned to capture consistent levels of hysteresis observed in biaxial experiments, while also minimizing the fractional viscoelastic weighting and opening angles to correctly capture opening angle dynamics. Results suggest that a substantial portion of the human abdominal aorta is viscoelastic, but exhibits a low fractional order (i.e. more elastically). Additionally, a significantly larger opening angle in the fully unloaded state is necessary to produce comparable hysteresis in biaxial testing. As a consequence, conventional estimates of residual stress using hyperelastic approaches over-estimate their viscoelastic counterparts by a factor of 2. Thus, a viscoelastic approach, such as the one illustrated in this study, in combination with an additional source of rate-controlled viscoelastic data is necessary to accurately analyze the residual stress distribution in soft biological tissues. STATEMENT OF SIGNIFICANCE: Residual stress plays a crucial role in achieving a homeostatic stress environment in soft biological tissues. However, the analysis of residual stress typically focuses on hyperelastic material models despite evidence of viscoelastic behavior. This work is the first attempt at analyzing the effects of viscoelasticity on residual stress. The application of viscoelasticity was crucial for producing realistic opening dynamics in arteries. The overall residual stresses were estimated to be 50% less than those from using hyperelastic material models, while the opening angles were projected to be 25% more than those measured after 16 hours, suggesting underestimated residual strain. This study highlights the importance viscoelasticity in the analysis of residual stress even in weakly dissipative materials like the human aorta.

Mechano-regulated cell-cell signaling in the context of cardiovascular tissue engineering

Cardiovascular tissue engineering (CVTE) aims to create living tissues, with the ability to grow and remodel, as replacements for diseased blood vessels and heart valves. Despite promising results, the (long-term) functionality of these engineered tissues still needs improvement to reach broad clinical application. The functionality of native tissues is ensured by their specific mechanical properties directly arising from tissue organization. We therefore hypothesize that establishing a native-like tissue organization is vital to overcome the limitations of current CVTE approaches. To achieve this aim, a better understanding of the growth and remodeling (G&R) mechanisms of cardiovascular tissues is necessary. Cells are the main mediators of tissue G&R, and their behavior is strongly influenced by both mechanical stimuli and cell-cell signaling. An increasing number of signaling pathways has also been identified as mechanosensitive. As such, they may have a key underlying role in regulating the G&R of tissues in response to mechanical stimuli. A more detailed understanding of mechano-regulated cell-cell signaling may thus be crucial to advance CVTE, as it could inspire new methods to control tissue G&R and improve the organization and functionality of engineered tissues, thereby accelerating clinical translation. In this review, we discuss the organization and biomechanics of native cardiovascular tissues; recent CVTE studies emphasizing the obtained engineered tissue organization; and the interplay between mechanical stimuli, cell behavior, and cell-cell signaling. In addition, we review past contributions of computational models in understanding and predicting mechano-regulated tissue G&R and cell-cell signaling to highlight their potential role in future CVTE strategies.

Constrained Mixture Models of Soft Tissue Growth and Remodeling - Twenty Years After

Soft biological tissues compromise diverse cell types and extracellular matrix constituents, each of which can possess individual natural configurations, material properties, and rates of turnover. For this reason, mixture-based models of growth (changes in mass) and remodeling (change in microstructure) are well-suited for studying tissue adaptations, disease progression, and responses to injury or clinical intervention. Such approaches also can be used to design improved tissue engineered constructs to repair, replace, or regenerate tissues. Focusing on blood vessels as archetypes of soft tissues, this paper reviews a constrained mixture theory introduced twenty years ago and explores its usage since by contrasting simulations of diverse vascular conditions. The discussion is framed within the concept of mechanical homeostasis, with consideration of solid-fluid interactions, inflammation, and cell signaling highlighting both past accomplishments and future opportunities as we seek to understand better the evolving composition, geometry, and material behaviors of soft tissues under complex conditions.

Vascular Mechanobiology: Homeostasis, Adaptation, and Disease

Cells of the vascular wall are exquisitely sensitive to changes in their mechanical environment. In healthy vessels, mechanical forces regulate signaling and gene expression to direct the remodeling needed for the vessel wall to maintain optimal function. Major diseases of arteries involve maladaptive remodeling with compromised or lost homeostatic mechanisms. Whereas homeostasis invokes negative feedback loops at multiple scales to mediate mechanobiological stability, disease progression often occurs via positive feedback that generates mechanobiological instabilities. In this review, we focus on the cell biology, wall mechanics, and regulatory pathways associated with arterial health and how changes in these processes lead to disease. We discuss how positive feedback loops arise via biomechanical and biochemical means. We conclude that inflammation plays a central role in overriding homeostatic pathways and suggest future directions for addressing therapeutic needs.

Spontaneous reversal of stenosis in tissue-engineered vascular grafts

We developed a tissue-engineered vascular graft (TEVG) for use in children and present results of a U.S. Food and Drug Administration (FDA)-approved clinical trial evaluating this graft in patients with single-ventricle cardiac anomalies. The TEVG was used as a Fontan conduit to connect the inferior vena cava and pulmonary artery, but a high incidence of graft narrowing manifested within the first 6 months, which was treated successfully with angioplasty. To elucidate mechanisms underlying this early stenosis, we used a data-informed, computational model to perform in silico parametric studies of TEVG development. The simulations predicted early stenosis as observed in our clinical trial but suggested further that such narrowing could reverse spontaneously through an inflammation-driven, mechano-mediated mechanism. We tested this unexpected, model-generated hypothesis by implanting TEVGs in an ovine inferior vena cava interposition graft model, which confirmed the prediction that TEVG stenosis resolved spontaneously and was typically well tolerated. These findings have important implications for our translational research because they suggest that angioplasty may be safely avoided in patients with asymptomatic early stenosis, although there will remain a need for appropriate medical monitoring. The simulations further predicted that the degree of reversible narrowing can be mitigated by altering the scaffold design to attenuate early inflammation and increase mechano-sensing by the synthetic cells, thus suggesting a new paradigm for optimizing next-generation TEVGs. We submit that there is considerable translational advantage to combined computational-experimental studies when designing cutting-edge technologies and their clinical management.

From Transcript to Tissue: Multiscale Modeling from Cell Signaling to Matrix Remodeling

Tissue-level biomechanical properties and function derive from underlying cell signaling, which regulates mass deposition, organization, and removal. Here, we couple two existing modeling frameworks to capture associated multiscale interactions-one for vessel-level growth and remodeling and one for cell-level signaling-and illustrate utility by simulating aortic remodeling. At the vessel level, we employ a constrained mixture model describing turnover of individual wall constituents (elastin, intramural cells, and collagen), which has proven useful in predicting diverse adaptations as well as disease progression using phenomenological constitutive relations. Nevertheless, we now seek an improved mechanistic understanding of these processes; we replace phenomenological relations in the mixture model with a logic-based signaling model, which yields a system of ordinary differential equations predicting changes in collagen synthesis, matrix metalloproteinases, and cell proliferation in response to altered intramural stress, wall shear stress, and exogenous angiotensin II. This coupled approach promises improved understanding of the role of cell signaling in achieving tissue homeostasis and allows us to model feedback between vessel mechanics and cell signaling. We verify our model predictions against data from the hypertensive murine infrarenal abdominal aorta as well as results from validated phenomenological models, and consider effects of noisy signaling and heterogeneous cell populations.

Numerical knockouts-In silico assessment of factors predisposing to thoracic aortic aneurysms

Myriad risk factors–including uncontrolled hypertension, aging, and diverse genetic mutations–contribute to the development and enlargement of thoracic aortic aneurysms. Detailed analyses of clinical data and longitudinal studies of murine models continue to provide insight into the natural history of these potentially lethal conditions. Yet, because of the co-existence of multiple risk factors in most cases, it has been difficult to isolate individual effects of the many different factors or to understand how they act in combination. In this paper, we use a data-informed computational model of the initiation and progression of thoracic aortic aneurysms to contrast key predisposing risk factors both in isolation and in combination; these factors include localized losses of elastic fiber integrity, aberrant collagen remodeling, reduced smooth muscle contractility, and dysfunctional mechanosensing or mechanoregulation of extracellular matrix along with superimposed hypertension and aortic aging. In most cases, mild-to-severe localized losses in cellular function or matrix integrity give rise to varying degrees of local dilatations of the thoracic aorta, with enlargement typically exacerbated in cases wherein predisposing risk factors co-exist. The simulations suggest, for the first time, that effects of compromised smooth muscle contractility are more important in terms of dysfunctional mechanosensing and mechanoregulation of matrix than in vessel-level control of diameter and, furthermore, that dysfunctional mechanobiological control can yield lesions comparable to those in cases of compromised elastic fiber integrity. Particularly concerning, therefore, is that loss of constituents such as fibrillin-1, as in Marfan syndrome, can compromise both elastic fiber integrity and mechanosensing.

Aneurysms are local dilatations of the arterial wall that are responsible for significant disability and death. Detailed analyses of clinical data continue to provide insight into the natural history of these potentially lethal conditions, with myriad risk factors–including uncontrolled hypertension, aging, and diverse genetic mutations–contributing to their development and enlargement. Yet, because of the co-existence of these risk factors in most cases, it has been difficult to isolate individual effects or to understand how they act in combination. In this paper, we use a computational model of the initiation and progression of thoracic aortic aneurysms to contrast key predisposing factors both in isolation and in combination as well as with superimposed hypertension and aging. The present study recovers many findings from mouse models but with new and important observations that promise to guide in vivo and ex vivo studies as we seek to understand and eventually better treat these complex, multi-factorial lesions, with data-informed patient-specific computations eventually the way forward.

Cell signaling model for arterial mechanobiology

Arterial growth and remodeling at the tissue level is driven by mechanobiological processes at cellular and sub-cellular levels. Although it is widely accepted that cells seek to promote tissue homeostasis in response to biochemical and biomechanical cues—such as increased wall stress in hypertension—the ways by which these cues translate into tissue maintenance, adaptation, or maladaptation are far from understood. In this paper, we present a logic-based computational model for cell signaling within the arterial wall, aiming to predict changes in extracellular matrix turnover and cell phenotype in response to pressure-induced wall stress, flow-induced wall shear stress, and exogenous sources of angiotensin II, with particular interest in mouse models of hypertension. We simulate a number of experiments from the literature at both the cell and tissue level, involving single or combined inputs, and achieve high qualitative agreement in most cases. Additionally, we demonstrate the utility of this modeling approach for simulating alterations (in this case knockdowns) of individual nodes within the signaling network. Continued modeling of cellular signaling will enable improved mechanistic understanding of arterial growth and remodeling in health and disease, and will be crucial when considering potential pharmacological interventions.

Biological soft tissues are characterized by continuous production and removal of material, which endows them with a remarkable ability to adapt to changes in their biochemical and biomechanical environments. For arteries, mechanical stimuli result primarily from changes in blood pressure or flow, and biochemical changes are induced by multiple factors, including pharmacological intervention. In order to understand how arterial properties are maintained in health, or how they adapt or fail to adapt in disease, we must understand better how these diverse stimuli affect material turnover. Extracellular matrix is tightly regulated by mechano-sensing and mechano-regulation, and therefore cell signaling, thus we present a computational model of relevant signaling pathways within the vascular wall, with the aim of predicting changes in wall composition and function in response to three main inputs: pressure-induced wall stress, flow-induced wall shear stress, and exogenous angiotensin II. We obtain qualitative agreement with a range of experimental studies from the literature, and provide illustrative examples demonstrating how such models can be used to further our understanding of arterial remodeling.

Corneal Stiffness and Collagen Cross-Linking Proteins in Glaucoma: Potential for Novel Therapeutic Strategy

Biomechanical properties of the cornea have recently emerged as clinically useful in risk assessment of diagnosing glaucoma and predicting disease progression. Corneal hysteresis (CH) is a dynamic tool, which measures viscoelasticity of the cornea. It represents the overall deformability of the cornea, and reduces significantly with age. Low CH has also been associated with optic nerve damage and progression of visual field loss in glaucoma. The extracellular matrix (ECM) constituents of the cornea, trabecular meshwork (TM), sclera, and lamina cribrosa (LC) are similar, as they are predominantly made of fibrillar collagen. This suggests that biomechanical changes in the cornea may also reflect optic nerve compliance in glaucomatous optic neuropathy, and in the known increase of TM tissue stiffness in glaucoma. Increased collagen cross-linking contributes to tissue stiffening throughout the body, which is observed in normal aging and occurs at an accelerated rate in systemic conditions such as fibrotic and cardiovascular diseases, cancer, and glaucoma. We reviewed 3 ECM cross-linking proteins that may have a potential role in the disease process of increased tissue stiffness in glaucoma, including lysyl oxidase (LOX)/lysyl oxidase-like 1 (LOXL1), tissue transglutaminase (TG2), and advanced glycation end products. We also report elevated messenger RNA (mRNA) levels of LOX and TG2 in glaucoma LC cells to support our proposed theory that increased levels of cross-linking proteins in glaucoma play a role in LC tissue stiffness. We highlight areas of research that are needed to better understand the role of cross-linking in glaucoma pathogenesis, leading potentially to a novel therapeutic strategy.

A Two-Dimensional Model of Hypertension-Induced Arterial Remodeling With Account for Stress Interaction Between Elastin and Collagen

We propose a novel structure-based two-dimensional (2D) mathematical model of hypertension-induced arterial remodeling. The model is built in the framework of the constrained mixture theory and global growth approach, utilizing a recently proposed structure-based constitutive model of arterial tissue that accounts for the individual natural configurations of and stress interaction between elastin and collagen. The basic novel predictive result is that provided remodeling causes a change in the elastin/collagen mass fraction ratio, it leads to a structural reorganization of collagen that manifests as an altered fiber undulation and a change in direction of the helically oriented fibers in the tissue natural state. Results obtained from the illustrative simulations for a porcine renal artery show that when remodeling is complete the collagen reorganization might have significant effects on the initial arterial geometry and mechanical properties of the arterial tissue. The proposed model has potential to describe and advance mechanistic understanding of adaptive arterial remodeling, promote the continual refinement of mathematical models of arterial remodeling, and provide motivation for new avenues of experimental investigation.

Computational modeling of cardiac growth and remodeling in pressure overloaded hearts-Linking microstructure to organ phenotype

Cardiac growth and remodeling (G&R) refers to structural changes in myocardial tissue in response to chronic alterations in loading conditions. One such condition is pressure overload where elevated wall stresses stimulate the growth in cardiomyocyte thickness, associated with a phenotype of concentric hypertrophy at the organ scale, and promote fibrosis. The initial hypertrophic response can be considered adaptive and beneficial by favoring myocyte survival, but over time if pressure overload conditions persist, maladaptive mechanisms favoring cell death and fibrosis start to dominate, ultimately mediating the transition towards an overt heart failure phenotype. The underlying mechanisms linking biological factors at the myocyte level to biomechanical factors at the systemic and organ level remain poorly understood. Computational models of G&R show high promise as a unique framework for providing a quantitative link between myocardial stresses and strains at the organ scale to biological regulatory processes at the cellular level which govern the hypertrophic response. However, microstructurally motivated, rigorously validated computational models of G&R are still in their infancy. This article provides an overview of the current state-of-the-art of computational models to study cardiac G&R. The microstructure and mechanosensing/mechanotransduction within cells of the myocardium is discussed and quantitative data from previous experimental and clinical studies is summarized. We conclude with a discussion of major challenges and possible directions of future research that can advance the current state of cardiac G&R computational modeling. STATEMENT OF SIGNIFICANCE: The mechanistic links between organ-scale biomechanics and biological factors at the cellular size scale remain poorly understood as these are largely elusive to investigations using experimental methodology alone. Computational G&R models show high promise to establish quantitative links which allow more mechanistic insight into adaptation mechanisms and may be used as a tool for stratifying the state and predict the progression of disease in the clinic. This review provides a comprehensive overview of research in this domain including a summary of experimental data. Thus, this study may serve as a basis for the further development of more advanced G&R models which are suitable for making clinical predictions on disease progression or for testing hypotheses on pathogenic mechanisms using in-silico models.

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.

Computational Modeling Predicts Immuno-Mechanical Mechanisms of Maladaptive Aortic Remodeling in Hypertension

Uncontrolled hypertension is a major risk factor for myriad cardiovascular diseases. Among its many effects, hypertension increases central artery stiffness which in turn is both an initiator and indicator of disease. Despite extensive clinical, animal, and basic science studies, the biochemomechanical mechanisms by which hypertension drives aortic stiffening remain unclear. In this paper, we show that a new computational model of aortic growth and remodeling can capture differential effects of induced hypertension on the thoracic and abdominal aorta in a common mouse model of disease. Because the simulations treat the aortic wall as a constrained mixture of different constituents having different material properties and rates of turnover, one can gain increased insight into underlying constituent-level mechanisms of aortic remodeling. Model results suggest that the aorta can mechano-adapt locally to blood pressure elevation in the absence of marked inflammation, but large increases in inflammation drive a persistent maladaptive phenotype characterized primarily by adventitial fibrosis. Moreover, this fibrosis appears to occur via a marked increase in the rate of deposition of collagen having different material properties in the absence of a compensatory increase in the rate of matrix degradation. Controlling inflammation thus appears to be key to reducing fibrosis, but therapeutic strategies should not compromise the proteolytic activity of the wall that is essential to mechanical homeostasis.

Mechanobiological Stability of Biological Soft Tissues

Like all other materials, biological soft tissues are subject to general laws of physics, including those governing mechanical equilibrium and stability. In addition, however, these tissues are able to respond actively to changes in their mechanical and chemical environment. There is, therefore, a pressing need to understand such processes theoretically. In this paper, we present a new rate-based constrained mixture formulation suitable for studying mechanobiological equilibrium and stability of soft tissues exposed to transient or sustained changes in material composition or applied loading. These concepts are illustrated for canonical problems in arterial mechanics, which distinguish possible stable versus unstable mechanobiological responses. Such analyses promise to yield insight into biological processes that govern both health and disease progression.

Coupled agent-based and hyperelastic modelling of the left ventricle post-myocardial infarction

Understanding the healing and remodelling processes induced by myocardial infarction (MI) of the heart is important, and the mechanical properties of the myocardium post-MI can be indicative for effective treatments aimed at avoiding eventual heart failure. MI remodelling is a multiscale feedback process between the mechanical loading and cellular adaptation. In this paper, we use an agent-based model to describe collagen remodelling by fibroblasts regulated by chemical and mechanical cues after acute MI, and upscale into a finite element 3D left ventricular model. We model the dispersed collagen fibre structure using the angular integration method and have incorporated a collagen fibre tension-compression switch in the left ventricle (LV) model. This enables us to study the scar healing (collagen deposition, degradation, and reorientation) of a rat heart post-MI. Our results, in terms of collagen accumulation and alignment, compare well with published experimental data. In addition, we show that different shapes of the MI region can affect the collagen remodelling, and in particular, the mechanical cue plays an important role in the healing process.

A Mechanobiologically Equilibrated Constrained Mixture Model for Growth and Remodeling of Soft Tissues

Growth and remodeling of soft tissues is a dynamic process and several theoretical frameworks have been developed to analyze the time-dependent, mechanobiological and/or biomechanical responses of these tissues to changes in external loads. Importantly, general processes can often be conveniently separated into truly non-steady contributions and steady-state ones. Depending on characteristic times over which the external loads are applied, time-dependent models can sometimes be specialized to respective time-independent formulations that simplify the mathematical treatment without compromising the goodness of the particularized solutions. Very few studies have analyzed the long-term, steady-state responses of soft tissue growth and remodeling following a direct approach. Here, we derive a mechanobiologically equilibrated formulation that arises from a general constrained mixture model. We see that integral-type evolution equations that characterize these general models can be written in terms of an equivalent set of time-independent, nonlinear algebraic equations that can be solved efficiently to yield long-term outcomes of growth and remodeling processes in response to sustained external stimuli. We discuss the mathematical conditions, in terms of orders of magnitude, that yield the particularized equations and illustrate results numerically for general arterial mechano-adaptations.

Modeling mechano-driven and immuno-mediated aortic maladaptation in hypertension

Uncontrolled hypertension is a primary risk factor for diverse cardiovascular diseases and thus remains responsible for significant morbidity and mortality. Hypertension leads to marked changes in the composition, structure, properties, and function of central arteries; hence, there has long been interest in quantifying the associated wall mechanics. Indeed, over the past 20 years there has been increasing interest in formulating mathematical models of the evolving geometry and biomechanical behavior of central arteries that occur during hypertension. In this paper, we introduce a new mathematical model of growth (changes in mass) and remodeling (changes in microstructure) of the aortic wall for an animal model of induced hypertension that exhibits both mechano-driven and immuno-mediated matrix turnover. In particular, we present a bilayered model of the aortic wall to account for differences in medial versus adventitial growth and remodeling and we include mechanical stress and inflammatory cell density as determinants of matrix turnover. Using this approach, we can capture results from a recent report of adventitial fibrosis that resulted in marked aortic maladaptation in hypertension. We submit that this model can also be used to identify novel hypotheses to guide future experimentation.

Critical roles of time-scales in soft tissue growth and remodeling

Most soft biological tissues exhibit a remarkable ability to adapt to sustained changes in mechanical loads. These macroscale adaptations, resulting from mechanobiological cellular responses, are important determinants of physiological behaviors and thus clinical outcomes. Given the complexity of such adaptations, computational models can significantly increase our understanding of how contributions of different cell types or matrix constituents, and their rates of turnover and evolving properties, ultimately change the geometry and biomechanical behavior at the tissue level. In this paper, we examine relative roles of the rates of tissue responses and external loading and present a new rate-independent approach for modeling the evolution of soft tissue growth and remodeling. For illustrative purposes, we also present numerical results for arterial adaptations. In particular, we show that, for problems defined by particular characteristic times, this approximate theory captures well the predictions of a fully general constrained mixture theory at a fraction of the computational cost.

Computer-Aided Evaluation of Blood Vessel Geometry From Acoustic Images

A method for computer-aided assessment of blood vessel geometries based on shape-fitting algorithms from metric vision was evaluated. Acoustic images of cross sections of the radial artery and cephalic vein were acquired, and medical practitioners used a computer application to measure the wall thickness and nominal diameter of these blood vessels with a caliper method and the shape-fitting method. The methods performed equally well for wall thickness measurements. The shape-fitting method was preferable for measuring the diameter, since it reduced systematic errors by up to 63% in the case of the cephalic vein because of its eccentricity.

Gradual loading ameliorates maladaptation in computational simulations of vein graft growth and remodelling

Vein graft failure is a prevalent problem in vascular surgeries, including bypass grafting and arteriovenous fistula procedures in which veins are subjected to severe changes in pressure and flow. Animal and clinical studies provide significant insight, but understanding the complex underlying coupled mechanisms can be advanced using computational models. Towards this end, we propose a new model of venous growth and remodelling (G&R) based on a constrained mixture theory. First, we identify constitutive relations and parameters that enable venous adaptations to moderate perturbations in haemodynamics. We then fix these relations and parameters, and subject the vein to a range of combined loads (pressure and flow), from moderate to severe, and identify plausible mechanisms of adaptation versus maladaptation. We also explore the beneficial effects of gradual increases in load on adaptation. A gradual change in flow over 3 days plus an initial step change in pressure results in fewer maladaptations compared with step changes in both flow and pressure, or even a gradual change in pressure and flow over 3 days. A gradual change in flow and pressure over 8 days also enabled a successful venous adaptation for loads as severe as the arterial loads. Optimization is used to accelerate parameter estimation and the proposed framework is general enough to provide a good starting point for parameter estimations in G&R simulations.

Computational Fluid Dynamics and Additive Manufacturing to Diagnose and Treat Cardiovascular Disease

Noninvasive engineering models are now being used for diagnosing and planning the treatment of cardiovascular disease. Techniques in computational modeling and additive manufacturing have matured concurrently, and results from simulations can inform and enable the design and optimization of therapeutic devices and treatment strategies. The emerging synergy between large-scale simulations and 3D printing is having a two-fold benefit: first, 3D printing can be used to validate the complex simulations, and second, the flow models can be used to improve treatment planning for cardiovascular disease. In this review, we summarize and discuss recent methods and findings for leveraging advances in both additive manufacturing and patient-specific computational modeling, with an emphasis on new directions in these fields and remaining open questions.

Mechanobiological model of arterial growth and remodeling

A coupled agent-based model (ABM) and finite element analysis (FEA) computational framework is developed to study the interplay of bio-chemo-mechanical factors in blood vessels and their role in maintaining homeostasis. The agent-based model implements the power of REPAST Simphony libraries and adapts its environment for biological simulations. Coupling a continuum-level model (FEA) to a cellular-level model (ABM) has enabled this computational framework to capture the response of blood vessels to increased or decreased levels of growth factors, proteases and other signaling molecules (on the micro scale) as well as altered blood pressure. Performance of the model is assessed by simulating porcine left anterior descending artery under normotensive conditions and transient increases in blood pressure and by analyzing sensitivity of the model to variations in the rule parameters of the ABM. These simulations proved that the model is stable under normotensive conditions and can recover from transient increases in blood pressure. Sensitivity studies revealed that the model is most sensitive to variations in the concentration of growth factors that affect cellular proliferation and regulate extracellular matrix composition (mainly collagen).

Continuous measurement of aortic dimensions in Turner syndrome: a cardiovascular magnetic resonance study

BACKGROUND

Severity of thoracic aortic disease in Turner syndrome (TS) patients is currently described through measures of aorta size and geometry at discrete locations. The objective of this study is to develop an improved measurement tool that quantifies changes in size and geometry over time, continuously along the length of the thoracic aorta.

METHODS

Cardiovascular magnetic resonance (CMR) scans for 15 TS patients [41 ± 9 years (mean age ± standard deviation (SD))] were acquired over a 10-year period and compared with ten healthy gender and age-matched controls. Three-dimensional aortic geometries were reconstructed, smoothed and clipped, which was followed by identification of centerlines and planes normal to the centerlines. Geometric variables, including maximum diameter and cross-sectional area, were evaluated continuously along the thoracic aorta. Distance maps were computed for TS and compared to the corresponding maps for controls, to highlight any asymmetry and dimensional differences between diseased and normal aortae. Furthermore, a registration scheme was proposed to estimate localized changes in aorta geometry between visits. The estimated maximum diameter from the continuous method was then compared with corresponding manual measurements at 7 discrete locations for each visit and for changes between visits.

RESULTS

Manual measures at the seven positions and the corresponding continuous measurements of maximum diameter for all visits considered, correlated highly (R-value = 0.77, P < 0.01). There was good agreement between manual and continuous measurement methods for visit-to-visit changes in maximum diameter. The continuous method was less sensitive to inter-user variability [0.2 ± 2.3 mm (mean difference in diameters ± SD)] and choice of smoothing software [0.3 ± 1.3 mm]. Aortic diameters were larger in TS than controls in the ascending [TS: 13.4 ± 2.1 mm (mean distance ± SD), Controls: 12.6 ± 1 mm] and descending [TS: 10.2 ± 1.3 mm (mean distance ± SD), Controls: 9.5 ± 0.9 mm] thoracic aorta as observed from the distance maps.

CONCLUSIONS

An automated methodology is presented that enables rapid and precise three-dimensional measurement of thoracic aortic geometry, which can serve as an improved tool to define disease severity and monitor disease progression.

TRIAL REGISTRATION

ClinicalTrials.gov Identifier - NCT01678274 . Registered - 08.30.2012.

Stability analysis of a continuum-based constrained mixture model for vascular growth and remodeling

A stabilizing criterion is derived for equations governing vascular growth and remodeling. We start from the integral state equations of the continuum-based constrained mixture theory of vascular growth and remodeling and obtain a system of time-delayed differential equations describing vascular growth. By employing an exponential form of the constituent survival function, the delayed differential equations can be reduced to a nonlinear ODE system. We demonstrate the degeneracy of the linearized system about the homeostatic state, which is a fundamental cause of the neutral stability observations reported in prior studies. Due to this degeneracy, stability conclusions for the original nonlinear system cannot be directly inferred. To resolve this problem, a sub-system is constructed by recognizing a linear relation between two states. Subsequently, Lyapunov's indirect method is used to connect stability properties between the linearized system and the original nonlinear system, to rigorously establish the neutral stability properties of the original system. In particular, this analysis leads to a stability criterion for vascular expansion in terms of growth and remodeling kinetic parameters, geometric quantities and material properties. Numerical simulations were conducted to evaluate the theoretical stability criterion under broader conditions, as well as study the influence of key parameters and physical factors on growth properties. The theoretical results are also compared with prior numerical and experimental findings in the literature.

A Finite Element Model for Mixed Porohyperelasticity with Transport, Swelling, and Growth

The purpose of this manuscript is to establish a unified theory of porohyperelasticity with transport and growth and to demonstrate the capability of this theory using a finite element model developed in MATLAB. We combine the theories of volumetric growth and mixed porohyperelasticity with transport and swelling (MPHETS) to derive a new method that models growth of biological soft tissues. The conservation equations and constitutive equations are developed for both solid-only growth and solid/fluid growth. An axisymmetric finite element framework is introduced for the new theory of growing MPHETS (GMPHETS). To illustrate the capabilities of this model, several example finite element test problems are considered using model geometry and material parameters based on experimental data from a porcine coronary artery. Multiple growth laws are considered, including time-driven, concentration-driven, and stress-driven growth. Time-driven growth is compared against an exact analytical solution to validate the model. For concentration-dependent growth, changing the diffusivity (representing a change in drug) fundamentally changes growth behavior. We further demonstrate that for stress-dependent, solid-only growth of an artery, growth of an MPHETS model results in a more uniform hoop stress than growth in a hyperelastic model for the same amount of growth time using the same growth law. This may have implications in the context of developing residual stresses in soft tissues under intraluminal pressure. To our knowledge, this manuscript provides the first full description of an MPHETS model with growth. The developed computational framework can be used in concert with novel in-vitro and in-vivo experimental approaches to identify the governing growth laws for various soft tissues.

On the effect of preload and pre-stretch on hemodynamic simulations: an integrative approach

In this work, we address the simulation of three-dimensional arterial blood flow and its effect on the stress state of arterial walls. The novel contribution is the unprecedented combination of several modeling techniques to account for (1) the fact that known configurations for the arterial wall are in a preloaded state, (2) the compliance of the vessel segments, (3) proper boundary data over the non-physical interfaces resulting from the isolation of an arterial district from the rest of the arterial tree, (4) the presence of surrounding tissues in which the vessel is embedded and (5) residual stress state due to pre-stretch. Firstly, we formulate both the forward mechanical problem when the reference (zero-load) configuration is assumed to be known and, the preload problem arising when the known domain is a configuration at equilibrium with a certain load state (typically due to internal pressure and tethering forces). Then, two additional complexities are faced: the fluid-structure interaction problem that follows when the compliant vessels are coupled with the blood flow, and the introduction of non-physical boundaries coming from the artificial isolation of the arterial district from the original vessel. This, in turn, posses the problem of coupling dimensionally heterogeneous models to incorporate the effect of upstream and downstream systemic impedances. Additionally, a viscoelastic support on the external surface of the vessel is also incorporated. Two examples are presented to quantify in a physiologically consistent scenario the differences in simulation results when either considering or not the preload state of arterial walls. These computational simulations shed light on the validity of simplifying hypotheses in most hemodynamic models.

Coupled Simulation of Hemodynamics and Vascular Growth and Remodeling in a Subject-Specific Geometry

A computational framework to couple vascular growth and remodeling (G&R) with blood flow simulation in a 3D patient-specific geometry is presented. Hyperelastic and anisotropic properties are considered for the vessel wall material and a constrained mixture model is used to represent multiple constituents in the vessel wall, which was modeled as a membrane. The coupled simulation is divided into two time scales-a longer time scale for G&R and a shorter time scale for fluid dynamics simulation. G&R is simulated to evolve the boundary of the fluid domain, and fluid simulation is in turn used to generate wall shear stress and transmural pressure data that regulates G&R. To minimize required computation cost, the fluid dynamics are only simulated when G&R causes significant vascular geometric change. For demonstration, this coupled model was used to study the influence of stress-mediated growth parameters, and blood flow mechanics, on the behavior of the vascular tissue growth in a model of the infrarenal aorta derived from medical image data.

Biomechanical diversity despite mechanobiological stability in tissue engineered vascular grafts two years post-implantation

Recent advances in vascular tissue engineering have enabled a paradigm shift from ensuring short-term graft survival to focusing on long-term stability and growth potential. We present the first experimental-computational study of a tissue-engineered vascular graft (TEVG) effectively over the full lifespan of the recipient. We show that grafts implanted within the venous circulation of mice remained patent over 2 years without thrombus, stenosis, or aneurysmal dilatation. Moreover, the gross appearance and mechanical properties of the grafts evolved to be similar to the host vein within 24 weeks, with mean neovessel geometry and properties remaining unchanged thereafter despite a continued turnover of extracellular matrix. Biomechanical diversity manifested after 24 weeks, however, via two subsets of grafts despite all procedures being the same. Computational modeling and associated immunohistological analyses suggested that this diversity likely resulted from a differential ratio of collagen types I and III, with lower I to III ratios promoting grafts having a compliance similar to the native vein. We submit that TEVGs can exhibit the desired long-term mechanobiological stability; hence, we must now focus on evaluating growth potential and optimizing scaffold properties to achieve compliance matching throughout neovessel development.

A hypothesis-driven parametric study of effects of polymeric scaffold properties on tissue engineered neovessel formation

Continued advances in the tissue engineering of vascular grafts have enabled a paradigm shift from the desire to design for adequate suture retention, burst pressure and thrombo-resistance to the goal of achieving grafts having near native properties, including growth potential. Achieving this far more ambitious outcome will require the identification of optimal, not just adequate, scaffold structure and material properties. Given the myriad possible combinations of scaffold parameters, there is a need for a new strategy for reducing the experimental search space. Toward this end, we present a new modeling framework for in vivo neovessel development that allows one to begin to assess in silico the potential consequences of different combinations of scaffold structure and material properties. To restrict the number of parameters considered, we also utilize a non-dimensionalization to identify key properties of interest. Using illustrative constitutive relations for both the evolving fibrous scaffold and the neotissue that develops in response to inflammatory and mechanobiological cues, we show that this combined non-dimensionalization computational approach predicts salient aspects of neotissue development that depend directly on two key scaffold parameters, porosity and fiber diameter. We suggest, therefore, that hypothesis-driven computational models should continue to be pursued given their potential to identify preferred combinations of scaffold parameters that have the promise of improving neovessel outcome. In this way, we can begin to move beyond a purely empirical trial-and-error search for optimal combinations of parameters and instead focus our experimental resources on those combinations that are predicted to have the most promise.

Mechanotransduction and extracellular matrix homeostasis

Soft connective tissues at steady state are dynamic; resident cells continually read environmental cues and respond to them to promote homeostasis, including maintenance of the mechanical properties of the extracellular matrix (ECM) that are fundamental to cellular and tissue health. The mechanosensing process involves assessment of the mechanics of the ECM by the cells through integrins and the actomyosin cytoskeleton, and is followed by a mechanoregulation process, which includes the deposition, rearrangement or removal of the ECM to maintain overall form and function. Progress towards understanding the molecular, cellular and tissue-level effects that promote mechanical homeostasis has helped to identify key questions for future research.

Extension of Murray's law including nonlinear mechanics of a composite artery wall

A goal function approach is used to derive an extension of Murray’s law that includes effects of nonlinear mechanics of the artery wall. The artery is modeled as a thin-walled tube composed of different species of nonlinear elastic materials that deform together. These materials grow and remodel in a process that is governed by a target state defined by a homeostatic radius and a homeostatic material composition. Following Murray’s original idea, this target state is defined by a principle of minimum work. We take this work to include that of pumping and maintaining blood, as well as maintaining the materials of the artery wall. The minimization is performed under a constraint imposed by mechanical equilibrium. We derive a condition for the existence of a cost-optimal homeostatic state. We also conduct parametric studies using this novel theoretical frame to investigate how the cost-optimal radius and composition of the artery wall depend on flow rate, blood pressure, and elastin content

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.

A goal function approach to remodeling of arteries uncovers mechanisms for growth instability

A novel, goal function-based formulation for the growth dynamics of arteries is introduced and used for investigating the development of growth instability in blood vessels. Such instabilities would lead to abnormal growth of the vessel, reminiscent of an aneurysm. The blood vessel is modeled as a thin-walled cylindrical tube, and the constituents that form the vessel wall are assumed to deform together as a constrained mixture. The growth dynamics of the composite material of the vessel wall are described by an evolution equation, where the effective area of each constituent changes in the direction of steepest descent of a goal function. This goal function is formulated in such way that the constituents grow toward a target potential energy and a target composition. The convergence of the simulated response of the evolution equation toward a target homeostatic state is investigated for a range of isotropic and orthotropic material models. These simulations suggest that elastin-deficient vessels are more prone to growth instability. Increased stiffness of the vessel wall, on the other hand, gives a more stable growth process. Another important finding is that an increased rate of degradation of materials impairs growth stability.

Computational model of the in vivo development of a tissue engineered vein from an implanted polymeric construct

Advances in vascular tissue engineering have been tremendous over the past 15 years, yet there remains a need to optimize current constructs to achieve vessels having true growth potential. Toward this end, it has been suggested that computational models may help hasten this process by enabling time-efficient parametric studies that can reduce the experimental search space. In this paper, we present a first generation computational model for describing the in vivo development of a tissue engineered vein from an implanted polymeric scaffold. The model was motivated by our recent data on the evolution of mechanical properties and microstructural composition over 24 weeks in a mouse inferior vena cava interposition graft. It is shown that these data can be captured well by including both an early inflammatory-mediated and a subsequent mechano-mediated production of extracellular matrix. There remains a pressing need, however, for more data to inform the development of next generation models, particularly the precise transition from the inflammatory to the mechanobiological dominated production of matrix having functional capability.

A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: theory and numerical verification

We implemented a constrained mixture model of arterial growth and remodeling in a nonlinear finite element framework to facilitate numerical analyses of diverse cases of arterial adaptation and maladaptation, including disease progression, resulting in complex evolving geometries and compositions. This model enables hypothesis testing by predicting consequences of postulated characteristics of cell and matrix turnover, including evolving quantities and orientations of fibrillar constituents and nonhomogenous degradation of elastin or loss of smooth muscle function. The nonlinear finite element formulation is general within the context of arterial mechanics, but we restricted our present numerical verification to cylindrical geometries to allow comparisons with prior results for two special cases: uniform transmural changes in mass and differential growth and remodeling within a two-layered cylindrical model of the human aorta. The present finite element model recovers the results of these simplified semi-inverse analyses with good agreement.

Multiscale computational models of complex biological systems

Integration of data across spatial, temporal, and functional scales is a primary focus of biomedical engineering efforts. The advent of powerful computing platforms, coupled with quantitative data from high-throughput experimental methodologies, has allowed multiscale modeling to expand as a means to more comprehensively investigate biological phenomena in experimentally relevant ways. This review aims to highlight recently published multiscale models of biological systems, using their successes to propose the best practices for future model development. We demonstrate that coupling continuous and discrete systems best captures biological information across spatial scales by selecting modeling techniques that are suited to the task. Further, we suggest how to leverage these multiscale models to gain insight into biological systems using quantitative biomedical engineering methods to analyze data in nonintuitive ways. These topics are discussed with a focus on the future of the field, current challenges encountered, and opportunities yet to be realized.

An efficient framework for optimization and parameter sensitivity analysis in arterial growth and remodeling computations

Computational models for vascular growth and remodeling (G&R) are used to predict the long-term response of vessels to changes in pressure, flow, and other mechanical loading conditions. Accurate predictions of these responses are essential for understanding numerous disease processes. Such models require reliable inputs of numerous parameters, including material properties and growth rates, which are often experimentally derived, and inherently uncertain. While earlier methods have used a brute force approach, systematic uncertainty quantification in G&R models promises to provide much better information. In this work, we introduce an efficient framework for uncertainty quantification and optimal parameter selection, and illustrate it via several examples. First, an adaptive sparse grid stochastic collocation scheme is implemented in an established G&R solver to quantify parameter sensitivities, and near-linear scaling with the number of parameters is demonstrated. This non-intrusive and parallelizable algorithm is compared with standard sampling algorithms such as Monte-Carlo. Second, we determine optimal arterial wall material properties by applying robust optimization. We couple the G&R simulator with an adaptive sparse grid collocation approach and a derivative-free optimization algorithm. We show that an artery can achieve optimal homeostatic conditions over a range of alterations in pressure and flow; robustness of the solution is enforced by including uncertainty in loading conditions in the objective function. We then show that homeostatic intramural and wall shear stress is maintained for a wide range of material properties, though the time it takes to achieve this state varies. We also show that the intramural stress is robust and lies within 5% of its mean value for realistic variability of the material parameters. We observe that prestretch of elastin and collagen are most critical to maintaining homeostasis, while values of the material properties are most critical in determining response time. Finally, we outline several challenges to the G&R community for future work. We suggest that these tools provide the first systematic and efficient framework to quantify uncertainties and optimally identify G&R model parameters.

Strain-induced tissue growth laws: applications to embryonic cardiovascular development

Hemodynamic conditions play an essential role in the cardiovascular system, with abnormal blood flow conditions leading to growth and remodeling of cardiovascular walls. During embryonic development, altered hemodynamic conditions lead to congenital heart disease, which affects about 1% of newborn babies in developed countries. However, the mechanisms by which hemodynamic conditions affect cardiovascular development have not been fully elucidated. In this paper, we propose a model of cardiac growth in response to hemodynamic conditions, in which growth is modulated by a combination of wall strains and wall shear stresses. This is in contrast to previous models that proposed stress-induced growth laws. Because during embryonic development blood pressure increases over time, and this increase in blood pressure produces an increase in wall stresses, stress-induced growth laws would require time-dependent parameters. While blood pressure increases during development, cardiovascular walls become stiffer and thicker, and thus we postulate that instead strains experienced by cells remain approximately the same during development. This assumption motivated our cardioavascular model of strain-induced growth in response to hemodynamic conditions, which we implemented using finite element methods. Model simulations show that the proposed model results in tissue growth that is physiologically reasonable. Further, our analyses demonstrate that mechanical coupling - that results from residual stresses originating from differential tissue growth - may play a more important role in the modulation of cardiovascular tissue growth and remodeling than currently acknowledged.

Parametric study of effects of collagen turnover on the natural history of abdominal aortic aneurysms

Abdominal aortic aneurysms (AAAs) are characterized by significant changes in the architecture of the aortic wall, notably, loss of functional elastin and smooth muscle. Because collagen is the principal remaining load-bearing constituent of the aneurysmal wall, its turnover must play a fundamental role in the natural history of the lesion. Nevertheless, detailed investigations of the effects of different aspects of collagen turnover on AAA development are lacking. A finite-element membrane model of the growth and remodelling of idealized AAAs was thus used to investigate parametrically four of the primary aspects of collagen turnover: rates of production, half-life, deposition stretch (prestretch) and material stiffness. The predicted rates of aneurysmal expansion and spatio-temporal changes in wall thickness, biaxial stresses and maximum collagen fibre stretch at the apex of the lesion depended strongly on all four factors, as did the predicted clinical endpoints (i.e. arrest, progressive expansion or rupture). Collagen turnover also affected the axial expansion, largely due to mechanical changes within the shoulder region of the lesion. We submit, therefore, that assessment of rupture risk could be improved by future experiments that delineate and quantify different aspects of patient-specific collagen turnover and that such understanding could lead to new targeted therapeutics.

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.