Parameter sensitivity study of a constrained mixture model of arterial growth and remodeling

Growth and remodeling of the dissected membrane in an idealized dissected aorta model

While transitioning from the acute to chronic phase, the wall of a dissected aorta often expands in diameter and adaptations in thickness and microstructure take place in the dissected membrane. Including the mechanisms, leading to these changes, in a computational model is expected to improve the accuracy of predictions of the long-term complications and optimal treatment timing of dissection patients. An idealized dissected wall was modeled to represent the elastin and collagen production and/or degradation imposed by stress- and inflammation-mediated growth and remodeling, using the homogenized constrained mixture theory. As no optimal growth and remodeling parameters have been defined for aortic dissections, a Latin hypercube sampling with 1000 parameter combinations was assessed for four inflammation patterns, with a varying spatial extent (full/local) and temporal evolution (permanent/transient). The dissected membrane thickening and microstructure was considered together with the diameter expansion over a period of 90 days. The highest success rate was found for the transient inflammation patterns, with about 15% of the samples leading to converged solutions after 90 days. Clinically observed thickening rates were found for 2-4% of the transient inflammation samples, which represented median total diameter expansion rates of about 5 mm/year. The dissected membrane microstructure showed an elastin decrease and, in most cases, a collagen increase. In conclusion, the model with the transient inflammation pattern allowed the reproduction of clinically observed dissected membrane thickening rates, diameter expansion rates and adaptations in microstructure, thus providing guidance in reducing the parameter space in growth and remodeling models of aortic dissections.

Persistent non-homeostatic remodeling of aortic collagen following a brief episode of hypertension: A computational study

The healthy adult aorta exhibits a remarkable homeostatic ability to respond to sustained changes in hemodynamic loads under many circumstances, but this mechanical homeostasis can be compromised or lost in natural aging and diverse pathological processes. Herein, we investigate persistent non-homeostatic changes in the composition and mechanical properties of the thoracic aorta in adult wild-type mice following 14 days of angiotensin II-induced hypertension. We employ a multiscale computational model of arterial growth and remodeling driven by mechanosensitive and angiotensin II-related cell signaling pathways. We find that experimentally observed findings can only be recapitulated computationally if the collagen deposited during the transient period of hypertension has altered properties (deposition stretch, fiber angle, crosslinking) compared with the collagen produced in the original homeostatic state. Some of these changes are predicted to persist for at least six months after blood pressure is restored to normal levels, consistent with the experimental findings.

The risk of rupture and abdominal aortic aneurysm morphology: A computational study

Prediction of rupture and optimal timing for abdominal aortic aneurysm (AAA) surgical intervention remain wanting even after decades of clinical, histological, and numerical research. Although studies estimating rupture from AAA geometrical features from CT imaging showed some promising results, they are still not being used in practice. Patient-specific numerical stress analysis introduced too many assumptions about wall structure for the related rupture potential index (RPI) to be considered reliable. Growth and remodeling (G&R) numerical models eliminate some of these assumptions and thus might have the most potential to calculate mural stresses and RPI and increase our understanding of rupture. To recognize numerical models as trustworthy, it is necessary to validate the computed results with results derived from imaging. Elastin degradation function is one of the main factors that determine idealized aneurysm sac shape. Using a hundred different combinations of variables defining AAA geometry or influences AAA stability (elastin degradation function parameters, collagen mechanics, and initial healthy aortic diameters), we investigated the relationship between AAA morphology and RPI and compared numerical results with clinical findings. Good agreement of numerical results with clinical expectations from the literature gives us confidence in the validity of the numerical model. We show that aneurysm morphology significantly influences the stability of aneurysms. Additionally, we propose new parameters, geometrical rupture potential index (GRPI) and normalized aneurysm length (NAL), that might predict rupture of aneurysms without thrombus better than currently used criteria (i.e., maximum diameter and growth rate). These parameters can be computed quickly, without the tedious processing of CT images.

Growth and remodeling in the pulmonary autograft: Computational evaluation using kinematic growth models and constrained mixture theory

Computational investigations of how soft tissues grow and remodel are gaining more and more interest and several growth and remodeling theories have been developed. Roughly, two main groups of theories for soft tissues can be distinguished: kinematic-based growth theory and theories based on constrained mixture theory. Our goal was to apply these two theories on the same experimental data. Within the experiment, a pulmonary artery was exposed to systemic conditions. The change in diameter was followed-up over time. A mechanical and microstructural analysis of native pulmonary artery and pulmonary autograft was conducted. Whereas the kinematic-based growth theory is able to accurately capture the growth of the tissue, it does not account for the mechanobiological processes causing this growth. The constrained mixture theory takes into account the mechanobiological processes including removal, deposition and adaptation of all structural constituents, allowing us to simulate a changing microstructure and mechanical behavior.

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.

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.

Differential cell-matrix mechanoadaptations and inflammation drive regional propensities to aortic fibrosis, aneurysm or dissection in hypertension

The embryonic lineage of intramural cells, microstructural organization of the extracellular matrix, local luminal and wall geometry, and haemodynamic loads vary along the length of the aorta. Yet, it remains unclear why certain diseases manifest differentially along the aorta. Toward this end, myriad animal models provide insight into diverse disease conditions-including fibrosis, aneurysm and dissection-but inherent differences across models impede general interpretations. We examined region-specific cellular, matrix, and biomechanical changes in a single experimental model of hypertension and atherosclerosis, which commonly coexist. Our findings suggest that (i) intramural cells within the ascending aorta are unable to maintain the intrinsic material stiffness of the wall, which ultimately drives aneurysmal dilatation, (ii) a mechanical stress-initiated, inflammation-driven remodelling within the descending aorta results in excessive fibrosis, and (iii) a transient loss of adventitial collagen within the suprarenal aorta contributes to dissection propensity. Smooth muscle contractility helps to control wall stress in the infrarenal aorta, which maintains mechanical properties near homeostatic levels despite elevated blood pressure. This early mechanoadaptation of the infrarenal aorta does not preclude subsequent acceleration of neointimal formation, however. Because region-specific conditions may be interdependent, as, for example, diffuse central arterial stiffening can increase cyclic haemodynamic loads on an aneurysm that is developing proximally, there is a clear need for more systematic assessments of aortic disease progression, not simply a singular focus on a particular region or condition.

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.

Evaluation of microstructurally motivated constitutive models to describe age-dependent tendon healing

Tendon injuries are common to all ages. Injured tendons typically do not recover full functionality. The amount and organization of tendon constituents dictate their mechanical properties. The impact of changes in these constituents during (patho)physiologic processes (e.g., aging and healing) are not fully understood. Toward this end, microstructurally motivated strain energy functions (SEFs) offer insight into underlying mechanisms of age-dependent healing. Several SEFs have been adapted for tendon; however, most are phenomenological. Therefore, the aims of this study are: (1) evaluate the descriptive capability of SEFs in age-dependent murine patellar tendon healing and (2) identify a SEF for implementation in a growth and remodeling (G&R) model. To accomplish these aims, models were fitted to patellar tendon tensile data from multiple age groups and post-injury timepoints. Model sensitivity to parameters and the determinability of the parameters were assessed. A two-way analysis of variance was used to identify changes in parameters and the feasibility of implementing each model into a G&R model is discussed. The evaluated SEFs exhibited adequate descriptive capability. Parameter determinability and sensitivity analysis, however, highlighted the need for additional data to inform and validate the models to increase physiologic relevance and enable G&R model formulation to determine underlying mechanisms of age-dependent healing. This work, as a first, evaluated changes in tendon mechanical properties both as functions of age and injury in an age-dependent manner using microstructurally motivated models, highlights inherent dependencies between parameters of widely used hyperelastic models, and identified unique post-injury behavior by the aging group compared to the mature and aged groups.

Potential biomechanical roles of risk factors in the evolution of thrombus-laden abdominal aortic aneurysms

Abdominal aortic aneurysms (AAAs) typically harbour an intraluminal thrombus (ILT), yet most prior computational models neglect biochemomechanical effects of thrombus on lesion evolution. We recently proposed a growth and remodelling model of thrombus-laden AAAs that introduced a number of new constitutive relations and associated model parameters. Because values of several of these parameters have yet to be elucidated by clinical data and could vary significantly from patient to patient, the aim of this study was to investigate the possible extent to which these parameters influence AAA evolution. Given that some of these parameters model potential effects of factors that influence the risk of rupture, this study also provides insight into possible roles of common risk factors on the natural history of AAAs. Despite geometrical limitations of a cylindrical domain, findings support current thought that smoking, hypertension, and female sex likely increase the risk of rupture. Although thrombus thickness is not a reliable risk factor for rupture, the model suggests that the presence of ILT may have a destabilizing effect on AAA evolution, consistent with histological findings from human samples. Finally, simulations support two hypotheses that should be tested on patient-specific geometries in the future. First, ILT is a potential source of the staccato enlargement observed in many AAAs. Second, ILT can influence rupture risk, positively or negatively, via competing biomechanical (eg, stress shielding) and biochemical (ie, proteolytic) effects. Although further computational and experimental studies are needed, the present findings highlight the importance of considering ILT when predicting aneurysmal enlargement and rupture risk.

A Computational Model of Biochemomechanical Effects of Intraluminal Thrombus on the Enlargement of Abdominal Aortic Aneurysms

Abdominal aortic aneurysms (AAAs) typically develop an intraluminal thrombus (ILT), yet most computational models of AAAs have focused on either the mechanics of the wall or the hemodynamics within the lesion, both in the absence of ILT. In the few cases wherein ILT has been modeled directly, as, for example, in static models that focus on the state of stress in the aortic wall and the associated rupture risk, thrombus has been modeled as an inert, homogeneous, load-bearing material. Given the biochemomechanical complexity of an ILT, there is a pressing need to consider its diverse effects on the evolving aneurysmal wall. Herein, we present the first growth and remodeling model that addresses together the biomechanics, mechanobiology, and biochemistry of thrombus-laden AAAs. Whereas it has been shown that aneurysmal enlargement in the absence of ILT depends primarily on the stiffness and turnover of fibrillar collagen, we show that the presence of a thrombus within lesions having otherwise the same initial wall composition and properties can lead to either arrest or rupture depending on the biochemical effects (e.g., release of proteases) and biomechanical properties (e.g., stiffness of fibrin) of the ILT. These computational results suggest that ILT should be accounted for when predicting the potential enlargement or rupture risk of AAAs and highlight specific needs for further experimental and computational research.

A novel photochemical cross-linking technology to improve luminal gain, vessel compliance, and buckling post-angioplasty in porcine arteries

Development of substituted 1,8-naphthalimides for photochemical cross-linking of biomolecules is the focus of this research. This study describes limited cross-linking of collagen in the artery wall to control recoil and buckling in arteries following balloon angioplasty. Isolated porcine arteries were overstretched (25%) with balloon angioplasty (BA) +/- light-activated naphthalimide treatment (NVS). Lumen size and recoil were measured as retention of stretch after angioplasty. Cross-sectional compliance and distensibility coefficients were measured as slope of cross-sectional area versus increasing hydrostatic pressure. Buckling was measured, with 30% axial pre-stretch and 200 mmHg, as deviation from the center line. Electron microscopy evaluation of collagen fibers was conducted.

RESULTS

Uninjured arteries have low compliance and low levels of buckling, whereas the BA-injured arteries demonstrated much greater compliance and buckling behavior. Treatment of the injured artery with NVS reduced buckling and demonstrated compliance midway between the two groups while retaining the increased luminal diameter imparted by angioplasty compared to untreated vessels. In summary, limited collagen cross-linking with NVS treatment resulted in lumen retention, as well as improved compliance without the accompanying rigidity and stiffness of conventional stent therapy or current cross-linking materials. This treatment shows great promise for dilation, repair and strengthening of arteries damaged by injury or vascular disease.

Contribution of collagen fiber undulation to regional biomechanical properties along porcine thoracic aorta

As major extracellular matrix components, elastin, and collagen play crucial roles in regulating the mechanical properties of the aortic wall and, thus, the normal cardiovascular function. The mechanical properties of aorta, known to vary with age and multitude of diseases as well as the proximity to the heart, have been attributed to the variations in the content and architecture of wall constituents. This study is focused on the role of layer-specific collagen undulation in the variation of mechanical properties along the porcine descending thoracic aorta. Planar biaxial tensile tests are performed to characterize the hyperelastic anisotropic mechanical behavior of tissues dissected from four locations along the thoracic aorta. Multiphoton microscopy is used to image the associated regional microstructure. Exponential-based and recruitment-based constitutive models are used to account for the observed mechanical behavior while considering the aortic wall as a composite of two layers with independent properties. An elevated stiffness is observed in distal regions compared to proximal regions of thoracic aorta, consistent with sharper and earlier collagen recruitment estimated for medial and adventitial layers in the models. Multiphoton images further support our prediction that higher stiffness in distal regions is associated with less undulation in collagen fibers. Recruitment-based models further reveal that regardless of the location, collagen in the media is recruited from the onset of stretching, whereas adventitial collagen starts to engage with a delay. A parameter sensitivity analysis is performed to discriminate between the models in terms of the confidence in the estimated model parameters.

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.

Computational model of matrix remodeling and entrenchment in the free-floating fibroblast-populated collagen lattice

Tissue equivalents represent excellent model systems for elucidating principles of mechanobiology and for exploring methods to improve the functionality of tissue-engineered constructs. The simplest tissue equivalent is the free-floating fibroblast-populated collagen lattice. Although introduced over 30 years ago, the associated mechanics of the cell-mediated compaction of this lattice was only recently analyzed in detail. The goal of this paper was to build on this recent stress analysis by developing a computational model of the evolving geometry, regionally varying material properties and cell stresses, and overall residual stress fields during the first two days of compaction. Baseline results were found to agree well with most experimental observations, namely evolving changes in radius, thickness, and material symmetry, yet hypothesis testing revealed aspects of the mechanobiology that require more experimental attention. Given the generality of the proposed framework, we submit that modifications and refinements can be used to study many similar systems and thereby help guide future experiments.

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.

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.

Extracellular matrix and the mechanics of large artery development

The large, elastic arteries, as their name suggests, provide elastic distention and recoil during the cardiac cycle in vertebrate animals. The arteries are distended from the pressure of ejecting blood during the active contraction of the left ventricle (LV) during systole and recoil to their original dimensions during relaxation of the LV during diastole. The cyclic distension occurs with minimal energy loss, due to the elastic properties of one of the major structural extracellular matrix (ECM) components, elastin. The maximum distension is limited to prevent damage to the artery by another major ECM component, collagen. The mix of ECM components in the wall largely determines the passive mechanical behavior of the arteries and the subsequent load on the heart during systole. While much research has focused on initial artery formation, there has been less attention on the continuing development of the artery to produce the mature composite wall complete with endothelial cells (ECs), smooth muscle cells (SMCs), and the necessary mix of ECM components for proper cardiovascular function. This review focuses on the physiology of large artery development, including SMC differentiation and ECM production. The effects of hemodynamic forces and ECM deposition on the evolving arterial structure and function are discussed. Human diseases and mouse models with genetic mutations in ECM proteins that affect large artery development are summarized. A review of constitutive models and growth and remodeling theories is presented, along with future directions to improve understanding of ECM and the mechanics of large artery development.

On integrating experimental and theoretical models to determine physical mechanisms of morphogenesis

Researchers in developmental biology are increasingly recognizing the value of theoretical models in studies of morphogenesis. However, creating and testing realistic quantitative models for morphogenetic processes can be an extremely challenging task. The focus of this paper is on models for the mechanics of morphogenesis. Models for these problems often must include large changes in geometry, leading to highly nonlinear problems with the possibility of multiple solutions that must be sorted out using experimental data. Here, we illustrate our approach to these problems using the specific example of head fold formation in the early chick embryo. The interplay between experimental and theoretical results is emphasized throughout, as the model is gradually refined. Some of the limitations inherent in theoretical/computational modeling of biological systems are also discussed.

Ensuring congruency in multiscale modeling: towards linking agent based and continuum biomechanical models of arterial adaptation

There is a need to develop multiscale models of vascular adaptations to understand tissue-level manifestations of cellular level mechanisms. Continuum-based biomechanical models are well suited for relating blood pressures and flows to stress-mediated changes in geometry and properties, but less so for describing underlying mechanobiological processes. Discrete stochastic agent-based models are well suited for representing biological processes at a cellular level, but not for describing tissue-level mechanical changes. We present here a conceptually new approach to facilitate the coupling of continuum and agent-based models. Because of ubiquitous limitations in both the tissue- and cell-level data from which one derives constitutive relations for continuum models and rule-sets for agent-based models, we suggest that model verification should enforce congruency across scales. That is, multiscale model parameters initially determined from data sets representing different scales should be refined, when possible, to ensure that common outputs are consistent. Potential advantages of this approach are illustrated by comparing simulated aortic responses to a sustained increase in blood pressure predicted by continuum and agent-based models both before and after instituting a genetic algorithm to refine 16 objectively bounded model parameters. We show that congruency-based parameter refinement not only yielded increased consistency across scales, it also yielded predictions that are closer to in vivo observations.

Toward a multi-scale computational model of arterial adaptation in hypertension: verification of a multi-cell agent based model

Agent-based models (ABMs) represent a novel approach to study and simulate complex mechano chemo-biological responses at the cellular level. Such models have been used to simulate a variety of emergent responses in the vasculature, including angiogenesis and vasculogenesis. Although not used previously to study large vessel adaptations, we submit that ABMs will prove equally useful in such studies when combined with well-established continuum models to form multi-scale models of tissue-level phenomena. In order to couple agent-based and continuum models, however, there is a need to ensure that each model faithfully represents the best data available at the relevant scale and that there is consistency between models under baseline conditions. Toward this end, we describe the development and verification of an ABM of endothelial and smooth muscle cell responses to mechanical stimuli in a large artery. A refined rule-set is proposed based on a broad literature search, a new scoring system for assigning confidence in the rules, and a parameter sensitivity study. To illustrate the utility of these new methods for rule selection, as well as the consistency achieved with continuum-level models, we simulate the behavior of a mouse aorta during homeostasis and in response to both transient and sustained increases in pressure. The simulated responses depend on the altered cellular production of seven key mitogenic, synthetic, and proteolytic biomolecules, which in turn control the turnover of intramural cells and extracellular matrix. These events are responsible for gross changes in vessel wall morphology. This new ABM is shown to be appropriately stable under homeostatic conditions, insensitive to transient elevations in blood pressure, and responsive to increased intramural wall stress in hypertension.

A multi-layered computational model of coupled elastin degradation, vasoactive dysfunction, and collagenous stiffening in aortic aging

Arterial responses to diverse pathologies and insults likely occur via similar mechanisms. For example, many studies suggest that the natural process of aging and isolated systolic hypertension share many characteristics in arteries, including loss of functional elastin, decreased smooth muscle tone, and altered rates of deposition, and/or crosslinking of fibrillar collagen. Our aim is to show computationally how these coupled effects can impact evolving aortic geometry and mechanical behavior. Employing a thick-walled, multi-layered constrained mixture model, we suggest that a coupled loss of elastin and vasoactive function are fundamental mechanisms by which aortic aging occurs. Moreover, it is suggested that collagenous stiffening, although itself generally an undesirable process, can play a key role in attenuating excessive dilatation, perhaps including the enlargement of abdominal aortic aneurysms.

Modelling carotid artery adaptations to dynamic alterations in pressure and flow over the cardiac cycle

Motivated by recent clinical and laboratory findings of important effects of pulsatile pressure and flow on arterial adaptations, we employ and extend an established constrained mixture framework of growth (change in mass) and remodelling (change in structure) to include such dynamical effects. New descriptors of cell and tissue behavior (constitutive relations) are postulated and refined based on new experimental data from a transverse aortic arch banding model in the mouse that increases pulsatile pressure and flow in one carotid artery. In particular, it is shown that there was a need to refine constitutive relations for the active stress generated by smooth muscle, to include both stress- and stress rate-mediated control of the turnover of cells and matrix and to account for a cyclic stress-mediated loss of elastic fibre integrity and decrease in collagen stiffness in order to capture the reported evolution, over 8 weeks, of luminal radius, wall thickness, axial force and in vivo axial stretch of the hypertensive mouse carotid artery. We submit, therefore, that complex aspects of adaptation by elastic arteries can be predicted by constrained mixture models wherein individual constituents are produced or removed at individual rates and to individual extents depending on changes in both stress and stress rate from normal values.

A 3-D Framework for Arterial Growth and Remodeling in Response to Altered Hemodynamics

We present a three-dimensional mathematical framework for modeling the evolving geometry, structure, and mechanical properties of a representative straight cylindrical artery subjected to changes in mean blood pressure and flow. We show that numerical predictions recover prior findings from a validated two-dimensional framework, but extend those findings by allowing effects of transmural gradients in wall constituents and vasoactive molecules to be simulated directly. Of particular note, we show that the predicted evolution of the residual stress related opening angle in response to an abrupt, sustained increase in blood pressure is qualitatively similar to measured changes when one accounts for a nonlinear transmural distribution of pre-stretched elastin. We submit that continuum-based constrained mixture models of arterial adaptation hold significant promise for deepening our basic understanding of arterial mechanobiology and thus for designing improved clinical interventions to treat many different types of arterial disease and injury.

Computational model of evolving lens capsule biomechanics following cataract-like surgery

Cataract surgery is an invasive procedure whereby lens fibers are removed through a permanent central hole, or capsulorhexis, in the surrounding lens capsule and replaced with an artificial intraocular lens (IOL). Remnant lens epithelial cells subsequently transdifferentiate to a more contractile and synthetic wound-healing phenotype, which causes significant structural and mechanical adaptations of the residual lens capsule. The goal of this study is to present a computational model capable of capturing salient features of the biomechanical evolution of the lens capsule following cataract-like surgery. The model is shown to predict marked long-term increases in thickness and stiffness of the lens capsule nearest the edge of the capsulorhexis comparable to reported measurements. Such models represent a first step toward understanding better the long-term interactions between the residual lens capsule and implanted IOL, thus initiating a new paradigm for the design of improved IOLs, including those having an accommodative feature.

Constitutive modelling of arteries

This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

Coupling hemodynamics with vascular wall mechanics and mechanobiology to understand intracranial aneurysms

Arteries exhibit a remarkable ability to adapt in response to sustained alterations in hemodynamic loading, to heal in response to injuries, and to compensate in response to diverse disease conditions. Nevertheless, such compensatory adaptations are limited and many vascular disorders, if untreated, lead to significant morbidity or mortality. Parallel advances in vascular biology, medical imaging, biomechanics, and computational methods promise to provide increased insight into many arterial diseases, including intracranial aneurysms. In particular, although it may be possible to identify useful clinical correlations between either the blood flow patterns within or the shape of aneurysms and their rupture-potential, our ultimate goal should be to couple studies of hemodynamics with those of wall mechanics and the underlying mechanobiology so that we can understand better the mechanisms by which aneurysms arise, enlarge, and rupture and thereby identify better methods of treatment. This paper presents one such approach to fluid-solid-growth (FSG) modeling of intracranial aneurysms.

Modeling effects of axial extension on arterial growth and remodeling

Diverse mechanical perturbations elicit arterial growth and remodeling responses that appear to optimize structure and function so as to promote mechanical homeostasis. For example, it is well known that functional adaptations to sustained changes in transmural pressure and blood flow primarily affect wall thickness and caliber to restore circumferential and wall shear stresses toward normal. More recently, however, it has been shown that changes in axial extension similarly prompt dramatic cell and matrix reorganization and turnover, resulting in marked changes in unloaded geometry and mechanical behavior that presumably restore axial stress toward normal. Because of the inability to infer axial stress from in vivo measurements, simulations are needed to examine this hypothesis and to guide the design of future experiments. In this paper, we show that a constrained mixture model predicts salient features of observed responses to step increases in axial extension, including marked increases in fibrous constituent production, leading to a compensatory lengthening that restores original mechanical behavior. Because axial extension can be modified via diverse surgical procedures, including bypass operations, and exploited in tissue regeneration research, there is a need for increased attention to this important aspect of arterial biomechanics and mechanobiology.