A Mathematical Model of Maladaptive Inward Eutrophic Remodeling of Muscular Arteries in Hypertension.

We propose a relatively simple two-dimensional mathematical model for maladaptive inward remodeling of resistive arteries in hypertension in terms of vascular solid mechanics. The main premises are: (i) maladaptive inward remodeling manifests as a reduced increase in the arterial mass compared to the case of adaptive remodeling under equivalent hypertensive pressures and (ii) the pressure-induced circumferential stress in the arterial wall is restored to its basal target value as happens in the case of adaptive remodeling. The rationale for these assumptions is the experimental findings that elevated tone in association with sustained hypertensive pressure down-regulate the normal differentiation of vascular smooth muscle cells from contractile to synthetic phenotype and the data for the calculated hoop stress before and after completion of remodeling. Results from illustrative simulations show that as the hypertensive pressure increases, remodeling causes a nonmonotonic variation of arterial mass, a decrease in inner arterial diameter, and an increase in wall thickness. These findings and the model prediction that inward eutrophic remodeling is preceded by inward hypertrophic remodeling are supported by published observations. Limitations and perspectives for refining the mathematical model are discussed.

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.

A structure-based constitutive model of arterial tissue considering individual natural configurations of elastin and collagen.

The study proposes a novel theoretical-experimental approach for structure-based constitutive modeling of the passive mechanical properties of arterial tissue. The major novelty is accounting for the existence of individual natural configurations of elastin and collagen and their mechanical interaction in terms of the constituents' individual prestretches in the tissue natural state. The structure-based modeling of collagen allows accounting for effects of change in constituents' prestretch in terms of the change in feasible microstructural parameters, such as range of collagen recruitment stretch, mode of collagen mass fraction intensity function, and fiber directions. The results from an illustrative example for a porcine renal artery show that the model is robust and can adequately describe pressure-radius response and the stress-stretch relationship. The predictive capability of the model is tested in simulations of an isolated change in collagen prestretch and of elastin degradation in an artery kept at constant length. We expect this model to advance understanding about arterial rheology and serve as a useful tool for interpreting experimental data and solving boundary value problems relevant to vascular physiology at normal and pathological states.

Mechanical and geometrical determinants of wall stress in abdominal aortic aneurysms: A computational study.

An aortic aneurysm (AA) is a focal dilatation of the aortic wall. Occurrence of AA rupture is an all too common event that is associated with high levels of patient morbidity and mortality. The decision to surgically intervene prior to AA rupture is made with recognition of significant procedural risks, and is primarily based on the maximal diameter and/or growth rate of the AA. Despite established thresholds for intervention, rupture occurs in a notable subset of patients exhibiting sub-critical maximal diameters and/or growth rates. Therefore, a pressing need remains to identify better predictors of rupture risk and ultimately integrate their measurement into clinical decision making. In this study, we use a series of finite element-based computational models that represent a range of plausible AA scenarios, and evaluate the relative sensitivity of wall stress to geometrical and mechanical properties of the aneurysmal tissue. Taken together, our findings encourage an expansion of geometrical parameters considered for rupture risk assessment, and provide perspective on the degree to which tissue mechanical properties may modulate peak stress values within aneurysmal tissue.

The perivascular environment along the vertebral artery governs segment-specific structural and mechanical properties.

The vertebral arteries (VAs) are anatomically divided into four segments (V1-V4), which cumulatively transport blood flow through neck and ultimately form the posterior circulation of the brain. The vital physiological function of these conduit vessels depends on their geometry, composition and mechanical properties, all of which may vary among the defined arterial segments. Despite their significant role in blood circulation and susceptibility to injury, few studies have focused on characterizing the mechanical properties of VAs, and none have investigated the potential for segmental variation that could arise due to distinct perivascular environments. In this study, we compare the passive mechanical response of the central, juxtaposed arterial segments of porcine VAs (V2 and V3) via inflation-extension mechanical testing. Obtained experimental data and histological measures of arterial wall composition were used to adjust parameters of structure-motivated constitutive models that quantify the passive mechanical properties of each arterial segment and enable prediction of wall stress distributions under physiologic loads and boundary conditions. Our findings reveal significant segmental differences in the arterial wall geometry and structure. Nevertheless, similar wall stress distributions are predicted in these neighboring arterial segments if calculations account for their specific perivascular environments. These findings allow speculation that segmental differences in wall structure and geometry are a consequence of a previously introduced principle of optimal operation of arteries, which ensures effective bearing of physiological load and a favorable mechanical environment for mechanosensitive vascular smooth muscle cells. STATEMENT OF SIGNIFICANCE: Among the numerous biomechanical investigations devoted to conduit blood vessels, only a few deal with vertebral arteries. While these studies provide useful information that describes the vessel mechanical response, they do not enable identification of a constitutive formulation of the mechanical properties of the vessel wall. This is an important distinction, as a constitutive material model is required to calculate the local stress environment of mechanosensitive vascular cells and fully understand the mechanical implications of both vascular injury and clinical intervention. Moreover, segmental differences in the mechanical properties of the vertebral arteries could be used to discriminate among distinct modes of injury and disease etiologies.

The biaxial active mechanical properties of the porcine primary renal artery.

The mechanical response of arteries under physiological loads can be delineated into passive and active components. The passive response is governed by the load-bearing constituents within the arterial wall, elastin, collagen, and water, while the active response is a result of vascular smooth muscle cell (SMC) contraction. In muscular blood vessels, such as the primary renal artery, high SMC wall content suggests an elevated importance of the active response in determining overall vessel behavior. This study is a continuation of our previous investigation, in which a four-fiber constitutive model of the passive response of the primary porcine renal artery was identified. Here we focus on the active response of this vessel, specifically in the case of maximal SMC contraction, and develop a constitutive model of the active stress-stretch relations. The results of this study demonstrate the existence of biaxial active stress in the vessel wall, and suggest the active mechanical response is a critical component of renal arterial performance.

Are geometrical and structural variations along the length of the aorta governed by a principle of "optimal mechanical operation"?

It is well-documented that the geometrical dimensions, the longitudinal stretch ratio in situ, certain structural mechanical descriptors such as compliance and pressure-diameter moduli, as well as the mass fractions of structural constituents, vary along the length of the descending aorta. The origins of and possible interrelations among these observed variations remain open questions. The central premise of this study is that having considered the variation of the deformed inner diameter, axial stretch ratio, and area compliance along the aorta to be governed by the systemic requirements for flow distribution and reduction of cardiac preload, the zero-stress state geometry and mass fractions of the basic structural constituents of aortic tissue meet a principle of optimal mechanical operation. The principle manifests as a uniform distribution of the circumferential stress in the aortic wall that ensures effective bearing of the physiological load and a favorable mechanical environment for mechanosensitive vascular smooth muscle cells. A mathematical model is proposed and inverse boundary value problems are solved for the equations that follow from finite elasticity, structure-based constitutive modeling within constrained mixture theory, and stress-induced control of aortic homeostasis, mediated by the synthetic activity of vascular smooth muscle cells. Published experimental data are used to illustrate the predictive power of the proposed model. The results obtained are in agreement with published experimental data and support the proposed principle of optimal mechanical operation for the descending aorta.

A preliminary analysis of the data from an in vitro inflation-extension test can validate the assumption of arterial tissue elasticity.

The objective of this study is to propose a method for preliminary processing of the experimental data from an inflation-extension test on tubular arterial specimens. The method is based on the condition for existence of a strain energy function (SEF) and can be used to verify whether the data from a certain experiment validate the assumption that the tissue can be considered as an elastic solid. As an illustrative example of the proposed method, experimental data for a porcine renal artery are used and the sources of the error in satisfying the condition of elasticity are analyzed. The results lead to the conclusion that the experimental data for a renal artery validate that the artery exhibits an elastic mechanical response and a constitutive formulation based on the existence of the SEF is justified. A modification of the proposed method for the case of an in-plane biaxial stretching test of mechanically isotropic and orthotropic tissues is considered.

Calculation of the outcomes of remodeling of arteries subjected to sustained hypertension using a 3D two-layered model.

Arteries manifest a remodeling response to long-term alterations in arterial pressure and blood flow by changing geometry, structure, and composition through processes driven by perturbations of the local stresses in the vascular wall from their baseline values. The objective of this study is twofold--to develop a general method for calculating the remodeling responses of an artery considered as a two-layered tube; and to provide results for adaptive and maladaptive remodeling of a coronary artery. By formulating an inverse problem of vascular mechanics, the geometrical dimensions and mechanical properties of an artery are calculated from a prescribed deformed configuration, stress field, structural stiffness, and applied load. As an illustrative example we consider a human LAD coronary artery in both a perfect and incomplete adaptive response to a sustained step-wise change in pressure and a maladaptive response due to impaired remodeling of adventitia. The results obtained show that adventitia plays an important role in vascular mechanics when an artery is subjected to high arterial pressure. In addition to its well-known short term function of preventing over-inflation of an artery, it seems reasonable to accept that the manner by which adventitia remodels in response to a chronic increase in pressure is essential for preserving normal arterial function or may lead to an increased risk of developing vascular disorders.

Design and fabrication of a mechanically matched vascular graft.

The study provides a pathway to design a mechanics-matching vascular graft for an end-to-end anastomosis to a host artery. For functional equivalence, we submit that the graft and a host artery should have equal inner deformed diameters, equal pressure-radius module, and experience equal axial forces when subjected to mean arterial pressure. These criteria for mechanical equivalence are valid for a large class of materials that can be considered as elastic incompressible and orthotropic solids. As an example, specific known artery properties were used to design or select a graft made from a new synthetic biomaterial to demonstrate that reliable and robust technology for graft fabrication is possible.

A constituent-based model of age-related changes in conduit arteries.

In the present report, a constituent-based theoretical model of age-related changes in geometry and mechanical properties of conduit arteries is proposed. The model was based on the premise that given the time course of the load on an artery and the accumulation of advanced glycation end-products in the arterial tissue, the initial geometric dimensions and properties of the arterial tissue can be predicted by a solution of a boundary value problem for the governing equations that follow from finite elasticity, structure-based constitutive modeling within the constrained mixture theory, continuum damage theory, and global growth approach for stress-induced structure-based remodeling. An illustrative example of the age-related changes in geometry, structure, composition, and mechanical properties of a human thoracic aorta is considered. Model predictions were in good qualitative agreement with available experimental data in the literature. Limitations and perspectives for refining the model are discussed.

Theoretical study on the effects of pressure-induced remodeling on geometry and mechanical non-homogeneity of conduit arteries.

A structure-based mathematical model for the remodeling of arteries in response to sustained hypertension is proposed. The model is based on the concepts of volumetric growth and constitutive modeling of the arterial tissue within the framework of the constrained mixture theory. The major novel result of this study is that remodeling is associated with a local change in the mass fractions of the wall constituents that ultimately leads to mechanical non-homogeneity of the arterial wall. In the new homeostatic state that develops after a sustained increase in arterial pressure, the mass fraction of elastin decreases from the intimal side to the adventitial side of arteries, while the collagen fraction manifests an opposite trend. The results obtained are supported by some experimental observations reported in the literature.

A structure-based model of arterial remodeling in response to sustained hypertension.

A novel structure-based mathematical model of arterial remodeling in response to a sustained increase in pressure is proposed. The model includes two major aspects of remodeling in a healthy matured vessel. First, the deviation of the wall stress and flow-induced shear stress from their normal physiological values drives the changes in the arterial geometry. Second, the new mass that is produced during remodeling results from an increase in the mass of smooth muscle cells and collagen fibers. The model additionally accounts for the effect of the average pulsatile strain on the recruitment of collagen fibers in load bearing. The model was used to simulate remodeling of a human thoracic aorta, and the results are in good agreement with previously published model predictions and experimental data. The model predicts that the total arterial volume rapidly increases during the early stages of remodeling and remains virtually constant thereafter, despite the continuing stress-driven geometrical remodeling. Moreover, the effects of a perfect or incomplete restoration of the arterial compliance on the remodeling outputs were analyzed. For instance, the model predicts that the pattern of the time course of the opening angle depends on the extent to which the average pulsatile strain is restored at the end of the remodeling process. Future experimental studies on the time course of compliance, opening angle, and mass fractions of collagen, elastin, and smooth muscle cells can validate and improve the introduced hypotheses of the model.

A phenomenological model for mechanically mediated growth, remodeling, damage, and plasticity of gel-derived tissue engineered blood vessels.

Mechanical stimulation has been shown to dramatically improve mechanical and functional properties of gel-derived tissue engineered blood vessels (TEBVs). Adjusting factors such as cell source, type of extracellular matrix, cross-linking, magnitude, frequency, and time course of mechanical stimuli (among many other factors) make interpretation of experimental results challenging. Interpretation of data from such multifactor experiments requires modeling. We present a modeling framework and simulations for mechanically mediated growth, remodeling, plasticity, and damage of gel-derived TEBVs that merge ideas from classical plasticity, volumetric growth, and continuum damage mechanics. Our results are compared with published data and suggest that this model framework can predict the evolution of geometry and material behavior under common experimental loading scenarios.

System and method for investigating arterial remodeling.

Organ culture systems are used to study remodeling of arteries and to fabricate tissue engineered vascular grafts. Investigations to date focused on changes in geometry and mechanical response of arteries or constructs associated with controlled sustained alterations in the global load parameters such as the arterial pressure, flow, or axial stretch. A new experimental paradigm is proposed, which is based on the simultaneous independent control of local mechanical parameters such as mean strain or stress in the arterial wall and flow-induced shear at the intima. An organ culture system and methodology were developed, which controls pressure, flow, and axial length of a specimen in order to maintain the local mechanical parameters at prescribed values. The operation of the system is illustrated by maintenance of elevated axial medial stress in porcine carotid artery, while keeping the mean circumferential stress and flow-induced shear stress at baseline values. Previously unknown aspects of remodeling that might be revealed by the novel approach are discussed.

A theoretical study of mechanical stability of arteries.

This study proposes a mathematical model for studying stability of arteries subjected to a longitudinal extension and a periodic pressure. An artery was considered as a straight composite beam comprised of an external thick-walled tube and a fluid core. The dynamic criterion for stability was used, based on analyzing the small transverse vibrations superposed on the finite deformation of the vessel under static load. In contrast to the case of a static pressurization, in which buckling is only possible if the load produces a critical axial compressive force, a loss of stability of arteries under periodic pressure occurs under many combinations of load parameters. Instability occurs as a parametric resonance characterized by an exponential increase in the amplitude of transverse vibrations over several bands of pressure frequencies. The effects of load parameters were analyzed on the basis of the results for a dynamic and static stability of a rabbit thoracic aorta. Under normal physiological loads the artery is in a stable configuration. Static instability occurs under high distending pressures and low longitudinal stretch ratios. When the artery is subjected to periodic pressure, an independent increase in the mean pressure, amplitude of the periodic pressure, or frequency, most often, but not always, increases the risk of stability loss. In contrary, an increase in longitudinal stretch ratio most likely, but not certain, stabilizes the vessel. It was shown that adaptive geometrical remodeling due to an increase in mean pressure and flow does not affect artery stability.

Markers of inflammation collocate with increased wall stress in human coronary arterial plaque.

In this study, we hypothesized that spatial relationships exist between the local mechanical environment and inflammatory marker expression in atherosclerotic plaques, and that these relationships are plaque-progression dependent. Histologic cross-sections were collected at regular intervals along the length of diseased human coronary arteries and classified as early, intermediate, advanced, or mature based on their morphological features. For each cross-section, the spatial distribution of stress was determined using a 2D heterogeneous finite element model, and the corresponding distribution of selected inflammatory markers (macrophages, matrix metalloproteinase-1 [MMP-1], and nuclear factor-kappa B [NF-κB]) were determined immunohistochemically. We found a monotonic spatial relationship between mechanical stress and activated NF-κB that was consistent in all stages of plaque progression. We also identified progression-dependent relationships between stress and both macrophage presence and MMP-1 expression. These findings add to our understanding of the role of mechanical stress in stimulating the inflammatory response, and help explain how mechanical factors may regulate complex biological changes in remodeling.

Matrix metalloproteinase-2 and -9 are associated with high stresses predicted using a nonlinear heterogeneous model of arteries.

Arteries adapt to their mechanical environment by undergoing remodeling of the structural scaffold via the action of matrix metalloproteinases (MMPs). Cell culture studies have shown that stretching vascular smooth muscle cells (VSMCs) positively correlates to the production of MMP-2 and -9. In tissue level studies, the expressions and activations of MMP-2 and -9 are generally higher in the outer media. However, homogeneous mechanical models of arteries predict lower stress and strain in the outer media, which appear inconsistent with experimental findings. The effects of heterogeneity may be important to our understanding of VSMC function since arteries exhibit structural heterogeneity across the wall. We hypothesized that local stresses, computed using a heterogeneous mechanical model of arteries, positively correlate to the levels of MMP-2 and -9 in situ. We developed a model of the arterial wall accounting for nonlinearity, residual strain, anisotropy, and structural heterogeneity. The distributions of elastin and collagen fibers in situ, measured in the media of porcine carotid arteries, showed significant nonuniformities. Anisotropy was represented by the direction of collagen fibers measured by the helical angle of VSMC nuclei. The points at which the collagen fibers became load bearing were computed, assuming a uniform fiber strain and orientation under physiological loading conditions, an assumption motivated by morphological measurements. The distributions of circumferential stresses, computed using both heterogeneous and homogeneous models, were correlated to the distributions of expressions and activations of MMP-2 and -9 in porcine common carotid arteries incubated in an ex vivo perfusion organ culture system under physiological conditions for 48 h. While strains computed using incompressibility were identical in both models, the heterogeneous model, unlike the homogeneous model, predicted higher circumferential stresses in the outer layer correlated to the expressions and activations of MMP-2 and -9. This implies that localized remodeling occurs in the areas of high stress and agrees with results from cell culture studies. The results support the role of mechanical stress in vascular remodeling and the importance of structural heterogeneity in understanding mechanobiological responses.

Arteries respond to independent control of circumferential and shear stress in organ culture.

Arteries respond to changes in global mechanical parameters (pressure, flow rate, and longitudinal stretching) by remodeling to restore local parameters (circumferential stress, shear stress, and axial strain) to baseline levels. Because a change in a single global parameter results in changes of multiple local parameters, the effects of individual local parameters on remodeling remain unknown. This study uses a novel approach to study remodeling in organ culture based on independent control of local mechanical parameters. The approach is illustrated by studying the short term effects of circumferential and shear stress on remodeling-related biological markers. Porcine carotid arteries were cultured for 3 days at a circumferential stress of 50 or 150 kPa or, in separate experiments, a shear stress of 0.75 or 2.25 Pa. At high circumferential stress, matrix synthesis, smooth muscle cell proliferation, and cell death are significantly greater, but matrix metalloproteinase-2 (MMP-2) and pro-MMP-2 activity are significantly less. In contrast, biological markers measured were unaffected by shear stress. Applications of the proposed approach for improved understanding of remodeling, optimizing mechanical conditioning of tissue engineered arteries, and selection of experimentally motivated growth laws are discussed.

Stented artery biomechanics and device design optimization.

The deployment of a vascular stent aims to increase lumen diameter for the restoration of blood flow, but the accompanied alterations in the mechanical environment possibly affect the long-term patency of these devices. The primary aim of this investigation was to develop an algorithm to optimize stent design, allowing for consideration of competing solid mechanical concerns (wall stress, lumen gain, and cyclic deflection). Finite element modeling (FEM) was used to estimate artery wall stress and systolic/diastolic geometries, from which single parameter outputs were derived expressing stress, lumen gain, and cyclic artery wall deflection. An optimization scheme was developed using Lagrangian interpolation elements that sought to minimize the sum of these outputs, with weighting coefficients. Varying the weighting coefficients results in stent designs that prioritize one output over another. The accuracy of the algorithm was confirmed by evaluating the resulting outputs of the optimized geometries using FEM. The capacity of the optimization algorithm to identify optimal geometries and their resulting mechanical measures was retained over a wide range of weighting coefficients. The variety of stent designs identified provides general guidelines that have potential clinical use (i.e., lesion-specific stenting).