Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions

doi:10.1016/S0021-9290(97)00032-8

Journal of Biomechanics

Volume 30, Issue 8, August 1997, Pages 819-827

https://doi.org/10.1016/S0021-9290(97)00032-8 Get rights and content

Abstract

Remodeling of arterial geometry was studied on the basis of a theoretical model. Sustained hypertension was simulated by a step increase in blood pressure. The artery was considered to be a thick-walled two-layer tube made of nonlinear elastic incompressible material. The basic hypothesis is that the artery remodels its zero-stress configuration in such a way that the strain and stress distributions in the arterial wall under hypertensive conditions are the same as under normotensive conditions. Using this hypothesis, a method for determining the geometrical dimensions of the zero-stress configuration of the hypertensive artery was proposed. To ensure uniqueness of the solution, two side conditions on the remodeling process are imposed: (a) the inner radius of the artery in the unloaded state remains unchanged; and (b) the ratio between the thickness of the inner and outer layer of the hypertensive artery in the zero-stress configuration is known. The model predicts that the arterial wall remodeling causes: (i) an increase of the wall thickness both in the unloaded and physiological states; (ii) an increase of the inner diameter of the hypertensive vessel under high pressure compared to the diameter of the normotensive artery under normal pressure; (iii) the opened-up configuration which arises when the unloaded arterial segment is cut radially still contains residual strains and stresses. These results are consistent with published experimental findings. It is speculated that the origin of residual stresses that exist in the unloaded and opened-up configurations is the stress-modulated growth.

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Cited by (124)

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2023, Journal of the Mechanics and Physics of Solids
In this paper, we present a large-deformation formulation of the mechanics of remodeling. Remodeling is anelasticity with an internal constraint—material evolutions that are mass and volume-preserving. In this special class of material evolutions, the explicit time dependence of the energy function is via one or more remodeling tensors that can be considered as internal variables of the theory. The governing equations of remodeling solids are derived using a two-potential approach and the Lagrange–d’Alembert principle. We consider both isotropic and anisotropic solids and derive their corresponding remodeling equations. We study a particular remodeling of fiber-reinforced solids in which the fiber orientation is time-dependent in the reference configuration— $S O (3)$ -remodeling. We define an additional remodeling energy, which is motivated by the energy spent in collagen fiber-reinforced living systems to remodel to enhance stiffness or strength in the direction of loading. We consider the examples of a solid reinforced with either one or two families of reorienting fibers and derive their remodeling equations. This is a generalization of some of the proposed remodeling equations in the literature. We study three examples of material remodeling, namely finite extensions and torsion of solid circular cylinders, which are universal deformations for incompressible isotropic solids and certain anisotropic solids. We consider both displacement and force-control loadings. Detailed parametric studies are included for the effects of various material and loading parameters on fiber remodeling. It is observed that during remodeling, there is a competition between the action of the internal strain energy function and the remodeling energy. For a given material, a remodeling process dominated by strain energy works to align fibers in a direction that minimizes strain energy. On the other hand, a remodeling process dominated by the remodeling energy aligns fibers in the direction of the maximum principal strain according to a constitutive choice. We finally linearize the governing equations of the remodeling theory and derive those of linear remodeling mechanics.
Fitting mechanical properties of the aortic wall and individual layers to experimental tensile tests including residual stresses
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The authors have observed that a stress-strain curve for uniaxial tension of an aortic intact wall cannot be simply obtained by combining the strain energy functions of the three individual aortic layers – intima, media and adventitia – even taking into account the interaction among the three layers; the strain energy functions of the three layers are obtained fitting tensile tests on strips from the individual layers. Due to the layer separation, the residual stresses are released and thus they do not affect the stress-strain curves of the individual layers. The present study shows that it is instead possible to fit the intact wall experimental curves with the combination of the strain energy functions of the three individual layers if residual strains are added. The residual strains are used as optimization parameters with specific constraints and allowing for the buckling (wrinkling) of the intima under unpressurized condition of the aortic wall, as experimentally observed. By varying these parameters in the experimentally observed range of values, it is possible to find a solution with the combined responses of the individual layers matching the experimental stress-strain curves of the intact wall.
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The theory of superimposed incremental elastic deformations is applied to a deformed configuration of a residually-stressed circular cylindrical tube. The tube is subject to internal or external pressure and an axial load that maintain its circular cylindrical shape, and then bifurcations from this shape are analysed. Detailed governing equations and boundary conditions are provided for axisymmetric, prismatic and asymmetric bifurcations for a general form of strain–energy function that incorporates radial and circumferential residual stress components. The theory is applied to a simple model strain–energy function with two material parameters and a parameter that reflects the magnitude of the residual stress. Numerical computations are used to illustrate the dependence of the results of the bifurcation analysis on these parameters and the axial and radial underlying deformation. It is shown that in general the presence of residual stress has a significant effect compared with the corresponding results without residual stress.
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While the effects of basal vascular tone (BVT) on the macroscopic stress-free configuration and on the intraparietal physiological stress/strain (PS/S) distribution in a healthy artery have been widely explored, the combined effects of residual stress/strain (RS/S), BVT, and in situ axial extension ratio on the PS/S distribution of healthy and pathological human coronary arteries remain unclear. As a result, they are largely ignored in the biomechanical and clinical analyses intended to diagnose the distensibility and vasodilatation performances of coronary arteries. The present study was therefore designed (1) to investigate ex vivo the macroscopic stress-free configurations of normal and atherosclerotic human coronary arteries, (2) to elucidate the combined effect of BVT and in situ axial stretch ratio on the PS/S distribution in healthy human coronary arteries by using an original hyperelastic homeostatic bilayered artery model accounting for the RS/S, and (3) to discuss the influence of RS/S on the biomechanical stability of human vulnerable coronary atherosclerotic plaques.
Modeling biological growth and remodeling: Contrasting methods, contrasting needs
2020, Current Opinion in Biomedical Engineering
Biological growth and remodeling processes are necessarily time-dependent due to the finite periods needed for the material to be synthesized, deposited, degraded, and/or reorganized and, hence, so have been predominantly modeled for the past 20+ years. However, a full-spectrum examination of the timescales present in these processes reveals the need to explore a new class of models for which time-dependent effects are negligible. These mechanobiologically (quasi-) equilibrated formulations not only appear to apply well in many cases but also provide the modeler with those additional pieces of information, and intuition, always needed when modeling complex time-dependent responses. Material model determination, optimization involving long-term adaptations, and mechanobiological stability analyses could be leveraged by the simplicity and computational efficiency of time-independent models. Although this concept is general, we address it by means of two particular theories for which we also highlight crucial differences entailed by their diametrically different material memory and heterogeneity descriptions.
A new analysis of stresses in arteries based on an Eulerian formulation of growth in tissues
2017, International Journal of Engineering Science
The simple analysis of stresses in arteries considers the artery as an incompressible elastic circular cylindrical tube. When the intact artery is unloaded and cut transmurally, it springs open to a nearly circular cylindrical sector. The resulting nonzero opening angle indicates the existence of residual stresses in the unloaded intact artery. This paper presents a new analysis of stresses in arteries based on an Eulerian formulation of elastic deformations in soft tissues that models the inelastic process of remodeling (homeostasis) towards its homeostatic state. Within the context of this new analysis it is not necessary to model details of homeostasis. Instead, it is possible to directly determine the loaded homeostatic state, which is considered as the reference configuration of the artery, and to specify the geometry and associated values of the circumferential and axial stresses by matching measurements of the geometry of the unloaded open artery. The predicted geometry of the unloaded intact artery is shown to be accurate relative to the measurements. Also, predictions of the stress distributions for remodeling due to hypertension are presented.

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^☆: Part of this paper was presented at the 13th Southern Biomedical Engineering Conference, Washington, DC, April 1994.

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Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions☆

Abstract

Journal of Biomechanics

Journal of Biomechanics

On residual stress in arteries

Journal of Biomechanical Engineering

Change of residual strains in arteries due to hypertrophy caused by aortic constriction

Circulation Research

Change of zero-stress state of rat pulmonary arteries in hypoxic hypertension

Journal of Applied Physiology

Intimal hyperplasia, vascular modeling, and the restenosis problem

Circulation