# A New Constitutive Framework For Arterial Wall Mechanics And A Comparative Study Of Material Models

G. Holzapfel, T. Gasser, R. Ogden

Published 2000 · Mathematics

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

This paper references

10.1016/0002-9149(84)90382-5

Percutaneous angioplasty of stenoses of bypass grafts or of bypass graft anastomotic sites.

P. Block (1984)

10.1113/jphysiol.1881.sp000088

The Elastic Properties of the Arterial Wall.

C. Roy (1881)

10.1016/S0010-4825(98)00022-5

Biomechanical behavior of the arterial wall and its numerical characterization.

G. Holzapfel (1998)

Changes in the mechanical behavior of arteries following balloon angioplasty

H. S. Oktay (1991)

10.2307/2008488

Mathematical Elasticity, Volume I: Three-Dimensional Elasticity

D. Arnold (1989)

10.1097/00004669-198310000-00014

Biomedical Engineering II Recent Developments

C. W. Hall (1983)

10.1016/S0021-9290(97)00025-0

Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation.

A. Delfino (1997)

10.1115/1.2895425

Species dependence of the zero-stress state of aorta: pig versus rat.

H. Han (1991)

Review and directions

J. D. Humphrey (1995)

10.1016/S0021-9290(97)00032-8

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

A. Rachev (1997)

10.1097/00003246-198505000-00025

Handbook of Physiology—The Cardiovascular System

A. Guyton (1985)

10.1115/1.2792261

Bending of blood vessel wall: stress-strain laws of the intima-media and adventitial layers.

J. Xie (1995)

Continuum formulation and finite element analysis

G. A. Holzapfel (2000)

Determination of an effective shear modulus of aorta

J. Vossoughi (1998)

10.1016/0021-9290(84)90142-8

Experimental measurements of elastic properties of media and adventitia of bovine carotid arteries.

W. von Maltzahn (1984)

10.1016/0021-9290(94)90021-3

Stress-dependent finite growth in soft elastic tissues.

E. K. Rodríguez (1994)

Application to rubberlike solids

R. W. Ogden (1978)

10.1115/1.3138417

Three-dimensional stress distribution in arteries.

C. Chuong (1983)

10.1007/978-3-7091-4336-0_1

Constitutive Theory for Strongly Anisotropic Solids

A. Spencer (1984)

10.1007/978-1-4612-4866-8_9

Residual Stress in Arteries

C. J. Chuong (1986)

The Finite Element Method: Linear Static and Dynamic Finite Element Analysis

Thomas J. R. Hughes (1987)

10.1039/TF9615700829

Thermodynamic relations for high elastic materials

P. J. Flory (1961)

Data book on mechanical properties of living cells, tissues, and organs

阿部 博之 (1996)

10.1152/ajpheart.1983.244.2.H298

Comparison of arterial wall mechanics using ring and cylindrical segments.

R. Cox (1983)

10.1161/01.RES.24.1.1

The Elastic Symmetry of Arterial Segments in Dogs

D. J. Patel (1969)

10.1152/ajpheart.1994.266.1.H1

New experiments on shear modulus of elasticity of arteries.

S. Deng (1994)

Warriyar, Experimental measurments of elastic properties of media and adventitia of bovine carotid arteries

R.G.W.-W. Von Maltzahn (1984)

Stress–strain laws of the intimamedia and adventitia layers

J. Xie (1995)

10.1016/0021-9290(81)90056-7

Elastic properties of arteries: a nonlinear two-layer cylindrical model.

W. von Maltzahn (1981)

Mechanical properties of the aorta: a review.

F. Silver (1989)

10.1159/000159104

Three-dimensional collagen organization of human brain arteries at different transmural pressures.

H. Finlay (1995)

10.1152/ajpheart.1979.237.5.H620

Pseudoelasticity of arteries and the choice of its mathematical expression.

Y. C. Fung (1979)

The Finite Element Method: Linear Static and Dynamic Finite Element

T.J.R. Hughes (1987)

Pig versus rat

H. C. Han (1991)

10.1016/0741-5214(86)90282-X

Percutaneous transluminal angioplasty. A suggested method for analysis of clinical, arteriographic, and hemodynamic factors affecting the results of treatment.

R. Walden (1986)

10.1161/01.RES.23.1.61

Compressibility of the Arterial Wall

T. E. Carew (1968)

10.1115/1.2835113

An evaluation of pseudoelastic descriptors used in arterial mechanics.

J. Humphrey (1999)

Continuum basis, computational aspects and applications

G. A. Holzapfel (2000)

Zur Physiologie und Wachstumsmechanik des Blutgefäßsystems

R. F. Fuchs (1900)

10.1016/0021-9290(88)90240-0

Isotropy and anisotropy of the arterial wall.

H. W. Weizsacker (1988)

10.1016/0021-9290(74)90067-0

Two-dimensional mechanical properties of rabbit skin. II. Experimental results.

Y. Lanir (1974)

10.1016/0264-682X(84)90061-3

Non-Linear Elastic Deformations

R. W. Ogden (1984)

Boriss, Intimal residual stress and strain in large arteries

J. Vossoughi (1993)

10.1016/0002-9149(84)90750-1

The mechanism of transluminal angioplasty.

P. Block (1982)

Nonlinear Solid Mechanics. A Continuum Approach for Engineering, Wiley

G. A. Holzapfel (2000)

10.1152/ajpheart.1993.265.1.H52

Neutral axis location in bending and Young's modulus of different layers of arterial wall.

Q. Yu (1993)

10.1115/1.3138479

Elastic properties of arteries and their influence on the cardiovascular system.

A. Tözeren (1984)

10.1139/o57-080

The reason for the shape of the distensibility curves of arteries.

M. R. Roach (1957)

10.5860/choice.28-2130

Biomechanics: Motion, Flow, Stress, and Growth

Carl Gans (1991)

10.1161/01.RES.20.3.354

Dependence of Wall Stress in the Human Thoracic Aorta on Age and Pressure

H. Bader (1967)

10.1016/0021-9290(70)90043-6

Large deformation of elastic tubes.

W. H. Hoppmann (1970)

10.1615/critrevbiomedeng.v23.i1-2.10

Mechanics of the arterial wall: review and directions.

J. Humphrey (1995)

10.1161/01.RES.18.3.278

Alterations with Age in the Viscoelastic Properties of Human Arterial Walls

B. Learoyd (1966)

10.1115/1.2798301

Identification and determination of material properties for porohyperelastic analysis of large arteries.

B. Simon (1998)

10.1115/1.3162171

Biomechanics. Mechanical Properties of Living Tissues

Y. Fung (1982)

Über die schraubenförmige Struktur der Arterienwand

B. S. Schultze-Jena (1939)

10.1016/0021-9290(91)90286-V

A two-phase finite element model of the diastolic left ventricle.

J. Huyghe (1991)

10.1007/BF00618816

Deformation of blood vessels upon stretching, internal pressure, and torsion

V. A. Kas'yanov (1980)

10.1016/0021-9290(94)00012-S

Identification of elastic properties of homogeneous, orthotropic vascular segments in distension.

D. Vorp (1995)

Mechanics of angioplasty: Wall, balloon and stent

G. A. Holzapfel (2000)

10.1152/ajpcell.1994.266.1.C1

Biological applications of atomic force microscopy.

R. Lal (1994)

Mechanism of transluminal angioplasty

P. C. Block (1984)

10.1115/1.2895528

Experimental approaches on measuring the mechanical properties and constitutive laws of arterial walls.

K. Hayashi (1993)

10.1016/0020-7225(91)90075-E

A layered cylindrical shell model for an aorta

H. Demiray (1991)

10.1016/0021-9290(87)90262-4

Strain energy density function and uniform strain hypothesis for arterial mechanics.

K. Takamizawa (1987)

10.1152/ajpheart.1978.234.5.H542

Regional variation of series elasticity in canine arterial smooth muscles.

R. Cox (1978)

Phenomenological and structural aspects of the mechanical response of arteries

R. Ogden (2000)

10.1088/0031-9155/40/10/002

Elastic properties of human aortas in relation to age and atherosclerosis: a structural model.

F. Wuyts (1995)

10.1007/978-1-4612-4866-8

Frontiers in Biomechanics

Y. Fung (1986)

10.1162/002409400552261

Part I

Louise Poissant (2000)

10.1002/CPHY.CP020201

Architecture of the Vessel Wall

J. Rhodin (1980)

Funktionelle Morphologie der Arterien, Venen und arterio-venösen Anastomosen

J. Staubesand (1959)

10.1161/01.RES.32.5.577

Distribution of Stresses and of Strain‐Energy Density through the Wall Thickness in a Canine Aortic Segment

J. Young (1973)

10.1115/1.3171730

Continuum Theory of the Mechanics of Fibre-Reinforced Composites

A. Spencer (1984)

10.1115/1.2798315

Porohyperelastic finite element analysis of large arteries using ABAQUS.

B. Simon (1998)

10.1016/B978-0-08-030145-7.50078-7

ESTIMATION OF RESIDUAL STRAINS IN AORTIC SEGMENTS

R. Vaishnav (1983)

10.1161/01.RES.65.5.1340

Change of Residual Strains in Arteries due to Hypertrophy Caused by Aortic Constriction

Y. C. Fung (1989)

10.1097/jnn.0b013e318274cc4d

A Review of the

Robert Wolpert (1985)

10.1016/0020-7101(82)90016-2

Biomechanics — Mechanical properties of living tissue

J. Lenihan (1982)

10.1016/0022-5096(78)90012-1

Nearly isochoric elastic deformations: Application to rubberlike solids

R. W. Ogden (1978)

10.1115/1.2894084

Passive material properties of intact ventricular myocardium determined from a cylindrical model.

J. Guccione (1991)

10.1023/A:1020843529530

Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science

G. Holzapfel (2000)

10.1016/S0045-7825(00)00323-6

A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications

G. Holzapfel (2001)

This paper is referenced by

MODELING BOVINE PERICARDIUM

R. D. Vita (2003)

10.1177/10812865030085004

Frame-Invariant Polyconvex Strain-Energy Functions for Some Anisotropic Solids

D. Steigmann (2003)

10.1142/9781860945403_0005

Assessment of plaque stability based on high-resolution magnetic resonance imaging of human atherosclerotic lesions and computational mechanical analysis

C. Schulze-Bauer (2004)

Modeling of anisotropic damage in arterial wallls based on polyconvex stored energy functions

D. Balzani (2005)

10.1007/3-540-31184-X_18

Simulation of In-stent Restenosis for the Design of Cardiovascular Stents

C. Lally (2006)

10.1007/3-540-31184-X_6

Modeling and Simulation of Remodeling in Soft Biological Tissues

E. Kuhl (2006)

10.1016/j.advengsoft.2006.01.001

Dynamic pre-processing software for the hyperviscoelastic modeling of complex anisotropic biological tissue materials

A. F. Saleeb (2006)

10.6092/UNINA/FEDOA/1493

Supramolecular assembly and mechanical properties of dermis

M. Ventre (2007)

10.1002/NME.1807

Computational method of inverse elastostatics for anisotropic hyperelastic solids

J. Lu (2007)

10.1016/j.jbiomech.2011.11.016

A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the artery wall.

M. R. Hill (2012)

10.1080/03091900600700533

Three-part passive constitutive laws for the aorta in simple elongation

D. Sokolis (2007)

Regulation of aortic wall mechanics and stress An experimental study in man

H. Åstrand (2008)

10.1007/978-3-211-99709-3_2

Modeling of Rubberlike Materials

A. Luis Dorfmann (2009)

Morphoelasticity: The Mechanics and Mathematics of Elastic Growth

Rebecca Marie Vandiver (2009)

10.1016/J.APM.2009.11.022

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

Rahmiye Ergün (2010)

10.1007/s10439-010-0236-7

Linear and Nonlinear Viscoelastic Modeling of Aorta and Carotid Pressure–Area Dynamics Under In Vivo and Ex Vivo Conditions

D. Valdez-Jasso (2010)

10.1007/8415_2011_106

Patient-Specific Modeling of the Cornea

Roy Asher (2011)

10.1007/s00348-020-02966-y

An ultrasound-based approach for the characterization of fluid–structure interaction of large arterial vessels

Sonja Pejcic (2020)

10.1007/978-1-4419-7765-6

Maximum dissipation non-equilibrium thermodynamics and its geometric structure / Henry W. Haslach Jr.

Henry W. Haslach (2011)

10.1007/978-94-007-4552-0_7

Patient-Specific Biomechanical Framework for Aiding Clinical Decisions in Eye Surgery

Elena Lanchares (2012)

Mechanical Characterization of the Thoracic Ascending Aorta

Aaron Romo (2012)

10.1016/j.jbiomech.2012.10.015

Biaxial and failure properties of passive rat middle cerebral arteries.

E. D. Bell (2013)

Microstructural modeling of cross-linked fiber network embedded in continuous matrix

Lijuan Zhang (2013)

10.4028/www.scientific.net/KEM.554-557.2414

Numerical Implementation and Finite Element Analysis of Anisotropic Hyperelastic Biomaterials - Influence of Fibers Orientation

Rachid Djeridi (2013)

10.1016/J.IJNONLINMEC.2013.03.002

Non-linear micromechanics of soft tissues.

Huan Chen (2013)

10.1007/978-3-319-06844-2_10

Stent Deployment Computer Based Simulations for Health Care Treatment of Diseased Arteries

Georgia S. Karanasiou (2014)

10.1007/978-3-319-06974-6_12

Simulation of Atherosclerotic Plaque Delamination Using the Cohesive Zone Model

Xiaochang Leng (2015)

10.1007/S11340-015-0042-0

Identifying Hyper-Viscoelastic Model Parameters from an Inflation-Extension Test and Ultrasound Images

E.-J. Courtial (2015)

10.1016/J.EUROMECHSOL.2015.03.007

Material-symmetries congruency in transversely isotropic and orthotropic hyperelastic materials

Marcos Latorre (2015)

10.1155/2015/720141

Relationship between Postmenopausal Estrogen Deficiency and Aneurysmal Subarachnoid Hemorrhage

Sadaharu Tabuchi (2015)

10.1016/J.JSV.2016.01.024

Fluid-structure interaction for nonlinear response of shells conveying pulsatile flow

Eleonora Tubaldi (2016)

10.1007/978-3-319-39904-1_26

Development of New Testing Method of Mechanical Properties of Porcine Coronary Arteries

Bozena Gzik-Zroska (2016)

See more