Mechanical Models of Artery Walls

Abstract The article presents the up to date review and discussion of approaches used to express mechanical behavior of artery walls. The physiology of artery walls and its relation to the models is discussed. Presented models include the simplest 0d and 1d ones but emphasis is put to the most sophisticated approaches which are based on the theory of 3d nonlinear elasticity. Also the alternative approach which consists in simple delinearization of the Koiter shell equations is presented.

[1]  J. Meister,et al.  Biomechanical and Physiological Aspects of Arterial Vasomotion , 1995 .

[2]  A. C. Burton Relation of structure to function of the tissues of the wall of blood vessels. , 1954, Physiological reviews.

[3]  F H Silver,et al.  Viscoelasticity of the vessel wall: the role of collagen and elastic fibers. , 2001, Critical reviews in biomedical engineering.

[4]  J W Melvin,et al.  Failure properties of passive human aortic tissue. II--Biaxial tension tests. , 1983, Journal of biomechanics.

[5]  Alexander Rachev,et al.  A Model of Arterial Adaptation to Alterations in Blood Flow , 2000 .

[6]  L J Brossollet,et al.  A new approach to mechanical testing and modeling of biological tissues, with application to blood vessels. , 1996, Journal of biomechanical engineering.

[7]  P. M. Anderson,et al.  Vascular mechanics of the coronary artery , 2000, Zeitschrift für Kardiologie.

[8]  S. Antman Nonlinear problems of elasticity , 1994 .

[9]  N. Westerhof,et al.  An artificial arterial system for pumping hearts. , 1971, Journal of applied physiology.

[10]  Ray W. Ogden,et al.  Nonlinear Elasticity, Anisotropy, Material Stability and Residual Stresses in Soft Tissue , 2003 .

[11]  W. Cascio,et al.  A LabVIEW™ Model Incorporating an Open-Loop Arterial Impedance and a Closed-Loop Circulatory System , 2005, Annals of Biomedical Engineering.

[12]  Gerhard A Holzapfel,et al.  Comparison of a multi-layer structural model for arterial walls with a fung-type model, and issues of material stability. , 2004, Journal of biomechanical engineering.

[13]  K. Takamizawa,et al.  Strain energy density function and uniform strain hypothesis for arterial mechanics. , 1987, Journal of biomechanics.

[14]  N. Stergiopulos,et al.  Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. , 1996, Journal of biomechanics.

[15]  H. Gajewski,et al.  Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen , 1974 .

[16]  Y. Lanir,et al.  Three-dimensional mechanical properties of porcine coronary arteries: a validated two-layer model. , 2006, American journal of physiology. Heart and circulatory physiology.

[17]  A. E. Ehret,et al.  A polyconvex anisotropic strain–energy function for soft collagenous tissues , 2006, Biomechanics and modeling in mechanobiology.

[18]  Jay D. Humphrey,et al.  Intracranial Saccular Aneurysms , 2003 .

[19]  J. Butler,et al.  Mechanical connections between elastin and collagen. , 1994, Connective tissue research.

[20]  R. Wulandana,et al.  An inelastic multi-mechanism constitutive equation for cerebral arterial tissue , 2005, Biomechanics and modeling in mechanobiology.

[21]  Y C Fung,et al.  Remodeling of the constitutive equation while a blood vessel remodels itself under stress. , 1993, Journal of biomechanical engineering.

[22]  Mehmet Toner,et al.  An Outline of Cardiovascular Structure and Function , 2003 .

[23]  R. Vito,et al.  Blood vessel constitutive models-1995-2002. , 2003, Annual review of biomedical engineering.

[24]  C. Kleinstreuer,et al.  A New Wall Stress Equation for Aneurysm-Rupture Prediction , 2005, Annals of Biomedical Engineering.

[25]  R. Ogden,et al.  Mechanics of biological tissue , 2006 .

[26]  S. Glagov,et al.  Transmural Organization of the Arterial Media: The Lamellar Unit Revisited , 1985, Arteriosclerosis.

[27]  J. Ball Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[28]  F P T Baaijens,et al.  A computational model for collagen fibre remodelling in the arterial wall. , 2004, Journal of theoretical biology.

[29]  W. Timmons,et al.  Cardiovascular Models and Control , 1999 .

[30]  Mette S. Olufsen,et al.  Modeling the arterial system with reference to an anesthesia simulator , 1998 .

[31]  M. Czerny,et al.  Fluid Dynamics, Wall Mechanics, and Oxygen Transfer in Peripheral Bypass Anastomoses , 2002, Annals of Biomedical Engineering.

[32]  Peter D. Richardson,et al.  Biomechanics of Plaque Rupture: Progress, Problems, and New Frontiers , 2002, Annals of Biomedical Engineering.

[33]  A Rachev,et al.  Theoretical study of the effect of stress-dependent remodeling on arterial geometry under hypertensive conditions. , 1997, Journal of biomechanics.

[34]  Gerhard A. Holzapfel,et al.  Structural and Numerical Models for the (Visco)elastic Response of Arterial Walls with Residual Stresses , 2003 .

[35]  E. Zeidler Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[36]  S. Ling,et al.  The mechanics of corrugated collagen fibrils in arteries. , 1977, Journal of biomechanics.

[37]  F. Migliavacca,et al.  Multiscale modeling of the cardiovascular system: application to the study of pulmonary and coronary perfusions in the univentricular circulation. , 2005, Journal of biomechanics.

[38]  Alfio Quarteroni,et al.  Computational vascular fluid dynamics: problems, models and methods , 2000 .

[39]  Freeman Dyson,et al.  A meeting with Enrico Fermi , 2004, Nature.

[40]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Blood Flow Model Based on the Coupling of ODEs and Hyperbolic PDEs , 2005, Multiscale Model. Simul..

[41]  B L Langille,et al.  Adaptations of carotid arteries of young and mature rabbits to reduced carotid blood flow. , 1989, The American journal of physiology.

[42]  H. Haslach,et al.  Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue , 2005, Biomechanics and modeling in mechanobiology.

[43]  Y. Fung,et al.  Pseudoelasticity of arteries and the choice of its mathematical expression. , 1979, The American journal of physiology.

[44]  P. Dobrin,et al.  Finite deformation analysis of the relaxed and contracted dog carotid artery. , 1971, Microvascular research.

[45]  W. K. Tucker,et al.  A device to test mechanical properties of tissues and transducers. , 1969, Journal of applied physiology.

[46]  Y. Fung,et al.  Biomechanics: Mechanical Properties of Living Tissues , 1981 .

[47]  Jeffrey E. Bischoff,et al.  Reduced Parameter Formulation for Incorporating Fiber Level Viscoelasticity into Tissue Level Biomechanical Models , 2006, Annals of Biomedical Engineering.

[48]  F. N. van de Vosse,et al.  Finite-element-based computational methods for cardiovascular fluid-structure interaction , 2003 .

[49]  R. Nerem Vascular fluid mechanics, the arterial wall, and atherosclerosis. , 1992, Journal of biomechanical engineering.

[50]  J D Humphrey,et al.  Finite strain elastodynamics of intracranial saccular aneurysms. , 1999, Journal of biomechanics.

[51]  Marc M. Budge,et al.  Terminology for Describing the Elastic Behavior of Arteries , 2003 .

[52]  L. R. John Forward electrical transmission line model of the human arterial system , 2006, Medical and Biological Engineering and Computing.

[53]  Ali Nadim,et al.  On deriving lumped models for blood flow and pressure in the systemic arteries. , 2004 .

[54]  Alexander Rachev,et al.  Remodeling of Arteries in Response to Changes in their Mechanical Environment , 2003 .

[55]  C. William Hall,et al.  Biomedical Engineering II Recent Developments , 1983 .

[56]  S. Glagov,et al.  Structural Basis for the Static Mechanical Properties of the Aortic Media , 1964, Circulation research.

[57]  J. Humphrey,et al.  Elastodynamics and Arterial Wall Stress , 2002, Annals of Biomedical Engineering.

[58]  J Stålhand,et al.  Aorta in vivo parameter identification using an axial force constraint , 2005, Biomechanics and modeling in mechanobiology.

[59]  A. Rachev,et al.  Deformation of blood vessels upon stretching, internal pressure, and torsion , 1980 .

[60]  Jeffrey E. Bischoff,et al.  A microstructurally based orthotropic hyperelastic constitutive law , 2002 .

[61]  Heinz-Otto Kreiss,et al.  Difference Approximations for the Second Order Wave Equation , 2002, SIAM J. Numer. Anal..

[62]  T. Olsson,et al.  Modeling initial strain distribution in soft tissues with application to arteries , 2006, Biomechanics and modeling in mechanobiology.

[63]  E. Zeidler Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics , 1997 .

[64]  Jay D. Humphrey,et al.  Structure, Mechanical Properties, and Mechanics of Intracranial Saccular Aneurysms , 2000 .

[65]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[66]  Donald L. Russell,et al.  Development of a Mathematical Model of the Human Circulatory System , 2006, Annals of Biomedical Engineering.

[67]  I. Yannas,et al.  Dependence of stress-strain nonlinearity of connective tissues on the geometry of collagen fibers. , 1976, Journal of biomechanics.

[68]  Hilmi Demiray,et al.  A layered cylindrical shell model for an aorta , 1991 .

[69]  Suncica Canic,et al.  Self-Consistent Effective Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries , 2005, Multiscale Model. Simul..

[70]  R. Schaefer,et al.  Filtration in cohesive soils : numerical approach , 1999 .

[71]  Nikos Stergiopulos,et al.  A constitutive formulation of arterial mechanics including vascular smooth muscle tone. , 2004, American journal of physiology. Heart and circulatory physiology.

[72]  Kozaburo Hayashi,et al.  Theoretical Study of the Effects of Vascular Smooth Muscle Contraction on Strain and Stress Distributions in Arteries , 1999, Annals of Biomedical Engineering.

[73]  Angela C. Biazutti,et al.  On a nonlinear evolution equation and its applications , 1995 .

[74]  G. Holzapfel,et al.  A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis , 2002 .

[75]  R. Contro,et al.  A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues , 2004, Biomechanics and modeling in mechanobiology.

[76]  Alfio Quarteroni,et al.  Multiscale modelling of the circulatory system: a preliminary analysis , 1999 .

[77]  Y C Fung,et al.  Three-dimensional stress distribution in arteries. , 1983, Journal of biomechanical engineering.

[78]  R. Ogden,et al.  Biomechanics of Soft Tissue in Cardiovascular Systems , 2003 .

[79]  S E Greenwald Pulse pressure and arterial elasticity. , 2002, QJM : monthly journal of the Association of Physicians.

[80]  Jeffrey E. Bischoff,et al.  Finite element simulations of orthotropic hyperelasticity , 2002 .

[81]  B. Lieber Arterial Macrocirculatory Hemodynamics , 1999 .

[82]  S. Glagov,et al.  A Lamellar Unit of Aortic Medial Structure and Function in Mammals , 1967, Circulation research.

[83]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[84]  P N Watton,et al.  A mathematical model for the growth of the abdominal aortic aneurysm , 2004, Biomechanics and modeling in mechanobiology.

[85]  C. Horgan,et al.  A description of arterial wall mechanics using limiting chain extensibility constitutive models , 2003, Biomechanics and modeling in mechanobiology.

[86]  T Matsumoto,et al.  Mechanical and dimensional adaptation of rat aorta to hypertension. , 1994, Journal of biomechanical engineering.

[87]  R. Ogden,et al.  Hyperelastic modelling of arterial layers with distributed collagen fibre orientations , 2006, Journal of The Royal Society Interface.

[88]  L. Taber A model for aortic growth based on fluid shear and fiber stresses. , 1998, Journal of biomechanical engineering.

[89]  H. Baumgärtel,et al.  Gajewski, H./Gröger, K./Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, VI, 281 S. Berlin. Akademie-Verlag. 1974. Preis 65,- M . , 1977 .

[90]  G. Mase,et al.  Continuum Mechanics for Engineers, Second Edition , 1999 .

[91]  Y C Fung,et al.  Compressibility and constitutive equation of arterial wall in radial compression experiments. , 1984, Journal of biomechanics.

[92]  Alfio Quarteroni,et al.  Mathematical Modelling and Numerical Simulation of the Cardiovascular System , 2004 .

[93]  Karl Grosh,et al.  A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue , 2004, Biomechanics and modeling in mechanobiology.

[94]  On large periodic motions of arteries. , 1983, Journal of biomechanics.

[95]  Robert Schaefer,et al.  The modified Fluid Particle Model for non-linear Casson fluid and its parallel distributed implementation , 2005 .

[96]  R. Shadwick,et al.  Mechanical design in arteries. , 1999, The Journal of experimental biology.

[97]  H S Borovetz,et al.  Identification of elastic properties of homogeneous, orthotropic vascular segments in distension. , 1995, Journal of biomechanics.

[98]  J D Humphrey,et al.  Mechanics of the arterial wall: review and directions. , 1995, Critical reviews in biomedical engineering.

[99]  Suncica Canic,et al.  Modeling Viscoelastic Behavior of Arterial Walls and Their Interaction with Pulsatile Blood Flow , 2006, SIAM J. Appl. Math..

[100]  G. E. Mase,et al.  Continuum Mechanics for Engineers , 1991 .

[101]  H. W. Weizsäcker,et al.  Biomechanical behavior of the arterial wall and its numerical characterization , 1998, Comput. Biol. Medicine.

[102]  Suncica Canic,et al.  Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries , 2003, SIAM J. Appl. Dyn. Syst..

[103]  Martin L. Dunn,et al.  A Microstructural Hyperelastic Model of Pulmonary Arteries Under Normo- and Hypertensive Conditions , 2005, Annals of Biomedical Engineering.

[104]  P. Kalita,et al.  NONLINEAR MODELS OF ARTERY DYNAMICS , 2006 .

[105]  S. Yazdani,et al.  A constitutive model of the artery with damage , 1997 .

[106]  W. Rutishauser,et al.  Arterial Fluid Dynamics: The Relationship to Atherosclerosis and Application in Diagnostics , 2000 .

[107]  J. C. Simo,et al.  Variational and projection methods for the volume constraint in finite deformation elasto-plasticity , 1985 .

[108]  Y. Fung Elasticity of soft tissues in simple elongation. , 1967, The American journal of physiology.

[109]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[110]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[111]  B R Simon,et al.  Identification and determination of material properties for porohyperelastic analysis of large arteries. , 1998, Journal of biomechanical engineering.

[112]  P. K. Mandal,et al.  Numerical simulation of unsteady two‐layered pulsatile blood flow in a stenosed flexible artery: Effect of peripheral layer viscosity 1 , 2005 .

[113]  Xiaoyi Wu,et al.  A constitutive model for protein-based materials. , 2006, Biomaterials.

[114]  Marcel C M Rutten,et al.  Determination of linear viscoelastic behavior of abdominal aortic aneurysm thrombus. , 2006, Biorheology.

[115]  Russell Heikes,et al.  Constitutive modeling of porcine coronary arteries using designed experiments. , 2003, Journal of biomechanical engineering.

[116]  Charles A. Taylor,et al.  A coupled momentum method for modeling blood flow in three-dimensional deformable arteries , 2006 .

[117]  A. Klarbring,et al.  Towards in vivo aorta material identification and stress estimation , 2004, Biomechanics and modeling in mechanobiology.

[118]  Peter F. Davies,et al.  Shear Stress Biology of the Endothelium , 2005, Annals of Biomedical Engineering.

[119]  A. Quarteroni,et al.  Coupling between lumped and distributed models for blood flow problems , 2001 .

[120]  Y Nose,et al.  Mechanical properties of aortas and pulmonary arteries of calves implanted with cardiac prostheses. , 1981, Journal of biomechanics.

[121]  G. Holzapfel,et al.  A polyconvex framework for soft biological tissues. Adjustment to experimental data , 2006 .

[122]  Jay D Humphrey,et al.  A 2D constrained mixture model for arterial adaptations to large changes in flow, pressure and axial stretch. , 2005, Mathematical medicine and biology : a journal of the IMA.

[123]  Fabio Nobile,et al.  Added-mass effect in the design of partitioned algorithms for fluid-structure problems , 2005 .

[124]  Gerhard A Holzapfel,et al.  Computational stress-deformation analysis of arterial walls including high-pressure response. , 2007, International journal of cardiology.

[125]  Andrew D. McCulloch,et al.  Computational Methods for Soft Tissue Biomechanics , 2003 .

[126]  Cornelius T. Leondes,et al.  Biofluid methods in vascular and pulmonary systems , 2001 .

[127]  Seymour Glagov,et al.  Micro‐architecture and composition of artery walls: relationship to location, diameter and the distribution of mechanical stress , 1992, Journal of hypertension. Supplement : official journal of the International Society of Hypertension.

[128]  Gerhard A. Holzapfel,et al.  A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications , 2001 .

[129]  N. Stergiopulos,et al.  Assessment of distributed arterial network models , 1997, Medical and Biological Engineering and Computing.

[130]  Jaroslav Hron,et al.  Fluid-structure interaction with applications in biomechanics , 2007, Nonlinear Analysis: Real World Applications.

[131]  Nikos Stergiopulos,et al.  Techniques in the Determination of the Mechanical Properties and Constitutive Laws of Arterial Walls , 2000 .

[132]  Y C Fung,et al.  Elastic and inelastic properties of the canine aorta and their variation along the aortic tree. , 1974, Journal of biomechanics.

[133]  M. Epstein,et al.  Cardiovascular Solid Mechanics: Cells, Tissues, and Organs , 2002 .

[134]  P B Dobrin,et al.  Distribution of lamellar deformations: implications for properties of the arterial media. , 1999, Hypertension.

[135]  Gerhard A. Holzapfel,et al.  A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation , 2002 .

[136]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[137]  Giuseppe Pontrelli,et al.  Numerical modelling of the pressure wave propagation in the arterial flow , 2003 .

[138]  N. Stergiopulos,et al.  Total arterial inertance as the fourth element of the windkessel model. , 1999, American journal of physiology. Heart and circulatory physiology.

[139]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations , 2003, Multiscale Model. Simul..

[140]  J. Z. Zhu,et al.  The finite element method , 1977 .

[141]  D. J. Patel,et al.  Distribution of Stresses and of Strain‐Energy Density through the Wall Thickness in a Canine Aortic Segment , 1973, Circulation research.

[142]  Joseph D. Bronzino,et al.  The Biomedical Engineering Handbook , 1995 .

[143]  Giovanna Guidoboni,et al.  Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics, and Experimental Validation , 2006, Annals of Biomedical Engineering.

[144]  R. Judd,et al.  Compressibility of perfused passive myocardium. , 1996, The American journal of physiology.

[145]  L. Horný,et al.  Age related constitutive laws and stress distribution in human main coronary arteries with reference to residual strain. , 2002, Bio-medical materials and engineering.

[146]  Cristina Cristalli,et al.  Techniques and Applications of Mathematical Modeling for Noninvasive Blood Pressure Estimation , 2000 .

[147]  M. Kaazempur-Mofrad,et al.  Hemodynamics and wall mechanics in human carotid bifurcation and its consequences for atherogenesis: investigation of inter-individual variation , 2004, Biomechanics and modeling in mechanobiology.

[148]  W. von Maltzahn,et al.  Experimental measurements of elastic properties of media and adventitia of bovine carotid arteries. , 1984, Journal of biomechanics.

[149]  James P. Keener,et al.  Mathematical physiology , 1998 .

[150]  S. Čanić,et al.  A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation , 2005 .

[151]  D. Bergel,et al.  The dynamic elastic properties of the arterial wall , 1961, The Journal of physiology.

[152]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[153]  K. Hayashi Mechanical Properties of Soft Tissues and Arterial Walls , 2003 .