A Numerical Implementation to Predict Residual Strains from the Homogeneous Stress Hypothesis with Application to Abdominal Aortic Aneurysms

Wall stress analysis of abdominal aortic aneurysm (AAA) is a promising method of identifying AAAs at high risk of rupture. However, neglecting residual strains (RS) in the load-free configuration of patient-specific finite element analysis models is a sever limitation that strongly affects the computed wall stresses. Although several methods for including RS have been proposed, they cannot be directly applied to patient-specific AAA simulations. RS in the AAA wall are predicted through volumetric tissue growth that aims at satisfying the homogeneous stress hypothesis at mean arterial pressure load. Tissue growth is interpolated linearly across the wall thickness and aneurysm tissues are described by isotropic constitutive formulations. The total deformation is multiplicatively split into elastic and growth contributions, and a staggered schema is used to solve the field variables. The algorithm is validated qualitatively at a cylindrical artery model and then applied to patient-specific AAAs (n = 5). The induced RS state is fully three-dimensional and in qualitative agreement with experimental observations, i.e., wall strips that were excised from the load-free wall showed stress-releasing-deformations that are typically seen in laboratory experiments. Compared to RS-free simulations, the proposed algorithm reduced the von Mises stress gradient across the wall by a tenfold. Accounting for RS leads to homogenized wall stresses, which apart from reducing the peak wall stress (PWS) also shifted its location in some cases. The present study demonstrated that the homogeneous stress hypothesis can be effectively used to predict RS in the load-free configuration of the vascular wall. The proposed algorithm leads to a fast and robust prediction of RS, which is fully capable for a patient-specific AAA rupture risk assessment. Neglecting RS leads to non-realistic wall stress values that severely overestimate the PWS.

[1]  R K Jain,et al.  Compatibility and the genesis of residual stress by volumetric growth , 1996, Journal of mathematical biology.

[2]  S. Baek,et al.  An Inverse Optimization Approach Toward Testing Different Hypotheses of Vascular Homeostasis Using Image-based Models , 2011 .

[3]  D. Brewster,et al.  Autopsy Study of Unoperated Abdominal Aortic Aneurysms: The Case for Early Resection , 1977, Circulation.

[4]  J. Rodríguez,et al.  A Pull-Back Algorithm to Determine the Unloaded Vascular Geometry in Anisotropic Hyperelastic AAA Passive Mechanics , 2013, Annals of Biomedical Engineering.

[5]  L. Taber.  Biomechanics of Growth, Remodeling, and Morphogenesis , 1995 .

[6]  A. R. Brady,et al.  Mortality results for randomised controlled trial of early elective surgery or ultrasonographic surveillance for small abdominal aortic aneurysms , 1998, The Lancet.

[7]  Daniel Balzani,et al.  Numerical simulation of residual stresses in arterial walls , 2007 .

[8]  A Rachev,et al.  A model for geometric and mechanical adaptation of arteries to sustained hypertension. , 1998, Journal of biomechanical engineering.

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

[10]  D. Vorp,et al.  The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta. , 2006, Journal of biomechanics.

[11]  S. Nicholls,et al.  Rupture in small abdominal aortic aneurysms. , 1998, Journal of vascular surgery.

[12]  T Christian Gasser,et al.  Failure properties of intraluminal thrombus in abdominal aortic aneurysm under static and pulsating mechanical loads. , 2008, Journal of vascular surgery.

[13]  A Rachev,et al.  Experimental investigation of the distribution of residual strains in the artery wall. , 1997, Journal of biomechanical engineering.

[14]  J D Humphrey,et al.  A theoretical model of enlarging intracranial fusiform aneurysms. , 2006, Journal of biomechanical engineering.

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

[16]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[17]  Martin Auer,et al.  Reconstruction and Finite Element Mesh Generation of Abdominal Aortic Aneurysms From Computerized Tomography Angiography Data With Minimal User Interactions , 2010, IEEE Transactions on Medical Imaging.

[18]  J. E. Adkins,et al.  Large Elastic Deformations , 1971 .

[19]  V. Alastrué,et al.  Modelling adaptative volumetric finite growth in patient-specific residually stressed arteries. , 2008, Journal of biomechanics.

[20]  M. Webster,et al.  Effect of intraluminal thrombus on wall stress in patient-specific models of abdominal aortic aneurysm. , 2002, Journal of vascular surgery.

[21]  Martin Kroon,et al.  A theoretical model for fibroblast-controlled growth of saccular cerebral aneurysms. , 2009, Journal of theoretical biology.

[22]  Jay D. Humphrey,et al.  A CONSTRAINED MIXTURE MODEL FOR GROWTH AND REMODELING OF SOFT TISSUES , 2002 .

[23]  J. Humphrey,et al.  Importance of initial aortic properties on the evolving regional anisotropy, stiffness and wall thickness of human abdominal aortic aneurysms , 2012, Journal of The Royal Society Interface.

[24]  W. Wall,et al.  A Comparison of Diameter, Wall Stress, and Rupture Potential Index for Abdominal Aortic Aneurysm Rupture Risk Prediction , 2010, Annals of Biomedical Engineering.

[25]  J Swedenborg,et al.  The impact of intraluminal thrombus failure on the mechanical stress in the wall of abdominal aortic aneurysms. , 2011, European journal of vascular and endovascular surgery : the official journal of the European Society for Vascular Surgery.

[26]  S. Anand,et al.  Immediate repair compared with surveillance of small abdominal aortic aneurysms. , 2002, Vascular medicine.

[27]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[28]  M. Newman.  Cells, Tissues, and Organs , 2014 .

[29]  F. N. van de Vosse,et al.  Patient-specific initial wall stress in abdominal aortic aneurysms with a backward incremental method. , 2007, Journal of biomechanics.

[30]  E. Kröner,et al.  Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen , 1959 .

[31]  J. Humphrey.  Cardiovascular solid mechanics , 2002 .

[32]  T Christian Gasser,et al.  Turnover of fibrillar collagen in soft biological tissue with application to the expansion of abdominal aortic aneurysms , 2012, Journal of The Royal Society Interface.

[33]  Mark F Fillinger,et al.  In vivo analysis of mechanical wall stress and abdominal aortic aneurysm rupture risk. , 2002, Journal of vascular surgery.

[34]  J. Salenius,et al.  Ruptured abdominal aortic aneurysm in a well-defined geographic area. , 2002, Journal of vascular surgery.

[35]  J. S. Yao,et al.  Collagen types and matrix protein content in human abdominal aortic aneurysms. , 1989, Journal of vascular surgery.

[36]  Y C Fung,et al.  On residual stresses in arteries. , 1986, Journal of biomechanical engineering.

[37]  J Swedenborg,et al.  Biomechanical rupture risk assessment of abdominal aortic aneurysms: model complexity versus predictability of finite element simulations. , 2010, European journal of vascular and endovascular surgery : the official journal of the European Society for Vascular Surgery.

[38]  Gerhard A Holzapfel,et al.  A three-dimensional finite element model for arterial clamping. , 2001, Journal of biomechanical engineering.

[39]  Y. C. Fung,et al.  What are the residual stresses doing in our blood vessels? , 2006, Annals of Biomedical Engineering.

[40]  Manuel Doblaré,et al.  Assessing the Use of the “Opening Angle Method” to Enforce Residual Stresses in Patient-Specific Arteries , 2007, Annals of Biomedical Engineering.

[41]  M. Balasubramaniam,et al.  Size and location of thrombus in intact and ruptured abdominal aortic aneurysms. , 2005, Journal of vascular surgery.

[42]  Per Eriksson,et al.  Influence of intraluminal thrombus on structural and cellular composition of abdominal aortic aneurysm wall. , 2003, Journal of vascular surgery.

[43]  M. L. Raghavan,et al.  Inverse elastostatic stress analysis in pre-deformed biological structures: Demonstration using abdominal aortic aneurysms. , 2007, Journal of biomechanics.

[44]  Jia Lu,et al.  Estimation of vascular open configuration using finite element inverse elastostatic method , 2009, Engineering with Computers.

[45]  M. L. Raghavan,et al.  Three-Dimensional Finite Element Analysis of Residual Stress in Arteries , 2004, Annals of Biomedical Engineering.

[46]  T Christian Gasser,et al.  Importance of material model in wall stress prediction in abdominal aortic aneurysms. , 2013, Medical engineering & physics.

[47]  A Rachev,et al.  Residual strains in conduit arteries. , 2003, Journal of biomechanics.

[48]  H Demiray.  Large deformation analysis of some soft biological tissues. , 1981, Journal of biomechanical engineering.

[49]  D. Bergel,et al.  The visco-elastic properties of the arterial wall. , 1960 .

[50]  M J Fagan,et al.  A comparative study of aortic wall stress using finite element analysis for ruptured and non-ruptured abdominal aortic aneurysms. , 2004, European journal of vascular and endovascular surgery : the official journal of the European Society for Vascular Surgery.

[51]  R. N. Vaishnav,et al.  Residual stress and strain in aortic segments. , 1987, Journal of biomechanics.

[52]  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.

[53]  Seungik Baek,et al.  Medical image-based simulation of abdominal aortic aneurysm growth , 2012 .

[54]  T Christian Gasser,et al.  Spatial orientation of collagen fibers in the abdominal aortic aneurysm's wall and its relation to wall mechanics. , 2012, Acta biomaterialia.