Abstract
Arterial tissue is commonly assumed to be incompressible. While this assumption is convenient for both experimentalists and theorists, the compressibility of arterial tissue has not been rigorously investigated. In the current study we present an experimental-computational methodology to determine the compressibility of aortic tissue and we demonstrate that specimens excised from an ovine descending aorta are significantly compressible. Specimens are stretched in the radial direction in order to fully characterise the mechanical behaviour of the tissue ground matrix. Additionally biaxial testing is performed to fully characterise the anisotropic contribution of reinforcing fibres. Due to the complexity of the experimental tests, which entail non-uniform finite deformation of a non-linear anisotropic material, it is necessary to implement an inverse finite element analysis scheme to characterise the mechanical behaviour of the arterial tissue. Results reveal that ovine aortic tissue is highly compressible; an effective Poisson’s ratio of 0.44 is determined for the ground matrix component of the tissue. It is also demonstrated that correct characterisation of material compressibility has important implications for the calibration of anisotropic fibre properties using biaxial tests. Finally it is demonstrated that correct treatment of material compressibility has significant implications for the accurate prediction of the stress state in an artery under in vivo type loading.
References
Anderson, T. L. Fracture mechanics: fundamentals and applications. Boca Raton: CRC press, 2005.
Boutouyrie, P., D. P. Germain, A.-I. Tropeano, B. Laloux, F. Carenzi, M. Zidi, X. Jeunemaitre, and S. Laurent. Compressibility of the carotid artery in patients with pseudoxanthoma elasticum. Hypertension. 38:1181–1184, 2001.
Canham, P. B., H. M. Finlay, J. G. Dixon, D. R. Boughner, and A. Chen. Measurements from light and polarised light microscopy of human coronary arteries fixed at distending pressure. Cardiovasc. Res. 23:973–982, 1989.
Carew, T. E., R. N. Vaishnav, and D. J. Patel. Compressibility of the arterial wall. Circ. Res. 23:61–68, 1968.
Chuong, C., and Y. Fung. Three-dimensional stress distribution in arteries. J. Biomech. Eng. 105:268–274, 1983.
Chuong, C., and Y. Fung. Compressibility and constitutive equation of arterial wall in radial compression experiments. J. Biomech. 17:35–40, 1984.
Conway, C., F. Sharif, J. McGarry, and P. McHugh. A computational test-bed to assess coronary stent implantation mechanics using a population-specific approach. Cardiovasc. Eng. Technol. 3:374–387, 2012.
Di Puccio, F., S. Celi, and P. Forte. Review of experimental investigations on compressibility of arteries and introduction of a new apparatus. Exp. Mech. 52:895–902, 2012.
Dobrin, P., and A. Rovick. Static elastic properties of dog carotid arterial wall. Fed. Proc. 26:439, 1967.
Dobrin, P. B. Biaxial anisotropy of dog carotid artery: estimation of circumferential elastic modulus. J. Biomech. 19:351–358, 1986.
Elliott, D. M., and L. A. Setton. Anisotropic and inhomogeneous tensile behavior of the human anulus fibrosus: experimental measurement and material model predictions. J. Biomech. Eng. 123:256–263, 2001.
Finlay, H., L. McCullough, and P. Canham. Three-dimensional collagen organization of human brain arteries at different transmural pressures. J. Vasc. Res. 32:301–312, 1995.
Finlay, H. M., P. Whittaker, and P. B. Canham. Collagen organization in the branching region of human brain arteries. Stroke 29: 1595–1601, 1998.
Gasser, T. C., R. W. Ogden, and G. A. Holzapfel. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface 3:15–35, 2006.
Ghriallais, R. N., and M. Bruzzi. Effects of knee flexion on the femoropopliteal artery: a computational study. Med. Eng. Phys. 35: 1620–1628, 2013.
Hayashi, K., H. Handa, S. Nagasawa, A. Okumura, and K. Moritake. Stiffness and elastic behavior of human intracranial and extracranial arteries, J. Biomech. 13:175–184, 1980.
Holzapfel, G. A. Nonlinear solid mechanics: a continuum approach for engineers. Wiley: Chichester, 2000.
Holzapfel, G. A., and R. W. Ogden. On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework. Math. Mech. Solids 14:474–489, 2009.
Holzapfel, G. A., T. C. Gasser, and R. W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast. Phys. Sci. Solids 61:1–48, 2000.
Huang, C.-Y., A. Stankiewicz, G. A. Ateshian, and V. C. Mow. Anisotropy, inhomogeneity, and tension–compression nonlinearity of human glenohumeral cartilage in finite deformation. J. Biomech. 38:799–809, 2005.
Humphrey, J., D. Vawter, and R. Vito. Pseudoelasticity of excised visceral pleura. J. Biomech. Eng. 109:115–120, 1987.
Humphrey, J. D. Cardiovascular solid mechanics: cells, tissues, and organs. Springer: New York, 2002.
Lagarias, J., J. Reeds, M. Wright, and P. Wright. Convergence properties of the nelder–mead simplex method in low dimensions. SIAM J. Optim. 9:112–147, 1998.
Lawton, R. W. The thermoelastic behavior of isolated aortic strips of the dog. Circ. Res. 2:344–353, 1954.
Misra, J., and S. Chakravarty. Study of compressibility in vascular rheology. Rheol. Acta 19:381–388, 1980.
Nelder, J. A., and R. Mead. A simplex method for function minimization. Comput. J. 7:308–313, 1965.
Nolan, D., and J. McGarry. On the correct interpretation of measured force and calculation of material stress in biaxial tests. J. Mech. Behav. Biomed. Mater. 2015. doi:10.1016/j.jmbbm.2015.08.019.
Nolan, D., A. Gower, M. Destrade, R. Ogden, and J. McGarry. A robust anisotropic hyperelastic formulation for the modelling of soft tissue. J. Mech. Behav. Biomed. Mater. 39:48–60, 2014.
Peña, E., A. P. Del Palomar, B. Calvo, M. Martínez, and M. Doblaré. Computational modelling of diarthrodial joints. physiological, pathological and pos-surgery simulations. Arch. Comput. Methods Eng. 14:47–91, 2007.
Raghavan, M., and D. A. Vorp. Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability. J. Biomech. 33:475–482, 2000.
Raghavan, M., S. Trivedi, A. Nagaraj, D. McPherson, and K. Chandran. Three-dimensional finite element analysis of residual stress in arteries. Ann. Biomed. Eng., 32, 257–263, 2004.
Sacks, M. S. Biaxial mechanical evaluation of planar biological materials. J. Elast. Phys. Sci. Solids 61:199–246, 2000.
Schneider, C. A., W. S. Rasband, and K. W. Eliceiri. NIH image to ImageJ: 25 years of image analysis. Nat. Methods 9:671–675, 2012.
Silver, F., D. Christiansen, and C. Buntin. Mechanical properties of the aorta: a review. Crit. Rev. Biomed. Eng. 17:323–358, 1988.
Silver, F. H., P. B. Snowhill, and D. J. Foran. Mechanical behavior of vessel wall: a comparative study of aorta, vena cava, and carotid artery. Ann. Biomed. Eng. 31:793–803, 2003.
Smith, H. E., T. J. Mosher, B. J. Dardzinski, B. G. Collins, C. M. Collins, Q. X. Yang, V. J. Schmithorst, and M. B. Smith. Spatial variation in cartilage t2 of the knee. J. Magn. Reson. Imaging 14:50–55, 2001.
Sokolis, D. P., E. M. Kefaloyannis, M. Kouloukoussa, E. Marinos, H. Boudoulas, and P. E. Karayannacos. A structural basis for the aortic stress–strain relation in uniaxial tension. J. Biomech. 39:1651–1662, 2006.
Storåkers, B. On material representation and constitutive branching in finite compressible elasticity. J. Mech. Phys. Solids 34:125–145, 1986.
Sun, W., and M. S. Sacks. Finite element implementation of a generalized fung-elastic constitutive model for planar soft tissues. Biomech. Model. Mechanobiol. 4:190–199, 2005.
Takamizawa, K., and K. Hayashi. Strain energy density function and uniform strain hypothesis for arterial mechanics. J. Biomech. 20:7–17, 1987.
Tickner, E. G., and A. H. Sacks. A theory for the static elastic behavior of blood vessels. Biorheology 4:151, 1967.
Vaishnav, R. N., and J. Vossoughi. Residual stress and strain in aortic segments. J. Biomech. 20:235–239, 1987.
Vilarta, R., and B. D. C. Vidal. Anisotropic and biomechanical properties of tendons modified by exercise and denervation: aggregation and macromolecular order in collagen bundles. Matrix 9:55–61, 1989.
Volokh, K. Compressibility of arterial wall in ring-cutting experiments. Mol. Cell. Biomech. 3:35, 2006.
Vorp, D. A., M. Raghavan, and M. W. Webster. Mechanical wall stress in abdominal aortic aneurysm: influence of diameter and asymmetry. J. Vasc. Surg. 27:632–639, 1998.
Yosibash, Z., I. Manor, I. Gilad, and U. Willentz. Experimental evidence of the compressibility of arteries. J. Mech. Behav. Biomed. Mater. 39:339–354, 2014.
Acknowledgments
The authors wish to acknowledge funding from Science Foundation Ireland under project SFI-12/IP/1723. Furthermore we acknowledge funding from the Irish Research Council and the College of Engineering and Informatics at NUI, Galway. The authors wish to thank Noel Reynods and Prof. Michel Destrade for insightful discussions on this topic.
Author information
Authors and Affiliations
Corresponding author
Additional information
Associate Editor Estefanía Peña oversaw the review of this article.
Appendices
Appendix A: Investigation of Different Forms of \(\Psi _\text {iso}\)
In the "Compressible Anisotropic Constitutive Model" section the strain energy potential for the MA model is additively split into isotropic and anisotropic parts \(\Psi = \Psi _\text {iso} + \Psi _\text {aniso}\). The MA model uses a common neo–Hookean strain energy potential, \(\Psi _\text {iso} = \frac{\mu }{2}(\bar{I}_1 -3) + \frac{\kappa }{2}(J - 1)^2\). Here the use of two additional hyperelastic strain energy potentials for use as the isotropic term \(\Psi _\text {iso}\) are investigated. Firstly a modified version of the standard neo–Hookean model which was developed for scenarios involving large compressive stresses is examined. Its strain energy potential is given as:
In this model the volumetric term has been modified to include the function \(\text {ln}(J)\). This has the effect of penalising very large volume changes and prevents \(J \rightarrow 0\).
Secondly a hyperfoam model38 developed for highly compressible hyperelastic polymer foams is examined. Its strain energy function is given as:
where N is the order of the function, \(\lambda _m\) (\(m=1,2,3\)) are the principal stretches, \(\alpha _i\) determines the non-linearity for each term in the function, \(\beta _i\) determines the compressiblity for each term in the strain energy function and is related to the Poisson’s ratio \(\nu _i\) through the expression \(\beta _i = \nu _i / (1-2 \nu _i)\), and the initial bulk modulus \(\kappa _0 = \sum _{i=1}^{N} 2\mu _i [(1-3\beta _i)/3]\).
Equation (A.1) is added to the anisotropic potential of the MA model to form the MA_LNH model and Eq. (A.2) is added to the anisotropic potential of the MA model to form the MA_HYF model. The simulations from "Assessment of Error Generated by the Incompressibility Assumption" are repeated using the MA_LNH and MA_HYF models for the artery wall. The MA_LNH model uses the mean bulk and shear modulus from Table 1 and the MA_HYF model uses \(N=1\), \(\alpha _i=2\), \(\nu _i=0.44\), and \(\mu _i=7.04\) kPa.
The mean radial strain – pressure curves for both of these models, as well as that of the original MA model are given in Fig. A.1. Use of the MA_LNH model results in no difference in arterial compliance compared to the MA model. The MA_HYF model computes a small difference in compliance. For the hyperfoam model it is recommended that the effective Poisson’s ratio does not exceed 0.45.
Appendix B: Calibration of Anisotropic Constitutive Models
Calibrations are performed for each of the six individual specimens tested using their unique force–displacement curve and the specimen thickness measured using the technique outlined in "Inverse FE Analysis for Calibration of Anisotropic Fibres" section. The results of these individual calibrations are presented in Table B.1.
Appendix C: Quasi-3D Method for Volume Change Measurement
Volume change is calculated using an alternative method by measuring two orthogonal axes/diameters of the specimen at each segmentation plane. The volume is computed by integrating the resultant cross-sectional areas over the height of the specimen using a trapezoidal type method. A schematic outlining this method is given in Fig. C.1 This alternative methodology results in a volume change of 10.11 ± 4.61% (mean ± SD). A t test indicates that there is no statistically significant difference between the mean volume change calculated using the axi-symmetric or the quasi-3D method.
Appendix D: Inclusion of Residual Stress
The influence of residual stresses on the stress-state in the vessel wall is assessed in this appendix. It is well established that residual stresses are present in the unpressurised vessel wall. Simulations of lumen inflation were performed, following the computational method outlined by Raghavan et al. 31 and using the experimental data of Vaishnav and Vossoughi42 to determine stress-free geometry. The vessel has a stress-free internal radius \(R_i=21.92\) mm, an external radius \(R_e=24.94\) mm, and an opening angle of \(80.22^{\circ }\). The material parameters for both the HGO and MA models are identical to those used in "Assessment of Error Generated by the Incompressibility Assumption" section.
Figure D.1 shows a contour plot of the residual von Mises stress in the vessel wall in the unpressurised configuration. The vessel moves from a state of compression on the inner face to tension on the external face. As the anisotropic component of both the HGO and MA models are inactive in compression, this explains the higher stress on the external face of the vessel. Figure D.2 shows a contour plot of the ratio of the axial stress to the circumferential stress (\(\sigma _{zz}/\sigma _{\theta \theta }\)) for the HGO and MA models. This figure further illustrates the dependence of the stress-state on the treatment of compressibility. Differences between the HGO and MA models are similar to those shown in Fig. 8 without residual stress. This illustrates that the important influence of compressibility on triaxial artery stress is significant regardless of the inclusion of residual stress.
Rights and permissions
About this article
Cite this article
Nolan, D.R., McGarry, J.P. On the Compressibility of Arterial Tissue. Ann Biomed Eng 44, 993–1007 (2016). https://doi.org/10.1007/s10439-015-1417-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10439-015-1417-1