Residual Stress Estimates from Multi-cut Opening Angles of the Left Ventricle

Springer Science and Business Media LLC - Tập 11 Số 4 - Trang 381-393 - 2020
Xin Zhuan1, Xiaoyu Luo1
1School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8SQ, UK

Tóm tắt

Abstract Purpose Residual stress tensor has an essential influence on the mechanical behaviour of soft tissues and can be particularly useful in evaluating growth and remodelling of the heart and arteries. It is currently unclear if one single radial cut using the opening angle method can accurately estimate the residual stress. In many previous models, it has been assumed that a single radial cut can release the residual stress in a ring of the artery or left ventricle. However, experiments by Omens et al. (Biomech Model Mechanobiol 1:267–277, 2003) on mouse hearts, have shown that this is not the case. The aim of this paper is to answer this question using a multiple-cut mathematical model. Methods In this work, we have developed models of multiple cuts to estimate the residual stress in the left ventricle and compared with the one-cut model. Both two and four-cut models are considered. Given that the collagen fibres are normally coiled in the absence of loading, we use the isotropic part of the Holzapfel-Ogden strain energy function to model the unloaded myocardium. Results The estimated residual hoop stress from our multiple-cut model is around 8 to 9 times greater than that of a single-cut model. Although in principle infinite cuts are required to release the residual stress, we find four cuts seem to be sufficient as the model agrees well with experimental measurements of the myocardial thickness. Indeed, even the two-cut model already gives a reasonable estimate of the maximum residual hoop stress. We show that the results are not significantly different using homogeneous or heterogeneous material models. Finally, we explain that the multiple cuts approach also applies to arteries. Conclusion We conclude that both radial and circumferential cuts are required to release the residual stress in the left ventricle; using multiple radial cuts alone is not sufficient. A multiple-cut model gives a marked increase of residual stress in a left ventricle ring compared to that of the commonly used single-cut model.

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Tài liệu tham khảo

Ambrosi, D., G. A. Ateshian, E. M. Arruda, S. C. Cowin, J. Dumais, A. Goriely, G. A. Holzapfel, J. D. Humphrey, R. Kemkemer, E. Kuhl, et al.. Perspectives on biological growth and remodeling. J. Mech. Phys. Solids. 59:863–883, 2011.

Chuong, C. J., and Y. C. Fung. Residual stress in arteries In: G.W. Schmid-Schönbein, S.L-Y. Woo and B.W. Zweifach (eds), Frontiers In Biomechanics 117–129, 1986.

Clark, John M and Glagov, Seymour Transmural organization of the arterial media. The lamellar unit revisited. Arteriosclerosis: An Official Journal of the American Heart Association, Inc. 5:19–34, 1985.

Costa, Kevin D and May-Newman, Karen and Farr, Dyan and O’Dell, Walter G and McCulloch, Andrew D and Omens, Jeffrey H Three-dimensional residual strain in midanterior canine left ventricle American Journal of Physiology-Heart and Circulatory Physiology. 273:H1968–H1976, 1997.

Dingemans, Koert P and Teeling, Peter and Lagendijk, Jaap H and Becker, Anton E Extracellular matrix of the human aortic media: an ultrastructural histochemical and immunohistochemical study of the adult aortic media The Anatomical Record: An Official Publication of the American Association of Anatomists. 258:1–14, 2000.

Fung, Y. C., and S. Q. Liu. Change of residual strains in arteries due to hypertrophy caused by aortic constriction. Circ. Res. 65:1340–1349, 1989.

Grobbel, Marissa R and Shavik, Sheikh Mohammad and Darios, Emma and Watts, Stephanie W and Lee, Lik Chuan and Roccabianca, Sara Contribution of left ventricular residual stress by myocytes and collagen: existence of inter-constituent mechanical interaction. Biomechanics and modeling in mechanobiology. 17: 985–999, 2018.

Guccione, Julius M and McCulloch, Andrew D and Waldman, LK Passive material properties of intact ventricular myocardium determined from a cylindrical model American Society of Mechanical Engineers, 113: 42–55, 1991.

Hoger, A. On the residual stress possible in an elastic body with material symmetry. Arch. Ration. Mech. Anal. 88:271–289, 1985.

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. Elasticity. 61:1–48, 2000.

Holzapfel, G. A., and R. W. Ogden. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos. Trans. R. Soc. A 367:3445–3475, 2009.

Holzapfel, G. A., and R. W. Ogden. Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta. J. R. Soc. Interface. 7:787–799, 2010.

Hsu, F. H. The influences of mechanical loads on the form of a growing elastic body. J. Biomech. 1:303–311, 1968.

Liu, S. Q., and Y. C. Fung. Zero-stress states of arteries. J. Biomech. Eng. 110:82–84, 1988.

Menzel, A., and E. Kuhl. Frontiers in growth and remodeling. Mech. Res. Commun. 42:1–14, 2012.

Merodio, J., R. W. Ogden, and J. Rodríguez. The influence of residual stress on finite deformation elastic response. Int. J. Non-Lin. Mech. 56:43–49, 2013.

Nevo, Erez and Lanir, Yoram The effect of residual strain on the diastolic function of the left ventricle as predicted by a structural model Journal of biomechanics. 27: 1433–1446, 1994.

Novak, V. P., F. C. P. Yin, and J. D. Humphrey. Regional mechanical properties of passive myocardium. J. Biomech. 27:403–412, 1994.

Ogden, R. W. Non-Linear Elastic Deformation. New York, NY: Dover Publications. 1997.

Omens, J. H., D. A. MacKenna, and A. D. McCulloch. Measurement of strain and analysis of stress in resting rat left ventricular myocardium. J. Biomech. 26:665–676, 1993.

Omens, J. H., A. D. McCulloch, and J. C. Criscione. Complex distributions of residual stress and strain in the mouse left ventricle: experimental and theoretical models. Biomech. Model. Mechanobiol. 1:267–277, 2003.

Omens, J. H., T. R, Miller, and J. W. Covell, Relationship between passive tissue strain and collagen uncoiling during healing of infarcted myocardium, Cardiovascular research, 33:351–358, 1997

Rodriguez, EDWARD K and Omens, JEFFREY H and Waldman, LK and McCulloch, AD Effect of residual stress on transmural sarcomere length distributions in rat left ventricle American Journal of Physiology-Heart and Circulatory Physiology 264 (4):H1048–H1056, 1993.

Shams, M., M. Destrade, and R. W. Ogden. Initial stresses in elastic solids: constitutive laws and acoustoelasticity. Wave Motion 48:552–567, 2011.

Sigaeva, T., M. Destrade, E. Di Martino. Multisector method for arteries: The residual stresses of circumferential rings with non-trivial openings, Journal of the Royal Society Interface, 20190023. 2019.

Taber, L. A., N. Hu, P. Tomas, E. B. Clark, and B. Keller. Residual strain in the ventricle of the stage 16–24 chick embryo. Circ. Res. 72:455–462, 1993.

Taber, L. A., and J. D. Humphrey. Stress-modulated growth, residual stress, and vascular heterogeneity. J. Biomech. Eng. 123:528–535, 2001.

Wang, H. M., X. Y. Luo, H. Gao, R. W. Ogden, B. E. Griffith, C. Berry, and T. J. Wang. A modified Holzapfel–Ogden law for a residually stressed finite strain model of the human left ventricle in diastole. Biomech. Model. Mechanobiol. 13:99–113, 2014.

Weis, Sara M and Emery, Jeffrey L and Becker, K David and McBride Jr, Daniel J and Omens, Jeffrey H and McCulloch, Andrew D Myocardial mechanics and collagen structure in the osteogenesis imperfecta murine (oim) Circulation research. 87:663–669, 2000.

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Springer Science and Business Media LLC

Tập 11 Số 4

381-393

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