A generic approach towards finite growth with examples of athlete’s heart, cardiac dilation, and cardiac wall thickening. (English) Zbl 1200.74109
Summary: The objective of this work is to establish a generic continuum-based computational concept for finite growth of living biological tissues. The underlying idea is the introduction of an incompatible growth configuration which naturally introduces a multiplicative decomposition of the deformation gradient into an elastic and a growth part. The two major challenges of finite growth are the kinematic characterization of the growth tensor and the identification of mechanical driving forces for its evolution. Motivated by morphological changes in cell geometry, we illustrate a micromechanically motivated ansatz for the growth tensor for cardiac tissue that can capture both strain-driven ventricular dilation and stress-driven wall thickening. Guided by clinical observations, we explore three distinct pathophysiological cases: athlete’s heart, cardiac dilation, and cardiac wall thickening. We demonstrate the computational solution of finite growth within a fully implicit incremental iterative Newton-Raphson based finite element solution scheme. The features of the proposed approach are illustrated and compared for the three different growth pathologies in terms of a generic bi-ventricular heart model.
Keywords:
biological growth; biological material; constitutive behavior; finite elements; numerical algorithmsReferences:
| [1] | Alastrue, V.; Martinez, M. A.; Doblare, M., Modelling adaptative volumetric finite growth in patient-specific residually stressed arteries, J. Biomech., 41, 1773-1781 (2009) |
| [2] | Alford, P. W.; Taber, L. A., Computational study of growth and remodelling in the aortic arch, Comput. Meth. Biomech. Biomed. Eng., 11, 525-538 (2008) |
| [3] | Ambrosi, D.; Mollica, F., On the mechanics of a growing tumor, Int. J. Eng. Sci., 40, 1297-1316 (2002) · Zbl 1211.74161 |
| [4] | Ambrosi, D.; Guillou, A.; Di Martino, E. S., Stress-modulated remodeling of a non-homogeneous body, Biomech. Model. Mechanobiol., 7, 63-76 (2008) |
| [5] | Ambrosi, D., Ateshian, G.A., Arruda, E.M., Ben Amar, M., Cowin, S.C., Dumais, J., Goriely, A., Holzapfel, G.A., Humphrey, J.D., Kemkemer, R., Kuhl, E., Ma, J., Olberding, J.E., Taber, L.A., Vandiver, R., Garikipati, K., under review. Perspectives on biological growth and remodeling.; Ambrosi, D., Ateshian, G.A., Arruda, E.M., Ben Amar, M., Cowin, S.C., Dumais, J., Goriely, A., Holzapfel, G.A., Humphrey, J.D., Kemkemer, R., Kuhl, E., Ma, J., Olberding, J.E., Taber, L.A., Vandiver, R., Garikipati, K., under review. Perspectives on biological growth and remodeling. |
| [6] | Ben Amar, M.; Goriely, A., Growth and instability in elastic tissues, J. Mech. Phys. Solids, 53, 2284-2319 (2005) · Zbl 1120.74336 |
| [7] | Berne, R.M., Levy, M.N., 2001. Cardiovascular Physiology. The Mosby Monograph Series.; Berne, R.M., Levy, M.N., 2001. Cardiovascular Physiology. The Mosby Monograph Series. |
| [8] | Boyce, M. C.; Weber, G. G.; Parks, D. M., On the kinematics of finite strain plasticity, J. Mech. Phys. Solids, 37, 647-665 (1989) · Zbl 0692.73043 |
| [9] | Carter, D. R.; Hayes, W. C., The behavior of bone as a twophase porous structure, J. Bone Jt. Surg., 59, 785-794 (1977) |
| [10] | Chen, Y. C.; Hoger, A., Constitutive functions of elastic materials in finite growth and deformation, J. Elast., 59, 175-193 (2000) · Zbl 0987.74009 |
| [11] | Cheng, A.; Nguyen, T. C.; Malinowski, M.; Ennis, D. B.; Daughters, G. T.; Miller, D. C.; Ingels, N. B., Transmural left ventricular shear strain alterations adjacent to and remote from infarcted myocardium, J. Heart Valve Dis., 15, 209-218 (2006) |
| [12] | Ciarletta, P.; Ben Amar, M., A finite dissipative theory of temporary interfibrillar bridges in the extracellular matrix of ligaments and tendons, J. R. Soc. Interface, 6, 909-924 (2009) |
| [13] | Cowin, S. C.; Hegedus, D. H., Bone remodeling. 1. Theory of adaptive elasticity, J. Elast., 6, 313-326 (1976) · Zbl 0335.73028 |
| [14] | Cowin, S. C., Tissue growth and remodeling, Ann. Rev. Biomed. Eng., 6, 77-107 (2004) |
| [15] | Dumais, J.; Shaw, S. L.; Steele, C. R.; Long, S. R.; Ray, P. M., An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth, Int. J. Dev. Biol., 50, 209-222 (2006) |
| [16] | Emmanouilides, G. C.; Riemenschneider, R. A.; Allen, H. D.; Gutgesell, H. P., Moss and Adams’ Heart Disease in Infants, Children, and Adolescents (1994), Lippincott Williams & Wilkins |
| [17] | Epstein, M.; Maugin, G. A., Thermomechanics of volumetric growth in uniform bodies, Int. J. Plast., 16, 951-978 (2000) · Zbl 0979.74006 |
| [18] | Ekblom, B.; Hermansen, L., Cardiac output in athletes, J. Appl. Physiol., 25, 619-625 (1968) |
| [19] | Figueroa, C. A.; Baek, S.; Taylor, C. A.; Humphrey, J. D., A computational framework for fluid-solid-growth modeling in cardiovascular simulations, Comput. Meth. Appl. Mech. Eng., 198, 3583-3602 (2009) · Zbl 1229.74097 |
| [20] | Garciarena, C. D.; Pinilla, O. A.; Nolly, M. B.; Laguens, R. P.; Escudero, E. M.; Cingolani, H. E.; Ennis, I. L., Endurance training in the spontaneously hypertensive rat: conversion of pathological into physiological cardiac hypertrophy, Hypertension, 53, 708-714 (2009) |
| [21] | Garikipati, K.; Arruda, E. M.; Grosh, K.; Narayanan, H.; Calve, S., A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics, J. Mech. Phys. Solids, 52, 1595-1625 (2004) · Zbl 1159.74381 |
| [22] | Garikipati, K., The kinematics of biological growth, Appl. Mech. Rev., 62, 030801-1-030801-7 (2009) |
| [23] | Göktepe, S.; Kuhl, E., Computational modeling of cardiac electrophysiology: a novel finite element approach, Int. J. Numer. Meth. Eng., 79, 156-178 (2009) · Zbl 1171.92310 |
| [24] | Göktepe, S.; Kuhl, E., Electromechanics of cardiac tissue: a unified approach to the fully coupled excitation-contraction problem, Comput. Mech., 45, 227-243 (2010) · Zbl 1183.78031 |
| [25] | Göktepe, S., Abilez, O.J., Parker, K.K., Kuhl, E., 2010a. A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis. J. Theor. Biol. 265, 433-442.; Göktepe, S., Abilez, O.J., Parker, K.K., Kuhl, E., 2010a. A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis. J. Theor. Biol. 265, 433-442. · Zbl 1461.92010 |
| [26] | Göktepe, S., Acharya, S.N.S., Wong, J., Kuhl, E., 2010b. Computational modeling of passive myocardium. Int. J. Numer. Meth. Biomed. Eng., doi:10.1002/cnm.1402; Göktepe, S., Acharya, S.N.S., Wong, J., Kuhl, E., 2010b. Computational modeling of passive myocardium. Int. J. Numer. Meth. Biomed. Eng., doi:10.1002/cnm.1402 · Zbl 1207.92006 |
| [27] | Göktepe, S.; Bothe, W.; Kvitting, J. P.; Swanson, J. C.; Ingels, N. B.; Miller, D. C.; Kuhl, E., Anterior mitral leaflet curvature in the beating ovine heart: a case study using video fluoroscopic markers and subdivision surfaces, Biomech. Model. Mechanobiol., 9, 281-293 (2010) |
| [28] | Goriely, A.; Ben Amar, M., On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity, Biomech. Model. Mechanobiol., 6, 289-296 (2007) |
| [29] | Himpel, G.; Kuhl, E.; Menzel, A.; Steinmann, P., Computational modeling of isotropic multiplicative growth, Comput. Mod. Eng. Sci., 8, 119-134 (2005) · Zbl 1188.74059 |
| [30] | Himpel, G.; Menzel, A.; Kuhl, E.; Steinmann, P., Time-dependent fibre reorientation of transversely isotropic continua—finite element formulation and consistent linearization, Int. J. Numer. Meth. Eng., 73, 1413-1433 (2008) · Zbl 1169.74043 |
| [31] | Holzapfel, G. A.; Ogden, R. W., Constitutive modelling of passive myocardium. A structurally-based framework for material characterization, Philos. Trans. R. Soc. London A, 367, 3445-3475 (2009) · Zbl 1185.74060 |
| [32] | Hsu, H. F., The influence of mechanical loads on the form of a growing elastic body, J. Biomech., 1, 303-311 (1968) |
| [33] | Humphrey, J. D., Cardiovascular Solid Mechanics (2002), Springer Verlag: Springer Verlag Berlin, Heidelberg, New York |
| [34] | Humphrey, J. D.; Rajagopal, K. R., A constrained mixture model for growth and remodeling of soft tissues, Math. Mod. Meth. Appl. Sci., 12, 407-430 (2002) · Zbl 1021.74026 |
| [35] | Humphrey, J. D., Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels, Cell. Biochem. Biophys., 50, 53-78 (2008) |
| [36] | Hunter, J. J.; Chien, K. R., Signaling pathways for cardiac hypertrophy and failure, New England J. Med., 341, 1276-1283 (1999) |
| [37] | Imatani, S.; Maugin, G. A., A constitutive model for material growth and its application to three-dimensional analysis, Mech. Res. Commun., 29, 477-483 (2002) · Zbl 1029.74004 |
| [38] | Klisch, S. M.; Chen, S. S.; Hoger, A., A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants, J. Biomech. Eng., 125, 169-179 (2003) |
| [39] | Kotikanyadanam, M.; Göktepe, S.; Kuhl, E., Computational modeling of electrocardiograms—a finite element approach towards cardiac excitation, Commun. Numer. Meth. Eng., 26, 524-533 (2010) · Zbl 1187.92062 |
| [40] | Kroon, W.; Delhaas, T.; Arts, T.; Bovendeerd, P., Computational modeling of volumetric soft tissue growth: application to the cardiac left ventricle, Biomech. Model. Mechanobiol., 8, 309-310 (2009) |
| [41] | Kuhl, E.; Steinmann, P., Mass- and volume specific views on thermodynamics for open systems, Proc. Royal Soc., 459, 2547-2568 (2003) · Zbl 1092.80500 |
| [42] | Kuhl, E.; Steinmann, P., On spatial and material settings of thermohyperelastodynamics for open systems, Acta Mech., 160, 179-217 (2003) · Zbl 1064.74007 |
| [43] | Kuhl, E.; Menzel, A.; Steinmann, P., Computational modeling of growth—a critical review, a classification of concepts and two new consistent approaches, Comp. Mech., 32, 71-88 (2003) · Zbl 1151.74385 |
| [44] | Kuhl, E.; Steinmann, P., Theory and numerics of geometrically non-linear open system mechanics, Int. J. Numer. Meth. Eng., 58, 1593-1615 (2003) · Zbl 1032.74504 |
| [45] | Kuhl, E.; Garikipati, K.; Arruda, E. M.; Grosh, K., Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network, J. Mech. Phys. Solids, 53, 1552-1573 (2005) · Zbl 1120.74635 |
| [46] | Kuhl, E.; Maas, R.; Himpel, G.; Menzel, A., Computational modeling of arterial wall growth: attempts towards patient-specific simulations based on computer tomography, Biomech. Mod. Mechanobiol., 6, 321-331 (2007) |
| [47] | Kuhl, E.; Holzapfel, G. A., A continuum model for remodeling in living structures, J. Mat. Sci., 2, 8811-8823 (2007) |
| [48] | Kumar, V.; Abbas, A. K.; Fausto, N., Robbins and Cotran Pathologic Basis of Disease (2005), Elsevier Saunders |
| [49] | Lee, E. H., Elastic-plastic deformation at finite strains, J. Appl. Mech., 36, 1-6 (1969) · Zbl 0179.55603 |
| [50] | Libby, P., Bonow, R.O., Mann, D.L., Zipes, D.P., 2007. Braunwald’s Heart Disease, Saunders.; Libby, P., Bonow, R.O., Mann, D.L., Zipes, D.P., 2007. Braunwald’s Heart Disease, Saunders. |
| [51] | Lubarda, A.; Hoger, A., On the mechanics of solids with a growing mass, Int. J. Solids Structures, 39, 4627-4664 (2002) · Zbl 1045.74035 |
| [52] | Maron, B. J.; McKenna, W. J., American College of Cardiology/European Society of Cardiology: clinical expert consensus document on hypertrophy cardiomyopathy, J. Am. College Cardiology, 42, 1687-1713 (2003) |
| [53] | Maron, B. J.; Pelliccia, A., The heart of trained athletes: cardiac remodeling and the risks of sports, including sudden death, Circ., 114, 1633-1644 (2006) |
| [54] | Menzel, A., Modelling of anisotropic growth in biological tissues—a new approach and computational aspects, Biomech. Model. Mechanobiol., 3, 147-171 (2005) |
| [55] | Mihl, C.; Dassen, W. R.M.; Kuipers, H., Cardiac remodelling: concentric versus eccentric hypertrophy in strength and endurance athletes, Nederlands Heart J., 16, 129-133 (2008) |
| [56] | Naghdi, P., A critical review of the state of finite plasticity, J. Appl. Math. Phys., 41, 315-394 (1990) · Zbl 0712.73032 |
| [57] | Opie, L. H., Heart Physiology: From Cell to Circulation (2003), Lippincott Williams & Wilkins |
| [58] | Pluim, B. M.; Zwinderman, A. H.; van der Laarse, A.; van der Wall, E. E., The athlete’s heart: a meta-analysis of cardiac structure and function, Circulation, 101, 336-344 (2000) |
| [59] | Rodriguez, E. K.; Hoger, A.; McCulloch, A. D., Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27, 455-467 (1994) |
| [60] | Sedehi, D.; Ashley, E. A., Defining the limits of athlete’s heart: implications for screening in diverse populations, Circulation, 121, 1066-1068 (2010) |
| [61] | Skalak, R., 1981. Growth as a finite displacement field. In: Carlson, D.E., Shield, R.T., (Eds.), IUTAM Symposium on Finite Elasticity. Martinus Nijhoff, pp. 347-355.; Skalak, R., 1981. Growth as a finite displacement field. In: Carlson, D.E., Shield, R.T., (Eds.), IUTAM Symposium on Finite Elasticity. Martinus Nijhoff, pp. 347-355. · Zbl 0543.73128 |
| [62] | Skalak, R.; Farrow, D. A.; Hoger, A., Kinematics of surface growth, J. Math. Biol., 35, 869-907 (1997) · Zbl 0883.92005 |
| [63] | Taber, L. A., Biomechanics of growth, remodeling and morphogenesis, Appl. Mech. Rev., 48, 487-545 (1995) |
| [64] | Taber, L. A.; Humphrey, J. D., Stress-modulated growth, residual stress, and vascular heterogeneity, J. Biomech. Eng., 123, 528-535 (2001) |
| [65] | Thompson, A. W., On Growth and Form (1917), Cambridge University Press |
| [66] | Xiao, H.; Bruhns, O. T.; Meyers, A., Elastoplasticity beyond small deformations, Acta. Mech., 182, 31-111 (2006) · Zbl 1116.74005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
