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Fast, rate-independent, finite element implementation of a 3D constrained mixture model of soft tissue growth and remodeling. (English) Zbl 07337981
Summary: Constrained mixture models of soft tissue growth and remodeling can simulate many evolving conditions in health as well as in disease and its treatment, but they can be computationally expensive. In this paper, we derive a new fast, robust finite element implementation based on a concept of mechanobiological equilibrium that yields fully resolved solutions and allows computation of quasi-equilibrated evolutions when imposed perturbations are slow relative to the adaptive process. We demonstrate quadratic convergence and verify the model via comparisons with semi-analytical solutions for arterial mechanics. We further examine the enlargement of aortic aneurysms for which we identify new mechanobiological insights into factors that affect the nearby non-aneurysmal segment as it responds to the changing mechanics within the diseased segment. Because this new 3D approach can be implemented within many existing finite element solvers, constrained mixture models of growth and remodeling can now be used more widely.
MSC:
76 Fluid mechanics
91 Game theory, economics, finance, and other social and behavioral sciences
Software:
FEBio
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[1] Cowin, S. C., Tissue growth remodeling, Annu. Rev. Biomed. Eng., 6, 77-107 (2004)
[2] Menzel, A.; Kuhl, E., Frontiers in growth and remodeling, Mech. Res. Commun., 42, 1-14 (2012)
[3] Myers, K.; Ateshian, G. A., Interstitial growth and remodeling of biological tissues: tissue composition as state variables, J. Mech. Behav. Biomed. Mater., 29, 544-556 (2014)
[4] Goriely, A., The Mathematics and Mechanics of Biological Growth, Vol. 45 (2017), Springer
[5] Ambrosi, D.; Ben Amar, M.; Cyron, C. J.; DeSimone, A.; Goriely, A.; Humphrey, J. D.; Kuhl, E., Growth and remodelling of living tissues: perspectives, challenges and opportunities, J. R. Soc. Interface, 16, 157, Article 20190233 pp. (2019)
[6] Humphrey, J. D.; Rajagopal, K. R., A constrained mixture model for growth and remodeling of soft tissues, Math. Models Methods Appl. Sci., 12, 03, 407-430 (2002) · Zbl 1021.74026
[7] Latorre, M.; Humphrey, J. D., A mechanobiologically equilibrated constrained mixture model for growth and remodeling of soft tissues, ZAMM-J. Appl. Math. Mech., 98, 2048-2071 (2018)
[8] Latorre, M.; Humphrey, J. D., Mechanobiological stability of biological soft tissues, J. Mech. Phys. Solids, 125, 298-325 (2019)
[9] Davies, K. J.A., Adaptive homeostasis, Mol. Asp. Med., 49, 1-7 (2016)
[10] Latorre, M.; Humphrey, J. D., Critical roles of time-scales in soft tissue growth and remodeling, APL Bioeng., 2, 2, Article 026108 pp. (2018)
[11] Latorre, M., Modeling biological growth and remodeling: contrasting methods, contrasting needs, Curr. Opin. Biomed. Eng., 15, 26-31 (2020)
[12] Baek, S.; Rajagopal, K. R.; Humphrey, J. D., A theoretical model of enlarging intracranial fusiform aneurysms, J. Biomech. Eng., 128, 1, 142-149 (2006)
[13] Humphrey, J. D.; Dufresne, E. R.; Schwartz, M. A., Mechanotransduction and extracellular matrix homeostasis, Nat. Rev. Mol. Cell. Biol., 15, 12, 802-812 (2014)
[14] Valentín, A.; Humphrey, J. D.; Holzapfel, G. A., A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: Theory and numerical verification, Int. J. Numer. Methods Biomed. Eng., 29, 8, 822-849 (2013)
[15] Latorre, M.; Humphrey, J. D., Modeling mechano-driven and immuno-mediated aortic maladaptation in hypertension, Biomech. Model. Mechanobiol., 17, 5, 1497-1511 (2018)
[16] Menzel, A., A fibre reorientation model for orthotropic multiplicative growth, Biomech. Model. Mechanobiol., 6, 5, 303-320 (2007)
[17] Fung, Y. C., Stress, strain, growth, and remodeling of living organisms, (Caseyand, J.; Crochet, M. J., Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids (1995), Birkhäuser: Birkhäuser Basel), 469-482
[18] Humphrey, J. D., Cardiovascular Solid Mechanics: Cells, Tissues and Organs (2002), Springer-Verlag: Springer-Verlag New York
[19] Holzapfel, G. A., Nonlinear Solid Mechanics a Continuum Approach for Engineering (2000), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0980.74001
[20] Epstein, M.; Maugin, G. A., Thermomechanics of volumetric growth in uniform bodies, Int. J. Plast., 16, 7, 951-978 (2000) · Zbl 0979.74006
[21] Himpel, G.; Kuhl, E.; Menzel, A.; Steinmann, P., Computational modelling of isotropic multiplicative growth, CMES Comput. Model. Eng. Sci., 8, 2, 119-134 (2005) · Zbl 1188.74059
[22] Bennett, K. C.; Regueiro, R. A.; Borja, R. I., Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change, Int. J. Plast., 77, 214-245 (2016)
[23] Simó, J. C., Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Comput. Methods Appl. Mech. Engrg., 99, 1, 61-112 (1992) · Zbl 0764.73089
[24] Sanz, M.Á.; Montáns, F. J.; Latorre, M., Computational anisotropic hardening multiplicative elastoplasticity based on the corrector elastic logarithmic strain rate, Comput. Methods Appl. Mech. Engrg., 320, 82-121 (2017) · Zbl 1439.74065
[25] Vignes, C.; Papadopoulos, P., Material growth in thermoelastic continua: theory, algorithmics, and simulation, Comput. Methods Appl. Mech. Engrg., 199, 17-20, 979-996 (2010) · Zbl 1227.74018
[26] Simó, J. C.; Taylor, R. L., Consistent tangent operators for rate-independent elastoplasticity, Comput. Methods Appl. Mech. Engrg., 48, 1, 101-118 (1985) · Zbl 0535.73025
[27] Schwlizerhof, K.; Ramm, E., Displacement dependent pressure loads in nonlinear finite element analyses, Comput. Struct., 18, 6, 1099-1114 (1984) · Zbl 0554.73069
[28] Bathe, K.-J., Finite Element Procedures (2014), Klaus-Jürgen Bathe
[29] Latorre, M.; Montáns, F. J., Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains, Comput. Struct., 163, 56-70 (2016)
[30] Bellini, C.; Ferruzzi, J.; Roccabianca, S.; Di Martino, E. S.; Humphrey, J. D., A microstructurally motivated model of arterial wall mechanics with mechanobiological implications, Ann. Biomed. Eng., 42, 3, 488-502 (2014)
[31] Baek, S.; Valentín, A.; Humphrey, J. D., Biochemomechanics of cerebral vasospasm and its resolution: II. constitutive relations and model simulations, Ann. Biomed. Eng., 35, 9, 1498 (2007)
[32] Miller, K. S.; Khosravi, R.; Breuer, C. K.; Humphrey, J. D., A hypothesis-driven parametric study of effects of polymeric scaffold properties on tissue engineered neovessel formation, Acta Biomater., 11, 283-294 (2015)
[33] Maas, S. A.; Ellis, B. J.; Ateshian, G. A.; Weiss, J. A., Febio: finite elements for biomechanics, J. Biomech. Eng., 134, 1, Article 011005 pp. (2012)
[34] Mousavi, S. J.; Avril, S., Patient-specific stress analyses in the ascending thoracic aorta using a finite-element implementation of the constrained mixture theory, Biomech. Model. Mechanobiol., 16, 5, 1765-1777 (2017)
[35] Zeinali-Davarani, S.; Sheidaei, A.; Baek, S., A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms, Comput. Methods Biomech. Biomed. Eng., 14, 9, 803-817 (2011)
[36] Horvat, N.; Virag, L.; Holzapfel, G. A.; Sorić, J.; Karšaj, I., A finite element implementation of a growth and remodeling model for soft biological tissues: Verification and application to abdominal aortic aneurysms, Comput. Methods Appl. Mech. Engrg., 352, 586-605 (2019) · Zbl 1441.74127
[37] Laubrie, J. D.; Mousavi, J. S.; Avril, S., A new finite-element shell model for arterial growth and remodeling after stent implantation, Int. J. Numer. Methods Biomed. Eng., 36, 1, Article e3282 pp. (2020)
[38] Sussman, T.; Bathe, K.-J., A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comput. Struct., 26, 1-2, 357-409 (1987) · Zbl 0609.73073
[39] Fok, P.-W.; Gou, K., Finite element simulation of intimal thickening in 2D multi-layered arterial cross sections by morphoelasticity, Comput. Methods Appl. Mech. Engrg., 363, Article 112860 pp. (2020) · Zbl 1436.74042
[40] Watton, P. N.; Hill, N. A.; Heil, M., A mathematical model for the growth of the abdominal aortic aneurysm, Biomech. Model. Mechanobiol., 3, 2, 98-113 (2004)
[41] Cyron, C. J.; Wilson, J. S.; Humphrey, J. D., Mechanobiological stability: a new paradigm to understand the enlargement of aneurysms?, J. R. Soc. Interface, 11, 100, Article 20140680 pp. (2014)
[42] Lin, W.; Iafrati, M.; Peattie, R.; Dorfmann, L., Growth and remodeling with application to abdominal aortic aneurysms, J. Eng. Math., 109, 1, 113-137 (2018) · Zbl 1408.74038
[43] Mousavi, S. J.; Farzaneh, S.; Avril, S., Patient-specific predictions of aneurysm growth and remodeling in the ascending thoracic aorta using the homogenized constrained mixture model, Biomech. Model. Mechanobiol., 18, 6, 1895-1913 (2019)
[44] Urabe, G.; Hoshina, K.; Shimanuki, T.; Nishimori, Y.; Miyata, T.; Deguchi, J., Structural analysis of adventitial collagen to feature aging and aneurysm formation in human aorta, J. Vasc. Surg., 63, 5, 1341-1350 (2016)
[45] Gasser, T. C.; Gallinetti, S.; Xing, X.; Forsell, C.; Swedenborg, J.; Roy, J., Spatial orientation of collagen fibers in the abdominal aortic aneurysm’s wall and its relation to wall mechanics, Acta Biomater., 8, 8, 3091-3103 (2012)
[46] Kuhl, E.; Maas, R.; Himpel, G.; Menzel, A., Computational modeling of arterial wall growth, Biomech. Model. Mechanobiol., 6, 5, 321-331 (2007)
[47] Ben Amar, M.; Goriely, A., Growth and instability in elastic tissues, J. Mech. Phys. Solids, 53, 10, 2284-2319 (2005) · Zbl 1120.74336
[48] Ambrosi, D.; Guillou, A.; Di Martino, E. S., Stress-modulated remodeling of a non-homogeneous body, Biomech. Model. Mechanobiol., 7, 1, 63-76 (2008)
[49] Wilson, J. S.; Baek, S.; Humphrey, J. D., Importance of initial aortic properties on the evolving regional anisotropy, stiffness and wall thickness of human abdominal aortic aneurysms, J. R. Soc. Interface, 9, 74, 2047-2058 (2012)
[50] Sheidaei, A.; Hunley, S. C.; Zeinali-Davarani, S.; Raguin, L. G.; Baek, S., Simulation of abdominal aortic aneurysm growth with updating hemodynamic loads using a realistic geometry, Med. Eng. Phys., 33, 1, 80-88 (2011)
[51] Aparicio, P.; Mandaltsi, A.; Boamah, J.; Chen, H.; Selimovic, A.; Bratby, M.; Uberoi, R.; Ventikos, Y.; Watton, P. N., Modelling the influence of endothelial heterogeneity on the progression of arterial disease: application to abdominal aortic aneurysm evolution, Int. J. Numer. Methods Biomed. Eng., 30, 5, 563-586 (2014)
[52] Grytsan, A.; Watton, P. N.; Holzapfel, G. A., A thick-walled fluid-solid-growth model of abdominal aortic aneurysm evolution: application to a patient-specific geometry, J. Biomech. Eng., 137, 3 (2015)
[53] Martufi, G.; Gasser, T. C., Turnover of fibrillar collagen in soft biological tissue with application to the expansion of abdominal aortic aneurysms, J. R. Soc. Interface, 9, 77, 3366-3377 (2012)
[54] Wilson, J. S.; Baek, S.; Humphrey, J. D., Parametric study of effects of collagen turnover on the natural history of abdominal aortic aneurysms, Proc. R. Soc. A, 469, 2150, Article 20120556 pp. (2013) · Zbl 1371.92069
[55] Grytsan, A.; Eriksson, T. S.E.; Watton, P. N.; Gasser, T. C., Growth description for vessel wall adaptation: A thick-walled mixture model of abdominal aortic aneurysm evolution, Materials, 10, 9, 994 (2017)
[56] Braeu, F. A.; Aydin, R. C.; Cyron, C. J., Anisotropic stiffness and tensional homeostasis induce a natural anisotropy of volumetric growth and remodeling in soft biological tissues, Biomech. Model. Mechanobiol., 18, 2, 327-345 (2019)
[57] Fillinger, M. F.; Racusin, J.; Baker, R. K.; Cronenwett, J. L.; Teutelink, A.; Schermerhorn, M. L.; Zwolak, R. M.; Powell, R. J.; Walsh, D. B.; Rzucidlo, E. M., Anatomic characteristics of ruptured abdominal aortic aneurysm on conventional ct scans: implications for rupture risk, J. Vasc. Surg., 39, 6, 1243-1252 (2004)
[58] Pappu, S.; Dardik, A.; Tagare, H.; Gusberg, R. J., Beyond fusiform and saccular: a novel quantitative tortuosity index may help classify aneurysm shape and predict aneurysm rupture potential, Ann. Vasc. Surg., 22, 1, 88-97 (2008)
[59] Hariton, I.; deBotton, G.; Gasser, T. C.; Holzapfel, G. A., Stress-driven collagen fiber remodeling in arterial walls, Biomech. Model. Mechanobiol., 6, 3, 163-175 (2007)
[60] Figueroa, C. A.; Baek, S.; Taylor, C. A.; Humphrey, J. D., A computational framework for fluid-solid-growth modeling in cardiovascular simulations, Comput. Methods Appl. Mech. Engrg., 198, 45, 3583-3602 (2009) · Zbl 1229.74097
[61] Ogden, R. W., Non-Linear Elastic Deformations (1997), Dover Publications: Dover Publications Mineola, New York
[62] Latorre, M.; Montáns, F. J., Stress and strain mapping tensors and general work-conjugacy in large strain continuum mechanics, Appl. Math. Model., 40, 5, 3938-3950 (2016) · Zbl 1459.74029
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