# Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 2: dispersed arrangement of collagen fibers

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## Abstract

This paper investigates on the relationship between the arrangement of collagen fibers within soft tissues and parameters of constitutive models. Starting from numerical experiments based on biaxial loading conditions, the study addresses both the direct (from structure to mechanics) and the inverse (from mechanics to structure) problems, solved introducing optimization problems for model calibration and regression analysis. A campaign of parametric analyses is conducted in order to consider fibers distributions with different main orientation and angular dispersion. Different anisotropic constitutive models are employed, accounting for fibers dispersion either with a generalized structural approach or with an increasing number of strain energy terms. Benchmark data sets, toward which constitutive models are fitted, are built by employing a multiscale description of fiber nonlinearities and accounting for fibers dispersion with an angular integration method. Results show how the optimal values of constitutive parameters obtained from model calibration vary as a function of fibers arrangement and testing protocol. Moreover, the fitting capabilities of constitutive models are discussed. A novel strategy for model calibration is also proposed, in order to obtain a robust accuracy with respect to different loading conditions starting from a low number of mechanical tests. Furthermore, novel results useful for the inverse determination of the mean angle and the variance of fibers distribution are obtained. Therefore, the study contributes: to better design procedures for model calibration; to account for mechanical alterations due to remodeling mechanisms; and to gain structural information in a nondestructive way.

## Keywords

Soft tissue mechanics Constitutive models Parameters identification Inverse analysis Fiber dispersion## Notes

### Acknowledgements

M. Marino acknowledges that this work has been carried out within the framework of the SMART BIOTECS alliance between the Technical University of Braunschweig and the Leibniz University of Hannover. This initiative is financially supported by the Ministry of Science and Culture (MWK) of Lower Saxony, Germany.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Auricchio F, Conti M, Ferrara A (2014) How constitutive model complexity can affect the capability to fit experimental data: a focus on human carotid arteries and extension/inflation data. Arch Comput Methods Eng 21(3):273–292MathSciNetCrossRefzbMATHGoogle Scholar
- Balzani D, Brinkhues S, Holzapfel GA (2012) Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls. Comput Methods Appl Mech Eng 213–216:139–151MathSciNetCrossRefzbMATHGoogle Scholar
- Cortes DH, Lake SP, Kadlowec JA, Soslowsky LJ, Elliott DM (2010) Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches. Biomech Model Mechanobiol 9(5):651–658CrossRefGoogle Scholar
- Cyron CJ, Humphrey JD (2017) Growth and remodeling of load-bearing biological soft tissues. Meccanica 52:645–664MathSciNetCrossRefGoogle Scholar
- Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3:15–35CrossRefGoogle Scholar
- Gasser TC, Gallinetti S, Xing X, Forsell C, Swedenborg J, Roy J (2012) Spatial orientation of collagen fibers in the abdominal aortic aneurysm’s wall and its relation to wall mechanics. Acta Biomater 8:3091–3103CrossRefGoogle Scholar
- Gilchrist MD, Murphy JG, Pierrat B, Saccomandi G (2017) Slight asymmetry in the winding angles of reinforcing collagen can cause large shear stresses in arteries and even induce buckling. Meccanica 52:3417–3429MathSciNetCrossRefzbMATHGoogle Scholar
- Gizzi A, Pandolfi A, Vasta M (2016) Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers. Mech Mater 92:119–138CrossRefGoogle Scholar
- Gizzi A, Pandolfi A, Vasta M (2018) A generalized statistical approach for modeling fiber-reinforced materials. J Eng Math 109:211–226MathSciNetCrossRefzbMATHGoogle Scholar
- Holzapfel GA, Ogden RW (2015) On the tension–compression switch in soft fibrous solids. Eur J Mech A/Solids 49:561–569MathSciNetCrossRefzbMATHGoogle Scholar
- Holzapfel GA, Ogden RW (2017) Comparison of two model frameworks for fiber dispersion in the elasticity of soft biological tissues. Eur J Mech A/Solids 66:193–200MathSciNetCrossRefzbMATHGoogle Scholar
- Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast Phys Sci Solids 61(1–3):1–48MathSciNetzbMATHGoogle Scholar
- Holzapfel GA, Niestrawska JA, Ogden RW, Reinisch AJ, Schriefl AJ (2015) Modelling non-symmetric collagen fibre dispersion in arterial walls. J R Soc Interface 12:20150188CrossRefGoogle Scholar
- Horgan CO, Murphy JG (2011) Simple shearing of soft biological tissues. Proc R Soc Lond A Math Phys Eng Sci 467(2127):760–777MathSciNetCrossRefzbMATHGoogle Scholar
- Joyce EM, Liao J, Schoen FJ, Mayer JE Jr, Sacks MS (2009) Functional collagen fiber architecture of the pulmonary heart valve cusp. Ann Thorac Surg 87:1240–1249CrossRefGoogle Scholar
- Kamensky AV, Dzenis YA, Jaffar Kazmi SA, Pemberton MA, Pipinos II, Phillips NY, Herber K, Woodford T, Bowen RE, Lomneth CS, MacTaggart JN (2014) Biaxial mechanical properties of the human thoracic and abdominal aorta, common carotid, subclavian, renal and common iliac arteries. Biomech Model Mechanobiol 13:1341–1359CrossRefGoogle Scholar
- Krasny W, Morin C, Magoariec H, Avril S (2017) A comprehensive study of layer-specific morphological changes in the microstructure of carotid arteries under uniaxial load. Acta Biomater 57:342–351CrossRefGoogle Scholar
- Lally C, Reid AJ, Prendergast PJ (2004) Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann Biomed Eng 32:1355–1364CrossRefGoogle Scholar
- Lanir Y (1983) Constitutive equations for fibrous connective tissues. J Biomech 16:1–12CrossRefGoogle Scholar
- Maceri F, Marino M, Vairo G (2010) A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement. J Biomech 43:355–363CrossRefGoogle Scholar
- Marino M, Vairo G (2014) Computational modelling of soft tissues and ligaments. In: Jin Z (ed) Computational modelling of biomechanics and biotribology in the musculoskeletal system, vol 81. Woodhead publishing series in biomaterials. Woodhead Publishing Limited, Cambridge, pp 141–172CrossRefGoogle Scholar
- Marino M, Wriggers P (2017) Finite strain response of crimped fibers under uniaxial traction: an analytical approach applied to collagen. J Mech Phys Solids 98:429–453MathSciNetCrossRefGoogle Scholar
- Marino M, von Hoegen M, Schröder J, Wriggers P (2018) Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 1: micro- and nano-structure of collagen fibers. Biomech Model Mechanobiol 17(4):1011–1036CrossRefGoogle Scholar
- Martufi G, Gasser TC (2011) A constitutive model for vascular tissue that integrates fibril, fiber and continuum levels with application to the isotropic and passive properties of the infrarenal aorta. J Biomech 44:2544–2550CrossRefGoogle Scholar
- Mathworks (2014) Global Optimization Toolbox User's Guide (r2014a). The MathWorks Inc., Natick, MA, USAGoogle Scholar
- Menzel A, Kuhl E (2012) Frontiers in growth and remodeling. Mech Res Commun 42:1–14CrossRefGoogle Scholar
- Montanino A, Angelillo M, Pandolfi A (2018a) Modelling with a meshfree approach the cornea-aqueous humor interaction during the air puff test. J Mech Behav Biomed Mater 77:205–216CrossRefGoogle Scholar
- Montanino A, Gizzi A, Vasta M, Angelillo M, Pandolfi A (2018b) Modeling the biomechanics of the human cornea accounting for local variations of the collagen fibril architecture. Z Angew Math Mech 00:1–13Google Scholar
- Niestrawska JA, Viertler C, Regitnig P, Cohnert TU, Sommer G, Holzapfel GA (2016) Microstructure and mechanics of healthy and aneurysmatic abdominal aortas: experimental analysis and modelling. J R Soc Interface 13:20160620CrossRefGoogle Scholar
- O’Connell MK, Murthy S, Phan S, Xu C, Buchanan J, Spilker R, Dalman RL, Zarins CK, Denk W, Taylor CA (2008) The three-dimensional micro- and nanostructure of the aortic medial lamellar unit measured using 3D confocal and electron microscopy imaging. Matrix Biol 27:171–181CrossRefGoogle Scholar
- Okamoto RJ, Wagenseil JE, DeLong WR, Peterson SJ, Kouchoukos NT, Sundt TM (2002) Mechanical properties of dilated human ascending aorta. Ann Biomed Eng 30:624–635CrossRefGoogle Scholar
- Pandolfi A, Holzapfel GA (2008) Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations. ASME J Biomech Eng 130:061006–12CrossRefGoogle Scholar
- Pandolfi A, Vasta M (2012) Fiber distributed hyperelastic modeling of biological tissues. Mech Mater 44:151–162CrossRefGoogle Scholar
- Polzer S, Gasser TC, Novak K, Man V, Tichy M, Skacel P, Bursa J (2015) Structure-based constitutive model can accurately predict planar biaxial properties of aortic wall tissue. Acta Biomater 14:133–145CrossRefGoogle Scholar
- Raghavan ML, Marshall MW, Webster W, Vorp DA (1996) Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model. Ann Biomed Eng 24:573–582CrossRefGoogle Scholar
- Sacks MS (2003) Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. J Biomech Eng 125:280–287CrossRefGoogle Scholar
- Sacks MS, Zhang W, Wognum S (2016) A novel fibre-ensemble level constitutive model for exogenous cross-linked collagenous tissues. Interface Focus 6(1):20150090CrossRefGoogle Scholar
- Schriefl AJ, Zeindlinger G, Pierce DM, Regitnig P, Holzapfel GA (2012) Determination of the layer-specific distributed collagen fibre orientations in human thoracic and abdominal aortas and common iliac arteries. J R Soc Interface 9:1275–1286CrossRefGoogle Scholar
- Schröder J (2010) Anisotropic polyconvex energies. In: Schröder J, Neff P (eds) Poly-, quasi- and rank-one convexity in applied mechanics, vol 516. Springer, Berlin, pp 53–105CrossRefGoogle Scholar
- Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445MathSciNetCrossRefzbMATHGoogle Scholar
- Sommer G, Haspinger DC, Andrä M, Sacherer M, Viertler C, Regitnig P, Holzapfel GA (2015) Quantification of shear deformations and corresponding stresses in the biaxially tested human myocardium. Ann Biomed Eng 43:2334–2348CrossRefGoogle Scholar
- Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physics, vol 1. Academic Press, New York, pp 239–353Google Scholar
- Spencer AJM (1987) Isotropic polynomial invariants and tensor functions. In: Boehler JP (ed) Applications of tensor functions in solid mechanics, volume 292 of CISM courses and lectures. Springer, Berlin, pp 141–170CrossRefGoogle Scholar
- Strijkers GJ, Bouts A, Blankesteijn WM, Peeters THJM, Vilanova A, van Prooijen MC, Sanders HMHF, Heijman E, Nicolay K (2009) Diffusion tensor imaging of left ventricular remodeling in response to myocardial infarction in the mouse. NRM Biomed 22:182–190Google Scholar
- Whittaker P, Boughner DR, Kloner RA (1989) Analysis of healing after myocardial infarction using polarized light microscopy. Am J Pathol 134:879–893Google Scholar
- Zemánek M, Burša J, Děták M (2009) Biaxial tension tests with soft tissues of arterial wall. Eng Mech 16:3–11Google Scholar
- Zimmerman SD, Karlon WJ, Holmes JW, Omens JH, Covell JW (2000) Structural and mechanical factors influencing infarct scar collagen organization. Am J Physiol Heart Circ Physiol 278:H194–H200CrossRefGoogle Scholar