Biomechanics and Modeling in Mechanobiology

, Volume 17, Issue 1, pp 31–53 | Cite as

An integrated inverse model-experimental approach to determine soft tissue three-dimensional constitutive parameters: application to post-infarcted myocardium

  • Reza Avazmohammadi
  • David S. Li
  • Thomas Leahy
  • Elizabeth Shih
  • João S. Soares
  • Joseph H. Gorman
  • Robert C. Gorman
  • Michael S. Sacks
Original Paper


Knowledge of the complete three-dimensional (3D) mechanical behavior of soft tissues is essential in understanding their pathophysiology and in developing novel therapies. Despite significant progress made in experimentation and modeling, a complete approach for the full characterization of soft tissue 3D behavior remains elusive. A major challenge is the complex architecture of soft tissues, such as myocardium, which endows them with strongly anisotropic and heterogeneous mechanical properties. Available experimental approaches for quantifying the 3D mechanical behavior of myocardium are limited to preselected planar biaxial and 3D cuboidal shear tests. These approaches fall short in pursuing a model-driven approach that operates over the full kinematic space. To address these limitations, we took the following approach. First, based on a kinematical analysis and using a given strain energy density function (SEDF), we obtained an optimal set of displacement paths based on the full 3D deformation gradient tensor. We then applied this optimal set to obtain novel experimental data from a 1-cm cube of post-infarcted left ventricular myocardium. Next, we developed an inverse finite element (FE) simulation of the experimental configuration embedded in a parameter optimization scheme for estimation of the SEDF parameters. Notable features of this approach include: (i) enhanced determinability and predictive capability of the estimated parameters following an optimal design of experiments, (ii) accurate simulation of the experimental setup and transmural variation of local fiber directions in the FE environment, and (iii) application of all displacement paths to a single specimen to minimize testing time so that tissue viability could be maintained. Our results indicated that, in contrast to the common approach of conducting preselected tests and choosing an SEDF a posteriori, the optimal design of experiments, integrated with a chosen SEDF and full 3D kinematics, leads to a more robust characterization of the mechanical behavior of myocardium and higher predictive capabilities of the SEDF. The methodology proposed and demonstrated herein will ultimately provide a means to reliably predict tissue-level behaviors, thus facilitating organ-level simulations for efficient diagnosis and evaluation of potential treatments. While applied to myocardium, such developments are also applicable to characterization of other types of soft tissues.


Soft tissue mechanics Inverse modeling Optimal design of experiments Constitutive models Myocardium Cardiac mechanics 



This work was supported in part by the US National Institutes of Health grants 1F32 HL132543 to R.A., and T32 EB007507 to D.S.L. We’d like to thank John Lesicko for helping in the development of the TRIAX device, MaiQuyen Nguyen for assistance in histological analysis of the post-infarcted myocardium specimen, and Samarth Raut for the development of the initial FE models.


  1. Atwood CL (1969) Optimal and efficient designs of experiments. Ann Math Stat 40:1570–1602MathSciNetCrossRefMATHGoogle Scholar
  2. Avazmohammadi R, Hill MR, Simon MA, Zhang W, Sacks MS (2016) A novel constitutive model for passive right ventricular myocardium: evidence for myofiber-collagen fiber mechanical coupling. Biomech Model Mechanobiol 16(2):561–581Google Scholar
  3. Beck JV, Arnold KJ (1977) Parameter estimation in engineering and science. Wiley, Hoboken, NJGoogle Scholar
  4. Chabiniok R et al (2016) Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics. Interface focus 6:20150083CrossRefGoogle Scholar
  5. Costa KD, Holmes JW, MeCulloch AD (2001) Modelling cardiac mechanical properties in three dimensions. Phil Trans R Soc Lond 359:1233–1250CrossRefMATHGoogle Scholar
  6. Dokos S, LeGrice IJ, Smaill BH, Kar J, Young AA (2000) A triaxial-measurement shear-test device for soft biological tissues. J Biomech Eng 122:471–478CrossRefGoogle Scholar
  7. Dokos S, Smaill BH, Young AA, LeGrice IJ (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:H2650–2659CrossRefGoogle Scholar
  8. Freed A, Srinivasa A (2015) Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient. Acta Mech 226:2645–2670MathSciNetCrossRefMATHGoogle Scholar
  9. Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  10. Göktepe S, Acharya S, Wong J, Kuhl E (2011) Computational modeling of passive myocardium. Int J Numer Methods Biomed Eng 27:1–12MathSciNetCrossRefMATHGoogle Scholar
  11. Green AE, Adkins JE (1960) Large elastic deformations and non-linear continuum mechanics. Clarendon Press, Wotton-under-EdgeMATHGoogle Scholar
  12. Guccione JM, McCulloch AD, Waldman LK (1991) Passive material properties of intact ventricular myocardium determined from a cylindrical model. J Biomech Eng 113:42–55CrossRefGoogle Scholar
  13. Gupta KB, Ratcliffe MB, Fallert MA, Edmunds L, Bogen DK (1994) Changes in passive mechanical stiffness of myocardial tissue with aneurysm formation. Circulation 89:2315–2326CrossRefGoogle Scholar
  14. Holzapfel GA, Ogden RW (2008) On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework. Math Mech Solids 14:474–489CrossRefMATHGoogle Scholar
  15. Holzapfel GA, Ogden RW (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans A Math Phys Eng Sci 367:3445–3475. doi: 10.1098/rsta.2009.0091 MathSciNetCrossRefMATHGoogle Scholar
  16. Hoppin F, Lee G, Dawson S (1975) Properties of lung parenchyma in distortion. J Appl Physiol 39:742–751CrossRefGoogle Scholar
  17. Humphrey J, Yin F (1987) A new constitutive formulation for characterizing the mechanical behavior of soft tissues. Biophys J 52:563–570CrossRefGoogle Scholar
  18. Humphrey JD, Strumpf RK, Yin FC (1990a) Determination of a constitutive relation for passive myocardium: I. A new functional form. J Biomech Eng 112:333–339CrossRefGoogle Scholar
  19. Humphrey JD, Strumpf RK, Yin FC (1990b) Determination of a constitutive relation for passive myocardium: II. Parameter estimation. J Biomech Eng 112:340–346CrossRefGoogle Scholar
  20. Intrigila B, Melatti I, Tofani A, Macchiarelli G (2007) Computational models of myocardial endomysial collagen arrangement. Comput Methods Programs Biomed 86:232–244. doi: 10.1016/j.cmpb.2007.03.004 CrossRefGoogle Scholar
  21. Lanir Y, Lichtenstein O, Imanuel O (1996) Optimal design of biaxial tests for structural material characterization of flat tissues. J Biomech Eng 118:41–47Google Scholar
  22. LeGrice IJ, Smaill B, Chai L, Edgar S, Gavin J, Hunter PJ (1995) Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Am J Physiol Heart Circul Physiol 269:H571–H582CrossRefGoogle Scholar
  23. Mehrabadi MM, Cowin SC (1990) Eigentensors of linear anisotropic elastic materials. Q J Mech Appl Math 43:15–41MathSciNetCrossRefMATHGoogle Scholar
  24. Morita M et al (2011) Modification of infarct material properties limits adverse ventricular remodeling. Ann Thorac Surg 92:617–624CrossRefGoogle Scholar
  25. Nathanson MH, Saidel GM (1985) Multiple-objective criteria for optimal experimental design: application to ferrokinetics. Am J Physiol Regul Integr Comp Physiol 248:R378–R386CrossRefGoogle Scholar
  26. Ogden RW (1997) Non-linear elastic deformations. Courier Corporation, North ChelmsfordGoogle Scholar
  27. Pukelsheim F (2006) Optimal design of experiments. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  28. Rivlin RS, Saunders D (1951) Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber. Philos Trans R Soc Lond A Math Phys Eng Sci 243:251–288Google Scholar
  29. Robinson TF, Cohen-Gould L, Factor SM (1983) Skeletal framework of mammalian heart muscle. Arrangement of inter-and pericellular connective tissue structures. Lab Invest 49:482–498Google Scholar
  30. Robinson TF, Factor SM, Sonnenblick EH (1986) The heart as a suction pump. Sci Am 254:84–91CrossRefGoogle Scholar
  31. Sacks MS, Chuong CJ (1993) A constitutive relation for passive right-ventricular free wall myocardium. J Biomech 26:1341–1345CrossRefGoogle Scholar
  32. Sacks MS, Smith DB, Hiester ED (1997) A small angle light scattering device for planar connective tissue microstructural analysis. Ann Biomed Eng 25:678–689CrossRefGoogle Scholar
  33. Schmid H, Nash MP, Young AA, Hunter PJ (2006) Myocardial material parameter estimation—a comparative study for simple shear. J Biomech Eng 128:742–750. doi: 10.1115/1.2244576 CrossRefGoogle Scholar
  34. Schmid H O’Callaghan P, Nash MP, Lin W, LeGrice IJ, Smaill BH, Young AA, Hunter PJ (2008) Myocardial material parameter estimation: a non-homogeneous finite element study from simple shear tests. Biomech Model Mechanobiol 7:161–173. doi: 10.1007/s10237-007-0083-0
  35. Scollan DF, Holmes A, Winslow R, Forder J (1998) Histological validation of myocardial microstructure obtained from diffusion tensor magnetic resonance imaging. Heart Circul Physiol 275:H2308–H2318CrossRefGoogle Scholar
  36. Sommer G, Schriefl AJ, Andrä M, Sacherer M, Viertler C, Wolinski H, Holzapfel GA (2015) Biomechanical properties and microstructure of human ventricular myocardium. Acta Biomater 24:172–192CrossRefGoogle Scholar
  37. Srinivasa A (2012) On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials. Int J Eng Sci 60:1–12MathSciNetCrossRefGoogle Scholar
  38. Vossoughi J, Vaishnav RN, Patel DJ (1980) Compressibility of myocardial tissue. In: 1980 ASME advances in bioengineering, pp 45–48Google Scholar
  39. Yin FC, Strumpf RK, Chew PH, Zeger SL (1987) Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading. J Biomech 20:577–589CrossRefGoogle Scholar
  40. Young AA, Legrice IJ, Young MA, Smaill BH (1998) Extended confocal microscopy of myocardial laminae and collagen network. J Microsc 192:139–150CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Reza Avazmohammadi
    • 1
  • David S. Li
    • 1
  • Thomas Leahy
    • 1
  • Elizabeth Shih
    • 1
  • João S. Soares
    • 1
  • Joseph H. Gorman
    • 2
  • Robert C. Gorman
    • 2
  • Michael S. Sacks
    • 1
  1. 1.Center for Cardiovascular Simulation, Institute for Computational Engineering and SciencesDepartment of Biomedical Engineering, The University of Texas at AustinAustinUSA
  2. 2.Gorman Cardiovascular Research Group, Smilow Center for Translational Research3400 Civic Center Blvd - Building 421 11th Floor, Room 112PhiladelphiaUSA

Personalised recommendations