Mathematical modeling of biomechanical elastic and hyperelastic properties of the myocardium

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Abstract

BACKGROUND: The study of mechanical properties of biological tissues is extremely informative and is one of the most important areas of biomechanics. Knowledge of these aspects of biological objects based on experimental data can become a source of new medical and technical solutions for the reconstruction of organs and the development of replacement materials.

AIM: Passive mechanical properties of isolated myocardium are compared with linear, bilinear, exponential and the most common hyperelastic models (neohookean, Mooney–Rivlin, Ogden, Yeoh, polynomial and Veronda–Westmann).

MATERIALS AND METHODS: Literature data on mechanical tests of autopsy material obtained from mongrel dogs were used as initial data. To search for the most advanced calculation algorithms the computer algebra system was used, the Mathcad 15.0 software package and the multifunctional finite element analysis application ANSYS 2022 R2 were used. Direct comparison of models was made based on mathematical statistics.

RESULTS: Among the first group of models, the results closest to the experimental data were demonstrated by the exponential model R = 0.9958/0.9984 (in the longitudinal/transverse direction with respect to the myocardial fibers), the lowest accuracy was demonstrated by the linear model R = 0.9813/0.9803. Young’s moduli of linear, bilinear and exponential models and material constants of hyperelastic models are determined. The coefficient of elastic anisotropy of the myocardium, defined as the ratio of the elastic moduli of the linear model measured along and across the direction of the fibers, is equal to 2.18, which is very different from the literature data for the myocardium of the human heart. Deformation along the fibers of the heart muscle is more energy-consuming in the direction along the fibers than in the transverse direction (3.81 and 2.52 mJ/cm3). The most accurate hyperelastic models turned out to be the 2nd order polynomial model R = 0.9971 and the 3rd order Yeoh model R = 0.997. The largest deviations and the lowest correlation coefficient between the experimental and model data were demonstrated by the simple neohookean model R = 0.974 with a single parameter μ. The numerical values of the parameters of hyperelastic models obtained by calculation methods used practically did not differ from each other (≤2.16%).

CONCLUSIONS: The study demonstrated the importance of selecting the correct mechanical model for isolated myocardium. The data obtained can be useful in virtual interventions (simulations) for predicting outcomes and supporting clinical decisions, developing replacement materials and structures made of them for reconstructive operations on heart structures.

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About the authors

Sergey A. Muslov

Russian University of Medicine

Author for correspondence.
Email: sergeymuslov@yandex.ru
ORCID iD: 0000-0002-9752-6804
SPIN-code: 7213-2852
Scopus Author ID: 6507596970
ResearcherId: AAK-9440-2020

Cand. Sci. (Physics and Mathematics), Dr. Sci. (Biol.), Professor of the Department of Normal Physiology and Medical Physics

Russian Federation, 20 building 1 Delegatskaya St., Moscow, 127473

Yury A. Vasyuk

Russian University of Medicine

Email: yvasyuk@yandex.ru
ORCID iD: 0000-0003-2913-9797
SPIN-code: 2265-5331
Scopus Author ID: 6508291333
ResearcherId: G-1772-2013

MD, Dr. Sci. (Med.), Professor, Scientific Secretary, Head of the Department of Hospital Therapy No. 1, Honored Doctor of the Russian Federation, Honored Worker of Higher Education of the Russian Federation

Russian Federation, 20 building 1 Delegatskaya St., Moscow, 127473

Alla I. Zavialova

Russian University of Medicine

Email: allaz05@list.ru
ORCID iD: 0009-0001-1727-4388
SPIN-code: 4883-7130

MD, Cand. Sci. (Med.), Assistant Professor, Department of Hospital Therapy No. 1, Head of the Therapeutic Department of the Kuskovo Medical Center

Russian Federation, 20 building 1 Delegatskaya St., Moscow, 127473

Elena Yu. Shupenina

Russian University of Medicine

Email: eshupenina@mail.ru
ORCID iD: 0000-0001-6188-4610
SPIN-code: 2090-9938

MD, Cand. Sci. (Med.), Professor of the Department of Hospital Therapy No. 1

Russian Federation, 20 building 1 Delegatskaya St., Moscow, 127473

Pavel Yu. Sukhochev

Lomonosov Moscow State University

Email: ps@moids.ru
ORCID iD: 0000-0002-8004-6011
SPIN-code: 7780-8694

Researcher at the Laboratory of Mathematical Support for Simulation Dynamic Systems, Department of Applied Research, Faculty of Mechanics and Mathematics

Russian Federation, Moscow

Layla Z. Guchukova

Odintsovo Regional Hospital

Email: Gucci.loca@mail.ru
ORCID iD: 0009-0007-2150-6034
SPIN-code: 8280-0970

general practitioner

Russian Federation, Odintsovo, Moscow Region

References

  1. Nemavhola F, Pandelani T, Ngwangwa H. Fitting of hyperelastic constitutive models in different sheep heart regions based on biaxial mechanical properties. bioRxiv preprint. 2021. doi: 10.1101/2021.10.28.466240
  2. Izakov VYa, Markhasin VS, Yasnikov GP, et al. Vvedenie v biomekhaniku passivnogo miokarda. Moscow: Nauka; 2000. (In Russ.)
  3. Skovoroda AR. Zadachi teorii uprugosti v probleme diagnostiki patologii myagkikh biologicheskikh tkanei. Moscow; 2006. (In Russ.)
  4. Ostrovsky NV, Chelnokova NO, Golyadkina AA, et al. Biomechanical parameters of the ventricles of the human heart. Fundamental research. 2015;(1–10):2070–2075. (In Russ.)
  5. Ovcharenko EA, Kalashnikov KYu, Glushkova TV, Barbarash LS. Modeling of implantation of a bioprosthesis by the finite element method. Complex problems of cardiovascular diseases. 2016;(1):6–11. (In Russ.) doi: 10.17802/2306-1278-2016-1-6-11
  6. Shilko SV, Kuzminsky YuG, Borisenko MV. Mathematical model and software implementation of monitoring of the cardiovascular system. Problems of Physics, Mathematics and Technics. 2011;3(8):104–112. (In Russ.)
  7. Demer Linda L, Yin Frank C. Passive biaxial mechanical properties of isolated canine myocardium. J Physiol. 1983;339(1):615–630. doi: 10.1113/jphysiol.1983.sp014738
  8. Fung YC. Elasticity of soft tissues in simple elongation. Am J Physiol. 1967;213(6):1532–1544. doi: 10.1152/ajplegacy.1967.213.6.1532
  9. Mirsky I. Assessment of passive, elastic stiffness of cardiac muscle: mathematical concepts, physiologic and clinical considerations, directions of future research. Prog Cardiovasc Dis. 1976;18(4):277–308. doi: 10.1016/0033-0620(76)90023-2
  10. Fung YC. Biorheology of soft tissues. Biorheology. 1973;10(2):139–155. doi: 10.3233/bir-1973-10208
  11. Panda SC, Natarajan R. Finite-element method of stress analysis in the human left ventricular layered wall structure. Med Biol Eng Comput. 1977;15(1):67–71. doi: 10.1007/bf02441577
  12. Smolyuk LT, Protsenko YL. Mechanical properties of passive myocardium: experiment and mathematical model. Biophysics. 2010;55(5):905–909. (In Russ.) doi: 10.1134/S0006350910050209
  13. Green AE, Adkins JE. Large Elastic Deformations and Nonlinear Continuum Mechanics. Oxford: Clarendon; 1960. doi: 10.2307/3613144
  14. Fung YC. Biomechanics, its scope, history, and some problems of continuum mechanics in physiology. Appl Mech Rev. 1973;21(1):1–20. doi: 10.1016/0043-1648(68)90345-1
  15. Muslov SA, Lotkov AI, Arutyunov SD, Albakova TM. Calculation of parameters of mechanical properties of the heart muscle. Perspective materials. 2020;(12):42–52. (In Russ.) doi: 10.30791/1028-978x-2020-12-42-52
  16. Anliker M. Direct measurements of the distensibility of heart ventricles. Presented at the 2nd Annual Workshop of the Basic Science Council of the American Heart Association, Ames Research Centre. Moffett Field, Calif., 1968 4-8th Aug.
  17. Papadacci C, Bunting EA, Wan EY, et al. 3D myocardial elastography in vivo. IEEE Trans Med Imaging. 2017;36(2):618–627. doi: 10.1109/TMI.2016.2623636
  18. da Silveira JS, Scansen BA, Wassenaar PA, et al. Quantification of myocardial stiffness using magnetic resonance elastography in right ventricular hypertrophy: initial feasibility in dogs. Magn Reson Imaging. 2016;34(1):26–34. doi: 10.1016/j.mri.2015.10.001
  19. Muslov SA, Albakova MB, Guchukova LZ. Constants of the hyperelastic Mooney–Rivlin model of the ventricular wall of the heart. Cardiological Bulletin. 2021;16(2–2):39. (In Russ.)
  20. Ren M, Ong CW, Buist ML, Yap CH. Biventricular biaxial mechanical testing and constitutive modelling of fetal porcine myocardium passive stiffness. J Mech Behav Biomed Mater. 2022;134:105383. doi: 10.1016/j.jmbbm.2022.105383
  21. Avazmohammadi R, Soares JS, Li DS, et al. A contemporary look at biomechanical models of myocardium. Annu Rev Biomed Eng. 2019;21:417–442. doi: 10.1146/annurev-bioeng-062117-121129
  22. Ogden RW, Saccomandi G, Sgura I. Fitting hyperelastic models to experimental data. Comput Mech. 2004;34(6):484–502. doi: 10.1007/s00466-004-0593-y
  23. Chen J, Ahmad R, Li W, et al. Biomechanics of oral mucosa. J R Soc Interface. 2015;12(109):20150325. doi: 10.1098/rsif.2015.0325
  24. Wertheim MG. Memoire sur l’elasticite et la cohesion des pricipaux tissus du corps humain. Ann Chimie Phys Paris (Ser. 3). 1847;21:385–414.
  25. Morgan FR. The mechanical properties of collagen fibers: stress-strain curves. J Soc Leather Trades Chem. 1960;44:171–182.
  26. Kenedi RM, Gibson T, Daly CH. Bioengineering study of the human skin. In: Structure and Function of Connective and Skeletal Tissue. S.F. Jackson, S.M. Harkness, R. Tristram (eds.) Scientific Comittee, St. Andrews, Scotland; 1964. P. 388–395. doi: 10.1016/b978-1-4831-6701-5.50022-x
  27. Ridge MD, Wright V. Mechanical properties of skin: A bioegineering study of skin texture. J Appl Physiol. 1966;21(5):1602–1606. doi: 10.1152/jappl.1966.21.5.1602
  28. Corporan D, Saadeh M, Yoldas A, et al. Passive mechanical properties of the left ventricular myocardium and extracellular matrix in hearts with chronic volume overload from mitral regurgitation. Physiol Rep. 2022;10(14):e15305. doi: 10.14814/phy2.15305
  29. Yamada H. Strength of Biological Materials. Baltimore; 1973. doi: 10.1126/science.171.3966.57-a
  30. Muslov SA, Pertsov SS, Arutyunov SD. Fiziko-mekhanicheskie svoistva biologicheskikh tkanei. Ed. by O.O. Yanushevich. Moscow; 2023. 457 p. (In Russ.)
  31. Fung Y.C. Biomechanics: Mechanical Properties of Living Tissues. Second edition. Springer; 1993. 586 p. doi: 10.1115/1.2901550
  32. Muslov SA, Pertsov SS, Chizhmakov EA, et al. Elastic linear, bilinear, nonlinear exponential and hyperelastic skin models. Russian Journal of Biomechanics. 2023;27(3):89–103. (In Russ.) doi: 10.15593/RZhBiomeh/2023.3.07
  33. Ivanov DV, Fomkina OA. Opredelenie postoyannykh dlya modelei Neo–Guka i Muni–Rivlina po rezul’tatam ehksperimentov na odnoosnoe rastyazhenie. Bulletin of the Saratov University. Mathematics. Mechanics. 2008;(10):114–117. (In Russ.)
  34. Shmurak MI, Kuchumov AG, Voronova NO. Analysis of hyperelastic models for describing the behavior of soft tissues of the human body. Master’s Journal. 2017;(1):230–243. (In Russ.)
  35. Yeoh OH. Some forms of the strain energy function for rubber. Rubber Chem Technol. 1993;66(5):754–771. doi: 10.5254/1.3538343
  36. Rivlin RS. Some applications of elasticity theory to rubber engineering. In: Collected Papers of R.S. Rivlin. 1997;1:9–16. doi: 10.1007/978-1-4612-2416-7_2
  37. Veronda DR, Westmann RA. Mechanical characterizations of skin-finite deformations. J Biomech. 1970;3(1):111–124. doi: 10.1016/0021-9290(70)90055-2
  38. Kanbara R, Nakamura Y, Ochiai KT, et al. Three-dimensional finite element stress analysis: the technique and methodology of nonlinear property simulation and soft tissue loading behavior for different partial denture designs. Dent Mater J. 2012;31(2):297–308. doi: 10.4012/dmj.2011-165
  39. Borak L, Florian Z, Bartakova S, et al. Bilinear elastic property of the periodontal ligament for simulation using a finite element mandible model. Dent Mater J. 2011;30(4):448–454. doi: 10.4012/dmj.2010-170
  40. Sacks M, Chuong C. Biaxial mechanical properties of passive right ventricular free wall myocardium. J Biomech Eng. 1993;115(2):202–205. doi: 10.1115/1.2894122
  41. Emig R, Zgierski-Johnston CM, Timmermann V, et al. Passive myocardial mechanical properties: meaning, measurement, models. Biophys Rev. 2021;13(5):587–610. doi: 10.1007/s12551-021-00838-1
  42. Sirry MS, Butler JR, Patnaik SS, et al. Characterisation of the mechanical properties of infarcted myocardium in the rat under biaxial tension and uniaxial compression. J Mech Behav Biomed Mater. 2016;63:252–264. doi: 10.1016/j.jmbbm.2016.06.029
  43. Hill R. A general theory of uniqueness and stability in elastic-plastic solids. J Mech Phys Solids. 1958;6(3):236–249. doi: 10.1016/0022-5096(58)90029-2
  44. Drucker DC. A definition of a stable inelastic material. J Appl Mech. 1959;26(1):101–195. doi: 10.1115/1.4011929
  45. Wang Y, Haynor DR, Kim Y. An investigation of the importance of myocardial anisotropy in finite-element modeling of the heart: methodology and application to the estimation of defibrillation efficacy. IEEE Trans Biomed Eng. 2001;48(12):1377–1389. doi: 10.1109/10.966597

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2. Fig. 1. Longitudinal direction. Stress-strain graphs of myocardial models: linear σlin, bilinear σbilin (with 2 elastic modules E1 and E2) and exponential σ (ε). The points σi represent experimental data, ε is the relative deformation

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3. Fig. 2. Cross myocardial fibers direction. Experimen- tal points (σi) and model curves of 6 hyperelastic models used: Neohookean (NH), Mooney–Rivlin (M–R), Ogden (O gden), Yeoh (Yeoh), polynomial (Polynom) and Veronda– Westmann (V–W). λ — strain coefficient

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4. Fig. 3. Myofibril, mitochondria and paravasal connective tissue. Longitudinal section of a cardiomyocyte fragment. Transmission electron microscopy, ×13,000. Arrow points to connective tissue next to a blood microvessel, the dotted line is the cut line in Fig. 4

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5. Fig. 4. Cross section of a myofibril. Transmission electron microscopy, ×60,000. Mutual hexagonal packing of thick and thin myofilaments. The cut was made approximately at the level indicated by the dotted line in Fig. 3

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