### Mathematics in Engineering

2020, Issue 4: 614-638. doi: 10.3934/mine.2020028
Research article Special Issues

# Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles

• Received: 17 January 2020 Accepted: 26 February 2020 Published: 25 May 2020
• This work concerns the mathematical and numerical modeling of the heart. The aim is to enhance the understanding of the cardiac function in both physiological and pathological conditions. Along this road, a challenge arises from the multi-scale and multi-physics nature of the mathematical problem at hand. In this paper, we propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium. Since the distribution of the electric signal is dependent on the fibres orientation of the ventricles, we use a Laplace-Dirichlet Rule-Based algorithm to determine the myocardial fibres and sheets configuration in the whole bi-ventricle. In this paper, we study the influence of different fibre directions and incompressibility constraint and penalization on the compressibility of the material (bulk modulus) on the pressure-volume relation simulating a full heart beat. The coupled electromechanical problem is addressed by means of a fully segregated scheme. The numerical discretization is based on the Finite Element Method for the spatial discretization and on Backward Differentiation Formulas for the time discretization. The arising non-linear algebraic system coming from application of the implicit scheme is solved through the Newton method. Numerical simulations are carried out in a patient-specific biventricle geometry to highlight the most relevant results of both electrophysiology and mechanics and to compare them with physiological data and measurements. We show how various fibre configurations and bulk modulus modify relevant clinical quantities such as stroke volume, ejection fraction and ventricle contractility.

Citation: Luca Azzolin, Luca Dedè, Antonello Gerbi, Alfio Quarteroni. Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles[J]. Mathematics in Engineering, 2020, 2(4): 614-638. doi: 10.3934/mine.2020028

### Related Papers:

• This work concerns the mathematical and numerical modeling of the heart. The aim is to enhance the understanding of the cardiac function in both physiological and pathological conditions. Along this road, a challenge arises from the multi-scale and multi-physics nature of the mathematical problem at hand. In this paper, we propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium. Since the distribution of the electric signal is dependent on the fibres orientation of the ventricles, we use a Laplace-Dirichlet Rule-Based algorithm to determine the myocardial fibres and sheets configuration in the whole bi-ventricle. In this paper, we study the influence of different fibre directions and incompressibility constraint and penalization on the compressibility of the material (bulk modulus) on the pressure-volume relation simulating a full heart beat. The coupled electromechanical problem is addressed by means of a fully segregated scheme. The numerical discretization is based on the Finite Element Method for the spatial discretization and on Backward Differentiation Formulas for the time discretization. The arising non-linear algebraic system coming from application of the implicit scheme is solved through the Newton method. Numerical simulations are carried out in a patient-specific biventricle geometry to highlight the most relevant results of both electrophysiology and mechanics and to compare them with physiological data and measurements. We show how various fibre configurations and bulk modulus modify relevant clinical quantities such as stroke volume, ejection fraction and ventricle contractility.

 [1] Ambrosi D, Arioli G, Nobile F, et al. (2011) Electromechanical coupling in cardiac dynamics: The active strain approach. SIAM J Appl Math 71: 605-621. doi: 10.1137/100788379 [2] Ambrosi D, Pezzuto S (2012) Active stress vs. active strain in mechanobiology: constitutive issues. J Elasticity 107: 199-212. [3] Arevalo HJ, Vadakkumpadan F, Guallar E, et al. (2016) Arrhythmia risk stratification of patients after myocardial infarction using personalized heart models. Nature Commun 7: 1-8. [4] Azzolin L, Quarteroni A, Dedè L, et al. (2018) Electromechanical modelling of the human heart in bi-ventricle geometries. MSc thesis, Politecnico di Milano, Italy. [5] Barbarotta L, Rossi S, Dedè L, et al. (2018) A transmurally heterogeneous orthotropic activation model for ventricular contraction and its numerical validation. Int J Numer Meth Bio 34: 2040-7939. [6] Bayer J, Blake R, Plank G, et al. (2012) A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models. Ann Biomed Eng 40: 2243-2254. doi: 10.1007/s10439-012-0593-5 [7] Boron W, Boulpaep E (2012) Medical Physiology, Saunders. [8] Bovendeerd PHM, Huyghe J, Arts T, et al. (1994) Influence of endocardial-epicardial crossover of muscle fibers on left ventricular wall mechanics. J Biomech 27: 941-951. doi: 10.1016/0021-9290(94)90266-6 [9] Brenner JI, Baker KR, Berman MA (1980) Prediction of left ventricular pressure in infants with aortic stenosis. Heart 44: 406-410. doi: 10.1136/hrt.44.4.406 [10] Bueno-Orovio A, Cherry E, Fenton F (2008) Minimal model for human ventricular action potentials in tissue. J Theor Biol 253: 544-560. doi: 10.1016/j.jtbi.2008.03.029 [11] Chabiniok R, Wang V, Hadjicharalambous M, et al. (2016) Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: Ventricular cardiac mechanics. Interface Focus 6: 15-83. [12] Colli Franzone P, Pavarino LF, Savaré G (2006) Computational electrocardiology: Mathematical and numerical modeling, In: Complex Systems in Biomedicine, Springer, 187-241. [13] Coupé P, Manjón JV, Fonov V, et al. (2011) Patch-based segmentation using expert priors: Application to hippocampus and ventricle segmentation. NeuroImage 54: 940-954. doi: 10.1016/j.neuroimage.2010.09.018 [14] Eriksson T, Prassl A, Plank G, et al. (2013) Influence of myocardial fiber/sheet orientations on left ventricular mechanical contraction. Math Mech Solids 18: 592-606. doi: 10.1177/1081286513485779 [15] Fedele M, Faggiano E, Dedè L, et al. (2017) A patient specific aortic valve model based on moving resistive immersed surfaces. Biomech Model Mechan 16: 1779-1803. doi: 10.1007/s10237-017-0919-1 [16] Formaggia L, Quarteroni A, Veneziani A (2010) Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer Science & Business Media. [17] Gerbi A (2018) Numerical approximation of cardiac electro-fluid-mechanical models: Coupling strategies for large-scale simulation. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland. [18] Gerbi A, Dedè L, Deparis S, et al. The lifev finite elements library: Recent developments and cardiovascular applications, In: ENUMATH 2017, Voss, Norway. [19] Gerbi A, Dedè L, Quarteroni A (2019) A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle. Mathematics in Engineering 1: 1-37. [20] Gerbi A, Dedè L, Quarteroni A (2018) Segregated algorithms for the numerical simulation of cardiac electromechanics in the left human ventricle, MOX Report No. 28. [21] Godunov S (1959) A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat Sbornik 89: 271-306. [22] Goldberger AL, Amaral LAN, Glass L, et al. (2000) Physiobank, physiotoolkit, and physionet: Components of a new research resource for complex physiologic signals. Circulation 101: e215-e220. [23] Holzapfel G, Ogden R (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos T Roy Soc A 367: 3445-3475. doi: 10.1098/rsta.2009.0091 [24] Hoogendoorn C, Duchateau N, Sanchez-Quintana D, et al. (2013) A high-resolution atlas and statistical model of the human heart from multislice CT. IEEE T Med Imaging 32: 28-44. doi: 10.1109/TMI.2012.2230015 [25] Hsu M, Bazilevs Y (2011) Blood vessel tissue prestress modeling for vascular fluid-structure interaction simulation. Finite Elem Anal Des 47: 593-599. doi: 10.1016/j.finel.2010.12.015 [26] Hunter P, Nash M, Sands G (1997) Computational electromechanics of the heart. Comput Biol Heart 12: 347-407. [27] Isgum I, Staring M, Rutten A, et al. (2009) Multi-atlas-based segmentation with local decision fusion-application to cardiac and aortic segmentation in CT scans. IEEE T Med Imaging 28: 1000-1010. doi: 10.1109/TMI.2008.2011480 [28] Krishnamurthy A, Villongco CT, Chuang J (2013) Patient-specific models of cardiac biomechanics. J Comput Phys 244: 4-21. doi: 10.1016/j.jcp.2012.09.015 [29] Lee H, Codella N, Cham M, et al. (2010) Automatic left ventricle segmentation using iterative thresholding and an active contour model with adaptation on short-axis cardiac mri. IEEE T Biomed Eng 57: 905-913. doi: 10.1109/TBME.2009.2014545 [30] Ogden RW (1997) Non-linear Elastic Deformations, Courier Corporation. [31] Organization WH, Cardiovascular diseases (cvds), 2017, Available from: https://www.who.int/news-room/fact-sheets/detail/cardiovascular-diseases-(cvds). [32] Otto CM (1995) Hurst's the Heart: Arteries and Veins. JAMA 274: 1640-1641. [33] Pennacchio M, Savaré G, Colli Franzone P (2005) Multiscale modeling for the bioelectric activity of the heart. SIAM J Math Anal 37: 1333-1370. doi: 10.1137/040615249 [34] Potse M, Dubé B, Richer J, et al. (2006) A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart. IEEE T Biomed Eng 53: 2425-2435. doi: 10.1109/TBME.2006.880875 [35] Quarteroni A, Dede' L, Manzoni A, et al. (2009) Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications, Cambridge University Press. [36] Quarteroni A, Lassila T, Rossi S, et al. (2017) Integrated heart-coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput Method Appl M 314: 345-407. doi: 10.1016/j.cma.2016.05.031 [37] Quarteroni A, Sacco R, Saleri F (2010) Numerical Mathematics, Springer Science & Business Media. [38] Redington AN, Gray HH, Hodson ME, et al. (1988) Characterisation of the normal right ventricular pressure-volume relation by biplane angiography and simultaneous micromanometer pressure measurements. Heart 59: 23-30. doi: 10.1136/hrt.59.1.23 [39] Rossi S (2014) Anisotropic modeling of cardiac mechanical activation. PhD thesis, EPFL, Switzerland. [40] Rossi S, Ruiz-Baier R, Pavarino L, et al. (2012) Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int J Numer Meth Bio 28: 761-788. doi: 10.1002/cnm.2473 [41] Ruiz-Baier R, Gizzi A, Rossi S, et al. (2014) Mathematical modelling of active contraction in isolated cardiomyocytes. Math Med Biol 31: 259-283. doi: 10.1093/imammb/dqt009 [42] Saffitz J, Kanter H, Green K, et al. (1994) Tissue-specific determinants of anisotropic conduction velocity in canine atrial and ventricular myocardium. Circ Res 74: 1065-1070. doi: 10.1161/01.RES.74.6.1065 [43] Sainte-Marie J, Chapelle D, Cimrman R, et al. (2006) Modeling and estimation of the cardiac electromechanical activity. Comput Struct 84: 1743-1759. doi: 10.1016/j.compstruc.2006.05.003 [44] Santiago A, Aguado-Sierra J, Zavala-Aké M, et al. (2018) Fully coupled fluid-electro-mechanical model of the human heart for supercomputers. Int J Numer Meth Bio 34: e3140. doi: 10.1002/cnm.3140 [45] Smith N, Nickerson D, Crampin E, et al. (2004) Multiscale computational modelling of the heart. Acta Numer 13: 371-431. doi: 10.1017/S0962492904000200 [46] Tagliabue A, Dedè L, Quarteroni A (2017) Complex blood flow patterns in an idealized left ventricle: A numerical study. Chaos 27: 093939. doi: 10.1063/1.5002120 [47] Tagliabue A, Dedè L, Quarteroni A (2017) Fluid dynamics of an idealized left ventricle: The extended Nitsche's method for the treatment of heart valves as mixed time varying boundary conditions. Int J Numer Meth Fl 85: 135-164. doi: 10.1002/fld.4375 [48] Takizawa K, Bazilevs Y, Tezduyar T (2012) Space-time and ale-vms techniques for patient-specific cardiovascular fluid-structure interaction modeling. Arch Comput Method E 19: 171-225. doi: 10.1007/s11831-012-9071-3 [49] Usyk T, LeGrice I, McCulloch A (2002) Computational model of three-dimensional cardiac electromechanics. Comput Visual Sci 4: 249-257. doi: 10.1007/s00791-002-0081-9 [50] Vadakkumpadan F, Arevalo H, Ceritoglu C, et al. (2012) Image-based estimation of ventricular fiber orientations for personalized modeling of cardiac electrophysiology. IEEE T Med Imaging 31: 1051-1060. doi: 10.1109/TMI.2012.2184799 [51] Westerhof N, Lankhaar J, Westerhof B (2009) The arterial windkessel. Med Biol Eng Comput 47: 131-141. doi: 10.1007/s11517-008-0359-2
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.0 2.2

Article outline

## Figures and Tables

Figures(12)  /  Tables(4)

• On This Site