American Institute of Mathematical Sciences

Advanced
August  2022, 15(8): 2391-2427. doi: 10.3934/dcdss.2022052

A geometric multiscale model for the numerical simulation of blood flow in the human left heart

Alberto Zingaro 1,, , Ivan Fumagalli 1, , Luca Dede 1, , Marco Fedele 1, , Pasquale C. Africa 1, , Antonio F. Corno 2, and  Alfio Quarteroni 1,3,

1. 

MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy
2. 

Children’s Heart Institute, Hermann Children’s Hospital, University of Texas Health, McGovern Medical School, Houston, TX, USA
3. 

Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, Av. Piccard, CH-1015 Lausanne, Switzerland (Professor Emeritus)

* Corresponding author: Alberto Zingaro

Received  January 2022 Published  August 2022 Early access  March 2022

We present a new computational model for the numerical simulation of blood flow in the human left heart. To this aim, we use the Navier-Stokes equations in an Arbitrary Lagrangian Eulerian formulation to account for the endocardium motion and we model the cardiac valves by means of the Resistive Immersed Implicit Surface method. To impose a physiological displacement of the domain boundary, we use a 3D cardiac electromechanical model of the left ventricle coupled to a lumped-parameter (0D) closed-loop model of the remaining circulation. We thus obtain a one-way coupled electromechanics-fluid dynamics model in the left ventricle. To extend the left ventricle motion to the endocardium of the left atrium and to that of the ascending aorta, we introduce a preprocessing procedure according to which an harmonic extension of the left ventricle displacement is combined with the motion of the left atrium based on the 0D model. To better match the 3D cardiac fluid flow with the external blood circulation, we couple the 3D Navier-Stokes equations to the 0D circulation model, obtaining a multiscale coupled 3D-0D fluid dynamics model that we solve via a segregated numerical scheme. We carry out numerical simulations for a healthy left heart and we validate our model by showing that meaningful hemodynamic indicators are correctly reproduced.

Keywords: Numerical analysis, computational fluid dynamics, cardiac modeling, left heart, circulation, cardiac valves, multiscale modeling.
Mathematics Subject Classification: Primary: 65-XX, 76-XX, 76-10, 76Z05, 92C50.
Citation: Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2391-2427. doi: 10.3934/dcdss.2022052
References:
[1]

http://www.vmtk.org

[2]

https://github.com/marco-fedele/vmtk

[3]

https://lifex.gitlab.io/

[4]

https://www.paraview.org/Wiki/PvPython_and_PvBatch

[5]

https://trilinos.github.io

[6]

P. C. Africa, R. Piersanti, M. Fedele, L. Dede' and A. Quarteroni, Life$^\mathrm x$ - heart module: A high-performance simulator for the cardiac function. Package 1: Fiber generation, arXiv preprint, arXiv: 2201.03303, 2022.

[7]

L. Antiga, M. Piccinelli, L. Botti, B. Ene-Iordache, A. Remuzzi and D. A. Steinman, An image-based modeling framework for patient-specific computational hemodynamics, Medical & Biological Engineering & Computing, 46 (2008), 1097-1112.  doi: 10.1007/s11517-008-0420-1.

[8]

C. P. Appleton, Evaluation of diastolic function by two-dimensional and doppler assessment of left ventricular filling including pulmonary venous flow, In Diastology, Elsevier (2008), 115–143. doi: 10.1016/B978-1-4160-3754-5.50016-0.

[9]

D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassmöller, T. Heister, L. Heltai, U. Köcher, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, S. Proell, K. Simon, B. Turcksin, D. Wells and J. Zhang, The $\mathsf{deal.II}$ library, version 9.3, Journal of Numerical Mathematics, 29 (2021), 171-186. 

[10]

P. M. Arvidsson, S. J. Kovács, J. Töger, R. Borgquist, E. Heiberg, M. Carlsson and H. Arheden, Vortex ring behavior provides the epigenetic blueprint for the human heart, Scientific Reports, 6 (2016), 1-9.  doi: 10.1038/srep22021.

[11]

C. M. Augustin, A. Crozier, A. Neic, A. J. Prassl, E. Karabelas, T. Ferreira da Silva, J. F. Fernandes, F. Campos, T. Kuehne and G. Plank, Patient-specific modeling of left ventricular electromechanics as a driver for haemodynamic analysis, EP Europace, 18 (2016), 121-129.  doi: 10.1093/europace/euw369.

[12]

B. Baccani, F. Domenichini and G. Pedrizzetti, Vortex dynamics in a model left ventricle during filling, European Journal of Mechanics-B/Fluids, 21 (2002), 527-543.  doi: 10.1016/S0997-7546(02)01200-1.

[13]

J. D. Bayer, R. C. Blake, G. Plank and N. A. Trayanova, A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models, Annals of Biomedical Engineering, 40 (2012), 2243-2254.  doi: 10.1007/s10439-012-0593-5.

[14]

Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J. R. Hughes, A. Reali and G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007), 173-201.  doi: 10.1016/j.cma.2007.07.016.

[15]

C. Bertoglio and A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, J. Comput. Phys., 261 (2014), 162-171.  doi: 10.1016/j.jcp.2013.12.057.

[16]

P. J. Blanco and R. A. Feijoo, A 3D-1D-0D computational model for the entire cardiovascular system, Mecanica Computacional, 24 (2010), 5887-5911. 

[17]

D. Bluestein and S. Einav, Transition to turbulence in pulsatile flow through heart valves-a modified stability approach, Journal of Biomechanical Engineering, 116 (1994), 477-487.  doi: 10.1115/1.2895799.

[18]

M. Bucelli, L. Dede', A. Quarteroni and C. Vergara, Partitioned and monolithic algorithms for the numerical solution of cardiac fluid-structure interaction, MOX Report, 78 (2021).

[19]

R. M. Carey and P. K. Whelton, Prevention, detection, evaluation, and management of high blood pressure in adults: Synopsis of the 2017 American college of cardiology/American heart association hypertension guideline, Annals of Internal Medicine, 168 (2018), 351-358.  doi: 10.7326/M17-3203.

[20]

C. Chnafa, S. Mendez and F. Nicoud, Image-based large-eddy simulation in a realistic left heart, Computers & Fluids, 94 (2014), 173-187.  doi: 10.1016/j.compfluid.2014.01.030.

[21]

Y. J. Choi, J. Constantino, V. Vedula, N. Trayanova and R. Mittal, A new MRI-based model of heart function with coupled hemodynamics and application to normal and diseased canine left ventricles, Frontiers in Bioengineering and Biotechnology, 3 (2015), 1-15.  doi: 10.3389/fbioe.2015.00140.

[22]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, 13, Springer, 2014. doi: 10.1007/978-3-319-04801-7.

[23]

M. Corti, A. Zingaro, L. Dede' and A. Quarteroni, Impact of atrial fibrillation on left atrium haemodynamics: A computational fluid dynamics study, arXiv preprint, arXiv: 2202.10893, 2022.

[24]

L. Dedè, F. Menghini and A. Quarteroni, Computational fluid dynamics of blood flow in an idealized left human heart, Int. J. Numer. Methods Biomed. Eng., 37 (2021), Paper No. e3287, 24 pp. doi: 10.1002/cnm.3287.

[25]

S. Deparis, G. Grandperrin and A. Quarteroni, Parallel preconditioners for the unsteady Navier-Stokes equations and applications to hemodynamics simulations, Computers & Fluids, 92 (2014), 253-273.  doi: 10.1016/j.compfluid.2013.10.034.

[26]

F. Domenichini, G. Pedrizzetti and B. Baccani, Three-dimensional filling flow into a model left ventricle, J. Fluid Mech., 539 (2005), 179-198.  doi: 10.1017/S0022112005005550.

[27]

F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian–Eulerian method for Navier–Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Engrg., 193 (2004), 4819-4836.  doi: 10.1016/j.cma.2004.05.003.

[28]

P. Dyverfeldt, M. Bissell, A. J. Barker, A. F. Bolger, C.-J. Carlhäll, T. Ebbers, C. J. Francios, A. Frydrychowicz, J. Geiger, D. Giese, M. D. Hope, P. J. Kilner, S. Kozerke, S. Myerson, S. Neubauer, O. Wieben and M. Markl, 4D flow cardiovascular magnetic resonance consensus statement, Journal of Cardiovascular Magnetic Resonance, 17 (2015), 1-19.  doi: 10.1186/s12968-015-0174-5.

[29]

M. Fedele, E. Faggiano, L. Dedè and A. Quarteroni, A patient-specific aortic valve model based on moving resistive immersed implicit surfaces, Biomechanics and Modeling in Mechanobiology, 16 (2017), 1779-1803.  doi: 10.1007/s10237-017-0919-1.

[30]

M. Fedele and A. Quarteroni, Polygonal surface processing and mesh generation tools for the numerical simulation of the cardiac function, Int. J. Numer. Methods Biomed. Eng., 37 (2021), Paper No. e3435, 34 pp. doi: 10.1002/cnm.3435.

[31]

L. Formaggia, J. F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels, Comput. Methods Appl. Mech. Engrg., 191 (2001), 561-582.  doi: 10.1016/S0045-7825(01)00302-4.

[32]

L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J. Engrg. Math., 47 (2003), 251-276.  doi: 10.1023/B:ENGI.0000007980.01347.29.

[33]

L. Formaggia and F. Nobile, A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 7 (1999), 105-131. 

[34]

D. Forti and L. Dedè, Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS-LES modeling in a high performance computing framework, Computers & Fluids, 117 (2015), 168-182.  doi: 10.1016/j.compfluid.2015.05.011.

[35]

I. Fumagalli, M. Fedele, C. Vergara, L. Dede', S. Ippolito, F. Nicolò, C. Antona, R. Scrofani and A. Quarteroni, An image-based computational hemodynamics study of the systolic anterior motion of the mitral valve, Computers in Biology and Medicine, 123 (2020), 103922.  doi: 10.1016/j.compbiomed.2020.103922.

[36]

A. Gerbi, Numerical Approximation of Cardiac Electro-Fluid-Mechanical Models: Coupling Strategies for Large-Scale Simulation, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2018.

[37]

J. M. Guccione and A. D. McCulloch, Finite element modeling of ventricular mechanics, In Theory of Heart, Springer, (1991), 121–144. doi: 10.1007/978-1-4612-3118-9_6.

[38]

G. S. Gulsin, A. Singh and G. P. McCann, Cardiovascular magnetic resonance in the evaluation of heart valve disease, BMC Medical Imaging,17 (2017), 1–14. doi: 10.1186/s12880-017-0238-0.

[39]

J. Hee Seo and R. Mittal, Effect of diastolic flow patterns on the function of the left ventricle, Physics of Fluids, 25 (2013), 110801, 22 pp. doi: 10.1063/1.4819067.

[40]

M. Hirschvogel, M. Bassilious, L. Jagschies, S. M. Wildhirt and M. W. Gee, A monolithic 3D-0D coupled closed-loop model of the heart and the vascular system: Experiment-based parameter estimation for patient-specific cardiac mechanics, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e2842, 22 pp. doi: 10.1002/cnm.2842.

[41]

D. I. Hollnagel, P. E. Summers, D. Poulikakos and S. S. Kollias, Comparative velocity investigations in cerebral arteries and aneurysms: 3d phase-contrast MR angiography, laser doppler velocimetry and computational fluid dynamics, NMR in Biomedicine: An International Journal Devoted to the Development and Application of Magnetic Resonance In vivo, 22 (2009), 795-808.  doi: 10.1002/nbm.1389.

[42]

J. C. R. Hunt, A. A. Wray and P. Moin, Eddies, stream, and convergence zones in turbulent flows, Center for Turbulence Research Report, CTR-S88, (1988), 193–208.

[43]

Zygote Media Group Inc, Zygote solid 3D heart generation ii developement report. tech. rep., 2014.,

[44]

E. Karabelas, M. A. F Gsell, C. M. Augustin, L. Marx, A. Neic, A. J. Prassl, L. Goubergrits, T. Kuehne and G. Plank, Towards a computational framework for modeling the impact of aortic coarctations upon left ventricular load, Frontiers in Physiology, 9 (2018), 1-20. 

[45]

H. J. Kim, I. E. Vignon-Clementel, C. A. Figueroa, J. F. LaDisa, K. E. Jansen, J. A. Feinstein and C. A. Taylor, On coupling a lumped parameter heart model and a three-dimensional finite element aorta model, Annals of Biomedical Engineering, 37 (2009), 2153-2169. 

[46]

W. Y. Kim, P. G. Walker, E. M. Pedersen, J. K. Poulsen, S. Oyre, K. Houlind and A. P. Yoganathan, Left ventricular blood flow patterns in normal subjects: A quantitative analysis by three-dimensional magnetic resonance velocity mapping, Journal of the American College of Cardiology, 26 (1995), 224-238.  doi: 10.1016/0735-1097(95)00141-L.

[47]

V. Kumar, A. K. Abbas, N. Fausto and J. C. Aster, Robbins and Cotran Pathologic Basis of Disease, Professional Edition E-Book., Elsevier health sciences, 2014.

[48]

P. K. Kundu, I. M. Cohen and D. R. Dowling, Fluid Mechanics, Academic Press, 5 edition, 2014.

[49]

A. M. Maceira, S. K. Prasad, M. Khan and D. J. Pennell, Normalized left ventricular systolic and diastolic function by steady state free precession cardiovascular magnetic resonance, Journal of Cardiovascular Magnetic Resonance, 8 (2006), 417-426.  doi: 10.1080/10976640600572889.

[50]

A. Masci, M. Alessandrini, D. Forti, F. Menghini, L. Dedé, C. Tomasi, A. Quarteroni and C. Corsi, A patient-specific computational fluid dynamics model of the left atrium in atrial fibrillation: Development and initial evaluation, In International Conference on Functional Imaging and Modeling of the Heart, Springer, (2017), 392–400. doi: 10.1007/978-3-319-59448-4_37.

[51]

A. Masci, M. Alessandrini, D. Forti, F. Menghini, L. Dedé, C. Tomasi, A. Quarteroni and C. Corsi, A proof of concept for computational fluid dynamic analysis of the left atrium in atrial fibrillation on a patient-specific basis, Journal of Biomechanical Engineering, 142 (2020), 011002, 11 pp. doi: 10.1115/1.4044583.

[52]

A. Masci, L. Barone, L. Dedè, M. Fedele, C. Tomasi, A. Quarteroni and C. Corsi, The impact of left atrium appendage morphology on stroke risk assessment in atrial fibrillation: A computational fluid dynamics study, Frontiers in Physiology, 9 (2019), 1-11.  doi: 10.3389/fphys.2018.01938.

[53]

V. Milišić and A. Quarteroni, Analysis of lumped parameter models for blood flow simulations and their relation with 1D models, M2AN Math. Model. Numer. Anal., 38 (2004), 613-632.  doi: 10.1051/m2an:2004036.

[54]

R. Mittal, J. H. Seo, V. Vedula, Y. J. Choi, H. Liu, H. H. Huang, S. Jain, L. Younes, T. Abraham and R. T. George, Computational modeling of cardiac hemodynamics: Current status and future outlook, J. Comput. Phys., 305 (2016), 1065-1082.  doi: 10.1016/j.jcp.2015.11.022.

[55]

M. T. Ngo, C. I. Kim, J. Jung, G. H. Chung, D. H. Lee and H. S. Kwak, Four-dimensional flow magnetic resonance imaging for assessment of velocity magnitudes and flow patterns in the human carotid artery bifurcation: Comparison with computational fluid dynamics, Diagnostics, 9 (2019), 1-12.  doi: 10.3390/diagnostics9040223.

[56]

K. Perktold, E. Thurner and Th. Kenner, Flow and stress characteristics in rigid walled and compliant carotid artery bifurcation models, Medical & Biological Engineering & Computing, 32 (1994), 19-26.  doi: 10.1007/BF02512474.

[57]

R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno and A. Quarteroni, Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Methods Appl. Mech. Engrg., 373 (2021), Paper No. 113468, 33 pp. doi: 10.1016/j.cma.2020.113468.

[58]

A. Quarteroni, Numerical Models for Differential Problems, volume 2., Springer, 2009. doi: 10.1007/978-88-470-1071-0.

[59]

A. Quarteroni, L. Dedè, A. Manzoni and C. Vergara, Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications, 33. Cambridge University Press, 2019. doi: 10.1017/9781108616096.

[60]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 37., Springer Science & Business Media, 2010.

[61]

A. Quarteroni, A. Veneziani and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Comput. Methods Appl. Mech. Engrg., 302 (2016), 193-252.  doi: 10.1016/j.cma.2016.01.007.

[62]

F. Regazzoni, L. Dedè and A. Quarteroni, Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics, Comput. Methods Appl. Mech. Engrg., 370 (2020), 113268, 30 pp. doi: 10.1016/j.cma.2020.113268.

[63]

F. Regazzoni, M. Salvador, P. C. Africa, M. Fedele, L. Dedè and A. Quarteroni, A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation, Journal of Computational Physics, 457 (2022), 111083.  doi: 10.1016/j.jcp.2022.111083.

[64]

M. Salvador, L. Dedè and A. Quarteroni, An intergrid transfer operator using radial basis functions with application to cardiac electromechanics, Comput. Mech., 66 (2020), 491-511.  doi: 10.1007/s00466-020-01861-x.

[65]

A. Santiago, Fluid-electro-mechanical model of the human heart for supercomputers, 2018.

[66]

A. Santiago, J. Aguado-Sierra, M. Zavala-Aké, R. Doste-Beltran, S. Gómez, R. Arís, J. C. Cajas, E. Casoni and M. Vázquez, Fully coupled fluid-electro-mechanical model of the human heart for supercomputers, Int. J. Numer. Methods Biomed. Eng., 34 (2018), e3140, 30 pp. doi: 10.1002/cnm.3140.

[67]

Y. Shi and T. Korakianitis, Numerical simulation of cardiovascular dynamics with left heart failure and in-series pulsatile ventricular assist device, Artificial Organs, 30 (2006), 929-948.  doi: 10.1111/j.1525-1594.2006.00326.x.

[68]

Y. Shi, P. Lawford and R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMedical Engineering Online, 10 (2011), 1-38.  doi: 10.1186/1475-925X-10-33.

[69]

T. Sugimoto, R. Dulgheru, A. Bernard, F. Ilardi, L. Contu, K. Addetia, L. Caballero, N. Akhaladze, G. D. Athanassopoulos, D. Barone, M. Baroni, N. Cardim, A. Hagendorff, K. Hristova, T. Lopez, G. de la Morena, B. A. Popescu, M. Moonen, M. Penicka, T. Ozyigit, J. D. Rodrigo Carbonero, N. van de Veire, R. S. von Bardeleben, D. Vinereanu, J. L. Zamorano, Y. Y. Go, M. Rosca, A. Calin, J. Magne, B. Cosyns, S. Marchetta, E. Donal, G. Habib, M. Galderisi, L. P. Badano, R. M. Lang and P. Lancellotti, Echocardiographic reference ranges for normal left ventricular 2D strain: Results from the EACVI NORRE study, European Heart Journal-Cardiovascular Imaging, 18 (2017), 833-840. 

[70]

A. Tagliabue, L. Dedè and A. Quarteroni, Complex blood flow patterns in an idealized left ventricle: A numerical study, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 093939, 26 pp. doi: 10.1063/1.5002120.

[71]

A. Tagliabue, L. Dedè and A. Quarteroni, Fluid dynamics of an idealized left ventricle: The extended Nitsche's method for the treatment of heart valves as mixed time varying boundary conditions, Internat. J. Numer. Methods Fluids, 85 (2017), 135-164.  doi: 10.1002/fld.4375.

[72]

C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite element analysis of pulsatile flow in the abdominal aorta under resting and exercise conditions, ASME-Publications-Bed, 33 (1996), 81-82. 

[73]

C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite element modeling of blood flow in arteries, Comput. Methods Appl. Mech. Engrg., 158 (1998), 155-196.  doi: 10.1016/S0045-7825(98)80008-X.

[74]

K. H. W. J. ten Tusscher and A. V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model, American Journal of Physiology-Heart and Circulatory Physiology, 291 (2006), 1088-1100.  doi: 10.1152/ajpheart.00109.2006.

[75]

A. This, L. Boilevin-Kayl, M. A. Fernández and J.-F. Gerbeau, Augmented resistive immersed surfaces valve model for the simulation of cardiac hemodynamics with isovolumetric phases, Int. J. Numer. Methods Biomed. Eng., 36 (2020), e3223, 26 pp. doi: 10.1002/cnm.3223.

[76]

A. This, H. G. Morales, O. Bonnefous, M. A. Fernández and J.-F. Gerbeau, A pipeline for image based intracardiac CFD modeling and application to the evaluation of the PISA method, Computer Methods in Applied Mechanics and Engineering, 358 (2020), 112627, 22 pp. doi: 10.1016/j.cma.2019.112627.

[77]

L. Thomas, E. Foster and N. B. Schiller, Peak mitral inflow velocity predicts mitral regurgitation severity, Journal of the American College of Cardiology, 31 (1998), 174-179. 

[78]

P. Tripathi, Pathophysiological mechanism behind diabetic cardiovascular disorders, International Journal of Current Research, 6 (2014), 4426-4436. 

[79]

F. N. van de Vosse and N. Stergiopulos, Pulse wave propagation in the arterial tree, Annual Review of Fluid Mechanics, 43 (2011), 467-499.  doi: 10.1146/annurev-fluid-122109-160730.

[80]

R. J. van der Geest and P. Garg, Advanced analysis techniques for intra-cardiac flow evaluation from 4D flow MRI, Current Radiology Reports, 4 (2016), 1-10. 

[81]

A. C. Verkaik, A. C. B. Bogaerds, F. Storti and F. N. van de Vosse., A coupled overlapping domain method for the computation of transitional flow through artificial heart valves, In ASME 2012 Summer Bioengineering Conference (SBC 2012), American Society of Mechanical Engineers, (2012), 217–218. doi: 10.1115/SBC2012-80257.

[82]

I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries, Computer Methods in Biomechanics and Biomedical Engineering, 13 (2010), 625-640.  doi: 10.1080/10255840903413565.

[83]

F. Viola, V. Meschini and R. Verzicco, Fluid–structure-electrophysiology interaction (FSEI) in the left-heart: A multi-way coupled computational model, Eur. J. Mech. B Fluids, 79 (2020), 212-232.  doi: 10.1016/j.euromechflu.2019.09.006.

[84]

F. Viola, V. Meschini and R. Verzicco, Effects of stenotic aortic valve on the left heart hemodynamics: A fluid-structure-electrophysiology approach, arXiv preprint, arXiv: 2103.14680, 2021.

[85]

F. Viola, V. Spandan, V. Meschini, J. Romero, M. Fatica, M. D. de Tullio and R. Verzicco, FSEI-GPU: GPU accelerated simulations of the fluid-structure-electrophysiology interaction in the left heart, Comput. Phys. Commun., 273 (2022), Paper No. 108248, 12 pp. doi: 10.1016/j.cpc.2021.108248.

[86]

S. Z. Zhao, P. Papathanasopoulou, Q. Long, I. Marshall and X. Y. Xu, Comparative study of magnetic resonance imaging and image-based computational fluid dynamics for quantification of pulsatile flow in a carotid bifurcation phantom, Annals of Biomedical Engineering, 31 (2003), 962-971.  doi: 10.1114/1.1590664.

[87]

X. Zheng, J. H. Seo, V. Vedula, T. Abraham and R. Mittal, Computational modeling and analysis of intracardiac flows in simple models of the left ventricle, Eur. J. Mech. B Fluids, 35 (2012), 31-39.  doi: 10.1016/j.euromechflu.2012.03.002.

[88]

A. Zingaro, L. Dede', F. Menghini and A. Quarteroni, Hemodynamics of the heart's left atrium based on a variational multiscale-LES numerical method, Eur. J. Mech. B Fluids, 89 (2021), 380-400.  doi: 10.1016/j.euromechflu.2021.06.014.

show all references

References:
[1]

http://www.vmtk.org

[2]

https://github.com/marco-fedele/vmtk

[3]

https://lifex.gitlab.io/

[4]

https://www.paraview.org/Wiki/PvPython_and_PvBatch

[5]

https://trilinos.github.io

[6]

P. C. Africa, R. Piersanti, M. Fedele, L. Dede' and A. Quarteroni, Life$^\mathrm x$ - heart module: A high-performance simulator for the cardiac function. Package 1: Fiber generation, arXiv preprint, arXiv: 2201.03303, 2022.

[7]

L. Antiga, M. Piccinelli, L. Botti, B. Ene-Iordache, A. Remuzzi and D. A. Steinman, An image-based modeling framework for patient-specific computational hemodynamics, Medical & Biological Engineering & Computing, 46 (2008), 1097-1112.  doi: 10.1007/s11517-008-0420-1.

[8]

C. P. Appleton, Evaluation of diastolic function by two-dimensional and doppler assessment of left ventricular filling including pulmonary venous flow, In Diastology, Elsevier (2008), 115–143. doi: 10.1016/B978-1-4160-3754-5.50016-0.

[9]

D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassmöller, T. Heister, L. Heltai, U. Köcher, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, S. Proell, K. Simon, B. Turcksin, D. Wells and J. Zhang, The $\mathsf{deal.II}$ library, version 9.3, Journal of Numerical Mathematics, 29 (2021), 171-186. 

[10]

P. M. Arvidsson, S. J. Kovács, J. Töger, R. Borgquist, E. Heiberg, M. Carlsson and H. Arheden, Vortex ring behavior provides the epigenetic blueprint for the human heart, Scientific Reports, 6 (2016), 1-9.  doi: 10.1038/srep22021.

[11]

C. M. Augustin, A. Crozier, A. Neic, A. J. Prassl, E. Karabelas, T. Ferreira da Silva, J. F. Fernandes, F. Campos, T. Kuehne and G. Plank, Patient-specific modeling of left ventricular electromechanics as a driver for haemodynamic analysis, EP Europace, 18 (2016), 121-129.  doi: 10.1093/europace/euw369.

[12]

B. Baccani, F. Domenichini and G. Pedrizzetti, Vortex dynamics in a model left ventricle during filling, European Journal of Mechanics-B/Fluids, 21 (2002), 527-543.  doi: 10.1016/S0997-7546(02)01200-1.

[13]

J. D. Bayer, R. C. Blake, G. Plank and N. A. Trayanova, A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models, Annals of Biomedical Engineering, 40 (2012), 2243-2254.  doi: 10.1007/s10439-012-0593-5.

[14]

Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J. R. Hughes, A. Reali and G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007), 173-201.  doi: 10.1016/j.cma.2007.07.016.

[15]

C. Bertoglio and A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, J. Comput. Phys., 261 (2014), 162-171.  doi: 10.1016/j.jcp.2013.12.057.

[16]

P. J. Blanco and R. A. Feijoo, A 3D-1D-0D computational model for the entire cardiovascular system, Mecanica Computacional, 24 (2010), 5887-5911. 

[17]

D. Bluestein and S. Einav, Transition to turbulence in pulsatile flow through heart valves-a modified stability approach, Journal of Biomechanical Engineering, 116 (1994), 477-487.  doi: 10.1115/1.2895799.

[18]

M. Bucelli, L. Dede', A. Quarteroni and C. Vergara, Partitioned and monolithic algorithms for the numerical solution of cardiac fluid-structure interaction, MOX Report, 78 (2021).

[19]

R. M. Carey and P. K. Whelton, Prevention, detection, evaluation, and management of high blood pressure in adults: Synopsis of the 2017 American college of cardiology/American heart association hypertension guideline, Annals of Internal Medicine, 168 (2018), 351-358.  doi: 10.7326/M17-3203.

[20]

C. Chnafa, S. Mendez and F. Nicoud, Image-based large-eddy simulation in a realistic left heart, Computers & Fluids, 94 (2014), 173-187.  doi: 10.1016/j.compfluid.2014.01.030.

[21]

Y. J. Choi, J. Constantino, V. Vedula, N. Trayanova and R. Mittal, A new MRI-based model of heart function with coupled hemodynamics and application to normal and diseased canine left ventricles, Frontiers in Bioengineering and Biotechnology, 3 (2015), 1-15.  doi: 10.3389/fbioe.2015.00140.

[22]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, 13, Springer, 2014. doi: 10.1007/978-3-319-04801-7.

[23]

M. Corti, A. Zingaro, L. Dede' and A. Quarteroni, Impact of atrial fibrillation on left atrium haemodynamics: A computational fluid dynamics study, arXiv preprint, arXiv: 2202.10893, 2022.

[24]

L. Dedè, F. Menghini and A. Quarteroni, Computational fluid dynamics of blood flow in an idealized left human heart, Int. J. Numer. Methods Biomed. Eng., 37 (2021), Paper No. e3287, 24 pp. doi: 10.1002/cnm.3287.

[25]

S. Deparis, G. Grandperrin and A. Quarteroni, Parallel preconditioners for the unsteady Navier-Stokes equations and applications to hemodynamics simulations, Computers & Fluids, 92 (2014), 253-273.  doi: 10.1016/j.compfluid.2013.10.034.

[26]

F. Domenichini, G. Pedrizzetti and B. Baccani, Three-dimensional filling flow into a model left ventricle, J. Fluid Mech., 539 (2005), 179-198.  doi: 10.1017/S0022112005005550.

[27]

F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian–Eulerian method for Navier–Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Engrg., 193 (2004), 4819-4836.  doi: 10.1016/j.cma.2004.05.003.

[28]

P. Dyverfeldt, M. Bissell, A. J. Barker, A. F. Bolger, C.-J. Carlhäll, T. Ebbers, C. J. Francios, A. Frydrychowicz, J. Geiger, D. Giese, M. D. Hope, P. J. Kilner, S. Kozerke, S. Myerson, S. Neubauer, O. Wieben and M. Markl, 4D flow cardiovascular magnetic resonance consensus statement, Journal of Cardiovascular Magnetic Resonance, 17 (2015), 1-19.  doi: 10.1186/s12968-015-0174-5.

[29]

M. Fedele, E. Faggiano, L. Dedè and A. Quarteroni, A patient-specific aortic valve model based on moving resistive immersed implicit surfaces, Biomechanics and Modeling in Mechanobiology, 16 (2017), 1779-1803.  doi: 10.1007/s10237-017-0919-1.

[30]

M. Fedele and A. Quarteroni, Polygonal surface processing and mesh generation tools for the numerical simulation of the cardiac function, Int. J. Numer. Methods Biomed. Eng., 37 (2021), Paper No. e3435, 34 pp. doi: 10.1002/cnm.3435.

[31]

L. Formaggia, J. F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels, Comput. Methods Appl. Mech. Engrg., 191 (2001), 561-582.  doi: 10.1016/S0045-7825(01)00302-4.

[32]

L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J. Engrg. Math., 47 (2003), 251-276.  doi: 10.1023/B:ENGI.0000007980.01347.29.

[33]

L. Formaggia and F. Nobile, A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 7 (1999), 105-131. 

[34]

D. Forti and L. Dedè, Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS-LES modeling in a high performance computing framework, Computers & Fluids, 117 (2015), 168-182.  doi: 10.1016/j.compfluid.2015.05.011.

[35]

I. Fumagalli, M. Fedele, C. Vergara, L. Dede', S. Ippolito, F. Nicolò, C. Antona, R. Scrofani and A. Quarteroni, An image-based computational hemodynamics study of the systolic anterior motion of the mitral valve, Computers in Biology and Medicine, 123 (2020), 103922.  doi: 10.1016/j.compbiomed.2020.103922.

[36]

A. Gerbi, Numerical Approximation of Cardiac Electro-Fluid-Mechanical Models: Coupling Strategies for Large-Scale Simulation, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2018.

[37]

J. M. Guccione and A. D. McCulloch, Finite element modeling of ventricular mechanics, In Theory of Heart, Springer, (1991), 121–144. doi: 10.1007/978-1-4612-3118-9_6.

[38]

G. S. Gulsin, A. Singh and G. P. McCann, Cardiovascular magnetic resonance in the evaluation of heart valve disease, BMC Medical Imaging,17 (2017), 1–14. doi: 10.1186/s12880-017-0238-0.

[39]

J. Hee Seo and R. Mittal, Effect of diastolic flow patterns on the function of the left ventricle, Physics of Fluids, 25 (2013), 110801, 22 pp. doi: 10.1063/1.4819067.

[40]

M. Hirschvogel, M. Bassilious, L. Jagschies, S. M. Wildhirt and M. W. Gee, A monolithic 3D-0D coupled closed-loop model of the heart and the vascular system: Experiment-based parameter estimation for patient-specific cardiac mechanics, Int. J. Numer. Methods Biomed. Eng., 33 (2017), e2842, 22 pp. doi: 10.1002/cnm.2842.

[41]

D. I. Hollnagel, P. E. Summers, D. Poulikakos and S. S. Kollias, Comparative velocity investigations in cerebral arteries and aneurysms: 3d phase-contrast MR angiography, laser doppler velocimetry and computational fluid dynamics, NMR in Biomedicine: An International Journal Devoted to the Development and Application of Magnetic Resonance In vivo, 22 (2009), 795-808.  doi: 10.1002/nbm.1389.

[42]

J. C. R. Hunt, A. A. Wray and P. Moin, Eddies, stream, and convergence zones in turbulent flows, Center for Turbulence Research Report, CTR-S88, (1988), 193–208.

[43]

Zygote Media Group Inc, Zygote solid 3D heart generation ii developement report. tech. rep., 2014.,

[44]

E. Karabelas, M. A. F Gsell, C. M. Augustin, L. Marx, A. Neic, A. J. Prassl, L. Goubergrits, T. Kuehne and G. Plank, Towards a computational framework for modeling the impact of aortic coarctations upon left ventricular load, Frontiers in Physiology, 9 (2018), 1-20. 

[45]

H. J. Kim, I. E. Vignon-Clementel, C. A. Figueroa, J. F. LaDisa, K. E. Jansen, J. A. Feinstein and C. A. Taylor, On coupling a lumped parameter heart model and a three-dimensional finite element aorta model, Annals of Biomedical Engineering, 37 (2009), 2153-2169. 

[46]

W. Y. Kim, P. G. Walker, E. M. Pedersen, J. K. Poulsen, S. Oyre, K. Houlind and A. P. Yoganathan, Left ventricular blood flow patterns in normal subjects: A quantitative analysis by three-dimensional magnetic resonance velocity mapping, Journal of the American College of Cardiology, 26 (1995), 224-238.  doi: 10.1016/0735-1097(95)00141-L.

[47]

V. Kumar, A. K. Abbas, N. Fausto and J. C. Aster, Robbins and Cotran Pathologic Basis of Disease, Professional Edition E-Book., Elsevier health sciences, 2014.

[48]

P. K. Kundu, I. M. Cohen and D. R. Dowling, Fluid Mechanics, Academic Press, 5 edition, 2014.

[49]

A. M. Maceira, S. K. Prasad, M. Khan and D. J. Pennell, Normalized left ventricular systolic and diastolic function by steady state free precession cardiovascular magnetic resonance, Journal of Cardiovascular Magnetic Resonance, 8 (2006), 417-426.  doi: 10.1080/10976640600572889.

[50]

A. Masci, M. Alessandrini, D. Forti, F. Menghini, L. Dedé, C. Tomasi, A. Quarteroni and C. Corsi, A patient-specific computational fluid dynamics model of the left atrium in atrial fibrillation: Development and initial evaluation, In International Conference on Functional Imaging and Modeling of the Heart, Springer, (2017), 392–400. doi: 10.1007/978-3-319-59448-4_37.

[51]

A. Masci, M. Alessandrini, D. Forti, F. Menghini, L. Dedé, C. Tomasi, A. Quarteroni and C. Corsi, A proof of concept for computational fluid dynamic analysis of the left atrium in atrial fibrillation on a patient-specific basis, Journal of Biomechanical Engineering, 142 (2020), 011002, 11 pp. doi: 10.1115/1.4044583.

[52]

A. Masci, L. Barone, L. Dedè, M. Fedele, C. Tomasi, A. Quarteroni and C. Corsi, The impact of left atrium appendage morphology on stroke risk assessment in atrial fibrillation: A computational fluid dynamics study, Frontiers in Physiology, 9 (2019), 1-11.  doi: 10.3389/fphys.2018.01938.

[53]

V. Milišić and A. Quarteroni, Analysis of lumped parameter models for blood flow simulations and their relation with 1D models, M2AN Math. Model. Numer. Anal., 38 (2004), 613-632.  doi: 10.1051/m2an:2004036.

[54]

R. Mittal, J. H. Seo, V. Vedula, Y. J. Choi, H. Liu, H. H. Huang, S. Jain, L. Younes, T. Abraham and R. T. George, Computational modeling of cardiac hemodynamics: Current status and future outlook, J. Comput. Phys., 305 (2016), 1065-1082.  doi: 10.1016/j.jcp.2015.11.022.

[55]

M. T. Ngo, C. I. Kim, J. Jung, G. H. Chung, D. H. Lee and H. S. Kwak, Four-dimensional flow magnetic resonance imaging for assessment of velocity magnitudes and flow patterns in the human carotid artery bifurcation: Comparison with computational fluid dynamics, Diagnostics, 9 (2019), 1-12.  doi: 10.3390/diagnostics9040223.

[56]

K. Perktold, E. Thurner and Th. Kenner, Flow and stress characteristics in rigid walled and compliant carotid artery bifurcation models, Medical & Biological Engineering & Computing, 32 (1994), 19-26.  doi: 10.1007/BF02512474.

[57]

R. Piersanti, P. C. Africa, M. Fedele, C. Vergara, L. Dedè, A. F. Corno and A. Quarteroni, Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Methods Appl. Mech. Engrg., 373 (2021), Paper No. 113468, 33 pp. doi: 10.1016/j.cma.2020.113468.

[58]

A. Quarteroni, Numerical Models for Differential Problems, volume 2., Springer, 2009. doi: 10.1007/978-88-470-1071-0.

[59]

A. Quarteroni, L. Dedè, A. Manzoni and C. Vergara, Mathematical Modelling of the Human Cardiovascular System: Data, Numerical Approximation, Clinical Applications, 33. Cambridge University Press, 2019. doi: 10.1017/9781108616096.

[60]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 37., Springer Science & Business Media, 2010.

[61]

A. Quarteroni, A. Veneziani and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Comput. Methods Appl. Mech. Engrg., 302 (2016), 193-252.  doi: 10.1016/j.cma.2016.01.007.

[62]

F. Regazzoni, L. Dedè and A. Quarteroni, Machine learning of multiscale active force generation models for the efficient simulation of cardiac electromechanics, Comput. Methods Appl. Mech. Engrg., 370 (2020), 113268, 30 pp. doi: 10.1016/j.cma.2020.113268.

[63]

F. Regazzoni, M. Salvador, P. C. Africa, M. Fedele, L. Dedè and A. Quarteroni, A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation, Journal of Computational Physics, 457 (2022), 111083.  doi: 10.1016/j.jcp.2022.111083.

[64]

M. Salvador, L. Dedè and A. Quarteroni, An intergrid transfer operator using radial basis functions with application to cardiac electromechanics, Comput. Mech., 66 (2020), 491-511.  doi: 10.1007/s00466-020-01861-x.

[65]

A. Santiago, Fluid-electro-mechanical model of the human heart for supercomputers, 2018.

[66]

A. Santiago, J. Aguado-Sierra, M. Zavala-Aké, R. Doste-Beltran, S. Gómez, R. Arís, J. C. Cajas, E. Casoni and M. Vázquez, Fully coupled fluid-electro-mechanical model of the human heart for supercomputers, Int. J. Numer. Methods Biomed. Eng., 34 (2018), e3140, 30 pp. doi: 10.1002/cnm.3140.

[67]

Y. Shi and T. Korakianitis, Numerical simulation of cardiovascular dynamics with left heart failure and in-series pulsatile ventricular assist device, Artificial Organs, 30 (2006), 929-948.  doi: 10.1111/j.1525-1594.2006.00326.x.

[68]

Y. Shi, P. Lawford and R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMedical Engineering Online, 10 (2011), 1-38.  doi: 10.1186/1475-925X-10-33.

[69]

T. Sugimoto, R. Dulgheru, A. Bernard, F. Ilardi, L. Contu, K. Addetia, L. Caballero, N. Akhaladze, G. D. Athanassopoulos, D. Barone, M. Baroni, N. Cardim, A. Hagendorff, K. Hristova, T. Lopez, G. de la Morena, B. A. Popescu, M. Moonen, M. Penicka, T. Ozyigit, J. D. Rodrigo Carbonero, N. van de Veire, R. S. von Bardeleben, D. Vinereanu, J. L. Zamorano, Y. Y. Go, M. Rosca, A. Calin, J. Magne, B. Cosyns, S. Marchetta, E. Donal, G. Habib, M. Galderisi, L. P. Badano, R. M. Lang and P. Lancellotti, Echocardiographic reference ranges for normal left ventricular 2D strain: Results from the EACVI NORRE study, European Heart Journal-Cardiovascular Imaging, 18 (2017), 833-840. 

[70]

A. Tagliabue, L. Dedè and A. Quarteroni, Complex blood flow patterns in an idealized left ventricle: A numerical study, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (2017), 093939, 26 pp. doi: 10.1063/1.5002120.

[71]

A. Tagliabue, L. Dedè and A. Quarteroni, Fluid dynamics of an idealized left ventricle: The extended Nitsche's method for the treatment of heart valves as mixed time varying boundary conditions, Internat. J. Numer. Methods Fluids, 85 (2017), 135-164.  doi: 10.1002/fld.4375.

[72]

C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite element analysis of pulsatile flow in the abdominal aorta under resting and exercise conditions, ASME-Publications-Bed, 33 (1996), 81-82. 

[73]

C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite element modeling of blood flow in arteries, Comput. Methods Appl. Mech. Engrg., 158 (1998), 155-196.  doi: 10.1016/S0045-7825(98)80008-X.

[74]

K. H. W. J. ten Tusscher and A. V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model, American Journal of Physiology-Heart and Circulatory Physiology, 291 (2006), 1088-1100.  doi: 10.1152/ajpheart.00109.2006.

[75]

A. This, L. Boilevin-Kayl, M. A. Fernández and J.-F. Gerbeau, Augmented resistive immersed surfaces valve model for the simulation of cardiac hemodynamics with isovolumetric phases, Int. J. Numer. Methods Biomed. Eng., 36 (2020), e3223, 26 pp. doi: 10.1002/cnm.3223.

[76]

A. This, H. G. Morales, O. Bonnefous, M. A. Fernández and J.-F. Gerbeau, A pipeline for image based intracardiac CFD modeling and application to the evaluation of the PISA method, Computer Methods in Applied Mechanics and Engineering, 358 (2020), 112627, 22 pp. doi: 10.1016/j.cma.2019.112627.

[77]

L. Thomas, E. Foster and N. B. Schiller, Peak mitral inflow velocity predicts mitral regurgitation severity, Journal of the American College of Cardiology, 31 (1998), 174-179. 

[78]

P. Tripathi, Pathophysiological mechanism behind diabetic cardiovascular disorders, International Journal of Current Research, 6 (2014), 4426-4436. 

[79]

F. N. van de Vosse and N. Stergiopulos, Pulse wave propagation in the arterial tree, Annual Review of Fluid Mechanics, 43 (2011), 467-499.  doi: 10.1146/annurev-fluid-122109-160730.

[80]

R. J. van der Geest and P. Garg, Advanced analysis techniques for intra-cardiac flow evaluation from 4D flow MRI, Current Radiology Reports, 4 (2016), 1-10. 

[81]

A. C. Verkaik, A. C. B. Bogaerds, F. Storti and F. N. van de Vosse., A coupled overlapping domain method for the computation of transitional flow through artificial heart valves, In ASME 2012 Summer Bioengineering Conference (SBC 2012), American Society of Mechanical Engineers, (2012), 217–218. doi: 10.1115/SBC2012-80257.

[82]

I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries, Computer Methods in Biomechanics and Biomedical Engineering, 13 (2010), 625-640.  doi: 10.1080/10255840903413565.

[83]

F. Viola, V. Meschini and R. Verzicco, Fluid–structure-electrophysiology interaction (FSEI) in the left-heart: A multi-way coupled computational model, Eur. J. Mech. B Fluids, 79 (2020), 212-232.  doi: 10.1016/j.euromechflu.2019.09.006.

[84]

F. Viola, V. Meschini and R. Verzicco, Effects of stenotic aortic valve on the left heart hemodynamics: A fluid-structure-electrophysiology approach, arXiv preprint, arXiv: 2103.14680, 2021.

[85]

F. Viola, V. Spandan, V. Meschini, J. Romero, M. Fatica, M. D. de Tullio and R. Verzicco, FSEI-GPU: GPU accelerated simulations of the fluid-structure-electrophysiology interaction in the left heart, Comput. Phys. Commun., 273 (2022), Paper No. 108248, 12 pp. doi: 10.1016/j.cpc.2021.108248.

[86]

S. Z. Zhao, P. Papathanasopoulou, Q. Long, I. Marshall and X. Y. Xu, Comparative study of magnetic resonance imaging and image-based computational fluid dynamics for quantification of pulsatile flow in a carotid bifurcation phantom, Annals of Biomedical Engineering, 31 (2003), 962-971.  doi: 10.1114/1.1590664.

[87]

X. Zheng, J. H. Seo, V. Vedula, T. Abraham and R. Mittal, Computational modeling and analysis of intracardiac flows in simple models of the left ventricle, Eur. J. Mech. B Fluids, 35 (2012), 31-39.  doi: 10.1016/j.euromechflu.2012.03.002.

[88]

A. Zingaro, L. Dede', F. Menghini and A. Quarteroni, Hemodynamics of the heart's left atrium based on a variational multiscale-LES numerical method, Eur. J. Mech. B Fluids, 89 (2021), 380-400.  doi: 10.1016/j.euromechflu.2021.06.014.

Figure 1.  The 0D circulation model
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Figure 2.  Fluid domain in reference configuration (left), ALE map $ {\boldsymbol{x}} = \mathcal{A}_t ( {\widehat{{\mathit{\boldsymbol{x}}}}}) $ and domain in current configuration (right). In the current configuration, the domain $ \Omega_t $ is bounded by $ \Gamma_t = \Gamma_t^\mathrm{D} \cup \Gamma_t^\mathrm{N} $; $ \Sigma_k $ is an immersed surface modeled by means of the RIIS method
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Figure 3.  The LH geometry: (A) the three subdomains $ \Omega_t = \overline \Omega_t^{{\rm{LA}}} \cup \overline \Omega_t^{_{\rm{LV}}} \cup \overline \Omega_t^{{\rm{AA}}} $; (B) the boundary portions of the LH geometry $ \Gamma_t = \left (\bigcup_{i=1}^5 \overline \Gamma^{{\rm{PVein}}_i} \right )\cup \overline \Gamma_t^{\rm{AA}} \cup \overline \Gamma_t^{{\rm{w}}} $; (C) in yellow, the immersed surfaces $ \Sigma_{\rm{MV}} $ and $ \Sigma_{\rm{AV}} $ (respectively in their open and closed configurations); in red, the Neumann data (for both inlet and outlet sections); in green the Dirichlet datum at wall
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Figure 4.  Boundary portions of the LH geometry in reference configuration. This splitting of the domain is used to define the Laplacian problem (13)
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Figure 5.  Displacement procedure. Boxes numbers are referred to lines in Algorithm 1
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Figure 6.  EM simulation of the LV: (A) LV in its reference configuration; (B), (C) LV during systole and diastole colored by displacement magnitude
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Figure 7.  LH geometry warped by $ \widehat{\boldsymbol{d}}_{\Gamma} $ at different times during a heart cycle
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Figure 8.  Volumes of LA, LV and AA achieved applying the displacement – $ \widehat{\boldsymbol{d}}_{\Gamma} $ defined in Eq. (12) – to $ \widehat \Gamma $
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Figure 9.  Immersed valve $ \Sigma_\mathrm k $ with upwind and downwind control volumes where average pressures are computed. $ Q_\mathrm k $ is the flowrate across $ \Sigma_\mathrm k $. This picture corresponds to a simple a two-dimensional fluid domain for the sake of simplicity
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Figure 10.  Cardiac valves in their fully closed and fully open configurations
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Figure 11.  The geometric multiscale model: coupling between the 3D CFD model of the LH and the 0D circulation model of the remaining cardiocirculatory system
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Figure 12.  Sketch of the algorithm for the CFD simulation of the LH. The coupling between the CFD and the circulation problem is solved via a segregated numerical scheme
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Figure 13.  The LH tetrahedral mesh made of three conforming meshes for the LA, LV and AA subdomains; a clip of the mesh showing the local mesh refinement near the MV and AV
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Figure 14.  Flowrates (top) and pressures (bottom) at the interfaces of the 3D-0D model
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Figure 15.  Flow properties to determine opening and closure of valves. Top: average pressure in LA, LV and AA; bottom: time derivative of LV volume. Average pressures in the chambers are computed in the control volumes in the LH displayed on the right
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Figure 16.  Volume rendering of velocity magnitude during the whole heartbeat
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Figure 17.  Pressure on a clip in the LV apico-basal direction during the whole heartbeat
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Figure 18.  AV section warped by velocity during the ejection phase. In trasparency: the AV represented by the RIIS method
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Figure 19.  Left: mean, first and third quartile of the space distribution of the velocity magnitude $ {\boldsymbol{u}}_\mathrm{MV} $ in a control volume $ \Omega_t^\mathrm{MV} $ downstream the MV section. Right: flowrate computed on a section downwind the AV ($ \int_{\Gamma_t^*} {\boldsymbol{u}} \cdot {\boldsymbol{n}} $)
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Figure 20.  Iso-contours of Q-criterion ($ \mathbb Q({\boldsymbol{u}}) = 40 \, \mathrm{s}^{-2} $) colored according to velocity magnitude during a whole heartbeat
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Figure 21.  Formation of ring shaped vortex during early diastole. Top: iso-contours of Q-criterion (with $ \mathbb Q({\boldsymbol{u}}) =1000\, \rm{s}^{-2} $) coloured according to velocity magnitude. Bottom: projection of the vorticity on the normal direction (pointing towards the reader) of a slice in the LV apico-basal direction
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Figure 22.  Vortex formation under the MV section at early diastolic peak: (A) Streamlines colored according to velocity magnitude ($ t=0.45 $ s); (B) Surface LIC visualization – with velocity as integrator – on a slice colored according to velocity magnitude ($ t=0.48 $ s)
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Table 2.  Parameters for the setup of the numerical simulations
$ \rho $ $ \mu $ $ {\boldsymbol{u}}_0 $ $ R_{\mathrm k} $ $ \varepsilon_\mathrm k $ $ T_f $ $ T_\mathrm{HB} $
$ [\mathrm{kg/m}^3] $ $ [\mathrm{kg/(m}\cdot\mathrm{s})] $ $ [\mathrm m / \mathrm s] $ [kg/(m$ ^2\cdot $s)] [mm] [s] [s]
MV AV MV AV
$ 1.06 \cdot 10^3 $ $ 3.5 \cdot 10^{-3} $ $ {\boldsymbol{0}} $ $ 10^4 $ $ 10^4 $ 0.6 0.6 2.0 1.0
 
$h$ cells DOFs ( $ \mathbb{P}_1 - \mathbb{P}_1 $) BDF $\Delta t$
[mm] [-] [-] [-] [s]
min avg max $\boldsymbol{u}$ $p$ total
0.4 1.2 4.1 1'627'795 806'295 268'765 1'075'060 1 $2.5 \cdot 10^{-4}$
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$ \rho $ $ \mu $ $ {\boldsymbol{u}}_0 $ $ R_{\mathrm k} $ $ \varepsilon_\mathrm k $ $ T_f $ $ T_\mathrm{HB} $
$ [\mathrm{kg/m}^3] $ $ [\mathrm{kg/(m}\cdot\mathrm{s})] $ $ [\mathrm m / \mathrm s] $ [kg/(m$ ^2\cdot $s)] [mm] [s] [s]
MV AV MV AV
$ 1.06 \cdot 10^3 $ $ 3.5 \cdot 10^{-3} $ $ {\boldsymbol{0}} $ $ 10^4 $ $ 10^4 $ 0.6 0.6 2.0 1.0
 
$h$ cells DOFs ( $ \mathbb{P}_1 - \mathbb{P}_1 $) BDF $\Delta t$
[mm] [-] [-] [-] [s]
min avg max $\boldsymbol{u}$ $p$ total
0.4 1.2 4.1 1'627'795 806'295 268'765 1'075'060 1 $2.5 \cdot 10^{-4}$
Table 3.  Biomarkers: comparison between numerical results and clinical values acquired in healthy individuals
Biomarker In-silico result In-vivo measurements Reference
LV stroke volume [ml] 82.6 $ 95 \pm 14 $ [49]
LV ejection fraction [%] 55.8 $ 57.5\pm 7.5 $ [47]
Peak AV flowrate [ml/s] $ 493.3 $ $ \approx 489 $ [38]
LV peak pressure [mmHg] 121.2 $ 119 \pm 13 $ [69]
Peak E-wave velocity [m/s] $ 0.96 $ $ 0.89 \pm 0.15 $ [77]
Peak A-wave velocity [m/s] $ 0.71 $ $ 0.78 \pm 0.26 $ [77]
EA ratio $ [-] $ $ 1.35 $ $ 1.30 \pm 0.57 $ [77]
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Biomarker In-silico result In-vivo measurements Reference
LV stroke volume [ml] 82.6 $ 95 \pm 14 $ [49]
LV ejection fraction [%] 55.8 $ 57.5\pm 7.5 $ [47]
Peak AV flowrate [ml/s] $ 493.3 $ $ \approx 489 $ [38]
LV peak pressure [mmHg] 121.2 $ 119 \pm 13 $ [69]
Peak E-wave velocity [m/s] $ 0.96 $ $ 0.89 \pm 0.15 $ [77]
Peak A-wave velocity [m/s] $ 0.71 $ $ 0.78 \pm 0.26 $ [77]
EA ratio $ [-] $ $ 1.35 $ $ 1.30 \pm 0.57 $ [77]
Table 4.  Parameters used in the circulation model
Compartment Parameter Description Unit of measure Value
Right atrium $ E_\mathrm{A} $ Active elastance [mmHg/ml] 0.06
$ E_\mathrm{B} $ Passive elastance [mmHg/ml] 0.07
$ d_\mathrm{c} $ Duration of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.335
$ d_\mathrm{r} $ Duration of relax. relative w.r.t. $ T_\mathrm{HB} $ $ 1.45 \cdot 10^{-2} $
$ t_\mathrm{c} $ Initial time of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.80
$ V_{0, \mathrm{RA}} $ Resting volume [ml] 4.00
Right ventricle $ E_\mathrm{A} $ Active elastance [mmHg/ml] 0.65
$ E_\mathrm{B} $ Passive elastance [mmHg/ml] 0.05
$ d_\mathrm{c} $ Duration of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.335
$ d_\mathrm{r} $ Duration of relax. relative w.r.t. $ T_\mathrm{HB} $ $ 1.45 \cdot 10^{-2} $
$ t_\mathrm{c} $ Initial time of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.00
$ V_{0, \mathrm{RV}} $ Resting volume [ml] 10.00
Pulmonary arterial system $ R^\mathrm{PUL}_\mathrm{AR} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.25
$ C^\mathrm{PUL}_\mathrm{AR} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 5.00
$ L^\mathrm{PUL}_\mathrm{AR} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-4} $
Pulmonary venous system $ R^\mathrm{PUL}_\mathrm{VEN} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.02
$ C^\mathrm{PUL}_\mathrm{VEN} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 100.00
$ L^\mathrm{PUL}_\mathrm{VEN} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-5} $
Systemic arterial system $ R^\mathrm{SYS}_\mathrm{AR} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 1.00
$ C^\mathrm{SYS}_\mathrm{AR} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 2.00
$ L^\mathrm{SYS}_\mathrm{AR} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-3} $
Systemic venous system $ R^\mathrm{SYS}_\mathrm{VEN} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.24
$ C^\mathrm{SYS}_\mathrm{VEN} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 60.00
$ L^\mathrm{SYS}_\mathrm{VEN} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-4} $
Tricuspid valve $ R_\mathrm{min} $ Minimum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 7.5 \cdot 10^{-3} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
Pulmonary valve $ R_\mathrm{min} $ Minimum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 7.5 \cdot 10^{-3} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
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Compartment Parameter Description Unit of measure Value
Right atrium $ E_\mathrm{A} $ Active elastance [mmHg/ml] 0.06
$ E_\mathrm{B} $ Passive elastance [mmHg/ml] 0.07
$ d_\mathrm{c} $ Duration of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.335
$ d_\mathrm{r} $ Duration of relax. relative w.r.t. $ T_\mathrm{HB} $ $ 1.45 \cdot 10^{-2} $
$ t_\mathrm{c} $ Initial time of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.80
$ V_{0, \mathrm{RA}} $ Resting volume [ml] 4.00
Right ventricle $ E_\mathrm{A} $ Active elastance [mmHg/ml] 0.65
$ E_\mathrm{B} $ Passive elastance [mmHg/ml] 0.05
$ d_\mathrm{c} $ Duration of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.335
$ d_\mathrm{r} $ Duration of relax. relative w.r.t. $ T_\mathrm{HB} $ $ 1.45 \cdot 10^{-2} $
$ t_\mathrm{c} $ Initial time of contract. relative w.r.t. $ T_\mathrm{HB} $ 0.00
$ V_{0, \mathrm{RV}} $ Resting volume [ml] 10.00
Pulmonary arterial system $ R^\mathrm{PUL}_\mathrm{AR} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.25
$ C^\mathrm{PUL}_\mathrm{AR} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 5.00
$ L^\mathrm{PUL}_\mathrm{AR} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-4} $
Pulmonary venous system $ R^\mathrm{PUL}_\mathrm{VEN} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.02
$ C^\mathrm{PUL}_\mathrm{VEN} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 100.00
$ L^\mathrm{PUL}_\mathrm{VEN} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-5} $
Systemic arterial system $ R^\mathrm{SYS}_\mathrm{AR} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 1.00
$ C^\mathrm{SYS}_\mathrm{AR} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 2.00
$ L^\mathrm{SYS}_\mathrm{AR} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-3} $
Systemic venous system $ R^\mathrm{SYS}_\mathrm{VEN} $ Resistance $ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $ 0.24
$ C^\mathrm{SYS}_\mathrm{VEN} $ Capacitance $ [\mathrm{ml}/\mathrm{mmHg}] $ 60.00
$ L^\mathrm{SYS}_\mathrm{VEN} $ Inductance $ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $ $ 5.00 \cdot 10^{-4} $
Tricuspid valve $ R_\mathrm{min} $ Minimum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 7.5 \cdot 10^{-3} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
Pulmonary valve $ R_\mathrm{min} $ Minimum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 7.5 \cdot 10^{-3} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
Table 5.  Initial state of the circulation model
Compartment Parameter Description Unit of measure Value
Right atrium $ V_\mathrm{RA} $ Volume [ml] 78.95
Right ventricle $ V_\mathrm{RV} $ Volume [ml] 154.00
Pulmonary arterial system $ p^\mathrm{PUL}_\mathrm{AR} $ Pressure [mmHg] 33.50
$ Q^\mathrm{PUL}_\mathrm{AR} $ Flowrate [ml/s] 69.44
Pulmonary venous system $ p^\mathrm{PUL}_\mathrm{VEN} $ Pressure [mmHg] 16.16
$ Q^\mathrm{PUL}_\mathrm{VEN} $ Flowrate [ml/s] 0.00
Systemic arterial system $ p^\mathrm{SYS}_\mathrm{AR} $ Pressure [mmHg] 91.68
$ Q^\mathrm{SYS}_\mathrm{AR} $ Flowrate [ml/s] 63.71
Systemic venous system $ p^\mathrm{SYS}_\mathrm{VEN} $ Pressure [mmHg] 23.99
$ Q^\mathrm{SYS}_\mathrm{VEN} $ Flowrate [ml/s] 65.40
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Compartment Parameter Description Unit of measure Value
Right atrium $ V_\mathrm{RA} $ Volume [ml] 78.95
Right ventricle $ V_\mathrm{RV} $ Volume [ml] 154.00
Pulmonary arterial system $ p^\mathrm{PUL}_\mathrm{AR} $ Pressure [mmHg] 33.50
$ Q^\mathrm{PUL}_\mathrm{AR} $ Flowrate [ml/s] 69.44
Pulmonary venous system $ p^\mathrm{PUL}_\mathrm{VEN} $ Pressure [mmHg] 16.16
$ Q^\mathrm{PUL}_\mathrm{VEN} $ Flowrate [ml/s] 0.00
Systemic arterial system $ p^\mathrm{SYS}_\mathrm{AR} $ Pressure [mmHg] 91.68
$ Q^\mathrm{SYS}_\mathrm{AR} $ Flowrate [ml/s] 63.71
Systemic venous system $ p^\mathrm{SYS}_\mathrm{VEN} $ Pressure [mmHg] 23.99
$ Q^\mathrm{SYS}_\mathrm{VEN} $ Flowrate [ml/s] 65.40
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