×

Multiscale modeling and numerical simulation of calcium cycling in cardiac myocytes. (English) Zbl 1400.92164

Summary: Simulation of electrophysiology and intracellular Ca\(^{2+}\) dynamics in cardiomyocytes comprises fast stochastic dynamics in tiny subcompartments, partial differential equations (PDEs) with stochastic source terms for concentration fields, and the globally coupling membrane potential. We use highly unstructured meshes appropriate for the spatial heterogeneity of intracellular Ca\(^{2+}\) release, and adaptive time-stepping algorithms appropriate for the simulation of stochastic channel opening and closing in subcompartments like the Ca\(^{2+}\) release units (CRUs). A set of reaction-diffusion equations describes the behavior of the intracellular concentration fields on length scales from tens of nanometers to cell size (tens of micrometers) and milliseconds to tens of seconds. Detailed highly stochastic CRU models drive source functions in the PDE model. These CRU models cover dynamics with time scales below 1 ms and length scales from a few to a few hundred nm. The spatially detailed Ca\(^{2+}\) dynamics and the cardiomyocyte membrane potential interact. Membrane potential introduces a global spatial coupling across the whole cell due to its large coupling length. Its dynamics are consequently represented by a set of ordinary differential equations (ODEs). We developed an efficient adaptive finite element simulator interface for the numerical simulation of this multiphysics and multiscale problem. The use of stationary Green functions within the CRU models and highly unstructured meshes for the PDE integration allows for bridging of many orders of magnitude of spatial scale, to represent accurately the Ca\(^{2+}\) concentration dynamics from within a single CRU up to the level of the whole cell. The time scale separation between fast stochastic CRU dynamics and slower PDE dynamics limits efficiency with traditional approaches. We present new methods to circumvent that problem. We demonstrate large-scale numerical results for a 436 CRU cellular subdomain using many-core parallel machines.

MSC:

92C37 Cell biology
92C40 Biochemistry, molecular biology
92C30 Physiology (general)
35K57 Reaction-diffusion equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Alfonsi, E. Cancès, G. Turinici, B. D. Ventura, and W. Huisinga, Exact Simulation of Hybrid Stochastic and Deterministic Models for Biochemical Systems, INRIA Rapport de Recherche 5435, Thèmes NUM et BIO, 2004. · Zbl 1070.92019
[2] M. Alkämper, A. Dedner, R. Klöfkorn, and M. Nolte, The DUNE-ALUGrid module, Arch. Numer. Software, 4 (2016), pp. 1–28, .
[3] D. Baddeley, I. Jayasinghe, L. Lam, S. Rossberger, M. Cannell, and C. Soeller, Optical single-channel resolution imaging of the ryanodine receptor distribution in rat cardiac myocyte, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 22275–22280.
[4] M. Blatt, A Parallel Algebraic Multigrid Method for Elliptic Problems with Highly Discontinuous Coefficients, Ph.D. thesis, Ruprechts-Karls-Universität Heidelberg, 2010. · Zbl 1194.65002
[5] M. Blatt and P. Bastian, On the generic parallelisation of iterative solvers for the finite element method, Int. J. Comput. Sci. Eng., 4 (2008), pp. 56–69, .
[6] V. E. Bondarenko, G. P. Szigeti, G. C. Bett, S. J. Kim, and R. Rasmusson, Computer model of action potential of mouse ventricular myocytes, Am. J. Physiol. Heart Circ. Physiol., 287 (2004), pp. 1378–1403.
[7] E. Chudin, J. Goldhaber, A. Garfinkel, J. Weiss, and B. Kogan, Intracellular \(Ca^{2+}\) dynamics and the stability of ventricular tachycardia, Biophys. J., 77 (1999), pp. 2930–2941, .
[8] A. Fabiato, Time and calcium dependence of activation and inactivation of calcium-induced release of calcium from the sarcoplasmic reticulum of a skinned canine cardiac purkinje cell, J. Gen. Physiol., 85 (1985), pp. 247–289.
[9] C. Franzini-Armstrong, F. Protasi, and V. Ramesh, Shape, size, and distribution of \(Ca^{2+}\) release units and couplons in skeletal and cardiac muscles, Biophys. J., 77 (1999), pp. 1528–1539.
[10] A. Glukhovsky, D. R. Adam, G. Amitzur, and S. Sideman, Mechanism of \(Ca^{++}\) release from the sarcoplasmic reticulum: A computer model, Ann. Biomed. Eng., 26 (1998), pp. 213–229, .
[11] M. K. Gobbert, Long-time simulations on high resolution meshes to model calcium waves in a heart cell, SIAM J. Sci. Comput., 30 (2008), pp. 2922–2947, . · Zbl 1178.92009
[12] J. Greenstein and R. Winslow, An integrative model of the cardiac ventricular myocyte incorporating local control of Ca\(^{2+}\) release, Biophys. J., 83 (2002), pp. 2918–2945, .
[13] J. Hake and G. Lines, Stochastic binding of \(Ca^{2+}\) ions in the dyadic cleft; continuous vs. random walk description of diffusion, Biophys. J., 94 (2008), pp. 4184–4201.
[14] T. J. Hund and Y. Rudy, Rate dependence and regulation of action potential and calcium transient in a canine cardiac ventricular cell model, Circ., 110 (2004), pp. 3168–3174.
[15] A. F. Huxley and R. Niedergerke, Structural changes in muscle during contraction; interference microscopy of living muscle fibres, Nature, 173 (1954), pp. 971–973.
[16] V. Iyer, R. Mazhari, and R. L. Winslow, A computational model of the human left-ventricular epicardial myocyte, Biophys. J., 87 (2004), pp. 1507–1525, .
[17] L. Izu, S. Means, J. Shadid, Y. Chen-Izu, and C. Balke, Interplay of ryanodine receptor distribution and calcium dynamics, Biophys. J., 91 (2006), pp. 95–112, .
[18] L. T. Izu, J. R. Mauban, C. Balke, and W. G. Wier, Large currents generate cardiac \(Ca^{2+}\) sparks, Biophys. J., 80 (2001), pp. 88–102, .
[19] M. Jafri, J. Rice, and R. Winslow, Cardiac \(Ca^{2+}\) dynamics: The roles of ryanodine receptor adaptation and sarcoplasmic reticulum load, Biophys. J., 74 (1998), pp. 1149–1168.
[20] G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359–392, . · Zbl 0915.68129
[21] J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2nd ed., Springer, New York, 2009. · Zbl 1273.92017
[22] C. Y. Ko, M. B. Liu, Z. Song, Z. Qu, and J. N. Weiss, Multiscale determinants of delayed afterdepolarization amplitude in cardiac tissue, Biophys. J., 112 (2017), pp. 1949–1961, .
[23] J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lect. Notes Comput. Sci. Eng. 16, Springer-Verlag, Berlin, 2001. · Zbl 0963.65102
[24] J. Lang and D. Teleaga, Towards a fully space-time adaptive FEM for magnetoquasistatics, IEEE Trans. Magn., 44 (2008), pp. 1238–1241, .
[25] P. Lipp and E. Niggli, Microscopic spiral waves reveal positive feedback in subcellular calcium signaling, Biophys. J., 65 (1993), pp. 2272–2276.
[26] C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes, Circ. Res., 74 (1994), pp. 1071–1096.
[27] C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation, Circ. Res., 74 (1994), pp. 1097–1113.
[28] A. Mahajan, Y. Shiferaw, D. Sato, A. Baher, R. Olcese, L.-H. Xie, M.-J. Yang, P.-S. Chen, J. G. Restrepo, A. Karma, A. Garfinkel, Z. Qu, and J. N. Weiss, A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates, Biophys. J., 94 (2008), pp. 392–410, .
[29] C. Nagaiah and S. Rüdiger, Whole-cell simulations of hybrid stochastic and deterministic calcium dynamics in 3d geometry, J. Comp. Interdisciplinary Sci. (JCIS), 3 (2012), pp. 3–18.
[30] C. Nagaiah, S. Rüdiger, G. Warnecke, and M. Falcke, Adaptive space and time numerical simulation of reaction-diffusion models for intracellular calcium dynamics, Appl. Math. Comput., 218 (2012), pp. 10194–10210, . · Zbl 1246.65017
[31] E. Niggli and W. Lederer, Voltage-independent calcium release in heart muscle, Science, 250 (1990), pp. 565–568.
[32] M. Nivala, E. de Lange, R. Rovetti, and Z. Qu, Computational modeling and numerical methods for spatiotemporal calcium cycling in ventricular myocytes, Front. Physiol., 3 (2012), pp. 1–12.
[33] M. Nivala, C. Y. Ko, M. Nivala, J. N. Weiss, and Z. Qu, The emergence of subcellular pacemaker sites for calcium waves and oscillations, J. Physiol., 591 (2013), pp. 5305–5320, .
[34] J.-i. Okada, S. Sugiura, S. Nishimura, and T. Hisada, Three-dimensional simulation of calcium waves and contraction in cardiomyocytes using the finite element method, Am. J. Physiol. Cell Physiol., 288 (2005), pp. C510–C522, .
[35] S. Pandit, R. Clark, W. R. Giles, and S. Demir, A mathematical model of action potential heterogeneity in adult rat left ventricular myocytes, Biophys. J., 81 (2001), pp. 3029–3051.
[36] Z. Qu, J. N. Weiss, and A. Garfinkel, Cardiac electrical restitution properties and stability of reentrant spiral waves: A simulation study, Am. J. Physiol. Heart Circ. Physiol., 276 (1999), pp. H269–H283,
[37] J. G. Restrepo and A. Karma, Spatiotemporal intracellular calcium dynamics during cardiac alternans, Chaos, 19 (2009), 037115.
[38] J. G. Restrepo, J. N. Weiss, and A. Karma, Calsequestrin-mediated mechanism for cellular calcium transient alternans, Biophys. J., 95 (2008), pp. 3767–3789.
[39] T. Schendel, R. Thul, J. Sneyd, and M. Falcke, How does the ryanodine receptor in the ventricular myocyte wake up - by a single or by multiple open l-type \(Ca^{2+}\) channels?, Eur. Biophys. J., 41 (2012), pp. 27–39.
[40] Y. Shiferaw, M. A. Watanabe, A. Garfinkel, J. N. Weiss, and A. Karma, Model of intracellular calcium cycling in ventricular myoctes, Biophys. J., 85 (2003), pp. 3666–3686.
[41] S. M. Snyder, B. M. Palmer, and R. L. Moore, A mathematical model of cardiocyte \(Ca^{2+}\) dynamics with a novel representation of sarcoplasmic reticular \(Ca^{2+}\) control, Biophys. J., 79 (2000), pp. 94–115, .
[42] E. A. Sobie, K. W. Dilly, J. dos Santos Cruz, W. J. Lederer, and M. S. Jafri, Termination of cardiac \(Ca^{2+}\) sparks: An investigative mathematical model of calcium-induced calcium release, Biophys. J., 83 (2002), pp. 59–78, .
[43] C. Soeller and M. Cannell, Numerical simulation of local calcium movements during L-type calcium channel gating in the cardiac diad, Biophys. J., 73 (1997), pp. 97–111.
[44] A. Takahashi, P. Camacho, J. D. Lechleiter, and B. Herman, Measurement of intracellular calcium, Physiol. Rev., 79 (1999), pp. 1089–1125, .
[45] A.-H. Tang and S.-Q. Wang, Transition of spiral calcium waves between multiple stable patterns can be triggered by a single calcium spark in a fire-diffuse-fire model, Chaos, 19 (2009), 037114, .
[46] K. H. ten Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov, A model for human ventricular tissue, Am. J. Physiol. Heart Circ. Physiol., 286 (2004), pp. H1573–H1589.
[47] H. E. D. J. ter Keurs and P. A. Boyden, Calcium and Arrhythmogenesis, Physiol. Rev., 87 (2007), pp. 457–506, .
[48] N. S. Torres, R. Larbig, A. Rock, J. I. Goldhaber, and J. H. B. Bridge, \(Na^{+}\) currents are required for efficient excitation–contraction coupling in rabbit ventricular myocytes: A possible contribution of neuronal \(Na^{+}\) channels, J. Physiol., 588 (2010), pp. 4249–4260, .
[49] H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631–644, . · Zbl 0761.65023
[50] J. Vierheller, W. Neubert, M. Falcke, S. Gilbert, and N. Chamakuri, A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: From detailed cleft dynamics to the whole cell concentration profiles, Front. Physiol., 6 (2015), 255, .
[51] M. Walker, G. Williams, T. Kohl, S. Lehnart, M. Jafri, J. Greenstein, W. Lederer, and R. Winslow, Superresolution modeling of calcium release in the heart, Biophys. J., 107 (2017), pp. 3018–3029, .
[52] S. Wang, L. Song, E. Lakatta, and H. Cheng, \(Ca^{2+}\) signalling between single l-type \(Ca^{2+}\) channels and ryanodine receptors in heart cells, Nature, 410 (2001), pp. 592–596.
[53] S. Wang, M. Stern, E. Rios, and H. Cheng, The quantal nature of \(Ca^{2+}\) sparks and in situ operation of the ryanodine receptor array in cardiac cell, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 3979–3984.
[54] C. R. Weber, V. Piacentino, K. S. Ginsburg, S. R. Houser, and D. M. Bers, \(Na^{+}\)-\(Ca^{2+}\) exchange current and submembrane [\(Ca^{2+}\)] during the cardiac action potential, Circ. Res., 90 (2002), pp. 182–189, .
[55] G. S. Williams, G. D. Smith, E. A. Sobie, and M. S. Jafri, Models of cardiac excitation-contraction coupling in ventricular myocytes, Math. Biosci., 226 (2010), pp. 1–15. · Zbl 1193.92026
[56] C. Y. Zhao, J. L. Greenstein, and R. L. Winslow, Mechanisms of the cyclic nucleotide cross-talk signaling network in cardiac L-type calcium channel regulation, J. Mol. Cell. Cardiol., 106 (2017), pp. 29–44, .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.