cbcbeat (cbcbeat 2017) is a Python-based software collection targeting computational cardiac electrophysiology problems.cbcbeat contains solvers of varying complexity and performance for the classical monodomain and bidomain equations coupled with cardiac cell models.The cbcbeat solvers are based on algorithms described in (Sundnes et al. 2006) and the core FEniCS Project software (Logg et al. 2012).All solvers allow for automated derivation and computation of adjoint and tangent linear solutions, functional derivatives and Hessians via the dolfin-adjoint software (Farrell et al. 2013).The computation of functional derivatives in turn allows for automated and efficient solution of optimization problems such as those encountered in data assimillation or other inverse problems.

Numerical simulation of subcellular Ca2+ dynamics with a resolution down to one nanometre can be an important tool for discovering the physiological cause of many heart diseases. The requirement of enormous computational power, however, has made such simulations prohibitive so far. By using up to 12,288 Intel Xeon Phi 31S1P coprocessors on the new hybrid cluster Tianhe-2, which is the new number one supercomputer of the world, we have achieved 1.27 Pflop/s in double precision, which brings us much closer to the nanometre resolution. This is the result of efficiently using the hardware on different levels: (1) a single Xeon Phi (2) a single compute node that consists of a host and three coprocessors, and (3) a huge number of interconnected nodes. To overcome the challenge of programming Intel's new many-integrated core (MIC) architecture, we have adopted techniques such as vectorization, hierarchical data blocking, register data reuse, offloading computations to the coprocessors, and pipelining computations with intra-/inter-node communications.

The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5.

Computational models are valuable tools for understanding the mechanical function of the heart. In particular, the prospect of doing patient–specific simulations of heart function may have a significant impact on clinical practice. However, patient–specific simulations give rise to severe challenges related to model choices, parameter fitting and model validation. In this study we investigate parameter variability in a model of left ventricular mechanics applied to four different canine heart cases. The mechanics is modeled by a transversely isotropic active strain model, with two parameters adjusted to fit end diastolic and end systolic pressures and volumes. The chosen model is able to accurately reproduce these data, and enables very efficient parameter fitting. Visual inspection of the resulting deformed geometries also shows a reasonable match with the image based reference.

Background— Early afterdepolarizations (EADs) are triggers of cardiac arrhythmia driven by L-type Ca 2+ current (I CaL ) reactivation or sarcoplasmic reticulum Ca 2+ release and Na + /Ca 2+ exchange. In large mammals the positive action potential plateau promotes I CaL reactivation, and the current paradigm holds that cardiac EAD dynamics are dominated by interaction between I CaL and the repolarizing K + currents. However, EADs are also frequent in the rapidly repolarizing mouse action potential, which should not readily permit I CaL reactivation. This suggests that murine EADs exhibit unique dynamics, which are key for interpreting arrhythmia mechanisms in this ubiquitous model organism. We investigated these dynamics in myocytes from arrhythmia-susceptible calcium calmodulin-dependent protein kinase II delta C (CaMKIIδC)-overexpressing mice (Tg), and via computational simulations. Methods and Results— In Tg myocytes, β-adrenergic challenge slowed late repolarization, potentiated sarcoplasmic reticulum Ca 2+ release, and initiated EADs below the I CaL activation range (–47±0.7 mV). These EADs were abolished by caffeine and tetrodotoxin (but not ranolazine), suggesting that sarcoplasmic reticulum Ca 2+ release and Na + current (I Na ), but not late I Na , are required for EAD initiation. Simulations suggest that potentiated sarcoplasmic reticulum Ca 2+ release and Na + /Ca 2+ exchange shape late action potential repolarization to favor nonequilibrium reactivation of I Na and thereby drive the EAD upstroke. Action potential clamp experiments suggest that lidocaine eliminates virtually all inward current elicited by EADs, and that this effect occurs at concentrations (40–60 μmol/L) for which lidocaine remains specific for inactivated Na + channels. This strongly suggests that previously inactive channels are recruited during the EAD upstroke, and that nonequilibrium I Na dynamics underlie murine EADs. Conclusions— Nonequilibrium reactivation of I Na drives murine EADs.

Compelling evidence suggests that reduced sarcoplasmic reticulum (SR) Ca2+ uptake via the SR Ca2+ ATPase (SERCA) is a major contributor to the development of Ca2+ signaling abnormalities in heart failure, and equally important as other known contributors to impaired Ca2+ handling. Mathematical models of Ca2+ uptake via SERCA have been proposed, including recent 'state-model' descriptions that represent distinct catalytic states of the ATPase. While these state models have answered important questions about the pump's dependence on Ca2+, Mg2+, and the free energy of ATP hydrolysis, to our knowledge state-based models that account for SERCA's molecular interaction with the endogenous phospholamban inhibitor are less well-developed. We thus propose a SERCA Ca2+ uptake model that is based on a sequence of crystallographically-determined conformational states. These distinct conformations have been shown to give rise to the cooperativity of cytosolic Ca2+ binding, and the kinetics of which are altered by phospholamban binding. We apply the new SERCA model to a 3-dimensional model of Ca2+ signaling in a realistic, confocal-microscopy derived cardiac ventricular myocyte, with which we demonstrate the effects of altered Ca2+ cooperativity on the Ca2+ transient.

Numerous signaling processes in the cell are controlled in microdomains that are defined by cellular structures ranging from nm to μm in size. Recent improvements in microscopy enable the resolution and reconstruction of these micro domains, while new computational methods provide the means to elucidate their functional roles. Collectively these tools allow for a biophysical understanding of the cellular environment and its pathological progression in disease. Here we review recent advancements in microscopy, and subcellular modeling on the basis of reconstructed geometries, with a special focus on signaling microdomains that are important for the excitation contraction coupling in cardiac myocytes.

High-order cubic Hermite finite elements have been valuable in modeling cardiac geometry, fiber orientations, biomechanics, and electrophysiology, but their use in solving three-dimensional problems has been limited to ventricular models with simple topologies. Here, we utilized a subdivision surface scheme and derived a generalization of the "local-to-global" derivative mapping scheme of cubic Hermite finite elements to construct bicubic and tricubic Hermite models of the human atria with extraordinary vertices from computed tomography images of a patient with atrial fibrillation. To an accuracy of 0.6 mm, we were able to capture the left atrial geometry with only 142 bicubic Hermite finite elements, and the right atrial geometry with only 90. The left and right atrial bicubic Hermite meshes were G1 continuous everywhere except in the one-neighborhood of extraordinary vertices, where the mean dot products of normals at adjacent elements were 0.928 and 0.925. We also constructed two biatrial tricubic Hermite models and defined fiber orientation fields in agreement with diagrammatic data from the literature using only 42 angle parameters. The meshes all have good quality metrics, uniform element sizes, and elements with aspect ratios near unity, and are shared with the public. These new methods will allow for more compact and efficient patient-specific models of human atrial and whole heart physiology.

Contractile function of cardiac cells is driven by the sliding displacement of myofilaments powered by the cycling myosin crossbridges. Critical to this process is the availability of ATP, which myosin hydrolyzes during the cross-bridge cycle. The diffusion of adenine nucleotides through the myofilament lattice has been shown to be anisotropic, with slower radial diffusion perpendicular to the filament axis relative to parallel, and is attributed to the periodic hexagonal arrangement of the thin (actin) and thick (myosin) filaments. We investigated whether atomistic-resolution details of myofilament proteins can refine coarse-grain estimates of diffusional anisotropy for adenine nucleotides in the cardiac myofibril, using homogenization theory and atomistic thin filament models from the Protein Data Bank. Our results demonstrate considerable anisotropy in ATP and ADP diffusion constants that is consistent with experimental measurements and dependent on lattice spacing and myofilament overlap. A reaction-diffusion model of the half-sarcomere further suggests that diffusional anisotropy may lead to modest adenine nucleotide gradients in the myoplasm under physiological conditions.

In heart failure, cardiomyocytes exhibit slowing of the rising phase of the Ca(2+) transient which contributes to the impaired contractility observed in this condition. We investigated whether alterations in ryanodine receptor function promote slowing of Ca(2+) release in a murine model of congestive heart failure (CHF). Myocardial infarction was induced by left coronary artery ligation. When chronic CHF had developed (10 weeks post-infarction), cardiomyocytes were isolated from viable regions of the septum. Septal myocytes from SHAM-operated mice served as controls. Ca(2+) transients rose markedly slower in CHF than SHAM myocytes with longer time to peak (CHF=152 ± 12% of SHAM, P<0.05). The rise time of Ca(2+) sparks was also increased in CHF (SHAM=9.6 ± 0.6 ms, CHF=13.2 ± 0.7 ms, P<0.05), due to a sub-population of sparks (≈20%) with markedly slowed kinetics. Regions of the cell associated with these slow spontaneous sparks also exhibited slowed Ca(2+) release during the action potential. Thus, greater variability in spark kinetics in CHF promoted less uniform Ca(2+) release across the cell. Dyssynchronous Ca(2+) transients in CHF additionally resulted from T-tubule disorganization, as indicated by fast Fourier transforms, but slow sparks were not associated with orphaned ryanodine receptors. Rather, mathematical modeling suggested that slow sparks could result from an altered composition of Ca(2+) release units, including a reduction in ryanodine receptor density and/or distribution of ryanodine receptors into sub-clusters. In conclusion, our findings indicate that slowed, dyssynchronous Ca(2+) transients in CHF result from alterations in Ca(2+) sparks, consistent with rearrangement of ryanodine receptors within Ca(2+) release units.

Calcium signaling in cardiomyocytes is strongly influenced by the topology of the sarcolemma (SL) and the distribution of sarcolemmal proteins, including the L-type calcium channel (LCC) and sodium-calcium exchanger (NCX). Peculiar to mammalian ventricular cardiomyocytes are sarcolemmal invaginations called transverse tubules (TT) exhibiting high densities of LCC clusters that trigger Ca2+ release from the sarcoplasmic reticulum (SR). Prior studies of pharmacologically-disabled SR release in rabbit ventricular myocytes have demonstrated that sub-micrometer resolution details of the SL geometry shape local Ca2+ dynamics and suggest a feedback between cytosolic [Ca2+] and SL ion channel and transporter activity. Here we investigate the hypothesis that the ordered spatial arrangement of TT and coupling between adjacent tubules leads to an organized Ca2+ transient. Moreover, we compare Ca2+ transients arising from clustered or continuously-distributed trigger fluxes for the propensity to produce Ca2+ waves. Our findings are that 1) the arrangement of TTs promotes a faster rise in [Ca2+] transversely relative to the longitudinal direction of cardiomyocytes and 2) clustering of SL transporters along the TT promotes an axially-uniform calcium transient. These results evidence contribution of structural detail at sub-micrometer resolution to excitation-contraction coupling and anomalous Ca2+ dynamics underlying cardiac arrhythmias. Supported by NBCR (NIH grant 2 P41 RR08605), NIH GM31749, NSF MCB-0506593, MCA93S013, Center for Theoretical Biological Physics, Howard Hughes Medical Institute, SDSC, W. M. Keck foundation, Richard A. and Nora Eccles Fund for Cardiovascular Research.

Key points We have developed a detailed computational model of a cardiac Ca2+ spark based on a three dimensional reconstruction of electron tomograms. Our model predicts near total junctional Ca2+ depletion after the spark, while regional Ca2+ reserve is preserved. The local Ca2+ gradient inferred by these findings reconciles previous model predictions with experimental measurements. Differences in local distribution of calsequestrin have a profound impact on spark termination time, as reported by Fluo5, solely based on its Ca2+ buffering capacity. The SERCA pump can prolong spark release time by pumping Ca2+ back into the junctional SR during the spark. Abstract Triggered release of Ca2+ from an individual sarcoplasmic reticulum (SR) Ca2+ release unit (CRU) is the fundamental event of cardiac excitationcontraction coupling, and spontaneous release events (sparks) are the major contributor to diastolic Ca2+ leak in cardiomyocytes. Previous model studies have predicted that the duration and magnitude of the spark is determined by the local CRU geometry, as well as the localization and density of Ca2+ handling proteins. We have created a detailed computational model of a CRU, and developed novel tools to generate the computational geometry from electron tomographic images. Ca2+ diffusion was modelled within the SR and the cytosol to examine the effects of localization and density of the Na+/Ca2+ exchanger, sarco/endoplasmic reticulum Ca2+-ATPase 2 (SERCA), and calsequestrin on spark dynamics. We reconcile previous model predictions of approximately 90% local Ca2+ depletion in junctional SR, with experimental reports of about 40%. This analysis supports the hypothesis that dye kinetics and optical averaging effects can have a significant impact on measures of spark dynamics. Our model also predicts that distributing calsequestrin within non-junctional Z-disc SR compartments, in addition to the junctional compartment, prolongs spark release time as reported by Fluo5. By pumping Ca2+ back into the SR during a release, SERCA is able to prolong a Ca2+ spark, and this may contribute to SERCA-dependent changes in Ca2+ wave speed. Finally, we show that including the Na+/Ca2+ exchanger inside the dyadic cleft does not alter local [Ca2+] during a spark.

The transverse tubular system of rabbit ventricular myocytes consists of cell membrane invaginations (t-tubules) that are essential for efficient cardiac excitation-contraction coupling. In this study, we investigate how t-tubule micro-anatomy, L-type Ca2+ channel (LCC) clustering, and allosteric activation of Na+/Ca2+ exchanger by L-type Ca2+ current affects intracellular Ca2+ dynamics. Our model includes a realistic 3D geometry of a single t-tubule and its surrounding half-sarcomeres for rabbit ventricular myocytes. The effects of spatially distributed membrane ion-transporters (LCC, Na+/Ca2+ exchanger, sarcolemmal Ca2+ pump, and sarcolemmal Ca2+ leak), and stationary and mobile Ca2+ buffers (troponin C, ATP, calmodulin, and Fluo-3) are also considered. We used a coupled reaction-diffusion system to describe the spatio-temporal concentration profiles of free and buffered intracellular Ca2+. We obtained parameters from voltage-clamp protocols of L-type Ca2+ current and line-scan recordings of Ca2+ concentration profiles in rabbit cells, in which the sarcoplasmic reticulum is disabled. Our model results agree with experimental measurements of global Ca2+ transient in myocytes loaded with 50 μM Fluo-3. We found that local Ca2+ concentrations within the cytosol and sub-sarcolemma, as well as the local trigger fluxes of Ca2+ crossing the cell membrane, are sensitive to details of t-tubule micro-structure and membrane Ca2+ flux distribution. The model additionally predicts that local Ca2+ trigger fluxes are at least threefold to eightfold higher than the whole-cell Ca2+ trigger flux. We found also that the activation of allosteric Ca2+-binding sites on the Na+/Ca2+ exchanger could provide a mechanism for regulating global and local Ca2+ trigger fluxes in vivo. Our studies indicate that improved structural and functional models could improve our understanding of the contributions of L-type and Na+/Ca2+ exchanger fluxes to intracellular Ca2+ dynamics.

DOLFIN is a C++/Python library that functions as the main user interface of FEniCS. In this 4806 chapter, we review the functionality of DOLFIN. We also discuss the implementation of some key 4807 features of DOLFIN in detail.

The uPy Python extension module provides a uniform abstraction of the APIs of several 3D computer graphics programs (called hosts), including Blender, Maya, Cinema 4D, and DejaVu. A plug-in written with uPy can run in all uPy-supported hosts. Using uPy, researchers have created complex plug-ins for molecular and cellular modeling and visualization. uPy can simplify programming for many types of projects (not solely science applications) intended for multihost distribution. It's available at http://upy.scripps.edu. The first featured Web extra is a video that shows interactive analysis of a calcium dynamics simulation. YouTube URL: http://youtu.be/wvs-nWE6ypo. The second featured Web extra is a video that shows rotation of the HIV virus. YouTube URL: http://youtu.be/vEOybMaRoKc.

This chapter describes decisions made and lessons learned in the implementation of the Python interface of DOLFIN. The chapter is quite technical, since we aim at giving the reader a thorough understanding of the implementation of the DOLFIN Python interface.

Intercellular calcium waves in cardiac myocytes are a well-recognized, if incompletely understood, phenomenon. In a variety of preparations, investigators have reported multi-cellular calcium waves or triggered propagated contractions, but the mechanisms of propagation and pathological importance of these events remain unclear. Here, we review existing experimental data and present a computational approach to investigate the mechanisms of multi-cellular calcium wave propagation.Over the past 50 years, the standard modeling paradigm for excitable cardiac tissue has seen increasingly detailed models of the dynamics of individual cells coupled in tissue solely by intercellular and interstitial current flow. Although very successful, this modeling regime has been unable to capture two important phenomena: 1) the slow intercellular calcium waves observed experimentally, and 2) how intercellular calcium events resulting in delayed after depolarizations at the cellular level could overcome a source-sink mismatch to initiate depolarization waves in tissue. In this paper, we introduce a mathematical model with subcellular spatial resolution, in which we allow both inter- and intracellular current flow and calcium diffusion. In simulations of coupled cells employing this model, we observe: a) slow inter-cellular calcium waves propagating at about 0.1 mm/s, b) faster Calcium-Depolarization-Calcium (CDC) waves, traveling at about 1 mm/s, and c) CDC-waves that can set off fast depolarization-waves (50 cm/s) in tissue with varying gap-junction conductivity. (C) 2012 Elsevier Ltd. All rights reserved.

Ca2+-Calmodulin dependent protein kinase II (CaMKII) is a nodal regulator of Ca2+-handling and electrophysiology in the ventricular myocyte, and transgenic mice expressing CaMKIIδc (CaMKIIδc TG) exhibit cellular afterdepolarizations and triggered arrhythmias. Here we studied myocytes isolated from these mice to determine: (1) how CaMKII-hyperactivity promotes electrophysiologic instability early in progression to HF, and (2) how βAR stimulation exacerbates this instability. METHODS: Myocytes were isolated, prior to overt HF, from cardiac-specific CaMKIIδc TG mice (TG, n = 18), and WT littermates (n = 16). Steady-state action potentials, Ca2+-handling, and electrophysiologic instability were assessed in whole-cell current clamp (1 Hz pacing), with simultaneous Ca2+ epifluorescence, and with and without βAR stimulation (100 nM Isoproterenol, Iso). RESULTS: EADs, but not DADs, were observed with Iso, and transgenic cells (8/18 cells, 19% of cycles) were more susceptible to EADs than WT (1/16 cells, 2% cycles; p < 0.05). Prior to EAD appearance, TG cells that later exhibited EADs (TG-EAD) also exhibited greater Iso-induced AP prolongation than cells that remained stable (52.3 ± 15.6 ms vs. 21.5 ± 3.9 ms; p < 0.05). TG-EAD cells also exhibited blunted baseline Ca2+ transient amplitude (CaT; 42 ± 9 nM vs. 123 ± 12 nM; p < 0.05), but a larger relative change in CaT with Iso (4.4 ± 0.47 fold vs. 2.8 ± 0.46 fold; p < 0.05). EADs could be abolished by caffeine, indicating a requirement for SR calcium release. CONCLUSIONS: Prior to overt HF, superimposing βAR stimulation upon CaMKIIδc overexpression elicits EADs without DADs. This EAD etiology requires SR calcium release, and given the larger relative effect of Iso on CaT in TG-EAD cells, it is likely that βAR stimulation combines withCaMKII overexpression to exceed the range of calcium-handling permissive of stable electrophysiology.