Modeling wave propagation in realistic heart geometries using the phase-field method

A smoothed boundary bidomain model for cardiac simulations in anatomically detailed geometries

This manuscript presents a novel finite difference method to solve cardiac bidomain equations in anatomical models of the heart. The proposed method employs a smoothed boundary approach that represents the boundaries between the heart and the surrounding medium as a spatially diffuse interface of finite thickness. The bidomain boundary conditions are implicitly implemented in the smoothed boundary bidomain equations presented in the manuscript without the need of a structured mesh that explicitly tracks the heart-torso boundaries. We reported some significant examples assessing the method's accuracy using nontrivial test geometries and demonstrating the applicability of the method to complex anatomically detailed human cardiac geometries. In particular, we showed that our approach could be employed to simulate cardiac defibrillation in a human left ventricle comprising fiber architecture. The main advantage of the proposed method is the possibility of implementing bidomain boundary conditions directly on voxel structures, which makes it attractive for three dimensional, patient specific simulations based on medical images. Moreover, given the ease of implementation, we believe that the proposed method could provide an interesting and feasible alternative to finite element methods, and could find application in future cardiac research guiding electrotherapy with computational models.

Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction using a circularly polarized electric field

We present the first experimental study of unpinning an excitation wave using a circularly polarized electric field. The experiments are conducted using the excitable chemical medium, the Belousov-Zhabotinsky (BZ) reaction, which is modeled with the Oregenator model. The excitation wave in the chemical medium is charged so that it can directly interact with the electric field. This is a unique feature of the chemical excitation wave. The mechanism of wave unpinning in the BZ reaction with a circularly polarized electric field is investigated by varying the pacing ratio, the initial phase of the wave, and field strength. The chemical wave in the BZ reaction unpins when the electric force opposite the direction of the spiral is equal to or above a threshold. We developed an analytical relation of the unpinning phase with the initial phase, the pacing ratio, and the field strength. This is then verified in experiments and simulations.

Homogenisation for the monodomain model in the presence of microscopic fibrotic structures

Computational models in cardiac electrophysiology are notorious for long runtimes, restricting the numbers of nodes and mesh elements in the numerical discretisations used for their solution. This makes it particularly challenging to incorporate structural heterogeneities on small spatial scales, preventing a full understanding of the critical arrhythmogenic effects of conditions such as cardiac fibrosis. In this work, we explore the technique of homogenisation by volume averaging for the inclusion of non-conductive micro-structures into larger-scale cardiac meshes with minor computational overhead. Importantly, our approach is not restricted to periodic patterns, enabling homogenised models to represent, for example, the intricate patterns of collagen deposition present in different types of fibrosis. We first highlight the importance of appropriate boundary condition choice for the closure problems that define the parameters of homogenised models. Then, we demonstrate the technique's ability to correctly upscale the effects of fibrotic patterns with a spatial resolution of 10 µm into much larger numerical mesh sizes of 100- 250 µm . The homogenised models using these coarser meshes correctly predict critical pro-arrhythmic effects of fibrosis, including slowed conduction, source/sink mismatch, and stabilisation of re-entrant activation patterns. As such, this approach to homogenisation represents a significant step towards whole organ simulations that unravel the effects of microscopic cardiac tissue heterogeneities.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Removal of spiral turbulence by virtual electrodes through the use of a circularly polarized electric field

Heart disease is the leading cause of death and is often accompanied by cardiac fibrillation. Defibrillation using the virtual electrode effects is a promising alternative compared to using the high-voltage electric shock in the clinic. Our earlier works [S. L. Murphy, K. D. Kochanek, J. Xu, and E. Arias, NCHS Data Brief 427 (2021); R. A. Gray, A. M. Pertsov, and J. Jalife, Nature 392, 75-78 (1998); F. X. Witkowski, L. J. Leon, P. A. Penkoske, W. R. Giles, M. L. Spano, W. L. Ditto, and A. T. Winfree, Nature 392, 78-82 (1998); M. Santini, C. Pandozi, G. Altamura, G. Gentilucci, M. Villani, M. C. Scianaro, A. Castro, F. Ammirati, and B. Magris, J. Interv. Card. Electrophysiol. 3, 45-51 (1999).] prove that, compared with other external electric fields, a low voltage circularly polarized electric field is more efficient in turning non-excitable defects in cardiac tissue into virtual electrodes. It, therefore, needs lower voltage to stimulate the excitation waves and causes less harm to reset the spiral turbulence of cardiac excitation for defibrillation. In this paper, we investigate the virtual electrode effect of multiple defects by the circularly polarized electric field for the removal of spiral turbulence. Considering different shapes, sizes, and distributions of multiple defects, we reveal the phase locking of stimulated excitations around multiple virtual electrodes. Furthermore, the circularly polarized electric field parameters are optimized to remove the spiral turbulence.

The role of pulse timing in cardiac defibrillation

Life-threatening cardiac arrhythmias require immediate defibrillation. For state-of-the-art shock treatments, a high field strength is required to achieve a sufficient success rate for terminating the complex spiral wave (rotor) dynamics underlying cardiac fibrillation. However, such high energy shocks have many adverse side effects due to the large electric currents applied. In this study, we show, using 2D simulations based on the Fenton-Karma model, that also pulses of relatively low energy may terminate the chaotic activity if applied at the right moment in time. In our simplified model for defibrillation, complex spiral waves are terminated by local perturbations corresponding to conductance heterogeneities acting as virtual electrodes in the presence of an external electric field. We demonstrate that time series of the success rate for low energy shocks exhibit pronounced peaks which correspond to short intervals in time during which perturbations aiming at terminating the chaotic fibrillation state are (much) more successful. Thus, the low energy shock regime, although yielding very low temporal average success rates, exhibits moments in time for which success rates are significantly higher than the average value shown in dose-response curves. This feature might be exploited in future defibrillation protocols for achieving high termination success rates with low or medium pulse energies.

Stochastic termination of spiral wave dynamics in cardiac tissue

Rotating spiral waves are self-organized features in spatially extended excitable media and may play an important role in cardiac arrhythmias including atrial fibrillation (AF). In homogeneous media, spiral wave dynamics are perpetuated through spiral wave breakup, leading to the continuous birth and death of spiral waves, but have a finite probability of termination. In non-homogeneous media, however, heterogeneities can act as anchoring sources that result in sustained spiral wave activity. It is thus unclear how and if AF may terminate following the removal of putative spiral wave sources in patients. Here, we address this question using computer simulations in which a stable spiral wave is trapped by an heterogeneity and is surrounded by spiral wave breakup. We show that, following ablation of spatial heterogeneity to render that region of the medium unexcitable, termination of spiral wave dynamics is stochastic and Poisson-distributed. Furthermore, we show that the dynamics can be accurately described by a master equation using birth and death rates. To validate these predictions in vivo, we mapped spiral wave activity in patients with AF and targeted the locations of spiral wave sources using radiofrequency ablation. Targeted ablation was indeed able to terminate AF, but only after a variable delay of up to several minutes. Furthermore, and consistent with numerical simulations, termination was not accompanied by gradual temporal or spatial organization. Our results suggest that spiral wave sources and tissue heterogeneities play a critical role in the maintenance of AF and that the removal of sources results in spiral wave dynamics with a finite termination time, which could have important clinical implications.

Optical Ultrastructure of Large Mammalian Hearts Recovers Discordant Alternans by In Silico Data Assimilation

Understanding and predicting the mechanisms promoting the onset and sustainability of cardiac arrhythmias represent a primary concern in the scientific and medical communities still today. Despite the long-lasting effort in clinical and physico-mathematical research, a critical aspect to be fully characterized and unveiled is represented by spatiotemporal alternans patterns of cardiac excitation. The identification of discordant alternans and higher-order alternating rhythms by advanced data analyses as well as their prediction by reliable mathematical models represents a major avenue of research for a broad and multidisciplinary scientific community. Current limitations concern two primary aspects: 1) robust and general-purpose feature extraction techniques and 2) in silico data assimilation within reliable and predictive mathematical models. Here, we address both aspects. At first, we extend our previous works on Fourier transformation imaging (FFI), applying the technique to whole-ventricle fluorescence optical mapping. Overall, we identify complex spatial patterns of voltage alternans and characterize higher-order rhythms by a frequency-series analysis. Then, we integrate the optical ultrastructure obtained by FFI analysis within a fine-tuned electrophysiological mathematical model of the cardiac action potential. We build up a novel data assimilation procedure demonstrating its reliability in reproducing complex alternans patterns in two-dimensional computational domains. Finally, we prove that the FFI approach applied to both experimental and simulated signals recovers the same information, thus closing the loop between the experiment, data analysis, and numerical simulations.

Interactive Simulation of the ECG: Effects of Cell Types, Distributions, Shapes and Duration

The shape of the ECG depends on the lead positions but also on the distribution and dispersion of different cell types and their action potential (AP) durations and shapes. We present an interactive JavaScript program that allows fast simulations of the ECG by solving and displaying the dynamics of cardiac cells in tissue using a web browser. We use physiologically accurate ODE models of cardiac cells of different types including SA node, right and left atria, AV node, Purkinje, and right and left ventricular cells with dispersion that accounts for apex-to-base and epi-to-endo variations. The software allows for real-time variations for each cell type and their spatial range so as to identify how the shape of the ECG varies as a function of cell type, distribution, excitation duration and AP shape. The propagation of the wave is visualized in real time through all the regions as parameters are kept fixed or varied to modify ECG morphology. The code solves thousands of simulated cells in real time and is independent of operating system, so it can run on PCs, laptops, tablets and cellphones. This program can be used to teach students, fellows and the general public how and why lead positions and the different cell physiology in the heart affect the various features of the ECG.

Electrophysiological Characterization of Human Atria: The Understated Role of Temperature

Ambient temperature has a profound influence on cellular electrophysiology through direct control over the gating mechanisms of different ion channels. In the heart, low temperature is known to favor prolongation of the action potential. However, not much is known about the influence of temperature on other important characterization parameters such as the resting membrane potential (RMP), excitability, morphology and characteristics of the action potential (AP), restitution properties, conduction velocity (CV) of signal propagation, etc. Here we present the first, detailed, systematic in silico study of the electrophysiological characterization of cardiomyocytes from different regions of the normal human atria, based on the effects of ambient temperature (5-50°C). We observe that RMP decreases with increasing temperature. At ~ 48°C, the cells lose their excitability. Our studies show that different parts of the atria react differently to the same changes in temperature. In tissue simulations a drop in temperature correlated positively with a decrease in CV, but the decrease was region-dependent, as expected. In this article we show how this heterogeneous response can provide an explanation for the development of a proarrhythmic substrate during mild hypothermia. We use the above concept to propose a treatment strategy for atrial fibrillation that involves severe hypothermia in specific regions of the heart for a duration of only ~ 200 ms.

Theory of unpinning of spiral waves using circularly polarized electric fields in mathematical models of excitable media

Spiral waves of excitation are common in many physical, chemical, and biological systems. In physiological systems like the heart, such waves can lead to cardiac arrhythmias and need to be eliminated. Spiral waves anchor to heterogeneities in the excitable medium, and to eliminate them they need to be unpinned first. Several groups focused on developing strategies to unpin such pinned waves using electric shocks, pulsed electric fields, and recently, circularly polarized electric fields (CPEF). It was shown that in many situations, CPEF is more efficient at unpinning the wave compared to other existing methods. Here, we study how the circularly polarized field acts on the pinned spiral waves and unpins it. We show that the termination always happens within the first rotation of the electric field. For a given obstacle size, there exists a threshold time period of the CPEF below which the spiral can always be terminated. Our analytical formulation accurately predicts this threshold and explains the absence of the traditional unpinning window with the CPEF. We hope our theoretical work will stimulate further experimental studies about CPEF and low energy methods to eliminate spiral waves.

Anisotropic shortening in the wavelength of electrical waves promotes onset of electrical turbulence in cardiac tissue: An in silico study

Several pathological conditions introduce spatial variations in the electrical properties of cardiac tissue. These variations occur as localized or distributed gradients in ion-channel functionality over extended tissue media. Electrical waves, propagating through such affected tissue, demonstrate distortions, depending on the nature of the ionic gradient in the diseased substrate. If the degree of distortion is large, reentrant activity may develop, in the form of rotating spiral (2d) and scroll (3d) waves of electrical activity. These reentrant waves are associated with the occurrence of lethal cardiac rhythm disorders, known as arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), which are believed to be common precursors of sudden cardiac arrest. By using state-of-the-art mathematical models for generic, and ionically-realistic (human) cardiac tissue, we study the detrimental effects of these ionic gradients on electrical wave propagation. We propose a possible mechanism for the development of instabilities in reentrant wave patterns, in the presence of ionic gradients in cardiac tissue, which may explain how one type of arrhythmia (VT) can degenerate into another (VF). Our proposed mechanism entails anisotropic reduction in the wavelength of the excitation waves because of anisotropic variation in its electrical properties, in particular the action potential duration (APD). We find that the variation in the APD, which we induce by varying ion-channel conductances, imposes a spatial variation in the spiral- or scroll-wave frequency ω. Such gradients in ω induce anisotropic shortening of wavelength of the spiral or scroll arms and eventually leads to instabilitites.

Multi-scale approaches for the simulation of cardiac electrophysiology: II - Tissue-level structure and function

Computational models of the heart, from cell-level models, through one-, two- and three-dimensional tissue-level simplifications, to biophysically-detailed three-dimensional models of the ventricles, atria or whole heart, allow the simulation of excitation and propagation of this excitation, and have provided remarkable insight into the normal and pathological functioning of the heart. In this article we present equations for modelling cellular excitation (i.e. the cell action potential) from both a phenomenological and a biophysical perspective. Hodgkin-Huxley formalism is discussed, along with the current generation of biophysically-detailed cardiac cell models. Alternative Markovian formulations for modelling ionic currents are also presented. Equations describing propagation of this cellular excitation, through one-, two- and three-dimensional idealised or realistic tissues, are then presented. For all types of model, from cell to tissue, methods for discretisation and integration of the underlying equations are discussed. The article finishes with a discussion of two tissue-level experimental imaging techniques - diffusion tensor magnetic resonance imaging and optical imaging - that can be used to provide data for parameterisation and validation of cell- and tissue-level cardiac models.

Spiral wave unpinning facilitated by wave emitting sites in cardiac monolayers

Rotating spiral waves of electrical activity in the heart can anchor to unexcitable tissue (an obstacle) and become stable pinned waves. A pinned rotating wave can be unpinned either by a local electrical stimulus applied close to the spiral core, or by an electric field pulse that excites the core of a pinned wave independently of its localization. The wave will be unpinned only when the pulse is delivered inside a narrow time interval called the unpinning window (UW) of the spiral. In experiments with cardiac monolayers, we found that other obstacles situated near the pinning centre of the spiral can facilitate unpinning. In numerical simulations, we found increasing or decreasing of the UW depending on the location, orientation and distance between the pinning centre and an obstacle. Our study indicates that multiple obstacles could contribute to unpinning in experiments with intact hearts.

Spatiotemporal correlation uncovers characteristic lengths in cardiac tissue

Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to period-doubling bifurcations that develop with increasing frequency of stimulation. Here, through high-resolution optical mapping experiments and mathematical modeling, we introduce a characteristic spatial length of cardiac activity in canine ventricular wedges via a spatiotemporal correlation analysis, at different stimulation frequencies and during fibrillation. We show that the characteristic length ranges from 40 to 20 cm during one-to-one responses and it decreases to a specific value of about 3 cm at the transition from period-doubling bifurcation to fibrillation. We further show that during fibrillation, the characteristic length is about 1 cm. Another significant outcome of our analysis is the finding of a constitutive phenomenological law obtained from a nonlinear fitting of experimental data which relates the conduction velocity restitution curve with the characteristic length of the system. The fractional exponent of 3/2 in our phenomenological law is in agreement with the domain size remapping required to reproduce experimental fibrillation dynamics within a realistic cardiac domain via accurate mathematical models.

Simultaneous unpinning of multiple vortices in two-dimensional excitable media

There are many examples of excitable media, such as the heart, that can show complex dynamics and where control is a challenging task. Heavy means like a strong electric shock are nowadays still necessary to control and terminate ventricular fibrillation (VF). It is known that heterogeneities in an excitable medium can stabilize the activity, e.g., spiral waves can pin to such obstacles. This might also be a reason for the persistence of VF and the difficulty to control it. Previous studies investigated systems with a single pinned spiral wave and demonstrated how the spiral can be unpinned. In this article, we extend this case and investigate a generic excitable system with multiple pinned spiral waves. We describe a control technique that allows the simultaneous unpinning of pinned spiral waves. Apart from theoretical considerations, we provide numerical evidence that the proposed technique is superior to the underdrive pacing method that has reportedly high success rates when applied to a single pinned spiral.

Real-time interactive simulations of large-scale systems on personal computers and cell phones: Toward patient-specific heart modeling and other applications

Cardiac dynamics modeling has been useful for studying and treating arrhythmias. However, it is a multiscale problem requiring the solution of billions of differential equations describing the complex electrophysiology of interconnected cells. Therefore, large-scale cardiac modeling has been limited to groups with access to supercomputers and clusters. Many areas of computational science face similar problems where computational costs are too high for personal computers so that supercomputers or clusters currently are necessary. Here, we introduce a new approach that makes high-performance simulation of cardiac dynamics and other large-scale systems like fluid flow and crystal growth accessible to virtually anyone with a modest computer. For cardiac dynamics, this approach will allow not only scientists and students but also physicians to use physiologically accurate modeling and simulation tools that are interactive in real time, thereby making diagnostics, research, and education available to a broader audience and pushing the boundaries of cardiac science.

Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of CT-based bone structures

The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

Computational assessment of the functional role of sinoatrial node exit pathways in the human heart

AIM

The human right atrium and sinoatrial node (SAN) anatomy is complex. Optical mapping experiments suggest that the SAN is functionally insulated from atrial tissue except at discrete SAN-atrial electrical junctions called SAN exit pathways, SEPs. Additionally, histological imaging suggests the presence of a secondary pacemaker close to the SAN. We hypothesise that a) an insulating border-SEP anatomical configuration is related to SAN arrhythmia; and b) a secondary pacemaker, the paranodal area, is an alternate pacemaker but accentuates tachycardia. A 3D electro-anatomical computational model was used to test these hypotheses.

METHODS

A detailed 3D human SAN electro-anatomical mathematical model was developed based on our previous anatomical reconstruction. Electrical activity was simulated using tissue specific variants of the Fenton-Karma action potential equations. Simulation experiments were designed to deploy this complex electro-anatomical system to assess the roles of border-SEPs and paranodal area by mimicking experimentally observed SAN arrhythmia. Robust and accurate numerical algorithms were implemented for solving the mono domain reaction-diffusion equation implicitly, calculating 3D filament traces, and computing dominant frequency among other quantitative measurements.

RESULTS

A centre to periphery gradient of increasing diffusion was sufficient to permit initiation of pacemaking at the centre of the 3D SAN. Re-entry within the SAN, micro re-entry, was possible by imposing significant SAN fibrosis in the presence of the insulating border. SEPs promoted the micro re-entry to generate more complex SAN-atrial tachycardia. Simulation of macro re-entry, i.e. re-entry around the SAN, was possible by inclusion of atrial fibrosis in the presence of the insulating border. The border shielded the SAN from atrial tachycardia. However, SAN micro-structure intercellular gap junctional coupling and the paranodal area contributed to prolonged atrial fibrillation. Finally, the micro-structure was found to be sufficient to explain shifts of leading pacemaker site location.

CONCLUSIONS

The simulations establish a relationship between anatomy and SAN electrical function. Microstructure, in the form of intercellular gap junction coupling, was found to regulate SAN function and arrhythmia.

Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential

This work reports the results of the theoretical investigation of nonlinear dynamics and spiral wave breakup in a generalized two-variable model of cardiac action potential accounting for thermo-electric coupling and diffusion nonlinearities. As customary in excitable media, the common Q₁₀ and Moore factors are used to describe thermo-electric feedback in a 10° range. Motivated by the porous nature of the cardiac tissue, in this study we also propose a nonlinear Fickian flux formulated by Taylor expanding the voltage dependent diffusion coefficient up to quadratic terms. A fine tuning of the diffusive parameters is performed a priori to match the conduction velocity of the equivalent cable model. The resulting combined effects are then studied by numerically simulating different stimulation protocols on a one-dimensional cable. Model features are compared in terms of action potential morphology, restitution curves, frequency spectra, and spatio-temporal phase differences. Two-dimensional long-run simulations are finally performed to characterize spiral breakup during sustained fibrillation at different thermal states. Temperature and nonlinear diffusion effects are found to impact the repolarization phase of the action potential wave with non-monotone patterns and to increase the propensity of arrhythmogenesis.

Critical phase transitions during ablation of atrial fibrillation

Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia with significant morbidity and mortality. Pharmacological agents are not very effective in the management of AF. Therefore, ablation procedures have become the mainstay of AF management. The irregular and seemingly chaotic atrial activity in AF is caused by one or more meandering spiral waves. Previously, we have shown the presence of sudden rhythm organization during ablation of persistent AF. We hypothesize that the observed transitions from a disorganized to an organized rhythm is a critical phase transition. Here, we explore this hypothesis by simulating ablation in an anatomically-correct 3D AF model. In 722 out of 2160 simulated ablation, at least one sudden transition from AF to an organized rhythm (flutter) was noted (33%). They were marked by a sudden decrease in the cycle length entropy and increase in the mean cycle length. At the same time, the number of reentrant wavelets decreased from 2.99 ± 0.06 in AF to 1.76 ± 0.05 during flutter, and the correlation length scale increased from 13.3 ± 1.0 mm to 196.5 ± 86.6 mm (both P < 0.0001). These findings are consistent with the hypothesis that transitions from AF to an anatomical flutter behave as phase transitions in complex non-equilibrium dynamical systems with flutter acting as an absorbing state. Clinically, the facilitation of phase transition should be considered a novel mechanism of ablation and may help to design effective ablation strategies.

Suppression of turbulence by heterogeneities in a cardiac model with fiber rotation

Electrical scroll wave turbulence in human ventricles is associated with ventricular fibrillation and sudden cardiac death. We perform three-dimensional simulations on the basis of the anisotropic Fenton-Karma model and show that macroscopic, insulating heterogeneities (e.g., blood vessels) can cause the spontaneous formation of pinned scroll waves. The wave field of these vortices is periodic, and their frequencies are sufficiently high to push the free, turbulent vortices into the system boundaries where they annihilate. Our study considers cylindrical heterogeneities with radii in the range of 0.1 to 2 cm that extend either in the transmural or a perpendicular direction. Thick cylinders cause the spontaneous formation of multi-armed rotors according to a radius-dependence that is explained in terms of two-dimensional dynamics. For long cylinders, local pinning contacts spread along the heterogeneity by fast and complex self-wrapping.

A composite visualization method for electrophysiology-morphous merging of human heart

BACKGROUND

Electrophysiological behavior is of great importance for analyzing the cardiac functional mechanism under cardiac physiological and pathological condition. Due to the complexity of cardiac structure and biophysiological function, visualization of a cardiac electrophysiological model compositively is still a challenge. The lack of either modality of the whole organ structure or cardiac electrophysiological behaviors makes analysis of the intricate mechanisms of cardiac dynamic function a difficult task. This study aims at exploring 3D conduction of stimulus and electrical excitation reactivity on the level of organ with the authentic fine cardiac anatomy structure.

METHODS

In this paper, a cardiac electrical excitation propagation model is established based on the human cardiac cross-sectional data to explore detailed cardiac electrical activities. A novel biophysical merging visualization method is then presented for biophysical integration of cardiac anatomy and electrophysiological properties in the form of the merging optical model, which provides the corresponding position, spatial relationship and the whole process in 3D space with the context of anatomical structure for representing the biophysical detailed electrophysiological activity.

RESULTS

The visualization result present the action potential propagation of the left ventricle within the excitation cycle with the authentic fine cardiac organ anatomy. In the visualized images, all vital organs are identified and distinguished without ambiguity. The three dimensional spatial position, relation and the process of cardiac excitation conduction and re-entry propagation in the anatomical structure during the phase of depolarization and repolarization is also shown in the result images, which exhibits the performance of a more detailed biophysical understanding of the electrophysiological kinetics of human heart in vivo.

CONCLUSIONS

Results suggest that the proposed merging optical model can merge cardiac electrophysiological activity with the anatomy structure. By specifying the respective opacity for the cardiac anatomy structure and the electrophysiological model in the merging attenuation function, the visualized images can provide an in-depth insight into the biophysical detailed cardiac functioning phenomena and the corresponding electrophysiological behavior mechanism, which is helpful for further speculating cardiac physiological and pathological responses and is fundamental to the cardiac research and clinical diagnoses.

BeatBox-HPC simulation environment for biophysically and anatomically realistic cardiac electrophysiology

The BeatBox simulation environment combines flexible script language user interface with the robust computational tools, in order to setup cardiac electrophysiology in-silico experiments without re-coding at low-level, so that cell excitation, tissue/anatomy models, stimulation protocols may be included into a BeatBox script, and simulation run either sequentially or in parallel (MPI) without re-compilation. BeatBox is a free software written in C language to be run on a Unix-based platform. It provides the whole spectrum of multi scale tissue modelling from 0-dimensional individual cell simulation, 1-dimensional fibre, 2-dimensional sheet and 3-dimensional slab of tissue, up to anatomically realistic whole heart simulations, with run time measurements including cardiac re-entry tip/filament tracing, ECG, local/global samples of any variables, etc. BeatBox solvers, cell, and tissue/anatomy models repositories are extended via robust and flexible interfaces, thus providing an open framework for new developments in the field. In this paper we give an overview of the BeatBox current state, together with a description of the main computational methods and MPI parallelisation approaches.

Mechanisms of Cell Polarization

Cell polarization is a key step in the migration, development, and organization of eukaryotic cells, both at the single cell and multicellular level. Research on the mechanisms that give rise to polarization of a given cell, and organization of polarity within a tissue has led to new understanding across cellular and developmental biology. In this review, we describe some of the history of theoretical and experimental aspects of the field, as well as some interesting questions and challenges for the future.

Dynamics and Molecular Mechanisms of Ventricular Fibrillation in Structurally Normal Hearts

Ventricular fibrillation (VF) is the most severe cardiac rhythm disturbance and one of the most important immediate causes of sudden cardiac death. In the structurally normal heart, a small number of stable reentrant sources, perhaps 1 or 2, underlie the mechanism of VF, and the stabilization of the sources, their frequency, and the complexity of the turbulent waves they generate depend on the expression, spatial distribution, and intermolecular interactions of the 2 most important ion channels that control cardiac excitability: the inward rectifier potassium channel, Kir2.1, and the alpha subunit of the main cardiac sodium channel, NaV1.5.

Comparison of Detailed and Simplified Models of Human Atrial Myocytes to Recapitulate Patient Specific Properties

Computer studies are often used to study mechanisms of cardiac arrhythmias, including atrial fibrillation (AF). A crucial component in these studies is the electrophysiological model that describes the membrane potential of myocytes. The models vary from detailed, describing numerous ion channels, to simplified, grouping ionic channels into a minimal set of variables. The parameters of these models, however, are determined across different experiments in varied species. Furthermore, a single set of parameters may not describe variations across patients, and models have rarely been shown to recapitulate critical features of AF in a given patient. In this study we develop physiologically accurate computational human atrial models by fitting parameters of a detailed and of a simplified model to clinical data for five patients undergoing ablation therapy. Parameters were simultaneously fitted to action potential (AP) morphology, action potential duration (APD) restitution and conduction velocity (CV) restitution curves in these patients. For both models, our fitting procedure generated parameter sets that accurately reproduced clinical data, but differed markedly from published sets and between patients, emphasizing the need for patient-specific adjustment. Both models produced two-dimensional spiral wave dynamics for that were similar for each patient. These results show that simplified, computationally efficient models are an attractive choice for simulations of human atrial electrophysiology in spatially extended domains. This study motivates the development and validation of patient-specific model-based mechanistic studies to target therapy.

Computational rabbit models to investigate the initiation, perpetuation, and termination of ventricular arrhythmia

Current understanding of cardiac electrophysiology has been greatly aided by computational work performed using rabbit ventricular models. This article reviews the contributions of multiscale models of rabbit ventricles in understanding cardiac arrhythmia mechanisms. This review will provide an overview of multiscale modeling of the rabbit ventricles. It will then highlight works that provide insights into the role of the conduction system, complex geometric structures, and heterogeneous cellular electrophysiology in diseased and healthy rabbit hearts to the initiation and maintenance of ventricular arrhythmia. Finally, it will provide an overview on the contributions of rabbit ventricular modeling on understanding the mechanisms underlying shock-induced defibrillation.

Mechanisms for the Termination of Atrial Fibrillation by Localized Ablation: Computational and Clinical Studies

BACKGROUND

Human atrial fibrillation (AF) can terminate after ablating localized regions, which supports the existence of localized rotors (spiral waves) or focal drivers. However, it is unclear why ablation near a spiral wave tip would terminate AF and not anchor reentry. We addressed this question by analyzing competing mechanisms for AF termination in numeric simulations, referenced to clinical observations.

METHODS AND RESULTS

Spiral wave reentry was simulated in monodomain 2-dimensional myocyte sheets using clinically realistic rate-dependent values for repolarization and conduction. Heterogeneous models were created by introduction of parameterized variations in tissue excitability. Ablation lesions were applied as nonconducting circular regions. Models confirmed that localized ablation may anchor spiral wave reentry, producing organized tachycardias. Several mechanisms referenced to clinical observations explained termination of AF to sinus rhythm. First, lesions may create an excitable gap vulnerable to invasion by fibrillatory waves. Second, ablation of rotors in regions of low-excitability (from remodeling) produced re-entry in more excitable tissue allowing collision of wavefront and back. Conversely, ablation of rotors in high-excitability regions migrated spiral waves to less excitable tissue, where they detached to collide with nonconducting boundaries. Third, ablation may connect rotors to nonconducting anatomic orifices. Fourth, reentry through slow-conducting channels may terminate if ablation closes these channels.

CONCLUSIONS

Limited ablation can terminate AF by several mechanisms. These data shed light on how clinical AF may be sustained in patients' atria, emphasizing heterogeneities in tissue excitability, slow-conducting channels, and obstacles that are increasingly detectable in patients and should be the focus of future translational studies.

Basis for the Induction of Tissue-Level Phase-2 Reentry as a Repolarization Disorder in the Brugada Syndrome

Aims. Human action potentials in the Brugada syndrome have been characterized by delayed or even complete loss of dome formation, especially in the right ventricular epicardial layers. Such a repolarization pattern is believed to trigger phase-2 reentry (P2R); however, little is known about the conditions necessary for its initiation. This study aims to determine the specific mechanisms that facilitate P2R induction in Brugada-affected cardiac tissue in humans. Methods. Ionic models for Brugada syndrome in human epicardial cells were developed and used to study the induction of P2R in cables, sheets, and a three-dimensional model of the right ventricular free wall. Results. In one-dimensional cables, P2R can be induced by adjoining lost-dome and delayed-dome regions, as mediated by tissue excitability and transmembrane voltage profiles, and reduced coupling facilitates its induction. In two and three dimensions, sustained reentry can arise when three regions (delayed-dome, lost-dome, and normal epicardium) are present. Conclusions. Not only does P2R induction by Brugada syndrome require regions of action potential with delayed-dome and lost-dome, but in order to generate a sustained reentry from a triggered waveback multiple factors are necessary, including heterogeneity in action potential distribution, tissue coupling, direction of stimulation, the shape of the late plateau, the duration of lost-dome action potentials, and recovery of tissue excitability, which is predominantly modulated by tissue coupling.

Image-Based Structural Modeling of the Cardiac Purkinje Network

The Purkinje network is a specialized conduction system within the heart that ensures the proper activation of the ventricles to produce effective contraction. Its role during ventricular arrhythmias is less clear, but some experimental studies have suggested that the Purkinje network may significantly affect the genesis and maintenance of ventricular arrhythmias. Despite its importance, few structural models of the Purkinje network have been developed, primarily because current physical limitations prevent examination of the intact Purkinje network. In previous modeling efforts Purkinje-like structures have been developed through either automated or hand-drawn procedures, but these networks have been created according to general principles rather than based on real networks. To allow for greater realism in Purkinje structural models, we present a method for creating three-dimensional Purkinje networks based directly on imaging data. Our approach uses Purkinje network structures extracted from photographs of dissected ventricles and projects these flat networks onto realistic endocardial surfaces. Using this method, we create models for the combined ventricle-Purkinje system that can fully activate the ventricles through a stimulus delivered to the Purkinje network and can produce simulated activation sequences that match experimental observations. The combined models have the potential to help elucidate Purkinje network contributions during ventricular arrhythmias.

Stabilization of three-dimensional scroll waves and suppression of spatiotemporal chaos by heterogeneities

Scroll waves in a three-dimensional medium with negative filament tension may break up and display spatiotemporal chaos. The presence of heterogeneities can influence the evolution of the medium, in particular scroll waves may pin to such heterogeneities. We show that as a result the medium may be stabilized by heterogeneities of a suitably chosen geometry. Thin rodlike heterogeneities suppress otherwise developing spatiotemporal chaos and additionally clear out already existing chaotic excitation patterns.

Coupling actin dynamics to phase-field in modeling neural growth

In this paper we model the growth of a neural cell together with the actin dynamics taking place at its growing region by constructing a phase-field model. This is done by assigning auxiliary fields to different constituents of the cell in order to differentiate them. Specifically, the inner and outer regions of the neural cell are described by ϕ = 1 and ϕ = 0 respectively, whereas the inside and outside of its leading edge are portrayed by ψ = 1 and ψ = 0. This formulation inherently locates the boundary, which is required to determine the evolution of the underlying actin dynamics. Therefore, it provides an alternative to boundary tracking algorithms. Then the equations governing the molecular workings of the cell specifically those of actin are modified in order to satisfy their corresponding boundary conditions.

Influence of excitability on unpinning and termination of spiral waves

Application of electrical forcing to release pinned spiral waves from unexcitable obstacles and to terminate the rotation of free spiral waves at the boundary of excitable media has been investigated in thin layers of the Belousov-Zhabotinsky (BZ) reaction, prepared with different initial concentrations of H_{2}SO_{4}. Increasing [H_{2}SO_{4}] raises the excitability of the reaction and reduces the core diameter of free spiral waves as well as the wave period. An electric current with density stronger than a critical value Junpin causes a pinned spiral wave to drift away from the obstacle. For a given obstacle size, Junpin increases with [H_{2}SO_{4}]. Under an applied electrical current, the rotation center of a free spiral wave drifts along a straight path to the boundary. When the current density is stronger than a critical value Jterm, the spiral tip is forced to hit the boundary, where the spiral wave is terminated. Similar to Junpin for releasing a pinned spiral wave, Jterm also increases with [H_{2}SO_{4}]. These experimental findings were confirmed by numerical simulations using the Oregonator model, in which the excitability was adjusted via the ratio of the excitation rate to the recovery rate of the BZ reaction. Therefore, our investigation shows that decreasing the excitability can facilitate elimination of spiral waves by electrical forcing, either in the presence of obstacles or not.

Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac tissue

Wave propagation around various geometric expansions, structures, and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten Tusscher-Noble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue. We show that, in a large range of parameters that account for the conductance of major inward and outward ionic currents of the model [fast inward Na+ current ( INa), L—type slow inward Ca2+ current ( ICaL), slow delayed-rectifier current ( IKs), rapid delayed-rectifier current ( IKr), inward rectifier K+ current ( IK1)], the critical period necessary for spiral formation is close to the period of a spiral wave rotating in the same tissue. We also show that there is a minimal size of the obstacle for which formation of spirals is possible; this size is ∼2.5 cm and decreases with a decrease in the excitability of cardiac tissue. We show that other factors, such as the obstacle thickness and direction of wave propagation in relation to the obstacle, are of secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes derived from experimental measurements of infarction scars and show that the formation of spiral waves there is facilitated by tissue remodeling around it. Overall, we demonstrate that the formation of reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast heart rate.

Mechanism of unpinning spirals by a series of stimuli

Antitachycardia pacing (ATP) is widely used to terminate tachycardia before it proceeds to lethal fibrillation. The important prerequisite for successful ATP is unpinning of the spirals anchored to the obstacle by a series of stimuli. Here, to understand the mechanism of unpinning spirals by ATP, we propose a theoretical explanation based on a nonlinear eikonal relation and a kinematical model. The theoretical results are quantitatively consistent with the numerical simulations in both weak and high excitabilities.

Theoretical considerations for mapping activation in human cardiac fibrillation

Defining mechanisms for cardiac fibrillation is challenging because, in contrast to other arrhythmias, fibrillation exhibits complex non-repeatability in spatiotemporal activation but paradoxically exhibits conserved spatial gradients in rate, dominant frequency, and electrical propagation. Unlike animal models, in which fibrillation can be mapped at high spatial and temporal resolution using optical dyes or arrays of contact electrodes, mapping of cardiac fibrillation in patients is constrained practically to lower resolutions or smaller fields-of-view. In many animal models, atrial fibrillation is maintained by localized electrical rotors and focal sources. However, until recently, few studies had revealed localized sources in human fibrillation, so that the impact of mapping constraints on the ability to identify rotors or focal sources in humans was not described. Here, we determine the minimum spatial and temporal resolutions theoretically required to detect rigidly rotating spiral waves and focal sources, then extend these requirements for spiral waves in computer simulations. Finally, we apply our results to clinical data acquired during human atrial fibrillation using a novel technique termed focal impulse and rotor mapping (FIRM). Our results provide theoretical justification and clinical demonstration that FIRM meets the spatio-temporal resolution requirements to reliably identify rotors and focal sources for human atrial fibrillation.

Termination of pinned vortices by high-frequency wave trains in heartlike excitable media with anisotropic fiber orientation

A variety of chemical and biological nonlinear excitable media, including heart tissue, exhibit vortices (spiral waves) that can anchor to nonexcitable obstacles. Such anchored vortices can be terminated by the application of high-frequency wave trains, as shown previously in isotropic excitable media. In this study, we examined the basic dependencies of the conduction velocities of planar waves and waves around curved obstacles as a function of anisotropy through numerical simulations of excitable media that mimic the fiber orientation in a real heart. We also investigated the unpinning of anchored spiral waves by high-frequency wave trains in an anisotropic excitable medium. Unlike the findings regarding the termination of spiral waves in isotropic excitable systems, we found a nonmonotonic relationship between the maximum unpinning period and the obstacle radius depending on the fiber orientation, where the formation of unwanted secondary pinned vortices or chaotic waves is seen over a wide range of parameters.

Modeling defibrillation of the heart: approaches and insights

Cardiac defibrillation, as accomplished nowadays by automatic, implantable devices (ICDs), constitutes the most important means of combating sudden cardiac death. While ICD therapy has proved to be efficient and reliable, defibrillation is a traumatic experience. Thus, research on defibrillation mechanisms, particularly aimed at lowering defibrillation voltage, remains an important topic. Advancing our understanding towards a full appreciation of the mechanisms by which a shock interacts with the heart is the most promising approach to achieve this goal. The aim of this paper is to assess the current state-of-the-art in ventricular defibrillation modeling, focusing on both numerical modeling approaches and major insights that have been obtained using defibrillation models, primarily those of realistic ventricular geometry. The paper showcases the contributions that modeling and simulation have made to our understanding of the defibrillation process. The review thus provides an example of biophysically based computational modeling of the heart (i.e., cardiac defibrillation) that has advanced the understanding of cardiac electrophysiological interaction at the organ level and has the potential to contribute to the betterment of the clinical practice of defibrillation.

Modeling atrial arrhythmias: impact on clinical diagnosis and therapies

Atrial arrhythmias are the most frequent sustained rhythm disorders in humans and often lead to severe complications such as heart failure and stroke. Despite the important insights provided by animal models into the mechanisms of atrial arrhythmias, direct translation of experimental findings to new therapies in patients has not been straightforward. With the advances in computer technology, large-scale electroanatomical computer models of the atria that integrate information from the molecular to organ scale have reached a level of sophistication that they can be used to interpret the outcome of experimental and clinical studies and aid in the rational design of therapies. This paper reviews the state-of-the-art of computer models of the electrical dynamics of the atria and discusses the evolving role of simulation in assisting the clinical diagnosis and treatment of atrial arrhythmias.

Inwardly rotating spirals in nonuniform excitable media

Inwardly rotating spirals (IRSs) have attracted great attention since their observation in an oscillatory reaction-diffusion system. However, IRSs have not yet been reported in planar excitable media. In the present work we investigate rotating waves in a nonuniform excitable medium, consisting of an inner disk part surrounded by an outer ring part with different excitabilities, by numerical simulations of a simple FitzHugh-Nagumo model. Depending on the excitability of the medium as well as the inhomogeneity, we find the occurrence of IRSs, of which the excitation propagates inwardly to the geometrical spiral tip.

Surfactant-induced gradients in the three-dimensional Belousov-Zhabotinsky reaction

Scroll waves are prominent patterns formed in three-dimensional excitable media, and they are frequently considered highly relevant for some types of cardiac arrhythmias. Experimentally, scroll wave dynamics is often studied by optical tomography in the Belousov-Zhabotinsky reaction, which produces CO(2) as an undesired product. Addition of small concentrations of a surfactant to the reaction medium is a popular method to suppress or retard CO(2) bubble formation. We show that in closed reactors even these low concentrations of surfactants are sufficient to generate vertical gradients of excitability which are due to gradients in CO(2) concentration. In reactors open to the atmosphere such gradients can be avoided. The gradients induce a twist on vertically oriented scroll waves, while a twist is absent in scroll waves in a gradient-free medium. The effects of the CO(2) gradients are reproduced by a numerical study, where we extend the Oregonator model to account for the production of CO(2) and for its advection against the direction of gravity. The numerical simulations confirm the role of solubilized CO(2) as the source of the vertical gradient of excitability in reactors closed to the atmosphere.

Tensorial mobilities for accurate solution of transport problems in models with diffuse interfaces

The general problem of two-phase transport in phase-field models is analyzed: the flux of a conserved quantity is driven by the gradient of a potential through a medium that consists of domains of two distinct phases which are separated by diffuse interfaces. It is shown that the finite thickness of the interfaces induces two effects that are not present in the analogous sharp-interface problem: a surface excess current and a potential jump at the interfaces. It is shown that both effects can be eliminated simultaneously only if the coefficient of proportionality between flux and potential gradient (mobility) is allowed to become a tensor in the interfaces. This opens the possibility for precise and efficient simulations of transport problems with finite interface thickness.

Effects of boundaries and geometry on the spatial distribution of action potential duration in cardiac tissue

Increased dispersion of action potential duration across cardiac tissue has long been considered an important substrate for the development of most electrical arrhythmias. Although this dispersion has been studied previously by characterizing the static intrinsic gradients in cellular electrophysiology and dynamical gradients generated by fast pacing, few studies have concentrated on dispersions generated solely by structural effects. Here we show how boundaries and geometry can produce spatially dependent changes in action potential duration (APD) in homogeneous and isotropic tissue, where all the cells have the same APD in the absence of diffusion. Electrotonic currents due to coupling within the tissue and at the tissue boundaries can generate dispersion, and the profile of this dispersion can change dramatically depending on tissue size and shape, action potential morphology, tissue dimensionality, and stimulus frequency and location. The dispersion generated by pure geometrical effects can be on the order of tens of milliseconds, enough under certain conditions to produce conduction blocks and initiate reentrant waves.

A diffuse-interface method for two-phase flows with soluble surfactants

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier-Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed.

Computational model for cell morphodynamics

We develop a computational model, based on the phase-field method, for cell morphodynamics and apply it to fish keratocytes. Our model incorporates the membrane bending force and the surface tension and enforces a constant area. Furthermore, it implements a cross-linked actin filament field and an actin bundle field that are responsible for the protrusion and retraction forces, respectively. We show that our model predicts steady state cell shapes with a wide range of aspect ratios, depending on system parameters. Furthermore, we find that the dependence of the cell speed on this aspect ratio matches experimentally observed data.

Models of cardiac tissue electrophysiology: progress, challenges and open questions

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

Adaptive macro finite elements for the numerical solution of monodomain equations in cardiac electrophysiology

Many problems in biology and engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction-diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

Effects of the purkinje system and cardiac geometry on biventricular pacing: a model study

Heart failure leads to gross cardiac structural changes. While cardiac resynchronization therapy (CRT) is a recognized treatment for restoring synchronous activation, it is not clear how changes in cardiac shape and size affect the electrical pacing therapy. This study used a human heart computer model which incorporated anatomical structures such as myofiber orientation and a Purkinje system (PS) to study how pacing affected failing hearts. The PS was modeled as a tree structure that reproduced its retrograde activation feature. In addition to a normal geometry, two cardiomyopathies were modeled: dilatation and hypertrophy. A biventricular pacing protocol was tested in the context of atrio-ventricular block. The contribution of the PS was examined by removing it, as well as by increasing endocardial conductivity. Results showed that retrograde conduction into the PS was a determining factor for achieving intraventricular synchrony. Omission of the PS led to an overestimate of the degree of electrical dyssynchrony while assessing CRT. The activation patterns for the three geometries showed local changes in the order of activation of the lateral wall in response to the same pacing strategy. These factors should be carefully considered when determining lead placement and optimizing device parameters in clinical practice.