Proper heart function results from the periodic execution of a series of coordinated interdependent mechanical, chemical, and electrical processes within the cardiac tissue. Central to these processes is the action potential - the electrochemical event that initiates contraction of the individual cardiac myocytes. Many models of the cardiac action potential exist with varying levels of complexity, but none account for the electrophysiological role played by caveolae - small invaginations of the cardiac cell plasma membrane. Recent electrophysiological studies regarding these microdomains reveal that cardiac caveolae function as reservoirs of 'recruitable' sodium ion channels. As such, caveolar channels constitute a substantial and previously unrecognized source of sodium current that can significantly influence action potential morphology. In this thesis, I formulate and analyze new models of cardiac action potential which account for these caveolar sodium currents and provide a computational venue in which to develop and test new hypotheses. My results provide insight into the role played by caveolar ionic currents in regulating the electrodynamics of cardiac myocytes and suggest that in certain pathological cases, caveolae may play an arrhythmogenic role.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations.<p> We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations.<p> We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances.

<p>To understand the underlying mechanisms of cardiac arrhythmias, computational models are used to study heart rhythm dynamics. The parameters of these models carry inherent uncertainty. Therefore, to interpret the results of these models, uncertainty quantification (UQ) and sensitivity analysis (SA) are important. Polynomial chaos (PC) is a computationally efficient method for UQ and SA in which a model output Y, dependent on some independent uncertain parameters represented by a random vector &xi;, is approximated as a spectral expansion in multidimensional orthogonal polynomials in &xi;. The expansion can then be used to characterize the uncertainty in Y.</p><p>PC methods were applied to UQ and SA of the dynamics of a two-dimensional return-map model of cardiac action potential duration (APD) restitution in a paced single cell. Uncertainty was considered in four parameters of the model: three time constants and the pacing stimulus strength. The basic cycle length (BCL) (the period between stimuli) was treated as the control parameter. Model dynamics was characterized with bifurcation analysis, which determines the APD and stability of fixed points of the model at a range of BCLs, and the BCLs at which bifurcations occur. These quantities can be plotted in a bifurcation diagram, which summarizes the dynamics of the model. PC UQ and SA were performed for these quantities. UQ results were summarized in a novel probabilistic bifurcation diagram that visualizes the APD and stability of fixed points as uncertain quantities.</p><p>Classical PC methods assume that model outputs exist and reasonably smooth over the full domain of &xi;. Because models of heart rhythm often exhibit bifurcations and discontinuities, their outputs may not obey the existence and smoothness assumptions on the full domain, but only on some subdomains which may be irregularly shaped. On these subdomains, the random variables representing the parameters may no longer be independent. PC methods therefore must be modified for analysis of these discontinuous quantities. The Rosenblatt transformation maps the variables on the subdomain onto a rectangular domain; the transformed variables are independent and uniformly distributed. A new numerical estimation of the Rosenblatt transformation was developed that improves accuracy and computational efficiency compared to existing kernel density estimation methods. PC representations of the outputs in the transformed variables were then constructed. Coefficients of the PC expansions were estimated using Bayesian inference methods. For discontinuous model outputs, SA was performed using a sampling-based variance-reduction method, with the PC estimation used as an efficient proxy for the full model.</p><p>To evaluate the accuracy of the PC methods, PC UQ and SA results were compared to large-sample Monte Carlo UQ and SA results. PC UQ and SA of the fixed point APDs, and of the probability that a stable fixed point existed at each BCL, was very close to MC UQ results for those quantities. However, PC UQ and SA of the bifurcation BCLs was less accurate compared to MC results.</p><p>The computational time required for PC and Monte Carlo methods was also compared. PC analysis (including Rosenblatt transformation and Bayesian inference) required less than 10 total hours of computational time, of which approximately 30 minutes was devoted to model evaluations, compared to approximately 65 hours required for Monte Carlo sampling of the model outputs at 1 &times; 10<super>6</super> &xi; points.</p><p>PC methods provide a useful framework for efficient UQ and SA of the bifurcation diagram of a model of cardiac APD dynamics. Model outputs with bifurcations and discontinuities can be analyzed using modified PC methods. The methods applied and developed in this study may be extended to other models of heart rhythm dynamics. These methods have potential for use for uncertainty and sensitivity analysis in many applications of these models, including simulation studies of heart rate variability, cardiac pathologies, and interventions.</p> / Dissertation

Modelling in cardiac electrophysiology results in a complex system of partial differential equations (PDE) describing the propagation of the electrical wave in the heart muscle coupled with a highly nonlinear system of ordinary differential equations (ODE) describing the ionic activity in the cardiac cells. This system forms the widely accepted bidomain model or its slightly simpler version, the monodomain model. To a large extent, the stiffness of the whole model depends on the choice of the ionic model, which varies in terms of complexity and realism. These simulations require accurate and, depending on the ionic model used, possibly very stable numerical methods. At this time, solving these models numerically requires CPU time of around one day per heartbeat. Therefore, it is necessary to use the most efficient method for these simulations. This research focuses on the comparison and analysis of several time-stepping methods: explicit or semi-implicit, operator splitting, deferred correction and Rush-Larsen methods. The goal is to find the optimal method for the ionic model used. For our analysis, we used the monodomain model but our results apply to the bidomain model as well. We compare the methods for three ionic models of varying complexity and stiffness: the Mitchell-Schaeffer models with only 2 variables, the more realistic Beeler-Reuter model with 8 variables, and the stiff and very complex ten Tuscher-Noble-Noble-Panfilov (TNNP) models with 17 variables. For each method, we derived absolute stability criteria of the spatially discretized monodomain model and verified that the theoretical critical time steps obtained closely match the ones in numerical experiments. Convergence tests were also conducted to verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. We looked at the efficiency of the different methods by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured respectively by the most negative eigenvalue of the model's Jacobian and the number of variables, and based on strict stability and accuracy criteria.

yes / Spatial dispersion of repolarization is known to play an important role in arrhythmogenesis. Electrotonic modulation of repolarization by the activation sequence has been observed in some species and tissue preparations, but to varying extents. Our study sought to determine the mechanisms underlying species- and tissue-dependent electrotonic modulation of repolarization in ventricles. Epi-fluorescence optical imaging of whole rat hearts and pig left ventricular wedges were used to assess epicardial spatial activation and repolarization characteristics. Experiments were supported by computer simulations using realistic geometries. Tight coupling between activation times (AT) and action potential duration (APD) were observed in rat experiments but not in pig. Linear correlation analysis found slopes of −1.03 ± 0.59 and −0.26 ± 0.13 for rat and pig, respectively (p < 0.0001). In rat, maximal dispersion of APD was 11.0 ± 3.1 ms but dispersion of repolarization time (RT) was relatively homogeneous (8.2 ± 2.7, p < 0.0001). However, in pig no such difference was observed between the dispersion of APD and RT (17.8 ± 6.1 vs. 17.7 ± 6.5, respectively). Localized elevations of APD (12.9 ± 8.3%) were identified at ventricular insertion sites of rat hearts both in experiments and simulations. Tissue geometry and action potential (AP) morphology contributed significantly to determining influence of electrotonic modulation. Simulations of a rat AP in a pig geometry decreased the slope of AT and APD relationships by 70.6% whereas slopes were increased by 75.0% when implementing a pig AP in a rat geometry. A modified pig AP, shortened to match the rat APD, showed little coupling between AT and APD with greatly reduced slope compared to the rat AP. Electrotonic modulation of repolarization by the activation sequence is especially pronounced in small hearts with murine-like APs. Tissue architecture and AP morphology play an important role in electrotonic modulation of repolarization.

The electrical activity of the heart is a well studied process. Mathematical modeling and computer simulations are used to study the cardiac electrical activity: several mathematical models exist, among them the microscopic model, which is based on the explicit representation of individual cells. The cardiac tissue is viewed as two separate domains: the intra-cellular and extra-cellular domains, Ωᵢ and Ωₑ, respectively, separated by cellular membranes Γ. The microscopic model consists of a set of Poisson equations, one for each sub-domain, Ωᵢ and Ωₑ, coupled on interfaces Γ with nonlinear transmission conditions involving a system of ODEs. The unusual transmission conditions on Γ make the model challenging to solve numerically. In this thesis, we first focus on the dimensional analysis of the microscopic model. We then reformulate the problem on the interface Γ using a Steklov-Poincaré operator. We discretize the model in space using finite element methods. We prove the existence of a semi-discrete solution using a reformulation of the model as an ODE system on the interface Γ. We derive stability and error estimates for the finite element method. Afterwards, we consider five numerical schemes including the Godunov splitting method, two implicit methods, (Backward Euler (BE) and second order Backward Differentiation Formula (BDF2)), and two semi-implicit methods (Forward Backward Euler (FBE), and second order Semi-implicit Backward Differentiation Formula (SBDF2)). A convergence analysis of the implicit and semi-implicit methods is performed and the results are compared with manufactured solutions that we have proposed. Numerical results are presented to compare the stability, accuracy and efficiency of the methods. CPU times needed to solve the problem over a single cell using FBE, SBDF2 and Godunov splitting methods are reported. The results show that FBE and Godunov splitting methods achieve better numerical accuracy and efficiency than implicit and SBDF2 schemes, for a given computational time. Finally, we solve the model using FBE and Domain Decomposition Method (DDM) for two cells connected to each other by a gap junction. We investigate the influence of the space discretization and we explore the differences between a conforming and nonconforming mesh on Γ. We compare the solutions obtained with both FBE and DDM methods. The results show that both methods give the same solution. Therefore, the DDM is capable of providing an accurate solution with a minimal number of sub-domain iterations.

<p>Cardiac arrhythmias triggered by both reentrant and focal sources are closely correlated with regions of tissue characterized by significant structural heterogeneity. Experimental and modeling studies of electrical activity in the heart have shown that local microscopic heterogeneities which average out at the macroscale in healthy tissue play a much more important role in diseased and aging cardiac tissue which have low levels of coupling and abnormal or reduced membrane excitability. However, it is still largely unknown how various combinations of microheterogeneity in the intracellular and interstitial spaces affect wavefront propagation in these critical regimes. </p> <p>This thesis uses biophysically realistic 1-D and 2-D computer models to investigate how heterogeneity in the interstitial and intracellular spaces influence both the initiation of ectopic beats and the escape of multiple ectopic beats from a poorly coupled region of tissue into surrounding well-coupled tissue. An approximate discrete monodomain model that incorporates local heterogeneity in both the interstitial and intracellular spaces was developed to represent the tissue domain. </p> <p>The results showed that increasing the effective interstitial resistivity in poorly coupled fibers alters the distribution of electrical load at the microscale and causes propagation to become more like that observed in continuous fibers. In poorly coupled domains, this nearly continuous state is modulated by cell length and is characterized by decreased gap junction delay, sustained conduction velocity, increased sodium current, reduced maximum upstroke velocity, and increased safety factor. In inhomogeneous fibers with adjacent well-coupled and poorly coupled regions, locally increasing the effective interstitial resistivity in the poorly coupled region reduces the size of the focal source needed to generate an ectopic beat, reduces dispersion of repolarization, and delays the onset of conduction block that is caused by source-load mismatch at the boundary between well-coupled and poorly-coupled regions. In 2-D tissue models, local increases in effective interstitial resistivity as well as microstructural variations in cell arrangement at the boundary between poorly coupled and well-coupled regions of tissue modulate the distribution of maximum sodium current which facilitates the unidirectional escape of focal beats. Variations in the distribution of sodium current as a function of cell length and width lead to directional differences in the response to increased effective interstitial resistivity. Propagation in critical regimes such as the ectopic substrate is very sensitive to source-load interactions and local increases in maximum sodium current caused by microheterogeneity in both intracellular and interstitial structure.</p> / Dissertation

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-08T11:17:37Z No. of bitstreams: 1 ricardosilvacampos.pdf: 8528381 bytes, checksum: 29e3f07b2a4b4215d4e42d012e0f5df3 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T13:24:53Z (GMT) No. of bitstreams: 1 ricardosilvacampos.pdf: 8528381 bytes, checksum: 29e3f07b2a4b4215d4e42d012e0f5df3 (MD5) / Made available in DSpace on 2016-07-13T13:24:53Z (GMT). No. of bitstreams: 1 ricardosilvacampos.pdf: 8528381 bytes, checksum: 29e3f07b2a4b4215d4e42d012e0f5df3 (MD5) Previous issue date: 2016-02-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / Este trabalho apresenta o simulador FisioPacer, que é um simulador que reproduz a propagação do pulso elétrico pelo tecido cardíaco e a sua deformação mecânica. Foi utilizado um autômato celular acoplado a um sistema massa-mola para que as simulações sejam realizadas rapidamente. Foi também utilizado um algoritmo genético para automaticamente determinar parâmetros do modelo de forma a reproduzir outros experimentos in silico e o comportamento de um ventrículo real. Com intuito de validar o modelo foram feitos setenta e dois experimentos e os resultados foram comparados com outro simulador robusto, baseado em equações diferenciais. As comparações mostraram que o FisioPacer reproduziu satisfatoriamente o comportamento do tecido, sendo até quinze mil vezes mais rápido. Além disto, foram simuladas as funcionalidades eletromecânicas de um ventrículo esquerdo a partir de dados de um paciente, obtidos via ressonância magnética. / This work proposes a computational heart model named FisioPacer, which aims to reproduce the electrical pulse propagation over the cardiac tissue and its mechanical deformation. In order to perform fast simulations, it was used a cellular automaton coupled with a mass-spring system. A genetic algorithm was also used to automatically adjust model parameters, in order to reproduce in silico experiments and a real left ventricle behavior. For the model validation, seventy two experiments were performed and the results were compared to another robust simulator, based on partial differential equations. The comparisons showed that the FisioPacer simulator could reproduce cardiac tissue electromechanics, with up to 15000-fold improvement in computational time. Furthermore, a real patient left ventricle was simulated, with data obtained via MRI.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-02-24T12:24:27Z No. of bitstreams: 1 brunogouveadebarros.pdf: 4637517 bytes, checksum: 0db5f859f17bd37484772dd26a331ce5 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-24T15:33:28Z (GMT) No. of bitstreams: 1 brunogouveadebarros.pdf: 4637517 bytes, checksum: 0db5f859f17bd37484772dd26a331ce5 (MD5) / Made available in DSpace on 2017-02-24T15:33:28Z (GMT). No. of bitstreams: 1 brunogouveadebarros.pdf: 4637517 bytes, checksum: 0db5f859f17bd37484772dd26a331ce5 (MD5) Previous issue date: 2013-02-25 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Os modelos computacionais tornaram-se ferramentas valiosas para o estudo e compreensão dos fenômenos da eletrofisiologia cardíaca. No entanto, a elevada complexidade dos processos biofísicos e o nível microscópico de detalhes exigem complexos modelos computacionais. Aspectos-chave da eletrofisiologia cardíaca, tais como condução lenta e bloqueio de condução tem sido tema de pesquisa de muitos estudos, uma vez que estão fortemente relacionados à arritmia cardíaca. No entanto, ao reproduzir estes fenômenos os modelos necessitam de uma discretização sub-celular para a solução das equações diferenciais e uma condutividade eléctrica do tecido não uniforme e heterogênea. Devido aos elevados custos computacionais de simulações que reproduzem a microestrutura fina do tecido cardíaco, estudos prévios têm considerado experimentos de tecido de pequenas dimensões e têm utilizados modelos simples de células cardíacas. Neste trabalho, desenvolvemos um modelo (modelo microscópico) da eletrofisiologia cardíaca que capta a microestrutura do tecido cardíaco usando uma discretização espacial muito fina (8µm) e utilizamos um modelo celular moderno e complexo baseado em Cadeias de Markov para a caracterização da estrutura e dinâmica dos canais iônicos. Para lidar com os desafios computacionais, o modelo foi paralelizado usando uma abordagem híbrida: a computação em cluster e GPGPUs (General-purpose computing on Graphics Processing Units). Nossa implementação paralela deste modelo, utilizando uma plataforma multi-GPU, foi capaz de reduzir os tempos de execução das simulações de mais de 6 dias (em um único processador) para 21 minutos (em um pequeno cluster de 8 nós equipado com 16 GPUs). Além disso, para diminuir ainda mais o custo computacional, foi desenvolvido um modelo discreto equivalente ao modelo microscópico. Este novo modelo foi paralelizado usando a mesma abordagem do modelo microscópico e foi capaz de executar simulações que demoravam 21 minutos em apenas 65 segundos. Acreditamos que esta nova implementação paralela abre caminho para a investigação de muitas questões em aberto associadas à natureza complexa e discreta da propagação dos potenciais de ação no tecido cardíaco. / Computer models have become valuable tools for the study and comprehension of the complex phenomena of cardiac electrophysiology. However, the high complexity of the biophysical processes and the microscopic level of details demand complex mathematical and computational models. Key aspects of cardiac electrophysiology, such as slow conduction, conduction block and saltatory effects have been the research topic of many studies since they are strongly related to cardiac arrhythmia. However, to reproduce these phenomena the numerical models need to use sub-cellular discretization for the solution of the PDEs and nonuniform, heterogeneous tissue electric conductivity. Due to the high computational costs of simulations that reproduce the fine microstructure of cardiac tissue, previous studies have considered tissue experiments of small or moderate sizes and used simple cardiac cell models. In this work we develop a cardiac electrophysiology model (microscopic model) that captures the microstructure of cardiac tissue by using a very fine spatial discretization (8µm) and uses a very modern and complex cell model based on Markov Chains for the characterization of ion channel's structure and dynamics. To cope with the computational challenges, the model was parallelized using a hybrid approach: cluster computing and GPGPUs (General-purpose computing on graphics processing units). Our parallel implementation of this model using a Multi-GPU platform was able to reduce the execution times of the simulations from more than 6 days (on a single processor) to 21 minutes (on a small 8-node cluster equipped with 16 GPUs). Furthermore, in order to decrease further the computational cost we have developed a discrete model equivalent to the microscopic one. This new model was also parallelized using the same approach as the microscopic model and was able to perform simulations that took 21 minutes to be executed in just 65 seconds. We believe that this new parallel implementation paves the way for the investigation of many open questions associated