RRoocchheesstteerr IInnssttiittuuttee ooff TTeecchhnnoollooggyy RRIITT SScchhoollaarr WWoorrkkss Theses 5-30-2014 AAnn AApppprrooaacchh ffoorr AAppppllyyiinngg DDaattaa AAssssiimmiillaattiioonn TTeecchhnniiqquueess ffoorr SSttuuddyyiinngg CCaarrddiiaacc AArrrrhhyytthhmmiiaass Stephen T. Scorse Follow this and additional works at: https://scholarworks.rit.edu/theses RReeccoommmmeennddeedd CCiittaattiioonn Scorse, Stephen T., "An Approach for Applying Data Assimilation Techniques for Studying Cardiac Arrhythmias" (2014). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. An Approach for Applying Data Assimilation Techniques for Studying Cardiac Arrhythmias by Stephen T. Scorse Submitted to the School of Mathematical Sciences, College of Science in partial fulfillment of the requirements for the degree of Masters of Science in Applied and Computational Mathematics at the Rochester Institute of Technology May 30, 2014 Accepted by: Matthew J. Hoffman, School of Mathematical Sciences Accepted by: Elizabeth M. Cherry, School of Mathematical Sciences Accepted by: Laura M. Munoz, School of Mathematical Sciences Committee Approval: Matthew J. Hoffman, D.Phil. May 30, 2014 School of Mathematical Sciences Thesis Advisor Elizabeth M. Cherry, D.Phil. May 30, 2014 School of Mathematical Sciences Committee Member Laura M. Munoz, D.Phil. May 30, 2014 School of Mathematical Sciences Committee Member Nathan Cahill, D.Phil. May 30, 2014 School of Mathematical Sciences Director of Graduate Programs Abstract Cardiac electrical waves are responsible for coordinating functionality in the heart muscle. When these waves are disrupted, it is referred to as a cardiac arrhythmia, which can be life threatening. Over the last two decades, experimental techniques to study electrical waves in the heart have been improved. Nevertheless, methods for observing electrical waves in the heart still do not provide enough data to draw conclusions about wave propagation. Observations about wave propagation can be made on the surface of the heart, but to collect data on the interior, the methods currently available either do not provide high enough spatial resolution or change the conduction properties of the region to be observed. The lack of depth information leaves us uncertainastotheprecisemechanismsbywhichwavescausearrhythmias. Numericalmodelshave beendevelopedforthesewavedynamics,buttheyareonlyapproximationsforqualitativebehavior. Fordecades,theweather-forecastingcommunityhasfacedsimilarproblemswiththeapproximate nature of numerical models and lack of available data, and tools known as data assimilation have beendevelopedwithinthatcommunitytodealwiththisproblem. Weaimtoapplythesemethods to cardiac wave propagation. By combining observations with numerical models through data- assimilation techniques from weather forecasting, we will attempt to constrain wave dynamics in theheartmuscleinterior. Inthisthesis,wepresentthefirstapproachforcouplingdata-assimilation techniqueswithacardiaccellularmodel. UsingtheFenton-Karma(FK)modelofcardiaccellular processestorepresentwavedynamicsandtheEnsembleTransformKalmanFilterdata-assimilation technique, synthetic observations have been integrated with the FK model. By analyzing the sensitivity of state-variable values in the FK model, we have developed initial conditions for data assimilationthatareabletoquantitativelyapproximatewavebehaviorina1-Dsetting. Thiswork will provide a proof of concept and give insight into challenges in the 2- and 3-D cases. i Contents 1 Introduction 1 1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Fenton-Karma Model 4 2.1 Cardiac Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Initialization of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Numerically Solving the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Analyzing Error and Instability Growth of the FK Model 11 3.1 Method and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Instabilities of the FK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Breeding Experiments on the FK Model 16 4.1 The Breeding Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Breeding with the FK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 An Introduction to Data Assimilation 19 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 The LETKF Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 Results from Data Assimilation 26 6.1 Methods of Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.2 Initial Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.3 Idealized Observation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.4 Ensemble Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.5 Realistic Observation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.6 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ii 7 Conclusions and Future Work 45 iii List of Figures 1 A cardiac action potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Interaction between the wave front and back due to discordant alternans. . . . . . . . . 6 3 Examples of the control and perturbed states of u 700 ms after a perturbation in w in part of the domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Differences between the control and perturbed states of all variables over time, after a perturbation in u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Differences between the control and perturbed states of all variables over time, after a perturbation in w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6 Differences between the control and perturbed states of all variables over time, after a perturbation in all variables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7 The breeding process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8 Growth rates from the breeding process with a breeding interval of 10 ms. . . . . . . . . 18 9 Model predictions for the path of Hurricane Sandy[19]. . . . . . . . . . . . . . . . . . . . 20 10 Beginning u ensemble obtained from breeding. . . . . . . . . . . . . . . . . . . . . . . . 23 11 The observation (at every grid point), the background estimate, and the analysis esti- mate for u after 1 iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 12 Idealized, almost-realistic observations, the background estimate, and the analysis esti- mate for u after 1 iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 13 The observation (at every 85 grid points) for just the u variable, the background esti- mate, and the analysis estimate for u after 1 iteration. . . . . . . . . . . . . . . . . . . . 31 14 The ”truth”, the background estimate, and the analysis estimate for v after 2200 ms. for idealized, realistic observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 15 Random initial ensemble members for u from breeding.. . . . . . . . . . . . . . . . . . . 33 16 Idealized, almost-realistic observations, the background estimate, and the analysis esti- mate for u after 1 iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 17 Idealized, almost-realistic observations, the background estimate, and the analysis esti- mate for u after 1300 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 18 Idealized, realistic observations, the background estimate, and the analysis estimate for u after 350 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 iv 19 RMS error over time using a random initial ensemble with 20 members and idealized, realistic observations compared with a non-random ensemble average free run.. . . . . . 36 20 Error in the analysis ensemble of size 10 from the truth over time for u with different observation distributions and initial ensembles compared with a non-random ensemble average free run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 21 Error in the analysis estimate for u with idealized, realistic observations with different ensemble sizes compared with a non-random ensemble average free run. . . . . . . . . . 38 22 Theidealized,realisticobservations,thebackgroundestimate,andtheanalysisestimate for u after 1300 ms with good initial background ensemble members and an ensemble size of 10 and ρ=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 23 Theidealized,realisticobservations,thebackgroundestimate,andtheanalysisestimate for u after 900 ms with ρ=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 24 The idealized, almost-realistic observations, the background estimate, and the analysis estimate for u after 1300 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 25 Idealized, realistic observations, the background estimate, and the analysis estimate for u after 1300 ms with 20 ensemble members and additive inflation every 250 ms. . . . . . 41 26 Error in the analysis ensemble from the truth over time for u with idealized, realistic observations with different methods to correct ensemble collapse compared with a non- random ensemble average free run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 27 Error in the analysis ensemble from the truth over time for u for realistic observations with varying ensemble sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 28 Wave dynamics from data assimilation for the u variable with parameter estimation for τ , τ , τ , τ+, and τ−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 r o si w w v List of Tables 1 FK model parameters used in this study.. . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Variable and parameter descriptions for the ETKF algorithm. . . . . . . . . . . . . . . . 21 vi 1 Introduction 1.1 The Problem Cardiovascular disease is the highest cause of death in the industrialized world. Approximately 10% of all deaths in the United States can be attributed to ventricular fibrillation, a fast-developing, unco- ordinated contraction of the cardiac muscle that leaves the heart unable to pump blood effectively[1]. For the heart to work properly, electrical waves travel throughout the muscle. But, when the heart is notfunctioningproperly, normalwavepropagationcanbedisturbed, compromisingtheheart’sability to pump effectively. This disruption is known as a cardiac arrhythmia. During an arrhythmia, the heart can beat too fast, too slow, or with an irregular rhythm [2]. Much remains to be understood about the dynamics of arrythmias due to a lack of understanding about the behavior of electrical waves. A large part of this lack of knowledge is because the thickness of the heart muscle obstructs us from viewing the waves deep in the interior of the heart. Over the last 20 years, new high-resolution optical mapping techniques have provided more information about wave dynamics on the heart’s surface, but not from within the muscle depth. Probes can be inserted into the tissue to record interior wave information, but using enough probes to properly see wave dynamicswouldultimatelychangethebehaviorwehopetoobservebyalteringtheelectrophysiological propertiesofthetissue[4]. Instead,wewillmodifydata-assimilationtechniquesfromnumericalweather forecasting to study these wave dynamics without disrupting the behavior of wave fronts. Forecasting a physical system generally requires both a model for the time evolution of the system and an estimate of the current state of the system. In some applications, the state of the system can be measured directly with high accuracy. In other applications, such as weather forecasting and cardiac modeling, direct measurement of the global system state is not feasible. Instead, the state must be inferred from available data. While a reasonable state estimate based on current data may be possible, in general one can obtain a better estimate by using both current and past data. Data assimilation provides such an estimate on an ongoing basis, alternating between a forecast step and a state estimation step, or analysis. The analysis step combines information from current data and from a prior short-term forecast (which is based on past data), producing a current state estimate. This estimate is used to initialize the next short-term forecast, which is subsequently used in the next analysis,andsoon. Thedata-assimilationprocedureisitselfadynamicalsystemdrivenbythephysical 1