UNIVERSITY OF CALGARY. Intercellular Coupling Abnormalities in the Heart:

Transcription

1 UNIVERSITY OF CALGARY Intercellular Coupling Abnormalities in the Heart: Quantification from Surface Measurements and Impact on Arrhythmia Vulnerability by Amin Ghazanfari A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING CALGARY, ALBERTA MAY, 2016 c Amin Ghazanfari 2016

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3 Abstract Cardiac conduction velocity is one of the most important electrophysiological characteristics of the heart. Several cardiac dysfunctions and arrhythmia are caused by slowed conduction velocity. Measurement of cardiac conduction velocity and other physiological characteristics of the heart such as anisotropy ratio are challenged by complex cardiac tissue structure and inaccurate measurement techniques. Diabetes mellitus is an example of a condition that can alter conduction velocity by reducing the electrical coupling between cardiac cells. Diabetes is also known to increase the risk of arrhythmia by increasing the action potential duration of cardiac myocytes. This thesis discusses a measurement method based on fitting ellipses to activation isochrones. Our results show that the intramural fiber rotation caused error in conventional measurement methods used to estimate fiber orientation and anisotropy ratio specially in thinner tissues. These errors are increased by optical mapping measurements specifically in thicker tissues. We developed a mathematical model for the diabetic rabbit ventricular action potential and also used an existing model of the diabetic rat ventricular action potential. We demonstrated the window of vulnerability to reentrant arrhythmia for healthy and diabetic models of both rabbit and rat. Connexin lateralization was modelled in the diabetic models by reducing the gap junction conductivity in the lateral direction. Results demonstrated that window of vulnerability in diabetic rat is smaller than in healthy rat. On the contrary, diabetic rabbit was more vulnerable to reentry than healthy rabbit. The ATP-dependent potassium channel was added to the models and the results demonstrated that diabetic models are more vulnerable to reentry when ischemia occurs and I KATP channels open consequently. ii

4 Acknowledgements A PhD thesis is truly a marathon journey which would not be possible without the support and guidance of many individuals. I would like to express my gratitude towards all of them who helped transform my research work into a successful PhD thesis. There is possibly no transformation capable of mapping my deep appreciation for my supervisor, Dr. Anders Nygren, into few lines of acknowledgements! He has provided me with invaluable continuous guidance and support, while offering full freedom of exploring new ideas and research directions. His insights and troubleshooting skills have strengthened this study significantly. I would like to extend my thanks to Dr. Vigmond, for his concise and precise inputs to my research. His software alongside with his insights and comments helped me develop major parts of this study. Over the years, I have also been fortunate to share a lab with some formidably intelligent fellow students, who will all go on to make a difference, whether it is in this field or another. In particular, thanks to Marcela Rodriguez for not only helping me with experimental studies in this study, but also for being the best friend I have had here. Over the past years, my roommates and friends were always beside me. I thank them for all kind of support they provided especially doing the dishes! Thank you Ehsan, Hesam, Mohsen and Alireza. iii

5 Dedication To my Family who I deprived myself from their presence beside me: Maman, your endless love for me helped me endure missing you. Baba, your support always helped me keep going on. Shiva, the smile you put on my face whenever I talked to you made my life beautiful. Hadi and Ali, I know how much we are proud of each other as brothers and friends. & to all other sacrifices I made.

6 Table of Contents Abstract ii Acknowledgements iii Dedication iv Table of Contents v List of Tables viii List of Figures ix List of Symbols xi 1 Motivation The Heart Anatomy of the Heart Heart chambers and circulation Cardiac cells Cardiac tissue Cardiac conduction system Electrical Activity of the Heart The cardiac action potential Propagation in cardiac tissue Excitation-Contraction coupling Electrocardiogram (ECG) Arrhythmia and Fibrillation Types of arrhythmia Reentry Gap junctions and arrhythmias Diabetes and the Heart Types of Diabetes Diabetic Heart Electrophysiological Complications of Diabetes ECG abnormalities Ion channel remodelling Connexin lateralization Calcium abnormalities Biophysical Mechanisms Cell Membrane The Nernst-Planck Equation Nernst Equilibrium Potential Electrical Model of Cell Membrane Ionic Current Models Hodgkin & Huxley Model Markov Models v

7 3.3 Signal Propagation in Cardiac Tissue Bidomain Modelling Monodomain modelling Numerical Methods Euler s method Finite Element Method Cardiac physiological characteristics measurements Chapter Specific Background Cardiac Optical Mapping Voltage-sensitive dye Imaging system Methods Computational Model Data Processing Estimation of Epicardial Fiber Orientation Estimation of Anisotropy Ratio (AR) Optical Mapping Model Experimental Methods Results Simulation Results Observations based on isochrones Conduction velocities Alternative estimates based on conduction velocities Effect of optical mapping Effect of photon scattering Effect of spatial discretization Experimental Results Discussion Effect of fiber rotation and wall thickness Effect of transmural conductivity Effect of optical mapping Implication for experimental measurements Study limitations Conclusion Diabetes and Vulnerability to Arrhythmias Chapter Specific Background Methods Rat Model Rabbit Model Summary of the rabbit mathematical model Changes in diabetic rabbit ATP-dependent potassium channels (I KATP ) vi

8 5.2.4 Spontaneous Calcium Release Connexin Lateralization Tissue Simulations Reentrant Arrhythmias Conduction Reserve Results Single cell APD is increased by diabetes I KATP shortens APD more in diabetic tissue than in healthy tissue Connexin lateralization does not affect source-sink relationship for an ectopic beat Vulnerability to reentry Rabbit and rat have the same conduction reserve Rabbit and rat have different APD restitution Discussion Diabetes decreases arrhythmia vulnerability in rat Diabetes increases arrhythmia vulnerability in rabbit Diabetes did not change the source-sink relationship Diabetes has different effects on arrhythmia vulnerability in different species Study limitations Conclusion Significant findings Cardiac measurements Diabetes and arrhythmia vulnerability Clinical relevance Modelling limitations and future work Future work Bibliography A Simulation Time B Copyright Permissions vii

9 List of Tables 3.1 Extra- and intracellular concentrations and Nernst potential for a ventricular myocyte Measured epicardial fiber direction and anisotropy Measured wave propagation parameters Angle of propagation for different spatial discretization Anisotropy ratio for different spatial discretization Longitudinal conduction velocity for different spatial discretization Transverse conduction velocity for different spatial discretization A.1 Simulation details: number of runs, simulation times, and total time for each simulation viii

10 List of Figures and Illustrations 2.1 Heart chambers, veins, and arteries Circulation system of the human body Cell membrane phospholipid bilayer with embedded proteins Cardiac conduction system Normal cardiac action potential and its phases Main ionic currents of the cardiac action potential Gap junction between two cells intracellular Ca 2+ dynamics The schematic representation of a human ECG Schematic representation of an anatomical reentry Simulation of functional reentry using S1-S2 stimulation Quantification of Cx43 in STZ-diabetic and healthy rat The fraction of immunofluorescence labelled Cx43 associated with intercalated discs (ICD) and that associated with lateralized Cx43 for both control and STZ-diabetic rat Analysis of Cx43 lateralization Electric circuit model of the cellular membrane A simple two-state Markov model Different configurations of an ion channel Equivalent Markov model of Hodgkin-Huxley sodium current Equivalent bidomain circuit diagram of single cell A solution region and its finite element discretization A triangular element used in FEM calculations. Local nodes are numbered as 1, 2, and Schematic diagram showing arrangement of the major components of the imaging system Fitting an ellipse to an isochrone Activation time isochrones on the epicardial surface Angle of wave propagation versus time Angle of wave propagation versus tissue thickness Anisotropy ratio versus tissue thickness for different transmural conductivities Transverse (θ t ) and longitudinal (θ l ) conduction velocity Different approaches to model emission light Isochrones based on the simulations of optical mapping with and without the scattering effect Experimental results ix

11 4.11 Experimentally obtained activation isochrones from a rat left ventricular free wall Schematic diagram of the rabbit model from Mahajan et al S1-S2 stimulation protocol Changes in diabetic rat ionic currents Changes in diabetic rabbit ionic currents Effect of I KATP opening on the rat action potential Effect of I KATP opening on the rabbit action potential Spontaneous Ca 2+ release propagation through the tissue Window of vulnerability for healthy and diabetic rat models Window of vulnerability for healthy and diabetic rabbit models Relationship between conduction velocity and conductivity for rat and rabbit models S1S2 APD restitution curves in single epicardial ventricular cells x

12 List of Symbols, Abbreviations and Nomenclature Symbol AP APD AF AR ATP AV CHD CHF CICR CV CVD Cx DAD DM DTMRI EAD ECC ECG GJ JSR LA LV Definition Action Potential Action Potential Duration Atrial Fibrillation Anisotropy Ratio Adenosin Tri-Phosphate Atrioventricular Coronary Heart Disease Congestive Heart Failure Ca 2+ -Induced Ca 2+ Release Conduction Velocity Cardiovascular Disease Connexin Delayed After Depolarization Diabetes Mellitus Diffusion Tensor Magnetic Resonance Imaging Early After Depolarization Excitation-Contraction Coupling Electrocardiogram Gap Junction Junctional Sarcoplasmic Reticulum Left Atrium Left Ventricle xi

13 MI V m NCX NSR OM PVC RA RV RyR SA Myocardial Infarction Membrane Potential Na + /Ca 2+ Exchanger Network Sarcoplasmic Reticulum Optical Mapping Premature Ventricular Complex Right Atrium Right Ventricle Ryanodine Receptor Sinoatrial SERCA Sarco-Endoplasmic Reticulum Ca 2+ SR STZ VF VT Sarcoplasmic Reticulum Streptozotocin Ventricular Fibrillation Ventricular Tachycardia xii

14 Chapter 1 Motivation Cardiac conduction velocity, the speed with which an electrical impulse propagates through the cardiac tissue, is one of the most important electrophysiological characteristics of the heart. The importance of conduction velocity arises in the context of cardiac rhythm and arrhythmia. Slowed myocardial conduction velocity is associated with an increased risk of re-entrant activities that can lead to cardiac arrhythmia. Conduction velocity in the tissue depends on the cell excitability and the electrical coupling between the cells. Excitability is controlled by sodium channel conductivity, while coupling is related to the gap junction conductivity. These determinants are altered by a wide range of pathophysiological conditions. Gap junctions are non-selective membrane channels that form low resistance cellto-cell connections to ease intercellular current as well as the transfer of ions, amino acids, and nucleotides. In general, cardiac cells express gap junctions at higher densities toward the ends of cells compared to their sides, resulting in higher conductivity in the longitudinal direction. These channels are composed of proteins called connexins (Cx). Changes in connexin distribution and functionality can alter gap junction function and thereby reduce intercellular coupling. Diabetes mellitus is known to cause migration of the connexins from cell ends to cell sides, which is known as connexin lateralization. This thesis mainly focuses on the intercellular coupling and cardiac conduction velocity. Chapter 2 is a medical and physiological background of the topics covered in this thesis. The biophysical mechanism and modelling techniques are discussed in 1

15 detail in Chapter 3. The mathematical requirements and the numerical methods are provided in this chapter. Chapter 4 provides simulations to discuss different methods of measuring elctrophysiological characteristics of the heart such as anisotropy ratio, fiber orientation, and conduction velocity. The effects of tissue thickness and intercellular coupling on these parameters are also discussed in this chapter. Experimental data are provided to support simulation results. Diabetes mellitus is a chronic progressive disease that results in microvascular and macrovascular complications. Other than the acute glucose level abnormalities, diabetes also causes chronic renal failure, retinal damage, nerve damage, micro vascular damage, cardiovascular disease and poor healing which can lead to gangrene and even amputation. Diabetes is a significant independent risk factor for heart failure and there is a substantial number of patients with both diabetes and heart failure. The most well documented electrophysiological dysfunction of diabetes is the QT interval prolongation in the electrocardiogram, which is a direct effect of the prolongation of the ventricle action potential. Conduction of electrical activation through the heart is one of the less thoroughly studied effects of diabetes. Chapter 5 is concerned with this aspect of cardiac electrophysiological effects and variations of diabetes. Different species have different cardiac action potential duration and morphology. These differences can cause species-dependent characteristics, which can lead to distinct behaviors in pathophysiological situations. This chapter provides simulations related to the electrophysiological differences in diabetic rat and rabbit models and how these differences affect the vulnerability of each species to the re-entrant arrhythmia. Finally, Chapter 6 provides concluding remarks on the results presented in this work. Future complementary research that can be conducted from this work is also 2

16 introduced in this chapter. 3

17 Chapter 2 The Heart The heart is a muscular pump responsible for circulating blood in the body. This vital organ beats approximately 100,000 times each day, pumping roughly 8,000 liters of blood per day [1]. It functions effectively thanks to a synchronized relation between its electrical, mechanical and fluidic systems. This chapter provides an overview of the anatomy of the heart, contraction, and its normal and disorderly electrical activities. 2.1 Anatomy of the Heart Heart chambers and circulation The heart is located between the lungs in the middle of the chest, behind and slightly to the left of the sternum. A double-layered membrane called the pericardium surrounds the heart like a sac. The inner layer of the pericardium is attached to the heart muscle, while the outer layer covers the heart s major blood vessels and is attached to the spinal column and diaphragm. A lubricant fluid separates the two layers of membrane, letting the heart move as it beats. As shown in Fig.2.1, the human heart is made up of four muscular chambers, the left and right atria (upper chambers) alongside with the left and right ventricles (lower chambers LV and RV respectively). The left and right chambers are separated by a muscular wall called the septum. The left ventricle is the largest and strongest chamber in the heart as it pumps the blood through the whole body. The blood is received at the right atrium by the superior and inferior vena cava 4

18 Superior vena cava Aorta Right Atrium Left Atrium Mitral valve Aortic valve Tricuspid valve Right Ventricle Left Ventricle Inferior vena cava Pulmonary valve Figure 2.1: Heart atria and ventricles, valves (shown by arrows), main arteries and veins. Blood is received from inferior and superior vena cava and pumped to lungs (pulmonary circulation shown in blue) and then returned to the heart and pumped to the whole body (main circulation shown in red); Adapted under a Creative Commons Attribution 2.5 Unported license from original work by Dcoetzee; Online: License: http: //goo.gl/tgfja veins. Blood travels from the right atrium to the right ventricle through the right atrioventricular valve known as the tricuspid valve. When the right ventricle contracts, the tricuspid valve closes to prevent the backflow of the blood to the right atrium. Blood traveling out of the right ventricle passes through the pulmonary valve to enter the pulmonary trunk. The blood then flows into left and right pulmonary arteries and into the lungs. The vessels branch into many thin walled vessels called capillaries where the gas exchange occurs (pulmonary circulation). Oxygenated blood is returned to the left atrium by the pulmonary veins. When the left atrium contracts, the blood flows into the left ventricle through the bicuspid valve, also known as the mitral valve. The mitral valve allows the blood to flow from 5

19 the left atrium to the left ventricle and prevents blood flow in opposite direction. The strong and thick walls of the left ventricle (approximately 3 times thicker than the right ventricle) provide a contraction with sufficient pressure to pump the blood through the whole body. Contraction of the left ventricle closes the mitral valve and pumps the blood out of the left ventricle to the aorta through aortic valve. The aorta connects to the arterial system and delivers oxygenated blood to the rest of the body (systemic circulation). Figure 2.2 shows pulmonary and systemic circulations in the human body. The heart itself receives the blood from the coronary arteries which are the first vessels that branch off the aorta. Like other arteries, they divide into a fine capillary network to perfuse the heart muscle with the oxygen-rich blood. This blood returns to the heart by the coronary sinus which drains into the lower part of the right atrium Cardiac cells The heart is constructed from electrically excitable cells like all muscle tissues. The cardiac muscle cells are called cardiomyocytes or briefly myocytes, and are surrounded by a cell membrane. The cell membrane of a myoycte, also known as sarcolemma, is mainly comprised of phospholipids, cholesterol, glycolipids, and membrane proteins. The cell membrane separates the intracellular fluid from extracellular space with a phospholipid bilayer. Phospholipids are molecules made up of a hydrophilic head and hydrophobic tail. As a result they form a bilayer as shown in Fig The resulting bilayer prevents molecules such as water and ions from traveling between intra and extracellullar fluids. However, there are specialized embedded proteins in the membrane that allow some small molecules and ions to pass the membrane. These proteins that are known as ion channels, function as highly selective pores that 6

20 Figure 2.2: Circulation system of the human body; the cardiovascular system has two distinct circulatory paths, pulmonary circulation and systemic circulation. Blood movement from the heart to the lungs which is done by right chambers is called pulmonary circulation. Systemic circulation is the movement of the blood to the rest of the body. Adapted under a Creative Commons Attribution 3.0 Unported license from original work by OpenStax College; Online: License: 7

21 Phospholipid Bilayer Membrane Proteins Figure 2.3: Cell membrane phospholipid bilayer with embedded proteins. can transport ions between cell interior and extracellular fluid without consuming or providing energy solely relying on electrical and chemical gradients of the ions. Ion channels open and close in different voltage, time, and chemically gated forms to allow specific ions to cross the cell membrane. Ion channels are not the only proteins embedded in the sarcolemma. Other types of proteins such as pumps, ion exchangers, and co-transporters consume energy to move ions in their own selective and regulatory manner. In contrast with ion channels, ion pumps consume energy from adenosine triphosphate (ATP) to move ions across the cell membrane against their electrical or chemical gradients. These proteins are essential in order to maintain the concentration gradients of K +, Na +, and Ca 2+ across the cell membrane. This gradient across the membrane results in a resting electric potential across the membrane. The electrical model of cell membrane and ionic currents will be discussed in detail in Section Cardiac tissue The heart pumps the blood continuously and strongly during the whole life without any rest, so cardiac muscle must have incredibly high contractility and endurance. 8

22 Cardiac muscle cells are striated and branched. Each cardiac myocyte is attached at its end to adjoining myocytes by a membrane called intercalated disc (ICD). These myocytes form long fiber bundles and in the atrial fibers they are arranged in concentric circles and wrapped around the atria. Over the past centuries, numerous efforts have been made to improve the understanding of the fiber structure of the heart. These efforts progressed with the development of more sophisticated analysis techniques. The term fiber has been used differently at different size scales. Throughout this work, fiber refers to the general axial direction of the myocytes in a specific cardiac site. Therefore, by this definition, fiber orientation is analogous to the direction of grain in wood. Initially, blunt-dissection techniques were applied by teasing apart the myocardium with fingers and blunt tools such as scissors. These techniques enable only basic qualitative description of myocardial fiber architecture. Lower reported that fiber orientation is not homogeneous in the wall in 1660 [2]. In 1849, Ludwig described the strands of myocytes in the wall as a figure-of-eighth as they proceed from base of right ventricle (RV) to the apex of RV, and then back to the base of left ventricle (LV) [3]. Modern histological analysis was employed in ultra-thin sliced tissue sections (less than 10µm thickness) to quantitatively measure two dimensional myocardial fiber architecture. In 1969, Streeter and colleagues performed the first quantitative measurement of fiber orientation in the LV wall of dog hearts [4]. They stated that the fiber angle transitioned smoothly from 60 at endocardial to -60 at epicardial surface. By their definition, fiber angle is the angle between fiber orientation and local circumferential axis. Caulfield and Brog used scanning electron microscopy and demonstrated that in- 9

23 dividual myocytes are grouped with other myocytes and surrounded by an extensive extracellular collagen network [5]. Quantitative characterization of the grouped myocyte structure showed that myocytes are integrated into laminar layers that typically are 4 cells thick and predominantly lie in the planes spanned by myofiber orientation and radial direction [6]. Diffusion tensor magnetic resonance imaging (DTMRI) has been proposed recently as an alternative method for non-invasive 3D characterization of myocardial fiber and sheet structure. The tree-dimensional fiber structure of the heart has been successfully reconstructed at high resolution using DTMRI [7]. The anisotropic (directionally-dependent) architecture of cardiac tissue is critical in coordinating electrical propagation and providing efficient force production in the heart. Anisotropy observed at the organ and tissue level, with the alignment of the cardiomyocytes into fibers and sheets, can also be traced down to the cellular level. Individual cardiomyocytes naturally exhibit an elongated morphology that contributes to the electrical and mechanical anisotropy. Electrical anisotropy at the cellular level is also attributed to the distribution of gap junctions. In cardiac myocytes, gap junctions are preferentially located at cell ends on the ICD Cardiac conduction system Cardiac myocytes are constantly responsible for contraction and relaxation. They are able to contract in response to an electric impulse. This property is known as cardiac excitability. In addition, some regions of the cardiac tissue are responsible for initiating and orchestrating the electrical impulses causing the mechanical contraction of the cardiac muscle. Unlike the rest of the tissue, these regions undergo spontaneous excitation at systematic frequent intervals, establishing a timely delivery of stimuli 10

24 to the rest of the heart. This property is called automaticity. Figure 2.4 shows a diagram of the cardiac conduction system. The sinoatrial (SA) node is embedded in the posterior wall of the right atrium near the superior vena cava. For normal contraction, excitation originates from the SA node. This region contains pacemaker cells capable of initiating the heart beats and is therefore known also as the cardiac pacemaker. The impulse generated by the SA node spreads through the atria and initiates their contraction. The action potential spreads through cell to cell contacts and reaches the atrioventricular (AV) node. The atria and ventricles are electrically insulated and the AV node is the only electrical connection between them. The conduction through the AV node is slow and as a result it causes a successive conduction delay from end to end for stimuli. This property inhibits uncontrolled cellto-cell activation of ventricles unlike the atria. This delay is also important to ensure the atrial contraction is complete before the ventricles begin to contract. Otherwise the powerful contraction of the ventricles would close the AV valves preventing the blood flow from the atria to the ventricles. As shown in Fig. 2.4, the ventricular conduction system consists of the bundle of His, that branches into the left and right bundle branches, and the Purkinje fibers. Once an impulse passes the AV node, it enters the bundle of His and travels to the interventricular septum to enter the left and right bundle branches. Both the branches descend along the left and right septal surfaces toward the apex of the heart, turn, and spread over the whole endocardial surface. As the branches bifurcate, the Purkinje fibers connect to the myocytes and conduct the action potential very rapidly. They are responsible for the synchronized depolarization of the ventricles. The Purkinje fibers radiate from the apex toward the base of the heart and therefor ventricular contraction begins at the apex toward the base to completely pump the blood and 11

25 Figure 2.4: Schematic of the cardiac conduction system; Impulse originates from the sinoatrial (SA) node, propagates through the atria to reach the atrioventricular (AV) node. After a delay the impulse passes through His bundle, branches into the left and right bundle branches to reach the Purkinje fibers causing a synchronized excitation of the ventricles. Adapted and modified under a Creative Commons Attribution 3.0 Unported license from original work by Madhero88; Online: License: haedo8 empty the ventricles into the aorta and pulmonary trunk. 2.2 Electrical Activity of the Heart The cardiac action potential Action potentials are generated by the movement of ions through the transmembrane ion channels in the cardiac cells. The cardiac myocyte has a negative membrane potential when at rest, which is caused by the difference in ionic concentrations and conductances across the membrane of the cell during the resting phase of the action potential. The normal resting membrane potential in the ventricular myocardium is 12

26 membrane potential (mv) time (ms) Figure 2.5: Normal cardiac action potential and its phases; Phase 0: depolarization, phase 1: early repolarization, phase 2: plateau, phase 3: repolarization, and phase 4: resting potential. Action potential is generated from simulation of a rabbit ventricular myocyte model [96]. about -75 to -85 mv. The different phases of an action potential are shown in Fig When the cell is stimulated and the membrane potential passes a threshold (approx. 20mV above the resting potential), it goes through the depolarization phase which is due to the opening of Na + channels. A massive influx of Na + raises the membrane potential and consequently causes the inactivation of sodium channels, resulting in a very shortlived inward current (phase 0). In the SA node and the AV node, phase 0 of an action potential is due to the inward Ca 2+ current through the L-type Ca 2+ channels (I CaL ). After the upstroke, there is an abrupt drop in the membrane potential (phase 1) due to the opening of the transient outward K + channels (I to ). The most important difference between cardiac action potentials and neural action potentials is the plateau 13

27 current (na/nf) current (na/nf) I Kr I Ks I to I K1 I -20 Na I CaL time (ms) (a) time (ms) (b) Figure 2.6: Main ionic currents of the cardiac action potential; major underlying inward (a) and outward (b) currents are generated from simulation of a rabbit ventricular myocyte model [96]. In (a) I Na is clipped: min I Na 160µA/cm 2. Stimulus applied at t = 50ms phase (phase 2). This plateau is due to the inward Ca 2+ current (I CaL ) which is in a relative balance with outward K + current. With inactivation of I to, rapid and slow delayed rectifier K + currents (I Ks and I Kr ) activate and gradually the membrane repolarization starts. With the closure of the L-type Ca 2+ channels and opening of inward rectifier K + current (I K1 ), the action potential goes to the rapid repolarization phase. I K1 also plays an important role in setting the resting potential (phase 4). The major inward and outward currents are shown in Fig The cardiac refractory period is when a cell has not completely recovered from the previous action potential and is not excitable so it does not respond to a new stimulus. The cardiac refractory period is divided into an absolute refractory period and a relative refractory period. In the absolute refractory period, a cardiac myocyte cannot fire a new action potential. During the relative refractory period, a new action potential can be elicited in certain circumstances (such as a large stimulus magnitude). 14

28 2.2.2 Propagation in cardiac tissue When a region of the cardiac tissue undergoes an action potential, the current flows into the neighboring cells through intercellular gap junctions. An action potential is generated from the depolarizing current flow of a neighboring cell. Then, the action potential propagates from an excited cell to the neighboring resting cells. This process is facilitated by low-resistance protein pathways called gap junctions. Gap junctions (GJs) regulate the amount of current that flows to the neighboring cells (sink) from depolarized cells (source). They form transmembrane channels that connect the cytoplasmic compartments of neighboring cells to facilitate highconductivity, non-selective conduction. This results in an orderly propagation of electrical stimulus, and hence, synchronized contraction of the heart. Each gap junction consists of two connexon hemi-channels, one embedded in each membrane of the neighboring cells. Each connexon consists of six protein subunits called connexins (Cx). Figure 2.7 shows the schematic representation of gap junctions and connexins. Almost 20 different types of connexins have been identified in mammalian cells [8], of which Cx40, Cx43, and Cx45 are the main isoforms in the myocardium. It is common to identify tissue region from the Cx type of the cell; i.e. the Cx40 can be found in atria, Cx43 is mainly abundant in ventricles, and Cx45 is exclusively found in the cardiac conduction system [9]. Gap junctions play an important role in the maintenance of normal conduction velocity (CV) of the action potential. As the GJ conductivity decreases (cells become less coupled), the depolarizing current becomes more confined to the cell and there is less electrical load and less current flow to the neighboring non-excited cells. 15

29 Figure 2.7: Each gap junction consists of two connexons. Connexons are made of six proteins called connexins; Adapted under a WikiMedia public domain license from original work by Mariana Ruiz; Online: Excitation-Contraction coupling The process of synchronization of mycocyte electrical excitation with the mechanical contraction of the cardiac muscle is called excitation-contraction coupling (ECC). A synchronized mechanical contraction is essential to pump the blood with a timely and maintained pressure. Ca 2+ is the main activator of myofilaments causing myocyte contraction. Therefore, Ca 2+ dynamics play an important role in ECC. The main structures involved in the ECC are: (1) sarcomere, which is a serial unit of myofilaments that form an extensive myofibrillar structure, and is responsible for the mechanical contraction and tension development; (2) sarcoplasmic reticulum (SR), which is an internal storage of Ca 2+ ions. During the diastole, Ca 2+ is stored in the SR; (3) T-tubules, which are deep invaginations of the sarcolemma and contain 16

30 the L-type calcium channels and (4) mitochondria, which provides the adenosine triphosphate (ATP) which is being used as an energy pack for contraction and other metabolic needs of cardiac myocytes. Calcium enters the cell during the cardiac action potential mainly through the L- type calcium channels in T-tubules responsible for the plateau phase of the AP. Entry of Ca 2+ increases the intracellular Ca 2+ concentration ([Ca 2+ ] i ) that activates ryanodine receptors (RyR), which triggers the release of Ca 2+ from the SR. This process is known as calcium induced calcium release (CICR) and results in a quick rise of [Ca 2+ ] i. The free Ca 2+ ions in the cytosol bind to the myofilament protein which reults in the cell contraction. Relaxation occurs when [Ca 2+ ] i declines, which allows Ca 2+ ions to dissociate from myofilament proteins (troponin C). Cytosolic Ca 2+ ions are extruded in four different ways: (1) almost 80% of the cytosolic Ca 2+ is taken up by the SR [10] through Ca 2+ -ATPase channel (J up ); (2) sarcolemmal Na + /Ca 2+ exchange current (NCX); (3) sarcolemmal Ca 2+ -ATPase and (4) mitochondrial Ca 2+ uniport as shown in Fig Electrocardiogram (ECG) The electrical conduction in the whole heart can be measured on the body surface in the form of the electrocardiogram (ECG). The propagation of the extracellular current in the heart generates an electrical field which can be measured by an electrode placed in the vicinity of the heart. The electrode will measure an attenuated and spatially averaged distribution of potentials generated by the current flow through the tissue in the torso until they reach the surface. An ECG is a routine clinical test that shows how fast the heart is beating, if the rhythm is steady or irregular, and the strength and timing of electrical signals as they 17

31 Figure 2.8: Ca 2+ enters the cell mainly through L-type Ca 2+ channels which are densely embedded in T-tubules. Ryanodine receptor channels ease the release of Ca 2+ from the SR in response to increased [Ca 2+ ] i. Different pumps and exchangers extrude Ca 2+ from the intracellular space. Reprinted by permission from [10]. pass through each part of the heart. ECG results are used to detect and study many health problems such as cardiac arrhythmias, heart attack, and heart failure. A schematic of one cardiac cycle of an idealized ECG is shown in Figure 2.9, in which each distinct waveform is labeled as P, Q, R, S, or T. Each waveform relates to the activation or repolarization of some part of the heart. The P wave is caused by the spreading activation over the atria, while the Q, R, and S waves are caused by the activation of the ventricles. The T wave is a result of the repolarization of the ventricles. The U wave is a small deflection immediately after the T wave, usually in the same direction as the T wave. 18

32 R P T U Q S Figure 2.9: The schematic representation of a human ECG. Individual waveforms are indicated by their associated names. 2.3 Arrhythmia and Fibrillation Arrhythmia is defined as an abnormal pattern of cardiac electrical activity. These abnormal activities can occur in different regions of the heart in various forms. Depending on the region of the arrhythmia, some types of arrhythmia are less fatal than others. For instance, disorderly atrial activation can result in irregular ventricular activation due to the electrical separation of the chambers and the delay of the AV node. In contrast, even small arrhythmogenic activities in ventricles can lead to more dangerous irregularities as they can alter or suppress the blood delivery of the heart Types of arrhythmia A successful heartbeat requires both impulse generation and propagation to be performed flawlessly. Abnormalities in either will result in an irregular heartbeat or dysrhythmia. Abnormal impulse generation and abnormal impulse conduction can result in irregular heart rhythms and therefore arrhythmia. Arrhythmia can be classified by the heart rate. If the heart rate is lower than the normal heartbeat, the arrhythmia is called bradycardia and if the heart rate is faster 19

33 than normal it is called tachycardia. Bradycardia may be due to slow SA firing rate (sinus bradycardia), or by damaged electrical pathways from atria to ventricles (AV node block). On the other hand, tachycardia mostly results from additional abnormal impulses to the normal sinus rhythm. The main mechanisms behind abnormal impulse generation are: 1) automaticity and 2) triggered beats. Automaticity is known as the mechanism of a myocyte firing action potential on its own. Every impulse that originates outside of the sinus node is called an ectopic focus. An ectopic focus, if successful to propagate, may cause a single premature beat or, if it fires at a higher rate than sinus rhythm, can produce abnormal rhythms (ectopic beats). Ectopic beats produced in the atria are less likely to be a dangerous arrhythmia but can reduce the cardiac pumping ability and efficiency. Triggered beats mainly happen due to abnormal depolarization of cardiac myocytes that can interrupt phase 2, phase 3, and phase 4 of the action potential and are known as afterdepolarizations. Deficiency in Ca 2+ or Na + channels can cause secondary depolarization of the action potential in phases 2 and 3 respectively, which is called an early afterdepolarization (EAD) [11]. If a secondary depolarization begins during phase 4 of the action potential just before repolarization is completed, it is called a delayed afterdepolarization (DAD)[12]. SR Ca 2+ overload may cause spontaneous Ca 2+ release during repolarization, resulting in high [Ca 2+ ] i. Released Ca 2+ exits through Na + /Ca 2+ exchanger (NCX) which has a net depolarizing current causing DADs. Abnormal impulse conduction is another cause of arrhythmia and can be divided into two possible categories: 1) conduction block, and 2) reentry (discussed in detail in next section). SA node block, AV node block, and bundle branch block are different arrhythmia caused by conduction block. 20

34 Fibrillation is a situation when an entire cardiac chamber is involved in a single or multiple electrophysiological abnormalities and therefore trembles with irregular chaotic electrical impulses. Fibrillation can occur in atria (atrial fibrillation AF) or ventricles (ventricular fibrillation VF). AF is not considered a medical emergency while VF is an imminent life-threatening situation and if left untreated can lead to death within minutes Reentry The electrical excitation generated at the SA node propagates through the heart surface in an orderly pattern and then vanishes; i.e. every excitation is initiated and generated by an impulse from the SA node in normal sinus rhythm. However, the activation wavefront may travel around a physical obstacle or unexcitable tissue in a self sustaining fashion instead of dying down. This robust recurring activity, which can be the source of abnormal cardiac electrical activities and arrhythmia, is known as reentry. This reentrant waveform continues to excite the heart because it always encounters excitable tissue in a loop. Reentry can be divided into two categories: anatomical and functional reentry. The anatomically reentrant wavefrom travels around an unexcitable tissue (such as dead tissue) or an object of reduced excitability. Figure 2.10 shows an anatomical reentry. When conduction blocks in one direction around an anatomical obstacle (unidirectional block), the activation waveform rotates around the block and establishes a macroscopic reentry. Functional reentry is in the form of a spiral wave that does not rotate around an obstacle, but propagates around the whole region of tissue based on the differences in the refractory properties of excitable tissue. The most common cause of functional 21

35 (a) (b) Figure 2.10: Schematic representation of an anatomical reentry; the anatomical obstacle is shown as the solid black circle. (a) Activation initiates from the red dot and propagates in both directions. Gray area shows a unidirectional block which prevents the activation to propagate counterclockwise but allows propagation in opposite direction. (b) Reentry propagates clockwise and sustained. reentry is the gradient in refractoriness in a cardiac region. As shown in Fig. 2.11, in order to simulate a reentrant activity, a plane wave is initiated by stimulating along the top part of the 2D sheet (S1). As S1 propagates downward through the tissue and repolarization wave vanishes in the upper part, the second stimulus (S2) is applied at the top right corner of the tissue. This results in propagation to the left side of the tissue but not downward because the upper side is either excitable or in the relative refractory state while the lower part is unexcitable and is in the absolute refractory state. The wavefront then rotates clockwise around a point near the center of the tissue (varies based on the timing difference between S1 and S2, stimulus strength, etc) as activation gradually propagates to the areas that recovering from repolarization and becoming excitable. It has to be noted that the S2 has to be applied at a certain 22

36 time interval after S1 to generate the reentry. This interval is called the window of vulnerability. This is explained in more detail in section This spiral wave can be the origin of many abnormal electrical activities such as fibrillation. S1 S2 5ms 30ms 32ms 34ms 40ms 45ms Figure 2.11: Simulation of functional reentry using S1-S2 stimulation; Reentry is initiated in a 2D 1cm 1cm tissue of rat ventricular myocyte [51]. S2 is applied in the top right corner. A sustained sprial wave is generated as the activation waveform propagates around the tissue. The concept of the wavelength of a circular impulse was introduced by Smeets and his colleagues in 1986 [13]. It is defined as the product of conduction velocity (CV) of the circulating wavefront and the effective refractory period of the tissue in which the reentry is propagating. Refractory period sometimes is approximated by the action potential duration. Wavelength quantifies the distance the wavefront travels during the refractory period. For reentry to occur and sustain, the wavelength 23

37 of the reentrant wavefront must be shorter than the length of the effective reentry pathway. That is, the time it takes for the impulse to travel around in one cycle must be longer than the refractory period to provide the myocardium sufficient time to recover excitability. In this work, functional reentry is the main type of reentry being discussed. The word reentry refers to functional reentry unless otherwise specified Gap junctions and arrhythmias Gap junction coupling plays an important role in the arrhythmogenic characteristics of cardiac tissue, mainly through three different mechanisms: 1) conduction velocity, 2) conduction block, and 3) source-sink relations. As explained above, the wavelength (λ) is an important factor for sustaining reentry which is a product of conduction velocity (CV) and refractory period approximated by action potential duration (APD): λ = CV APD (2.1) Reduced λ increases the chances of reentrant activities as shorter wavelength increases the recovery time in a given path length. Therefore, action potential shortening and reduced CV result in more vulnerability to reentrant activities. Conditions such as gap junction uncoupling, which reduces the CV, increases the likelihood of arrhythmia. Conduction block itself, if it happens in a conduction system, can be an arrhythmogenic phenomenon. Conduction block is known to happen mostly in the tissue boundary regions where CV heterogeneity exists such as SA and AV nodes. However, as explained in section 2.3.2, one of the basic requirements for initiation of an anatomical reentry is the formation of conduction block. In the regions with more 24

38 homogeneous CV, conduction block may occur due to changes in ionic currents or gap junction remodelling which leads to longer APD and reduced CV. Hence, heterogeneities of CV or APD can be pathological and increase the vulnerability to arrhythmia in the corresponding substrate. Initiation of the reentry also plays a key role in vulnerability to arrhythmia as well as reentry sustainability. As explained before, EADs and DADs are arrhythmogenic triggered activities that can lead to ectopic beats. The occurrence of EAD and DAD does not always result in reentry. However, they can generate premature beats (commonly known as premature ventricular complexes PVCs) if they can propagate into a large number of myocytes. Therefore, afterdepolarizations are arrhythmogenic activities not only because they provide regions with APD heterogeneity in the tissue, but also because they develop PVCs that can propagate into the tissue and cause a reentry. Considering an EAD in a region of tissue as a source of the electrical current and the neighbouring region as a sink of that current, certain levels of coupling are required for current propagation. These are known as source-sink relationships. It is known that normal levels of coupling result in synchronization of the tissue with EAD without developing PVCs and APD heterogeneity [14]. On the other hand, reduced coupling can allow a smaller size of tissue exhibiting an EAD to generate a PVC and increases the likelihood of arrhythmia. 2.4 Diabetes and the Heart Diabetes Mellitus (DM), which is also commonly known as diabetes, is a syndrome of dysfunctional metabolism resulting in abnormally high levels of glucose in the blood (hyperglycemia). The levels of blood glucose in the body are controlled mainly by 25

39 insulin which is secreted by a group of beta cells in the pancreas Types of Diabetes If the pancreas fails to produce enough insulin, the DM is called type 1 diabetes. The patients then require using insulin replacement. Type 2 diabetes results from the body developing resistance to the effects of insulin. Both types result in increased blood sugar level which if left untreated can cause long term acute complications such as cardiovascular diseases, stroke, kidney failure, and damage to the eyes. In this study we mainly focus on type 1 diabetes which is an autoimmune disease that results in the destruction of the insulin producing beta cells in the pancreas [15] Diabetic Heart Complications related to DM affect many organ systems and are responsible for the majority of morbidities and mortalities associated with the disease. These complications can be divided into vascular and nonvascular, and are similar for type 1 and type 2 DM. The vascular complications of DM are further subdivided into microvascular (retinopathy, neuropathy, nephropathy) and macrovascular complications (coronary heart disease CHD, peripheral arterial disease PAD, and stroke). Microvascular complications are diabetes-specific, whereas macrovascular complications are similar to those in nondiabetics but occur at greater frequency in individuals with diabetes. Nonvascular complications include gastroparesis, infections, skin changes, and hearing loss [16]. Cardiovascular diseases (CVD) are increased in individuals with type 1 or type 2 DM. The Framingham Heart Study revealed a one- to fivefold risk increase in PAD, coronary artery disease, myocardial infarction (MI), and congestive heart fail- 26

40 ure (CHF) in DM. Diabetes can affect cardiac structure and ventricular function, a condition which is called diabetic cardiomyopathy. This makes diabetes a risk factor for developing systolic and diastolic dysfunction and heart failure. Heart failure is a common disorder with high rates of mortality which is widespread in many patients with diabetes mellitus. In addition, the prognosis for individuals with diabetes who have coronary artery disease or MI is worse than for nondiabetics. CHD is more likely to involve multiple vessels in individuals with DM. In addition to CHD, cerebrovascular disease is increased in individuals with DM (threefold increase in stroke). DM is responsible for increased atherosclerosis in aorta and coronary arteries which increases the risk of myocardial infarction and stroke. Thus, after checking for all known cardiovascular risk factors, type 2 DM increases the cardiovascular death rate twofold in men and fourfold in women [16]. The American Heart Association has designated DM as a CHD risk equivalent, and type 2 DM patients without a prior MI have a similar risk for coronary artery-related events as nondiabetic individuals who have had a prior MI. The increase in cardiovascular morbidity and mortality rates in diabetes appears to relate to the synergism of hyperglycemia with other cardiovascular risk factors. Risk factors for macrovascular disease in diabetic individuals include dyslipidemia, hypertension, obesity, reduced physical activity, and cigarette smoking [16]. Dyslipidemia is an abnormal amount of lipids in the blood which mostly appears as an elevation of lipids in the blood in developed countries (hyperlipidemia). 27

41 2.4.3 Electrophysiological Complications of Diabetes ECG abnormalities Prolongation of the QT interval in the ECG is one of the most common electrophysiological effects of type 1 diabetes [17]. QT interval mainly represents the APD of ventricular myocytes and therefore, QT prolongation corresponds to the prolongation of APD. T wave abnormalities and QRS prolongation are also suggested in other studies [18, 19]. These abnormalities are partially due to reduced impulse propagation conduction velocity Ion channel remodelling It has been shown in many studies that the APD is prolonged in diabetic hearts. The prolongation of AP is mainly due to reduced outward repolarization currents such as calcium independent transient outward potassium current (I to ), rapid and slow delayed rectifier currents (I Kr and I Ks respectively). It has been shown that the expression of corresponding potassium channel subunits is reduced in the diabetic cardiac tissue [20]. Smaller amount of I to results in delayed early repolarization phase and reduced I Kr and I Ks results in prolonged repolarization phase of the action potential Connexin lateralization Alterations in GJ expression as well as GJ localization lead to impaired impulse conduction. Slowed CV reflects in broadening of QRS complex of the ECG, results in disorderly ventricular contraction and forms a potential arrhythmogenic substrate. Nygren et al. [21] obtained the organization of Cx43 the main connexin protein in ventricular myocytes and compared streptozotocin-induced diabetic (STZdiabetic) with healthy rat. Total expression of Cx43 was not altered in diabetic rats 28

42 as depicted in Fig However, immunofluoresence labeling of Cx43 confirmed that the organization of Cx43 was significantly altered 7 days after injection of STZ. These alterations not only include a reduction in the amount of Cx43 at the intercalated discs (Fig.2.13), but also a redistribution of Cx43 from the ends of the cell to the cell sides which is known as connexin lateralization (illustrated in Fig.2.13 and Fig.2.14 respectively)[21]. Nygren et al. showed that Cx43 is dissociated from other major components of gap junctions suggesting that the lateralization results in nonfunctional junctions. This reduces the conductivity of GJs and altered anisotropy of the tissue which is also a cause of QT prolongation. (A) (B) Figure 2.12: Quantification of Cx43 in STZ-diabetic and healthy rat; Western blot of ventricular cardiac myocyte for control and diabetic rats(a) and its analysis (B) is shown. The releative density of the Cx43 in control and diabetic samples were not significantly different from one another. Reprinted by permission from [21]. Lateralization of Cx43 is also a common feature of surviving myocytes around the infarct region in human ventricle [22]. A similar redistribution has been observed in some rat models of ventricular hypertrophy and has been shown to be correlated with reduced longitudinal conduction velocity [23]. At four days post-infarction in a dog model, lateralized Cx43 correlated spatially with re-entrant electrophysiological 29

43 Figure 2.13: The fraction of immunofluorescence labelled Cx43 associated with intercalated discs (ICD) and that associated with lateralized Cx43 for both control and STZ-diabetic rat; The fraction of lateralized Cx43 was significantly higher in the STZ-diabetic myocytes. Reprinted by permission from [21]. activities [24]. In addition to observations made at the infarct border zone, lateralization has been reported in end-stage human heart failure [25] and in the ventricles of patients with compensated hypertrophy due to valvular aortic stenosis (a disease in which the aortic valve narrows and prevents the valve from fully opening [26]). Cx43 is particularly redistributed in hypertrophic cardiomyopathy, which is the most common cause of sudden cardiac death in young adults due to cardiac arrhythmia [9]. Another different form of gap junction remodelling had been observed in patients with ischaemic heart disease. When the blood flow to the heart is reduced, some regions of ventricular myocardium lose their ability to contract properly, but are able to retain their contractile functionality if the blood flow is restored. These regions are called hibernating myocardium. In hibernating myocardium, the overall amount of 30

44 Figure 2.14: Analysis of Cx43 lateralization; immunofluorescence images of an isolated ventricular myocyte from STZ-diabetic rat heart (left) and healthy rat heart (right); The images show Cx43 migrated from the cell ends to the cell sides in the diabetic myocyte. Reprinted by permission from [21]. Cx43 per intercalated disc is reduced compared to the normally perfused myocardial regions of the same heart [27]. Apart from disturbances in gap junction remodelling discussed above, connexin expression may be altered in human heart disease. Several studies [28, 29] also demonstrated reduced Cx43 transcript and protein levels in the left ventricles of transplant patients with congestive heart failure Calcium abnormalities Calcium is the main ionic regulator in excitation-contraction coupling and is essential for normal cardiac function. High concentration of Ca 2+ ion can be arrhythmogenic as explained in Sections and 2.3.1, where it is shown that activities of all of the currents, exchangers, and pumps involved in the Ca 2+ dynamics and excitation contraction coupling (ECC) such as NCX, J up, SERCA, and RyR are reduced, which 31

45 is explained in detail in Sections to Some of these alterations result in reduced [Ca 2+ ] i while the others cause increased [Ca 2+ ] i. The overall effect is increased overload of cytosolic calcium, impaired relaxation and diastolic dysfunction. 32

46 Chapter 3 Biophysical Mechanisms This chapter briefly reviews the basics of cellular electrophysiology, the mathematical modelling techniques used in cellular electrophysiology, and the numerical methods to overcome the complexity of the mathematical models without too much simplification. 3.1 Cell Membrane The cellular membrane is a lipid bilayer in which proteins are inserted. These proteinlined pores are called ionic channels, which allow the flow of specific ions, mainly Na +, K +, Cl, and Ca 2+. The membrane does not allow the ions to flow freely and maintains concentration differences of these ions across the membrane. The concentration gradients of ions produce a potential difference across the membrane, the transmembrane potential, drives the ionic currents The Nernst-Planck Equation Ions move across the membrane for two reasons. The concentration gradients itself causes the diffusion flux J diff which satisfies Fick s law: J diff = D c, where D is the diffusion coefficient (cm 2 s 1 ) and c is the concentration (mol cm 3 ). The electric field generated by the concentration gradients also causes the electric 33

47 flux of ions J elect which satisfies Planck s equation: J elect = z z µc u, where u is the electric potential (V ), µ is the mobility of the ion (cm 2 V 1 s 1 ), z is the valence of the ion so that z = +1, for positive ions. z 1, for negative ions. The total ionic flux J can be calculated by adding the electric flux and diffusion flux: J = J diff + J elect = D c z µc u. (3.1) z Considering Einstein s formula, we can relate the diffusion coefficient D and ion mobility µ via: D = µrt z F, (3.2) where R is the universal gas constant (8.314 J/(K.mol)), T is the absolute temperature (K), and F is the Faraday s constant ( C/mol). Hence substituting 3.2 in 3.1 we can obtain the Nernst-Planck equation: ( J = D c + zf ) RT c u. (3.3) Nernst Equilibrium Potential We can assume that the electric potential only depends on x across the membrane from intracellular (x = 0) to extracellular (x = L) and the diffusion coefficient is constant. Therefore we can re-write as: dc zf du (x) + dx RT dx (x)c(x) + J D = 0 (3.4) 34

48 as: Equation 3.4 is a linear differential equation for c(x) and the solution can be found ( c(x) = exp zf ( ) )( u(x) u(0) c(0) J RT D x 0 exp ( zf ) RT (u(s) u(0))) ds (3.5) At thermodynamical equilibrium, each local process and its reverse proceed at the same rate, therefore the flux across the membrane of the generic ion K with valence z is zero (J = 0). The solution of the Nernst-Planck equation then becomes: from which it follows that, for x = L, ( c(x) = exp zf ( ) ) u(x) u(0) c(0), RT log ( c(l) ) = zf (u(l) u(0)). c(0) RT Denoting i and e to show intra- and extracellular spaces respectively, we can obtain the Nernst equilibrium potential, v K := u i u e = RT zf log ( ci c e ). (3.6) Table 3.1 shows the typical values of extra- and intracellular concentrations and Nernst potential values of a ventricular myocytes Electrical Model of Cell Membrane The cell membrane separates ions (electrical charges) between intracellular and extracellular medium. Therefore, it can be modelled as a capacitor with lipid bilayer dielectric (C m 1µF/cm 2 ). Each ionic channel can also be modelled as a branch with variable conductance in series with the Nernst equilibrium potential of that specific ion. Different ionic channels are connected to each other in parallel between intracellular and extracellular media. Figure 3.1 shows this configuration of membrane behaviour, which is known as a parallel conductance model. 35

49 Extracellular Intracellular Nernst concentration (mm) concentration (mm) potential (mv) Na K Ca e Cl H + 1e-4 2e-4-18 Table 3.1: Extra- and intracellular concentrations and Nernst potential for a ventricular myocyte. Considering Kirchoff s current law, the transmembrane current given by sum of capacitive and ionic currents must be equal to applied current I app (or stimulation current I stim ). Therefore we can write: C m dv dt + I ion = I stim, (3.7) where I ion is the sum of all ionic currents. In order to describe each ionic current we need to discuss different modelling approaches for these currents. 3.2 Ionic Current Models In this section we discuss two out of many different mathematical techniques that are used to model ionic currents through the cellular membrane. 36

50 Figure 3.1: Electric circuit model of the cellular membrane Hodgkin & Huxley Model In 1963, Alan Hodgkin and Andrew Huxley received the Nobel Prize for their experiments on squid giant axon published in They developed a conceptual model based on the studies on the squid giant axon and proposed a formalization to mathematically represent the nonlinear dependency of channel conductance on membrane potential and time. For example in this model, the movement of sodium ions is explained by activation and inactivation gates denoted by m and h respectively. At the resting membrane potential the sodium channel is in a closed state. The m gate is closed and the h gate is open. Voltage sensitive gates undergo conformational changes when the membrane potential changes. As the membrane potential increases, the m gates rapidly activate and open the sodium channels. The electrochemical gradient of sodium drives sodium ions to enter the cell. As the m gates open and membrane potential increases, the h gates begin to close however m gates open more rapidly than the h gates close. The difference between the rates of opening and closing of two gates causes the sodium to enter the cell for few milliseconds. During the repolarization phase of the action potential, sodium channels stay inactivated. Near the end of the repolarization, the negative membrane potential closes the m gates and opens h 37

51 gates. The ionic current through a population of ionic channel is generally given in a unit area of membrane surface and can be modelled as: I Na = g Na (V m, t)ψ(v m ) (3.8) where g Na (V m, t) is the proportion of open sodium channels on the membrane and ψ(v m ) is the current-voltage relation of a single open sodium channel. There are three similar activation and one inactivation independent subunits for the sodium channel, therefore equation 3.8 can be rewritten as: I Na = G Na m 3 h(v m E Na ) (3.9) where G Na and E Na are the maximum conductance and the Nernst potential of sodium channels respectively, and m and h are the probability of activation and inactivation gates to be open respectively. For a generic gating variable w, we can write a first order kinetics equation as: w t = α w(1 w) β w w = w (V m ) w. τ w (V m ) Hodgkin and Huxley examined the response of Na + and K + currents separately to different membrane voltage protocols and used curve fitting to match their results with derived exponential equations: α m = 0.1(25 v m) e 0.1(25 vm) 1 ; α h = 0.07e vm 20 ; βh = 1 e 30 vm β m = 4e vm 18 (3.10) (3.11) where v m is the difference between V m and its resting level (i.e. resting potential is shifted to zero). We can also find the steady state and time constant of each gating variable from the above equations as 38

52 α m m = ; τ m = 1 α α m + β m+β m (3.12) m h = α h ; τ h = 1 α α h + β h +β h (3.13) h Hodgkin and Huxley followed a similar procedure to develop a model for the K + current which is the main outward current. The difference between potassium and sodium models is that the K + channel does not have the inactivation gate. When the membrane potential increases the K + current gradually rises to its steady state value without an inactivation mechanism similar to that of the Na + current. The K + channels have four activation subunits shown by n so the formulation for I K similarly is I K = G K n 4 (V m E K ), (3.14) where G K and E K are the maximum conductance and the Nernst potential of potassium channels respectively. Kinetics of n were calculated by isolating potassium current from total current and using curve fitting techniques to match the results when different voltage protocols are applied, resulting in the following equations: Markov Models α n = 0.01(10 v m) e 0.1(10 vm) 1 ; n = β n = 0.125e vm 80 (3.15) α n α n + β n ; τ n = 1 α n+β n (3.16) The Hodgkin-Huxley formalism describes some ionic channel activities clearly and provides relatively simple models with little computation effort. However, many recent experiments have shown that the Hodgkin-Huxley gating parameters do not represent specific kinetic states of ion channels and cannot describe various aspects 39

53 of channel behavior. For example, inactivation of a Na + channel is more likely when the channel is open. In other words, inactivation depends on activation and the assumption of independent gating that gives the Hodgkin Huxley conductance m 3 h is not valid. In case of modelling ionic currents with this level of detail, we can use a Markovian formalism to represent each physical state of a specific channel with one state of related Markov model. A Markov model consists of a list of possible states of a system, the possible transitions between those states, and the rate parameters of those transitions. The transition rates between Markov states are governed by first order kinetics. Figure 3.2 shows a single ion channel with only one opening gate that results in a two-state Markov model: open state (O) and closed state (C). Then the governing equations of the model is: dc dt do dt = γc + δo = δo + γc O and C denote the probability of the channel to be in open and closed states respectively and hence we can write O + C = 1. C γ δ O Figure 3.2: A simple two-state Markov model Now consider adding an inactivation mechanism to the channel. Depending on the configuration of the channel, this inactivation state can be modelled differently as shown in figure 3.3. In the subfigure 3.3(a), the inactivation state I is only accessible from the open state O while there is a transition to inactivation from closed state in configuration 3.3(b). Configuration 3.3(c) has two inactivation states shown by I 0 and I 1 which are reachable from the closed and open state, respectively. This model 40

54 can explain the slow inactivation state of sodium that can only be accessed while the activation gate is conducting and the fast inactivation gates are non-conducting. I I I 0 I 1 C O C O C O (a) (b) (c) Figure 3.3: Different configurations of an ion channel with one activation and one inactivation gate. Figure 3.4 shows the equivalent Markov model of the Hodgkin-Huxley sodium channel where α m, β m, α h, and β h explained in section Inactivation and closed states are shown by I X and C X respectively where X denotes the number of activation gates being conductive. For example, I 0 is the state where all activation gates are closed and the inactivation gate is non-conductive. I 1 is accessible from I 0 when one activation gate opens. The probability of this transition is 3α since there are three closed activation gates available to be opened. Similarly, I 1 is going to I 2 with a probability of 2α, since one of the activation gates is already open in I 1. C 0 is reached from I 0 when inactivation gate opens with probability of α h and all activation gates are closed. In this configuration the only conducting state is O which is the term m 3 h in equation 3.9. Each state is the probability of the channel to be in that configuration therefore they have a value between 0 and 1 and hence the sum of all states is 1 at any given time. The rate of change in each state can be determined by the difference between 41

55 3α m 2α m α m I 0 I 1 I 2 I 3 β m 2β m 3β m α h β h α h β h α h β h α h β h 3α m 2α m α m C 0 C 1 C 2 O β m 2β m 3β m Figure 3.4: Equivalent Markov model of Hodgkin-Huxley sodium current incoming and outgoing arrows. For example for state O we can write: do dt = α mc 2 + α h I 3 (3β m + β h )O Another advantage of Markov models becomes evident when a drug or a channel blocker acts on a specific state of the channel. A new state can be added to the related Markov model to account for the drug binding to the channel. However, in Markovian modelling, the price to pay for for the highly detailed description of ionic channels and other sub-cellular processes is an increase in complexity and computational cost. 3.3 Signal Propagation in Cardiac Tissue Bidomain Modelling The bidomain representation of cardiac tissue describes both extra- and intracellular potential fields (φ e and φ i respectively) and relates them to membrane behaviors and transmembrane current density I m. Consider the equivalent circuit representation of cardiac tissue in one dimension as shown in figure 3.5. The intracellular space has a conductivity of σ i (or resistivity of r i ) while the extracellular conductivity is σ e (or resistivity of r e ). 42

56 r e ie r e i ion i m i c r m... V m C m Em r i r i i i Figure 3.5: Equivalent bidomain circuit diagram of single cell. The current density J can be calculated from electrical field E based on Ohm s law: J = σe (3.17) where the electric field can be calculated as the negative of the gradient of scalar potential field φ (E = φ), therefore equation 3.17 becomes J i = σ i φ i (3.18) J e = σ e φ e (3.19) When there is no stimulation current and if the cardiac cell assumed to be isolated, any changes in current density in one domain implies a current flow through the cell membrane into the other domain: J i = J e = βi m (3.20) (σ i φ i ) = (σ e φ e ) = βi m (3.21) where β is surface to volume ratio of the cell membrane. 43

57 The media are considered to be continuous and material properties are averaged. The structural features of cell to cell connections are homogenized. By expressing the tissue properties as continuous we can use numerical methods to discretize domains and solve the differential equations. Therefore, the conductivities are formalized as tensors ( σ e and σ i ). If there are any external stimuli applied to extracellular medium we can rewrite the equations as follows: ( σ i φ i ) = βi m (3.22) ( σ e φ e ) = βi m I e (3.23) where I e is external stimulus applied to extracellular medium and I m is calculated as: I m = C m V m t + I ion (V m, v). (3.24) In this equation, I ion is a function of membrane potential V m and all other gating variables and time which are denoted by v. This formulation is a set of coupled equations because both extra- and intracellular potentials have to be calculated simultaneously. These two domains are linear, and nonlinearity arises from the voltage-current relation of membrane current which is described by a set of nonlinear ordinary differential equations (ODEs). Combining the equations 3.22 and 3.23 and considering that V m = φ i φ e will result in the following equations: ( σ i + σ e ) φ e = σ i V m I e (3.25) ( σ i V m ) = σ i φ e + βi m (3.26) Equation 3.25 is an elliptic partial differential equation relating membrane potential and stimulation current to extracellular potential. Equation 3.26 is a parabolic 44

58 partial differential equation relating the extracellular potential and membrane current to membrane potential. I m (as described in equation 3.24) includes the changes of membrane potential over the time. The bidomain equations are necessary when modelling extracellular stimulus and virtual electrodes Monodomain modelling The elliptic equation is computationally expensive and if the extracellular field is ignored, the solution of equation 3.25 can be avoided and only the parabolic PDE needs to be solved. Alternatively if the ratios of the longitudinal and transverse conductivities in the intracellular and extracellular domains are equal (i.e. σ e = kσ i ), then equations 3.22 and 3.23 can be combined to obtain single equation: σ i V m = (1 + 1 k )βi m (3.27) By replacing the intracellular conductivity tensor with the monodomain conductivity tensor (i.e. σ m = σ i ( σ i + σ e ) 1 σ e ), the monodomain equation is obtained as σ m V m = βi m (3.28) which is equivalent to Equation 3.26 when φ e is ignored. Monodomain equations are less computationally expensive and are very fast to solve comparing to bidomain equations. However, the monodomain model generally cannot include external stimulus (except when there is linear relation between extracellular and intracellular conductivities) and thus cannot be used to describe the response to an electric shock. In this thesis, a monodomain model was used throughout. 45

59 3.4 Numerical Methods Euler s method Each dynamic system consists of state variables that describe the mathematical state of the system. Solving a system of differential equations is updating a set of state variables at a specific time instance to the their values at the next point of time denoted by x n and x n+1 respectively. The time difference between two consecutive time instances (n and n + 1) is denoted by t. Explicit numerical methods compute the next value of a state variable based on the current values. For example if a dynamic system is defined as: x t = f(x), (3.29) then the explicit forward Euler method would approximate the next value of x as: x n+1 = x n + t f(x n ). (3.30) On the other hand, the implicit backward Euler methods include x n+1 in the formulation and the discretization of the system 3.29 will be: x n+1 = x n + t f(x n+1 ). (3.31) The explicit method is easy to solve because it does not require any equations to update x n+1. However, the use of explicit methods requires impractically small time steps to be stable and convergent specially when solving stiff dynamic systems. A dynamic system is called stiff when the system has a very fast dynamic and a slow dynamic. Therefore, the numerical methods for solving the stiff system requires an extremely small step size in order to have a stable solution. In contrast, implicit 46

60 methods are unconditionally stable for PDEs used in cardiac modelling but they are more time consuming since equation 3.31 needs to be solved for each iteration. For example, consider a Hodgkin-Huxley gating variable ζ for ion current dynamics: ζ t = ζ ζ τ ζ At a given time n, the forward Euler method would approximate the next value of ζ as: ζ n+1 = ζ n + t ζ ( n t = ζ ζ ζ ) n n + t. τ ζ The implicit backward Euler method would use: ζ n+1 = ζ n + t ζ n+1 t ( ζ ζ ) n+1 = ζ n + t = ζ n + t τ ζ ( ζ ζ n τ ζ + t If the rate of change depends on other state variables rather than ζ then this method becomes noticeably more complicated and a set of equations are required to be solved simultaneously. Both forward and backward Euler methods are first order accurate in time, meaning that the error of approximations are subject to error in order of t. This is very significant when large time steps are used. ). Note that ζ and τ ζ are functions of V m and considered to be constant in an interval of a time step. This assumption is very common and well suited in biophysical solutions. If V m is considered to be constant in a time step, then the solution to the ordinary differential equation (ODE) of m can be approximated with a simple exponential function varying from ζ n to ζ by the time constant τ ζ : ζ n+1 = ζ ( ζ ζ n ) e t/τ ζ (3.32) This solution is known as Rush-Larsen method and popularly used in biophysical phenomena, especially in gating variables of cardiac and neural cells. This method is 47

61 simple to implement and produces solutions as accurate as those produced via more expensive techniques [30]. Sometimes additional accuracy can be obtained by combining both forward and backward Euler methods, i.e., ζ n+1 = ζ n + t 2 [ ζn + ζ ] n+1 t (3.33) This approach, known as Crank-Nicolson method, is widely used in spatial problems. Consider the diffusion equation which is a parabolic PDE as follows, ( ) u t = D u x x (3.34) where D is the diffusion constant. By applying the Crank-Nicolson method we have: u n+1 j u n j t = D 2 [ (u n+1 j+1 2un+1 j + u n+1 j 1 ) + (un j+1 2u n j + u n ] j 1). (3.35) ( x) 2 This model is second order accurate in time and is stable for any t Finite Element Method Finite Element Method is a numerical technique to solve boundary value problems (BVPs) for partial differential equations. This method involves five main steps: 1) dividing the whole domain (solution region) into a finite number of elements, 2) finding related equations for a typical element, 3) assembling all elements together, 4) applying boundary conditions, and 5) compute the solution of resultant system of equations. As an example, consider we want to solve Poisson s equation to calculate the potential distribution V (x, y) in a connected open region Ω in the (x, y) plane with 48

62 Figure 3.6: A solution region and its finite element discretization. Region Ω is divided into four elements shown by circled numbers. Boundary conditions on Ω must satisfy on the approximated boundary. boundary conditions on Ω: 2 V = f(x, y) on Ω, V (x, y) = 0 in Ω. (3.36) The solution region is divided into a number of finite elements as illustrated in figure 3.6. Here all elements are considered to be triangles which is a practical approach for the ease of computations but it is also possible to have different shape elements such as quadrilaterals, hexagons, etc. An approximation of the potential on the whole surface is given by: N V (x, y) = V e (x, y) (3.37) where V e is the potential within element e and N is the number of elements. e=1 can be approximated using basis functions which are called element shape function. Polynomial approximations are used as basis functions to interpolate the potential V e 49

63 Figure 3.7: A triangular element used in FEM calculations. Local nodes are numbered as 1, 2, and 3. within element e. The most common polynomial function for a triangle element is: V e (x, y) = a + bx + cy (3.38) where a, b, and c are related to the geometry of the element and nodal potentials. Consider a typical triangular element as shown in figure 3.7. Potentials at each local node (V e1,v e2, and V e3 ), can be calculated using equation 3.38 as follows: V e1 1 x 1 y 1 a V e2 = 1 x 2 y 2 b V e3 1 x 3 y 3 c (3.39) If we solve this equation for a, b, and c in terms of coordinates of local nodes, then we can rewrite equation 3.38 as follows: V e (x, y) = 3 α ei (x, y)v ei (3.40) i=1 50

64 Similarly the source term f(x, y) in equation 3.36can be written as: 3 f(x, y) = α ei (x, y)f ei (3.41) i=1 where α ei are the shape functions considered as a set of basis function for interpolation. Each α ei is 1 for the corresponding local node i, and 0 at the other two local nodes and varies linearly between: α e1 = 1 [ ] (x e2 y e3 x e3 y e2 ) + (y e2 y e3 )x + (x e3 x e2 )y 2A α e2 = 1 [ ] (x e3 y e1 x e1 y e3 ) + (y e3 y e1 )x + (x e1 x e3 )y 2A α e3 = 1 [ ] (x e1 y e2 x e2 y e1 ) + (y e1 y e2 )x + (x e2 x e1 )y 2A (3.42) (3.43) (3.44) where A is the area of element e. Note that the above equations are equivalent to equation Equations 3.40 and 3.41 give the potentials and sources at any point within each element considering that the potentials and sources are known at all vertices. The functional form for the potential energy in the element e is given by: F (V e ) = 1 2 Ω [ V 2 + 2f e V e ] ds (3.45) The V e that minimizes F is the approximate solution of Poisson s equation. Substituting equation 3.40 and 3.41 results in: F (V e ) = i=1 3 [ V ei j=1 Ω ] α i. α j ds V ej + which can be written in matrix form as: 3 i=1 3 [ V ei j=1 Ω ] α i α j ds f ej (3.46) F (V e ) = 1 2 [V e] T [C (e) ][V e ] + [V e ] T [M (e) ][f e ] (3.47) where C (e) and M (e) are the local stiffness and mass matrices of element e respectively. The elements of these matrices which represent the coupling between the nodes of i 51

65 and j of element e can be calculated as: C (e) i,j = α i. α j ds (3.48) Ω M (e) i,j = α i α j ds (3.49) Ω The final step is to use each local 3 3 stiffness and mass matrices calculated above (C (e) and M (e) ) and assemble them to obtain global equation F (V ) = 1 2 [V ]T [C][V ] + [V ] T [M][f]. (3.50) In this equation, each index i relates to the node i in the global numbering scheme and not element-wise. In large problems, such as a discretized cardiac geometry, this results into a huge coefficient matrix [C] which is solved by iterative methods. The coefficient matrix [C] however is symmetrical and sparse since many nodes have no interaction with each other and the corresponding entry of matrix is zero. The method above, where F (V ) is minimized, is known as the Ritz variational formulation. In practice, a simpler form based on Galerkin s weighted residual method, is used [31]. First, given Poisson s problem, it is assumed that V and f can be approximated by linear combinations of N basis functions α i, chosen in a similar manner as the element shape functions discussed above. Both sides of the equation can be multiplied by some arbitrary weighting function and integrated over the spatial domain to obtain a new, equivalent equation; in Galerkin s method, the N basis functions are thus employed to generate new equations equivalent to the original problem: Ω α j (x, y) 2 V (x, y)ds = α j (x, y). V (x, y) = Ω Ω Ω α j (x, y)f(x, y)ds j = 1, 2,..., N (3.51) α j (x, y)f(x, y)ds j = 1, 2,..., N (3.52) 52

66 where the left-hand side has been simplified using integration by parts. When the approximate forms of V and f are substituted in the equation, results in the following: N [ i=1 α i α j ds Ω ] V i = N [ i=1 α i α j ds Ω ] f i j = 1, 2,..., N (3.53) Similar to Equation 3.47, this can be converted to matrix form and solved to obtain an approximation of the solution V (x, y). In the tissue simulations of this thesis, a robust tool for performing biophysical simulations (Cardiac Arrhythmia Research Package CARP) was used. CARP is developed by Vigmond et al. [32] and is designed to run in both shared memory and clustered computing environments. CARP solves bidomain elliptical and parabolic equations as well as monodomain equations. It consists of several packages like bidomain simulations, ionic model libraries, visualization of structured and non-structured grids which are dedicated to specific tasks. 2D and 3D simulations in this thesis were solved using monodomain modelling. Space discretization was implemented by mesher, a simple mesh generator for slabs and surfaces. Space elements are triangular or quadrilateral in the 2D simulations, and hexahedral or tetrahedral in the 3D simulations. 53

67 Chapter 4 Cardiac physiological characteristics measurements The majority of the contents of the following chapter has been previously published in [33] c [2012] IEEE and [34] c [2014] IEEE. 4.1 Chapter Specific Background Physiological characteristics of heart tissue such as fiber rotation, anisotropy, cell coupling, and fiber curvature are known to have major effects on wave propagation patterns in the heart. One way to obtain information about these structural properties is to measure epicardial potential patterns. There have been several studies designed to determine myocardial anisotropy and conduction velocity based on epicardial potential measurements. While some studies have accounted for the effect of fiber rotation [35], many have assumed that epicardial conduction patterns directly reflect epicardial fiber orientation [36, 37]. Known effects of fiber curvature [38] and transmural fiber rotation on epicardial surface wavefront propagation patterns [39, 40] have frequently not been accounted for. Cells are connected internally by gap junctions, which are mainly located at intercalated disks in healthy tissue [21], leading to a higher conductivity in the direction of myocardial fibers. Therefore, wave propagation is anisotropic and faster in the myocardial fiber direction (longitudinal) than in the direction across the fibers within the sheet (transverse) and across sheets (transmural). Also, fiber orientation is known to change from epicardium to endocardium [41, 42]. Therefore, anisotropy and fiber ori- 54

68 entation have to be considered in designing measurement approaches for conduction velocity. Taccardi et al [43] studied the effect of the myocardial structure on wave propagation patterns in canine left ventricle and showed that the activation isochrones resulting from epicardial stimuli differed from activation patterns when the wavefront was limited to only the epicardium. They also showed that both superficial and transmural fiber directions were reflected in potential distributions [44]. Finally, they analyzed how features of epicardial potential distribution were affected by pacing site, pacing depth, and time elapsed after the stimulus. Nygren et al. [45, 46] used Optical Mapping (OM) to record high resolution optical maps of activation times and conduction in the mouse heart. They showed that central stimulation resulted in an elliptical pattern demonstrating that conduction velocities are anisotropic. Knisely [47] used OM and showed that changes in transmembrane voltage depend on fiber orientation. However OM does not measure the transmembrane potential on the surface, but a weighted average of potential over the depth of the tissue due to light intensity decay in the tissue. Fluorescent photon scattering is known to distort OM. Bishop et al. [48] provided a model that accurately synthesizes the 3D photon scattering effect over the irregular geometry of the rabbit ventricles. They also studied the effect of photon scattering in optical signal distortion during ventricular tachycardia and defibrillation, and showed that photon scattering causes the difference in amplitude of optically recorded and simulated virtual electrode polarization induced by a defibrillation-strength shock [49]. 55

69 4.2 Cardiac Optical Mapping To study the electrical activity and impulse propagation in the heart, it is important to record the membrane potential by some means. There are different ways to measure electrical activity of the heart: A. Patch-clamp is a technique that allows the study of the ionic current or membrane potential of a single or multiple cells or ion channels. Sharp electrodes are also used to measure the electrical potential inside the cell membrane. B. Extracellular electrodes can be used to record electrical activity of the heart. Microelectrodes have high temporal resolution and can be useful to measure the electric field caused by the electrical activity of the heart [50]. A single electrode will record information of a single region of the heart and their use for large regions activity mapping results in the use of a vast number of electrodes. This method requires complex electrode arrays to achieve good levels of spatial resolution. C. In order to non-invasively measure the membrane potential of the heart rather than the electrical field caused by the heart (can be measured by extracellular electrodes), we can use optical mapping. Cardiac optical mapping is an alternative method that can record surface action potential with higher spatial resolution. A voltage sensitive dye is perfused in the heart and binds to the cell membrane. The dye fluoresces based on the membrane potential. In this thesis, optical mapping (OM) was used to record electrical activity of the heart from the surface. OM provides a direct membrane potential measurement with high spatial resolution in return of lower temporal resolution than electrode measurements. 56

70 4.2.1 Voltage-sensitive dye In this study Di-4-ANEPPS was used as the fluorescent dye due to its ability to follow the voltage changes across the cell membrane on a time scale of milliseconds. When Di-4-ANEPPS is illuminated at appropriate wavelength (500±25 nm), it emits a fluorescent signal proportional to the membrane potential of cardiac myocytes Imaging system Figure 4.1 shows the block diagram of the imaging system used in this study. Illumination is provided by a 250 W Quartz Tungsten halogen light source. The light is reflected off a mirror and filtered with a 500±25 nm band pass filter. The filtered light (green light) is then reflected off a dichroic mirror to the epicardial surface of the heart. The fluorescent light (red light) emitted from the cardiac surface is filtered through the dichroic mirror and then a long pass filter (>590 nm) and reaches the charge-coupled device (CCD) sensor. The shutter of the camera is controlled by the acquisition software and opens automatically before image acquisition and closes after recording. The camera is connected to a National Instruments PCI-1422 image acquisition board. 4.3 Methods Computational Model The Pandit [51] mathematical model of rat ventricular myocytes was used as the base single cell model. 2 2 centimeter tissue slabs were created with different thicknesses (1mm, 2mm, 4mm, and 1cm). The myocardial tissue included fiber orientation that varied continuously and linearly 120 degrees from epicardium to endocardium 57

71 Figure 4.1: Schematic diagram showing arrangement of the major components of the imaging system [52, 53, 54]. Simulations were run assuming a monodomain model using the Cardiac Arrhythmia Research Package [32] with a time step of 1µs and spatial discretization step of 100µm. The tissue was stimulated with a 50µA/cm 2 current density injected into a mm cube for 5ms at the center of epicardial surface. Simulations were run for all tissue thickness and the membrane potential of each point was recorded. The transmural conductivity (denoted by g n ) is defined as the conductivity between different layers of the tissue and is perpendicular to the plane of longitudinal and transverse conductivity directions. This conductivity affects wave propagation speed from epicardium to endocardium. Higher transmural conductivity results in more electrotonic loading from underlying layers of cardiac tissue on the epicardial surface. Coli Franzone et al. [53] considered transmural conductivity to be equal to one tenth of the transverse conductivity so the associated propagation velocity (θ n ) was in accordance with the findings reported by LeGrice et al. [6]. Therefore, to 58

72 determine the effect of transmural electrotonic interactions on the epicardial surface, simulations were run with different transmural conductivities (g n = g t, g t /2, g t /4, and g t /8) Data Processing Activation time was defined for each point as the moment when membrane potential crossed zero in the action potential upstroke. Isochrones were defined as contour plots of the activation time which represent the points that have the same activation time. Since the action potential propagates faster in the fiber direction, isochrones were expected to have an elliptic shape. A least square error algorithm was implemented in Matlab to fit ellipses to each isochrone. Coordinates of all the points (X and Y) in the isochrones are used to find the parameters in 4.1 A(X h) 2 + B(X h)(y k) + C(Y k) 2 = 1 (4.1) where h and k are the coordinates of the ellipse center. The angle of rotation, along with the major and minor axes of each ellipse, were determined. Conduction velocity vectors for each point were calculated using the gradient of the activation times as described in detail by Morley et al. [55]. Briefly, a plane was fitted by the least squares method to a 7 7 point neighborhood of the activation times to obtain a smooth approximation to the activation times in the neighborhood. The direction of the gradient of this approximating plane was taken to be the local direction of propagation, while the inverse of the magnitude of this gradient was used as an estimate of the local velocity [55]. This approach is similar to that used by Bayly et al. [56], with the exception that these authors fitted a quadratic surface, rather than a linear plane, to the activation times. These methods share the advantage that 59

73 they factor in activation times over a neighborhood in a least-squares sense, and thus can be expected to be relatively robust in the presence of measurement noise in a small number of points. This is in contrast to simpler methods such as the one proposed by Mazeh et al. [57], which relies on activation time measurements in only four or five points. Another robust approach for conduction velocity measurement is that of Kay and Gray [58]. However, this method depends on absolute measurements of membrane potential and is thus not directly applicable to optical mapping data (which only provides relative measurements of membrane voltage). Velocity vectors at the apices of the fitted ellipse (intersection between the long and short axes and the ellipse itself) were considered to be the longitudinal and transverse conduction velocities, denoted θ l and, θ t, respectively in this chapter. The maximum and minimum velocities along isochrones (not necessarily on the fitted ellipse) were also determined as an alternative estimate of longitudinal and transverse conduction velocities and are denoted θ max and θ min, respectively. In all cases, velocities were calculated using the method described above. The largest conduction velocity vector in any point along the isochrone was considered to be equal to the longitudinal conduction velocity, θ max. The minimum conduction velocity in any point along the isochrone was used as an estimate for transverse propagation velocity, θ min Estimation of Epicardial Fiber Orientation The angle of wave propagation was considered to be the angle of the major axis of the ellipse relative to the epicardial fiber orientation. This angle provides one possible estimate of epicardial fiber orientation. Figure 4.2 depicts an ellipse fit to an isochrone (in 1mm thick tissue) and its major and minor axes. Isochrones in thin tissues are not completely elliptical, but are skewed and have notches along the 60

74 sides. As a result, the best-fit ellipse is not completely aligned with the perceived long axis of the isochrone, potentially contributing to the error in this estimate (see Discussion). As an alternative, fiber orientation may be estimated based on the direction of maximum conduction velocity along a particular isochrone, i.e., at a particular time post stimulus. In this study, this time was selected to be 20ms to ensure that measurements were obtained well after the end of the stimulus pulse (to ensure that they represent conduction rather than the passive response to stimulation) and before the activation wavefront reached the edge of the tissue slab. In this alternative measure, fiber direction was considered to be the direction of maximum conduction velocity along the 20ms isochrones (θ max, as defined above) Figure 4.2: An isochrone (black solid line) and the corresponding least square fitted ellipse (dashed line). Major and minor axes are shown with a and b respectively. The angle of rotation is φ. The anisotropy ratio is equal to a/b. 61

75 4.3.4 Estimation of Anisotropy Ratio (AR) The ratio of longitudinal to transverse conduction velocity is a dimensionless parameter called the anisotropy ratio (AR). Three measures of AR were considered in this study: 1. The anisotropy ratio can be calculated based on the distance traveled by the activation wavefront, as the ratio of the major to minor axis of the best-fit ellipse to an isochrone (i.e., a b in Fig. 4.2). In this chapter, we used the 20ms isochrone unless otherwise noted. 2. Alternatively, the AR can be estimated as the ratio of the conduction velocities observed at the apices of the best-fit ellipse, i.e., the AR is estimated as θ l /θ t with θ l and θ t as defined above. 3. Finally, we considered a third estimate of AR based on the ratio of maximum to minimum conduction velocity along the 20ms isochrone (θ max /θ min ) Optical Mapping Model Fluorescent photon scattering distorts the recordings of cardiac the membrane potential [59, 60]. Excitation light intensity decays as it penetrates through the tissue and as it goes back out through the tissue to the camera. Due to penetration and emission, OM does not measure only surface membrane potential. The simplest way to model and implement this effect is to assume that the excitation light decays exponentially on the way in and that emitted light also decays exponentially on the way out, modeling the process as the product of two exponentials. After electrical simulations were completed, measured membrane potential with the effect of OM (V OM ) was calculated based on V OM = l 0 V m(x, y, z)w(z)dz l 0 w(z)dz (4.2) 62

76 where l is the tissue thickness, V m (x, y, z) is the membrane potential at each point, z is the depth below the surface, and w(z) is the weighting function which can be considered as a product of two exponentials for excitation (φ ex ) and emission lights (φ em ) derived from Baxter et al.[60]: w(z) = φ ex (z)φ em (z) (4.3) φ ex = 927e z/0.8mm 702e z/0.44mm (4.4) φ em = e z/1.34mm (4.5) The second term in equation 4.4 accounts for photons leaking out of the tissue near the surface [60]. Using equations 4.2 to 4.5 on the membrane potential of the tissue, the measured membrane potential on the epicardium was calculated offline for analysis Experimental Methods All experiments followed the guidelines of the University of Calgary Animal Care and the Canadian Council for Animal Care and were performed by Marcela Rodriguez. Rat hearts were isolated as described by Nygren et al. [46]. For OM protocols, the optical mapping dye Di-4-ANEPPS was added after a 20-minute normalization period. After normalization, the dye was perfused for 5 minutes, followed by a 10 minute wash out. Di-4-ANEPPS was diluted from a 10mM stock solution in DMSO (dimethyl sulfoxide) to a 1µM concentration in Krebs-Henseleit buffer solution. To override sinus activation of the heart pacing with a pulse cycle of 200ms was used. The pacing current was initiated at 1µA, and increased until the tissue was activated, overriding sinus rhythm. Once the minimum pacing current needed (threshold) was determined, the pacing current was set to twice threshold for the remainder of the experiment. 63

77 The objective is to use a low stimulus to capture a small area of the cardiac tissue so the isochrones reflect the active propagation in the tissue rather than the passive depolarization caused by the stimulus. Twice threshold was considered large enough to excite the tissue and small enough not to depolarize a large portion of the heart passively. If the current became insufficient (happened once) to initiate tissue activation then the current was again increased until activation occurred. The threshold might change due to the changes in the tissue health and also due to the changes in the electrode contact to the tissue. The tissue health was constantly monitored by monitoring the heart temperature, perfusion pressure, and the electrocardiogram. The electrode contact to the heart surface is more likely to change in a short time period during the experiments due to the heartbeats. Six male rats (n=6) were used for the experiments to investigate the rotation of isochrones qualitatively. The hearts were cut open after the experiments and tissue thicknesses of left ventricle were measured under the microscope. The average tissue thickness was 3.5 ± 0.3mm. However this measure is not accurate since the tissue has cleavages and does not have the same thickness over the heart and left ventricle. The recorded images were processed off line to obtain the membrane potential of each pixel. Light intensity mapped to membrane voltage and action potentials were obtained. Activation time for each pixel was detected and averaged over 10 cycles. Ellipses were fitted to each isochrone using the same method used for simulation data. The angle of each ellipse was found and the angle difference of the isochrones was calculated for each activation time. 64

78 4.4 Results Simulation Results Observations based on isochrones Figure 4.3 shows the contour plot of activation time (isochrones) for different tissue thicknesses. Direction and speed of wave propagation can be seen from the isochrones. Fibers on the epicardial surface are oriented in the horizontal direction in all slabs. It can be seen from Fig. 4.3 that the direction of wave propagation is not aligned with epicardial fiber orientation. This misalignment is more evident in thinner tissues. It can also be noted that for a specific thickness, subsequent isochrones are not oriented in the same direction and the angle of propagation increases with time post-stimulus. Figure 4.3: Activation time isochrones on the epicardial surface. Fibers are oriented in the horizontal direction. Isochrones are less aligned with the fiber direction in thinner tissues. 65

79 Figure 4.4 shows the angle of wave propagation, as determined by the orientation of the long axis of the best-fit ellipse to the 20ms isochrone, for different values of transmural electrical coupling. The angle of propagation increased with the time elapsed from the stimulus application. The difference between the angle of propagation and fiber direction was smaller for thicker tissues. Also, as electrical coupling between layers decreased, this angle discrepancy decreased. It can be noted that for the tissues with the lowest transmural conductivity (g n = g t /8) the wavefront tended to propagate in a constant direction (the slope of the blue line is almost zero in all thicknesses). Figure 4.4: Angle of wave propagation versus time for different transmural conductivities and thicknesses. As electrical coupling between layers decreases (g n decreases), impulse propagation is more aligned with the epicardial fiber orientation. The angle of propagation is increasing with the time elapsed since the stimulus. This increment is less for thicker tissues and smaller couplings. Angle of propagation was estimated as the orientation of the long axis of the best-fit ellipse to the activation isochrone at the time indicated post stimulus. 66

80 Figure 4.5 shows angle of wave propagation 20 ms after stimulation for different thicknesses and different transmural conductivities (different g n /g t ratios). Lower g n /g t ratios resulted in less misalignment of wave propagation with epicardial fiber orientation for all thicknesses. Figure 4.6 shows the diagram of measured anisotropy ratio, determined as the ratio of the long axis to the short axis of the best-fit ellipse, versus tissue thickness for different transmural conductivities. With thicker slabs, the AR increased and approached the true AR value of tissue with no fiber rotation [61]. It can be noted from Fig. 4.6 that less transmural conductivity caused less electrical interaction between the tissue layers. Therefore, measurements in cardiac tissues would be more accurate under these conditions. However, errors are still substantial for thin tissues. Figure 4.5: Angle of wave propagation versus tissue thickness for different transmural conductivities (based on the best-fit ellipse, 20ms after stimulation). 67

81 Figure 4.6: Anisotropy ratio versus tissue thickness for different transmural conductivities. Anisotropy ratios are measured based on the best-fit ellipse to the activation isochrone 20ms after the stimulation. Measurements are more accurate for thicker tissues. Electrical coupling between the layers are less effective for thicker tissues Conduction velocities Figure 4.7 shows the longitudinal and transverse conduction velocities (θ l and θ t ) for different tissue thicknesses and transmural conductivities. Nominal values of longitudinal and transverse conduction velocities are 51 and 17cm/s respectively (θ l 3-fold higher than θ t ). These values are calculated for a tissue with no fiber rotation. Estimated longitudinal conduction velocities are smaller than the nominal value of θ l and transverse ones are greater than the true value of θ t. As tissues become thicker, measurements for both longitudinal and transverse velocities approach the true value. The effect of transmural conductivity on measurements is clearly visible in Fig Measurements from the tissues with smaller transmural conductivity are closer to the nominal values for both θ l and θ t. 68

82 Figure 4.7: Transverse (θ t ) and longitudinal (θ l ) conduction velocity versus tissue thickness for different transmural to transverse conductivity ratios. As tissues get thicker and transmural conductivity decreases, measurements approach the nominal conduction velocity (which is 17 and 51 cm/s for transverse and longitudinal conduction velocity respectively). θ l and θ t were obtained at the apices of the 20ms activation isochrones as described in Section Alternative estimates based on conduction velocities The estimates of epicardial fiber orientation and anisotropy ratio presented in the previous section were determined based on the best-fit ellipse to the isochrones. To determine whether the alternative measures based on conduction velocities discussed in Section 4.3 can be expected to produce more accurate estimates, the same simulation results were subjected to additional analysis. Table 4.1 shows the values of 69

83 Thickness 1mm 2mm 4mm 1cm Ellipse AR θ l /θ t θ max /θ min φ (degrees) Ellipse θ max Table 4.1: Measured epicardial fiber direction and anisotropy ratio for different methods: direction of 1) fitted ellipse (Ellipse) and 2) maximum conduction velocity (θ max ). AR is calculated based on: the ratio of 1) major and minor axis of fitted ellipse (Ellipse), 2) longitudinal to transverse conduction velocities of fitted ellipse (θ l /θ t ), and 3) maximum to minimum conduction velocity (θ max /θ min ). estimated AR for the different methods considered. Table 4.1 shows that using the θ max /θ min ratio results in the most accurate measurements (see Discussion Section 4.5). Similarly, Table 4.1 shows that measurement of fiber orientation based on θ max yields a more accurate estimate than that based on the long axis of the best-fit ellipse to the isochrone. However, attempts to apply these alternative measures to experimental data demonstrated that they are highly sensitive to experimental noise and thus unsuitable for practical application. The remainder of the analysis in this chapter is therefore based on fitting of ellipses to isochrones Effect of optical mapping The measured membrane potential on the epicardium was calculated with the effect of OM. Angle of propagation, anisotropy ratio, and conduction velocities were measured 70

84 and compared to the ones with no OM effect considered. Table 4.2 shows a complete comparison between all the parameters with and without considering OM. Table 4.2 shows that the OM caused more error (compared to the nominal AR of 3) in AR measurement for thicker tissues, i.e., for the 1mm slab, AR is measured as 1.63 and 1.72 with and without effect of OM respectively. These values are 2.76 and 2.94 for the 10mm slab (see Discussion section 4.5). This shows that OM is the dominant cause of error in the thick tissues while the fiber rotation causes the major part of error in the thin tissues. The angle of wave propagation is also affected by the OM. The error is increased from 24 degrees to 32 degrees for the thinnest tissue and from 2 degrees to 5 degrees for the thickest tissue. OM did not significantly change the measurements of θ l (2-3% additional error) for any tissue thickness. Transverse conduction velocities were affected more by the OM (1-10% additional error) but the main cause of error was the fiber rotation. The difference between simulations that did or did not account for OM was again more pronounced in the thicker slabs. However, for all parameters considered, the errors observed in thick tissues (due to either fiber rotation or OM) are relatively small and unlikely to substantially affect experimental conclusions. 71

85 AR φ wit(degrees) θ l (cm/s) θ t (cm/s) OM NOM OM NOM OM NOM OM NOM 1mm mm mm cm Nominal Table 4.2: Anisotropy ratio (AR), fiber orientation (φ), longitudinal and transverse conduction velocities (θ l and θ t ) with and without considering the effect of optical mapping (shown in the table by OM and NOM respectively). It is evident that optical mapping causes more error in measurements Effect of photon scattering Equation 4.5 considers that the emission light travels in a linear path perpendicular to the tissue to hit the surface. Hence, the calculated V OM is a weighted average over the depth of the tissue. Another approach to model emission light is to consider the effect of photon scattering. The emission light spread from the emission source to reach the epicardial surface. Figure 4.8 shows the difference between these two methods. 72

86 Figure 4.8: Different approaches to model emission light; exponential decay without (a) and with (b) the effect of photon scattering. The effect of photon scattering can be modelled by adding Γ(z, ρ) to be the pointspread function, describing the flux Γ of photons emitted by the fluorescent dye at point (x, y, z ) and exiting the tissue at point (x, y), with ρ the radial coordinate on the imaged xy surface, centered around (x, y ), i.e. ρ = (x x ) 2 + (y y ) 2. The fluorescence optical signal V OM recorded from the surface point (x, y) can then be obtained by V OM (x, y) = V V m ( r ).Φ e ( r ).Γ(z, ρ).d r (4.6) One possible estimation of the spread function Γ is spherical photon scattering. Therefore Γ(z, ρ) can be written as: ( Γ(z, ρ) = exp z 2 + ρ 2 δ ), (4.7) where δ = 1.34 mm is the attenuation constant of emission light in the cardiac tissue [62, 39]. If the Eqn. 4.7 is used to calculate the measured membrane potential V OM on the epicardium (z = 0), then Eqn. 4.7 can be re-written as: ( Γ( r ) = exp (x x ) 2 + (y y ) 2 + (z z ) 2 ) ( r = exp δ δ ). (4.8) 73

87 This equation is similar to Eqn.4.5 except that the variable z is substituted with r to model the scattering effects of emitted fluorescent light. Measured V OM on the epicardium was calculated based on the above equations. The results are shown in Fig.4.9 below. This figure clearly demonstrates that the addition of photon scattering does not affect the results significantly. Isochrones are completely parallel to each other and anisotropy and conduction velocity measurements are thus comparable. Figure 4.9: Isochrones (5 ms intervals) based on the simulations of optical mapping without the scattering effect ( OM, dashed lines) compared to isochrones obtained for the same tissue using the photon diffusion approach ( OM+Scattering, solid line). The times corresponding to each isochrone have been offset slightly between the two simulations to allow a side-by-side comparison. 74

88 Effect of spatial discretization To study the effect of space discretization, we have repeated three simulations with a spatial discretization step of 50µm in all three directions (rather than the 100µm step presented above). We chose a tissue thickness of 1 mm to ensure that we addressed the worst-case scenario in which the effects of transmural fiber rotation are maximal. Three scenarios were considered: 1) Baseline refers to the case without transmural fiber rotation and with transmural (perpendicular to the epicardial surface) conductivity equal to transverse (parallel to the epicardial surface) conductivity (g n = g t ). 2) g n = g t refers to the case with transmural fiber rotation and with transmural conductivity equal to transverse conductivity (g n = g t ). 3) g n = 1/8 g t refers to the case with transmural fiber rotation and with transmural conductivity reduced by a factor of 8 compared to transverse conductivity (g n = 1/8 g t ). The results of these simulations are summarized below. Table 4.3 reports the results for angle of propagation measurements. The anisotropy ratio is presented in table 4.4. The effect of spatial discretization on longitudinal and transverse CV is represented in tables 4.5 and 4.6 respectively. Discretization Baseline g n = g t g n = 1/8 g t 100µm µm Table 4.3: Angle of propagation for different spatial discretization 75

89 Discretization Baseline g n = g t (cm/s) g n = 1/8 g t (cm/s) 100µm (59%) 2.2 (76%) 50µm (61%) 2.1 (75%) Table 4.4: Anisotropy ratio for different spatial discretization (percentages are percentage difference from baseline) Discretization Baseline g n = g t (cm/s) g n = 1/8 g t (cm/s) 100µm (91%) 48.2 (94%) 50µm (91%) 49.8 (95%) Table 4.5: Longitudinal conduction velocity for different spatial discretization (percentages are percentage difference from baseline) These results demonstrate that the values of these four measures are affected by the spatial discretization step. This effect is more pronounced for transverse conduction velocity under conditions of strong transmural uncoupling (g n = 1/8 g t ). However, the results show that even under these extreme conditions, the difference in transverse conduction velocity between the two spatial step sizes is 5% or less. Numerical simulations always involve a trade-off between computational speed and accuracy. Due to the 8-fold increase in the number of elements for the 50µm simulations (compared to 100µm), the three simulations with 50µm discretization presented here each required nearly 40 hours of computer time (sixteen core Intel CPU 2.8 GHz, 32 GB RAM). This would increase even further for thicker tissues. 76

90 Discretization Baseline g n = g t (cm/s) g n = 1/8 g t (cm/s) 100µm (261%) 22.5 (139%) 50µm (251%) 23.6 (129%) Table 4.6: Transverse conduction velocity for different spatial discretization (percentages are percentage difference from baseline) Experimental Results Figure 4.10 shows an activation map of a rat heart in the left panel. The right panel shows the measurements of the angle of rotation of each isochrone versus time elapsed after stimulation. The time scale starts 7ms after stimulus application to allow accurate detection of activation and wave propagation. Since the exact epicardial fiber direction is unknown, the angle difference is calculated by finding the angle of the fitted ellipse relative to the angle of the 7ms isochrone. The rotation appeared to continue at a similar rate, but isochrones reached to the boundary of the field of view after 12ms and made it difficult to measure the angle of propagation reliably. The angle of wave propagation changed with elapsed time, consistent with the simulation results in Fig Experiments were performed on rat left ventricle with a thickness of 3.5 ± 0.3mm. The change in angle difference between the 7ms and 10ms isochrones was 6.8 ± 2.1 degrees. The change in angle was 6 and 2.3 degrees in simulation results with the OM effect included for 2mm and 4mm tissues. The experiments were designed to qualitatively observe the rotation of the isochrones. The experimental results are closer to the simulation results for the 2mm tissue rather than to the 4mm tissue. This can be caused by the significant measurement error in how the thickness of the tissue was estimated. The cardiac tissue does not 77

91 have a uniform thickness in the left ventricle and endocardium has irregular formation of trabeculae branches which signify the measurement errors. However, rather than the tissue thickness, intramural conductivity (g n ) affects the angle of propagation as shown in Figure 4.4. Therefore, even accurate measuring of the tissue thickness will not provide precise information about the isochrone rotation since the changes in the g n is equivalent to the changes in the tissue thickness and the exact value of g n in the experiments is unknown. (a) (b) Figure 4.10: (a) Activation map of a rat heart. The isochrones are described by black lines. (b) Angle difference of wave propagation versus elapsed time after stimulus. The angle of wave propagation has an increasing trend as the time passes after the stimulation (n=6). Simulation results for 2mm and 4mm tissues are also provided. The angle of wave propagation was determined based on the orientation of the long axis of the best-fit ellipse to the activation time isochrones at the time indicated post stimulus. Measurement of conduction velocities for experimental data based on the θ max method, is sensitive to the noise. Figure 4.11(a) illustrates how noise in experimental data can cause significant variability in measurements based on the direction of max- 78

92 imum conduction velocity. We cannot rule out that further refinement of this method to smooth the results might improve this situation. However, it is worth noting that the velocity estimates are based on a 7 7 pixel neighborhood (about mm with the magnification used for the experimental results shown here) and thus already contain a significant element of smoothing. This method uses a moving average technique by moving a 7 7 window over the image and substituting the middle pixel by the average of the window. (a) (b) Figure 4.11: (a) Experimentally obtained activation isochrones from a rat left ventricular free wall and the corresponding local velocity estimates (arrows, obtained based on a 7 7 pixel neighbourhood centered on the pixel of interest) in each pixel. (b) For easier visibility, the figure has been edited to show only three isochrones and the corresponding velocity estimates along each isochrone. Note that velocity estimates near the apex of the isochrone (highlighted) can vary significantly in direction between adjacent points. Using the direction of the largest local velocity as an estimate of fiber orientation can thus lead to significant variability from isochrone to isochrone. 79

93 4.5 Discussion Effect of fiber rotation and wall thickness It is clear that the cardiac tissue transmural fiber rotation has a significant effect on the wave propagation patterns observed on the epicardium [61]. Thinner tissues have fewer layers so there is a larger angle difference between layers. Electrical coupling between nonaligned layers causes surface wave propagation not to follow fibre orientation. This misalignment is more significant in thinner tissues due to a larger angle difference between the layers. Figure 4.2 illustrates an ellipse fitted to the 20ms isochrone in the 1mm-thick tissue (largest error). The fitted ellipse is not exactly directed to the direction of wave propagation. The reason is in notches in the shape of isochrone which causes the deviation between the direction of the isochrone and the ellipse. However, this mismatch actually reduces the error, since the fitted ellipse is better aligned with the epicardial fiber direction than the perceived long axis of the isochrone. As the thickness of the tissue increases, adjacent layers have a smaller angle difference. Deeper layers are less electrically coupled to the epicardium and, thus, have a smaller effect on surface wave propagation. This is demonstrated in Figure 4.3. Figure 4.4 shows that the angle of propagation not only depends on thickness but also varies with time after stimulation. It takes some time for activity to propagate through the tissue to reach deeper layers. As the epicardial stimulus is delivered, the electrotonic loading from deeper layers that have yet to activate initially limits the epicardial conduction velocity. As deeper layers activate, the current contributed by the deeper layers interacts with epicardial conduction and the wavefront observed on the epicardium begins to rotate toward the fiber orientation of deeper layers. This 80

94 causes the epicardial wave front to change direction over time. The angle of propagation approaches steady state as the wavefront propagates through the whole tissue. The experimental recordings confirmed that the isochrones rotate with time after the stimulus application (Fig. 4.10). These recordings are qualitatively consistent with the simulations as depicted in Fig The simulations appear to underestimate the size of this response, i.e., the results of a simulation using a 2 mm thick tissue agree quantitatively with experimental results obtained in 3.5 mm thick rat left ventricular free wall. Possible reasons include the difference between the idealized model with uniform epicardial fiber orientation and the real heart, which most likely exhibits some change in epicardial fiber orientation over the field of view. It is also possible that the thickness measurement over-estimated the left ventricular wall thickness, as these measurements were obtained by slicing the heart open after the experiment, which may have allowed the heart to contract somewhat compared to its state during perfusion Effect of transmural conductivity Another parameter that affects electrotonic loading is the amount of electrical coupling between the layers, which is described by transmural conductivity (g n ). Lower transmural conductivity results in a weaker coupling between the layers and therefore, less electrotonic effect from the layers beneath the epicardium. As depicted in Fig. 4.4 and Fig. 4.5, in all thicknesses, tissues with lower transmural conductivity have smaller angle of deviation from the epicardial fiber direction. However, even in the extreme case of g n = 1g 8 t, thinner tissues still show substantial deviation. The effect of transmural conductivity is more pronounced with time. As time passes and other layers become excited, underlying layers have more electrotonic loading effect on the 81

95 epicardium so the effect of transmural conductivity is more visible. Reducing transmural conductivity is mathematically equivalent to increasing tissue thickness. It is therefore not surprising that the effects of changing transmural conductivity parallel the effects of changing tissue thickness Effect of optical mapping Table 4.2 shows a complete comparison between all the parameters with and without considering the effect of OM. It can be seen from Table 4.2 that when taking OM effects into account, measurement errors in structural parameters of cardiac tissue (AR, angle of propagation, θ l, and θ t ) are exacerbated due to light decay effects. Therefore, in general, OM effects increase error in measurements. It can be also noted from Table 4.2 that for thicker tissues, the dominant cause of error is OM and in thinner tissue, the dominant cause of error is electrotonic loading from underlying layers. In 1mm thick tissue, OM has a negligible effect on the measurements and the measurement error arises primarily from electrotonic effects due to fiber rotation. In a 1cm slab, fiber rotation causes small error when OM is not considered. Accounting for OM increased measurement error significantly. This is due to the fact that in thin tissues, electrotonic loading effects are stronger and, also, photons decay less and recorded light intensity is affected more by the light intensity of underlying layers Implication for experimental measurements Distortion of wave propagation causes errors in estimates of epicardial fiber direction, as well as in measurements of conduction velocities and anisotropy ratio [61]. Our simulations and experimental observations show that the error in the estimated fiber 82

96 orientation increases as time elapses post-stimulus. Measurements should therefore be done as soon as possible after stimulation to ensure that electrotonic effects are minimized. Simulation results confirmed that this observation applies not only to the estimated fiber orientation, but to measurements of AR as well (whether based on the ratio of long to short axis of the fitted ellipse or based on conduction velocities). However, it is important to recognize that measurements obtained during, or immediately after, the stimulus pulse mainly reflect passive depolarization due to the injected stimulus rather than true propagation of the action potential. Also, during the early stages of propagation, there is a significant component of transmural propagation (perpendicular to the plane of imaging) as deeper layers are activated. For both these reasons, sufficient time must be allowed between the stimulus pulse and measurement of fiber orientation, conduction velocities, and AR. The nominal AR value of the tissue in Fig. 4.6 is equal to 3, calculated in a tissue with no fiber rotation. Fitting ellipses to the activation time isochrones and considering the direction of propagation equal to the direction of those ellipses, not only yields an inaccurate estimate of the fiber orientation (Fig. 4.3, 4.4, and 4.5) but also results in an error of the AR (Fig. 4.6). As previously explained, when transmural conductivity decreases, epicardial wave propagation is less distorted. Therefore, AR measurements are more accurate for tissues with smaller transmural conductivity, especially for thicker tissues. Measurements in thinner tissues are more accurate when transmural conductivity decreases but there is still significant error (even in the extreme case of g n = 1g 8 t) due to high electrical coupling. Errors in 1cm thick tissue are not large enough to be considered physiologically relevant and are presented in this chapter for the sake of comparison and to support the observation that fiber rotation has more effect in thinner tissues. However, errors in thinner tissues are 83

97 relevant to experiments in rat or mouse ventricles. Measurement of conduction velocities is also affected by the tissue fiber rotation. As the fitted ellipse is not aligned with fiber orientation on the epicardium, measured velocities are not in the longitudinal and transverse directions either so the measured θ l has a component of transverse conduction and is smaller than the true θ l. The measured θ t has a component of longitudinal conduction and is larger than the nominal θ t. The measured θ t in thinner tissues with higher g n has significant error that can be attributed to the notch in the isochrones of activation maps for these tissues. This notch is clearly shown in Fig. 4.3, in 1 and 2mm-thick tissues, and has been observed in previous studies [52, 53]. The larger the notch is, the less accurate are the measured transverse conduction velocities. In an attempt to reduce the errors incurred in estimating fiber orientation and AR experimentally, we investigated alternative measurement approaches based on conduction velocity measurements rather than isochrones. In particular, estimates based on maximum and minimum conduction velocities along an isochrone (20 ms in this study), appeared to yield significantly improved estimates of fiber orientation (based on the direction of θ max ) and AR (based on the ratio θ max /θ min ) when applied to simulated (noise free) data. However, in practice, these measures proved to be too sensitive to experimental noise and low pixel resolution to be of practical use. Among the methods considered in this work, our recommendation is thus that experimental measurements of fiber orientation and AR should be based on the best-fit ellipse to a suitable activation isochrone. The measurement errors identified should be taken into account when interpreting these observations. We cannot rule out that alternative methods, providing a smoother estimate of conduction velocities, may yield a better estimate of fiber orientation and/or AR. However, it is worth noting that our estimate 84

98 of conduction velocities, being based on a 7 7 pixel (approximately mm) area, already employ significant smoothing. Our results also showed that reduced coupling between the layers (in the direction normal to the epicardial surface) results in more accurate measurements. One may therefore be tempted to use an uncoupler (e.g. Heptanol) during experiments. However, application of a gap junction uncoupler will clearly affect coupling in all directions. In addition, the effect of an uncoupler may be somewhat different in different directions since the conductivity along the fibers is due to both myoplasm and gap junction conductivity, while across the fibers, the conductivity is mainly due to the gap junctions. Thus, using an uncoupler is not likely to improve estimates of AR, θ l, θ t, and fiber orientation in a predictable manner. It is also worth noting that using near-infrared dyes as described by Walton et al. [63] will cause increased optical contributions from the underlying layers in the recordings and, therefore, will result in less accurate measurements in this context. These dyes do have useful applications due to their ability to penetrate deeper into the tissue. However, for the measurements discussed in this chapter, a shorter penetration depth (shorter-wavelength dyes) is preferable Study limitations The experimental optical mapping data has limited spatial and temporal resolution due to the characteristics of the optical system which are not considered in the simulations. Derived images also have motion artifacts that can cause inaccuracy in the data processing step, in particular for the repolarization phase. Since our calculation and measurements are based solely on the activation times, the impact of these motion artifacts on our results expected to be minimal. 85

99 The 100µm spatial step size used in this work is a trade-off between computational speed and accuracy as in any numerical simulation. To assess the sensitivity of the results presented in this chapter to the spatial step size, we repeated three simulations with a refined step size of 50µm. We chose a tissue thickness of 1mm and repeated the simulations for a) no fiber rotation and g n = g t, b) with fiber rotation and g n = g t, and c) with fiber rotation and g n = 1g 8 t. This ensured that the worst-case scenario, with the most pronounced transmural voltage gradients (case c), was included. Values for angle of propagation, anisotropy ratio, and conduction velocities remained within 5% of the values obtained with 100µm step size, confirming that the conclusions of this chapter do not depend on the exact spatial discretization. As expected, the largest differences were observed for transverse conduction velocity in case c). Our model of tissue does not include all the structural details such as curvature, cleavage planes, and Purkinje system. The tissue slabs used in our simulations are rectangular parallelepipeds which have no curvature. Cleavage planes, which are located between muscle layers, are not included in the model. Purkinje fibers, which are located in ventricular walls and conduct action potential more quickly, are not included in our model. These limitations can be addressed by considering a model with more detail but this would result in higher computational cost. 4.6 Conclusion This chapter showed that fiber rotation causes error in conventional measurement methods used to estimate fiber orientation and anisotropy ratio. These errors are affected by the tissue thickness and intramural conductivity and are increased by optical effects during optical mapping recordings. Despite significant errors in thin 86

100 tissue preparations, the most appropriate method (at least of the ones considered here) for estimating fiber orientation and AR experimentally appears to be least-squares fitting of an ellipse to an activation time isochrone. 87

101 Chapter 5 Diabetes and Vulnerability to Arrhythmias The majority of the contents of the following chapter has been prepared to be submitted in c PLOSone 5.1 Chapter Specific Background In recent years many studies have been performed of the electrophysiological differences between healthy and diabetic cardiac myocytes [64, 65, 66] and the vulnerability of diabetic hearts to cardiovascular disease such as arrhythmias [67, 68]. In the setting of type I diabetes, the heart is often considered to be at higher risk of developing arrhythmias or other cardiovascular diseases such as myocardial ischemia [69]. Myocardial ischemia results in changes in biochemical and electrophysiological characteristics of the heart that can lead to an unstable electrical substrate which can be able to initiate and sustain arrhythmias. Ischemia can also cause infarction leading to unexcitable tissue and conduction block, which increases the chances of reentrant activities in the heart that is arrhythmogenic. Many studies have shown that diabetic hearts are more susceptible to ischemic injuries compared to healthy models [70, 71, 72, 73]. However, some studies indicated no difference [74, 75, 76], or even less sensitivity [77, 78, 79, 80, 81, 82] of diabetic hearts to ischemia in comparison to healthy hearts. Cardiac ischemia results in the opening of ATP dependent potassium channels (KATP) which is due to the lack of ATP in the heart [83]. Activation of the KATP current results in a significant 88

102 decrease in action potential duration (APD) [84]. Shortening of the action potential could introduce an additional risk of developing arrhythmias due to shortening of the refractory period. Some studies directly discussed the effect of diabetes on susceptibility to arrhythmias. However, contradictory results were presented by different groups. For example on one hand, Ravingerova et al. [79] compared the susceptibility to ventricular arrhythmias in rats with prolonged duration of STZ-induced diabetes using both open chest rats in vivo and isolated Langendorff-perfused hearts. Following 8 weeks of STZ-induced diabetes, they subjected both models to 30 minute regional zero flow ischemia induced by occlusion of the left anterior descending (LAD) coronary artery. Their observations suggested that rat hearts with chronic diabetes are less sensitive to ischemic injuries and less vulnerable to ventricular arrhythmias in both models. On the other hand Zhang et al. [67] performed whole cell patch-clamp studies in a rabbit model of alloxan-induced diabetes mellitus. They demonstrated changes in ionic currents and observed a 20% prolongation of action potential duration. They also showed that the QT prolongation in a diabetic rabbit model resulted in occurrence of arrhythmias, mainly of ventricular tachycardia (VT) lasting longer than 30s, which was otherwise absent in their control model. These observed VT lead to ventricular fibrillation (VF) causing sudden death. In this chapter, the effects of diabetes on reentrant arrhythmias are studied using computer modelling. Reentry is a circular or spiral propagation of an action potential which is happening due to different refractory period and conductivities of different pathways. Reentry is a common cause of arrhythmias such as ventricular tachycardia [85] and atrioventricular node reentry [86]. It can occur when a conduction path is partly slowed down or completely blocked in one direction (uni-directional block) due 89

103 to reduced electrical conductivity. Conductivity of a tissue is dependent on intercellular protein channels called connexins located on gap junctions. A major component of cardiac gap junctions is connexin 43 (Cx43) which is known to migrate from cell ends to cell sides in diabetic hearts (known as connexin lateralization) [21]. This change from regular arrangement of gap junctions affects propagation, increasing risk of arrhythmias [87, 88]. The gap junctions can be affected by a variety of factors, including pharmaceuticals [89] and diseases [90]. Seidel et al [91] simulated the connexin lateralization in a rat ventricle computer model and showed that the susceptibility to conduction block at tissue expansion becomes significantly higher as a negative effect of connexin lateralization. Diabetes also affects the calcium handling inside the cardiac cell [92, 93]. Changes in the sarcoplasmic reticulum (SR) calcium load ([Ca 2+ ] SR ), decreased rates of calcium depletion from and uptake to the SR, and the depressed efflux of calcium through Na + /Ca 2+ exchanger current (NCX) can alter the intracellular calcium concentration ([Ca 2+ ] i ). However, these abnormalities cause opposite effects on [Ca 2+ ] i. Decreased [Ca 2+ ] SR and reduced depletion of Ca 2+ from SR reduce [Ca 2+ ] i while reduced uptake of Ca 2+ to SR and regulation of NCX results in higher [Ca 2+ ] i. High levels of [Ca 2+ ] i can result in spontaneous calcium release current into the sarcoplasm through calcium induced calcium release (CICR) mechanism. This can increase depolarizing NCX current which itself is arrhythmogenic and can lead to early (EAD) or delayed after depolarization (DAD). Rabbit ventricular action potentials are significantly longer than those of rat ventricular myocytes. Unlike rabbit myocytes, rat ventricular myocytes lack a welldefined plateau phase during repolarization. This is due to relatively large outward repolarization currents in rat that bring the membrane potential rapidly close to rest- 90

104 ing potential [94]. The plateau phase in rabbit AP is due to the balance between inward I CaL and outward delayed rectifier potassium current. In this chapter, we simulated the vulnerability to reentrant arrhythmias in healthy and diabetic myocardium for rat and rabbit models. Connexin lateralization was implemented in diabetic tissue and the effect of I KATP opening in ischemic conditions was also discussed for both control and diabetic models. 5.2 Methods Rat Model The Pandit mathematical models of rat ventricular myocytes were used for healthy [51] and diabetic [95] single cell models. A summary of the differences between these models is provided in Fig Single cell action potentials were simulated in MAT- LAB. A 1cm 1cm two dimensional tissue was created using CARP (cardiac arrhythmia research package [32]) Rabbit Model In this chapter we used a rabbit ventricular action potential mathematical model of cardiac single cell from Mahajan et al. [96] as a baseline control model. Changes from Zhang et al [67] were applied to this model and single cell simulations were performed with MATLAB Summary of the rabbit mathematical model Figure 5.1 shows a schematic diagram of the rabbit model from Mahajan et al [96]. Sodium current (I Na ) and L-type calcium current (I CaL ) are the only inward ionic channels. I CaL is modelled by a seven-state Markovian mechanism. Outward potas- 91

105 sium currents consist of fast and slow transient outward currents (I to,f and I to,s respectively), slow and rapid delayed rectifiers (I Ks and I Kr respectively) and inward rectifier current (I K1 ). The sodium-potassium pump is also included in the model. A basic calcium cycling mechanism is included in the model. Ca 2+ enters the cell in the dyadic junction through I CaL. The dyadic junction is a subset of submembrane space. Sodium-calcium exchanger (Na + /Ca 2+ ) and I Na are connected to the submembrane space. Ca 2+ depletes from the junctional sarcoplasmic reticulum (JSR) to the dyadic junction by J rel. Ca 2+ is uptaken to the network sarcoplasmic reticulum (NSR) by J up. Some Ca 2+ ions leak from NSR to cytoplasm, which is modelled by J leak. Figure 5.1: Schematic diagram of the rabbit model from Mahajan et al [96], featuring whole-cell model showing basic elements of Ca cycling machinery and membrane ion currents, as well as a seven-state Markovian model of the L-type Ca channel. Adapted under a Creative Commons Attribution 3.0 Unported license; Online: License: 92

106 Changes in diabetic rabbit Zhang et al [67] performed whole cell patch-clamp studies in a rabbit model of alloxaninduced diabetes. They observed a 20% increase in the action potential duration (APD) of diabetic heart and changes in several ionic currents in diabetic rabbit single cell. Transient outward potassium current (I to ) reduced by 60%, rapid and slow delayed rectifier K + currents reduced 70% and 40% (I Kr and I Ks ) respectively. However, they did not observe any changes in time-dependent kinetics of these currents. L-type calcium current (I CaL ) peak amplitude was decreased by 22% and inactivation kinetics were slowed resulting in a total reduction of 15% in I CaL. Other ionic currents and kinetics did not change such as the inward rectifier K + current (I K1 ) and sodium current (I Na ). Calcium handling abnormalities such as those affecting sodium calcium exchanger current (NCX), the uptake current (J up ) via SERCA pumps in the sarcoplasmic reticulum (SR) and the release of Ca 2+ from SR were also included in the model as discussed in [93]. A summary of these changes are provided in Fig ATP-dependent potassium channels (I KATP ) The ATP-dependent potassium channels are inactive in healthy cells, but an outward current through them increases as the ATP levels decrease [83]. The formulation for this current was acquired from Shaw and Rudy [84], and the current was added to the single cell model implementation. The amplitude of ATP-dependent potassium channel (I KATP ) was controlled by the conductance value, g KATP. Single cell simulations were performed in MATLAB to study the effect of I KATP opening on healthy and diabetic single cell action potential. I KATP conductance (g KATP ) was set to zero when it was considered to be close. g KATP was considered 1nS and 2nS in healthy and diabetic rat single cell models to consider two different amplitudes of I KATP. The 93

107 maximum value of g KATP that resulted in generation of an AP in single cell was 2.4nS and 2.6nS for the healthy and diabetic rat models respectively. These values were 7.2nS and 7.9 for the healthy and diabetic rabbit models. Three different values of g KATP (3, 4 and 5nS) were used in rabbit simulations to account for the effects of I KATP on the vulnerability to reentrant arrhythmias Spontaneous Calcium Release Spontaneous calcium release (J spon ) in a single cell was modelled by a calcium release from the sarcoplasmic reticulum (SR) to the cytoplasm using the approach introduced by Xie et al. [97]: J spon = G spon g 1 g 2 ( v s c j c s ) v SR (5.1) 1 g 1 = 1 + exp( (t t 0 )/τ 1 ) (5.2) 1 g 2 = 1 + exp((t t 0 )/τ 2 ), (5.3) where G spon is the maximum conductance with the value of ms 1, τ 1 = 10ms and τ 2 = 30ms, v s and v SR are volumes of submembrane space and SR respectively. c j and c s are the calcium concentration in the junctional and SR respectively. The release time is controlled by t 0 in the equation Connexin Lateralization In the healthy tissue the longitudinal conductivity (g l = S/m) is almost 9 times greater than transverse conductivity (g t = S/m, and g l /g t 9) [98]. Connexin lateralization in diabetic tissue was modelled by reducing the longitudinal conductivity as connexins travel from cell ends to cell sides. Lateralized connexins 94

108 were considered not to be functional, therefore g t was not changed and g l was reduced so that g l /g t 5. We performed simulations considering functional lateralized connexins in diabetic rat tissue. Results showed that the vulnerability was significantly reduced in these models (window of vulnerability was 1ms and 5ms for diabetic rat with and without I KATP respectively) not comparable to modeling and clinical literature [21] therefore we didnt pursue the assumption of functional lateralized connexins Tissue Simulations Using ionic models for healthy and diabetic rat ventricular myocytes from Pandit et al. [51, 95] and healthy and diabetic rabbit ventricular myocytes explained previously in 5.2.2, simulations were performed to study the effect of diabetes on arrhythmia vulnerability. Single cell models of healthy and diabetic rat and rabbit cardiac myocytes were used to create a 1cm 1cm and 2cm 2cm two dimensional tissue for rat and rabbit respectively using CARP [32]. The time step was 1µs and spatial discretization of 100µm was used to solve mono-domain equations in the aforementioned 2D tissue (1cm 1cm tissue for rat and 2cm 2cm for rabbit simulations). The larger physical dimensions of the rabbit 2D model accounts for the difference in size and action potential duration (APD) between rat and rabbit hearts. There was no reentry in a 1cm 1cm tissue for rabbit (see Section 5.4.2) Reentrant Arrhythmias An S1-S2 stimulation protocol was applied to generate reentrant activity. As depicted in Fig. 5.2, S1 occurred at the top edge of the tissue and propagated down (transversely) through the tissue. S2 occurred a certain time later in the top right corner 95

109 of the tissue. S2 occupied 1 of the tissue in area. Reentry was considered successful 4 if the wave traveled around at least twice to its starting point following generation of S2. In this case the delay between S1 and S2 was noted. The range of S1-S2 delays that resulted in successful reentry was called the window of vulnerability (WoV). The larger the WoV, the more vulnerable the tissue is to reentrant arrhythmias. Figure 5.2: S1-S2 stimulation protocol (left to right) - In (a) delay is too short and second stimulus cannot propagate at all. Reentrant activity happened in (b) and stimulus rotates through the tissue permanently. If the delay is too large then the stimulus propagates through the tissue and vanishes (c) Conduction Reserve Conduction velocity is mainly dependent on two major parameters in the heart: excitability and coupling. There is a significant redundancy in these parameters which 96

110 is known as conduction reserve. Conduction reserve means that a small reduction in conduction velocity requires significant reduction in excitability or coupling; or moderate reduction in both factors [99]. Conduction velocity was measured for rabbit and rat model on a 5mm simulated cable composed of 50 cells. Each cell is 100µm long. Activation time is the time when a cell is excited and was considered to be the time when membrane potential crossed 0V. This was implemented by a linear interpolation between the discrete time points. Conduction velocity was measured from the 10 th cell to 40 th in the model in order to avoid the effect of sealed boundary conditions. 5.3 Results Single cell APD is increased by diabetes Figure 5.3 shows the single cell simulation of action potential in the healthy and diabetic rat model. The diabetic action potential is longer mainly due to reduced potassium currents as explained by [95]. Changes in diabetic ionic currents and calcium handling characteristics are presented in the table of Fig Figure 5.4 shows the simulation results of action potentials of normal and diabetic rabbit model. APD was increased by 20% in diabetic rabbit model simulation results consistent with observations in [67]. The changes in diabetic ionic currents are provided in the table of Fig I KATP shortens APD more in diabetic tissue than in healthy tissue Opening of I KATP results in more outward potassium current and shorter action potential consequently as depicted in Figures 5.5 and 5.6. However, the effect of I KATP 97

111 Currents diabetic rat I t -32% I ss -23% I CaL -24% I BCa -50% I BNa +25% NCX -40% I NaK -37% J up -20% J rel -45% Figure 5.3: Changes in diabetic rat ionic currents based on [95] and simulation results of normal and diabetic action potential in a rat single cell. Action potential duration is increased in diabetic rat. 98

112 Currents diabetic rat I to -60% I Kr -70% I Ks -40% I CaL -15% I K1 I Na NA NA NCX -40% J up -20% J rel -30% Figure 5.4: Changes in diabetic rabbit ionic currents based on [67, 93] and simulation results of normal and diabetic action potential in a single cell. Action potential duration is increased by 20% in diabetic model. 99

113 (a) (b) Figure 5.5: Effect of I KATP opening on action potential in (a) healthy and (b) diabetic single cell rat model- Opening of ATP potassium channels reduces the diabetic action potential duration by 29% but the same amount of I KATP results in 20% reduction of healthy APD. is not identical in healthy and diabetic cells. In rat, opening of I KATP, when g KATP is 1nS, results in 29% and 20% APD reduction for diabetic and healthy models respectively. In the rabbit model, diabetic APD was decreased by 15% by opening of I KATP whereas this resulted in 8% APD shortening for the healthy model Connexin lateralization does not affect source-sink relationship for an ectopic beat We simulated a 2D sheet with only the cells at the top right corner region exhibiting spontaneous calcium release in the rabbit model. The size of this region was changed to find the minimum size of the tissue for which spontaneous release resulted in wave propagation in the whole tissue (Fig. 5.7). The results showed that despite connexin lateralization and current changes in diabetic tissue, the minimum size was similar for 100

114 (a) (b) Figure 5.6: Effect of I KATP opening on action potential in (a) healthy and (b) diabetic single cell rabbit model- Opening of ATP potassium channels reduces the diabetic action potential duration by 15% but the same amount of I KATP results in 8% reduction of healthy APD. 101

115 both healthy and diabetic tissue. The moderate uncoupling resulting from connexin lateralization thus does not appreciably affect source-sink relationship for an ectopic beat. Figure 5.7: A 2 2 cm tissue was used in simulations. The top right corner square regions are used to find the minimum size of the tissue with spontaneous Ca release which results in wave propagation in the whole tissue. The minimum square was cm for both healthy and diabetic rabbit model Vulnerability to reentry Simulations were performed with and without I KATP and for different levels of coupling. For the rat model g KATP was either 1nS or 2nS while g KATP was considered 3, 4, and 5nS for rabbit. Gap junction conductivities were considered to be at 100%, 75%, 50%, and 25% of their baseline values. Simulation results are shown in Fig 5.8 and 5.9. In the rat model (Fig. 5.8), WoV was larger in the healthy tissue than the diabetic tissue in all cases. WoV increased 102

116 as the coupling was decreased in healthy and diabetic models. Opening of I KATP resulted in larger WoV. In the rabbit model (Fig. 5.9), the WoV is significantly larger in the diabetic tissue than the healthy tissue which is the complete opposite of the results for the rat model. Reduced coupling and increased I KATP resulted in a larger WoV, which is similar to the rat model. In Fig. 5.9, when there is no I KATP, no reentrant activity was observed in rabbit models. Similarly when gap junction conductivities were not changed, there was no reentrant arrhythmia in any healthy and diabetic rabbit models. As gap junction conductivities are decreased and I KATP current increases, reentry was observed in rabbit models. The WoV was larger in diabetic rabbit than in healthy one Rabbit and rat have the same conduction reserve Figure 5.10 shows the conduction velocity versus coupling for rat and rabbit models. The relationship between CV and coupling saturates as coupling increases and becomes steeper as cells are uncoupled. This robustness of CV against reduced coupling is called the conduction reserve. In healthy models, coupling is in the saturated region and moderate uncoupling does not affect CV significantly. With increased degrees of uncoupling (such as connexin lateralization in diabetes), a small reduction in coupling results in a significant decrease in CV. Figure 5.10 shows that the rabbit and rat models have similar conduction reserve plots and a particular change in coupling thus results in the same change in CV in both species. These simulations were done for healthy models of rat and rabbit. Results for the diabetic models are identical to the results of the healthy models. Conduction reserve is mainly dependent on the coupling and excitability. Coupling depends on the 103

117 (a) (b) Figure 5.8: Window of vulnerability for healthy (a) and diabetic (b) rat models. As the conductivity is reduced in both models, WoV increases. Adding I KATP increases the WoV in both cases. 104