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From cardiac electrical activity to the ECG A finite element model Master thesis A.M.J Lenssen 535227 BMTE 08.26 Committee: Prof. Dr. Ir. F.N. v.d. Vosse Dr. Ir. P.H.M. Bovendeerd Dr. Ir. G.J. Strijkers Ir. N.H.L. Kuijpers Drs. P.M. v. Dam (Medtronic, Arnhem) Eindhoven University of Technology Department: Biomedical Engineering Division: Biomechanics and tissue engineering Group: Cardiovascular Biomechanics Eindhoven, May 2008 Abstract In the group of Cardiovascular Biomechanics, finite element models of cardiac electromechanics are developed, that ultimately might be used as a tool to assist in diagnosis of cardiac disease. In such a diagnosis, with the use of the model, clinically available data such as the ECG of MR tagging images would be translated into an estimation of cardiac tissue properties. The current models lack the possibility to translate the ECG into a prediction of cardiac electrical activity. As a first step towards this possibility, in the present thesis the goal is to develop and implement a model to solve the forward relation between cardiac electrical activity and the ECG. First a few test problems were analyzed in a two dimensional domain, where various descriptions for the cardiac source were considered. From this it was concluded that modeling cardiac activity through electric potentials yielded a more robust estimation of the potential field, than a model in which cardiac activity was modeled through electric current sources. Next, the relation between cardiac electrical activity and the ECG was investigated in a geometrically more realistic 3D model. The geometry of the left and right ventricle of a dog heart was placed inside a stylized torso model. Activation times were determined from a wave speed model, the Eikonal equation. Action potential dynamics were estimated and coupled to the activation times. The potential distribution caused by cardiac potentials, was calculated throughout the torso and body surface potentials were determined on the front of the torso model. A 12 lead ECG was calculated and results were compared to model results obtained by Miller et al. The body surface potential pattern and the ECG showed many differences with the results of Miller et al. Therefore a few parameters that could influence the body surface potentials were analyzed. Making the action potential duration heterogeneous throughout the myocardium was found to have an effect on the calculated T waves in the ECG only. Changing the orientation of the heart with respect to the thorax affected the body surface potentials only slightly. Since the variations did not yet lead to a more realistic result, the parameters need more study. In conclusion, a model was developed that provides the ability to calculate body surface potentials from a description of cardiac activation. To obtain realistic results more parameter studies are needed. ii Contents 1 General introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Theoretical Background 5 2.1 Anatomy of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Electrophysiology of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 The 12 lead ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2 The ECG signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Modeling electrical activity 3.1 3.2 11 The bidomain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 The heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 The torso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.3 Summary of the bidomain approach . . . . . . . . . . . . . . . . . . . 14 Decoupling the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.1 Eikonal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.2 Common methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Two dimensional test problems 4.1 4.2 21 Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.1 Analytical description . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Dipole layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 iii 4.3 4.4 4.2.1 Analytical description . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.3 Simulations performed . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Volume source versus surface source . . . . . . . . . . . . . . . . . . . . . . . 30 4.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 From cardiac activation to body surface potentials 5.1 33 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1.1 Activation of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1.2 Heart inside the torso . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1.3 Determining the ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 Parameter study 6.1 6.2 45 Heterogeneous action potential duration . . . . . . . . . . . . . . . . . . . . . 45 6.1.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Orientation of the heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 General discussion 51 A Results of variations in orientation of the heart. 57 iv Chapter 1 General introduction 1.1 Background In the clinic, information about the functioning of the heart is mainly obtained by electrocardiography and various non invasive imaging techniques such as magnetic resonance tagging (MRT). The ECG and the deformation pattern obtained with these clinical measurements give information about the status of the myocardium and about abnormalities in the tissue, such as a conduction disorder or cardiac ischaemia. Diagnosing a patient involves the translation of the ECG and the deformation pattern into a conclusion on the status of the myocardium. Traditional diagnosis is often based on clinical experience and not all information available from measurements is used. Mathematical models of cardiac function can play a role in studying the relation between clinical measurements and the status of the myocardium and might be used to assist in diagnosis. At the group of Cardiovascular Biomechanics, mechanical models of the heart have been studied extensively. Initially, the influence of parameter variations on local wall mechanics was studied with models that are geometrically simple. With the use of clinical data about the wall mechanics obtained with for instance MRT, these models are tuned to gradually make them useful for diagnostics. Cardiac muscle contractions are triggered by electrical stimulation of the cardiac muscle. Therefore the electrical functioning of the heart is of influence on the mechanical characteristics. In the current models, the initiation of muscle contraction is regulated by an electrical stimulation sequence which is calculated using a wave speed model [25]. Studies showed that this model is capable of simulating left ventricular mechanics for hearts with normal and abnormal pulse conduction[20]. However, there were still some differences with respect to clinical measurements when comparing local deformation patterns. Furthermore, simulations were performed to study whether the model is able to simulate cardiac wall strain for normal activation and in case of a left bundle branch block (LBBB)[29]. The activation sequence was adjusted to what was found in literature. From this simulations could be concluded that the model is not yet capable of simulating physiologic strain patterns in a left ventricle with a LBBB. Information about the electrical status of the myocardium can be of additional value to the mechanical models described. Since clinically cardiac activation cannot be measured directly, the ECG is used as an indirect measure for the electric functioning of the heart. The ECG 1 1.2 Outline of the thesis General introduction represents the effect of distribution of potentials in the cardiac muscle on the body surface. Since cardiac potentials are difficult to assess, mathematical models play an essential role in studying the genesis of the ECG. Mathematical models used to calculate body surface potentials from a description of the cardiac activation sequence, are called forward models. In the situation where cardiac activation is determined from potentials on the body surface, the inverse problem is solved. To obtain more insight in the relation between cardiac electrical activity and body surface potentials, the aim of this study is to develop a model to calculate body surface potentials from cardiac electrical activity. Over the years, many studies have been performed on the forward problem of electrocardiography. Since the first numerically solved forward model by Gelernter and Swihart in 1964 [18], the development in numerical methods and computer hardware have increased the complexity of recent mathematical models. Although the possibilities of computational power have increased, there are still limits in the level of detail one is able to incorporate in the model. Microscopic descriptions of myocardial tissue are used by many research groups [9, 17]. The level of detail in these models brings them closer to the physiological situation, however modeling a complete heart built from blocks the size of a single cell can give problems with respect to computation time and memory. Therefore, in modeling electrical activity of the complete heart, more macroscopic approaches are more convenient. The heart, the torso, and possible other conductive regions that play a role in the problem are represented by either boundary surfaces or volumes. When the torso is represented as a conducting volume, a volume method is used. A distinction can be made between models where the cardiac wall volume is taken into account and approaches where the activation of the heart is described only on the surface. If the model only comprises the heart and torso boundary surfaces a surface method is used. In this situation, often a boundary element method (BEM) is used to represent the geometries numerically. The activation of the heart is represented on the surface of the heart. Homogeneous and isotropic conducting properties are assumed in each compartment to calculate body surface potentials. In many cases, regions with different conductive properties inside the torso, for example the lungs, are taken into account [3, 28]. The volume method needs more computational time than a surface method because a larger amount of elements is needed, but a great advantage of the use of volume elements is the ability to incorporate the different conducting properties of the various regions, such as the lungs, that are present between the epicardium and the body surface. Models of cardiac mechanics, like the models developed by Kerckhoffs et al. [25], are generally implemented in finite element (FE) packages. To be able to eventually couple electrical modeling with mechanical modeling, a model of cardiac electrical activity is most valuable when solved with a FE model. To be able to incorporate anisotropic conduction properties and to connect with mechanical models, the aim is to use a FE model to describe the forward problem numerically. 1.2 Outline of the thesis In chapter 2 the anatomy and (electro-)physiology of the heart will be briefly discussed, as well as the concept of electrocardiography. Subsequently, the modeling of electrical activity of the heart to the body surface will be discussed in chapter 3. In chapter 4 a few two dimensional test problems are performed where various models are considered to represent the cardiac source. A simplified three dimensional forward model is analyzed in chapter 5. In 2 1.2 Outline of the thesis General introduction chapter 6 the influence of different parameters on the solution determined with the model is studied. Finally, in chapter 7 an overall discussion and conclusion of the study is presented. 3 1.2 Outline of the thesis General introduction 4 Chapter 2 Theoretical Background 2.1 Anatomy of the heart The heart consists of four chambers, the left and right ventricle and the left and right atrium. In figure 2.1 a drawing of the anatomy of the heart is shown. The atria are separated from the ventricles by the base, where the the four cardiac valves are situated; the tricuspid valve between the right atrium and the right ventricle, the mitral valve between the left atrium and the left ventricle, the aortic valve between the left ventricle and the aorta and the pulmonary valve between the right ventricle and the pulmonary artery. The atria are relatively thin walled cavities that function as a weak filling pump for the ventricles. The right and left ventricle are separated by the septum. The walls of the ventricles are thicker. The right ventricle (RV) pumps blood into the pulmonary system and the left ventricle (LV) pumps blood through the aortic valve into the aorta providing the rest of the body with oxygenated blood. The LV has a thicker wall than the right ventricle, since the RV only has to serve the pulmonary system and the LV needs to work against a higher pressure to pump blood into the systemic circulation. The heart tissue, the myocardium, is composed out of muscle cells, the myocytes. The muscle cells have the ability to contract to fulfill the pumping function of the heart. To initiate contraction the myocytes are electrically stimulated. The myocytes are mechanically and electrically connected through intercalated discs. The overall alignment of the myocytes is anisotropic throughout the cardiac tissue. This anisotropic alignment gives rise to anisotropy in both contraction and conduction of the electrical stimulus. 2.2 Electrophysiology of the heart In a normal muscle cell at rest an electric potential difference is present over the membrane. This potential difference is called the transmembrane potential and is given by: Vm = φ i − φ e . (2.1) φi represents the intracellular potential, φ e the extracellular potential. At rest the transmembrane potential remains approximately -85mV. Excitation of cardiac tissue occurs when 5 2.2 Electrophysiology of the heart Theoretical Background Figure 2.1: Anatomy of the heart, which consists of four chambers: the right and left atrium and the right and left ventricle. the transmembrane potential of the myocytes reaches a certain threshold value caused by a stimulus. A typical myocardial action potential can be seen in figure 2.2. In phase 0 of the action potential, the transmembrane potential rises to a positive value due to a rapid inflow of sodium ions. Then, caused by an outward potassium current, in phase 1 a fast decrease in the membrane potential occurs. The plateau phase of the action potential, phase 2, results from a slow influx of calcium ions. The outflow of potassium ions finally brings back the membrane potential to its resting value. The amplitude of the action potential in cardiac tissue is in the order of 100mV. The shape of the action potential differs locally in the myocardium. Especially the initial repolarization in phase 1 is known to vary in different regions of the heart. The depolarization will pass on to the neighbour cells through the gap junctions that connect cardiac cells. In this way the action potential travels over the myocardium. The electrical activity of the cardiac muscle starts in the sinoatrial (SA) node. The SA node consists of the so-called pacemaker cells, that have the ability to generate a stimulus. From the SA node an electric stimulus is generated at a rate of 60 to 100 per minute and conducted over the right and left atria to the atrioventricular (AV) node. From the AV node the electric pulse is conducted to the ventricles through the bundle of His. The bundle separates into two bundles along both sides of the septum, the left and right bundle branches. Both bundle branches lead to the Purkinje fibers that merge into the ventricular endocardial walls. From there, the propagation of the depolarization continues over the ventricular walls from cell to cell at a much slower rate (0.5-1 m/s) than through the conduction system described above The wave form of the cardiac action potential is not constant. The shape of the action potential varies in different regions of the heart tissue. The cells in the sinoatrial and atrioventricular node differ from the cells in Purkinje fibers and also differ from the myocardial cells. This causes changes in the shape of the action potential when traveling from sinoatrial 6 2.2 Electrophysiology of the heart Theoretical Background 4 0 1 3 4 Vm (mV) 2 Time (ms) Figure 2.2: Typical form of a myocardial action potential. The dashed lines divide the action potential in four phases. In phase 0 the cell depolarizes rapidly. Then, in phase 1 a quick initial repolarization occurs. Phase 2 is the plateau phase of the action potential and in phase 3 the cell repolarizes to the resting state in phase 4. (adapted from [1]) node throughout the myocardium. Also in the myocardial tissue differences in the shape of the action potential can be found. The duration of the action potential varies throughout the heart. In figure 2.3 action potential dynamics for different regions in the heart are shown. Figure 2.3: The shape of the cardiac action potential in different regions of the heart. From [19] 7 2.3 The electrocardiogram 2.3 Theoretical Background The electrocardiogram As a consequence of the depolarization and repolarization of the cardiac muscle cells, potential differences can be measured with electrodes placed on the skin. Potential differences are registered for a period of time in a collection of recording electrodes. The signals measured by these electrodes are collected in an electrocardiogram (ECG). As a potential difference is measured, at least two electrodes are needed to measure the ECG. A recording electrode measures the potential difference with a reference electrode. The connection between the reference and the recording electrode is called a lead. If the depolarization wave is moving towards the recording electrode, the lead will record a positive signal. When the recording electrode is placed at a point where the depolarization waves departs, the signal will be negative. If the lead is perpendicular to the direction of the traveling depolarization, the lead will give a biphasic signal or no signal at all. The amplitude of the recorded signal is dependent on the thickness of the muscle. 2.3.1 The 12 lead ECG The first clinical important ECG measurement setup was developed by Willem Einthoven [30], called Einthoven’s triangle. The equilateral triangle consists of three limb electrodes, one on each arm and one electrode on the left leg. The heart can be seen as an electrical source in the center of this triangle. The leads are assigned I, II and III, and are bipolar since a potential difference can be calculated between two electrodes. Wilson developed a way to use unipolar leads for the electrocardiogram. Normally these are determined with respect to a reference in infinity. Wilson and his colleagues had the idea to use a mean value of the three limb electrodes as a reference for unipolar leads on the human torso [11, 12]. With Wilsons central terminal as a reference three unipolar leads can be formed with each limb electrode, the three additional limb leads. Golberger later introduced the augmented limb leads, the aVR , aVL and aVF where not Wilsons central terminal is the reference, but the mean of the remaining limb potentials. To measure potentials more closely to the heart, Wilson introduced six so called precordial leads [13], V 1 to V6 . These are measured with respect to the central terminal and are placed on the left side of the chest. In the clinic the 12 lead ECG is most commonly used. The 12 lead system consists out of the three standard leads I II and III, the unipolar leads aV R , aVL , aVF and also contains the six precordial leads. All the electrodes with lead connections are shown in figure 2.4. 2.3.2 The ECG signal The standard ECG signal consists of six peak signals each defined with a different letter, the P, Q, R, S, T and U peaks (see figure 2.5). The P peak results from the depolarization of the atria. The P-R interval is the time between the depolarization of the atria and the depolarization of the ventricles. The QRS-complex results from the depolarization of the ventricles. The T wave displays the repolarization of the ventricles, and the U wave is usually not present or not important resulting from a rest potential. The origin of the U wave is not clear but it probably represents ”afterdepolarizations” in the ventricles [19]. 8 2.3 The electrocardiogram Theoretical Background Figure 2.4: 12 lead ECG electrode placement (from Mybiomedical) R T P Q S Figure 2.5: An example of a typical ECG recording 9 2.3 The electrocardiogram Theoretical Background 10 Chapter 3 Modeling the electrical activity of the heart through the torso In order to model the forward problem of electrocardiography a representation of the actual cardiac source is used to calculate the body surface potentials. The representations of the cardiac tissue differ from very detailed microscopic descriptions to the more phenomenological descriptions, where the active tissue is not described at cellular level but is represented by a dipole or a collection of dipoles. Modeling the cardiac tissue as a collection of individual active cells is done by many research groups [9, 17, 5]. The first model to describe the activity of a single cell was made by Hodgkin and Huxley [2]. The model uses three ionic currents to describe the flow of ions over the cell membrane. The currents are regulated by gating variables which makes them flow voltage and time-dependent. Later, other even more detailed models were developed [16, 7]. In the forward problem of cardiac electrophysiology, changes in the ECG to be calculated are rather determined by the action of a group of cells than by the action of a single cell. Furthermore to model the complete heart built out of single cells could give problems in computation time and memory. A model that is able to describe the complete cardiac excitation, is the bidomain model. The bidomain model does not make use of single cells but uses a so called continuum cell model to describe the properties of the intra- and extracellular space of the myocardium. The intra- and extracellular spaces are coupled through a membrane where a transmembrane potential is present. The description of the electrical activation of a single cardiac cell is used for the continuum cell. 3.1 The bidomain model The modeling of the forward problem starts with defining a geometry. A schematic drawing of the parts incorporated in the geometry is shown in figure 3.1 If the assumption is made that there is no accumulation of charge in a volume, the current flux across the boundary must be equal to the current produced by the source. Therefore, in general, for every region the conservation of current is described as: ~ · J~ = Iv ∇ 11 (3.1) 3.1 The bidomain model Modeling electrical activity ∂T H ∂H σe ,φ e σi ,φ i T σo ,φo Figure 3.1: Schematic overview of the regions considered in the forward problem, using a bidomain approach. H is the cardiac volume and T the torso. ∂H is the cardiac boundary, ∂T the torso surface, σ i,e the intra- and extracellular conductivity tensors respectively and φi,e are the intra- and extracellular potentials. For the torso T the conductivity is given by σ o and the potential by φo . where J~ equals the current density and Iv a produced volume current. From Ohm’s law a ~ constitutive equation can be derived for the current density J. ~ J~ = −σ · ∇φ (3.2) with σ the conductivity and φ the potentials of the volume considered. 3.1.1 The heart The bidomain equations couple two domains in the cardiac volume, the intracellular and extracellular domains, as a continuous medium, connected by a cellular membrane. The intracellular domain has a conductivity given by σ i and a potential field given by φi . For the extracellular domain these properties are indicated by σ e and φe . The conductivity tensors contain conductivity values in fiber direction and in cross fiber direction, allowing anisotropic conductivity properties of the myocardium. During an action potential there is an exchange of ions between the intra- and extracellular space. This transmembrane current is the current source Iv in excitable media. The derivation of the equations governing the bidomain model starts with the conservation of current for both domains. From 3.1 for the intracellular domain the conservation of current gives: ~ · J~i = −Iv ∇ (3.3) Where Iv is the membrane current per unit volume and J~i the intracellular current density. This equation describes that the change in current in the intracellular region is equal to the current Iv leaving the volume. The minus sign indicates that a positive current is defined to leave the intracellular domain and to flow into the extracellular domain. For the extracellular region a similar equation can be derived. 12 3.1 The bidomain model Modeling electrical activity ~ · J~e = Iv ∇ (3.4) Combining the expressions for the continuity of current for both domains gives: ~ · J~i + ∇ ~ · J~e = 0 ∇ (3.5) For the two domains in the cardiac region, the constitutive equation 3.2 is given by: ~ e,i . J~e,i = −σ · ∇φ (3.6) Equation 3.5 is often given in terms of the transmembrane potential V m which is defined as: Vm = φ i − φ e (3.7) Using equation 3.7 and equation 3.6, equation 3.5 can be rewritten to the first equation of the bidomain model ~ · ((σ i + σ e ) · ∇φ ~ e ) = −∇ ~ · (σ i · ∇V ~ m ). ∇ (3.8) The volume current Iv can be specified by Iv = A m Im (3.9) where Am is the surface to volume ratio of the cell membrane, and I m is the transmembrane current density per unit area defined as: Im = C m ∂Vm + Iion . ∂t (3.10) Where Cm represents the membrane capacitance per unit area and I ion is the sum of the membrane currents of different ions and is defined per unit area of membrane. Combining equations 3.3, 3.6, 3.9 and 3.10 the second equation for the bidomain model can be derived: ~ · (σ i · ∇φ ~ i ) = Am (Cm ∂Vm + Iion ) ∇ (3.11) ∂t This equation can also be written with respect to the transmembrane potential V m . As the transmembrane potential is related to the transmembrane current, the source term, it is more convenient to express the equation in terms of V m . This gives the final representation for the second equation of the bidomain model. ~ · (σ i · ∇V ~ m) + ∇ ~ · (σ i · ∇φ ~ e ) = Am (Cm ∂Vm + Iion ) ∇ ∂t (3.12) The ion membrane currents Iion depend on gating variables. These are collected in a column s and their relation is described by first order differential equations. ˜ ∂s (3.13) ˜ = F (t, s, Vm ) in H ∂t ˜ 13 3.1 The bidomain model 3.1.2 Modeling electrical activity The torso For the torso the governing equations can be derived in a similar way as for the intra- and extracellular domains. Conservation of current gives ~ · J~o = Iv ∇ (3.14) In passive tissue, where no sources are present, the equation above can be simplified using expression 3.2 derived from Ohm’s law. This results into ~ · σ o · ∇φ ~ o=0 ∇ (3.15) This equation, known as the volume conductor equation or Laplace’s equation, is the general equation to describe the potential distribution inside a passive medium. 3.1.3 Summary of the bidomain approach From cell to body surface the forward problem can be solved using the bidomain model. Summarized the model consists of the following equations: Vm = φ i − φ e (3.16) ~ · ((σ i + σ e ) · ∇φ ~ e ) = −∇ ~ · (σ i · ∇V ~ m) ∇ (3.17) ~ · (σ i ∇V ~ m) + ∇ ~ · (σ i · ∇φ ~ e ) = Am (Cm ∂Vm + Iion ) ∇ ∂t (3.18) The gating variables, regulating the ion membrane currents I ion , are collected in the column s. They are described by first order differential equations. ˜ ∂s in H (3.19) ˜ = F (t, s, Vm ) ∂t ˜ The potential distribution inside the torso is described with Laplace’s equation. ~ · σ o · ∇φ ~ o=0 ∇ (3.20) The boundary conditions needed to solve the set of equation are listed below. From the intracellular region no current flows into the torso. ~ i ) · ~n = 0 (σ i · ∇φ on ∂H (3.21) Between the extracellular and torso region a current balance is kept through the following boundary condition ~ e ) · ~n = −(σ o · ∇φ ~ 0) · n (σ e · ∇φ on ∂H. (3.22) Because no potential drop is possible across a boundary the potentials of the extracellular domain and the torso have to be equal at the boundary. 14 3.2 Decoupling the problem Modeling electrical activity φe = φ o on ∂H (3.23) Considering the torso to be surrounded by air, with conductivity zero, no current leaves the torso through the surface. (σ o · ∇φo ) · nT = 0 on ∂T (3.24) 3.2 Decoupling the problem Within the cardiac region, the bidomain equations are generally used to calculate the propagation of action potentials. The bidomain model describes the membrane potential as a function of the location ~x and of time t. A typical cardiac action potential at a certain ~x = ~x 0 is given in figure 3.2. potential (mV) 10 0 -90 0 250 time (ms) Figure 3.2: A typical shape of the membrane potential V m (~x0 , t) described by the bidomain model Because of the fact that the upstroke of the action potential is so steep it will only cover a few cells at the same moment. Therefore, to describe the propagation of the action potential realistically, a spatial grid size in the order of a cell is needed. Furthermore the action potential has a duration of approximately 250 ms and the upstroke has a duration in the order of a millisecond. To be able to closely describe the dynamics of the action potential a timescale in the order of a microsecond is needed. In conclusion, the problem is computationally very expensive, both in time and in spatial domain. Therefore the aim is to find a less time and memory consuming method. Several research groups have tried to achieve a faster and more simplified way to the solution by decoupling the problem [31, 22]. In these works the model of activation of the myocardium is divided into two steps. The first step is to determine the activation sequence in the heart. The second step involves the modeling of the action potential dynamics. For the first step for example the moment of depolarization can be used to simplify the activation model. The activation sequence can be determined in several ways, Miller used an estimation from literature based on measurements by Durrer et al. [31, 10], Van Dam uses a shortest path algorithm [24]. Another method is to use the Eikonal equation, derived from the bidomain model by Colli-Franzone et al. [23]. This equation describes the motion of the depolarization front through the myocardium. The Eikonal equation describes the moment of the upstroke of the action potential as a moving front without using a detailed description of 15 3.2 Decoupling the problem Modeling electrical activity the ionic current. It also provides the ability to incorporate anisotropic conduction properties of the cardiac tissue. 3.2.1 Eikonal equation The original description of the Eikonal equation in Franzone et al. [23] has been rewritten by Kerckhoffs [25] by assuming the ratio between conductivities in longitudinal and transverse direction to be equal for intra- and extracellular media. The anisotropy ratio m is given by σti,e =m σli,e with subscript t denoting the transverse direction, and subscript direction. The conductivity tensors can be written as: (3.25) l the longitudinal or fiber σ i = σli (~el~el + m(~et1~et1 + ~et2~et2 )) = σli M (3.26) σ e = σle M (3.27) with σ i the intracellular conductivity tensor and σ e the extracellular conductivity tensor. Vector ~el is a unit vector in fiber direction and ~e t1 and ~et2 are unit vectors in transverse direction. These assumptions result in the following equation that has to be solved for t dep : cf q ~ dep · M · ∇t ~ dep − k0 ∇ ~ · (M · ∇t ~ dep ) = 1 ∇t (3.28) where cf represents the velocity of the depolarization wave along myofiber direction, k 0 is a constant that determines the influence of wave front curvature on the wave velocity. Boundary conditions used to solve the equation are tdep = 0 at Γstim (3.29) and ~n · M · p~ = 0 on Γext (3.30) where Γstim is the site where the depolarization is initiated, the stimulation site, and Γ ext is the boundary of the domain assumed that it is electrically insulated. The advantages of the Eikonal model over the bidomain approach are the reduction in calculation time, the only parameters are the activation times of the cardiac fibers, and that local velocity differences can be modeled. However, the advantages of the Eikonal model also bring some disadvantages. The model only describes the moment of the upstroke of the action potential, this means that the repolarization is not included. To be able to describe the complete action potential dynamics an additional description of the action potential dynamics has to be coupled to the activation times. Because the shape of the action potential is not constant through the cardiac tissue it has to be estimated. Furthermore the model is a more phenomenological description and is more difficult to relate to the physiological basis of the problem. 16 3.2 Decoupling the problem 3.2.2 Modeling electrical activity Common methods When the activation sequence on the myocardium is known there are different ways to model the forward problem. The conductive regions incorporated in the model can either be described by volumes or by surfaces. If volumes are considered, a volume method is used. If only interfaces between different regions are comprised, a surface method is applied. Here a short overview of common methods to solve the forward problem is given. A more complete overview can be found in [27] and [15]. Volume method Volume methods can be subdivided into two approaches. The cardiac activity can be represented by either equivalent current sources, or by potentials prescribed on the cardiac surface. Volume methods considering current sources are based on solving the governing equation for the cardiac source ~ · σ e · ∇φ ~ e = −Iv . ∇ in H (3.31) And for the torso, where no sources are present the following equation is used ~ · σ o · ∇φ ~ o = 0. ∇ in T (3.32) With the volume method the activation can be defined through the complete heart. This can be done by defining the volume current given in equation 3.31 and switching it on according to the predefined activation sequence through the myocardium. Solving the complete volume model To calculate the activation of the heart through the torso to the body surface, the finite element method (FEM) can be used. The governing equations to solve the problem using the FEM are given below. For the complete volume, H and T , equation 3.31 is solved. The weak form of this equation is given by Z ~ · σ · ∇φ(~ ~ x, t)dV = − w(~x)∇ H T T Z H w(~x)Iv (~x, t)dV T (3.33) T with w(~x) a weighting function, the potential φ is φ e in H and φo in T. The conductivity tensor σ is σe in H and σ o in T. For convenience the dependence on ~x and t is left out in the remainder of this section. When applying Greens theorem to this, the following equation is derived: Z ∂T ~ · ~ndS − wσ ∇φ Z H ~ · (∇w)dV ~ (σ e ∇φ) =− T T Z wIv dV H T (3.34) T For the torso T there is no source term, so the integral on the right side only holds for the heart H. Together with a zero flux boundary condition at ∂T , which cancels out the integral on the left, the equation becomes 17 3.2 Decoupling the problem Modeling electrical activity Z ~ · (∇w)dV ~ (σ ∇φ) = H T Z wIv dV (3.35) H T To make the problem well-posed the potential is prescribed at one point on the outer boundary ∂T φ(xref ) = 0 for xref at ∂T (3.36) Equations 3.35 the basis for solving the forward problem using the FEM. Volume method for thorax only When using the finite element method, the torso volume is filled with volume elements. In this situation, the cardiac volume is left out of the solution, only the space between the epicardium and the torso surface is considered. Equation 3.31 disappears in this case, because in the torso no sources are present. Equation 3.32 needs to be solved. The derivation of the equations is similar as in the volume method. The weak form becomes Z Z ~ · (∇w)dV ~ ~ · ~ndS − (σ o · ∇φ) =0 (3.37) wσ o · ∇φ T ∂T The integral on the left disappears because of a zero flux boundary condition at ∂T , resulting in Z ~ · (∇w)dV ~ (σ o · ∇φ) =0 (3.38) T For the heart surface ∂H the following boundary condition is given φe = φe,0 on ∂H (3.39) Surface method If the forward problem is solved using a surface approach only the boundary of the heart, ∂H, is taken into account. The activation of the heart is represented on the cardiac surface. To solve the surface method, a boundary element method (BEM) is used to solve the problem numerically. Solving the surface model When the BEM is used, the torso volume T is not described, only the heart and torso surface ∂H and ∂T are numerically represented. In surface methods isotropic conduction properties are assumed for the volume of the cardiac wall and the torso. This means that the tensor σ becomes a scalar σ for every domain. The BEM reduces the amount of nodes that is needed to calculate the solution by expressing the problem as function of the boundary instead of describing it in space using Laplace’s equation. By assuming isotropic conduction properties and homogeneity of the thorax, the governing equation for potential distribution due to the cardiac electrical activity can be expressed in terms of surface integrals only. A representation 18 3.2 Decoupling the problem Modeling electrical activity of the source Iv is derived on the heart surface ∂H, also called an equivalent surface source. For a detailed derivation of the governing equations describing the surface method, see [21]. Since all equations are expressed in terms of surface integrals, they can be combined into a single equation. The equation contains a function A, the transfer function which transforms the intracellular potential φi originating from the sources on the cardiac wall, into potentials on the boundary ∂T . This function contains information on the conductivity values of the different parts of the volume conductor model and of the geometries of the compartments. φ(~x0 , t) = Z A(~x, ~x0 , t)φi (~x, t)dS S with x~0 a point on the torso boundary ∂T . 19 (3.40) 3.2 Decoupling the problem Modeling electrical activity 20 Chapter 4 Two dimensional test problems In chapter 3, the volume method to calculate the forward problem was described. With the volume method the potentials inside a volume conductor caused by a volume current source are calculated. Here, the volume method will be analyzed and results will be compared with the results of simulations where the source is represented by surface potentials. The equations governing both solution methods are given in section 3.2.2. To gain insight in the functioning of the volume method, a few test problems are defined. The domain that is used in these test problems, a square, is a simple representation of a thorax. Various source descriptions are considered, which represent cardiac activity. Potentials distributed from these sources are considered at the right boundary of the domain, representing the body surface. At first, the field of a simple dipole source is calculated analytically and a numerical approximation is made. In a second test problem the source is extended to a dipole layer. And a third test problem contains the comparison of solutions of the volume method and the surface method. 4.1 4.1.1 Test 1 (dipole) Analytical description A combination of two point sources with opposite current strength I 0 and −I0 is called a dipole. In the previous chapter it was explained that dipoles are often used as a source model in electrocardiographic modeling. The analytical solution for the potential produced by a dipole in a 2D domain starts with the description for a single point source. The potential generated by a point source has spherical symmetry and is usually derived in a three dimensional domain. However since the concept of this chapter is to give more insight, here, a two dimensional description is given. The current density caused by a monopole with current strength I 0 in a two dimensional space is given by the current passing the surface. I0 ~er J~ = 2πr (4.1) with ~er a unit vector in the radial direction. In chapter 3 the following constitutive equation was assumed using Ohm’s law. J~ = −σ∇φ (4.2) 21 4.1 Dipole Two dimensional test problems Because of the fact that the problem is rotationally symmetric only the radial component is non-zero, this leads to dφ I0 = (4.3) dr 2πr After integration with respect to r and defining the potential at infinity as zero, it follows for the potential generated by a monopole in 2D −σ I0 ln(r) (4.4) 2πσ For a dipole consisting of two opposite point charges separated by a distance 2d the potential in a point P (x, y) is found by superposition of the potentials of the two point sources. In figure 4.1 the configuration of the dipole is drawn schematically. φ= y + I0 R1 x r 2d - I0 R2 P(x,y) Figure 4.1: The position of a dipole consisting of two opposite charges I 0 and −I0 separated by a distance 2d Using equation 4.4 for the potential for a single point source, the superposition of the two opposite sources in figure 4.1 gives φ(x, y) = I0 R1 ln( ) 2πσ R2 (4.5) For a line x = x0 the potential is given by p (y − d)2 + (x0 )2 I0 φ(x0 , y) = ln p 2π (y + d)2 + (x0 )2 (4.6) The result of equation 4.6 can be seen in figure 4.3. 4.1.2 Numerical Implementation A finite element approach is made using SEPRAN, and compared to the analytical solution described above. A dipole was simulated by assigning opposite current strengths I 0 and −I0 to two points in a homogeneous mesh. When using finite elements the value of the current strength is interpolated over the elements connected to the active node. 22 4.1 Dipole Two dimensional test problems In chapter 3 the weak form for active region in the volume method was derived from Poisson’s equation for active tissue. Z ~ · ~ndS − wσ ∇φ ∂T Z ~ · (∇w)dV ~ (σ e ∇φ) =− H∩T Z wIv dV (4.7) H∩T The right integral of this equation describes the source. The current that was prescribed at the active node is interpolated with the use of linear shape functions. Within each element an approximation of the current strength I v is given by Ive = 4 X Ni (x, y)Ii (4.8) i=1 With Ni the shape functions of each element node in a global coordinate system. Generally a local coordinate system is used, with coordinates −1 < ξ < 1 and −1 < η < 1. The shape functions are for instance given by N1 = 1 (1 − ξ)(1 − η) 4 (4.9) Since only one node in the element is given a value I v0 for Iv the following can be derived Ive (ξ, η) = N1 (ξ, η)Iv0 = I v0 (1 − ξ − η + ηξ) 4 (4.10) The total source strength I0 of one element can be calculated using the expression for I ve and is given by Z Ive (x, y)dΩ (4.11) I0 (x, y) = Ωe Filling in the terms and writing ξ = x/L 1 and η = y/L2 , with L1 and L2 the element dimensions of Ωe in x and y direction, gives the following equation I0 (x, y) = Z Ωe x y xy I v0 (1 − − + )dΩe 4 L1 L2 L1 L2 (4.12) This equation has the following solution I v0 L1 L2 (4.13) 4 In this test problem a two dimensional mesh is used. Because quadrilateral elements are used with a bilinear interpolation, the current given to the node is integrated over 4 elements with equal side lengths Le . This gives a surface source of I0 = I0 = Iv0 L2e (4.14) with L2e the surface of one element. This means that when the element size changes, the dipole strength changes. For illustration, two simulations are performed with different element sizes. A total current strength of 4 · 10−4 [A · m−2 ] is used in both simulations. Simulation 1 is performed with elements with sides L e = 0.02 m and in simulation 2 the same domain is used 23 4.1 Dipole Two dimensional test problems but elements with sides Le = 0.01 m are used. Using equation 4.14 the current strength I 0 is 1 A in simulation 1 and 4 A in simulation 2. Poisson’s equation is solved to derive the potential throughout the domain. A zero flux boundary condition is applied at the outer boundary ∂T . In a third simulation the original domain is made two times larger. The settings used for the three simulations are displayed in table 4.2. Simulation 1 2 3 element size[m] 0.02 0.01 0.02 domain 2x2 2x2 4x4 I 0 [A] 1 4 1 Iv0 [A · m−2 ] 4e−4 4e−4 4e−4 Figure 4.2: Settings for the simulations in SEPRAN 4.1.3 Results The results of the three simulations can be seen in figure 4.3. −5 1.5 x 10 Potentials generated by a current dipole 1 potential (V) 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 y coordinate 0.5 1 Figure 4.3: Potentials generated by a dipole derived with three simulations and the analytical solution. Simulation 1 is given with the dashed and dotted line (− · − · −) , simulation 2 with the dashed line (− − − −), simulation 3 with the dotted line (· · ·) and the analytical solution is given with the solid line ( ). In the center of the figure all the results look similar, however, at the boundaries of the domain differences in the potential are visible. The solutions from simulation 1 and 2 are similar, the result of simulation 3 approaches the analytical solution. Note that the domain of simulation 3 is in fact larger than displayed in figure 4.3. 4.1.4 Discussion As can be seen in figure 4.3, the results of the simulations differ from the analytical solution at the boundary of the domain. This is probably a result of the boundary conditions that are used in the simulations. The zero flux boundary condition forces the gradient of the potential to be aligned along the boundary. In the analytical solution the potential is assumed to 24 4.2 Dipole layer Two dimensional test problems be zero in infinity. Therefore it can be expected that at the boundary of the domain the solution of the simulations differs from the analytical solution. And when the dimensions of the domain are increased, the solution is expected to move closer to the analytical solution. This is shown by the third simulation where a larger domain is used. 4.2 Test 2 (Dipole layer) A source that is often used to model the activity of tissue, is the dipole layer. The depolarization wave front spreading throughout the myocardium can be represented by a dipole layer with the vector normal to the curvature of the wave front. 4.2.1 Analytical description A dipole layer can be seen as a line of dipoles. In this test the dipoles are oriented in the same direction. The solution of the dipole potential can be used to formulate the potential generated by dipoles arranged in a line along x. The dipole layer consists of a area A of length L and width d with positive sources and an area B with negative sources, separated by a distance 2a. In figure 4.4 the dipole layer is drawn schematically. The potentials are determined in a line along y located at x = x 0 . The total current strength I per area is 153[A]. The potential generated by a dipole layer at a point (x,y) with distances R1 and R2 from the poles and a current strength I 0 for every pole is given by φ(x, y) = 1 2πσ Z Z y I0 ln( x R1 )dxdy R2 The integration domain is the surface of the dipole layer. 25 (4.15) 4.2 Dipole layer 4.2.2 Two dimensional test problems Numerical implementation To simulate the potential generated by a dipole layer, simulations were performed using SEPRAN. First a simulation is performed using the volume method described in chapter 3. To obtain a dipole layer source, nodes in area A are given a current strength I 0 while in area B the nodes have a current strength of −I 0 . This results in a total current strength equal to the current strength used in the analytical problem. As was explained in the previous chapter, for the passive domain Laplace’s equation is solved to obtain the potentials that are caused by the source. A zero flux boundary condition is used at the outer boundary ∂T . Furthermore the potential was set to zero at one point on ∂T . P0 A d 2a B L Y X0 X Figure 4.4: Schematic drawing of the geometry used for the dipole layer. With the dipole layer consisting of area A and area B which are prescribed opposite current strengths, L = 1 m, a = 0.04 m, d = 0.06 m and the domain has dimensions 2 m by 2 m. In the node in the top left corner of the domain (P0 ) the potential is set to zero. 4.2.3 Simulations performed In the first simulation the domain is filled with 200 rectangular shaped elements, and in the second simulation the same grid size was used but the elements shape is triangular and this generates an irregular mesh. In figure 4.5 the two different element types are plotted in the problem domain. Figure 4.5: The mesh formed by rectangular elements (top) and triangular elements selected from one half of the dipole layer. In a third simulation the effect of an unequal source size on the solution is shown. For area 26 4.2 Dipole layer Two dimensional test problems A the current size is kept at 1, but for area B a decrease in current strength of 0.5% and 1% was applied. In this simulation rectangular elements are used. For the numerical approximation made with a rectangular grid, potentials are determined in a line along y at x = x0 (see figure 4.4). 4.2.4 Results In figure 4.6, the results of the potentials determined in a line along y at x = x 0 are shown for the analytical solution and the numerical approximation. The result of the numerical approximation is shifted 1.05 · 10−3 V to make it symmetric around zero, and to be able to compare with the analytical solution. -3 1.5 x 10 1 potential (V) 0.5 0 -0.5 -1 -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 y coordinate 0.4 0.6 0.8 1 Figure 4.6: Plot of the potential determined alon line x = x 0 with the solid line representing the analytical solution and the dashed line the numerical solution. As can be seen the solutions are similar in the center and vary at the boundary of the domain. In figure 4.7 surface equipotential plots of the potentials calculated in the first and the second simulation are shown. 27 4.2 Dipole layer Two dimensional test problems LEVELS -3.2978E-03 -3.0724E-03 -2.8470E-03 -2.6216E-03 -2.3963E-03 -2.1709E-03 -1.9455E-03 -1.7201E-03 LEVELS LEVELS -1.4947E-03 -1.2693E-03 Contour levels of solution Regular mesh -3.2978E-03 -1.3130E-02 -3.0724E-03 -1.2474E-02 -2.8470E-03 -1.1817E-02 -2.6216E-03 -1.1161E-02 -2.3963E-03 -1.0504E-02 -2.1709E-03 -9.8475E-03 -1.9455E-03 -9.1910E-03 -1.7201E-03 -8.5345E-03 -1.4947E-03 -7.8780E-03 -1.2693E-03 -7.2215E-03 -1.0439E-03 -6.5650E-03 -8.1855E-04 -5.9085E-03 -5.9317E-04 -5.2520E-03 -3.6778E-04 -4.5955E-03 -1.4239E-04 -3.9390E-03 8.2991E-05 -3.2825E-03 3.0838E-04 -2.6260E-03 5.3376E-04 -1.9695E-03 7.5915E-04 -1.3130E-03 9.8453E-04 -6.5650E-04 1.2099E-03 0.0000E+00 scaley: 7.500 scalex: 7.500 time t: 0.000 Contour levels of solution Irregular mesh scaley: 7.500 scalex: 7.500 scaley: 0.000 time t: scalex: time t: Contour levels of solution -1.0439E-03 -8.1855E-04 -5.9317E-04 -3.6778E-04 -1.4239E-04 8.2991E-05 3.0838E-04 5.3376E-04 7.5915E-04 9.8453E-04 1.2099E-03 7.500 7.500 0.000 Figure 4.7: Contour plot of potentials calculated using the volume method with a regular mesh (left) and an irregular mesh (right) LEVELS -3.0000E-03 -2.5650E-03 -2.1300E-03 -1.6950E-03 -1.2600E-03 -8.2500E-04 -3.9000E-04 4.5000E-05 LEVELS LEVELS LEVELS 4.8000E-04 9.1500E-04 Contour levels of solution -3.0000E-03 -3.0000E-03 -3.0000E-03 -2.5650E-03 -2.5650E-03 -2.5650E-03 -2.1300E-03 -2.1300E-03 -2.1300E-03 -1.6950E-03 -1.6950E-03 -1.6950E-03 -1.2600E-03 -1.2600E-03 -1.2600E-03 -8.2500E-04 -8.2500E-04 -8.2500E-04 -3.9000E-04 -3.9000E-04 -3.9000E-04 4.5000E-05 4.5000E-05 4.5000E-05 4.8000E-04 4.8000E-04 4.8000E-04 9.1500E-04 9.1500E-04 9.1500E-04 1.3500E-03 1.3500E-03 1.3500E-03 1.7850E-03 1.7850E-03 1.7850E-03 2.2200E-03 2.2200E-03 2.2200E-03 2.6550E-03 2.6550E-03 2.6550E-03 3.0900E-03 3.0900E-03 3.0900E-03 3.5250E-03 3.5250E-03 3.5250E-03 3.9600E-03 3.9600E-03 3.9600E-03 4.3950E-03 4.3950E-03 4.3950E-03 4.8300E-03 4.8300E-03 4.8300E-03 5.2650E-03 5.2650E-03 5.2650E-03 5.7000E-03 5.7000E-03 5.7000E-03 scaley: 7.500 scalex: 7.500 time t: 0.000 Contour levels of solution scaley: 7.500 scaley: 7.500 scalex: 7.500 scalex: 7.500 time t: 0.000 Contour levels of solution scaley: 0.000 time t: scalex: time t: Contour levels of solution 1.3500E-03 1.7850E-03 2.2200E-03 2.6550E-03 3.0900E-03 3.5250E-03 3.9600E-03 4.3950E-03 4.8300E-03 5.2650E-03 5.7000E-03 7.500 7.500 0.000 Figure 4.8: Contour plot of potentials generated by a dipole layer with left: I 0 = −I0 , middle: 0.5% decrease in area B and right: 1% decrease in current strength I 0 in area B. What can be noticed from figure 4.7 is that when a regular mesh is used the potential field is symmetric, and when the elements are irregularly shaped, the potential field is asymmetric. The solution seems to be out of balance in the second simulation in figure 4.7. The result of experiment three can be seen in figure 4.8. The picture on the left in this figure is the same as the left picture in figure 4.7, however the scale is altered to be able to compare the solution with the other results in the figure. From figure 4.8 it can be noticed that when the source strength is unequal for the two layers, the potential distribution is asymmetric. Vector plots were made from the top left quarter of the domain, these can be seen in figure 4.9. In figure 4.9 there is a flux visible through the point in the upper left corner where the potential is prescribed. 28 4.2 Dipole layer Two dimensional test problems factor: 4.000 factor: scaley: 15.000 scaley: scalex: 15.000 scalex: 15.000 time t: 0.000 time t: 0.000 Vector plot of gradient Vector plot of gradient factor: scaley: Vector plot of gradient 4.000 15.000 scalex: 15.000 time t: 0.000 Figure 4.9: Vector plots made from the gradient in potential in the top left quarter of the domain. Top left :I0 = −I0 , top right: 0.5% decrease in current in area B, bottom left: 1% decrease in current strength I0 in area B. 29 4.000 15.000 4.3 Volume source versus surface source 4.2.5 Two dimensional test problems Discussion In figure 4.6 the analytical solution is compared with a numerical approximation. The results are similar in the center but vary at the boundary of the domain. This can be expected because of the zero flux boundary condition applied in the numerical problem. In figure 4.7 and 4.8 the results of test 2 can be seen. In figure 4.8 the positive and negative part of the dipole layer are made unequal in current strength. As can be seen, already a small deviation gives a change in the potential distribution. When the source strength is out of balance there will be flow possible out of the domain through the top left corner of the outer boundary where the potential is set to zero. The asymmetric potential distribution seen in figure 4.7 in case of an irregular mesh can therefore be declared by the fact that the irregular mesh creates an imbalance in current strength between the layers. 4.3 Test 3 (Volume source versus surface source) To compare the solution obtained using a volume source with the solution obtained using a surface source, the solution of simulation 1, in part Test 2, is used as input for the present simulation. A mesh was created with the same dimensions, and a circular saving in the center. This hole has a radius of 0.5m and represents the outer surface of the heart. In Matlab a value for the potential for every node on the circular boundary is determined from the solution in the closest node in the volume method. These values for the potential are used as boundary condition on the circular boundary, the cardiac surface ∂T . A solution for the potential was determined solving Laplace’s equation. In an additional simulation the hole is filled with elements, representing a heart of which the volume is filled. On the boundary of the hole potentials are prescribed, and the area inside the hole is conductive. 4.3.1 Results Contour plots of the potentials in the three simulations are shown in figure 4.10. Potentials along the right boundary of the domain are plotted for both methods in figure 4.11, to be able to compare the two solutions more directly. 4.3.2 Discussion In the third test situation the volume method is compared with the surface method. From figure 4.9 it can be seen that the approaches show similar potential patterns throughout the domain. Also when the area inside the hole is filled with elements, the pattern looks similar. Inside the hole the potentials show differences, this can be explained by the fact that that area is modeled as a passive conducting region. However, in the forward problem, the interest lies in the potential distribution from the heart surface to the torso surface. 30 4.3 Volume source versus surface source Two dimensional test problems LEVELS -3.2978E-03 -3.0724E-03 -2.8470E-03 -2.6216E-03 -2.3963E-03 -2.1709E-03 -1.9455E-03 -1.7201E-03 LEVELS LEVELS LEVELS -1.4947E-03 -1.2693E-03 Volume source Contour levels of solution -3.2978E-03 -3.2978E-03 -3.2978E-03 -3.0724E-03 -3.0724E-03 -3.0724E-03 -2.8470E-03 -2.8470E-03 -2.8470E-03 -2.6216E-03 -2.6216E-03 -2.6216E-03 -2.3963E-03 -2.3963E-03 -2.3963E-03 -2.1709E-03 -2.1709E-03 -2.1709E-03 -1.9455E-03 -1.9455E-03 -1.9455E-03 -1.7201E-03 -1.7201E-03 -1.7201E-03 -1.4947E-03 -1.4947E-03 -1.4947E-03 -1.2693E-03 -1.2693E-03 -1.2693E-03 -1.0439E-03 -1.0440E-03 -1.0440E-03 -8.1855E-04 -8.1857E-04 -8.1857E-04 -5.9317E-04 -5.9318E-04 -5.9318E-04 -3.6778E-04 -3.6780E-04 -3.6780E-04 -1.4239E-04 -1.4241E-04 -1.4241E-04 8.2991E-05 8.2975E-05 8.2975E-05 3.0838E-04 3.0836E-04 3.0836E-04 5.3376E-04 5.3374E-04 5.3374E-04 7.5915E-04 7.5913E-04 7.5913E-04 9.8453E-04 9.8452E-04 9.8452E-04 1.2099E-03 1.2099E-03 1.2099E-03 scaley: 7.500 scalex: 7.500 time t: 0.000 Contour levels of solution Surface source scaley: 7.500 scalex: 7.500 time t: 0.000 Contour levels of solution scaley: 7.500 scalex: 7.500 Surface source, filled scaley: 0.000 time t: scalex: time t: Contour levels of solution -1.0439E-03 -8.1855E-04 -5.9317E-04 -3.6778E-04 -1.4239E-04 8.2991E-05 3.0838E-04 5.3376E-04 7.5915E-04 9.8453E-04 1.2099E-03 7.500 7.500 0.000 Figure 4.10: Contour plot of potentials calculated using the volume method (left), the surface method (middle) and the surface method with the cavity filled with elements (right) -3 1 x 10 0 potential -1 -2 -3 Volume source Surface source -4 Surface source filled -5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 y coordinate 0.4 0.6 0.8 1 Figure 4.11: Absolute potentials along the right boundary of the domain for the volume source (dashed and dotted line), the surface source (solid line) and the filled surface source (dashed line) 31 4.4 General Discussion 4.4 Two dimensional test problems General Discussion The three tests problems described above were performed to obtain more insight into modeling the potential generated by current dipole sources inside a volume conductor. From the first two tests it can be seen that when the source is in balance, the volume source generates a potential distribution as can be expected from the analytical solutions. However, as soon as there is an imbalance in the source strength, the potential distribution becomes asymmetric. In test three it can be seen that the solutions obtained with a surface source and with the volume source match closely in the thorax domain (T). The aim is to describe the potentials generated by the activity of a realistically shaped heart in a realistic torso geometry. Since the solution based on current dipole sources has problems with an irregular mesh, it was chosen to use a source represented by potentials on the cardiac wall to model a 3D situation in the following chapter. 32 Chapter 5 From cardiac activation to body surface potentials In chapter 4 it was motivated that the volume method with the use of current dipole sources, is sensitive to imbalances in current sources and sinks due to, for example, irregular meshing. Prescribing cardiac electrical activity through voltage sources seems more robust. Therefore, in this chapter potentials are prescribed throughout the cardiac wall. A mesh of a dog heart described in Kerckhoffs et al. [26] is placed inside a stylized torso. Activation times are determined in the wall of the heart using the Eikonal equation. The activation times are coupled to an action potential and by using Laplace’s equation for the potential distribution, potentials are calculated on the surface of the torso. The results are compared to simulations performed by Miller et al. [31] who used a multiple dipole approach. 5.1 5.1.1 Methods Activation of the heart The geometry of the heart that is used is taken from Kerckhoffs et al. [26]. It contains the right and left ventricle of a dog heart, the atria are not taken into account. The Eikonal curvature equation described in chapter 3 is used to calculate the activation times in each node of the heart mesh. The activation is initiated at four points on the cardiac wall central at the left septal surface, posterior paraseptal at one third the distance from apex to base, at the anterior paraseptal wall below the mitral valve and near the right anterior papillary muscle. The mesh that is made of this geometry is build out of 33536 (hexahedral) elements. The activation times that are calculated for the heart mesh are shown in figure 5.1. The total depolarization of the ventricles lasts approximately 60ms. For comparison, the activation times measured by Durrer et al. [10] in isolated human hearts are shown in figure 5.2. 33 5.1 Methods From cardiac activation to body surface potentials 60 50 40 60 50 40 30 30 20 20 10 10 0 0 Figure 5.1: Activation sequence according to Kerckhoffs et al.[25] Figure 5.2: Isochronic surfaces of cardiac activation, measured by Durrer et al. in isolated human hearts [10]. 5.1.2 Heart inside the torso The smooth mesh of the heart is placed inside a stylized torso. The torso is constructed from two eccentric half cylinders, one with a radius of 340 mm and one with a radius of 385 mm with a shifted center to create a boundary with less curvature to represent the back of the torso. A front and top view of the torso contour containing the heart is shown in figure 5.3. The height of the resulting torso is 500 mm, the width is 300 mm and the depth is approximately 200 mm. The measures of the thorax are estimated to be comparable to the measures used by Miller et al. The mesh that forms the torso is build out of 242190 elements, a tetrahedral element shape is used. The anatomical position and orientation of the heart inside the torso differs for every individual, and since in this situation no realistic geometries are used, the right position and orientation of the heart is difficult to motivate. To be able to make comparisons with Miller et al. the heart is placed in a similar position 34 5.1 Methods From cardiac activation to body surface potentials as described in their work. The coordinate system contains orthogonal vectors ~e x , ~ey , ~ez , where ~ez coincides with the LV long axis, ~ex points at the LV free wall. ~ey is orientated perpendicular to ~ex and ~ez such that ~ey = ~ez × ~ex . The origin was chosen on the LV long axis at 25 mm below the basal plane. With respect to the torso, the heart is rotated around the z, y and x axis with angles π/8, π/4 and π/4 respectively. Subsequently the heart is translated to be placed at a more anatomic location inside the torso. From the center of the bottom plane of the torso (x,y,z) = (0,0,0) the origin of the heart is placed at z = 300 mm, y = -90 mm and x = 40 mm. The result of the location and orientation of the heart inside the torso can be seen in figure 5.3. Front view Top view Figure 5.3: Left: front view of the torso boundaries containing the heart. Right: a top view of the torso containing the heart. The electrode positions of the standard 12 lead system are indicated with the squared markers. R, L and F are the three limb electrodes and V 1 to V6 the precordial electrodes. 35 5.1 Methods From cardiac activation to body surface potentials In the torso, the nodes that lie inside the wall of the heart are assigned a value for the activation time. Because the torso nodes do not coincide with the heart nodes, the activation time has to be interpolated for the torso. This is done by looking in a region with a radius 3 mm around every node in the torso and selecting the nodes of the heart mesh that are within that radius. The value of the activation time for the node in the torso mesh is determined from a weighted value of the activation times of the surrounding nodes in the heart mesh. PN i=1 (1 tdep = PN ri rmax )ti ri − rmax ) − i=1 (1 (5.1) with tdep the activation time to be calculated for the torso node, N the surrounding nodes of the heart mesh, ri the distance of node i to the node in the torso mesh, r max the maximum distance of the surrounding nodes and t i the activation time of node i in the heart mesh. Now only the time of depolarization is known in the nodes describing the cardiac wall, however to describe the action potential dynamics, an action potential has to be coupled to these depolarization times. The action potential shape is shown in figure 5.4. The action potential starts at a resting potential of -90 mV and rises to 10.45 mV to repolarize back to the resting potential in approximately 260 ms. Potential (mV) 20 0 -20 -40 -60 -80 -100 0 50 100 150 200 250 300 Time (ms) Figure 5.4: The shape of the actionpotential coupled to the depolarization times in the simulations. 5.1.3 Determining the ECG A simulation is performed where the potentials are prescribed on 1430 nodes in the torso, representing the myocardium. Potentials are calculated on the torso surface using Laplace’s equation. A no flux boundary condition is applied to the torso surface. Surface equipotential maps are made of the front side of the thorax. The voltages are calculated with respect to Wilsons central terminal which is the mean value of the potential of the three limb electrodes: φCT = φR + φ L + φ F 3 (5.2) The potentials were normalized to obtain a scale from -1 mV to 1 mV, the multiplication factor 36 5.2 Results From cardiac activation to body surface potentials is given above each map. During depolarization 8 time steps are shown, during repolarization 4 time steps are considered. To determine an ECG signal, in the clinic, electrodes are placed on the body surface. To reproduce ECG signals, potentials were measured at the outer surface of the torso. Nine surface sites were chosen, in comparison to the sites chosen in Miller et al., to calculate a 12 lead ECG. The surface sites used for the calculation of the ECG are sketched in figure 5.3. R, L and F represent the limb leads and V 1 to V6 form the set of precordial electrodes. As was explained in chapter 2, the 12 lead ECG contains the three limb leads, three augmented limb leads and 6 precordial leads. The potentials differences recorded by these leads can be determined according to VLeadI = φL − φR (5.3) VLeadII = φF − φR (5.4) VLeadIII = φF − φL (5.5) The signals recorded with the precordial leads follow from Vi = φVi − φCT for i = 1, 2, 3, 4, 5, 6 (5.6) The three augmented limb leads are calculated with respect to the mean of the remaining limb leads which gives for example for the aV R lead VaVR = φR − φF + φ L 2 (5.7) For the aVL and aVF lead a similar expression can be derived. 5.2 Results Surface equipotential maps of the front side of the thorax can be seen in figure 5.5. The results will be discussed seen from the torso, which means left and right are switched from our point of view. Eight surface equipotential plots were made during activation. During early activation, 10 ms, the equipotential maps show a dipolar pattern, with a minimum on the left side of the torso and a maximum slightly below the center. The maximum has a much larger amplitude than the minimum. At 20 ms the maximum and the minimum have increased in amplitude and a second maximum appears at the top left. During the middle phase of the depolarization, 30 to 40 ms, the minimum increases with respect to the maximum. The maximum spreads over the left side of the thorax. During late activation, 50 to 60 ms, the minimum is concentrated at the right side and the maximum at the left side of the torso. The amplitude of both extremes drops in time. At the end of the depolarization phase, 70 ms, the amplitude of the potential is around zero. Four equipotential surface plots were made for the repolarization. At 180 ms, early depolarization, two major peaks can be seen. A minimum on the right and a maximum on the left side. A second, much smaller maximum is visible on the left and a small minimum on the top left. Later during repolarization, 240 to 280 ms the pattern remains virtually unchanged but the amplitude decreases. 37 5.3 Discussion From cardiac activation to body surface potentials ECG signals determined in 12 leads are shown in figure 5.7. The lead signals all show an early signal from 10 ms to 75 ms after initiation of activation, and a later peak around 200 ms after the start of activation. The first signal seems to be a typical QRS signal, and the late signal could be a T-peak present in normal ECG recordings (see chapter 2). Lead I and aVL show a positive QRS peak, Leads III shows a negative ORS. Lead II, V 1 , V2 and aVF show an oscillating signal. In the signals determined in aV R V2 , V4 , V5 and V6 more clearly a biphasic signal is present. The T-peaks of Lead I, aV L , V1 and V2 are negative, Lead II, III, aVF , V3 and V4 show a positive T-peak. In leads aVR , V5 and V6 the T-peaks are biphasic, aVR shows an opposite signal with respect to the other two leads. The amplitudes of most lead signals are below 5 mV, lead V3 and V4 have an amplitude of more than 10 mV. 5.3 Discussion In the model described in this chapter, activation times are determined using an Eikonal equation and coupled to an action potential shape. Laplace’s equation is solved to obtain a potential distribution throughout the torso. With this model we are able to determine potentials on the outer surface of the torso. To assess how realistic these surface potentials are we compare them to body surface potentials determined by Miller et al. The surface potential maps shown in figure 5.5 differ from the surface equipotential maps calculated by Miller et al.(see figure 5.6). They see a central maximum at 10, 20 and 30 ms which increases from 0.25 mV to 1 mV. In our simulations also a maximum is visible in those time steps, of which the amplitude increases from approximately 4 mV at 10 ms to 13 mV at 30 ms. We also clearly see a minimum on the left where Miller sees it on the right side. At 35 and 40 ms the maps show some similarities, in both simulations a maximum is visible on the left which is curled around a minimum on the right. However, when looking at the amplitudes of the extremes we see that in our results the minimum has a two times higher amplitude than the maximum, where in the results of Miller, the amplitude of the minimum stays below the maximum amplitude until at 50 ms. At 50 ms and 60 ms the results are difficult to relate, in the results of Miller a strong minimum can be seen in the center of the thorax, while in our results besides a small minimum, a maximum is constantly present in the center. The amplitudes drop until they are close to zero at 70 ms. During early repolarization, 180 ms, in our results a maximum and a minimum of equal strength are present central at the torso, while Miller sees a central maximum and a relatively small minimum at the top right. At 240 ms, in our results the amplitude of both extremes becomes approximately 2 times higher, where in the simulations of Miller the amplitudes become more than 4 times higher. In both simulations the pattern does not change to the end of depolarization (280 ms) while the amplitudes drop. Also the ECG signals calculated with the model show differences with the results of Miller et al (see figure 5.8). The QRS complex only matches for Lead I. The T tops for Leads I, aV L , V2 , and V6 , are opposite in sign. In both the surface plots and the ECG simulations, the amplitude of the potential differences is not equivalent with what Miller found. This may be caused by the fact that Miller et al. uses an unknown scaling factor to make sure all potentials have a maximum of 2 mV. When looking at the relation between the amplitudes our results seem to be within the same limits. 38 5.3 Discussion From cardiac activation to body surface potentials Since the determined potentials are relative potentials, the amplitude does not say as much as the sign of the signal does. Overall, the results of the present simulation are quite different from what Miller et al described. Miller et al. found similarity in their results with body surface potential measurements performed on a human subject by Taccardi [4], therefore we conclude that our simulation is not realistic yet. Some aspects that may be a cause of the dissimilarities between the results could be the positioning of the heart in the thorax and the shape of the thorax itself. The shape of the thorax used in the present simulation was simplified and although the dimensions were kept close to the ones used in Miller et al., this could result in a different potential pattern on the surface. In previous studies it is concluded that the geometry of the torso must be accurately modeled in order to obtain an accurate solution [14], [8]. The orientation and position of the heart was estimated according to sketches made by Miller et al., perhaps the heart is positioned slightly different and therefore a different potential distribution is seen. The influence of the orientation and location of the heart in relation to the torso was studied by Huiskamp et al. [14]. They showed the effect of small changes in rotation and translation of the heart on the calculation of the ECG from cardiac activity. They found that small changes in geometry can lead to changes in QRS amplitude. Furthermore they stated that in solving the forward problem, the effect of rotation of the heart is of greater influence on the QRS than the location. Between the results of our simulation and the results of Miller et al, major differences were seen in the T waves of the ECG signals which are known to result from the repolarization of the ventricles. A possible reason for this difference could therefore be the fact that a constant action potential duration was used throughout the heart while Miller uses a heterogeneous action potential duration. An other factor that could influence the results is the exclusion of the activity of the atria in our simulations. It is known that the depolarization of the atria gives rise to the P peak, therefore, the exclusion of the atria is probably the reason why there are no P peaks present in the ECG simulations. Since Miller did not consider the atria either, this will probably not be a reason why our results do not match with those obtained by Miller. However, this does make the ECG signals hard to relate to physiological measurements. To get more insight in the influence of several aspects discussed above, in the next chapter, a parameter validation study is performed. 39 5.3 Discussion From cardiac activation to body surface potentials Factor = 4.19 Factor = 10.45 Factor = 13.08 Factor = 15.7 500 500 1 500 1 500 450 450 0.8 450 0.8 450 1 1 0.8 0.8 Factor = 15.7 500 1 400 400 0.6 400 0.6 400 350 350 0.4 350 0.4 350 400 0.4 0.4 0.6 300 300 0.2 300 0.2 300 350 0.2 0.4 0.2 300 0.2 250 250 0 250 0 250 450 0.6 0.6 0.8 0 0 250 200 200 −0.2 200 −0.2 200 0 −0.2 −0.2 200 −0.2 150 150 −0.4 150 −0.4 150 100 100 −0.6 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −100 0 0 100 10 ms −1 −100 0 0 100 20 ms −1 −100 Factor = 4.96 Factor = 13.9 0 0 100 30 ms 150 −0.4 −0.4 −0.4 0 −1 −100 −100 Factor = 1.51 0 0 100 100 35 ms −1 −1 levels Factor = 0.069 500 500 1 500 1 500 450 450 0.8 450 0.8 450 1 1 0.8 0.8 Factor = 0.069 500 1 400 400 0.6 400 0.6 400 350 350 0.4 350 0.4 350 400 0.4 0.4 0.6 300 300 0.2 300 0.2 300 350 0.2 0.4 0.2 250 250 0 250 0 250 300 0.2 450 0.6 0.6 0.8 0 0 250 200 200 200 −0.2 −0.2 200 0 −0.2 −0.2 200 −0.2 150 150 −0.4 150 −0.4 150 100 100 −0.6 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −100 0 0 100 40 ms −1 −100 Factor = 1.44 0 0 100 50 ms −1 −100 Factor = 2.97 0 0 100 60 ms 150 −0.4 0 −1 −100 −100 Factor = 0.9387 −0.4 −0.4 0 0 100 100 70 ms −1 −1 levels Factor = 0.2 500 500 1 500 1 500 450 450 0.8 450 0.8 450 1 0.8 1 0.8 Factor = 0.2 500 1 400 400 0.6 400 0.6 400 350 350 0.4 350 0.4 350 400 0.4 0.4 0.6 300 300 0.2 300 0.2 300 350 0.2 0.4 0.2 300 0.2 250 250 0 250 0 250 450 0.6 0.6 0.8 0 0 250 200 200 −0.2 200 −0.2 200 0 −0.2 −0.2 200 −0.2 150 150 −0.4 150 −0.4 150 100 100 −0.6 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −100 0 100 180 ms 0 −1 −100 0 100 240 ms 0 −1 −100 0 100 260 ms 0 150 −0.4 −0.4 −0.4 0 −1 −100 −100 0 0 100 100 280 ms −1 −1 levels Figure 5.5: Equipotential lines of potentials determined on the outer surface of the torso, seen from the front. 8 time steps during depolarization and 4 during repolarization are shown. 40 5.3 Discussion From cardiac activation to body surface potentials 10 ms 20 ms 30 ms 35 ms 40 ms 50 ms 60 ms 70 ms 180 ms 240 ms 260 ms 280 ms Figure 5.6: Surface equipotential lines on the front of the torso determined by Miller et al. [31] 41 5.3 Discussion From cardiac activation to body surface potentials Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 Potential (mV) 10 5 0 0 200 400 Time (ms) Figure 5.7: ECG determined in 12 leads on the torso surface, with the heart in the reference position. 42 5.3 Discussion From cardiac activation to body surface potentials Figure 5.8: The ECG determined in 12 leads by Miller et al.[31] 43 5.3 Discussion From cardiac activation to body surface potentials 44 Chapter 6 Parameter study In the previous chapter a simulation was performed with the heart in a reference orientation placed inside a torso geometry. The results were not directly resembling the results described by Miller et al. In this chapter several adjustments are made to the settings of the model, to investigate the influence of different parameters. First, the action potential duration was made variable throughout the cardiac wall. Furthermore, the influence of the orientation of the heart is studied by rotating the heart in three directions with varying angles. 6.1 6.1.1 Heterogeneous action potential duration Methods The action potential duration is known to vary between regions in the myocardium. In Miller et al. the repolarization time was altered from endocardium to epicardium and from apex to base. This could mean that later activated regions are repolarized before the early activated regions. In the previous simulation the action potential duration was kept constant. This means a that the repolarization sequence through the heart lasts just as long as the depolarization, approximately 60 ms. Here, a simulation is performed where the later activated regions have an action potential duration that is 40 ms shorter than the early activated regions. This means that the duration of the repolarization sequence is decreased to 20 ms from early activated regions to late activated regions. In a second simulation the repolarization is made simultaneous throughout the heart by decreasing the action potential duration for late activated regions with 60 ms with respect to the action potential duration in early activated regions. The longest and shortest action potentials are shown in figure 6.1. Since the repolarization of the ventricles is known to give rise to the T wave, possible effects of this variation are expected to be visible in the T waves. 6.1.2 Results Since the effects are expected to be visible in the T waves of the ECG signal, a 12 lead ECG is calculated for both simulations. The results can be seen in figure 6.2 and 6.3, the ECG determined in the previous simulation is added to serve as a reference. 45 6.1 Heterogeneous action potential duration Parameter study 20 0 Potential (mV) −20 −40 −60 −80 −100 0 50 100 150 Time (ms) 200 250 300 Figure 6.1: Action potential dynamics used in the simulation with the solid line representing the longest action potential duration applied in the early activated regions, the dotted line showing the action potential with a duration 40 ms shorter than in early activated regions and the dashed and dotted line showing the action potential with a duration 60 ms shorter than in early activated regions applied in the late activated regions. Lead I v1 Lead II v2 Lead III aVR aVL aVF v3 v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) 400 Figure 6.2: 12 Lead ECG calculated for simulation with heterogeneous action potential duration, resulting in a repolarization sequence of 20 ms, the thin line represents the reference simulation and the thick line represents the present simulation. In all leads in figure 6.2 there is a change visible with respect to the reference simulation. In all leads the amplitude of the T-wave has decreased. In figure 6.3 in all leads the effect has increased with respect to the results in figure 6.2. In most leads, the amplitude of the T wave is close to zero. 46 6.2 Orientation of the heart Lead I v1 Lead II v2 Parameter study Lead III v3 aVR aVL aVF v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) 400 Figure 6.3: 12 Lead ECG calculated for simulation with heterogeneous action potential duration resulting in a simultaneous repolarization, the thin line represents the reference simulation and the thick line represents the present simulation. 6.1.3 Discussion In both simulations the T waves of the ECG signals differ with the reference. The main change in figure 6.2 was a decrease in amplitude. When the repolarization was made simultaneous, the effect increases and the T wave disappears in most leads. Since the T wave reflects the repolarization, it could be expected that with a simultaneous repolarization the T waves disappear. A heterogeneous action potential duration clearly affects the T waves in the ECG, therefore it should be considered to incorporate this in the model. However, the ECG signals still are not yet realistic, this means that this parameter needs more study. 6.2 Orientation of the heart To study the influence of the orientation of the heart, the heart is rotated in steps around the x, y and z axes with respect to the torso. 6.2.1 Methods Rotations of π/18 radians are performed clockwise and counter clockwise around the x, y and z axis with respect to the torso. This means that 6 additional simulations are performed. 47 6.2 Orientation of the heart 6.2.2 Parameter study Results In figure 6.4 equipotential maps are shown in three time-steps during depolarization for rotation of π/18 rad around the x axis. This result is shown since it produces the largest change with respect to the reference simulation. The results of the other simulations can be seen in appendix 7. One time step during early activation is shown (10ms), one in the middle of the depolarization, 40 ms, and one at the end of the depolarization phase at 70 ms. The equipotential maps of the reference simulations are added for comparison. Factor = 1.56 Factor = 15.92 Factor==0.06 0.06 Factor 500 500 1 500 500 1 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 250 250 0 250 0 00 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 −1 −100 0 0 −1 −100 −100 0 −100 0 100 0 10 ms Factor = 4.19 0 100 40 ms Factor = 13.9 0 0 11 100 100 70 ms Factor==0.069 0.069 Factor −1 −1 levels 500 500 1 500 500 1 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 250 250 0 250 0 00 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −1 −100 0 0 −1 −100 −100 0 −100 0 100 10 ms 0 100 40 ms 0 0 11 100 100 70 ms −1 −1 levels Figure 6.4: Surface equipotential lines on the front side of the torso. Top: Results of the simulation with the heart rotated 10 degrees around the x axis in three time steps after the start of depolarization. Bottom: results of the reference simulation. 48 6.2 Orientation of the heart Parameter study In the results in figure 6.4 the main difference with the reference simulation is visible at 10 ms. Here the amplitude is lower and the minimum has a higher amplitude in relation to the maximum than in the reference. At 40 ms the amplitude in the present simulation is higher but the pattern is similar. At the end of depolarization (70 ms) the amplitude is close to zero, similar as in the results of the reference simulation. The ECG signals calculated from a simulation with the heart rotated π/18 radians around the x axis is shown in figure 6.5. Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) 400 Figure 6.5: 12 lead ECG calculated from potentials determined with the heart rotated π/18 rad around the x axis. The thin line represents the reference signal the thick line shows the result of the present simulation. In the ECG signals the main differences of the simulation with the reference can be seen in the V1 and V2 leads, where the T-wave has changed sign. In the other leads no or not much difference between both simulations can be found. 6.2.3 Discussion The result of only one simulation is described here, because the other simulations do not provide extra information. Huiskamp et al. found that the main change in the ECG after a 49 6.2 Orientation of the heart Parameter study rotation of 0.03π resulted in amplitude changes in the QRS complexes in the V 2 and V3 lead [14]. Our results also show amplitude changes in the precordial leads, however mainly the T waves are affected. Since the results presented here are the results of the simulation with the most influence, it can be concluded that the rotation of the heart is not the most important factor influencing the resulting body surface potentials. 50 Chapter 7 General discussion The aim of this study was to develop and implement a model to solve the forward problem of electrocardiography. A model is presented that is capable of producing body surface potentials from a representation of cardiac electrical activity. A 12 lead ECG can be calculated which shows a QRS peak and a T wave that are typically present in clinical ECG measurements. The results obtained with the model are compared to results obtained by Miller et al. who used a multiple dipole approach to simulate body surface potentials. Reproducing the body surface potentials and ECG’s from Miller et al. seemed not achievable with this model. Some parameters that could be of influence on the body surface potentials were further analyzed. Heterogeneity of the action potential duration throughout the cardiac wall was found to have an effect on the T-waves only in the ECG determined in all 12 leads. The effect of the orientation of the heart inside the torso was studied by rotating the heart in various directions. The rotations over an angle of π/18 rad had a minor effect on the ECG’s calculated with the model. None of the variations resulted in an improved agreement with the results of Miller et al., this means improvement of the model is needed to be able to obtain more realistic results. Improvements of the model could be made at various points. First of all, a heterogeneous action potential duration throughout the cardiac tissue should be incorporated in the model. The distribution of the heterogeneity however, has to be studied in more detail. Furthermore the geometry of the torso could have an effect on the body surface potential pattern. The geometry of the torso used in this study is highly simplified. Studies have shown that the geometry of both heart and torso should be modeled with sufficient accuracy to obtain an accurate forward solution [14]. This means that if we want to simulate realistic body surface potentials, realistic geometries should be used. Since rotation of the heart did not have major effects on the ECG, it is not expected that this will be a solution for the difficulties encountered in reproducing the ECG’s of Miller. An other improvement that could be made in the model is in the amount of nodes used to describe the cardiac wall. Since the activation of the heart is translated from an activation described on 37405 nodes in the activation model to 1430 nodes in the present model, there could be deviations in the activation sequence. To validate the effect of mesh resolution on 51 General discussion the activation pattern, further analysis of this parameter is needed. To make the model more realistic it could be considered to incorporate torso inhomogeneities into the model. It is known that especially the low conductivity of the lungs has a major effect on the smoothing of the body surface potentials [28], [6], [3]. The use of finite elements makes it relatively easy to incorporate inhomogeneities when the geometry of the concerning region is known. Despite the fact that results of Miller et al. could not be reproduced with the model, it was shown that our model is capable of producing ECG-like signals from a representation of cardiac activation. Additional analysis of parameters influencing the solution calculated with the model is needed to eventually be able to produce realistic body surface potential maps and ECG’s. After that a next step would be to model an inverse analysis of the ECG. 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IEEE Engineering in Medicine and Biology, 17:84–101, 1998. 54 BIBLIOGRAPHY BIBLIOGRAPHY [28] Klepfer RN, Johnson CR, and Macleod RS. The effect of torso inhomogeneities and anisotropies on electrocardiographic fields: A 3-d finite element study. IEEE Trans. Biomed. Eng., 44:706–719, 1997. [29] Ooij P v. Electromechanical behavior of the cardiac left ventricle with left bundle branch block. Master Thesis, Eindhoven University of Technology, 2007. [30] Einthoven W. Weiteres über das elektrokardiogram. Pflger Arch. ges. Physiol., 122:517– 548, 1908. [31] Miller WT and Geselowitz DB. Simulation studies of the electrocardiogram i. the normal heart. Circulation Research, 43:301–314, 1978. 55 BIBLIOGRAPHY BIBLIOGRAPHY 56 Appendix A Results of variations in orientation of the heart. Below, the results belonging to the simulations described in section 6.2 are shown. Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) 400 Figure A.1: 12 lead ECG signal determined with the heart rotated π/18 rad counterclockwise around the x axis. The thin line represents the reference simulation, the thick line the present simulation. 57 Results of variations in orientation of the heart. Factor = 14.5 Factor = 2.86 Factor==0.08 0.08 Factor 500 500 1 500 500 1 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 250 250 0 250 0 00 200 200 -0.2 200 200 −0.2 −0.2 −0.2 150 150 -0.4 150 150 −0.4 −0.4 −0.4 100 100 -0.6 100 100 −0.6 −0.6 −0.6 50 50 -0.8 50 50 −0.8 −0.8 −0.8 0 0 −1 −100 −100 0 -100 0 100 10 ms 0 -1 −100 0 100 40 ms 0 0 11 100 100 70 ms −1 −1 levels Figure A.2: Surface equipotential lines on the front side of the torso with the heart rotated π/18 rad counterclockwise around the x axis. Three time steps after the start of depolarization are shown. 58 Results of variations in orientation of the heart. Lead I v1 Lead II v2 Lead III v3 aVR aVL aVF v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 400 5 Potential (mV) 0 0 200 Time (ms) 400 Figure A.3: 12 lead ECG signal determined with the heart rotated clockwise (top) and counterclockwise (bottom) around the y axis. The thin line represents the reference simulation, the thick lines the present simulations. 59 Results of variations in orientation of the heart. Factor = 1.64 Factor = 17 Factor==0.068 0.068 Factor 500 500 1 500 500 1 11 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 0 00 250 250 0 250 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −1 −100 0 0 −1 −100 −100 0 −100 0 100 10 ms Factor = 3.27 0 100 20 ms Factor = 15.77 0 0 100 100 30 ms Factor==0.07 0.07 Factor −1 −1 levels 500 500 1 500 500 1 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 250 250 0 250 0 00 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −1 −100 0 0 −1 −100 −100 0 −100 0 100 10 ms 0 100 20 ms 0 0 11 100 100 30 ms −1 −1 levels Figure A.4: Surface equipotential lines on the front side of the torso with the heart rotated π/18 rad clockwise (top) and counterclockwise (bottom) around the y axis. Three time steps after the start of depolarization are shown. 60 Results of variations in orientation of the heart. Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 5 Potential (mV) 0 0 200 Time (ms) Lead I Lead II Lead III aVR aVL aVF v1 v2 v3 v4 v5 v6 400 5 (mV) 0 0 200 Time (ms) 400 Figure A.5: 12 lead ECG signal determined with the heart rotated clockwise (top) and counterclockwise (bottom) around the z axis. The thin line represents the reference simulation, the thick lines the present simulations. 61 Results of variations in orientation of the heart. Factor = 4.02 Factor = 11.85 Factor==0.09 0.09 Factor 500 500 1 500 500 1 11 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 0 00 250 250 0 250 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −1 −100 0 0 −1 −100 −100 0 −100 0 100 10 ms Factor = 0.98 0 100 40 ms Factor = 13.56 0 0 100 100 70 ms Factor==0.055 0.055 Factor −1 −1 levels 500 500 1 500 500 1 450 450 0.8 450 0.8 0.8 0.8 400 400 0.6 400 0.6 0.6 0.6 350 350 0.4 350 0.4 0.4 0.4 300 300 0.2 300 0.2 0.2 0.2 250 250 0 250 0 00 200 200 −0.2 200 200 −0.2 −0.2 −0.2 150 150 −0.4 150 150 −0.4 −0.4 −0.4 100 100 −0.6 100 100 −0.6 −0.6 −0.6 50 50 −0.8 50 50 −0.8 −0.8 −0.8 0 −1 −100 0 0 −1 −100 −100 0 −100 0 100 10 ms 0 100 40 ms 0 0 11 100 100 70 ms −1 −1 levels Figure A.6: Surface equipotential lines on the front side of the torso with the heart rotated π/18 rad clockwise (top) and counterclockwise (bottom) around the z axis. Three time steps after the start of depolarization are shown. 62