Simulations of magnetocardiographic signals using realistic geometry ...

Adv. Radio Sci., 10, 85–91, 2012 www.adv-radio-sci.net/10/85/2012/ doi:10.5194/ars-10-85-2012 © Author(s) 2012. CC Attribution 3.0 License.

Advances in Radio Science

Simulations of magnetocardiographic signals using realistic geometry models of the heart and torso C. V. Motrescu and L. Klinkenbusch Institut für Elektrotechnik und Informationstechnik, Christian-Albrechts-Universität zu Kiel, Germany Correspondence to: L. Klinkenbusch ([email protected])

Abstract. Although the first measurement of the cardiac magnetic field was reported almost half a century ago magnetocardiography (MCG) is not yet widely used as a clinical diagnostic technique. With the development of a new generation of magnetoelectric sensors it is believed that MCG will become widely accepted in the clinical diagnosis. Our goal is to build a computer-based tool for medical diagnosis and to use it for the clarification of open electro-physiological questions. Here we present results from modelling of the cardiac electrical activity and computation of the generated magnetic field. For the simulations we use MRT-based anatomical models of the human atria and ventricles where the shape of the action potential is determined by ionic currents passing through the cardiac cell membranes. The monodomain reaction-diffusion equation is chosen for the description of the heart’s electrical activity. This equation is solved for the transmembrane voltage which is in turn used to calculate current densities at discrete time instants. In subsequent simulations these current densities represent primary sources of magnetostatic fields arising from a volume conduction problem. In these simulations the heart is placed in a realistic torso model where the lungs are also considered. Both, the volume conduction problem as well as the reaction-diffusion problem are modelled using Finite-Element techniques.

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the variation in time of the magnetic fields produced by the heart outside the body and the graphical representation of the recorded magnetic signals is called magnetocardiogram. Whereas the ECG is a widely used technique in the clinical diagnosis, the MCG is not yet. The reasoning for this may lay in the fact that currently used magnetic sensors (Superconducting QUantum Interference Devices – SQUIDs) require extremely low temperatures as well as magnetically shielded environments. Therefore, electrocardiography (ECG) is considerably more attractive for clinicians, though MCG and ECG supposedly supply disjunctive information and MCG offers several advantages (e.g. no electrodes needed, magnetic transparency of tissue). Efforts are nowadays directed to the overcoming of the above mentioned disadvantages of MCG by using magnetoelectric composite materials (Jahns et al., 2011) with giant magnetoelectric effect. In the same time computational tools for modelling the electrical activity of the heart as well as for solving the inverse problem of magnetocardiography are developed as a framework for studies on open electrophysiological questions such as the non-response of some patients to bi-ventricular stimulation. In this paper we present a modelling of the electrical activity of the heart at both, cellular and organ level and a method to compute the associated magnetic field produced by a healthy heart.

Introduction 2

The human body can be seen as a volume conductor where the electrical activity of the heart gives rise to small electric currents in the body with associated potential distributions. The electrocardiography (ECG) records the variation in time of the electrical potential produced by the heart at the surface of the body. The graphical representation of the recorded potentials depicts the well known electrocardiogram. However, the same currents in the body give also rise to weak magnetic fields. The magnetocardiography (MCG) records

Biological and electro-physiological background

The heart is a hollow muscular organ being structured on four chambers i.e. two atria and two ventricles. The function of the heart is to pump blood. This function is achieved by contractions of the muscles composing the atria and ventricles (working myocardium). The contraction of the cardiac muscles is electrically controlled by the excitation-conduction system of the heart. This system is made of cells that are able to fire impulses – a property called autorhythmicity.

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

In contrast to the atrial and ventricular myocytes (of the working myocardium), the cells belonging to the excitationFig. 1. Schematic representation of a section through the human heart showing the excitation-conduction system. Picture adapted conduction system of the heart do not need an external stimKlinkenbusch: Simulations of Magnetocardiographic Signals from (Malmivuo and Plonsey , 1995). ulus for the depolarization because they are able to depolar86 C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals ize themselves initiating thus electrical impulses (pacemaker cells). The transmembrane voltage propagates from a myocyte to the neighboring myocytes in the form of an action potential. The spread of the action potential is supported by some rapid intercellular channels called gap junctions which couple the interiors of adjacent cells. Thus, at the organ level the excitation propagates in the form of isochronal surfaces also called wave-fronts. Electrically active cells of the heart Fig. 2. Stylized form of the normal electrocardiogram. are located within these wave-fronts and they are usually asFig. 2. Stylized form of the normal electrocardiogram. sociated to current dipoles. It is these moving current dipoles with their accompanying conduction currents that give rise to 3 The normal electrocardiogram the magnetic field of the heart. gram and the repolarization of the ventricles that determines 120 theThe T-wave. excitation-conduction system of the heart (see Fig. 1) The atrial repolarization place during the(AV) QRSconsists of the sinoatrial (SA)takes node, the atrioventricular 3 The normal electrocardiogram complex butcommon it is masked much node, the bundle by alsothe called the stronger bundle ofprocess His, the of left and right bundle branches and the Purkinje fibres. ventricular depolarization. The excitation-conduction system of the heart (see Fig. 1) Under normal conditions, the SA node generates action consists of the sinoatrial (SA) node, the atrioventricular (AV) potentials at the rate of about 70 pulses/min (Malmivuo and 4 Plonsey, Mathematical modelling at the cellular 1995) which spread through the right level and left atria node, the common bundle also called the bundle of His, the depolarizing them and determining the occurrence of the so left and right bundle branches and the Purkinje fibres. 125 As already mentioned in Sect. 2 the electrical activity called P-wave in the recorded electrocardiogram (see Fig.of 2).the UnderFig. normal conditions, the SAofnode generates 1. Schematic representation a section through action the human The only conduction path between the atria and the ventricardiac myocyte is mainly the result of the ionic currents that potentials at the rate representation of about 70 pulses/min (Malmivuo and heart showing the excitation-conduction Picture adapted Fig. 1. Schematic of a system. section through the human cles is provided by the AV node which has a low conduction are crossing the cell membrane through ion channels and (Malmivuo and Plonsey, 1995).the right and left atria Plonsey ,from 1995) which spread through heart showing the excitation-conduction system. Picture adapted velocity andThese consequently introduces a delay in the propagatransporters. currents determine the shape of the acdepolarizing them and determining the occurrence of the so tion of the excitation that corresponds to the PQ-segment of at from (Malmivuo and Plonsey , 1995). tion potential of a cell. The available cellular ionic models called P-wave in the recorded electrocardiogram (see Fig. 2). the electrocardiogram. Once the AV node has been passed, Electrical impulses result from ionic activity at the cellular 130 present can be divided into three classes (Clayton and PanThe onlylevel. conduction pathcell between theofatria and themuscle ventri-cells the excitation propagates through a high conduction-velocity At rest, the membrane the cardiac filov , 2008): cles is provided by the AV node which has a low conduction system (the His-Purkinje system) to the inner sides of the (myocytes) composing the atria and ventricles, is electrically velocity polarized and consequently introduces a delay in the propagawall. The ventricular depolarization in the sense that the inner side of the membrane simplified two-variable models that reduce determines the behavior –ventricular the QRS complex on the electrocardiogram. After the comtion of the excitation that corresponds to the PQ-segment of is more electronegative than the outer side of the membrane. of all the ionic currents to two variables that describe plete ventricular depolarization, it follows a plateau phase the electrocardiogram. the AV node has been passed, This produces theOnce so called transmembrane voltage. When a excitation and recovery; which corresponds to the ST-segment on the electrocardiostrongpropagates enough stimulus is applied to the membrane, certain the excitation through a high conduction-velocity and the repolarization of the ventricles determinesinprotein channels will open letting ionic of species biophysically detailed models that use that experimental –gram system (the His-Purkinje system) to theparticular inner sides the 135to the T-wave. cross the membrane along their electro-chemical gradients. formation about the voltage and time dependence of ion ventricular wall. The ventricular depolarization determines The atrial repolarization takes place during the QRSThis depolarization of the membrane is followed by a rechannel conductances (first generation) as well as inthe QRS complex on the electrocardiogram. After the comcomplex but it is masked by the much stronger process of polarization which brings the myocyte in the resting state. tracellular ionic concentrations (second generation) in plete ventricular depolarization, it follows a plateau phase ventricular depolarization. In contrast to the atrial and ventricular myocytes (of the order to reproduce the action potential; which corresponds to the ST-segment on the electrocardioFig. 2. Stylized of the normal electrocardiogram. working form myocardium), the cells belonging to the excitationconduction system of the heart do not need an external stim4 Mathematical modelling at the cellular level ulus for the depolarization because they are able to depolarize themselves initiating thus electrical impulses (pacemaker gram andcells). the The repolarization the ventricles thata mydetermines As already mentioned in Sect. 2 the electrical activity of transmembraneof voltage propagates from the cardiac myocyte is mainly the result of the ionic curocyte to the neighboring myocytes in the form of an action the T-wave. rents that are crossing the cell membrane through ion chanpotential. The spread of the action potential is supported by The atrial repolarization takes place during the QRSnels and transporters. These currents determine the shape some rapid intercellular channels called gap junctions which of the complexcouple but the it is masked by the much stronger process of action potential of a cell. The available cellular interiors of adjacent cells. Thus, at the organ level ionic models at present can be divided into three classes the excitation propagates in the form of isochronal surfaces ventricular depolarization. (Clayton and Panfilov, 2008): also called wave-fronts. Electrically active cells of the heart are located within these wave-fronts and they are usually as– simplified two-variable models that reduce the behavior sociated to current dipoles. It is these moving current dipoles of all the ionic currents to two variables that describe 4 Mathematical modelling at thecurrents cellular levelrise with their accompanying conduction that gives excitation and recovery; to the magnetic field of the heart.

As already mentioned in Sect. 2 the electrical activity of the – biophysically detailed models that use experimental incardiac myocyte is mainly the result of the ionic currents thatformation about the voltage and time dependence of ion are crossing the cell membrane through ion channels and Adv. Radio Sci., 10, 85–91, 2012 www.adv-radio-sci.net/10/85/2012/ transporters. These currents determine the shape of the action potential of a cell. The available cellular ionic models at present can be divided into three classes (Clayton and Pan-

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C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals

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Fig. 4. MRT-based anatomical mesh models provided with the freely available anatomical software code ECGsim (van Oosterom, 2004). with th Fig. 4. MRT-based mesh models provided Left: heart and lungs inside the torso. Right: detailed view of the heart. software code ECGsim (van Oosterom, 2004). Left Fig. 3. Action potential shape of the Luo-Rudy phase freely 1 cellularavailable ionic model. heart and lungs inside the torso. Right: detailed view of the heart.

Action potential shape of the Luo-Rudy phase 1 cellular [0,1] and describing the activation, inactivation and recovery of the ion channel and VS being the membrane voltage channel conductances (first generation) as well as inmodel. of the species “S” alone. The gating variables are obtained tracellular ionic concentrations (second generation) in order to reproduce the action potential;

by solving a system of eight nonlinear ordinary differential equations (ODEs) of the general form: 7 Method – reduced cardiac models that simplify the underlying dxS xS∞ (Vm ) − xS electrophysiology to a minimum number of currents for = (3) dt τS (Vm ) The byprocedure adopted by us for computing the magneti which the activation and inactivation is controlled some empirical variables. where xby theheart steady-state value of single gating variS∞ is field produced the consists ofa three subsequent step able xS and τS is the time constant for the return of the variwhere output one step is after useda voltage as an perturbation. input for the nex Choosing the appropriate cellular model for 225 a certain prob- theable x to itsof steady-state value lem is a matter of balancing its benefits and limitations. For voltage dependencies of xS∞ and τS are step. TheThe steps are also differentiated byexperimentally the involved tim this work we have chosen the Luo-Rudy phase 1 model of determined. space scales. mammalian myocytes (Luo and Rudy, 1991). Thisand cellular The shape of the action potential as given by the Luo-Rudy model belongs to the first generation of the biophysically dephase 1 cellular and used in this work is shownofin electrica In the first step wemodel simulate the propagation tailed models and it is considered that this model offers a Fig. 3. excitation in the isolated heart shown in the right-hand sid good balance between electrophysiological details and computational efficiency. Within this model the Hodgkin-Huxley 230 of Fig. 4. As a result we obtain the distribution of the trans 5 Mathematical modelling at the tissue level type of formalism is used to describe the action potential. membrane voltage in the heart at discrete time instants during The rate of change of the membrane voltage Vm is given by: At the tissue level the granular structure may be neglected a complete heart beat. dVm and the cardiac muscle may be modelled as a functional synCm = −(Iion + Ist ) (1) In the second step the a volume problem cytium in which membrane conduction voltage propagates smoothly. is solved dt Under this assumption the action potential can be considwhere Cm is the membrane capacitance, Ist is anwhere applied the driving force is represented by impressed curren ered to propagate by diffusion in an excitable medium that stimulus current and Iion is the sum of six ionic currents: a 235 densities in theofheart areThe computed using consists one or which two phases. bi-domain model con-the trans fast sodium current (INa ), a slow inward current (Isi ), a timesiders the cardiac muscle to be a two-phase medium conmembrane voltage resulted at each time instant considered dependent potassium current (Ik ), a time-independent potassisting of intracellular and extracellular spaces separated at sium current (Ik1 ), a plateau potassium current (Ikp in) and thea simulations the (first) step. The each point in of space by previous the cellular membrane. Although thiscomputa time-independent background current (Ib ). In general, each is a very detailed model, it is also computationally expentional current IS carried by an ionic species “S” through an ion domain consists of the heart and lungs inside the torso sive. The simpler monodomain model is based on a single channel is expressed as: as shown phase on the left-hand ofmuscle Fig. 4. output description of theside cardiac but The it is still able of thes to reproduce many of the by experimentally phenom- and con 240 simulations is represented the totalobserved (impressed (2) IS = GS + xs1 xs2 ...xsn (Vm − VS ) ena with a much higher numerical efficiency. The simulation duction) current density in the entire torso. with GS being the maximal conductance of the ion channel, of a problem with the monodomain model can be by a facxs being gating variables with values varying in the interval tor of ten or more than the field same problem simulated In the third step the faster magnetic outside the body pro

ts of one or two phases. The bi-domain model conthe cardiac muscle to be a two-phase medium cong of intracellular and extracellular spaces separated at point in space by the cellular membrane. Although this ery detailed model, it is also computationally expenThe simpler monodomain model is based on a single description of the cardiac muscle but it is still able roduce many of the experimentally observed phenomith a much higher numerical efficiency. The simulation roblem with the monodomain model can be by a facten or more faster than the same problem simulated he bi-domain model (Clayton and Panfilov , 2008). In ork we consider that the action potential propagates fusion in the monodomain model of the cardiac tisith local excitation of the membrane voltage. The so monodomain reaction-diffusion model consists of a olic partial differential equation (PDE) to a density in the torso, is computed duced bycoupled the total current www.adv-radio-sci.net/10/85/2012/ Radio Sci., 10, 85–91, 2012 m of ODEs (Whiteley , 2006): as a magnetostatic problemAdv. at each time instant considered [ ] in the previous (second) step. ∂V 245

C.V. Motrescu and L. Klinkenbusch: Simula and L. Klinkenbusch: Simulations of Magnetocardiographic Signals 88


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Fig. 5. The spread of electrical excitation in the heart during the atrial row), and ventricular (bot-the Fig. 5. depolarization The spread of(top electrical excitation in depolarization the heart during tom row).

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with the bi-domain model (Clayton and Panfilov, 2008). In this work we consider that the action potential propagates meshes by using the monodomain ”Tetgen” - amodel freelyofavailable software by diffusion in the the cardiac tissuefor withresearch local excitation of the membrane voltage.(Si, The2006). so code and non-commercial purposes called reaction-diffusion model consists ofwith a The atriamonodomain are discretized with 1.23 millions mesh nodes partial differential equation coupledcontain to a an parabolic average edge length of 0.8 mm and (PDE) the ventricles system of ODEs 2006): 1.43 millions mesh(Whiteley, nodes with an average edge length of 0.95 Fig. 6. Position of the magnetic sensor array (red color) relative to mm. ∂Vm ∇ · (σ ∇Vm ) = β Cm (4) + I + Ist The simulations are ∂tbased ion on the monodomain reaction- the body.

diffusion model−1described by Eq. (4). Each node of the where σ (Sm ) is the electrical conductivity tensor, Vm computational domain contains the Luo-Rudy ionic cellular (V) is the transmembrane voltage, β (m−1 ) is the surface model described in Sect. 4. This cellular model withspecon−2 to volume ratio of the cardiac cells, Cm (Fm ) is the stant action potential duration (APD) is used for both, atria −2 cific membrane capacitance, Iion (Am ) is the current flow andthrough ventricles. Theofcardiac areand assumed to−2be) is homounit area the cellmuscles membrane Ist (Am an geneous isotropic with the electricalcurrent conductivity appliedand stimulation current. Thescalar net membrane Sm−1 . Other parameter values used in the simulaσ = 0.25 ∂V tion Cmm =m1+µFcm Imare: = C Iion −2 and β = 1400 cm−1 . (5) ∂t The simulations are performed within ”CHASTE” - the is givenHeart by the cellular ionic models such as -the one ”Cancer, And Soft-Tissue Environment” a multidescribed in Sect. 4. purpose software library for simulations in the area of biology (Pitt-Francis et al., 2009) provided under the LGPL license by a teammodels from the Computer Science Department 6 Anatomical of the University of Oxford, the UK. Examples showing the Anatomical models of the heart, torso used this membrane voltage distribution atlungs threeand different timeininstants work are those provided with the freely available software during the atrial and ventricular depolarization are provided forDue simulations of electrocardiograms called “ECGsim” in code Fig. 5. to the constant action potential durations con(van Oosterom, 2004). The models are based on magnetic sidered in the simulations, the repolarizations follow a simiresonance tomography (MRT) data of a healthy young male laroftime course. Rectangular current pulses of 2 ms duration 22 yr. They are available in the form of triangulated surarefaces usedasfor the initial The areas of initial stimshown in Fig.stimulations. 4. ulations are taken from the ECGsim (van Oosterom, 2004) software where depolarization isochrones on the surfaces en7 Method veloping the atria and ventricles are obtained from the potentials on the surface of the body by following an inverse The procedure adopted by us for computing the magnetic procedure. The atria stimulated inthree an area corresponding field produced by theare heart consists of subsequent steps to where the position of the SA-node. In the case the output of one step is used as an input of forthe the ventrinext cles, three areas inside the left ventricle and one area inside the right ventricle are initially stimulated with some small Adv. Radio Sci., 10, 85–91, 2012 time delays between them. Animated representations of the spread of electrical excitation during the depolarization and

step. The steps are also differentiated by the involved time and space scales. In the first step we simulate the propagation of electrical excitation in the isolated heart shown in the right-hand side of Fig. 4. As a result we obtain the distribution of the transmembrane voltage in the heart at discrete time instants during a complete heart beat. In the second step a volume conduction problem is solved where the driving force is represented by impressed current densities in the heart which are computed using the transmembrane voltage resulted at each time instant considered in the simulations of the previous (first) step. The computational domain consists of the heart and lungs inside the torso as shown on the left-hand side of Fig. 4. The output of these simulations is represented by the total (impressed and conduction) current density in the entire torso. In the third step the magnetic field outside the body produced by the total current density in the torso, is computed as a magnetostatic problem at each time instant considered in the previous (second) step. Thus slowly varying magnetocardiographic signals with time are obtained from a sequence of stationary computations.

Fig. 6. Position of the magnetic sensor array (red the body.

repolarization of the atria and ventricles are a the form of short films. 9 Computation of currents in the body

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hand side of the Fig. 4 are transformed to tetrahedral volume meshes by using the “Tetgen” – a freely available software code for research and non-commercial purposes (Si, 2006). −11 The atria are discretized x 10with 1.23 millions mesh nodes with 2 an average edge length of 0.8 mm and the ventricles6 contain 1.43 millions mesh nodes with an average edge length of 0.95 mm. 1 The simulations are based on the monodomain reactiondiffusion model described by Eq. (4). Each node of the 0 computational domain contains the Luo-Rudy ionic cellular model described in Sect. 4. This cellular model with con−1 duration (APD) is used for both, atria stant action potential and ventricles. The cardiac muscles are assumed to be homogeneous and isotropic with the scalar electrical conduc−2 tivity σ = 0.25 Sm−1 . Other parameter values used in the simulation are: Cm = 1 µFcm−2 and β = 1400 cm−1 . The simulations are performed within “CHASTE” – the 0 200 400 600 800 “Cancer, Heart And Soft-Tissue Time Environment” – a multi[ms] purpose software library for simulations in the area of biology (Pitt-Francis et al., 2009) provided under the LGPL license by a team from the Computer Science Department of the University of Oxford, the UK.

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Fig. 7. Magnetic flux density component onduring the chest threeperpendicular different time instants the atrial and ventricuThe spread of excitation in the heart is simulated with Finitelar depolarization are provided in Fig. 5. Due to the conrecorded as a function of time at the position of each node of the Element methods. For this purpose the surface meshes of the stant action potential durations considered in the simulations, atria and ventricles composing the heart shown 6. on the rightsensor array shown in Fig. the repolarizations follow a similar time course. Rectangular

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Fig. 7. Magnetic flux density component perpendicular on the chest recorded as a function of time at the position of each node of the sensor array shown in Fig. 6.

current pulses of 2 ms duration are used for the initial stimulations. The areas of initial stimulations are taken from the ECGsim (van Oosterom, 2004) software where depolariza−12 tionx isochrones on the surfaces enveloping the atria and ven10 10 tricles are obtained from the potentials 12 on the surface of the body by following an inverse procedure. The atria are stim8 ulated in an area corresponding to the position of the SAnode. In the case of the ventricles, three areas inside the left 6 ventricle and one area inside the right ventricle are initially 4 stimulated with some small time delays between them. Animated representations of the spread of electrical excitation 2 during the depolarization and repolarization of the atria and ventricles are also provided in the form of short films. 0 −2

9 Computation of currents in the body −4 0 200 400 600 800 Time [ms] The electrical excitation in the heart gives rise to currents in the entire body. Impressed current densities in the heart can be computed as Je = −σ ∇Vm (Aoki et al., 1987) using the membrane voltage Vm already computed at discrete time instants during the signals spread of excitation in the heart. The asmagnetocardiographic recorded sociated conduction currents Jc can then be determined by

Fig. 8. Detailed view of the at the positions of the sensors denoted by 6 (left) and 12 (right) in Fig. 7. www.adv-radio-sci.net/10/85/2012/ Adv. Radio Sci., 10, 85–91, 2012

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Fig. 8. Detailed view of the magnetocardiographic signals signals recorded Fig. 8. Detailed view of the magnetocardiographic recorded at the positions of the sensors denoted by 6 (left) and 12 (right) at the positions of the sensors denoted by 6 (left) and 12 (right)inin Fig. 7. Fig. 7.

solving a stationary volume conduction problem described Fig. 7. figure∇J allt the are by In thethis equation = 0 magnetocardiographic where Jt = Je + Jc is thesignals total current density. Wesame compute the Detailed total current density in the enrepresented on the scale. views of the signals tire torso shown on the6 left-hand sidegiven of Fig. by using the recorded by the sensors and 12 are in4Fig. 8. When commercially software for sensors Finite-Element analyplotting the signalsavailable recorded by all the in a single figsis called COMSOL Multiphysics (COMSOL Multiphysics ure, the so called butterfly plot is obtained as shown in Fig. 9. v. 4.2, 2011). The electrical conductivity values used for the torso and345 the lungs are σtorso = 0.2 Sm−1 and σlung = 0.04 Sm−1 , respectively.

11 Conclusions

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11 Conclusions puting conduction currents in the inhomogeneous and irregularly shaped torso. Future work will focus on the reduction 395 The method used in this work proved to be able to provide of the number of unknowns in order to reduce the computarealistic values for the computed magnetic fields of the heart. tion time as well as for more conveniently solving the correWe coupled different state-of-the-art existent software codes sponding inverse problem. and realistic anatomical models in order to compute the magnetic field produced by the heart. We used the magneto-static approximation to compute slowly varying magnetic field 12 withAcknowledgements time at successive time instants. The Finite-Element method (FEM) with tetrahedral elements was used for comWe areconduction thankful currents to all the persons who contributed puting in the inhomogeneous and irreg- to the freely available software codes (ECGsim, TetGen, ularly shaped torso. Future work will focus on the reduction CHASTE) usedofinunknowns this work.inThis work was supported by the of the number order to reduce the computation time as well as for more conveniently solving the correDeutsche Forschungsgemeinschaft (DFG) within the SFB sponding inverse problem. 855: Magnetoelectric composite material - Future biomag-

Computation of magnetic fieldsto (MCG signals) The 10 method used in this work proved be able to provide realistic values for the computed magnetic fields of the heart. The currents in the torso produce magnetic fields inside and We coupled different existent software codes outside the body. state-of-the-art We compute these magnetic fields using and realistic anatomical models in order to compute the mag350 Ampere’s law ∇ ×H = Jt at each time instant previously conneticsidered field produced by the heart. Wetotal used the magneto-static in the computation of the current density in the approximation to compute varying magneticMulfield torso Jt . These simulationsslowly are coupled in COMSOL netic interfaces. v. 4.2, with those with tiphysics time at (COMSOL successive Multiphysics time instants. The2011) Finite-Element for the solution of the volume conduction problems. All bimethod (FEM) with tetrahedral elements was used for com- Acknowledgements. We are thankful to all the persons who conological tissues are considered to be non-magnetic with the relative magnetic permeability µr = 1. For the results we consider a four by four magnetic sensor array positioned just in front of the torso as shown in Fig. 6.355 The nodes of the magnetic sensor array are spaced 8.2 cm and 7.2 cm apart from each other on the horizontal and vertical directions, respectively. The variation in time of the magnetic field component perpendicular to the chest computed in each node of the mag-360 netic sensor array is shown in Fig. 7. In this figure all the magnetocardiographic signals are represented on the same scale. Detailed views of the signals recorded by the sensors 6 and 12 are given in Fig. 8. When plotting the signals recorded by all the sensors in a single figure, the so called butterfly plot365 is obtained as shown in Fig. 9.

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tributed to the freely available software codes (ECGsim, TetGen, CHASTE) used in this work. This work was supported References by the Deutsche Forschungsgemeinschaft (DFG) within the SFB 855: Magnetoelectric composite material – Future biomagnetic Aoki, M. et al.: Three-Dimensional Simulation of the Ventricular interfaces.

Depolarisation and Repolarisation Processes and Body Surface Potentials: Normal Heart and Bundle Branch Block, IEEE T. Bio-Med. Eng., 34, 6:454-462, 1987 References Clayton, R. H. and Panfilov, A. V.: A guide to modelling cardiac electrical activity in anatomically detailed ventricles, Prog. BioAoki, M., Okamoto, Y., Musha, T., and Harumi, K.-I.: Threephys. Mol. Bio., 96, 19-43, 2008 Dimensional Simulation of the Ventricular Depolarisation and r COMSOL AB:Processes COMSOL 4.2, Repolarisation and BodyMultiphysics Surface Potentials: v. Normal http://www.comsol.se, 2011 Heart and Bundle Branch Block, IEEE T. Bio-Med. Eng., 34, Dössel, O.: Inverse 454–462, 1987. Problem of Electro- and Magnetocardiography: ReviewR. and Progress, Journal of BioelectroClayton, H.Recent and Panfilov, A. International V.: A guide to modelling cardiac magnetism, Volume 2, Number 2,detailed 2000 ventricles, Prog. Bioelectrical activity in anatomically Jahns, R. Mol. et al.:Bio., Noise phys. 96,performance 19–43, 2008.of magnetometers with resonant r v. 4.2, thin-film magnetoelectric sensors, IEEE T. Instrum. COMSOL AB: COMSOL Multiphysics availableMeas., at: http:60, 8:2995-3001, 2011 last access date: July 7, 2012. //www.comsol.com, Luo, C. and Rudy, Y.: A Model of the Ventricular Cardiac Action Potential - Depolarisation, Repolarisation and Their Interaction, www.adv-radio-sci.net/10/85/2012/ Circ. Res., 68, 1501-1526, 1991

Malm an ed Naka io Pitt-F de 18 Sato, gr 30 Si, B Th Ja Smit tro of van O fo 16 Whit of En Zaza tro Lo

C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals Dössel, O.: Inverse Problem of Electro- and Magnetocardiography: Review and Recent Progress, International Journal of Bioelectromagnetism, vol. 2, no. 2. Online available at: http://ijbem.k. hosei.ac.jp/volume2/number2/toc.htm (last access date: July 7, 2012), 2000 Jahns, R., Greve, H., Woltermann, E., Quandt, E., and Knöchel, R.H. : Noise performance of magnetometers with resonant thinfilm magnetoelectric sensors, IEEE T. Instrum. Meas., 60, 2995– 3001, 2011. Luo, C. and Rudy, Y.: A Model of the Ventricular Cardiac Action Potential – Depolarisation, Repolarisation and Their Interaction, Circ. Res., 68, 1501–1526, 1991. Malmivuo, J. and Plonsey, R.: Bioelectromagnetism – Principles and Applications of Bioelectric and Biomagnetic Fields, Webedition, Oxford University Press, New York, 1995. Nakaya, Y. and Mori, H.: Magnetocardiography, Clin. Phys. Physiol. M., 13, 191–229, 1992. Pitt-Francis, J., Pathmanathan, P., Bernabeu, M. O., Bordas, R., Cooper, J., Fletcher, A. G., Mirams, G. R., Murray, P. J., Osbourne, J., Walter, A., Chapman, S. J., Garny, A., van Leeuwen, I. M. M., Maini, P. K., Rodriguez, B., Waters, S. L., Whiteley, J. P., Byrne, H. M., and Gavaghan, D. J.: Chaste: A test-driven approach to software development for biological modelling, Comput. Phys. Commun., 180, 2452–2471, 2009.

www.adv-radio-sci.net/10/85/2012/

91

Sato, M., Terada, Y., Mitsui,T., Miyashita,T., Kandori, A., and Tsukada K.: Visualization of atrial excitation by magnetocardiogram, The International Journal of Cardiovascular Imaging, 18, 305–312, 2002 Si, B. H.: TetGen, A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator, User’s manual v1.4, 18 January 2006, available at: http://tetgen.berlios.de(last access date: July 7, 2012) 2006. Smith, F. E., Langley, P., van Leeuwen, P., Hailer, B., Trahms, L., Steinhoff, U., Bourke, J.P., and Murray, A.: Comparison of magnetocardiography and electrocardiography: a study of automatic measurement of dispersion of ventricular repolarization, Europace Advance Access, Vol. 8 (10), 887-893, 2006. van Oosterom, A. and Oostendorp, T.: ECGSIM; an interactive tool for studying the genesis of QRST waveforms, Heart, available at: http://www.ecgsim.org, 90, 165–168, 2004. Whiteley, J. P.: An efficient numerical technique for the solution of the monodomain and bidomain equations, IEEE T. Bio-Med. Eng., 53, 2139–2147, 2006. Zaza, A., Wilders, R., and Opthof, T.: Cellular electrophysiology, in: Comprehensive Electrocardiography, 2nd Edn., edited by: Macfarlane, P.W., van Oosterom, A., Pahlm, O., Kligfield, P., Janse, M., and Camm, J., Springer-Verlag London, 107–144, 2011.

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