Inverse Problem and Modelling of Cardiac Sources in Cardiography

Gozd Martuljek, Slovenia
September 29-30, 1995

Invited papers

Forward and inverse computation in ECG and MCG. (T. Oostendorp, Nijmegen)
Modelling of current sources in cardiology. (J. Nenonen, Helsinki)
Volume conductors, volume currents in cardiology - modelling. (V. Jazbinsek, Ljubljana)
Clinical multichannel measurements of cardiac magnetic field and electric potential. (M. Burghoff, Berlin)
Anatomic picture: 3-D cardiac image processing. (I. Magnin, Lyon)
Software for MCG. (S.N. Erné , Ulm)
Localization of arrhythmogenic substrate - validation by successful catheter ablation procedures. (M. Oeff, Berlin)
Analysis of depolarization and repolarization for the characterization of the arrhythmogenic substrate of the myocardium (P. Stevens, Berlin)

Contributed papers (not complete)

The software package CURRY TM and its application in cardiography (U. Katscher, Hamburg)
Effects of source extension on electric and magnetic location: A simulation study (C. Del Gratta, Chieti)

Forward and inverse computations in ECG and MCG

Thom Oostendorp

Department of Medical Physics Biophysics
University of Nijmegen, Netherlands

In order to evaluate the condition of the heart from measured ECG or MCG signals, the relation between the current sources at the heart and the electric and magnetic fields they generate has to be known. The problem of the estimation of the source from measured fields is called the inverse problem. In order to solve the inverse problem, an expression is needed for the fields generated by current sources of known properties, as obtained by forward computations.

The forward problem can only be solved analytically for simple volume conductor geometries (semi-infinite space, spheres etc.). The two most commonly used methods to solve the forward problem numerically for volume conductors of arbitrary shape are the Boundary Element Method (BEM) and the Finite Element Method (FEM). The BEM [1] is only applicable to isotropic, piecewise homogeneous volume conductors. The computations involve discretization nodes at the interfaces of the homogeneous compartments only. This makes the BEM much more computational efficient than the FEM [2], in which the complete 3D volume conductor is discretized. The FEM however can handle anisotropic conductivities.

There is no unique solution to the inverse problem. As a consequence, the current source distribution generating the external field can not be determined without the use of additional information. Commonly, this additional information follows implicitly from the choice of a source model. For example, if a dipolar source model is used, the source distribution is implicitly constrained to a point-like, dipolar nature, and the solution is unique. For the solution to make sense, however, it is vital that the additional information corresponds to physiological properties of the source. As the current source generating the ECG and MCG is essentially distributed, the dipole source model is not an appropriate model for the cardiac sources.

The two most popular models for the cardiac sources are the epicardial potential model [1] and the double layer model [3]. In the first model, the source is described as the equivalent boundary potential impressed at the epicardium for each time instant separately. The solution of the inverse problem for this source model is a series of epicardial potential distributions. In the double layer model the spreading of depolarization over the epicardium is modeled. The epicardium is discretized into a large number of elements that, from the moment the depolarization wave front reaches them, act as dipole layer sources of a certain strength. For this model, the inverse computations results in a map of the activation sequence at the epicardium.

Both the epicardial potential and the double layer model are two dimensional distributed source models, with many degrees of freedom. In the external field, not all of the degrees of freedom are expressed, as many different source patterns generate fields that differ less the noise level (remote sensing problem). As a consequence, the inverse problem involving these source model is mathematically ill-posed, i.e: small differences in the recorded fields will yield completely different solutions. The ill-posedness is overcome by - again - including additional information by means of regularization. In regularization a penalty function, expressing some undesired property of the solution, is included in the inverse procedure. Popular penalty functions (commonly called regularization operators) are the norm of the norm of the solution (i.e. small source strength is prefferd) and the surface Laplacian of the solution (i.e. smooth solution are preferred).

Regularization is effectively a method to select the most desirable solution from the range of possible solutions. Obviously, this only makes sense if the regularization operator expresses a realistic physiological property of the source. For instance, for the double layer model Laplacian regularization is quite suitable, as the mean Laplacian of the activation times is a measure for the velocity with which the depolarization spreads over the epicardium, which is a well-described physiological quantity. The amount of regularization can be chosen such that this quantity reaches a realistic value.


[1] R.C. Barr et al. Determining Surface Potentials from Current Dipoles with Application to Electrocardiography, IEEE Trans. Biomed. Eng., BME-13:88--92, 1966.
[2] Y. Yamashita et al. Source-field relationships for cardiac generators on the heart surface based on their transfer coefficients. IEEE Trans. Biomed. Eng., BME-32: 964-970
[3] G.J.M. Huiskamp et al. The depolarization sequence of the human heart surface computed from measured body surface potentials. IEEE Trans. Biomed. Eng., BME-35: 1047-1058, 1988.
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Modelling of current sources in cardiology

Jukka Nenonen

Helsinki University of Technology, Laboratory of Biomedical Engineering
Rakentajanaukio 2C, 02150 Espoo, Finland

dvi source of this abstract (TeX), ps source of this abstract (Post Script)

Electrocardiograms (ECGs) and magnetocardiograms (MCGs) arise primarily from electric currents in the body associated with flow of charged ions (such as ${\rm Na}^+,\;{\rm K}^+,\;{\rm Ca}^{2+}$) in the myocardium. Ionic currents across the membrane of a myocardial muscle cell give rise to a time-dependent {\it impressed} current density ${\bf J}^i$, which could be described in terms of modified Hodgkin-Huxley differential equations. Impressed currents give rise to an electric potential $\phi({\bf r}')$, which in turn generates ohmic volume currents ${\bf J}_v({\bf r}') = -\sigma'\nabla\phi$ everywhere in the body ($\sigma$' is the microscopic conductivity inside $v'$). Because the volume of the actual cell membranes is very small as compared to the whole myocardium, their direct contribution to the external electromagnetic fields is also small. In fact, the active source regions are usually characterized with the concept of the {\it primary} current density ${\bf J}_p$. It can be expressed as ${\bf J}_p({\bf r}') = {\bf J} + \sigma\nabla\phi$, i.e., as a difference between the total current density ${\bf J}$ and the passive volume currents ${\bf J}_v$. The time variation of ${\bf J}^i$ is relatively slow (below 100~Hz), which means that the source currents and external electromagnetic fields can be treated in a {\it quasistatic} approximation (the true time-dependent terms in Maxwell's equations vanish). A very large number of cells has to be excited to get a measurable signal; thus it is convenient to introduce a macroscopic equivalent current dipole ${\bf P} = \int_{v'} {\bf J}_p({\bf r}')dv'$. One can also derive higher order equivalent generators by using various multipole expansions.

In the ECG and MCG {\it inverse} problem the aim is to characterize the primary current density on the basis of the measured signals. For this purpose various equivalent generators are usually applied. The current dipole ${\bf P}$ or quadrupole generators are applicable only to sources which are confined to a small restricted volume, such as the initial atrial or ventricular activation or the starting point of a focal arrhythmia. A more realistic description of actual excitation wavefronts can be achieved by using uniform dipole (or double) layers, e.g. confined onto the endo- and epicardial surfaces. The influence of anisotropic electric conductivity can be accounted for by adding an oblique component to the dipole layer. Different techniques have been developed to find the parameters of the equivalent dipole and/or quadrupole sources, multiple dipoles, double layers, epicardial potential distributions, and (continuous or discrete) current densities characterizing the primary currents.

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Volume conductors, volume currents in cardiology - modelling

Vojko Jazbinsek

Institute of Mathematics, Physics and Mechanics
University of Ljubljana

Deriving the information on the current sources from the measured electric potential on the body surface and magnetic field distribution in vicinity of the body is the subject of the electrocardiographic (ECG) and magnetocardiographic (MCG) inverse problem. In order to solve the inverse problem, one has to introduce models for the current sources as well as models of the human body. In this paper the volume conductor modelling is discussed. Human body is a complex volume conductor consisting of several compartments, characterized by different conductivities, which can be in general anisotropic, such as myocardium and skeletal muscle. Properties of volume conductor model are determined by its shape and conductivity distribution. Volume currents, driven by the primary current sources within the heart, are influenced by the volume conductor structure which in turn affect the measured ECG and MCG signals. First attempts to the inverse problem solutions have neglected the contribution of irregular torso shape and its internal structure. This approach still plays an important role in qualitative interpretation of measured ECG and MCG data, however, for quantitative evaluation more realistic volume conductor model has to be introduced. The most common approach is a piece-wise homogeneous volume conductor composed of torso and some internal inhomogeneities, like lungs and heart cavities. In this model, bioelectric current sources and measured quantities (ECG, MCG) are related by integral equations, which can be solved numerically by the Boundary Element Method (BEM). When using BEM, surfaces between different compartments are tessellated. The integral equation for the electric potential becomes in discrete form a system of linear equations which is defined by a large matrix. This matrix is determined by the material and geometric properties of the volume conductor model. The electric potential distribution on the body surface due to an arbitrary source within the body is then calculated rather fast once the inverse of the matrix is found. The contribution of volume currents to the magnetic field is determined by the geometrical transfer matrix connecting the geometry of the volume conductor model with the position of magnetic sensors which is also computationally efficient what is necessary for the inverse problem solution.

Examples and simulations of different volume conductor models will be discussed. Further information of volume conductor studies will be illustrated with Wolff-Parkinson-White (WPW) case study of 12 patients.

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Clinical multichannel measurements of cardiac magnetic field and electric potential

Martin Burghoff

Physikalische-Tehnische Bundesanstalt, Institut Berlin
Abbestr. 2-12, 15879 Berlin, Germany

Requirements for clinical multichannel cardiac measurements are:

Today we have more than 5 multichannel systems in Europe for cardiac magnetic field measurements in a clinical setting, e.g., in Jena, Hamburg, Erlangen, Helsinki, Berlin Steglitz.

Each system has another configuration --> the signals and calculated parameters such as QRS length cannot be compared without recalculation into a virtual sensor configuration --> programs for recalculation are designed in Helsinki and Berlin

Berlin experience with the Steglitz MCG multichannel system:

Trends for the near future:

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Anatomic picture: 3-D cardiac image processing

Isabelle E. Magnin

CREATIS, Lyon, France

A large variety of techniques exist to investigate the anatomic shape and the functional behaviour of the moving heart. The information content issued from them is roughly complementary and directly related to the physical phenomena involved. They can be divided into two classes called passive and active techniques.

Passive techniques record the natural activity of the beating heart. It mainly concerns the electrocardiography (ECG) which measures the variations of the electric field (along the cardiac cycle) and the magnetocardiography (MCG) which provides the magnetic field variations. The cardiac signals are recorded using electrodes (ECG) or SQUIDs (MCG) located on the body. When placed in particular positions, they can provide orthogonal projections of the ECG (vectorcardiography) or spatio-temporal information (MCG). The resulting recorded signals give functional information. Active techniques consist in using some external wave or excitation to get information back. It mainly includes imaging techniques using X-Rays, magnetic resonance imaging (MRI), ultrasound (US) and isotopic imaging (SPECT and PET). Both anatomical and functional information cam be extracted from such 2-D or 3-D sequences.

The heart experiences a complex motion that is only partially captured by the previously mentioned acquisition techniques. The inverse problem which consists in extracting global and local motion and metabolism information from these data is a quite difficult task because it is an ill-posed inverse problem. Moreover, according to the diversity of data, various mathematical approaches will be necessary. Studying the true 4-D behaviour of the heart, implies to be able to efficiently mix such heterogeneous information in a fusion process.

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Software for MCG

Sergio N. Erné

Zentralinstitut für Biomedizinische Technik
Universität Ulm, Ulm, Germany

This talk has been designed as an introduction to a fruitful discussion to define what is needed, what is desirable, and what is available in the MCG analysis. This talk will be organized in the following sections:

One of the main aspects discussed will be the problem of computational speed or analysis time, being this one of the main problems in the design of biomagnetic cardiac systems for clinical use.

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Localization of the arrhythmogenic substrate - validation by successful cathater ablation procedures

Michael Oeff, Philipp Stevens

Cardiopulmonary Department, Klinikum Benjamin Franklin
Freie Universität Berlin, Germany

High-resolution magnetocardiographic mapping is used for non-invasive measurements of the the dynamic magnetic field distribution generated by the electrical activation of the heart. The magnetic field propagation is much less influenced by the tissue conductive inhomogeneities than the electrical propagation. The isomagnetic line of the precordial magnetic field represents the position and the direction of the current source generated by the electric myocardial activity, the equivalent current dipole (ECD). The analysis of the distance of the extrema allows the determination of the sources depth and the quantitative three-dimensional calculation of the ECD position. In addition, with cardiac imaging techniques (Magnetic Resonance Imaging) the functional abnormalities as measured magnetocardiographycally can be projected into anatomical structures which may be of particular value for ablative techniques.

For localization of the accessory pathway, 21 patients with Wolff-Parkinson-White syndrome were investigated. Patients with dual pathways were excluded. The accessory pathway position was localized according to standard electrophysiologic techniques or by successful catheter ablation (9 consecutive patients). The intrathoracic position of the catheter at the pathway was measured in the biplane cine-fluoroscopy. For analysis of the magnetocardiographic "late fields" 12 patients with coronary heart disease and spontaneous and inducible ventricular tachycardia were investigated.

Localization of ventricular preexcitation
The dynamic isofield contours of the high resolution magnetocardiogram were used to determine the position and strength of the three-dimensional propagation of the equivalent current source, which represents the center of the electrical activity for the chosen time and is by definition the current source whose magnetic field distribution at a specified time instant matches according to least square fit the measured field distribution. The initial 1 to 3 msec of the magnetocardiographic delta wave were analyzed; the progression of the ventricular preexcitation during the delta-wave in space was followed by the calculation of the ECD in steps of 1 - 2 msec. Standardized values of the overlay of the atrial repolarization were subtracted.

Comparison of magnetocardiographic and electrophysiologic localization
The differences between magnetocardiographic and invasive localization was (0.7 cm,1.0 cm,0.9 cm) in the X-plane, (0.9 cm,1.0 cm,1.0 cm) in the Y-plane and (1.2 cm,1.0 cm,1.0 cm) in depth (Z-plane). The correlation coefficients of these positions were 0.89 (p < 0.0001), 0.92 (p < 0.0001) and 0.84 (p < 0.0003), respectively. The difference between both positions in space was (1.9 cm,1.0 cm,1.3 cm).

Magnetocardiographic localization and catheter ablation
Magnetocardiographic localization of the accessory pathway was made in 9 consecutive patients before catheter ablation. This position was projected into the MR image, and the ablation catheter was advanced to that site. The final catheter position was chosen according to criteria of the endocardial electrograms i.e. the recording of the accessory pathway potential, with changes of catheter position in the range of millimeters. However, the gross position was reliably predicted by the magnetocardiographic mapping even in differentiating right and left paraseptal pathways.

In conclusion, the high-resolution magnetocardiographic mapping is a new and promising diagnostic method for quantitative 3-dimensional localization of ventricular preexcitation. The localization of other abnormal arrhythmogenic activation of the heart may also be suitable. In combination with cardiac imaging it provides an accurate functional and anatomical picture of the heart.

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Analysis of depolarization and repolarization for the characterization of the arrhythmogenic substrate of the myocardium

Philipp Stevens, Michael Oeff

Cardiopulmonary Department, Klinikum Benjamin Franklin
Freie Universität Berlin, Germany

Post-myocardial infarction mortality is largaely related to the incidence of malignant tachyarrhythmias like sustained ventricular tachycardia or ventricular fibrillation; reinfarction does not play an important part in its causation. Their prevalence is dependant on multiple factors including abnormal de- and repolarization, which can be identified by various tests characterizing pathological electrophysiological states.

Concept of arrhythmogenesis
It is important to recognize the interdependency of the factors that lead to the appearance of ventricular tachycardia: The cellular electrophysiology of the organic substrate (the diseased myocardium) is irritated by repetitive ventricular premature beats (trigger events) towards a (further) slowing of conduction and a change in the membrane potential, which in turn leads to a micro-reentry circuit, the origin of ventricular tachycardia. This process itself is influenced by modulating factors.

The organic substrate
Most of the organic cardiac diseases carry an elevated risk of ventricular tachycardia and it is debated whether the so-called idiopathic vetricular tachycardia is the symptom of an undiagnosed underlying condition. Left ventricular ejection fraction quantifies the left ventricular function and is well recognized as predictor of outcome. In electrophysiologic testing, premature beats are applied in a standardized fashion to mimic trigger events that may lead to the induction of ventricular tachycardia and consequently to the identification of the patients at risk. Ventricular late potentials (as recorded by high resolution signal averaged ECG) represent areas of slow conduction due to myocardial damage and are of prognostic significance. Magnetocardiographic mapping is a more sensitive and sophisticated method for quantification and even localization of abnormal cardiac electrophysiological phenomena. Its value in diagnostics of various states of different cardiac diseases is well recognized, but needs further evaluation.

Modulating factors
The extent of coronary artery disease is a main predictor of outcome in patients with this condition. But detection of silent or overt ischemia by noninvasive techniques (resting ECG, holter monitoring, exercise ECG) did not show to be useful in stratifying high risk patients. Stress-magnetocardiography may particularily be suitable in detecting ischemic conditions. More recently, research has focussed on the autonomous innervation of the heart, i.e. the system responsible for heart-rate and pressure control. Reduced heart rate variability as well as reduced baroreflex sensitivity as indicators of vago-sympathetical dysbalance seem to proof promising in stratifying patients at risk. QT-dispersion indicates an imbalance of the heart's sympathetic innervation which results in abnormal spatial distribution of repolarization and, thus, can be recorded excellently by mapping techniques, including magnetocardiographic mapping.

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The software package CURRY TM and its application in cardiography

Ulrich Katscher

Philips GmbH Forschungslaboratorien
Forschungsabteilung Technische Systeme Hamburg
Hamburg, Germany

CURRY TM is a software package for scanning and analysing magnetic and/or electric signals of the human body, for example, the human heart. Supported reconstruction modes are the fitting of single dipole(s) and the current density reconstruction. Results can be overlayed with the anatomic information.

Besides scientific research, possible applications of CURRY TM in cardiography are the family of arrhythmias and all kind of myocardial infarctions. In the first case, a single dipole can be fitted using Nelder-Mead simplex to localize arrhythmic center and to determine the strength of the arrhythmic impulse. In the case of infarction, a current density can be reconstructed on the myocardium yielding information about position, size, and vitality of the injured tissue. Several concepts to restrict the underdetermination of the problem are offered to the user (e.g., L1/L2 norm criterion, SVD regularization).

CURRY TM is able to read and visualize anatomic images, e.g., from the magnetic resonance or computer tomography measurements. A toolkit helps the user to segment anatomic surfaces like thorax, lungs, or myocardium out of these images. The surfaces can be used to determine the position of the current sources to be reconstructed and to illustrate the anatomic context of the reconstruction result. Moreover, the surfaces can be used to model a realistic volume conductor for the forward problem using the boundary element method.

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Effects of source extension on electric and magnetic location: A simulation study

S.N. Erné1 and C. Del Gratta2

1 Central Institute for Biomedical Technology, Ulm University, Ulm, Germany.
2 Institute of Medical Physics, Gabriele D'Annunzio University, Chieti, Italy.

One of the difficulties in source localisation lies in the choice of a model for the source. Due to the fact that the real source is extended, the use of a pointlike model source (e.g. a current dipole) in an inverse calculation may lead to localisation errors due to incorrectness of the model. Reconstruction of current distribution does not assume a pointlike source but still requires that the sources be constrained in an a priori determined surface. On the other hand, dipole localisation is still useful because it is fast, easy to interpret, and provides an error estimate. We have compared the location of the ECD obtained by means of a standard single dipole localization technique, when the source is an array of dipoles simulating an extended source, both for electric and magnetic data. The forward calculations were performed according to a simple mathematical model for the conductor: the homogeneous half-space. The sensor was a square array of 11 x 11 = 121 sensors both for the electric and for the magnetic data. Magnetic sensors were in turn: magnetometers, first order vertical gradiometers, and first order planar gradiometers. Sources consisted of arrays of dipoles in various geometrical configurations: pointlike sources like multiple dipoles, linear sources like stripes of dipoles. Gaussian noise was added to the field and potential distribution at a constant level (independent of source depth) and without correlation among channels. The inverse calculations were performed using the single current dipole model by minimising the usual cost functions i.e. the sum of squared differences between theoretical and experimental electric and magnetic data. In the case of combination of electric and magnetic data, the respective cost functions were added together after separate normalisation to take into account differences in value and in signal-to-noise ratio. The best localisation error was taken as the mean of 100 localisations; localisation error was taken as the standard deviation. Comparison of electric and magnetic localisations shows that the two techniques give markedly different source location, the mismatch of which may be as large as the source dimensions. Combination of electric and magnetic data does not necessarily improve the localisation accuracy compared to either one of the data type alone.

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