Gerald Fischer

A Bidomain Model Based BEM-FEM Coupling Formulation for Anisotropic Cardiac Tissue

AbstractA hybrid boundary element method (BEM)/finite element method (FEM) approach is proposed in order to properly consider the anisotropic properties of the cardiac muscle in the magneto- and electrocardiographic forward problem. Within the anisotropic myocardium a bidomain model based FEM formulation is applied. In the surrounding isotropic volume conductor the BEM is adopted. Coupling is enabled by requesting continuity of the electric potential and the normal of the current density across the boundary of the heart. Here, the BEM part is coupled as an equivalent finite element to the finite element stiffness matrix, thus preserving in part its sparse property. First, continuous convergence of the coupling scheme is shown for a spherical model comparing the computed results to an analytic reference solution. Then, the method is extended to the depolarization phase in a fibrous model of a dog ventricle. A precomputed activation sequence obtained using a fine mesh of the heart was downsampled and used to calculate body surface potentials and extracorporal magnetic fields considering the anisotropic bidomain conductivities. Results are compared to those obtained by neglecting in part or totally (oblique or uniform dipole layer model) anisotropic properties. The relatively large errors computed indicate that the cardiac muscle is one of the major torso inhomogeneities.

Application of high-order boundary elements to the electrocardiographic inverse problem.

Eight-noded quadrilateral boundary elements are applied to the electrocardiographic inverse problem as an example for high-order boundary elements. It is shown that the choice of the shape functions used for approximation of the potentials has a remarkable influence on the solution obtained if the number of electrodes is smaller than the number of primary source points (under-determined equation system). Three different formulations are investigated considering a concentric spheres problem where an analytic solution is available: (a) the isoparametric formulation; (b) the quasi-first-order formulation; and (c) the pseudo-subparametric formulation as a new method. In a second step the pseudo-subparametric formulation (which provided the best results in the test problem) is applied to real word data. The transmembrane potential pattern of a 40 years old female suffering from severe heart failure and ventricular tachycardia after large anterior wall myocardial infarction is reconstructed for one time instant. Furthermore, an algorithm for the calculation of the transfer matrix is presented which avoids restrictions to the boundary element mesh caused by the placement of the electrodes.

An iterative algorithm for myocardial activation time imaging

An iterative algorithm based on a general regularization scheme for nonlinear ill-posed problems in Hilbert scales (method A) is applied to the magnetocardiographic inverse problem imaging the surface myocardial activation time map. This approach is compared to an algorithm using an optimization routine for nonlinear ill-posed problems based on Tikhonovs approach of second order (method B). Method A showed good computational performance and the scheme for determining the proper regularization parameter lambda was found to be easier than in case of method B. The formulation is applied to magnetocardiographic recordings from a patient suffering from idiopathic ventricular tachycardia in which a sinus rhythm sequence was followed by a ventricular extrasystolic beat.

Analytical validation of the BEM : application of the BEM to the electrocardiographic forward and inverse problem

The objective of this study is to analytically validate a boundary element (BE) formulation for the relationship between the transmembrane potential on the hearts surface and the potential on the body surface applying a concentric spherical test geometry. The relative difference (reldif) between the potential on the outer sphere of the test geometry computed analytically and numerically is determined by 3.59% for the coarse discretization (48 BEs) and by 0.46% in the case of the finer subdivision (192 BEs). In the inverse problem, the transmembrane potential on the inner sphere is estimated numerically from the electric potential on the outer sphere by using a minimum-norm least-square approach. The relative differences found are 20.2% when no measurement noise is added and 26.4% in the presence of 2% additional Gaussian noise. The BE formulation is also applied to real world data for solving the electrocardiographic inverse problem. A normal volunteers inhomogeneous thorax (outer thorax surface, surfaces of the lungs, epicardial heart surface) is modelled by 424 BEs. The same inverse method is then applied in order to reconstruct the transmembrane potential on the epicardium from the measured body surface potential (BSP) data during normal ventricular depolarisation.

An iterative linearized optimization technique for non‐linear ill‐posed problems applied to cardiac activation time imaging

A promising approach for the solution of the electrocardiographic inverse problem is the calculation of the cardiac activation sequence from body surface potential (BSP) mapping data. Here, a two‐fold regularization scheme is applied in order to stabilize the inverse solution of this intrinsically ill‐posed problem. The solution of the inverse problem is defined by the minimum of a non‐linear cost function. The L‐curve method can be applied for regularization parameter determination. Solving the optimization problem by a Newton‐like method, the L‐curve may be of pronged shape. Then a numerically unique determination of the optimal regularization parameter will become difficult. This problem can be avoided applying an iterative linearized algorithm. It is shown that activation time imaging due to temporal and spatial regularization is stable with respect to large model errors. Even neglecting cardiac anisotropy in activation time imaging results in an acceptable inverse solution.

Atrial myocardium model extraction

We present two approaches for reconstructing a patient’s atrial myocardium from morphological image data. nBoth approaches are based on a segmentation of the left and right atrial blood masses which mark the inner nborder of the atrial myocardium. The outer border of the atrial myocardium is reconstructed differently by the ntwo approaches. The surface manipulation approach is based on a triangle manipulation procedure while the nlabel-voxel-field approach adds or deletes label-voxels of the segmented blood mass labelset. Both approaches nyield models of a patient’s atrial myocardium that qualify for further applications. The obtained atrial models nhave been implemented many times in the construction of a patient’s volume conductor model needed for solving nthe electrocardiographic inverse problem. The label-voxel-field approach has to be favored because of its superior nperformance and ability of implementation in a segmentation pipeline.

Cardiac modeling using active appearance models and morphological operators

We present an approach for fast reconstructing of cardiac myocardium and blood masses of a patients heart from morphological image data, acquired either MRI or CT, in order to estimate numerically the spread of electrical excitation in the patients atria and ventricles. The approach can be divided into two main steps. During the first step the ventricular and atrial blood masses are extracted employing Active Appearance Models (AAM). The left and right ventricular blood masses are segmented automatically after providing the positions of the apex cordis and the base of the heart. Because of the complex geometry of the atria the segmentation process of the atrial blood masses requires more information as the ventricular blood mass segmentation process of the ventricles. We divided, for this reason, the left and right atrium into three divisions of appearance. This proved sufficient for the 2D AAM model to extract the target blood masses. The base of the heart, the left upper and left lower pulmonary vein from its first up to its last appearance in the image stack, and the right upper and lower pulmonary vein have to be marked. After separating the volume data into these divisions the 2D AAM search procedure extracts the blood masses which are the main input for the second and last step in the myocardium extraction pipeline. This step uses morphologically-based operations in order to extract the ventricular and atrial myocardium either directly by detecting the myocardium in the volume block or by reconstructing the myocardium using mean model information, in case the algorithm fails to detect the myocardium.

Noninvasive cardiac source imaging from potential and magnetic field data

Ventricular surface activation time maps are estimated from simulated and measured body surface potential (BSP) maps and extra‐corporal magnetic field maps. In a first step the transfer matrix, relating the primary cardiac sources to the measured potential and/or magnetic field data, is calculated applying the boundary element method. Activation times are determined by minimizing a cost function which is based on this transfer matrix. This optimization method is solved by a quasi Newton method. The critical point theorem is used in order to estimate the starting column matrix.

Modeling Mother Rotor Anchoring in Branching Atrial Tissue

Background: It has been hypothesized that stable mother rotors may increase the stability of atrial fibrillation. Local shortening of action potential duration due to vagotonic activity may promote the formation of mother rotors. However, prior studies have shown that meandering rotors typically drift out of the region of short action potential duration (APD), making the formation of stable rotors unlikely. Thus, for the formation of mother rotors in vagotonic AF a mechanism must be involved which increases rotor stability. Hypothesis: Local variation of electrotonic load due to branching muscle sleeves in the atrial tissue may avoid rotor drift and ease the formation of stable vagotonic AF. In this chapter the underlying mechanisms are illustrated by means of computer modeling. Model formulation: Anchoring is illustrated in a square patch of tissue and a monolayer model of the atria. Vagotonic activity was modeled by increasing ACh-concentration in the vicinity of the pulmonary veins (PVs). Close to the right lower PV, a muscle sleeve branching into the vein was included in the model. A model of canine atrial membrane kinetics was used providing detailed data on electrical remodeling and vagal activity. AF was analyzed computing dominant frequency (DF) maps and phase singularities (PSs). Model predictions: In the case of a missing anchor site AF was self terminating within 2 s of simulated activity. The rotor drifted out of the short APD region. After including a branching tissue structure, mother rotor formation was observed (anchoring and almost periodic activity over the entire simulated interval of 4.2 s). Pull and push currents were identified as the mechanism stabilizing the rotor trajectory. Push currents in the bundle reduced the DF at the branching by about 10%with respect to the highest DF (15 Hz). The computed DFmapmainly reflected the underlying ACh concentration (correlation 0.9). Summary: Branching tissue structures in regions of high ACh-concentration may constitute the substrate underlying vagotonic AF. Future experimental studies are needed to further confirm motor rotor anchoring in branching tissue.

Simulation of cardiac activation patterns for checking suggestions about the suitability of multi-lead ECG electrode arrays

In this study results gained for different electrode array schemes are compared which are used to image the activation time of the heart in a noninvasive way. The tested arrays have been selected based on sensitivity and effort-gain analysis. A cellular automaton was used for generating 6 different patterns resembling a sinus rhythm overlaid by an accessory pathway. The BSP was computed using a finite element approach. For generating the AT maps a boundary element model was used. It was found that methods for noninvasive imaging of the cardiac electrophysiology can profit from the increased details and features contained in the BSP maps recorded by a 125 lead array