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OBJECTIVES: Using computer models for the study of complex atrial arrhythmias such as atrial fibrillation is computationally demanding as long observation periods in the order of tens of seconds are required. A well established approach for reducing computational workload is to approximate the thin atrial walls by curved monolayers. On the other hand, the finite element method (FEM) is a well established approach to solve the underlying partial differential equations. METHODS: A generalized 2D finite element method (FEM) is presented which computes the corresponding stiffness and coupling matrix for arbitrarily shaped monolayers (ML). Compared to standard 2D FEM, only one additional coordinate transformation is required. This allows the use of existing FEM software with minor modifications. The algorithm was tested to simulate wave propagation in benchmark geometries and in a model of atrial anatomy. RESULTS: The ML model was able to simulate electric activation in curved tissue with anisotropic conductivity. Simulations in branching tissue yielded slightly different patterns when compared to a volumetric model with finite thickness. In the model of atrial anatomy the computed activation times for five different pacing protocols displayed a correlation of 0.88 compared to clinical data. CONCLUSIONS: The presented method provides a useful and easily implemented approach to model wave propagation in MLs with a few restrictions to volumetric models.

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OBJECTIVES: Presently, the protein interaction information concerning different signaling pathways is available in a qualitative manner in different online protein interaction databases. The challenge here is to derive a quantitative way of modeling signaling pathways from qualitative way of modeling signaling pathways from a qualitative level. To address this issue we developed a database that includes mathematical modeling knowledge and biological knowledge about different signaling pathways. METHODS: The database is part of an integrative environment that includes environments for pathway design, visualization, simulation and a knowledge base that combines biological and modeling information concerning pathways. The system is designed as a client-server architecture. It contains a pathway designing environment and a simulation environment as upper layers with a relational knowledge base as the underlying layer. RESULTS: DMSP--Database for Modeling Signaling Pathways incorporates biological datasets from online databases like BIND, DIP, PIP, and SPiD. The modeling knowledge that has been incorporated is based on a literature study. Pathway models can be designed, visualized and simulated based on the knowledge stored in the DMSP. The user can download the whole dataset and build pathway models using the knowledge stored in our database. As an example, the TNFalpha pathway model was implemented and tested using this approach. CONCLUSION: DMSP is an initial step towards the aim of combining modeling and biological knowledge concerning signaling pathways. It helps in understanding pathways in a qualitative manner from a qualitative level. Simulation results enable the interpretation of a biological system from a quantitative and system-theoretic point of view.

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We propose a general workflow to numerically estimate the spread of electrical excitation in the patients' hearts. To this end, a semi-automatic segmentation pipeline for extracting the volume conductor model of structurally normal hearts is presented. The cardiac electrical source imaging technique aims to provide information about the spread of electrical excitation in order to assist the cardiologist in developing strategies for the treatment of cardiac arrhythmias. The volume conductor models of eight patients were extracted from cine-gated short-axis magnetic resonance imaging (MRI) data. The non-invasive estimation of electrical excitation was compared with the CARTO maps. The development of a volume conductor modeling pipeline for constructing a patient-specific volume conductor model in a fast and accurate way is one essential step to make the technique clinically applicable.

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OBJECTIVES: Phase singularities have become a key marker in animal and computer models of atrial and ventricular fibrillation. However, existing algorithms for the automatic detection of phase singularities are limited to regular, quadratic mesh grids. We present an algorithm to automatically and exactly detect phase singularities in triangular meshes. METHODS: For each node an oriented path inscribing the node with one unit of spatial discretization is identified. At each time step the phase information is calculated for all nodes. The so-called topological charge is also computed for each node. A non-zero (+/- 2pi) charge is obtained for all nodes with a path enclosing a phase singularity. Thus all charged nodes belonging to the same phase singularity have to be clustered. RESULTS: With the use of the developed algorithm, phase singularities can be detected in triangular meshes with an accuracy of below 0.2 mm - independent of the type of membrane kinetics used. CONCLUSIONS: With the possibility to detect phase singularities automatically and exactly, important quantitative data on cardiac fibrillation can be gained.

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Increased local load in branching atrial tissue (muscle fibers and bundle insertions) influences wave propagation during atrial fibrillation (AF). This computer model study reveals two principal phenomena: if the branching is distant from the driving rotor (>19 mm), the load causes local slowing of conduction or wavebreaks. If the driving rotor is close to the branching, the increased load causes first a slow drift of the rotor towards the branching. Finally, the rotor anchors, and a stable, repeatable pattern of activation can be observed. Variation of the bundle geometry from a cylindrical, volumetric structure to a flat strip of a comparable load in a monolayer model changed the local activation sequence in the proximity of the bundle. However, the global behavior and the basic effects are similar in all models. Wavebreaks in branching tissue contribute to the chaotic nature of AF (fibrillatory conduction). The stabilization (anchoring) of driving rotors by branching tissue might contribute to maintain sustained AF.

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OBJECTIVES: Activation time (AT) imaging from electrocardiographic (ECG) mapping data has been developing for several years. By coupling 4-dimensional volume data (3D + time) the electrical sequence can be computed non-invasively. In this paper an approach for extracting the ventricular and atrial blood masses for structurally normal hearts by using cine-gated short-axis data obtained via magnetic resonance imaging (MRI) is introduced. METHODS: The blood masses are extracted by employing Active Appearance Models (AAMs). The ventricular blood masses are segmented, applying the AAMs after providing apex cordis and base of the heart in the volume data, whereas the more complex geometry of the atria requires a more specific attempt. On account of this the atrium was divided into three divisions of appearance, where the images of the volume data in the related divisions have a maximum affinity. The first division reaches from the base of the heart to initial visibility of the upper and left lower pulmonary vein. The second division up from there to the last occurrence and the third division from there to the end of the visibility of the right upper and lower pulmonary vein. After extracting the cardiac blood masses the result gets triangulated and remeshed for activation time imaging. RESULTS: With this method the cardiac models of eight patients were extracted and the AT imaging approach was applied to single-beat ECG data of atrial and ventricular depolarization. CONCLUSION: The advantage of the proposed AAM approach is that only a few initial parameters have to be set. Therefore, the approach can be integrated into a processing pipeline that works semi-automatically. The extracted models can be used for further investigations.

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OBJECTIVES: This paper presents an efficient approach for extracting myocardial structures from given atrial and ventricular blood masses to enable non-invasive estimation of electrical excitation in human atria and ventricles. METHODS: Based on given segmented atrial and ventricular blood masses, the approach constructs the myocardial structure directly, in the case that the myocardium can be detected in the volume data, or by using mean model information, in the case that the myocardium cannot be seen in the volume data due to image modalities or artefacts. The approach employs mathematical and gray-value morphology operations. Regulated by the spatial visibility of the myocardial structure in the medical image data especially the atrial myocardium needs to be estimated repeatedly using the a-priori knowledge given by the anatomy. RESULTS: The approach was tested using eight patient data sets. The reconstruction process yielded satisfying results with respect to an efficient generation of a volume conductor model which is essential when trying to implement the estimation of electrical excitation in clinical application. CONCLUSION: The approach yields ventricular and atrial models that qualify for cardiac source imaging in a clinical setting.

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OBJECTIVES: Noninvasive imaging of the cardiac activation sequence in humans could guide interventional curative treatment of cardiac arrhythmias by catheter ablation. Highly automated signal processing tools are desirable for clinical acceptance. The developed signal processing pipeline reduces user interactions to a minimum, which eases the operation by the staff in the catheter laboratory and increases the reproducibility of the results. METHODS: A previously described R-peak detector was modified for automatic detection of all possible targets (beats) using the information of all leads in the ECG map. A direct method was applied for signal classification. The algorithm was tuned for distinguishing beats with an adenosine induced AV-nodal block from baseline morphology in Wolff-Parkinson-White (WPW) patients. Furthermore, an automatic identification of the QRS-interval borders was implemented. RESULTS: The software was tested with data from eight patients having overt ventricular preexcitation. The R-peak detector captured all QRS-complexes with no false positive detection. The automatic classification was verified by demonstrating adenosine-induced prolongation of ventricular activation with statistical significance (p <0.001) in all patients. This also demonstrates the performance of the automatic detection of QRS-interval borders. Furthermore, all ectopic or paced beats were automatically separated from sinus rhythm. Computed activation maps are shown for one patient localizing the accessory pathway with an accuracy of 1 cm. CONCLUSIONS: The implemented signal processing pipeline is a powerful tool for selecting target beats for noninvasive activation imaging in WPW patients. It robustly identifies and classifies beats. The small beat to beat variations in the automatic QRS-interval detection indicate accurate identification of the time window of interest.

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In order to be able to solve the inverse problem of electrocardiography, the lead field matrix (transfer matrix) has to be calculated. The two methods applied for computing this matrix, which are compared in this study, are the boundary element method (BEM) and the finite element method (FEM). The performance of both methods using a spherical model was investigated. For a comparable discretization level, the BEM yields smaller relative errors compared to analytical solutions. The BEM needs less computation time, but a larger amount of memory. Inversely calculated myocardial activation times using either the FEM or BEM computed lead field matrices give similar activation time patterns. The FEM, however, is also capable of considering anisotropic conductivities. This property might have an impact for future development, when also individual myocardial fiber architecture can be considered in the inverse formulation.

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OBJECTIVE: The computer model-based computation of the cardiac activation sequence in humans has been recently subject of successful clinical validation. This method is of potential interest for guiding ablation therapy of arrhythmogenic substrates. However, computation times of almost an hour are unattractive in a clinical setting. Thus, the objective is the development of a method which performs the computation in a few minutes run time. METHODS: The computationally most expensive part is the product of the lead field matrix with a matrix containing the source pattern on the cardiac surface. The particular biophysical properties of both matrices are used for speeding up this operation by more than an order of magnitude. A conjugate gradient optimizer was developed using C++ for computing the activation map. RESULTS: The software was tested on synthetic and clinical data. The increase in speed with respect to the previously used Fortran 77 implementation was a factor of 30 at a comparable quality of the results. As an additional finding the coupled regularization strategy, originally introduced for saving computation time, also reduced the sensitivity of the method to the choice of the regularization parameter. CONCLUSIONS: As it was shown for data from a WPWpatient the developed software can deliver diagnostically valuable information at a much shorter span of time than current clinical routine methods. Its main application could be the localization of focal arrhythmogenic substrates.

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Non-invasive imaging of cardiac electrophysiology provides a non-invasive way of obtaining information about electrical excitation. An iterative algorithm based on a general regularisation scheme for non-linear, ill-posed problems in Hilbert scales was applied to the electrocardiographic inverse problem, imaging the ventricular surface activation time (AT) map. This method was applied to electrocardiographic data from a 31-year-old healthy volunteer and a 24-year-old patient suffering from a Wolff-Parkinson-White (WPW) syndrome. The objective was to evaluate non-invasive AT imaging of an autonomous sinus rhythm and to quantify the localisation error of non-invasive AT imaging by localising the accessory pathway of the WPW syndrome and a pacing site for left ventricle pacing. The distances between the invasive and non-invasive localisation of the pacing site and the accessory pathway were 8 mm and 5 mm. The clinical case presented, shows that this non-invasive AT imaging approach may enable the reconstruction of single focal events with sufficient accuracy for potential clinical application.

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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 Tikhonov's 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.

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A 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.

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The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.

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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.

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The objective of this study is to analytically validate a boundary element (BE) formulation for the relationship between the transmembrane potential on the heart's 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 volunteer's 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.

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The estimation of pseudo primary current dipoles on a 2D-manifold in the atrial and ventricular myocardium and septum, and of the transmembrane potential on the endocardium and epicardium, from the magnetic heart field is investigated. The human thorax surrounding the heart is modelled by an inhomogeneous boundary element volume conductor model, including the outer thorax surface and the surfaces of the lungs. The influence of the blood mass is neglected. In the inverse problem Tikhonov's regularisation is applied. The regularisation parameter is determined by the L-curve method. An algorithm for iterative improvement is applied to estimate the pseudo primary current dipoles. Synthetic magnetic field and electric potential data are generated using a cellular automaton model of the entire human heart. Real world magnetic field data for a normal subject are analysed to demonstrate the practicability and effectiveness of the presented method.

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A simulation study on magnetic source imaging within the human brain from a synthetic evoked magnetic field is presented. An inhomogeneous boundary element (BE) head model (cortex cerebrospinal fluid) was built up from real magnetic resonance imaging (MRI) cross-sections. In the forward problem, one or two rotating primary current dipoles (PCDs) are located at arbitrary sites within the auditory cortices. The PCDs should represent focal and distributed neural activities, respectively. The reconstruction space (predefined by a priori morphological information) is defined as a surface within the three-dimensional cortical volume, with an averaged distance of 0.005 m to the outer cortex surface. The reconstructed pseudo primary current dipoles (PPCDs) are not restricted to any particular direction. The observation space consists of two concave surfaces closely above the scalp. Each observation surface contains 37 observation points. An iterative Wiener filter estimation (WFE) is applied in order to reconstruct PPCD distribution from simulated magnetic field data. This iterative WFE approach enables the simultaneous localization of focal and distributed activities. Aspects on the correlation between neural activities are not investigated within this paper.

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A computer model study on magnetic source imaging from magnetocardiographic data is presented using a cellular automaton model of the entire human heart in the so-called forward problem. A homogeneous boundary element (BE) torso is built up from real magnetic resonance imaging (MRI) cross-sections. The heart model, which has a realistic anatomical shape, is positioned inside the BE torso. In the forward problem the spread of excitation is simulated by applying a modified Huygen's propagation principle. The magnetocardiogram (MCG) and electrocardiogram (ECG) can then be computed following the bidomain theory. From the simulated MCG data, pseudo primary current dipole (PPCD) estimation within the electrically active tissue is performed. The reconstruction space is defined as a surface in the middle of the atrial and ventricular myocardium and septum. The observation space consists of two mutually perpendicular planes closely above the torso surface on the frontal and the left lateral side, respectively. An iterative minimum-norm approach is applied in order to reconstruct PPCD distributions. The errors in PPCD estimation arising from noisy data and regularization algorithms are investigated in more detail.

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