This book offers a concise and practical introduction to Python programming, specifically tailored for Electrical Engineering students. It begins with the fundamentals to help readers get started, covering essential concepts such as variables, operations, control structures, and plotting. As the material progresses, students are introduced to core topics in numerical computation and the basics of signal processing using powerful Python libraries such as NumPy and SciPy. To support learning, the book provides numerous clear examples and an optional step-by-step guide designed to help readers work through the material at their own pace. This structured approach makes it easy for beginners to build confidence and develop practical coding skills. Designed as a fast and accessible entry point, the book equips students with the programming foundation needed to support and enhance their learning in other engineering courses.

Atrial fibrillation (AF) is a common age-related cardiac arrhythmia. AF is characterized by rapid and irregular electrical activity of the heart leading to a higher risk of stroke and heart failure. During AF, the upper chambers of the heart, called atria, experience chaotic electrical wave propagation. However, despite the various mechanisms introduced in the literature, there is still an ongoing debate on a precise and consistent mechanism underlying the initiation and perpetuation of AF. Some studies show that AF is rooted in impaired electrical conduction and structural damage of atrial tissue, known as electropathology. Atrial electrograms (EGMs) recorded directly from heart’s surface, provide an important diagnostic tool to localize and quantify the degree of electropathology in the tissue. However, the analysis of the electrograms is currently constrained by the lack of suitable methods that can reveal the hidden electrophysiological parameters of the tissue. These parameters can be used as local indication of electropathology in the tissue. We believe that understanding AF and improving AF therapy starts with developing a proper forward model that is accurate enough (from a physiological point of view) and simultaneously simple enough to allow for subsequent parameter estimation. Therefore, the main focus of this thesis is on developing a simplified forward model that can efficiently explain the observed EGM based on AF relevant tissue parameters. An initial step before performing any analysis on the data is to remove noise and artefacts. All atrial electrogram recordings suffer from strong far-field ventricular activities (VA). Therefore, as the first step, we propose a new framework for removal of VA from atrial electrograms, which is based on interpolation and subtraction followed by low-rank and sparse matrix decomposition. The proposed framework is of low complexity, does not require high resolution multi-channel recordings, or a calibration step for each individual patient. In the next step, we develop a simplified electrogram model. We represent the model in a compact matrix form and show its linear dependence on the conductivity vector, enabling the estimation of this parameter from the recorded electrograms. The results show that despite the low resolution and all simplifying assumptions, the model can efficiently estimate the conductivity map and regenerate realistic electrograms, especially during sinus rhythm. In the next contribution of this dissertation, we propose a new approach for a better estimation of local activation times for atrial mapping by reducing the spatial blurring effect that is inherent to electrogram recordings using deconvolution. Employing sparsity based regularization and first-order time derivatives in formulating the deconvolution problem, improved performance of transmembrane current estimation is obtained. In the final part, we focus on translating our findings from research to clinical application. Therefore, we studied the effect of electrode size on electrogram properties including the length of the block line observed on the resulting activation map, percentage of observed low voltage areas, percentage of electrograms with low maximum steepness, and the number of deflections in the recorded electrograms.

Background: Atrial electrograms recorded from the epicardium provide an important tool for studying the initiation, perpetuation, and treatment of AF. However, the properties of these electrograms depend largely on the properties of the electrode arrays that are used for recording these signals. Method: In this study, we use the electrode's transfer function to model and analyze the effect of electrode size on the properties of measured electrograms. To do so, we use both simulated as well as clinical data. To simulate electrogram arrays we use a two-dimensional (2D) electrogram model as well as an action propagation model. For clinical data, however, we first estimate the trans-membrane current for a higher resolution 2D modeled cell grid and later use these values to interpolate and model electrograms with different electrode sizes. Results: We simulate electrogram arrays for 2D tissues with 3 different levels of heterogeneity in the conduction and stimulation pattern to model the inhomogeneous wave propagation observed during atrial fibrillation. Four measures are used to characterize the properties of the simulated electrogram arrays of different electrode sizes. The results show that increasing the electrode size increases the error in LAT estimation and decreases the length of conduction block lines. Moreover, visual inspection also shows that the activation maps generated by larger electrodes are more homogeneous with a lower number of observed wavelets. The increase in electrode size also increases the low voltage areas in the tissue while decreasing the slopes and the number of detected deflections. The effect is more pronounced for a tissue with a higher level of heterogeneity in the conduction pattern. Similar conclusions hold for the measurements performed on clinical data. Conclusion: The electrode size affects the properties of recorded electrogram arrays which can respectively complicate our understanding of atrial fibrillation. This needs to be considered while performing any analysis on the electrograms or comparing the results of different electrogram arrays.

Atrial electrograms are often used to gain understanding on the development of atrial fibrillation (AF). Using such electrograms, cardiologists can reconstruct how the depolarization wave-front propagates across the atrium. Knowing the exact moment at which the depolarization wavefront in the tissue reaches each electrode is an important aspect of such reconstruction. A common way to determine the LAT is based on the steepest deflection (SD) of the individual electrograms. However, the SD annotates each electrogram individually and is expected to be more prone to errors compared to approaches that would employ the data from the surrounding electrodes to estimate the LAT. As electrograms from neighboring electrodes tend to have rather similar morphology up to a delay, we propose in this paper to use the cross-correlation to find the pair-wise relative delays between electrograms. Instead of only using the direct neighbors we consider the array as a graph and involve higher order neighbors as well. Using a least-squares method, the absolute LATs can then be estimated from the calculated pair-wise relative delays. Simulated and clinically recorded electrograms are used to evaluate the proposed approach. From the simulated data it follows that the proposed approach outperforms the SD approach.

Background: Local activation time (LAT) annotation in unipolar electrograms is complicated by interference from nonlocal atrial activities of neighboring tissue. This happens due to the spatial blurring that is inherent to electrogram recordings. In this study, we aim to exploit multi-electrode electrogram recordings to amplify the local activity in each electrogram and subsequently improve the annotation of LATs. Methods: An electrogram array can be modeled as a spatial convolution of per cell transmembrane currents with an appropriate distance kernel, which depends on the cells’ distances to the electrodes. By deconvolving the effect of the distance kernel from the electrogram array, we undo the blurring and estimate the underlying transmembrane currents as our desired local activities. However, deconvolution problems are typically highly ill-posed and result in unstable solutions. To overcome this issue, we propose to use a regularization term that exploits the sparsity of the first-order time derivative of the transmembrane currents. Results: We perform experiments on simulated two-dimensional tissues, as well as clinically recorded electrograms during paroxysmal atrial fibrillation. The results show that the proposed approach for deconvolution can improve the annotation of the true LAT in the electrograms. We also discuss, in summary, the required electrode array specifications for an appropriate recording and subsequent deconvolution. Conclusion: By ignoring small but local deflections, algorithms based on steepest descent are prone to generate smoother activation maps. However, by exploiting multi-electrode recordings, we can efficiently amplify small but local deflections and reveal new details in the activation maps that were previously missed.

In this study, we propose a novel approach for estimation of local activation times (LATs) in fractionated electrograms. Using an electrophysiological tissue model, we first formulate the electrogram array as a convolution of transmembrane currents with a distance kernel. These currents are more local activities and less affected by the heterogeneity in the tissue compared to electrograms. We then deconvolve the distance kernel with the electrograms to reconstruct the transmembrane current. To stabilize the solution of this ill-posed deconvolution, we use spatio-temporal total variation as a regularization. This helps to preserve sharp spatial and temporal deflections in the currents that are of higher importance in LAT estimation. Finally, the maximum negative slope of the reconstructed transmembrane currents are used to estimate the LATs. Instrumental comparison to two reference approaches shows that the proposed approach performs better in estimating the LATs in fractionated electrograms.

Finding the hidden parameters of the cardiac electrophysiological model would help to gain more insight on the mechanisms underlying atrial fibrillation, and subsequently, facilitate the diagnosis and treatment of the disease in later stages. In this work, we aim to estimate tissue conductivity from recorded electrograms as an indication of tissue (mal)functioning. To do so, we first develop a simple but effective forward model to replace the computationally intensive reaction-diffusion equations governing the electrical propagation in tissue. Using the simplified model, we present a compact matrix model for electrograms based on conductivity. Subsequently, we exploit the simplicity of the compact model to solve the ill-posed inverse problem of estimating tissue conductivity. The algorithm is demonstrated on simulated data as well as on clinically recorded data. The results show that the model allows to efficiently estimate the conductivity map. In addition, based on the estimated conductivity, realistic electrograms can be regenerated demonstrating the validity of the model.

An electrogram array can be modeled as a spatial convolution of transmembrane currents generated by each activated atrial cell, with an appropriate measurement function that depends on the cells' distance to the electrodes. Compared to electrograms, transmembrane currents suffer less from the superposition of far-field atrial activities as they are not affected by the spatial convolution. As a result, transmembrane currents represent more local information and estimation of the local activation time using the steepest deflection will be more accurate than when using fractionated electrograms. However, transmembrane current estimation from electrogram array recordings is an under-determined problem, having infinite solutions, among which the desired solution. To constrain the solution space, additional prior information can be used. As the temporal derivative of transmembrane currents are typically sparse, we use this as prior information. We use a Split Bregman method with a quick convergence rate to iteratively solve the problem. Our implementation of circulant operations in the formulation of the solution of each step, makes it possible to perform them very fast using FFTs. Using simulated electrograms, we show that the proposed approach outperforms the conventional approach of annotating the steepest deflection of electrograms as the activation time.

Diagnosis and treatment of atrial fibrillation can benefit from various signal processing approaches employed on atrial electrograms. However, the performance and interpretation of these approaches get highly degraded by far-field ventricular activities (VAs) that distort the morphology of the pure atrial activities (AAs). In this study, we aim to remove VAs from the recorded unipolar electrogram while preserving the AA components. To do so, we have developed a framework which first removes the VA-containing segments and interpolates the remaining samples. This will also partly remove the atrial components that overlap with VA signals, e.g., during atrial fibrillation. To reconstruct the AA components, we estimate them from the removed VA-containing segments based on a low-rank and sparse matrix decomposition and add them back to the electrograms. The presented framework is of rather low complexity, preserves AA components, and requires only a single EGM recording. Instrumental comparison to template matching and subtraction and independent component analysis shows that the proposed approach leads to smoother results with better similarity to the true atrial signal.