The Inverse Problem in Electrocardiography: A Model Study of the Effects of Geometry and Conductivity Parameters on the Reconstruction of Epicardial Potentials

An idealized, analytic model using spherical harmonics was developed to analyze the effects of variations in torso geometry and volume conductivity parameters on the recovery of epicardial potentials from torso potentials. The model was also used to analyze the effects of these variations on individual terms in the orthogonal series expansion. The ability to reconstruct separate, local electrical events on the epicardium was examined under the following simulated situations: 1) all conductivity and geometry parameters were known accurately, 2) the conductivity of individual torso tissue layers was varied, 3) the torso-air boundary was eliminated (the "infinite medium" assumption), 4) the heart position was not accurately known, and 5) the heart size was not accurately known. Variation in conductivity and geometry parameters was found to exert a quantitative and qualitative effect on the amplitude, resolution, and position of the reconstructed epicardial maxima and minima. Significant differences were found in the ability of the inverse procedure to recover epicardial potentials resulting from posterior as opposed to anterior myocardial sources. Important conclusions regarding the narrow allowance for error in heart size and position, and the relative contributions of the torso tissue layer conductivities can provide guidelines for inverse reconstruction of epicardial potentials with a realistic model utilizing the true geometry.     

Quasi-Linearization and Inverse Electrocardiography

The feasibility of using quasi-linearization for obtaining solutions to the inverse problem in electrocardiography is investigated. Results suggest that convergence is unlikely when using real electrocardiographic data.     

Statistically Constrained Inverse Electrocardiography

This paper examines the feasibility of utilizing statistical constraints on the inverse potential model to determine the potential distribution over a 4 cm sphere surrounding the heart from perturbed torso potentials. These perturbed torso potentials reflect instrumentation, quadrature, electrode placement, and heart position uncertainties. This work is an extension of the authors' previous work which concluded that it is not feasible to determine this same potential distribution using unconstrained solutions. However, the results of the present work indicate that with the use of approximate signal and noise covariance matrices, it is possible to achieve estimates of this potential distribution with an average sum squared error of twenty-five percent. Further, the estimation of the signal and noise covariance matrices can be accomplished with a knowledge of heart geometry, torso geometry, The approximate measurement error, and a rough estimate of the time an average section of myocardium is depolarized, but without an a priori specification of the activation sequence.     

Free-Moment Current Dipoles in Inverse Electrocardiography

A feasibility study is made of the multiple free-moment dipole equivalent cardiac source concept as applied to the inverse problem in electrocardiology. The study deals with a bounded homogeneous model of a dog from which extensive geometry and potential measurements were taken. Torso surface potentials are related to the dipole sources through a set of 268 overdetermined linear algebraic equations. Three source configurations are modeled, 76, 20, and 9 dipoles, and simulation studies are made to evaluate the performance of the models and the feasibility of the concept. Inverse solutions are determined subject to a least-squares error fit of the infinite medium potentials. It is found that the number of free dipoles that can be used as an inverse source model is limited by the solution noise sensitivity, and that the upper limit is in the neighborhood of twenty. The results show that the solutions are unstable in the presence of small geometry perturbations. Localized activation simulated by the 76-dipole source cannot be detected in either the 20- or the 9-dipole solutions, even in the absence of noise and perturbations. Solutions calculated from potentials measured in vivo yield results that are unphysiologic.     

The Use of Time Dependent Models in Inverse Electrocardiography

This study investigates the feasibility of utilizing multiple dipole cardiac generators with time dependent dipole moments to obtain physiologically feasible inverse electrocardiographic solutions.     

Determining Surface Potentials from Current Dipoles, with Application to Electrocardiography

This paper presents a method for determining the potentials over the surface of a three-dimensional volume due to internal current sources. The volume may be inhomogeneous and irregularly shaped. The method for determining the potentials uses N simultaneous equations which when solved produce the potentials at N different surface points. The N simultaneous equations are solved by an iterative technique on an IBM computer.     

The Use of Unipolar Epicardial QRS Potentials to Estimate Myocardial Infarction

This canine study assesses the ability to predict whether the myocardium lying directly below epicardial recording sites is infarcted or normal. The results indicate that sufficient information is present in unipolar QRS waveforms to predict with 90 percent accuracy the state of the myocardium. A simplified classifier based only on the magnitude of the Q wave also provides sufficient information for reliable prediction.     

Effect of Cardiac Motion on Solution of the Electrocardiography Inverse Problem

Previous studies of the ECG inverse problem often assumed that the heart was static during the cardiac cycle; consequently, a time-dependent geometrical error was thought to be unavoidably introduced. In this paper, cardiac motion is included in solutions to the electrocardiographic inverse problem. Cardiac dynamics are simulated based on a previously developed biventricular model that coupled the electrical and mechanical properties of the heart, and simulated the ventricular wall motion and deformation. In the forward computation, the heart surface source model method is employed to calculate the epicardial potentials from the action potentials, and then, the simulated epicardial potentials are used to calculate body surface potentials. With the inclusion of cardiac motion, the calculated body surface potentials are more reasonable than those in the case of static assumption. In the epicardial potential-based inverse studies, the Tikhonov regularization method is used to handle ill-posedness of the ECG inverse problem. The simulation results demonstrate that the solutions obtained from both the static ECG inverse problem and the dynamic ECG inverse problem approaches are approximately the same during the QRS complex period, due to the minimal deformation of the heart in this period. However, with the most obvious deformation occurring during the ST-T segment, the static assumption of heart always generates something akin to geometry noise in the ECG inverse problem causing the inverse solutions to have large errors. This study suggests that the inclusion of cardiac motion in solving the ECG inverse problem can lead to more accurate and acceptable inverse solutions.      

The Effects of Thoracic Inhomogeneities on the Relationship Between Epicardial and Torso Potentials

This study examines the effects of the lungs, spine, sternum, and the anisotropic skeletal muscle layer on the relationship between torso and epicardial potentials. Boundary integral equations representing potentials on the epicardial surface, the torso surface, and the internal conductivity interfaces were solved yielding a set of transfer coefficients valid for any source inside the epicardium and for any conductivity configuration outside the epicardial surface. These transfer coefficients relate potentials on the torso to potentials on the epicardial surface. Calculated torso potentials are generated via the transfer coefficients and measured epicardial potentials for comparison to measured torso potentials. This comparison indicates whether including the thoracic inhomogeneities improves attainable accuracy in calculations relating torso potentials to epicardial potentials.     

Theoretical Studies on the Inverse Problem in Electrocardiography and the Uniqueness of the Solution

This paper focuses on the theoretical formulation of the inverse problem in electrocardiography under a general inhomogeneous and anisotropic configuration of the torso conductor. The following points are especially considered: 1) derivation of the Fredholm integral equation of the first kind which relates epicardial to body surface potentials, 2) specification of the conditions which its kernel must satisfy, 3) general solution method for the inverse problem by using the vector space theory, taking into consideration the ill condition of this inverse problem, and 4) proof of the uniqueness of the epicardial potentials.     

Epicardial cardiac source-field behavior

The accurate determination of the spatial distribution of cardiac electrophysiological state is essential for the mechanistic assessment of cardiac arrhythmias in both clinical and experimental cardiac electrophysiological laboratories. The authors describe three fundamental cardiac source-field relationships: 1) activation fields; 2) electrotonic fields; and 3) volume conductor fields. The three cases are described analytically and illustrated with experimentally obtained canine cardiac recordings that capitalize on a recently formulated technique for in vivo cardiac transmembrane current estimation     

Effects of Geometrical Uncertainties on Electrocardiography

This paper investigates the influence of uncertainties in geometrical coordinates on calculated torso and epicardial potentials. Two types of sources, spherical epicardial and multipolar, are both considered. A concentric sphere model with the epicardial surface represented by a unit sphere and the torso surface represented by a spherical surface of radius 1.4 is utilized. While the representation of biological surfaces with spheres is a highly idealized situation, the dimensions chosen and the approximations utilized are such that the results should be indicative of those using measured biological coordinates.     

A Circuit for Contact Monitoring in Electrocardiography

A circuit for contact impedance monitoring in ECG is presented. An ac current in the same frequency band as the ECG signal is used for the measurement. This makes the measurement independent of polarization potentials and gives the correct weight to the impedances in the skin-to-electrode junction. The measurement of the contact impedance is made continuously during ECG monitoring and causes no interference in the ECG signal.     

Capability and Limitations of Electrocardiography and Magnetocardiography

The capabilities and limitations of electrocardiography and magnetocardiography are discussed. Representing the electrical activity of the heart by an impressed current density ji, electrocardiography determines the spherical harmonic multipole expansion of its divergence (flux source), while magnetocardiography determines the spherical harmonic multipole expansion of the radial component of its curl (vortex source).     

模拟技术任重道远

不知从什么时候开始,我们的生活似乎完全被数字产品所主宰了：数字电视,数码相机,数字音乐,数字手机……我们逐渐习惯了数字技术带给我们的快捷、生动的工作和生活感受,而“模拟技术”这个字眼仿佛带着落后和守旧的味道。     

求解无约束优化问题的类电磁机制算法

 针对标准类电磁机制算法中电荷溢出和参数敏感的问题,提出了新的电荷计算公式;基于电磁场中的吸引-排斥原理,引导粒子沿着合力方向向较优的区域移动;为提高算法的局部搜索能力,结合邻域搜索技术来改进种群中的粒子.在此基础上,提出了求解无约束优化问题的类电磁机制算法.理论分析表明新算法以概率1收敛到问题的ε-最优解集.对28个标准测试函数进行了仿真实验,并和已有算法对比,结果表明新算法具有收敛快、求解性能好的优点.     

Two Theorems Concerning the Quadrupole Applicable to Electrocardiography

The first part of the paper discusses various representations for the electric scalar potential, and develops a procedure for determining the magnitude of the quadrupole term of the multipole expansion from a knowledge of its five components. The terms in the multipole expansion for an equivalent cardiac generator depend on the location in the heart region chosen as the origin. In the second part of the paper, an optimum dipole location is defined as the point where the dipole term alone gives the best fit to the potential on a least squares basis. If the heart is well represented by a dipole plus quadrupole, with the z-axis chosen to coincide with the dipole moment p, the optimum dipole location is given by     

基于小波和扩展卡尔曼的心电检测

文中将小波变换和扩展卡尔曼滤波器相结合,利用小波变换多尺度多分辨的特点,将心电信号进行分解。然后对心电信号在各尺度上进行扩展卡尔曼滤波。最后在扩展卡尔曼滤波的输出结果上进行QRS波形检测。文中方法经MIT-BIH心电数据库检验,QRS波Se（探测灵敏度）在99.40%以上,同时,QRS＋P（正探测率）在99.39%以上,提高了心电信号检测的正确率。