A model study of instability of the inverse problem in electrocardiography.

The inverse problem in electrocardiography is studied analytically using a concentric spheres model with no symmetry assumptions on the potential distribution. The mathematical formulation is presented, and existence and uniqueness of the solution are briefly discussed. Solution to the inverse problem is inherently very unstable. The magnitude of this instability is demonstrated using the derived analytical inverse solution for the spherical model. Regularization methods used to date are based on a regularization parameter that does not relate to any measurable physiological parameters. This paper presents a regularization method that is based on a parameter in the form of an a priori bound on the L2 norm of the inverse solution. Such a bound can be obtained from the theoretical estimates based on the measured values of the body surface potentials together with experimental knowledge about the magnitudes of the epicardial potentials. Based on the presented regularization, an exact form of the regularized solution and estimates of its accuracy are derived.     

A new method for regularization of the inverse problem of electrocardiography.

The inverse problem of electrocardiography (specifically, that part concerned with the computation of the ventricular surface activation isochrones) is shown to be formally equivalent to the problem of identification and measurement of discontinuities in derivatives of body surface potentials. This is based on the demonstration that such measurements allow localization of the relative extrema of the ventricular surface activation map (given a forward problem solution), which in turn restricts the space of admissible solution maps to a compact set. Although the inverse problem and the problem of identifying derivative discontinuities are both ill-posed, it is possible that the latter may be more easily or justifiably resolved with available information, particularly as current methods for regularizing the inverse problem typically rely on a regularization parameter chosen in an a posteriori fashion. An example of the power of the approach is the demonstration that a recent Uniform Dipole Layer Hypothesis-based method for producing the ventricular surface activation map is largely independent on that hypothesis and capable in principle of generating maps that are very similar in a precise sense to those that would result from the usual epicardial potential formulation (assuming the latter were capable of producing intrinsic deflections in computed epicardial electrograms sufficiently steep to accurately compute the activation map). This is consistent with the preliminary success of the former method, despite the significant inaccuracy of its underlying assumption.     

Inverse electrocardiographic transformations: dependence on the number of epicardial regions and body surface data points

The inverse problem of electrocardiography, the computation of epicardial potentials from body surface potentials, is influenced by the desired resolution on the epicardium, the number of recording points on the body surface, and the method of limiting the inversion process. To examine the role of these variables in the computation of the inverse transform, Tikhonov's zero-order regularization and singular value decomposition (SVD) have been used to invert the forward transfer matrix. The inverses have been compared in a data-independent manner using the resolution and the noise amplification as endpoints. Sets of 32, 50, 192, and 384 leads were chosen as sets of body surface data, and 26, 50, 74, and 98 regions were chosen to represent the epicardium.The resolution and noise were both improved by using a greater number of electrodes on the body surface. When 60% of the singular values are retained, the results show a trade-off between noise and resolution, with typical maximal epicardial noise levels of less than 0.5% of maximum epicardial potentials for 26 epicardial regions, 2.5% for 50 epicardial regions, 7.5% for 74 epicardial regions, and 50% for 98 epicardial regions. As the number of epicardial regions is increased, the regularization technique effectively fixes the noise amplification but markedly decreases the resolution, whereas SVD results in an increase in noise and a moderate decrease in resolution. Overall the regularization technique performs slightly better than SVD in the noise-resolution relationship.There is a region at the posterior of the heart that was poorly resolved regardless of the number of regions chosen. The variance of the resolution was such as to suggest the use of variable-size epicardial regions based on the resolution.     

Improving Tikhonov regularization with linearly constrained optimization: application to the inverse epicardial potential solution.

Two methods to improve on the accuracy of the Tikhonov regularization technique commonly used for the stable recovery of solutions to ill-posed problems are presented. These methods do not require a priori knowledge of the properties of the solution or of the error. Rather they exploit the observed properties of overregularized and underregularized Tikhonov solutions so as to impose linear constraints on the sought-after solution. The two methods were applied to the inverse problem of electrocardiography using a spherical heart-torso model and simulated inner-sphere (epicardial) and outer-sphere (body) potential distributions. It is shown that if the overregularized and underregularized Tikhonov solutions are chosen properly, the two methods yield epicardial solutions that are not only more accurate than the optimal Tikhonov solution but also provide other qualitative information, such as correct position of the extrema, not obtainable using ordinary Tikhonov regularization. A heuristic method to select the overregularized and underregularized solutions is discussed.     

Comparison and testing of least-squares time domain inverse solutions in electrocardiography

The use of several mathematical methods for estimating epicardial ECG potentials from arrays of body surface potentials has been reported in the literature; most of these methods are based on least-squares reconstruction principles and operate in the time-space domain. In this paper we introduce a general Bayesian maximum a posteriori (MAP) framework for time domain inverse solutions in the presence of noise. The two most popular previously applied least-squares methods, constrained (regularized) least-squares and low-rank approximation through the singular value decomposition, are placed in this framework, each of them requiring the a priori knowledge of a ‘regularization parameter’, which defines the degree of smoothing to be applied to the inversion. Results of simulations using these two methods are presented; they compare the ability of each method to reconstruct epicardial potentials. We used the geometric configuration of the torso and internal organs of an individual subject as reconstructed from CT scans. The accuracy of each method at each epicardial location was tested as a function of measurement noise, the size and shape of the subarray of torso sensors, and the regularization parameter. We paid particular attention to an assessment of the potential of these methods for clinical use by testing the effect of using compact, small-size subarrays of torso potentials while maintaining a high degree of resolution on the epicardium.     

Adaptive local regularization methods for the inverse ECG problem

One of the fundamental problems in theoretical electrocardiography can be characterized by an inverse problem. We present new methods for achieving better estimates of heart surface potential distributions in terms of torso potentials through an inverse procedure. First, we outline an automatic adaptive refinement algorithm that minimizes the spatial discretization error in the transfer matrix, increasing the accuracy of the inverse solution. Second, we introduce a new local regularization procedure, which works by partitioning the global transfer matrix into sub-matrices, allowing for varying amounts of smoothing. Each submatrix represents a region within the underlying geometric model in which regularization can be specifically ‘tuned’ using an a priori scheme based on the L-curve method. This local regularization method can provide a substantial increase in accuracy compared to global regularization schemes. Within this context of local regularization, we show that a generalized version of the singular value decomposition (GSVD) can further improve the accuracy of ECG inverse solutions compared to standard SVD and Tikhonov approaches. We conclude with specific examples of these techniques using geometric models of the human thorax derived from MRI data.     

人体躯干模型中肺的存在对体表电位分布的影响

在所建三维人体躯干模型的基础上,给出了如何应用边界元方法对非均匀人体场域进行求解。在设定心外膜电位分别呈现为单偶极子和双偶极子时,求出相应非均匀场域中的体表电位分布,并将它们与相同情况下均匀场域的体表电位分布进行分析比较。结果表明：躯干模型中肺的存在虽然对体表电位中极值的大小和位置没有太大影响,但却较大程序地改变了整个体表电位的分布状况,具体地说,就是由于肺的存在使得体表电位值较均匀时的相对误差高     

改善心电图逆问题病态条件的两个有效方法

本文对应用数值方法求解三维心电图逆问题时出现的解的不稳定现象,提出了两种有效的解决办法,即提高电导率值和选用包含适当的正则因子的阻尼最小二乘法。文中应用有限元和边界元结合的方法,在一个包含各向异性导电性肌肉层的三维人体模型下进行了心外膜电位的重构计算。其结果证明这种方法对提高心电图逆问题的数值稳定性和解的精度非常有效。     

On the computation of molecular surface correlations for protein docking using fourier techniques

The computation of surface correlations using a variety of molecular models has been applied to the unbound protein docking problem. Because of the computational complexity involved in examining all possible molecular orientations, the fast Fourier transform (FFT) (a fast numerical implementation of the discrete Fourier transform (DFT)) is generally applied to minimize the number of calculations. This approach is rooted in the convolution theorem which allows one to inverse transform the product of two DFTs in order to perform the correlation calculation. However, such a DFT calculation results in a cyclic or "circular" correlation which, in general, does not lead to the same result as the linear correlation desired for the docking problem. In this work, we provide computational bounds for constructing molecular models used in the molecular surface correlation problem. The derived bounds are then shown to be consistent with various intuitive guidelines previously reported in the protein docking literature. Finally, these bounds are applied to different molecular models in order to investigate their effect on the correlation calculation.     

Image reconstruction of fluorescent molecular tomography based on the tree structured Schur complement decomposition

Background  

The inverse problem of fluorescent molecular tomography (FMT) often involves complex large-scale matrix operations, which may lead to unacceptable computational errors and complexity. In this research, a tree structured Schur complement decomposition strategy is proposed to accelerate the reconstruction process and reduce the computational complexity. Additionally, an adaptive regularization scheme is developed to improve the ill-posedness of the inverse problem.     

Electrotonic load triggers remodeling of repolarizing current Ito in ventricle

A change in activation sequence electrically remodels ventricular myocardium, causing persistent changes in repolarizing currents (T-wave memory). However, the underlying mechanism for triggering activation sequence-dependent remodeling is unknown. Optical action potentials were mapped with high resolution from the epicardial surface of the arterially perfused canine wedge preparation (n = 23) during 30 min of baseline endocardial stimulation, followed by 40 min of epicardial stimulation, and, finally, restoration of endocardial stimulation. Immediately after the change from endocardial to epicardial stimulation, phase 1 notch amplitude of epicardial cells was attenuated by 74 +/- 8% (P < 0.001) compared with baseline and continued to diminish during the period of epicardial pacing, suggesting progressive remodeling of the transient outward current (Ito). When endocardial pacing was restored, notch amplitude did not immediately recover but remained attenuated by 23 +/- 10% (P < 0.001), also consistent with a remodeling effect. Peak Ito current measured from isolated epicardial myocytes changed by 12 +/- 4% (P < 0.025), providing direct evidence for Ito remodeling occurring on a surprisingly short time scale. The mechanism for triggering remodeling of Ito was a significant reduction (by 14 +/- 4%, P < 0.001) of upstroke amplitude in epicardial cells during epicardial stimulation. Reduction in upstroke amplitude during epicardial pacing was explained by electrotonic load on epicardial cells by fully repolarized downstream endocardial cells. These data suggest a novel mechanism for triggering electrical remodeling in the ventricle. Electrotonic load imposed by a change in activation sequence reduces upstroke amplitude, which, in turn, attenuates Ito according to its known voltage-dependent properties, triggering downregulation of current.     

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log₁₀-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate α(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize α(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.      

Permeability,phase-boundary potential and conductance in a cholinergic channel without constant field.

A potassium-selective, chemically excitable channel, whose characteristics cannot be accurately described by constant-field theory, is studied by a new approach based on diffusion theory but with no need for the classical assumptions of constant field, homogeneous membrane, and equal phase-boundary potentials at both interfaces. Permeability is defined, free of these constraints, and the Goldman coefficient is demonstrated to be a special case useful only when the constraints apply. Permeability can be evaluated directly from current-voltage data, and it is found not to be a parameter in this channel, but rather a function of both the voltage and the concentration of the permeant ion. However, it becomes concentration-independent when the membrane voltage is equal to the sum of the phase-boundary potentials. That sum can therefore be determined from these data, and it is -65 mV in this channel. The permeability at that potential is a channel parameter, and equal to 8.66 X 10(-6) cm/s for this channel. A constant field is shown not to exist in this channel and the Goldman coefficient not to be a parameter but a function of potential and concentration. Although errors introduced into this coefficient by nonconstant field and unequal surface potentials partially cancel each other, the coefficient is nevertheless not a correct measure of permeability.     

What you feel is what you see: inverse dynamics estimation underlies the resistive sensation of a delayed cursor

How our central nervous system (CNS) learns and exploits relationships between force and motion is a fundamental issue in computational neuroscience. While several lines of evidence have suggested that the CNS predicts motion states and signals from motor commands for control and perception (forward dynamics), it remains controversial whether it also performs the ‘inverse’ computation, i.e. the estimation of force from motion (inverse dynamics). Here, we show that the resistive sensation we experience while moving a delayed cursor, perceived purely from the change in visual motion, provides evidence of the inverse computation. To clearly specify the computational process underlying the sensation, we systematically varied the visual feedback and examined its effect on the strength of the sensation. In contrast to the prevailing theory that sensory prediction errors modulate our perception, the sensation did not correlate with errors in cursor motion due to the delay. Instead, it correlated with the amount of exposure to the forward acceleration of the cursor. This indicates that the delayed cursor is interpreted as a mechanical load, and the sensation represents its visually implied reaction force. Namely, the CNS automatically computes inverse dynamics, using visually detected motions, to monitor the dynamic forces involved in our actions.     

Cardiac electric field on the epicardial and body surfaces during the period of depolarization and repolarization of ventricular myocardium in pikes

Body surface and ventricular epicardial potential distributions during the electrocardiographic QRST interval were studied in pikes with the aid of potential mapping. The earliest epicardial activation was observed at the posterior base near the atrioventricular orifice. The areas of the earliest repolarization were found at the apex and the posterior base, whereas the area of the latest repolarization was detected at the anterior base. In the initial period of the QRS, the minimum was developed in the middle third of the right lateral body surface, and the maximum in the middle third of the ventral body surface. The body surface potential distribution during the ST-Twas characterized by the clear-cut negative potential zone in the cranial ventral area with the rest of the body surface having positive potentials, a pattern being largely unchanged throughout the period of the T-wave. The ventricular epicardial repolarization sequence differed from the activation sequence. The ventricular epicardial depolarization and repolarization sequences as well as epicardial potential distributions are expressed in the cardiac electric field on the body surface during the QRS and ST-T complexes.     

A new method for choosing sample size for confidence interval-based inferences

Scientists often need to test hypotheses and construct corresponding confidence intervals. In designing a study to test a particular null hypothesis, traditional methods lead to a sample size large enough to provide sufficient statistical power. In contrast, traditional methods based on constructing a confidence interval lead to a sample size likely to control the width of the interval. With either approach, a sample size so large as to waste resources or introduce ethical concerns is undesirable. This work was motivated by the concern that existing sample size methods often make it difficult for scientists to achieve their actual goals. We focus on situations which involve a fixed, unknown scalar parameter representing the true state of nature. The width of the confidence interval is defined as the difference between the (random) upper and lower bounds. An event width is said to occur if the observed confidence interval width is less than a fixed constant chosen a priori. An event validity is said to occur if the parameter of interest is contained between the observed upper and lower confidence interval bounds. An event rejection is said to occur if the confidence interval excludes the null value of the parameter. In our opinion, scientists often implicitly seek to have all three occur: width, validity, and rejection. New results illustrate that neglecting rejection or width (and less so validity) often provides a sample size with a low probability of the simultaneous occurrence of all three events. We recommend considering all three events simultaneously when choosing a criterion for determining a sample size. We provide new theoretical results for any scalar (mean) parameter in a general linear model with Gaussian errors and fixed predictors. Convenient computational forms are included, as well as numerical examples to illustrate our methods.     

Changes in the amplitude and direction of goal-directed hand movements in the lack of visual information

The goal of the present investigation was to determine the precision of goal-directed hand movements in the lack of visual information. The movement amplitude and direction was examined under different experimental conditions. Subjects were ten female and ten male university students. The motor test was drawing 10 cm long straight line and 24 cm long zigzag line in four different experimental conditions. 1) The drawing with open eyes was followed immediately with drawing with closed eyes. 2) The drawing was executed from memory in the lack of visual information. 3) Drawing with restricted amplitude or direction. 4) Drawing with verbal feedback. The errors of the target distance and the lateral deviations from the target were different under the different experimental conditions. The largest errors and underestimation of the target distance occurred in drawing horizontal straight line with closed eyes. No statistically significant gender differences were found. It is concluded that the practice, adjustment of single movement parameter to the target, and the verbal feedback assist better the accuracy of unseen goal-directed hand movement than the recent visual memory.     

Recovery of action potentials and twitches after K-contractures in frog skeletal muscle

To give information about intracellular Ca2+ translocation during and after K-contractures in vertebrate skeletal muscle fibers, we examined recovery of action potentials and twitches after interruption and spontaneous relaxation of K-contractures at low temperature (3 degrees C) that greatly reduced the rate of Ca2+ reuptake by the sarcoplasmic reticulum. On membrane repolarization interrupting K-contractures, the amplitude of both action potentials and twitches recovered quickly, while the falling phase of action potential was markedly slowed at first to prolong its refractory period, so that repetitive stimulation (20 Hz) did not produce a complete tetanus. Meanwhile, on membrane repolarization after spontaneous relaxation of K-contractures, the action potentials were markedly reduced in amplitude and prolonged in duration at first, also resulting in prolonged refractory period. These results are discussed in connection with Ca2+ absorption to the surface and transverse tubule membranes, producing changes in action potential kinetics.     

Detection of motor unit action potentials with surface electrodes: influence of electrode size and spacing

A model of the motor unit action potential was developed to investigate the amplitude and frequency spectrum contributions of motor units, located at various depths within muscle, to the surface detected electromyographic (EMG) signal. A dipole representation of the transmembrane current in a three-dimensional muscle volume was used to estimate detected individual muscle fiber action potentials. The effects of anisotropic muscle conductance, innervation zone location, propagation velocity, fiber length, electrode area, and electrode configuration were included in the fiber action potential model. A motor unit action potential was assumed to be the sum of the individual muscle fiber action potentials. A computational procedure, based on the notion of isopotential layers, was developed which substantially reduced the calculation time required to estimate motor unit action potentials. The simulations indicated that: 1) only those motor units with muscle fibers located within 10–12 mm of the electrodes would contribute significant signal energy to the surface EMG, 2) variation in surface area of electrodes has little effect on the detection depth of motor unit action potentials, 3) increased interelectrode spacing moderately increases detection depth, and 4) the frequency content of action potentials decreases steeply with increased electrode-motor unit territory distance.     

Optical mapping of cryoinjured rat myocardium grafted with mesenchymal stem cells

Mesenchymal stem cells (MSCs) have been shown to improve cardiac electrophysiology when administered in the setting of acute myocardial infarction. However, the electrophysiological phenotype of MSCs in situ is not clear. We hypothesize that MSCs delivered intramyocardially to cryoinjured myocardium can engraft, but will not actively generate, action potentials. Cryoinjury-induced scar was created on the left ventricular epicardial surface of adult rat hearts. Within 30 min, hearts were injected with saline (sham, n = 11) or bone marrow-derived MSCs (2 × 10(6)) labeled with 1,1'-dioctadecyl-3,3,3,3'-tetramethylindocarbocyanine percholate (DiI; n = 16). At 3 wk, optical mapping and cell isolation were used to measure optical action potentials and calcium transients, respectively. Histological analysis confirmed subepicardial scar thickness and the presence of DiI-positive cells that express connexin-43. Optical action potential amplitude within the scar at MSC-positive sites (53.8 ± 14.3%) was larger compared with sites devoid of MSCs (35.3 ± 14.2%, P < 0.05) and sites within the scar of shams (33.5 ± 6.9%, P < 0.05). Evidence of simultaneous action potential upstroke, the loss of action potential activity following ablation of adjacent viable myocardium, and no rapid calcium transient response in isolated DiI+ cells suggest that the electrophysiological influence of engrafted MSCs is electrotonic. MSCs can engraft when directly injected into a cryoinjury and are associated with evidence of action potential activity. However, our results suggest that this activity is not due to generation of action potentials, but rather passive influence coupled from neighboring viable myocardium.