AimsUse of a non-invasive electrocardiographic mapping system may aid in rapid diagnosis of atrial or ventricular arrhythmias or the detection of ventricular dyssynchrony. The aim of the present study was to validate the mapping accuracy of a novel non-invasive epi- and endocardial electrophysiology system (NEEES).Methods and resultsPatients underwent pre-procedural computed tomography or magnetic resonance imaging of the heart and torso. Radiographic data were merged with the data obtained from the NEEES during pacing from implanted pacemaker leads or pacing from endocardial sites using an electroanatomical mapping system (CARTO 3, Biosense Webster). The earliest activation as denoted on the NEEES three-dimensional heart model was compared with the true anatomic location of the tip of the pacemaker lead or the annotated pacing site on the CARTO 3 map. Twenty-nine patients [mean age: 62 ± 11 years, 6/29 (11%) female, 21/29 (72%) with ischaemic cardiomyopathy] were enrolled into the pacemaker verification group. The mean distance from the non-invasively predicted pacing site to the anatomic reference site was 10.8 ± 5.4 mm for the right atrium, 7.7 ± 5.8 mm for the right ventricle, and 7.9 ± 5.7 mm for the left ventricle activated via the coronary sinus lead. Five patients [mean age 65 ± 4 years, 2 (33%) females] underwent CARTO 3 verification study. The mean distance between non-invasively reconstructed pacing site and the reference pacing site was 7.4 ± 2.7 mm for the right atrium, 6.9 ± 2.3 mm for the left atrium, 6.5 ± 2.1 mm for the right ventricle, and 6.4 ± 2.2 for the left ventricle, respectively.ConclusionThe novel NEEES was able to correctly identify the site of pacing from various endo- and epicardial sites with high accuracy.

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of the solutions of a system of path-dependent (ordinary) differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on the Functional Ito calculus. MSC2010 classification: 60H10 ; 28C20 ; 34K50.

The inverse problem of electrocardiography consists in reconstructing cardiac electrical activity from given body surface electrocardiographic measurements. Despite tremendous progress in the field over the last decades, the solution of this problem in terms of electrical potentials on both epi- and the endocardial heart surfaces with acceptable accuracy remains challenging. This paper presents a novel numerical approach aimed at improving the solution quality on the endocardium. Our method exploits the solution representation in the form of electrical single layer densities on the myocardial surface. We demonstrate that this representation brings twofold benefits: first, the inverse problem can be solved for the physiologically meaningful single layer densities. Secondly, a conventional transfer matrix for electrical potentials can be split into two parts, one of which turned out to posess regularizing properties leading to improved endocardial reconstructions. The method was tested in-silico for ventricular pacings utilizing realistic CT-based heart and torso geometries. The proposed approach provided more accurate solution on the ventricular endocardium compared to the conventional potential-based solutions with Tikhonov regularization of the 0th, 1st, and 2nd orders. Furthermore, we show a uniform spatio-temporal behavior of the single layer densities over the heart surface, which could be conveniently employed in the regularization procedure.

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We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic pathdependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman-Kac formula and a one-dimensional global existence-and uniqueness result. MSC2010 classification: 45G15, 60H30, 60J25, 60J68, 35K40, 35K59.for all r, t ∈ [0, T ] with r ≤ t and each x ∈ S. Here, we assume implicitly that k ∈ N, D ∈ B(R k ) has non-empty interior, f : [0, T ] × S × D → R k is product measurable, µ is an atomless Borel measure on [0, T ], and g : S → D is Borel measurable and bounded.We first remark that for D = R k a Picard iteration and Banach's fixed-point theorem produce existence of solutions to (M) locally in time. This can be found, for example, in Pazy [14, Theorem 6.1.4] when X is a diffusion process. Regarding existence, we will suppose more generally that D is convex. By modifying analytical methods from the classical theory of ordinary differential equations (ODEs), we will derive unique non-extendible solutions to (M) that are admissible in an appropriate topological sense. Moreover, weak conditions ensuring the continuity of the derived solutions will be provided. In the particular case when D = R k and f is an affine map in the third variable w ∈ R k , we will prove a representation for solutions to (M). This gives a multidimensional generalization to the Feynman-Kac formula in Dynkin [6, Theorem 4.1.1].Let us also emphasize that non-negative solutions to one-dimensional Markovian integral equations are well-studied. Namely, for k = 1 and D = R + , solutions to (M) have been deduced by a Picard iteration approach. For instance, the classical references are Watanabe [19, Proposition 2.2], Fitzsimmons [8, Proposition 2.3], and Iscoe [10, Theorem A]. In these works the existence of solutions to (M) is used for the construction of superprocesses. Dynkin [2, 3, 6] establishes superprocesses with probabilistic methods by means of branching particle systems, which in turn yields another existence result to our Markovian integral equations.These treatments of (M) in one dimension require that the function f admits a representation that is related to measure-valued branching processes. To give one of the main examples, the following case is included in [2,3,6]:Borel measurable and bounded, and α 1 , . . . , α n ∈ [1, 2]. Here, the bound α i ≤ 2 for all i ∈ {1, . . . , n} is strict. However, this paper intends to derive solutions without imposing a specific form of f . Rather, as in the multidimensional case, we will intr...

The article presents a modification of the algorithm for the inverse problem of electrocardiography originally proposed in [6]. The modification is intended to improve the computation accuracy and to reduce the computing time.Keywords: inverse problem of electrocardiography, Cauchy problem for the Laplace equation, boundary integral equations, Tikhonov regularization method, iterative algorithm.The inverse problem of electrocardiography in potential form reconstructs the potential on the outer surface of the heart from potential measurements on the surface of the chest [1, 2]. The relevance of this inverse problem is understandable in view of the adoption in clinical practice of new techniques for the treatment of cardiac arrhythmia.The algorithm proposed in [3] solves the inverse problem of electrocardiography for a highly schematic geometry of the trunk and the heart; a more realistic geometry is used in [4], but a homogeneous thorax is assumed; an algorithm for a piecewise-homogeneous model of the thorax has been developed in [5,6]. In the present study we modify the algorithm of [6] with the aim of increasing the computation accuracy and reducing the computation time.Consider the region (see Fig. 1) in the space R 3 , bounded from the outside by the closed surface B and from the inside by the closed surface H . The surface B is the union of two surfaces, T and E . Given are two nonintersecting regions i with the boundaries i , i = 1, 2 . This geometrical configuration is amenable to the following interpretation: H is the surface of the heart, E is the part of the surface of the trunk on which the electric potential is measured, T are the upper and lower sections of the trunk, i , i = 1, 2 , are the regions of nonhomogeneity of the human thorax (the left and the right lung). Define 0 = \ ( 1 1 ) , 0 = E