Adaptive placement of the pseudo:boundaries improves the conditioning of the inverse problem

Abstract

Meshing the heart and measurement surfaces can be time consuming, especially when dealing with complicated geometries or cardiac motion. To overcome this, a meshless method based on the method of fundamental solutions (MFS) has been adapted to non-invasive electrocardiographic imaging (ECGI). In the MFS, potentials are expressed as a summation over a discrete set of virtual point sources placed outside of the domain of interest (named 'pseudo-boundary'). It is well-known that optimal placement of the pseudoboundary can improve the efficacy of the MFS. Despite this, there have been no attempts to optimize their placement in the ECGI problem as far as we are aware. In the standard MFS, the sources are placed in two pseudo-boundaries constructed by inflating and deflating the heart and torso surfaces with respect to the geometric center of the heart. However, for some heart-torso geometries, this geometric center is a poor reference. We here show that adaptive placement of the pseudoboundaries (depending on the distance between the torso electrodes and the nearest heart locations) improves the conditioning of the inverse problem, making it less sensitive to the regularization process.

Improving the Spatial Solution of Electrocardiographic Imaging: A New Regularization Parameter Choice Technique for the Tikhonov Method

“…We want to remark that while here we only plot the DPC for a certain spiral wave data and for a fixed instant of time, , (as example to explain the details of a DPC plot) the SVs for the ECGI MFS matrix as formulated in [9,10] usually decrease slowly initially, and they start to highly decrease for larger values of the index (commonly ~). The same SV decay behavior for ECGI MFS for other real torso-heart geometries can be observed in [24]. Figure 3 shows the reconstructed potentials in a random location on the epicardium through the different time instants, (ms) for the different regularization parameters together with the simulated ones.…”

Exploring Possible Choices of the Tikhonov Regularization Parameter for the Method of Fundamental Solution in Electrocardiography

“…In [6] we showed that the influence of the regularization parameter can decrease by optimizing the transfer matrix used (e.g. decreasing its ill-conditioning by adjusting some of its key elements [6]).…”

“…In [6] we showed that the influence of the regularization parameter can decrease by optimizing the transfer matrix used (e.g. decreasing its ill-conditioning by adjusting some of its key elements [6]). This result links with the conclusion of [3], where the authors stated that no major difference was found by changing the regularization parameters, but show also notably degradation of the reconstruction performance, when errors were introduced in the transfer matrix.…”

Accuracy of electrocardiographic imaging using the method of fundamental solutions

“…A previous application of the MFS to ECGI [24] created the source points by moving the surface nodes along a ray, either inward or outward, as appropriate, joining the nodes to the centre of gravity of the heart. Other approaches to specifying the source nodes have also been presented elsewhere [28]. It is worth noting that it is possible to determine the number and position of the source nodes as part of the solution process, using non-linear optimisation routines [25].…”

“…The MFS approach has been applied to various inverse problems previously [26,27] and, as mentioned above, in particular to the inverse problem of electrocardiology [24]. Recent applications of the MFS to ECGI include studies of the locations of the heart and torso boundaries [28], application of the U-curve and the discrete Picard condition [29,30].…”