**Links**

(Section Under Developmentā¦ See Publications)

**Example:** Variations in ECG Propagation Paths and Velocities

Sequential heartbeats were paced with stimuli delivered to the same location on the endocardial surface of the ventricles. Measurements were made using 120 electrodes on the torso surface. After time-alignment (according to known timing of stimuli) and nonlinear dimensionality reduction (using Laplacian eigenmaps), we were able to see potentially important variations across beats in the propagation of electrical excitation. Large points (one for each beat) in this visualization represent the data at the current time sample. We use color to show the progression of time within the QRS complex.

Cardiac electrical imaging is a tool with the potential to be of significant use in the clinic. My research on this topic has been related to addressing several potential impediments to widespread adoption of this technology, including

- the inability to image electric potentials on both the endocardial and epicardial surfaces of the ventricles, and
- the need for full anatomical imaging of the patient's thorax to build an accurate patient-specific geometry.

We applied dynamic analysis to the inverse problem of electrocardiography -- the problem of imaging parameters of cardiac electrical sources from electrical measurements on the body surface, given a model of the intervening torso volume conductor. This problem is ill-posed and requires regularization to achieve stable solutions. We have extended current approaches which estimate the potentials on the outer surfaces of the cardiac ventricles (the epicardium) -- potential-based inverse ECG -- to also estimate potentials on the inner surface (endocardium), which are more challenging to recover from the body surface but arguably of greater medical importance. We do so while using a less precise model of the subject's thorax geometry, compared to standard methods, constructed from a reduced set of acquired CT slices augmented by a coarse morphing of a single standardized anatomical model. The key innovation is a non-linear low-order temporal model of the signal trajectories.

My research also includes work on another common formulation of the inverse problem of electrocardiography, in which the goal is to estimate the spatial distribution of electrical activation times during a cardiac cycle. My work addresses the challenge of understanding the robustness of solutions to this formulation. This formulation poses a non-convex, non-linear least squares optimization problem. I have shown that it can be relaxed to be convex, and that this can be used as a framework to study the robustness of solutions to model uncertainties.

Below are some figures from work on this topic. In these figures, the term "Nearest Neighbor" refers to a set of activation times extracted from the convex relaxation (see publications for more details).