Sensitivity of Forward Potential Calculations to Errors in Geometry using Boundary Elements

by
Andrew Shafer

Advised by
Rob Macleod

Many physiological events result in measurable bioelectricity, for example the beating of the heart and the transmission of information in the brain (ECG, EEG, etc). Mathematical models and simulation can be useful tools to study and explore these bioelectrical events. One common model describes the behavior of electric fields in the region surrounding a bioelectric source with Poisson's equation. Typical error analysis of the numerical methods required to solve such problems in realistic geometries solve the discrete form of the differential equation on a known and simplified geometry and compare the computed result to an analytic solution. The driving question in this project was to what extent and in which manner geometric uncertainties affect the accuracy of the solution to Poisson's equation applied to electrocardiography. For this, we used the Boundary Element Method (BEM) and determined the effects of geometric errors on computed potentials in a homogeneous conductor. We first established baseline levels of accuracy for our BEM implementation using analytic solutions to dipolar sources on spherical geometries. We then applied correlated and uncorrelated geometric errors to the geometry and compared the resulting computations with the original analytic solutions. We then extended the results from the spherical geometry to similar calculations on realistic torso geometry. Results indicate that solution error is, indeed, highly dependent on geometric error, both with regard to the source location relative to the surfaces in the model and the locations of the measurement electrodes.