Abstract

Cardiac arrhythmias are irregularities in the electrical activity in the heart; the electrical impulses in the heart become chaotic or disorganized, which can cause a possibly lethal problem to the contraction of the heart. In order to understand the dynamics of arrhythmias and to be able to predict and treat them, numerical models have been developed to capture the dynamics of the electrical impulses in the heart. In a clinical setting, optical mapping technologies — using cameras and voltage-sensitive dyes to capture the electrical impulses propagating across the heart — have been used to capture the dynamics of the electrical activity along the surface of the heart with high spatial and temporal resolution. Despite the high resolution provided by the optical mapping technologies along the surface of the heart, the techniques are unable to capture measurements of the voltage in the interior of the heart. Kalman Filters attempt to solve this problem by combining experimental data — that which is obtained by direct measurement, such as by optical mapping methods — with a mathematical model. This has been shown using synthetic data to be an effective method of reconstructing the dynamics of certain cardiac arrhythmias in tissue. It is desirable to be able to obtain the values of the parameters that guide the dynamical behavior of the cardiac arrhythmias in a given mathematical model. Knowledge of the values of the model parameters can be used to retroactively explain why dynamical effects occurred or to predict future behavior of the electrical impulses propagating throughout the cardiac tissue. In this thesis, we utilize a state-augmentation method of estimating model parameters using a nonlinear extension of the general Kalman Filter. We use a three-variable model of the cardiac action potential in conjunction with the Local Ensemble Transform Kalman Filter (LETKF) in order to estimate the state of the electrical impulses traveling along cardiac tissue. We show the viability of the state-augmentation methods of parameter estimation with the LETKF and determine three criteria that can be used to explain the effectiveness of the parameter estimation algorithm. We first establish the results by estimating a single parameter, and then expand our results by showing the same criteria hold when estimating multiple model parameters simultaneously. The results provide evidence that this method of parameter estimation is useful for cardiac models — both by a good estimation of the state and the predictable estimation of the model parameters — and suggest additional avenues of research for the preliminary work presented in this thesis.

Library of Congress Subject Headings

Arrhythmia--Mathematical models; Kalman filtering

Publication Date

5-7-2019

Document Type

Thesis

Student Type

Graduate

Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

School of Mathematical Sciences (COS)

Advisor

Elizabeth Cherry

Advisor/Committee Member

Matthew Hoffman

Advisor/Committee Member

Laura Munoz

Campus

RIT – Main Campus

Plan Codes

ACMTH-MS

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