On the possibility for computing the transmembrane potential in the heart with a one shot method: an inverse problem

Area of Science:

• Biophysics
• Computational Biology
• Medical Imaging

Background:

• Accurate computation of transmembrane potential is crucial for understanding cardiac function and disease.
• Existing methods face challenges with stability and uniqueness, especially in complex scenarios like regional ischemia.

Purpose of the Study:

• To investigate the feasibility of using body surface potential maps (BSPMs) and a priori information with bidomain equations to compute myocardial transmembrane potential.
• To develop a stable and unique solution for this inverse problem using an output least squares framework and PDE constraints.

Main Methods:

• Formulated the problem as an inverse problem for elliptic partial differential equations (PDEs).
• Employed an output least squares framework with PDE constraints for minimization.
• Developed a fully implicit, one-shot method solving a system of three linear elliptic PDEs for stability and uniqueness.

Main Results:

• Demonstrated a methodology for stabilizing and enforcing uniqueness in the inverse problem.
• Numerical experiments showed potential for approximating transmembrane potential during resting and plateau phases in regional ischemia.
• Accurate a priori information appears key for successful recovery during specific cardiac cycle phases.

Conclusions:

• The proposed method offers a stable approach to compute transmembrane potential using BSPMs and bidomain equations.
• Successful recovery of transmembrane potential is feasible during specific phases (resting, plateau) in conditions like regional ischemia.
• Computing transmembrane potential at arbitrary time instances during a heart beat remains an open challenge.