On the inverse problem of detecting cardiac ischemias: theoretical analysis and numerical reconstruction
This work addresses the detection of myocardial ischemia, a clinically relevant problem, but the method is tested only on idealized geometries and may be incremental.
The paper develops a theoretical framework and numerical method for detecting small ischemic regions in the heart from boundary potential measurements, using an asymptotic expansion and topological gradient optimization. Numerical tests on a 3D left ventricle model demonstrate feasibility and robustness.
In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of a small inhomogeneity $ω_\varepsilon$ (where the coefficients of the equation are altered) located inside a domain $Ω$ starting from observations of the potential on the boundary $\partial Ω$. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a reconstruction procedure based on the topological gradient of a suitable cost functional. Numerical results obtained on an idealized three-dimensional left ventricle geometry for different measurement settings assess the feasibility and robustness of the algorithm.