On the identification of multiple space dependent ionic parameters in cardiac electrophysiology modelling

Abstract

In this paper, we consider the inverse problem of space dependent multiple ionic parameters identification in cardiac electrophysiology modelling from a set of observations. We use the monodomain system known as a state-of-the-art model in cardiac electrophysiology and we consider a general Hodgkin-Huxley formalism to describe the ionic exchanges at the microscopic level. This formalism covers many physiological transmembrane potential models including those in cardiac electrophysiology. Our main result is the proof of the uniqueness and a Lipschitz stability estimate of ion channels conductance parameters based on some observations on an arbitrary subdomain. The key idea is a Carleman estimate for a parabolic operator with multiple coefficients and an ordinary differential equation system.

Ionic parameters identification of an inverse problem of strongly coupled PDE’s system in cardiac electrophysiology using Carleman estimates

“…Theorem 3.1. Let t 0 ∈ (0, T ), Γ 0 be a subdomain of ∂Ω and let¯ (2) ∈ A be arbitrary fixed. We assume that…”

“…Here v (2) , w (2) and z (2) are the state variables of (2.13) with¯ =¯ (2) and I s i,e = I s, i,e . Furthermore, we assume that v…”

“…Despite their importance, ion channel maximal conductances have not been subject of abondant theoretical studies. The only theoretical work that we know is [2], which proves under certain conditions the stability of the ionic conductance parameters identification problem in case of the monodomain approximation. Two other works has been dedicated to the parameters of phenomenological ionic models still in the case of the monodomain approximation.…”

“…and we consider the difference (2) , u e = u (1) e − u (2) e , w = w (1) − w (2) , z = z (1) − z (2) ,¯ =¯ (1) −¯ (2) . (4.9)…”

Functional Imaging and Modeling of the Heart

“…In an other work [7], authors propose a strategy to optimize a non differentiable cost function obtained from a fit of activation times map. Recently, theoretical studies of the stability of the maximal conductances identification problem in the monodomain [8] and bidomain [9] models have been carried out. Yan and Veneziani proposed a variational procedure for the estimation of cardiac conductivities from measures of the transmembrane and extracellular potentials available at some sites of the tissue [10].…”