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

Abstract

In this paper, we consider an inverse problem of determining multiple ionic parameters of a 2 × 2 strongly coupled parabolic–elliptic reaction–diffusion system arising in cardiac electrophysiology modeling. We use the bidomain model coupled to an ordinary differential equation (ODE) system and we consider a general formalism of physiologically detailed cellular membrane models to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate of the ion channels conductance parameters of the model using subboundary observations over an interval of time. The key ingredients are a global Carleman-type estimate with a suitable observations acting on a part of the boundary.

Analysis of the heart-torso conductivity parameters recovery inverse problem in cardiac electrophysiology ECG modelling

“…In the computational electrophysiology community, the monodomain model is the most used in order that is computationally much cheaper than the bidomain model. In this context, there are some works focused to the study of the cardiac parameters identifiability problem by using Carleman estimates [1,2,29] and in the framework of the cardiac conductivities recovery problem by means of Carleman estimates, Aniseba, Bendahmane and Yuan in [3] obtains the stability results for the conductivities diffusion-coefficients and Wu, Yan, Gao and Chen in [41] proved a Holder stability result for the inverse conductivities problem.…”

Global Lipschitz stability of multiple space dependent ionic parameters for the MEA/hiPSC-CM assays

“…Despite their importance, a few theoretical studies dealt with the estimation of ionic parameters in a bidomain model. We state the work of [3], which proves the stability of the ionic conductance parameters identification problem in case of the bidomain model but with the conductivities are scalar functions in order to transform the strongly coupled term in terms of a gradient. Two other works have been dedicated to the parameters of phenomenological ionic models still in the case of the monodomain approximation.…”

“…C e λ∥ψ∥ ∞ (e λ∥ψ∥ ∞ − 1)3 . Combining (B.4) and (B.8), for all λ ⩾ λ 0 , for all s ⩾ s 0 (λ) := max(s 1 , e C1T ) we obtainˆQ |u(x)| φ 0 (x, t) 2 e −2sη0(x,t) dx dt = |u(x)| |ω| 2 (x, t)dx dt ⩽ ˆΩ ˆIεs |u(x)| ω 2 (x, t)dx dt + 2 ˆΩ ˆ(0, T 2 )\Iε s |u(x)| ω 2 (x, t)dx dt ⩽ ˆΩ |u(x)| 2εsφ 2 0 (x, t 0 ) + T C0 ln(s) 2 e −2sη0(x,t0) dx ⩽ ˆΩ |u(x)| ln(s) 2e −2sη0(x,t0) dx ⩽ C1 (λ) ln(s) ˆΩ |u(x)| e −2sη0(x,t0) dx, (B.9)where C1 (λ) := 1 + 2e 2λ∥ψ∥ ∞ t Which completes the proof.…”

Mathematical analysis of a 1D-3D coupled problem in cardiac electrophysiology

“…The electrical behaviour of the heart has been extensively studied in various mathematical works [1,5,11,12,17,18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1,5].…”