An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Elena Beretta, Cecilia Cavaterra, M. Cristina Cerutti, Andrea Manzoni, Luca Ratti

Research output: Contribution to journalArticle

Abstract

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 an inhomogeneity ω∈ (where the coefficients of the equation are altered) located inside a domain φ starting from observations of the potential on the boundary ∂φ. 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 in 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.

Original languageEnglish (US)
Article number105008
JournalInverse Problems
Volume33
Issue number10
DOIs
StatePublished - Sep 20 2017

Fingerprint

Cardiac Electrophysiology
Semilinear Parabolic Equation
Electric Potential
Inverse problems
Inhomogeneity
Conductivity
Inverse Problem
Inverse Boundary Value Problem
Three-dimensional
Left Ventricle
Electric potential
Parabolic Problems
Asymptotic Formula
Volume Fraction
Boundary value problems
Blood
Volume fraction
Theoretical Analysis
Tissue
Gradient

Keywords

  • cardiac electrophysiology
  • inverse boundary value problem
  • semilinear parabolic equation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics

Cite this

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology. / Beretta, Elena; Cavaterra, Cecilia; Cerutti, M. Cristina; Manzoni, Andrea; Ratti, Luca.

In: Inverse Problems, Vol. 33, No. 10, 105008, 20.09.2017.

Research output: Contribution to journalArticle

Beretta, Elena ; Cavaterra, Cecilia ; Cerutti, M. Cristina ; Manzoni, Andrea ; Ratti, Luca. / An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology. In: Inverse Problems. 2017 ; Vol. 33, No. 10.
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