### 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 language | English (US) |
---|---|

Article number | 105008 |

Journal | Inverse Problems |

Volume | 33 |

Issue number | 10 |

DOIs | |

State | Published - Sep 20 2017 |

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### 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

*Inverse Problems*,

*33*(10), [105008]. https://doi.org/10.1088/1361-6420/aa8737

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

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 33, no. 10, 105008. https://doi.org/10.1088/1361-6420/aa8737

}

TY - JOUR

T1 - An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

AU - Beretta, Elena

AU - Cavaterra, Cecilia

AU - Cerutti, M. Cristina

AU - Manzoni, Andrea

AU - Ratti, Luca

PY - 2017/9/20

Y1 - 2017/9/20

N2 - 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.

AB - 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.

KW - cardiac electrophysiology

KW - inverse boundary value problem

KW - semilinear parabolic equation

UR - http://www.scopus.com/inward/record.url?scp=85029880307&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029880307&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/aa8737

DO - 10.1088/1361-6420/aa8737

M3 - Article

VL - 33

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 10

M1 - 105008

ER -