### Abstract

Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging technique that aims to provide electrical conductivity images with sufficiently high spatial resolution and accuracy. A new MREIT image reconstruction method called the harmonic B_{z} algorithm was proposed in 2002, and it is based on the measurement of B_{z} that is a single component of an induced magnetic flux density B = (B_{x}, B_{y}, B_{z}) subject to an injection current. Since then, MREIT imaging techniques have made significant progress, and recent published numerical simulations and phantom experiments show that we can produce high-quality conductivity images when the conductivity contrast is not very high. Though numerical simulations can explain why we could successfully distinguish different tissues with small conductivity differences, a rigorous mathematical analysis is required to better understand the underlying physical and mathematical principle. The purpose of this paper is to provide such a mathematical analysis of those numerical simulations and experimental results. By using a uniform a priori estimate for the solution of the elliptic equation in the divergent form and an induction argument, we show that, for a relatively small contrast of the target conductivity, the iterative harmonic B_{z} algorithm with a good initial guess is stable and exponentially convergent in the continuous norm. Both two- and three-dimensional versions of the algorithm are considered, and the difference in the convergence property of these two cases is analyzed. Some numerical results are also given to show the expected exponential convergence behavior.

Original language | English |
---|---|

Pages (from-to) | 1259-1282 |

Number of pages | 24 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 67 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 Oct 26 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

_{Z}algorithm in magnetic resonance electrical impedance tomography.

*SIAM Journal on Applied Mathematics*,

*67*(5), 1259-1282. https://doi.org/10.1137/060661892

}

_{Z}algorithm in magnetic resonance electrical impedance tomography',

*SIAM Journal on Applied Mathematics*, vol. 67, no. 5, pp. 1259-1282. https://doi.org/10.1137/060661892

**On the convergence of the harmonic B _{Z} algorithm in magnetic resonance electrical impedance tomography.** / Liu, J. J.; Seo, J. K.; Sinl, M.; Woo, E. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the convergence of the harmonic BZ algorithm in magnetic resonance electrical impedance tomography

AU - Liu, J. J.

AU - Seo, J. K.

AU - Sinl, M.

AU - Woo, E. J.

PY - 2007/10/26

Y1 - 2007/10/26

N2 - Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging technique that aims to provide electrical conductivity images with sufficiently high spatial resolution and accuracy. A new MREIT image reconstruction method called the harmonic Bz algorithm was proposed in 2002, and it is based on the measurement of Bz that is a single component of an induced magnetic flux density B = (Bx, By, Bz) subject to an injection current. Since then, MREIT imaging techniques have made significant progress, and recent published numerical simulations and phantom experiments show that we can produce high-quality conductivity images when the conductivity contrast is not very high. Though numerical simulations can explain why we could successfully distinguish different tissues with small conductivity differences, a rigorous mathematical analysis is required to better understand the underlying physical and mathematical principle. The purpose of this paper is to provide such a mathematical analysis of those numerical simulations and experimental results. By using a uniform a priori estimate for the solution of the elliptic equation in the divergent form and an induction argument, we show that, for a relatively small contrast of the target conductivity, the iterative harmonic Bz algorithm with a good initial guess is stable and exponentially convergent in the continuous norm. Both two- and three-dimensional versions of the algorithm are considered, and the difference in the convergence property of these two cases is analyzed. Some numerical results are also given to show the expected exponential convergence behavior.

AB - Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging technique that aims to provide electrical conductivity images with sufficiently high spatial resolution and accuracy. A new MREIT image reconstruction method called the harmonic Bz algorithm was proposed in 2002, and it is based on the measurement of Bz that is a single component of an induced magnetic flux density B = (Bx, By, Bz) subject to an injection current. Since then, MREIT imaging techniques have made significant progress, and recent published numerical simulations and phantom experiments show that we can produce high-quality conductivity images when the conductivity contrast is not very high. Though numerical simulations can explain why we could successfully distinguish different tissues with small conductivity differences, a rigorous mathematical analysis is required to better understand the underlying physical and mathematical principle. The purpose of this paper is to provide such a mathematical analysis of those numerical simulations and experimental results. By using a uniform a priori estimate for the solution of the elliptic equation in the divergent form and an induction argument, we show that, for a relatively small contrast of the target conductivity, the iterative harmonic Bz algorithm with a good initial guess is stable and exponentially convergent in the continuous norm. Both two- and three-dimensional versions of the algorithm are considered, and the difference in the convergence property of these two cases is analyzed. Some numerical results are also given to show the expected exponential convergence behavior.

UR - http://www.scopus.com/inward/record.url?scp=35348986496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348986496&partnerID=8YFLogxK

U2 - 10.1137/060661892

DO - 10.1137/060661892

M3 - Article

AN - SCOPUS:35348986496

VL - 67

SP - 1259

EP - 1282

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -

_{Z}algorithm in magnetic resonance electrical impedance tomography. SIAM Journal on Applied Mathematics. 2007 Oct 26;67(5):1259-1282. https://doi.org/10.1137/060661892