### Abstract

This paper presents a new method of providing both conductivity (θ) and permittivity (-) images at the MR Larmor frequency in magnetic resonance electrical property tomography (MREPT), a relatively new MR-based electrical tissue property imaging modality. In MREPT, the RF coil of the MR scanner is used to feed a sinusoidal current at the Larmor frequency, ε/2π (approximately 128 MHz for a 3 Tesla MRI machine), to an imaging object within the MR scanner. Inside the object, this injection current induces a time-harmonic magnetic field, H = (H_{x},H_{y},H_{z}), that is influenced by the body's admittivity distribution, δ = θ + iε, via Maxwell's equations. Currently, the positive rotating field, H _{+} = (H_{x} + iH_{y})/2, is the only measurable quantity which can be obtained from B1 mapping techniques. The inverse problem of MREPT is to reconstruct δ from the H_{+} data. Existing MREPT reconstruction methods use the local homogeneity assumption (δ = 0) to simplify the relation between δ and H_{+}, so that the admittivity is proportional to the ratio of the Laplacian _{2}H_{+} to H_{+}. However, the conventional reconstruction method based on this local homogeneity assumption gives rise to serious reconstruction errors at large |δ||δ|. Hence, the major issue in MREPT is to find a reconstruction formula without assuming the local homogeneity condition. The proposed method removes the assumption of ( θ θx δ, θ θy δ) = 0 (but still requires θ θz δ ? 0). We have found an exact relation between δ and H+, so that δ is a solution of a semilinear elliptic PDE with coefficients depending only upon H+. Numerical simulations show that the proposed method successfully reconstructs discontinuous admittivities.

Original language | English |
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Pages (from-to) | 2262-2280 |

Number of pages | 19 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 73 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2013 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

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*SIAM Journal on Applied Mathematics*, vol. 73, no. 6, pp. 2262-2280. https://doi.org/10.1137/130906842

**Conductivity and permittivity image reconstruction at the larmor frequency using MRI.** / Song, Yizhuang; Seo, Jin Keun.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Conductivity and permittivity image reconstruction at the larmor frequency using MRI

AU - Song, Yizhuang

AU - Seo, Jin Keun

PY - 2013/12/1

Y1 - 2013/12/1

N2 - This paper presents a new method of providing both conductivity (θ) and permittivity (-) images at the MR Larmor frequency in magnetic resonance electrical property tomography (MREPT), a relatively new MR-based electrical tissue property imaging modality. In MREPT, the RF coil of the MR scanner is used to feed a sinusoidal current at the Larmor frequency, ε/2π (approximately 128 MHz for a 3 Tesla MRI machine), to an imaging object within the MR scanner. Inside the object, this injection current induces a time-harmonic magnetic field, H = (Hx,Hy,Hz), that is influenced by the body's admittivity distribution, δ = θ + iε, via Maxwell's equations. Currently, the positive rotating field, H + = (Hx + iHy)/2, is the only measurable quantity which can be obtained from B1 mapping techniques. The inverse problem of MREPT is to reconstruct δ from the H+ data. Existing MREPT reconstruction methods use the local homogeneity assumption (δ = 0) to simplify the relation between δ and H+, so that the admittivity is proportional to the ratio of the Laplacian 2H+ to H+. However, the conventional reconstruction method based on this local homogeneity assumption gives rise to serious reconstruction errors at large |δ||δ|. Hence, the major issue in MREPT is to find a reconstruction formula without assuming the local homogeneity condition. The proposed method removes the assumption of ( θ θx δ, θ θy δ) = 0 (but still requires θ θz δ ? 0). We have found an exact relation between δ and H+, so that δ is a solution of a semilinear elliptic PDE with coefficients depending only upon H+. Numerical simulations show that the proposed method successfully reconstructs discontinuous admittivities.

AB - This paper presents a new method of providing both conductivity (θ) and permittivity (-) images at the MR Larmor frequency in magnetic resonance electrical property tomography (MREPT), a relatively new MR-based electrical tissue property imaging modality. In MREPT, the RF coil of the MR scanner is used to feed a sinusoidal current at the Larmor frequency, ε/2π (approximately 128 MHz for a 3 Tesla MRI machine), to an imaging object within the MR scanner. Inside the object, this injection current induces a time-harmonic magnetic field, H = (Hx,Hy,Hz), that is influenced by the body's admittivity distribution, δ = θ + iε, via Maxwell's equations. Currently, the positive rotating field, H + = (Hx + iHy)/2, is the only measurable quantity which can be obtained from B1 mapping techniques. The inverse problem of MREPT is to reconstruct δ from the H+ data. Existing MREPT reconstruction methods use the local homogeneity assumption (δ = 0) to simplify the relation between δ and H+, so that the admittivity is proportional to the ratio of the Laplacian 2H+ to H+. However, the conventional reconstruction method based on this local homogeneity assumption gives rise to serious reconstruction errors at large |δ||δ|. Hence, the major issue in MREPT is to find a reconstruction formula without assuming the local homogeneity condition. The proposed method removes the assumption of ( θ θx δ, θ θy δ) = 0 (but still requires θ θz δ ? 0). We have found an exact relation between δ and H+, so that δ is a solution of a semilinear elliptic PDE with coefficients depending only upon H+. Numerical simulations show that the proposed method successfully reconstructs discontinuous admittivities.

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U2 - 10.1137/130906842

DO - 10.1137/130906842

M3 - Article

VL - 73

SP - 2262

EP - 2280

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 6

ER -