Stability analysis for Magnetic Resonance Elastography

Ammari, Habib; Waters, Alden; Zhang, Hai

doi:10.1016/j.jmaa.2015.05.029

Cited by 15 publications

(20 citation statements)

References 29 publications

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“…The modification between this theorem and the result in [4] is the reduction of the necessary regularity to practically relevant values. The discrepancy functional in d = 3 must be modified to include both sets of data as in [4]. We can then conclude that: Theorem 7.…”

Section: Stability Of the Landweber Iteration Schemementioning

confidence: 97%

“…We consider as in [4] the reconstruction of the true shear modulus distribution µ tr . The treatment follows the book [2] closely.…”

Section: Stability Of the Landweber Iteration Schemementioning

confidence: 99%

“…Hybrid modalities involve exciting the system with more than one wave or modality. In 3d it was already shown in [4] using elliptic regularity theory that hybrid modalities are necessary for reconstruction of the shear modulus. This necessity is in contrast to dimension 2.…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis in magnetic resonance elastography II

Gimperlein

Waters

2016

Journal of Mathematical Analysis and Applications

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We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to [4], we use pseudodifferential methods for the problem of recovering the shear modulus for Stokes systems from internal data. We prove stability estimates in d = 2, 3 with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed. This implies the convergence of the Landweber numerical iteration scheme. We also show that these hypotheses are natural for experimental use in constructing shear modulus distributions.Mathematics Subject Classification (MSC2000): 35B30.

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Section: Stability Of the Landweber Iteration Schemementioning

confidence: 97%

“…We consider as in [4] the reconstruction of the true shear modulus distribution µ tr . The treatment follows the book [2] closely.…”

Section: Stability Of the Landweber Iteration Schemementioning

confidence: 99%

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Stability analysis in magnetic resonance elastography II

Gimperlein

Waters

2016

Journal of Mathematical Analysis and Applications

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“…Further ρ describes the density of the tissue which can be taken equal to that of water i.e. ρ = 1000 kg/m 3 and ω, g are the frequency and a component of the displacement vector input to the human body. In the rest of this paper we assume that ρ is equal to the above constant for simplicity.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Levenberg–Marquardt method for the inverse problem with an interior measurement

Jiang

Nakamura

2018

Journal of Inverse and Ill-Posed Problems

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The convergence of Levenberg-Marquard method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so called scalar model by single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two layered picewise homogneneous scalar model in a rectangular domain.

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“…The reader is referred to [11,16,25,26] and references therein for recent works on uniqueness and stability results on inverse problems from internal data. The use of multiple frequencies to enforce non-zero constraints in PDE, and to obtain well-posedness for several hybrid problems, has been discussed in [1,2,3,4,5,9].…”

Section: Introductionmentioning

confidence: 99%

Multi-frequency acousto-electromagnetic tomography

Alberti¹,

Ammari²,

Ruan³

2016

A Panorama of Mathematics: Pure and Applied

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Abstract. This paper focuses on the acousto-electromagnetic tomography, a recently introduced hybrid imaging technique. In a previous work, the reconstruction of the electric permittivity of the medium from internal data was achieved under the Born approximation assumption. In this work, we tackle the general problem by a Landweber iteration algorithm. The convergence of such scheme is guaranteed with the use of a multiple frequency approach, that ensures uniqueness and stability for the corresponding linearized inverse problem. Numerical simulations are presented.

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Stability analysis for Magnetic Resonance Elastography

Cited by 15 publications

References 29 publications

Stability analysis in magnetic resonance elastography II

Stability analysis in magnetic resonance elastography II

Convergence of Levenberg–Marquardt method for the inverse problem with an interior measurement

Multi-frequency acousto-electromagnetic tomography

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