Analysis of the Magnetoacoustic Tomography with Magnetic Induction

Magnetoacoustic tomography with magnetic induction (MAT-MI) is a coupled-physics medical imaging modality for determining conductivity distribution in biological tissue. The capability of MAT-MI to provide high-resolution images has been demonstrated experimentally. MAT-MI involves two steps. The first step is a well-posed inverse source problem for acoustic wave equations, which has been well studied in the literature. This paper concerns mathematical analysis of the second step, a quantitative reconstruction of the conductivity from knowledge of the internal data recovered in the first step, using techniques such as time reversal. The problem is modeled by a system derived from Maxwell's equations. We show that a single internal data determines the conductivity. A global Lipschitz-type stability estimate is obtained. A numerical approach for recovering the conductivity is proposed, and results from computational experiments are presented.

[1]  Bin He,et al.  A Reconstruction Algorithm of Magnetoacoustic Tomography With Magnetic Induction for an Acoustically Inhomogeneous Tissue , 2014, IEEE Transactions on Biomedical Engineering.

[2]  Bin He,et al.  Imaging biological tissues with electrical conductivity contrast below 1 S m by means of magnetoacoustic tomography with magnetic induction. , 2010, Applied physics letters.

[3]  Xu Li,et al.  Multi-excitation Magnetoacoustic Tomography With Magnetic Induction for Bioimpedance Imaging , 2010, IEEE Transactions on Medical Imaging.

[4]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[5]  G. Uhlmann,et al.  Thermoacoustic tomography with variable sound speed , 2009, 0902.1973.

[6]  Otmar Scherzer,et al.  Filtered backprojection for thermoacoustic computed tomography in spherical geometry , 2005, Mathematical Methods in the Applied Sciences.

[7]  Linh V. Nguyen,et al.  Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media , 2008 .

[8]  Bin He,et al.  Magnetoacoustic tomography with magnetic induction for imaging electrical impedance of biological tissue , 2006 .

[9]  G. Bal.  Hybrid inverse problems and redundant systems of partial differential equations , 2012, 1210.0265.

[10]  P. Kuchment,et al.  Mathematics of thermoacoustic tomography , 2007, European Journal of Applied Mathematics.

[11]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[12]  Guillaume Bal,et al.  Hybrid inverse problems and internal functionals , 2011, 1110.4733.

[13]  Hongkai Zhao,et al.  An Efficient Neumann Series-Based Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed , 2011, SIAM J. Imaging Sci..

[14]  Simon Boulier,et al.  A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction , 2015, 1501.04803.

[15]  Bradley J. Roth,et al.  The Effect of Electrical Anisotropy During Magnetoacoustic Tomography With Magnetic Induction , 2008, IEEE Transactions on Biomedical Engineering.

[16]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[17]  O. Scherzer,et al.  Hybrid tomography for conductivity imaging , 2011, 1112.2958.

[18]  Otmar Scherzer,et al.  Local analysis of inverse problems: Hölder stability and iterative reconstruction , 2011, 1108.3570.

[19]  Bin He,et al.  Magnetoacoustic Tomography With Magnetic Induction: Bioimepedance Reconstruction Through Vector Source Imaging , 2013, IEEE Transactions on Medical Imaging.

[20]  Bin He,et al.  Magnetoacoustic tomography with magnetic induction (MAT-MI) , 2005, Physics in medicine and biology.

[21]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[22]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[23]  Otmar Scherzer,et al.  An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints , 2015, Numerische Mathematik.

[24]  Carlos Montalto,et al.  Stability of coupled-physics inverse problems with one internal measurement , 2013, 1306.1978.

[25]  Laurent Seppecher,et al.  A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography☆ , 2014 .

[26]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[27]  Peter Kuchment,et al.  Mathematics of Hybrid Imaging: A Brief Review , 2011, 1107.2447.

[28]  Faouzi Triki,et al.  Uniqueness and stability for the inverse medium problem with internal data , 2010 .

[29]  Habib Ammari,et al.  An Introduction to Mathematics of Emerging Biomedical Imaging , 2008 .

[30]  Rakesh,et al.  Recovering a Function from Its Spherical Mean Values in Two and Three Dimensions , 2017 .

[31]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .