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**Harvard**

Wall, D., Olsson, P. och van Houten, E. (2011) *On an inverse problem from magnetic resonance elastic imaging*.

** BibTeX **

@article{

Wall2011,

author={Wall, David J.N. and Olsson, Peter and van Houten, Elijah E. W.},

title={On an inverse problem from magnetic resonance elastic imaging},

journal={SIAM Journal on Applied Mathematics},

issn={0036-1399},

volume={71},

issue={5},

pages={1578-1605},

abstract={The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Frechet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify.},

year={2011},

keywords={magnetic resonance elastography, MRE, inverse problem, Frechet differentiability, shear stiffness reconstruction, noninvasive palpation, tumor imaging},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 146401

A1 Wall, David J.N.

A1 Olsson, Peter

A1 van Houten, Elijah E. W.

T1 On an inverse problem from magnetic resonance elastic imaging

YR 2011

JF SIAM Journal on Applied Mathematics

SN 0036-1399

VO 71

IS 5

SP 1578

OP 1605

AB The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Frechet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify.

LA eng

DO 10.1137/110832082

LK http://dx.doi.org/10.1137/110832082

LK http://publications.lib.chalmers.se/records/fulltext/local_146401.pdf

OL 30