On an inverse problem from magnetic resonance elastic imaging
Journal article, 2011

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.

shear stiffness reconstruction

tumor imaging

magnetic resonance elastography

inverse problem

Frechet differentiability

noninvasive palpation

MRE

Author

David J.N. Wall

University of Canterbury

Peter Olsson

Chalmers, Applied Mechanics, Dynamics

Elijah E. W. van Houten

University of Canterbury

SIAM Journal on Applied Mathematics

0036-1399 (ISSN) 1095-712X (eISSN)

Vol. 71 1578-1605

Subject Categories

Mechanical Engineering

Materials Engineering

Applied Mechanics

Computational Mathematics

Radiology, Nuclear Medicine and Medical Imaging

Roots

Basic sciences

Areas of Advance

Life Science Engineering

Materials Science

DOI

10.1137/110832082