Levenberg–Marquardt method for solving inverse problem of MRE based on the modified stationary Stokes system

Abstract

Magnetic resonance elastography (MRE) is a developing medical imaging technique which can successfully diagnose liver diseases such as cirrhotic and fibrosis. As a hardware MRE gives an image of a viscoelastic wave in a region of interest (ROI) of a human body. Here the wave is generated by an external vibration system and further transmitted to the human body through a probe. MRE also has a software usually called elastogram which reconstructs viscoelastic modulus from the wave displacement vector. An appropriate model equation or system which connects the wave image to the modulus is known as modified stationary Stokes system if viscoelastic property of the ROI is isotropic (Ammari et al 2008 Q. Appl. Math. 66 139–175; Jiang et al 2011 SIAM J. Appl. Math. 71 1965–1989). Comparing this system with the original stationary Stokes system, it has an extra term with frequency. In most cases the elastogram has been using a more simpler model equation such as scalar model (Manduca et al 2003 Med. Image Anal. 5 237–254). In this paper we discuss about a scheme of solving the equation (*)F(A) = u, where u is the measured wave displacement and F(A) is the solution to the boundary value problem for the modified stationary Stokes system with the modulus A (see subsection for more details). More precisely, we will show the convergence of a Newton type iteration scheme called the Levenberg–Marquardt method by proving that the nonlinear operator F satisfies the tangential cone condition. We will also provide several numerical tests even with noisy data.