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IMAGE RECONSTRUCTION OF ANISOTROPIC ELASTIC MODULI IN MRE Jiah SONG 1 , Oh In KWON 2 and Jin Keun SEO 1 1) Department of Computational Science and Engineering, Yonsei University, Seoul, KOREA 2) Department of Mathematics, Konkuk University, Seoul, KOREA Corresponding Author : : Jiah SONG, firstname.lastname@example.org ABSTRACT Magnetic Resonance Elastography(MRE) is a recent medical imaging modality to provide a cross-sectional image of mechanical elastic properties of tissue using special phase encoding sequence in MR system. Last two decades, MRE techniques have been made marked progress in terms of reconstruction methods and measurement techniques, and now become successfully commercialized as a routinely used clinical device for evaluation of liver disease. In MRE, we apply a harmonically oscillating mechanical vibration using active and passive driver placed on a surface of the object, and measure the displacement of the induced transverse acoustic strain waves inside the body using MRI. The corresponding inverse problem is to recover shear mod- ulus using measured displacement and elasticity equation. Most of research outcomes are based on the assumption of isotropy and local homogeneity. However, several tissues such as mus- cle show strong anisotropy. Hence, challenging issue in MRE is to identify Young’s modulus and Poisson ratio as well as shear modulus that are different depending on directions. In this work, we proposed a new MRE technique for recovering anisotropic elastic moduli under the assumption of transversely isotropy. The proposed method decomposes the measured displace- ment wave vector ﬁelds into curl-free and divergence-free components via Helmholtz-Hodge decomposition and extract the three unknown elastic coefﬁcients from a careful use of two main components. Various numerical simulations show that the proposed method successfully reconstruct transversely isotropic elastic moduli. MODEL PROBLEM AND NUMERICAL SIMULATION Elastography is based on numerous ﬁnding that tissue stiffness is closely related to the velocity of wave, and the shear modulus (or modulus of rigidity) varies over a wide range differentiating various pathological states of tissues. Hence, the speed of the harmonic elastic wave provides quantitative information for describing malignant tissues that are typically known to be much stiffer than normal tissues. In elastography, we apply a time-harmonic excitation at a frequency ω in the range of 50 - 200 Hz through the surface of the object to be imaged. This produce harmonic mechanical displacements inside an object. We can measure the resulting displacements using either ultra- sound or MR system. The corresponding inverse problem is to identify elastic moduli from the measured displacement vector, denoted by u, and the time-harmonic elasticity equation ( ) 1 ∇ · C : ( (∇u + ut )) + ρω 2 u = 0 (1) 2 where ρ is density, ω the angular frequency, and C is the elastic 4th order tensor. In this work, we assume that the tissue is transversely isotropic and incompressible so that C can be reduced by 6 by 6 matrix form C C12 C13 0 11 0 0 C11 C13 0 0 0 C11 = 4µ12 = 2C13 , C33 0 0 0 C= (2) C44 0 0 C33 = 4µ12 E3 , C55 = 2µ13 sym C44 0 E1 C11 −C12 2 By projecting the image domain to x-z plane, the equation (1) can be simpliﬁed into C11 C11 ∂x u1 + ∂z u3 C2 ∂z u1 55 + C55 ∂x u3 u 2 1 ∇ · 2 2 + ρω = 0. (3) C55 C55 2 ∂z u1 + 2 ∂x u3 C2 ∂x u1 11 + C33 ∂z u3 u3 Applying Helmholtz-Hodge decomposition, we obtain ⊥ −∇φ1 + ∇ ψ1 C11 C11 ∂x u1 + 2 ∂z u3 C2 ∂z u1 55 + C55 2 ∂x u3 = . (4) C55 2 ∂z u1 + C55 2 ∂x u3 C2 ∂x u1 11 + C33 ∂z u3 −∇φ3 + ∇⊥ ψ3 Taking divergence operator to (4) leads to −∇2 φj = −ρω 2 uj (j = 1, 3) which enable us to determine φ1 and φ3 . Neglecting ∇⊥ ψj (j = 1, 3) in the right side of (4), ∗ ∗ ∗ we can get principal components C11 , C33 , C55 which satisfy ∗ ∗ C11 C11 ∂x u1 + 2 ∂z u3 = −∂x φ1 ∗ C55 ∗ C55 2 ∂z u1 + 2 ∂x u3 = (−∂z φ1 − ∂x φ3 )/2 ∗ C11 ∗ 2 ∂x u1 + C33 ∂z u3 = −∂z φ3 . ∗ ∗ ∗ We can estimate ϕj (j = 1, 3) by substituting principal components C11 , C33 , C55 and taking n+1 n+1 n+1 curl operator on the both side of (4). Finally we can update C11 , C33 , C55 by solving n+1 C11 n+1 C11 ∂x u1 + 2 ∂z u3 = −∂x φ1 − ∂y ψ1 n+1 n+1 C55 C55 2 ∂z u1 + 2 ∂x u3 = (−∂z φ1 + ∂x ψ1 − ∂x φ3 − ∂y ψ3 )/2 n+1 C11 2 ∂x u1 + C33 ∂z u3 = −∂z φ3 + ∂x ψ3 n+1 Figure 1. Displacement data generated from two different places 5 5 5 Figure 2. The results of reconstruction C11 , C33 , and C55 with an iteration method Figure 3. The results of reconstruction C11 , C33 , and C55 without an iteration method Fig 1 illustrates the measured displacement data obtained from two different positions. Fig 2 de- scribes the reconstruction images of 5th iterations using the proposed algorithm. Additionally, by using the symmetry, we can reconstruct them without iteration in the following way. −∂z φ1 + ∂x ψ1 = −∂x φ3 − ∂z ψ3 The symmetry leads us to ∂x φ3 − ∂z φ1 = −∇ · Ψ where Ψ = [ψ1 ψ3 ]. We take gradient then use the operator identity, ∇2 = ∇∇ · −∇ × ∇×, then −∇2 Ψ = ∇(∂x φ3 − ∂z φ1 ) assuming ∇×∇×Ψ ≈ 0. In this way, we can approximately determine curl-free and divergence- free components so that we can reconstruct three unknown components without an iteration method. Fig 3 depicts the results of C11 , C33 , and C55 without an iteration method. REFERENCES 1. Muthupillai R; Lomas, DJ; Rossman, PJ; Greenleaf, JF; Manduca, A; Ehman, RL, ”Magnetic resonance elastography by direct visualization of propagating acoustic strain waves”, Science 269, (5232): 1854-1857, 1995. 2. A.Manduca, T.E.Oliphant, M.A.Dresner, J.L.Mahowald, S.A.Kruse, E.Amromin, J.P.Felmlee, J.F.Greenleaf and R.L.Ehman,”Magnetic resonance elastography: non-invasive mapping of tissue elasticity”, Medical Image Analysis ,Vol 5:237-254, 2001. 3. T.H. Lee, C.Y. Ahn, O.I. Kwon and J.K. Seo , “A hybrid one-step inversion method for shear modulus imaging using time-harmonic vibrations”, Inverse Problems , Vol. 26, 2010. 4. Sebatian Papazoglou, Jurgen Braun, Uwe Hamhaber and Ingolf Sack , “Two-dimensional waveform analysis in MR elastography of skeletal muscles”, Physics In Medicine And Biology , Vol. 50, 2005.
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