EE 364B Project Final Report: SCP Solver for a Nonconvex Quantitative Susceptibility Mapping Formulation

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1 EE 364B Project Final Report: SCP Solver for a Nonconvex Quantitative Susceptibility Mapping Formulation Grant Yang Thanchanok Teeraratkul June 4, Background Changes in iron concentration in brain tissue are of interest in the study of neurodegenerative diseases such as Alzheimer s and multiple sclerosis [MPY + 13]. Magnetic resonance imaging (MRI) is inherently sensitive to changes in tissue magnetic susceptibility, which can be used as a direct measure of iron concentration [Sch96]. Changes in tissue magnetic susceptibility result in phase shifts in the MR image. Calculating quantitative susceptibility maps (QSM) from the MR image gives researchers a noninvasive method for studying iron concentrations in the brain. 2 Problem Formulation The relationship between the phase of the image and the tissue magnetic susceptibility can be described as a 3D convolution with a unit dipole function [SdSM03]. The convolution relationship between tissue magnetic susceptibility and the phase of the MR signal can be expressed as a linear system, Ax = b, where A R n n is a known Toeplitz matrix representing the convolution kernel, x R n is a vector of magnetic susceptibilities and b R n is a vector of phase angles. The matrix, A, can be decomposed into F H DF, where F C n n is the discrete Fourier transform matrix, and D R n n is a diagonal matrix αkz containing the Fourier transform of the unit dipole function, 2 (kx 2+k2 y +k2 z ). Here, k denotes the spatial frequency coordinates, and α is a constant that accounts for the imaging parameters. Since A contains low singular values, direct inversion of A results in amplification of noise and artifacts in the reconstructed image. Therefore, a regularization scheme is needed to calculate the susceptibility maps. The simplest form of regularization is to threshold the singular values of A before inversion [WSB10]. However this approach does a poor job of suppressing artifacts. More sophisticated formulations have been developed to take advantage of prior knowledge of the location of tissue boundaries. The most popular of these formulations 1

2 employs a l 2 norm penalty on the gradient of the image, minimize M 0 (Ax b) λ 1 [W x G x W y G y W zgz] T x λ 2 M 1 x 2 2, where M 0, M 1 R n n are diagonal weighting matrices, W x, W y, W z R n n are diagonal weighting matrices specifying tissue boundaries, G x, G y, G z R n n are gradient matrices, and λ 1, λ 2 R + adjust the weight of the regularization [LLdR + 12]. However, this formulation has two deficiencies: the l 2 constraint encourages a piecewise constant solution which obscures small tissue structures, and the linear relationship between susceptibility and phase is sensitive to phase unwrapping errors. In order to improve the robustness and detail preservation in the susceptibility maps, we employ a total variation norm-based constraint on a complex valued QSM formulation, minimize M(e iax e ib ) 2 2 subject to (W x G x x) 2 + (W y G y x) 2 + (W z G z x) 2 1 ɛ, where M R n n is a diagonal matrix containing the magnitude of the complex MRI. This formulation should be an improvement over current methods, because the total variation norm improves retention of fine tissue structures, while the complex formulation increases robustness to phase unwrapping errors. Because the objective function is nonconvex and the problem size is very large, we are writing a Sequential Convex Program (SCP) which takes advantage of Toeplitz matrix structure to solve the minimization problem [Mur97]. 3 SCP Algorithm We implemented a Sequential Convex Program (SCP) to solve our problem in MATLAB. Our solver takes advantage of the Toeplitz structure of A, and an affine approximation of the nonconvex problem in order to achieve fast convergence. Denoting the nonconvex objective function as f(x), and the total variation norm-based constraint function as c(x), the SCP algorithm can be written as follows, given x 0 := 0, λ 0 := 1, k := 0. repeat until convergence 1. minimize f(x k ) T p pt p subject to c(x k ) + c(x k ) T p 0 2. Compute step size α by backtracking line search. 3. Update x k+1 := x k + αp, λ k+1 := λ, k := k + 1. return x. The gradient of the objective function and the constraint in each iteration were evaluated 2

3 analytically: f(x) = 2AM 2 (diag(sin(b)) cos(ax) + diag(cos(b)) sin(ax)) c(x) = n i=1 C T i C ix C i x 2, where C i R 3 n contains the i th row of W x G x, W y G y and W z G z. The QP in step 1 was solved directly by maximizing the dual function. This results in the update formulae, ( ) λ = max 0, c(x)t f(x) c(x) c(x) T c(x) p = λ c(x) f(x). By choosing a convex approximation which allows direct calculation of the search direction, we expect to reduce the computation time of the proposed SCP algorithm. 4 Results Quantitative susceptibility maps were calculated from an MR data set acquired on a 7T system of the brain of a healthy volunteer. The dimension of the image was voxels with mm resolution. The phase images were unwrapped using FSL PRELUDE, and the external phase not resulting from changes in tissue susceptibility was removed using projection onto dipole fields [LKdR + 11]. The nonconvex formulation of QSM was solved using the proposed algorithm with zero as the initial guess and ɛ chosen heuristically to be Computations were conducted on the Sherlock computing cluster, which uses an Intel(R) Xeon(R) CPU E GHz with 24 GB of RAM. The proposed SCP algorithm was compared to the current method for solving the nonconvex QSM formulation based on a Taylor expansion of the argument of the norm in the objective function, and a conjugate gradient solver for each iteration [LWL + 13]. Although the conjugate gradient based solver converged in fewer steps, the direct method for calculating each step direction in the proposed SCP was much faster than performing conjugate gradients, resulting in a reduction of the total calculation time from 46 minutes to 4.5 minutes. The proposed SCP algorithm also converged to a lower objective value. This improvement may occur because the smaller step size generated by the proposed SCP results in a more accurate local approximation of the objective function. The convergence of the proposed SCP and the current nonconvex solver can be seen in Figure 1. 3

4 2.5 3 x 105 Convergence of SCP of SCP Nonconvex CG based Algorithm Proposed Implementation Nonconvex CG-based Algorithm Proposed Implementation 2 f (k) (x) time (min) time Figure 1: The SCP implementation requires significantly less computation than the current method based on conjugate gradients. The SCP algorithm generated QSM with the expected contrast and range of susceptibilities based on known tissue properties. For example, the basal ganglia, which are known to have large deposits of paramagnetic ferritin, have high susceptibility, and tissue boundaries between white matter and grey matter are well delineated. 4

5 ppm ppm 0.2 Figure 2: QSM generated by SCP solver on nonconvex formulation The QSM also demonstrated an increase in detail and artifact reduction over conventional l 2 and thresholded inversion methods. As seen in Figure 3, the SCP algorithm result shows few reconstruction artifacts while retaining fine tissue structures. The streaking artifacts emanating from the blood vessels in the coronal view of the brain are significantly reduced compared to direct thresholding, while the image does not display the smoothing effect in the l 2 norm regularized image. TV l 2 TKD TV l 2 TKD Figure 3: The nonconvex formulation with a total variation based constraint shows improved tissue detail retention and artifact suppression compared to the popular l 2 norm and direct inversion methods. 5

6 5 Conclusions Our SCP implementation for solving the nonconvex QSM formulation represents a significant increase in speed over current nonconvex solvers. In addition, the total variation norm-based constraint limits image artifacts while retaining more tissue detail than the current l 2 norm or direct inversion methods. While more work is needed to validate performance on pathological tissue, our SCP shows significant promise as a robust method for studying changes in tissue magnetic susceptibility caused by neurodegenerative diseases. References [LKdR + 11] Tian Liu, Ildar Khalidov, Ludovic de Rochefort, Pascal Spinc le, Jing Liu, AJ Tsiouris, and Yi Wang. A novel background field removal method for mri using projection onto dipole fields (pdf). NMR Biomed., 9: , [LLdR + 12] Jing Liu, Tian Liu, Ludovic de Rochefort, James Ledoux, Ildar Khalidov, Weiwei Chen, AJ Tsiouris, Cynthia Wisnieff, Pascal Spinc le, Martin R. Prince, and Yi Wang. Morphology enabled dipole inversion for quantitative susceptibility mapping using structural consistency between the magnitude image and the susceptibility map. 69: , [LWL + 13] [MPY + 13] [Mur97] [Sch96] [SdSM03] [WSB10] Tian Liu, Cynthia Wisnieff, Min Lous, Weiwei Chen, Pascal Spinc le, and Yi Wang. Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping. Magnetic Resonance in Medicine, 69: , Veela Mehta, Wei Pei, Grant Yang, Suyang Li, Eashwar Swamy, Aaron Boster, Petra Schmalbrock, and David Pitt. Iron is a sensitive biomarker for inflammation in multiple sclerosis lesions. PLOS ONE, 8:1 10, Walter Murray. Sequential quadratic programming methods for large-scale problems. Computational Optimization and Applications, 7: , J. F. Schenck. The role of magnetic susceptibility in magnetic resonance imaging: Mri magnetic compatibility of the first and second kinds. Med. Phys, 26: , Rares Salomir, Baudouin Denis de Senneville, and Chrit TW Moonen. A fast calculation method for magnetic field inhomogeneity due to an arbitrary distribution of bulk susceptibility. ERT Imagerie Moleculaire et Fonctionnelle, 117:26 34, Samuel Wharton, Andreas Schafer, and Richard Bowtell. Susceptibility mapping in the human brain using threhsold-based k-space division. Magnetic Resonance in Medicine, 63: ,