Using spatial prior knowledge in the spectral fitting of MRS images

Abstract

We propose a Bayesian smoothness prior in the spectral fitting of MRS images which can be used in addition to commonly employed prior knowledge. By combining a frequency-domain model for the free induction decay with a Gaussian Markov random field prior, a new optimization objective is derived that encourages smooth parameter maps. Using a particular parameterization of the prior, smooth damping, frequency and phase maps can be obtained whilst preserving sharp spatial features in the amplitude map. A Monte Carlo study based on two sets of simulated data demonstrates that the variance of the estimated parameter maps can be reduced considerably, even below the Cramér-Rao lower bound, when using spatial prior knowledge. Long-TE (1)H MRSI at 1.5 T of a patient with a brain tumor shows that the use of the spatial prior resolves the overlapping peaks of choline and creatine when a single voxel method fails to do so. Improved and detailed metabolic maps can be derived from high-spatial-resolution, short-TE (1)H MRSI at 3 T. Finally, the evaluation of four series of long-TE brain MRSI data with various signal-to-noise ratios shows the general benefit of the proposed approach.

A Subspace Approach to Spectral Quantification for MR Spectroscopic Imaging

“…We solve this problem for all the spatial locations (or voxels) jointly, incorporating any spatial priors available (e.g., spatial smoothness constraints). Incorporation of spatial priors about the spectral parameters has been demonstrated to be useful for improving spectral quantification [10, 11] but at the expense of significantly increased computational complexity. The proposed subspace model in Eq.…”

“…To simplify notation, let a ℓ, q = [ a ℓ, q ( x 1 ), a ℓ, q ( x 2 ), …, a ℓ, q ( x P )] T denote the linear coefficients for a particular basis and $\mathrm{bold-italica}={false[{\mathit{a}}_{1,1}^T,{\mathit{a}}_{1,2}^T,\dots ,{\mathit{a}}_{1,Q_1}^T,\dots ,{\mathit{a}}_{L,1}^T,{\mathit{a}}_{L,2}^T,\dots ,{\mathit{a}}_{L,Q_L}^{T}false]}^T$ denote the collection of all the coefficients. Following [10, 11], we formulate the problem as a regularization problem: $${\mathit{a}}^{\ast }=\text{arg }\underset{\mathit{a}}{\text{min}}\sum _{p,n}{true\Vert dfalse({\mathit{x}}_p,t_{n}false)-\sum _{\ell ,q}{a}_{\ell ,q}false({\mathit{x}}_{p}false)b_{\ell ,q}false(t_{n}false)true\Vert }_2^2+Rfalse(\mathrm{bold-italica}false),$$where R (·) represents a regularization functional imposing any desired spatial constraints. Two types of regularizations have been used for spectral quantification [10, 11]: a) weighted- L 2 regularization, and b) total variation regularization, both of which aim at imposing edge-preserving spatial smoothness on the linear coefficients.…”

“…Following [10, 11], we formulate the problem as a regularization problem: $${\mathit{a}}^{\ast }=\text{arg }\underset{\mathit{a}}{\text{min}}\sum _{p,n}{true\Vert dfalse({\mathit{x}}_p,t_{n}false)-\sum _{\ell ,q}{a}_{\ell ,q}false({\mathit{x}}_{p}false)b_{\ell ,q}false(t_{n}false)true\Vert }_2^2+Rfalse(\mathrm{bold-italica}false),$$where R (·) represents a regularization functional imposing any desired spatial constraints. Two types of regularizations have been used for spectral quantification [10, 11]: a) weighted- L 2 regularization, and b) total variation regularization, both of which aim at imposing edge-preserving spatial smoothness on the linear coefficients. For weighted- L 2 regularization, R (·) can be expressed as $$Rfalse(\mathrm{bold-italica}false)=\sum _{\ell =1}^L{\lambda }_{\ell }\sum _{q=1}^{Q_{\ell }}{false\Vert \mathrm{boldW}{\mathit{a}}_{\ell ,q}false\Vert }_2^2,$$where the λ ℓ is a tunable regularization parameter and W is a weighting matrix derived from reference anatomical images [15].…”

Spectral Quantification for High-Resolution MR Spectroscopic Imaging With Spatiospectral Constraints

“…Recent methods incorporate spatial prior knowledge as a regularization term, penalizing the parameter variation within neighboring voxels [18], [19]. The proposed method differs from these methods in the following key aspects.…”

“…Early efforts (e.g., Soher et al [17] and LCModel [11]) make use of parameters estimated from neighboring voxels to obtain a better initial guess or a stronger constraint to mitigate the local minimum problem. Recently, methods incorporating spatial prior knowledge through regularization have also been reported, promoting smoothness of the parameter maps (e.g., AQSES-MRSI [18] and Kelm et al [19]), which have led to significant improvements.…”