Spectral quantification is a key step in practical applications of MRSI. Many methods have been proposed to solve this problem, which can be roughly classified into three groups: 1) basic frequency integration methods; 2) classical linear prediction methods exploiting the linear predictability of spectral signals;^1,2 3) modern methods incorporating strong spectral constraints in the form of basis functions.^3-5 In this work, we propose a new method that enables the use of both spectral and spatial constraints. In contrast to existing methods that perform quantification point-by-point independently, the proposed method solves the spectral quantification jointly, incorporating spatial prior knowledge to further improve the quantification performance.⁶

The proposed method has two features: 1) the use of prior spectral information in the form of spectral basis functions, and 2) the use of prior spatial information in the form of spatial regularization. More specifically, we express the MRSI signal as $$$d(\mathbf{x},t)=\sum_{n=1}^{N}a_n(\mathbf{x})e^{-\frac{t}{T_2^{*}(\mathbf{x})}}\varphi_n(t)+n(\mathbf{x},t)$$$, where $$$\varphi_n(t)$$$ is the basis function for the $$$n$$$-th metabolite, and noise $$$n(\mathbf{x},t)$$$ follows a white Gaussian distribution, or more compactly, $$\mathbf{d}=\mathbf{K}(\mathbf{T}_2^*) \mathbf{a} + \mathbf{n},$$ where $$$\mathbf{d}$$$, $$$\mathbf{T}_2^*$$$, $$$\mathbf{a}$$$, and $$$\mathbf{n}$$$ are each discretized and flattened into a vector, and operator $$$\mathbf{K}$$$ is linear and dependent on the nonlinear parameters, $$$\mathbf{T}_2^*$$$. The model parameters are then determined by solving the following optimization problem.$$(\hat{\mathbf{a}},\mathbf{\hat{T}}_2^*) = \textrm{arg}\min_{\mathbf{a},\mathbf{T}_2^*}{\|\mathbf{d}-\mathbf{K}(\mathbf{T}_2^*)\mathbf{a}\|_2^2 + \sum_{m=1}^{M}{\sum_{n=1}^{N}{\lambda_{m,n}\|\nabla \Omega_m \mathbf{a}^{(n)}}\|_2^2}+\sum_{n=1}^{N}{\mu_n\|\nabla \mathbf{a}^{(n)}\|_1}},$$ where $$$\lambda_{m,n}>0$$$ and $$$\mu_{n}>0$$$ are regularization parameters, operator $$$\nabla$$$ denotes the first order finite difference, $$$\Omega_m$$$ is the spatial mask for the $$$m$$$-th tissue, and the superscript "(n)" of $$$\mathbf{a}$$$ represents the $$$n$$$-th metabolite. The first regularization term penalizes the regional spatial variation (e.g., within gray matter, white matter, and CSF, respectively), and the second regularization term penalizes the global spatial variation (e.g., to avoid the artificial boundaries created by strict regional constraints). This formulation is motivated by the fact that in biological imaging samples (the human brain, calf muscles, etc.), the spatial distribution of a metabolite is usually similar within each type of tissue, but may also be very different across different tissue types (e.g., CSF is known to have negligible metabolites).

In practice, the spectral basis functions are readily available by either quantum mechanics simulation or in vitro experiments, and the spatial masks for different tissues can be obtained through segmentation of corresponding anatomic images. To solve the joint quantification problem efficiently, we can firstly solve the nonlinear parameters (i.e., $$$\mathbf{T}_2^*$$$) without incorporating spatial constraints using state-of-the-art methods, and then fix all the nonlinear parameters and solve the resulting problem. If nonlinear terms are fixed, the resulting problem is an $$$\ell_1$$$ regularized least-squares problem and can be solved using the alternating direction method for multipliers (ADMM) algorithm.⁷

The proposed method has been evaluated using both simulated and experimental data. The simulated MRSI dataset (Fig. 1a) was generated from a computational phantom which contains six common metabolites, NAA, Cre, Cho, Glu, mIn, and Gln. Metabolite distributions were designed such that they are smooth within each type of tissue, but different across different tissues (Fig. 1b). White Gaussian noise was also added to the dataset (Fig. 1c), with three typical spectra shown in Fig. 1d. In Fig. 2, the proposed quantification method is compared with one state-of-the-art method, QUEST,³ which performs quantification point-by-point independently. As can be seen, QUEST still had large spatial variations, but the proposed method significantly improved the quantification and were much closer to the true distribution. Spectral estimation at some representative points are also evaluated as in Fig. 3. Both methods had residuals close to the noise floor (column 4 and 5), indicating comparable performance in terms of data fitting, but the spectral errors (difference between the true spectra) of QUEST were very large, which was reduced by the proposed method.

Experimental results from a healthy volunteer on a 3T MRI scanner are also shown in Fig. 4-5. The data were acquired using an echo-planar spectroscopic imaging (EPSI) sequence with bipolar acquisition. The estimated concentration maps and fitting results are compared with QUEST, which clearly show the performance improvement of the proposed method over QUEST.

A novel quantification method is proposed to address the spectral quantification problem for MRSI. In the proposed method, both spectral and spatial prior information is incorporated in a joint estimation framework to improve quantitation accuracy, and its effectiveness has been demonstrated via experimental results. The proposed method should prove useful in various quantitative MRSI studies.