Synopsis

This work presents a novel spectral model for spectral quantification, which represents each spectral component using a subspace instead of a parametric basis function with unknown parameters. The proposed model enables efficient and effective incorporation of both spectral and spatial prior information to improve the quantification performance. The proposed method is validated using both simulation and experimental data, demonstrating superior performance to existing methods using parametric spectral bases. This method is expected to be useful for processing noisy MRSI data.

Introduction

Method

Subspace spectral model:

The current spectral models express the measured data from $$$L$$$ compounds (or spectral components) as

\begin{equation}s(t)=\sum^L_{l=1}C_l\phi_l(\vec{\beta},t) \tag{1}\end{equation}

where $$$\phi_l(\vec{\beta},t)$$$ is a spectral basis function generated using quantum-simulation for a particular compound and $$$\vec{\beta}$$$ are spectral parameters used to accommodate spectral variations under practical experimental conditions (e.g., lineshape, etc.). Determining the $$$\vec{\beta}$$$ often entails solving a nonlinear optimization problem. Here we propose a novel subspace model to represent the entire set of functions that are now represented by a single parametric function $$$\phi_l(\vec{\beta},t)$$$ with unknown $$$\vec{\beta}$$$. More specifically, let $$$\vec{\beta}$$$ have prior distribution $$$p(\vec{\beta})$$$ and $$$\{\phi_l(\vec{\beta_m},t)\}^M_{m=1}$$$ be the set of functions drawn according to $$$p(\vec{\beta})$$$. In practice, $$$\{\phi_l(\vec{\beta_m},t)\}^M_{m=1}$$$ often occupy a low-dimensional subspace. In other words, $$$\phi_l(\vec{\beta},t)$$$ can be expressed as

$$\phi_l(\vec{\beta},t)=\sum^{Q_l}_{q=1}a_{l,q} b_{l,q}(t) \tag{2}$$

where the basis functions $$$\{b_{l,q}(t)\}^{Q_l}_{q=1}$$$ can be determined from the eigenvectors of the following Casorati matrix,

$$ \begin{bmatrix} \phi_l(\beta_1,t_1) & \phi_l(\beta_1,t_2) & \phi_l(\beta_1,t_3) & \dots & \phi_l(\beta_1,t_n) \\ \phi_l(\beta_2,t_1) & \phi_l(\beta_2,t_2) & \phi_l(\beta_2,t_3) & \dots & \phi_l(\beta_2,t_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \phi_l(\beta_M,t_1) & \phi_l(\beta_M,t_2) & \phi_l(\beta_M,t_3) & \dots & \phi_l(\beta_M,t_n) \end{bmatrix} $$

With (2), we will obtain a subspace model for $$$s(t)$$$ or for $$$s(\vec{x},t)$$$ (in the case of spectroscopic imaging) as:

$$ s(t)=\sum^L_{l=1}\sum^{Q_l}_{q=1}a_{l,q}b_{l,q}(t) \quad or \quad s(\vec{x},t)=\sum^L_{l=1}\sum^{Q_l}_{q=1}a_{l,q}(\vec{x})\cdot b_{l,q}(t) \tag{3}$$

where $$$a_{l,q}$$$ absorbs the $$$C_l$$$ in (1).

Spectral quantification with spatiospectral constraints:

The linear coefficients $$$\{a_{l,q}(\vec{x})\}$$$ are obtained by solving the following optimization problem:

$$ \{a_{l,q}\} = arg \underset{\{a_{l,q}\}}{min} ||s(\vec{x},t)-\sum_l \sum_q a_{l,q}(\vec{x})\cdot b_{l,q}(t)||_2^2+R(\{a_{l,q}(\vec{x})\}) \tag{4}$$

where $$$s(\vec{x},t)$$$ is the acquired MRSI data, $$$\{ a_{l,q}(\vec{x})\}$$$ are the desired coefficients containing concentration information, $$$\{ b_{l,q}(t)\}$$$ are the predetermined basis functions that span the subspace for the $$$l^{th}$$$ spectral component and $$$R(\cdot )$$$ is a regularization function. Here, in this work we use a weighted-L2 regularization for $$$R(\cdot )$$$.

Results

The proposed method has been evaluated using both simulation and experimental data. The simulation phantom is composed of six common metabolite signals: NAA, Cre, Cho, mIn, Glu and Gln. Their concentrations are set to be smooth within tissues of the same type but vary across them. Strong white Gaussian noise was also added to the data. Figure 1 shows the comparison of the proposed method to one of the state-of-the-art methods called QUEST [5]. Figure 2 shows the spectra from three locations comparing the quantification accuracy among different methods. As the results show, the proposed method performed better than QUEST. In-vivo data acquired from a healthy subject on a 3T scanner using an echo-planar spectroscopic imaging (EPSI) sequence were also used for evaluating the proposed method. Figure 3 shows that the concentration maps obtained using QUEST have much higher spatial variations than those obtained using the proposed method.

Conclusions

Acknowledgements

References

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[6] Ning, Q., Ma, C., Lam, F., & Liang, Z. P. (2016). Spectral quantification for high-resolution MR spectroscopic imaging with spatiospectral constraints. IEEE Transactions on Biomedical Engineering, In Press.