The water sideband artifact is a major obstacle to proton MR spectroscopic imaging (1H-MRSI) without water suppression (WS). This work presents a novel method to remove the sideband artifacts from non-WS MRSI data, characterized by the use of a reference-based parametric model to represent the artifacts. Our method obtains the reference signal from two auxiliary scans and subsequently estimates the sideband signals from a particular MRSI data. The proposed method has been validated using both phantom and in vivo experimental data, demonstrating that it can effectively remove the sideband artifacts without introducing spectral distortion. This method is expected to be useful for many non-WS MRSI studies.
Signal model for sideband artifacts
The sideband artifacts at spatial location $$$\boldsymbol{x}$$$ can be modeled as the sum of the frequency modulations of the associated unsuppressed water signal1. More specifically, we express the sideband signal with $$$N$$$ prominent components as:
$$\hspace{13.5em}S_{side}(\boldsymbol{x},t)=\sum_{n=1}^{N}\tilde{S}_{n}(\boldsymbol{x},t){e^{-\beta_{n}(\boldsymbol{x})t}}S_{w}(\boldsymbol{x},t),\hspace{13.5em}(1)$$
where $$$\{\tilde{S}_n\}$$$ are the carrier functions, $$$\{\beta_n\}$$$ the damping factors and $$$S_w$$$ is the unsuppressed water signal. One possible form of $$$\tilde{S}_n$$$ is an FIR filter with unknown frequency locations. However, this representation would require the estimation of the sideband signals directly from the measured MRSI data which may contain significant spectral distortions because of the overlap between metabolites and sideband signals. To avoid this, we propose a reference-based signal model for sideband signals which represents the carrier functions as the convolution of a reference signal $$$S_{ref,n}(t)$$$ and FIR filters with fixed frequencies:
$$\hspace{13.5em}\tilde{S}_n(\boldsymbol{x},t)=S_{ref,n}(t)\sum_{p=-P}^{P}\tilde{c}_{p,n}(\boldsymbol{x}){e^{i2{\pi}pΔft}},\hspace{13.5em}(2)$$
where $$$Δf$$$ is the frequency resolution and $$$\{\tilde{c}_{p,n}\}$$$ are spatially dependent coefficients. $$$S_{ref,n}(t)$$$ can be estimated from auxiliary scans and is chosen to have to the following form:
$$\hspace{12.65em}S_{ref,n}(t)=({e^{i2{\pi}f_{n}t}}-{e^{-i2{\pi}f_{n}t}})\sum_{l=-L}^{L}c_{l,n}{e^{i2{\pi}lΔft}}.\hspace{12.65em}(3)$$
This model is motivated by the fact that the prominent sideband peaks are symmetrically located on both sides of the main water peak with opposite phases.
Algorithm
The proposed model enables removal of the sideband artifacts in two steps: 1) estimation of $$$S_{ref,n}(t)$$$ from auxiliary scans, and 2) estimation of $$$S_{side}(\boldsymbol{x},t)$$$ for a particular MRSI dataset.
In this work, we estimate the reference signal from two navigator signals $$$S_{nav,1}(t)$$$ and $$$S_{nav,2}(t)$$$ acquired at the k-space origin with and without WS respectively. After removal of the residual water signals (e.g., using HSVD2), their difference signal $$$S_{diff}(t)$$$ should contain negligible metabolites but keep the sideband artifacts. We then solve the following optimization problem:
$$\hspace{7.75em}\min_{f_n,c_{l,n},\beta_n}||S_{diff}(t)-\sum_{n=1}^N\{({e^{i2{\pi}f_nt}}-{e^{-i2{\pi}f_nt}})\sum_{l=-L}^{L}c_{l,n}{e^{i2{\pi}lΔft}}\}{e^{-\beta_n(\boldsymbol{x})t}}S_w(t)||_2^2.\hspace{7.75em}(4)$$
The optimal solution to the above problem will be used to synthesize the reference signal based on Eq. (3).
Even with the reference signal $$$S_{ref,n}(t)$$$, estimation of the sideband artifacts from the original MRSI data is still possible to include some metabolites due to determination of the FIR filter in Eq. (2) (even though $$$P$$$ is usually small). To overcome this, inspired by QUEST3, we fit the metabolite signals after truncating an appropriate number of initial points and then subtract the back-extrapolation of these estimated metabolites from the original data to obtain the metabolite reduced signal $$$\hat{S}(\boldsymbol{x},t)$$$. This strategy is appropriate since the sideband signals usually have much higher decay rates than common metabolites1. The final estimates of the unknown parameters in Eq. (1) and Eq. (2) are obtained by solving the following optimization problem:
$$\hspace{9em}\min_{\tilde{c}_{p,n},\beta_{n}}||\hat{S}(\boldsymbol{x},t)-\sum_{n=1}^{N}S_{ref,n}(t)\sum_{p=-P}^{P}\tilde{c}_{p,n}(\boldsymbol{x}){e^{i2{\pi}pΔft}}{e^{-\beta_n(\boldsymbol{x})t}}S_w(\boldsymbol{x},t)||_2^2.\hspace{9em}(5)$$
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