Magnetic resonance (MR) susceptibility weighted venography (SWV) is an imaging technique which provides high-contrast vein structures, with little radiation exposure to patients, unlike other imaging techniques such as X-rays and computed tomography (CT). In the SWV technique, magnitude and phase data decomposed from complex MR raw data are combined with each other to improve the contrast of visible veins. However, a low-SNR problem distinct in the phase data hinders visualizing low-contrast and fine vein structures from SWV images and has limited the multiplication number of a phase mask to less than four. [1] Efforts to improve SNR of phase data were previously made such as by using a linear phase model in temporal domain for multi-echo MR images. [2] However, it is extremely difficult to unwrap the phase at long echo times (TEs) correctly before fitting with linear phase model due to high noise which alters phase up to 180° on some cases. The inaccurate phase unwrapping results in a serious problem making many scattered points just like noise in the phase masks and corresponding SWV images. In this study, we propose a complex signal modeling method for multi-echo MR images which does not need phase unwrapping. This method can significantly improve SNR of the phase data without any other artifacts or noise-like scattered points. High-SNR and multi-contrast SWV images were successfully produced with high visibility of detailed vein structures, some of which were not visible in the conventional single-echo SWV images.

For in vivo experiments, a normal brain was scanned by a multi-echo gradient-recalled-echo (MGRE) sequence using a 3T SIEMENS MRI system (Erlangen, Germany). The parameters were the first echo time (TE₁)=5.67ms, echo spacing (ES)=5.51ms, repetition time (TR)=95ms, flip angle (FA)=27°, slice thickness=1.6mm, bandwidth=444 Hz/Px, field of view (FOV)=215×215×51.2mm³, the number of echoes=16, the number of slices=32, matrix size=1024×1024×32×16 and in-plane resolution=0.21×0.21mm². The modeled phase $$$\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)$$$ used for fitting in the previous study [2] is represented as:$$\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)=\omega\left(\overrightarrow{r}\right){TE}_{i}+{\varphi}_{0}\left(\overrightarrow{r}\right)$$

where $$$\overrightarrow{r}$$$ is a position of a pixel, $$${TE}_{i}$$$ is the ith TE, $$$\omega\left(\overrightarrow{r}\right)$$$ is the rate of phase change with time of pixel $$$\overrightarrow{r}$$$ and $$${\varphi}_{0}\left(\overrightarrow{r}\right)$$$ is the phase at TE=0 ms of pixel $$$\overrightarrow{r}$$$. Then the following minimization equation is solved using a least squares (LS) algorithm to denoise the measured phase signal.$$\min_{\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)}{\sum_{i=1}^{N}{{\left\{{\varphi}_{u,m}\left(\overrightarrow{r},{TE}_{i}\right)-\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right){e}^{i\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)}\right\}}^{2}}}\\\xrightarrow{\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)=\omega\left(\overrightarrow{r}\right){TE}_{i}+{\varphi}_{0}\left(\overrightarrow{r}\right)}\min_{\omega\left(\overrightarrow{r}\right),{\varphi}_{0}\left(\overrightarrow{r}\right)}{\sum_{i=1}^{N}{{\left\{{\varphi}_{u,m}\left(\overrightarrow{r},{TE}_{i}\right)-\left(\omega\left(\overrightarrow{r}\right){TE}_{i}+{\varphi}_{0}\left(\overrightarrow{r}\right)\right)\right\}}^{2}}}$$

where N is the number of echoes, $$${\varphi}_{u,m}\left(\overrightarrow{r},{TE}_{i}\right)$$$ is the unwrapped phase of the measured phase of pixel $$$\overrightarrow{r}$$$ at $$${TE}_{i}$$$. However, $$${\varphi}_{u,m}\left(\overrightarrow{r},{TE}_{i}\right)$$$ has to be unwrapped correctly. Otherwise, large errors caused in $$$\omega\left(\overrightarrow{r}\right)$$$ and $$${\varphi}_{0}\left(\overrightarrow{r}\right)$$$ due to the inaccurate phase unwrapping introduce scattered artifacts over phase masks and SWV images. To eliminate the phase unwrapping procedure, the proposed complex signal modeling was used, whose corresponding minimization formula is as follows:$$\min_{\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)}{\sum_{i=1}^{N}{{\left\{{S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)-{\left|{S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)\right|}_{d}{e}^{i\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)}\right\}}^{2}}}\\\xrightarrow{\hat{\varphi}\left(\overrightarrow{r},{TE}_{i}\right)=\omega\left(\overrightarrow{r}\right){TE}_{i}+{\varphi}_{0}\left(\overrightarrow{r}\right)}\min_{\omega\left(\overrightarrow{r}\right),{\varphi}_{0}\left(\overrightarrow{r}\right)}{\sum_{i=1}^{N}{{\left\{{S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)-{\left|{S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)\right|}_{d}{e}^{i\left(\omega\left(\overrightarrow{r}\right){TE}_{i}+{\varphi}_{0}\left(\overrightarrow{r}\right)\right)}\right\}}^{2}}}$$

where $$${S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)$$$ is the measured complex value of $$$\overrightarrow{r}$$$ at $$${TE}_{i}$$$ and $$${\left|{S}_{m}\left(\overrightarrow{r},{TE}_{i}\right)\right|}_{d}$$$ is the denoised magnitude value of $$$\overrightarrow{r}$$$ at $$${TE}_{i}$$$ . The used denoising method for magnitude data was the model-based denoising method using multiple T₂(*) compartments intrinsic in a temporal decay signal of each pixel. [3] The final high-quality phase masks were generated from the denoised phase images by the proposed complex signal modeling. 128×128 hamming filter and threshold value of 0.2π were used for the generation of phase mask. Finally, high-SNR multiple SWV images were obtained by merging the denoised phase masks and magnitude images at all TEs.

Fig. 1 shows the phase mask⁴ images and SWV images obtained with the conventional single-echo GRE (a-e), the MGRE with the linear phase model (f-m) and the MGRE with the proposed complex signal model (n-u). The magnitude images for the both cases of MGRE were denoised using the model-based denosing method. Scattered noise-like artifacts largely occurred in the red-circled areas in (h). The degree of these scattered artifacts increased as M and TE increased. On the other hand, the denoised images with the proposed method do not such artifacts, and provide clear vein structures with high SNR even when M is 16 or TE is long(49.8ms)(p,s,t,u). The final denoised SWV images with the proposed method (r,s,t,u) successfully visualized more vein structures with high visibility than the SWV images with the conventional single-echo GRE (d,e).(The total scanning time was almost the same for both GRE and MGRE acquisitions)The multi-echo SWV with the proposed complex signal modeling method can provide high-SNR and multi-contrast phase masks and SWV images. The multiplication number of the phase mask for SWV was increased up to 16 without image degradation even at the long TE of 49.8ms. More detailed vein structures were visualized with higher- and multiple contrasts than the conventional single-echo GRE SWV.