Kosuke Ito^{1} and Masahiro Takizawa^{1}

Parallel imaging is widely used in clinical routine practice. However, SNR degradation occurs due to undersampling and higher g-factor in higher acceleration factor. In this study, a new algorithm of parallel imaging reconstruction mitigating noise enhancement for fast spin echo sequence was proposed. The algorithm uses information of phase distribution of unaliased image, aliasing image, and folded image. SNR was compared in vivo T2 weighted image between full sampling, conventional parallel imaging, and proposed method. And higher SNR was demonstrated.

Method

**Reconstruction Algorithm**

i) Calculation of phase distribution

Sampling pattern of proposed algorithm was same as conventional parallel imaging. Central 64 lines of k-space (ACS) was fully sampled to estimate phase distribution. Phase distribution of unaliased image ( $$$\theta^i$$$ ), aliased image ( $$$\phi^i$$$ ), and folded image ( $$$\psi^i$$$ ) were estimated as equation 1 using ACS data. $$\begin{align}\theta^i&=angle(FFT(ACS^i))\\\phi^i&=angle(\frac{FFT(ACS^i_{pi})}{FFT(ACS^i)})\ \ (1)\\\psi^i&=angle(FFT(ACS^i_{pi}))\end{align}$$ Where i is index of receiver channel, FFT is Fourier transformation, $$$ACS^i$$$ is central 64 lines of k-space of channel i, and $$$ACS^i_{pi}$$$ is under sampled data of $$$ACS^i$$$.

ii) Unfolding algorithm

Figure 1 shows relation of unaliased image, aliasing image and folded image in a pixel. To calculate unaliased image of channel i, data of channel j ($$$i \neq j$$$) was used. By using phase of each image and absolute value of folded image, unaliased image was decomposed as equation 2. $$S_i=\begin{vmatrix}R\end{vmatrix}\begin{vmatrix}\frac{sin(\psi^{i+j}-\phi^{i+j})sin(\theta^{i+j}-\theta^j)}{sin(\theta^j-\phi^{i+j})sin(\theta^i-\theta^j)}\end{vmatrix}e^{i\theta^i}\ \ (2)$$ Where $$$R=R_i+R_je^{i\alpha_j}$$$ is sum of folded image of channel i and j, $$$R_{i,j}$$$ is folded image of channel i and j, $$$\theta^{i+j}, \phi^{i+j}, \psi^{i+j}$$$ are phase of sum of channel i and j of unaliased, aliasing, and folded image, and $$$\alpha_j$$$ is determined to minimize enhancement of noise as follows. In this study, noise of $$$R_i$$$ and $$$R_j$$$ were considered. By using propagation law of error, noise of decomposed unaliased image was estimated as equation 3, and $$$\alpha_j$$$ was determined as minimizing equation 3. $$(\delta S_i)^2=(\frac{\partial S_i}{\partial R_i})^2(\delta R_i)^2+(\frac{\partial S_i}{\partial R_j})^2(\delta R_j)^2=\frac{2sin^2(\theta^{i+j}-\theta^j)}{sin^2(\theta^{i+j}-\phi^{i+j})sin^2(\theta^i-\theta^j)}(\delta R)^2\ \ (3)$$ Where $$$\delta R_{i,j}$$$ are noise of $$$R_{i,j}$$$ . For simplicity, $$$\delta R_i=\delta R_j=\delta R$$$ was assumed. $$$\theta^{i+j},\ \phi^{i+j},$$$ and $$$\theta^j$$$ depend on $$$\alpha_j$$$ . This calculation was done for each pixel and each channel, and unaliased image of each channel was calculated.

iii) Combining multi channel data

In the region of edge between brain and air, phase estimation was not accurate and unfolding procedure described above failed, so conventional parallel imaging was applied in such region. In other regions, sensitivity map was used as equation 4. $$ I(x,y) = \frac{\sum_{i=1}^{Ch} c_i^*(x,y) S_i(x,y)}{\sum_{i=1}^{Ch} \begin{vmatrix}c_i(x,y)\end{vmatrix}^2}\ \ (4)$$ Where I(x,y) is combined signal, Ch is number of receiver channels, and $$$c_i(x,y)$$$ is coil sensitivity of channel i.

**Experiment**

Experiment was conducted on 3 Tesla whole body MRI system (Hitachi, Ltd.). A healthy volunteer was imaged. This study was approved by the ethics committee of Hitachi group headquarters. 15 channel receiving coil was used. Scan parameter of T2 weighted image was follows; TR / TE = 4500 msec / 90 msec, Freq# x Phase# = 256 x 256, slice# = 22, echo train length = 14, scan time = 1 min 25 sec. Full sampled image was acquired and under sampling was applied retrospectively. Acceleration factor was 3.

**Evaluation**

SNR of white matter was compared between full sampled image, conventional parallel imaging, and proposed method.

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5. Martin Blaimer, et al. Comparison of Phase-Constrained Parallel MRI Approaches: Analogies and Differences. MRM 75: 1086-1099 (2016)

Figure 1. Relation of unaliased image, aliasing
image, and folded image. In (), phase estimated from ACS data is showen. Unaliased
image can be decomposed from sum of fold image of channel i and j using phase
information.

Figure 2. Phase distribution of full sample (a), ACS
data (b) of a channel and line profile (c) .

Figure
3. Image of a channel with full sample (a) and reconstructed using proposed
method (b). (c) is profile on red and purple line. (d) is noise enhancement
factor calculated using equation 3.

Figure 4. FSE image of (a): full sample,
(b): conventional PI, and (c): proposed method
(d): difference of (a) and (b), (e):
difference of (a) and (c)
Gray
scale of (d) and (e) is 0.1 times that of (a) to (c).

Table 1. SNR of white matter