Motofumi Fushimi^{1,2}, Thanh Nguyen^{2}, and Yi Wang^{2,3}

^{1}Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan, ^{2}Radiology, Weill Cornell Medical College, New York, NY, United States, ^{3}Biomedical Engineering, Cornell University, Ithaca, NY, United States

We propose a simultaneous conductivity and susceptibility reconstruction method by estimating B1 phase and B0 distributions from a multi-echo gradient echo (mGRE) signal. B1 phase and B0 maps are simultaneously determined by applying nonlinear least squares method on the complex signal equation of the mGRE signal. The poor conditioned inversion of field (B1/B0) to source (conductivity/susceptibility) is regularized using anatomical information. This morphology enabled quantitative conductivity and susceptibility mapping (QCSM) was performed on healthy subjects and patients with brain tumors. Our preliminary in-vivo experiments demonstrated that the proposed QCSM method can reconstruct conductivity and susceptibility from a single mGRE acquisition.

$$c_{j}(\boldsymbol{r})=\mathrm{argmin}_{c_{j}(\boldsymbol{r})}\|\sum_{j = 1}^{N}B_{1,j}^{-}c_{j}(\boldsymbol{r})-1\|_{2}^{2}+\lambda\sum_{j = 1}^{N}|c_{j}(\boldsymbol{r})|^{2},$$

where norm was taken inside the neighboring region of each point . Next, B1 phase as well as B0 field were estimated according to Gaussian noise model in complex mGRE data:

$$\phi,\Delta B_{0}=\mathrm{argmin}_{\phi,\Delta B_{0}}\|S(T_{E})-\rho\exp(-R_{2}^{\ast}T_{E})\exp(\mathrm{i}(\phi-\gamma\Delta B_{0}T_{E}))\|_{2}^{2},$$

where $$$\rho$$$ is proton density, $$$R_{2}^{\ast}$$$ is relaxation rate, $$$\gamma$$$ is gyromagnetic ratio, and $$$T_{E}$$$ is echo time. The summation is over the all echoes. Minimization was conducted by the Levenberg–Marquardt method. To compare B1 phase estimated from mGRE with SE phase, two FSE images with opposite readout gradients were also acquired and averaged to yield the gold standard phase image.

Once coil-combined B1 phase image was obtained, its Laplacian was calculated using Savitzky-Golay filter, which is based on the second order weighted polynomial fitting in each local region around the voxel of interest. Kernel size was 15x15x5 voxels and the weighting factor was determined from magnitude image as follows: $$$w(\boldsymbol{r}) = G(|I(\boldsymbol{r}) - I(\boldsymbol{r}_{0})|)$$$, where $$$G$$$ represents Gaussian function [7]. Finally, conductivity $$$\sigma$$$ was reconstructed as follows:

$$\sigma=\mathrm{argmin}_{\sigma}\|\sigma-\Delta\phi/(2\omega_{0}\mu_{0})\|_{2}^{2}+\lambda\|M(\nabla I)\nabla\sigma\|_{1},$$

where $$$M(\nabla I)$$$ represents the mask that removes boundaries of different anatomical regions and is used in morphology enabled dipole inversion (MEDI) method [9] in QSM. QSM was also reconstructed using nonlinear MEDI with automatic uniform cerebrospinal fluid zero reference (MEDI+0) [10]. This electromagnetic tissue property estimation from mGRE complex signal is referred to as quantitative conductivity and susceptibility mapping (QCSM).

MRI acquisitions were performed on 5 healthy human subjects and brain data were obtained using 2D FSE and 3D mGRE in a 3T clinical scanner (MR750, GE Healthcare, Waukesha, WI). A 32-channel head coil was used as the receiver coil. The imaging parameters for mGRE were as follows: TR: 53.2 ms, first TE: 4.4 ms, echo spacing: 4.9 ms, flip angle: 15 deg; and for FSE : TR: 5350 ms, eff. TE: 87.9 ms, echo train length: 24, flip angle: 111 deg, NEX: 2. FOV was 240x240x144 mm

Figure 2 shows the conductivity maps reconstructed from B1 phase estimated by the nonlinear least square method. We varied the regularization parameter and chose the one that maximizes the curvature of L-curve plot shown in Fig.2 (bottom). When no regularization is adopted (middle), the conductivity map is suffered from severe noise. When the anatomical image is utilized as a regularizer (top), the conductivity results are less noisy and anatomical structures are discernible.

Simultaneously, we obtained B0 field map by solving Eq.2 and reconstructed QSM (Fig.3). This means that electromagnetic tissue properties can be successfully obtained from a single mGRE acquisition. Figure 4 shows the reconstructed conductivity and susceptibility maps along with T1w (3

- Tobias Voigt, Ulrich Katscher, and Olaf Doessel, Magnetic Resonance in Medicine, 2011; 66(2): 456–466.
- Ulrich Katscher, Cornelius A.T. van den Berg, NMR in Biomedicine, 2017; 30(8): 1–15.
- Yi Wang and Tian Liu, Magnetic Resonance in Medicine, 2015; 73(2): 82–101.
- Dong-Hyun Kim, Naraeand Choi, Sung-Min Gho, Jaewookand Shin, and Chunlei Liu, Magnetic Resonance in Medicine, 2014; 71(3): 1144–1150.
- Motofumi Fushimi, Pascal Spincemaille, and Yi Wang, 5th International Workshop on MRI Phase Contrast & Quantitative Susceptibility Mapping, 2019.
- Joonsung Lee, Narae Choi, Jaewook Shin, and Dong-Hyun Kim, Proc. Intl. Soc. Magnetic Resonance in Medicine 21, 2013; p. 4179.
- Joonsung Lee, Jaewook Shin, and Dong-Hyun Kim, Magnetic Resonance in Medicine, 2016; 76(2): 530–539.
- Martin Uecker, Peng Lai, Mark J. Murphy, Patrick Virtue, Michael Elad, John M. Pauly, Shreyas S. Vasanawala, and Michael Lustig, Magnetic Resonance in Medicine, 2014; 71(3): 990–1001.
- Tian Liu, Cynthia Wisnieff, Min Lou, Weiwei Chen, Pascal Spincemaille, and Yi Wang, Magnetic Resonance in Medicine, 2013; 69(2): 467–476.
- Zhe Liu, Pascal Spincemaille, Yihao Yao, Yan Zhang, and Yi Wang, Magnetic Resonance in Medicine, 2018; 79(2): 2795–2803.

Figure 1: Transceive
B1 phase maps estimated from multi-echo GRE data by the linear extrapolation
method (left), by the proposed nonlinear least square method (center), and transceive
B1 phase map directly measured by FSE acquisition (right).

Figure 2: Conductivity maps reconstructed using Eq. 2 with
(top) and without (middle) the morphology-based regularization. The regularization
parameter was chosen by the L-curve method (bottom).

Figure 3: Susceptibility map reconstructed from B0 field
map simultaneously estimated from the multi-echo GRE data.

Figure 4: Conductivity (top) and susceptibility (2^{nd}
row) results for 5 tumor patients. T1w (3^{rd} row) and T2w (bottom)
images are included for references. The structures of lesions are consistent
between conductivity and susceptibility maps, but the conductivity maps have
higher values in all cases.