Phase Modulated Array Coil Image Reconstruction:  A means to correct B1+/- signal modulations for sodium Tx/Rx Array Coil at 3T
Yongxian Qian1, Tiejun Zhao2, Steven Baete1, Karthik Lakshmanan1, Graham Wiggins1, and Fernando E Boada1

1Radiology, New York University, New York, NY, United States, 2Siemens Medical Solutions USA, New York, NY, United States

### Synopsis

A transmit/receive (Tx/Rx) array coil has high efficiency in transmission and thus reduces specific absorption rate (SAR), which is highly desired in clinical applications for sodium (23Na) MR imaging. However, a Tx/Rx array coil at 3T usually produces a spatially-varying excitation (B1+) field and does not provide a uniform sum-of-squares of coil sensitivities (B1- fields) for quantitative sodium MRI. Here we present a solution to this problem and demonstrate its effectiveness with studies on phantoms and subjects with neurological disorders such as multiple sclerosis (MS), epilepsy, and mild traumatic brain injury (mTBI).

### INTRODUCTION

A transmit/receive (Tx/Rx) array coil uses the same individual elements for both excitation and reception to increase transmit efficiency and decrease specific absorption rate (SAR) 1. This is highly desired in clinical applications for sodium (23Na) MR imaging that requires high power input for a very short (~0.5ms) RF pulse excitation to reach a flip angle of 90°. A Tx/Rx array sodium coil at 3T typically splits transmitted power across individual coil elements to generate a nearly uniform excitation (B1+) field, and then uses different independent channels to receive the MR signal. Images from individual channels are usually combined via sum-of-squares (SOS) to form a final image that is modulated in intensity by the coil sensitivities (B1- fields) and flip angle (B1+ field). For quantitative sodium imaging, spatially-varying modulations of the B1+/- fields have to be corrected. This is challenging when using Tx/Rx array coils because they have no uniform mode available for estimation of the coil element sensitivities and the sum-of-squares of these element sensitivities is usually not uniform. Here we present a solution to this problem and demonstrate its effectiveness with studies on phantoms and human subjects.

### METHODS

Individual channel images, mi(r), i=1, 2, …, N, of an object ρ(r) are squared and summed in Eq. [1] to form an SOS image, msos(r), which is modulated by unknown coil sensitivities, Ci(r), and B1+ field at flip angle θ.

Eq. [1a] $$m_{sos}(r)\equiv\sqrt{\sum_{i=1}^N|m_{i}(r)|^2}=|\rho(r)\sin(\theta(r))|\sqrt{\sum_{i=1}^N|C_{i}(r)|^2}$$

Eq. [1b] $$m_{i}(r)=\rho(r)C_{i}(r)\sin(\theta(r))$$

Eq. [1c] $$\theta(r)=\gamma\tau B_1^+(r)=vb_1^+(r)$$

Eq. [1d] $$b_1^+(r)=\sum_{i=1}^Nb_{1,i}^+(r)$$

To find the combined B1+ field, $b_1^+(r)$, a separate low-resolution scan (~1.5min) is performed at a number of flip angles θn corresponding to voltages vn, n=1,2,…,M. The SOS of the low-resolution images are then used to fit $\sin(vb_1^+(r))$ on a pixel-by-pixel basis in a region of interest (ROI) via a nonlinear curve fitting of y = p1 + p2 * sin(p3*x).

To correct for the effects of coil element sensitivities (B1- fields), we first calculate the sum-of-squares and a weighted complex sum (WCS) for the low-resolution (LR) individual images as described in Eq. [2].

Eq. [2a] $$\sqrt{\sum_{i=1}^N|C_{i}(r)|^2}=b_1^+(r)m_{LR,SOS}(r)/m_{LR,WCS}(r)$$

Eq. [2b] $$m_{LR,SOS}(r)\equiv\sqrt{\sum_{i=1}^N|m_{LR,i}(r)|^2}=|\rho_{LR}(r)\sin(\theta(r))|\sqrt{\sum_{i=1}^N|C_{i}(r)|^2}$$

Eq. [2c]$$m_{LR,WCS}(r)\equiv|\sum_{i=1}^Nw_{i}m_{LR,i}(r)|=|\rho_{LR}(r)\sin(\theta(r))||\sum_{i=1}^Nw_{i}C_{i}(r)|$$

Eq. [2d] $$w_{i}\equiv\exp(-j\triangle\phi_{i})$$

Eq. [2e] $$\sum_{i=1}^Nw_{i}C_{i}(r)=b_1^+(r)$$

The weights, wi, are phase corrections for the receive channels to establish the relation Eq. [2e]. The principle of reciprocity between the B1+ and B1- fields for a coil element is then applied to Eq. [2e] because sodium has a low Larmor frequency at 3T 2. The phase difference, $\triangle\phi_i$ between a receive channel and a reference channel (any one of the receive channels) is then measured at the isocenter of the low-resolution images.

The signal modulation is then removed using Eq. [3] below,

Eq. [3] $$\rho(r)=m_{SOS}(r)/\sin(vb_1^+(r))/\sqrt{\sum_{i=1}^N|C_{i}(r)|^2}$$

### Experiments

Sodium MRI scans were performed on phantoms and patients on a clinical 3T scanner of multi-nuclear option (Prisma, Siemens), with a custom-built 8-channel dual-tuned (1H-23Na) Tx/Rx head coil 1,3. Images from a uniform phantom (2 littler bottle, 140mM NaCl) and five patients (ages 13-48 years, 3 female) with neurological disorders (2 multiple sclerosis, 2 epilepsy, and 1 mild TBI) were studied using the twisted projection imaging (TPI) sequence (a research prototype) 4, with a 10min scan for regular resolution (FOV=220mm, matrix size=64, 3D isotropic, RF duration=0.5ms, TE/TR=0.3/100ms, flip angle=90°, rings=28, p=0.4, averages=4) and a 1.5min scan for low resolution at 6 nominal flip angles in the range 18.7-112.4° corresponding to 6 voltages in the range 46.8-281V, respectively. The images were reconstructed offline with custom-developed programs in C++. The nonlinear curve fitting for B1+ field and overall B1 correction scheme were implemented in MATLAB (R2015b, The MathWorks) using a Levenberg-Marquardt algorithm.

### Results and Discussion

Figure 1 shows images and profiles from the phantom study. The profiles through the center of phantom from left to right detail the improvement in uniformity of image intensity before and after corrections for B1+/- modulation. The standard deviation along the profile was reduced by 72.5%, from 16.0 to 4.4 % after the correction. Figure 2 shows results from a patient study (the mTBI case). The b1+ map shows that the combined transmit field was fairly uniform while the SOS of coil sensitivities was not. After the corrections for B1+/- fields, intensity of the sos image became uniform on both sides of the brain as expected. These phantom and patient results demonstrated that proposed methodology can effectively remove the image intensity modulation introduced by the Tx/Rx array coil.

### Acknowledgements

This work was financially supported in part by NIH grants R01 MH088370, R01 CA111996 and R01NS082436.

### References

1. Lakshmanan K, et al. ISMRM 2014; p.4879.

2. Hoult DI. Concepts Magn Reson 2000; 12:173-187.

3. Wiggins GC, et al. NMR Biomed 2015; Sep 24 Epub.

4. Boada FE, et al. MRM 1997; 37:706-715.

### Figures

Fig. 1. Phantom sum-of-squares (sos) images before (a) and after (b) corrections for B1+ field (c) and coil sensitivity (d). Profiles (e-h) through the center of phantom from left to right show the details of improved homogeneity in the central region (e, f) and show the inhomogeneity of estimated B1+ field (g) and coil sensitivity (h).

Fig. 2. Patient (in the mild TBI case) images before (a) and after (b) corrections for B1+ field (c) and coil sensitivity (d). Profiles (e-h) through the middle of the brain from left to right show the improvement of homogeneity in the central region (e, f) and show the inhomogeneity of estimated B1+ field (g) and coil sensitivity (h).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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