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.

Individual channel images, m_i(r), i=1, 2, …, N, of an object ρ(r) are squared and summed in Eq. [1] to form an SOS image, m_sos(r), which is modulated by unknown coil sensitivities, C_i(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 θ_n corresponding to voltages v_n, 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, w_i, 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] $$ \rho(r)=m_{SOS}(r)/\sin(vb_1^+(r))/\sqrt{\sum_{i=1}^N|C_{i}(r)|^2}$$

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.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.