Phase Modulated Array Coil Image Reconstruction: A means to correct B1+/- signal modulations for sodium Tx/Rx Array Coil at 3T

Yongxian Qian^{1}, Tiejun Zhao^{2}, Steven Baete^{1}, Karthik Lakshmanan^{1}, Graham Wiggins^{1}, and Fernando E Boada^{1}

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

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.

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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