Maxwell’s equations dictate that the linear spatial encoding gradient fields used in MRI must be accompanied by spatially-varying higher-order fields known as concomitant fields¹. These additional high-order fields introduce spatial-dependent phase accumulation throughout data acquisition^2,3. Conventional whole-body MRI typically employ symmetric gradient design, and its concomitant fields contain only second-order and higher spatial dependent terms¹. For specialized MR systems with asymmetric gradient design⁴, the concomitant fields contain additional zeroth-order/first-order terms⁵. Although the zeroth-order term can be readily corrected by adjusting demodulation frequency⁶, the additional first-order terms can cause spatially-dependent blurring/ghosting in spiral acquisition and echo-shifting in EPI^5,7. We have previously described a gradient pre-emphasis scheme that simultaneously corrects all the first-order concomitant fields for arbitrary gradient waveforms⁷. In that method, the exact, closed-form solution of a cubic equation is solved on a point-by-point basis to obtain the pre-emphasized gradient waveforms, which then prospectively correct all the first-order terms. Initial tests were performed by modifying waveforms on host computer before data acquisition. Although readily solvable on host computer, solving a cubic equation on a point-by-point basis may generate considerable computational load for the gradient board, and thus impairing its potential to be implemented in real-time, e.g., along with the gradient eddy current compensation and logical-to-physical axis rotation. Here, a fast but accurate approximation of our previous solution is developed to provide an effective pre-emphasis scheme and is demonstrated in a real-time implementation.

For an asymmetric gradient system, its concomitant fields are summarized as: $$B_c=\left[G_x\left(z+z_{0x}\right)-G_z\alpha\left(x+x_0\right)\right]^2/\left(2B_0\right)+\left[G_y\left(z+z_{0y}\right)-G_z\left(1-\alpha\right)\left(y+y_0\right)\right]^2/\left(2B_0\right),$$ from which the first-order terms can be extracted as: $$G_x^2zz_{0x}/B_0+G_y^2zz_{0y}/B_0+\left(1-\alpha\right)^2G_z^2yy_0/B_0+\alpha^2G_z^2xx_0/B_0-\alpha G_xG_z\left(xz_{0x}+zx_0\right)/B_0 - \left(1-\alpha \right)G_yG_z\left(yz_{0y}+zy_0\right)/B_0,$$ where $$$\alpha$$$ (dimensionless) describes the relative strength of z gradient-induced concomitant field along the x and y axes, $$$x_0,y_0$$$, denote the offset of z gradient relative to magnet isocenter, $$$z_{0x},z_{0y}$$$, denote the offset of x, y gradient relative to isocenter⁵. The first-order terms can be eliminated by constraining the physical/actual gradient field ($$$G_x,G_y,G_z$$$) to satisfy the following conditions⁷: $$G_z^0=G_z+G_x^2z_{0x}/B_0+G_y^2z_{0y}/B_0-\alpha G_xG_zx_0/B_0-(1-\alpha)G_yG_zy_0/B_0,$$ $$G_x^0=G_x+\alpha^2G_z^2x_0/B_0-\alpha G_xG_zz_{0x}/B_0,$$ $$G_y^0=G_y+(1-\alpha)^2G_z^2y_0/B_0-(1-\alpha)G_yG_zz_{0y}/B_0,$$ where $$$G_x^0,G_y^0,G_z^0$$$ are target gradient waveforms determined by pulse sequence. As shown in previous work⁷, such a system of equations forms a cubic equation whose closed-form solution can be used to obtain the pre-emphasized gradients $$$(G_x,G_y,G_z)$$$. To the first order, the pre-emphasized gradients can be approximated to yield: $$G_z=G_z^0-(G_x^0)^2z_{0x}/B_0-(G_y^0)^2z_{0y}/B_0+\alpha G_x^0G_z^0x_0/B_0+(1-\alpha)G_y^0G_z^0y_0/B_0+o((G^2z_0/B_0)^2),$$ $$G_x=G_x^0-\alpha^2(G_z^0)^2x_0/B_0+\alpha G_x^0G_z^0z_{0x}/B_0+o((G^2z_0/B_0)^2),$$ $$G_y=G_y^0-(1-\alpha)^2(G_z^0)^2y_0/B_0+(1-\alpha)G_y^0G_z^0z_{0y}/B_0+o((G^2z_0/B_0)^2)$$ with approximation error on the order of $$$o\left((G^2z_0/B_0)^2\right)$$$. Hence, the pre-emphasized gradients can be approximated via simple arithmetic operations, which are well-suited to real-time (e.g., 4us update rate) implementation on the gradient driver sub-system with moderate computational power. For a head-only asymmetric gradient system of our interest⁴, $$$z_{0x}=z_{0y}=z_0$$$(typically~12cm), $$$\alpha=0.5$$$, $$$x_0=y_0=0$$$, which yields: $$G_z=G_z^0-(G_x^0)^2z_{0}/B_0-(G_y^0)^2z_0/B_0,$$ $$G_x=G_x^0+G_x^0G_z^0z_0/(2B_0),$$ $$G_y=G_y^0+G_y^0G_z^0z_0/(2B_0).$$ To test this method, 2D spiral acquisitions⁸ were simulated (details in Table 1), and two pre-emphasis update rates (4 and 16us) were used. The imaging plane in axial scan is shifted in S/I direction to simulate strong concomitant field effect. To examine its compatibility with the computational power of a standard gradient driver, the proposed method was implemented on a standard whole-body gradient system (GE 750w) with symmetric gradients. A firmware patch was applied to the gradient driver to allow the pre-emphasis to be easily enabled/disabled, with flexibility to support a variety of gradient coils and geometry parameters. The first-order concomitant term was artificially introduced by modifying the spiral gradient waveforms. Multiple 2D-axial spiral acquisitions were then performed on the ACR phantom using: (a)standard spiral, (b)first-order concomitant field corrupted spiral, and (c)first-order concomitant field corrupted and firmware pre-emphasized spiral.Figure 1 shows the ideal gradient waveforms of a coronal spiral scan(a), the pre-emphasis components calculated from the previous analytical solution-based method(b), the proposed approximate solution(c), and their difference(d). Figure 2 shows the images (and line profiles) before and after the proposed pre-emphasis at high(4us) and low(16us) update rates. Shown in Fig. 3 are the ACR phantom results from standard spiral scan, and the first-order concomitant field corrupted spiral scans before and after the proposed correction.Phantom scan and simulation results show that the proposed gradient pre-emphasis effectively corrects for first-order concomitant fields, and can be implemented in real-time during data acquisition. Further reduction of computational load can be achieved by decreasing pre-emphasis update rate. Although tested on 2D spiral, the proposed method is general and can be applied to any gradient waveforms including EPI, phase-contrast acquisitions, etc. It can readily be extended to other asymmetric gradients⁵.A gradient pre-emphasis method suitable for real-time implementation provides effective correction for the first-order concomitant fields in the asymmetric gradient systems. The difference between the arithmetic approximation and the exact cubic solution is negligible at 3T.