Standard approaches to generating MR images require a highly uniform magnetic field (B0) and thus necessitate the use of magnets that are physically large relative to the object to be imaged, leading to the high cost of magnet construction and siting. Previously we proposed an approach called STEREO (for STEering REsonance over the Object) as a means to generate images using magnets with drastically reduced B0 uniformity [1]. Unlike standard Fourier imaging techniques, STEREO exploits full spatiotemporal encoding, and images are reconstructed by a non-standard approach (e.g., by solving the inverse problem). In the original STEREO method, spatial variations in B0 and radiofrequency field (B1) are compensated by adjusting pulse amplitudes and frequencies in a temporal manner. In the present study, we investigated whether it is possible to use the dynamic multi-coil technique (DYNAMITE) [2] as a means to create spatiotemporal excitation for STEREO. We aimed to design a limited set of well-defined MC arrays able to strongly reduce the magnetic field distortions across the object of interest

After selection of center positions (x_0C, y_0C, z_0C) of the current carrying circular coils on the cylindrical surface (Fig 1), the z-component of the magnetic field generated by the circular coils (B_Z) per unit current was then derived analytically by use of the Biot-Savart law, which is adopted from Juchem’s work [3].

$$B_{Z}(x,y,z)= \frac{\mu_0.n}{2\pi}.\sum_c\left(\frac{(y-y_{0C})}{\sqrt{((x-x_{0C})^2 + (z-z_{0C})^2 )[(r + \sqrt{(x-x_{0C})^2 + (z-z_{0C})^2})^2 +(y - y_{0C})^2)}} \right)$$

$$\times \left((-K(\kappa^2) +\frac{(r^2 + (x-x_{0C})^2 + (y-y_{0C})^2 + (z-z_{0C})^2)}{[(r - \sqrt{(x-x_{0C})^2 + (z-z_{0C})^2})^2 +(y - y_{0C})^2)]}.E(\kappa^2)\right)$$ (1)

Here μ₀ is the magnetic field constant, n is the number of turns, r is the radius of the coil, c is the coil number, and x, y, and z are the Cartesian coordinates describing three-dimensional space [4]. K(κ²) and E(κ²) refer to the complete elliptical integrals of the first and second kind, respectively. Equivalent values in Hertz were derived using the Larmor equation for protons (¹H). A simulated, uncorrected magnetic induction field, ΔH_Z, was generated within the 256x256x256mm FOV and 64x64x64 matrix. Assuming constant coil currents, ΔH_Z was then projected into the magnetic field basis, B_Z, to obtain the corresponding current values, I_C, in each individual coil to compensate ΔH_Z, within a spherical region of interest (ROI) with a 10-mm radius. The estimation of I_C (see Eq.(2)) was based on the Least-squares fitting based on the Levenberg-Marquardt method. The current values in each coil is,

$$I_{C} = [B_{Z}(x,y,z)^{T}. B_{Z}(x,y,z)]^{-1}. B_{Z}(x,y,z)^{T}. \triangle H_{Z}(x,y,z)$$ (2)

The compensation of field homogeneity was extended by moving a ROI along the spiral trajectory (Fig 2A), in clock-wise direction at 4000 different time points during the period of 16 ms in synchrony with traversing shimmed region. A uniform B1 pulse of 5.9 μT was introduced simultaneously with modulated coil currents for duration of 16 ms. The Bloch equations were carried out to predict the spatiotemporal excitation profile of the B1 pulse. The maximum transverse magnetization (Figure 2B) and maximum flip angle were evaluated during the whole B1 pulse for each pixel. Further, by introducing a linearly varying B1 pulse (Figure 2D), the Bloch simulations were carried out to evaluate the maximum transverse magnetization, M_XY (Figure 2C) [5]. Potential relaxation effects were not considered in the simulations.

Fig 2(A) shows selected spherical ROIs along the spiral trajectory. Fig 2(B) illustrates the maximum M_XY produced in the 64x64 matrix in the presence of a uniform 5.9 uT B1 field. It can be seen that uniform excitation along the traversed spiral trajectory is achieved. Above a threshold B1 amplitude, the method is insensitive to B1 field inhomogeneity, as demonstrated in Fig 2(C) which shows the M_XY produced by the linearly varying B1 shown in Fig2(D). In this work, simulations have provided proof-of-concept for a unique way of exciting spins uniformly by dynamically driving the coil currents in a multi-coil array with DYNAMITE while applying RF irradiation. The next step will involve experimental validation of this spatiotemporal excitation as a means to accomplish MR imaging in extreme field inhomogeneity.