Modern MR scanners employ shim coils in order to make the B0 field homogeneous. However, susceptibility differences at the interfaces, intentional motion of the subject or involuntary physiological motion such as breathing can alter the shimmed fields and lead to image distortions and signal loss. To solve such problems, it is common to use dynamic shimming (real-time prospective shim correction), which updates the shim settings concurrently with the acquisition. This requires a periodic evaluation of the field fluctuations, which can be done by using the sensitivity information of coil elements ¹, shim navigators ², or external field sensors attached to the subject or the coil ³. The existing approaches, however, have limitations such as increased scan time duration, slow performance or the need for dedicated external field monitoring hardware. In this work, we employ a radial acquisition both for imaging and estimating the first order inhomogeneity of the B0 field, bypassing the need of dedicated navigators or hardware.

Recently it was proposed to use cloverleaf navigators for combined real-time motion correction and shimming ⁴. Assuming a perfect shim and neglecting the phase of the receiver coil, the strongest signal intensity is found in the center of k-space which represents DC component of k-space. First order shim errors lead to a shift of the DC component away from the center. The three traversals of the cloverleaf navigator through the center of k-space provide the projections of this shift onto the X, Y, and Z axis. Based on this idea, we hypothesize that in radial acquisitions the spokes can be used not only for imaging but also for estimation of the first-order shim errors. Since each spoke is acquired in a different direction, the information provided by several projections can be used to determine the first order shim error in X, Y and Z axis. In Figure 1, we show how position of DC component changes in the XY plane given a good shim (Fig. 1A) and a small linear shim offset in X direction (Fig. 1B). Traversing k-space along a given axis in the presence of a first order field inhomogeneity oriented in the same direction can be mathematically described by Eq. [1].

$$k(t)=(\frac{3}{2}t_{ramp}+t_{deph}+t){\triangle G}_{}+(t_{ramp}+t_{deph})G_{deph}+(\frac{t_{ramp}}{2}+t)G_{readout} \quad\quad [1]$$

where t=0 is a starting time of the data acquisition, G_deph and G_readout are the amplitudes of the dephasing and readout gradients, t_ramp the gradient ramp-up or ramp-down time (assumed to be identical for all gradients) , t_deph the flat-top time of the dephasing gradient, and ΔG the first order component of a field inhomogeneity modeled as a gradient offset. In 2D k-space, the signal will be maximum at t_peak when K_x²+K_y² is minimum. Applying the derivative operator to K_x²+K_y² with respect to time results in a circle (see Eq. [2]).

$$a_{3}(\triangle G_x^2 + \triangle G_y^2)+a_{2}\triangle G_{y}+a_{1}\triangle G_{x}+a_{0} = 0 \quad\quad [2]$$

$$a_{0}=(G_{xreadout}^2 +G_{yreadout}^2)(\frac{t_{ramp}}{2}+t_{peak})+(G_{xdeph}G_{xreadout}+G_{ydeph}G_{yreadout})(t_{ramp}+t_{deph})$$

$$a_{1} = G_{xreadout}(t_{deph}+2t_{peak}+2t_{ramp}) + G_{xdeph}(t_{ramp} +t_{deph} )$$

$$a_{2} = G_{yreadout}(t_{deph}+2t_{peak}+2t_{ramp})+ G_{ydeph}(t_{ramp} +t_{deph} )$$

$$a_{3} = \frac{3}{2}t_{ramp}+t_{deph}+t_{peak}$$

This circle is a set of solutions associated with all possible gradient offsets which result in a maximum intensity peak at t_peak. Finding a unique solution (the gradient offsets) requires several circles, and thus several projections. At least 3 non-identical circles are required to find the gradient offsets, as shown in Figure 2.

We have evaluated the proposed method in simulation and in vivo. A spoiled GRE sequence was modified to allow a projection-based acquisition of k-space. While scanning a subject, we altered the first order component of the shim in small steps of -20uT/m to +20uT/m in X and Y directions in each measurement. All experiments were performed on a MAGNETOM Prisma 3T scanner (SIEMENS Healthcare, Erlangen, Germany) using a two-channel head coil. Additionally, we computed gradient delays of the scanner and applied them to the measured samples. Figure 3 shows a comparison of applied shim offsets and the offsets estimated by our method. The precision depends on how well it is possible to pinpoint the position of the peak, which requires to maximally shorten the dwell-time of analogue-to-digital converter (ADC). As shown in Table 1, there is a good agreement between applied and the measured offsets.We propose an approach for first order shim error estimation that does not require navigators, reference scans or external hardware, and is appropriate for radial imaging with short TRs. Shim gradients can be updated from three consecutive projection measurements allowing to perform the imaging with real-time shim correction. An extension of the method to 3D imaging is possible and is planned as a future work.