Synopsis

B₀ field inhomogeneities lead to distortions in gradient echo acquired field maps. In particular, these distortions are present when mapping the spatial distribution of a system's shim coils. Here, we assess the effect of these distortions using in-vivo data and computer simulations, and suggest an iterative algorithm which completely removes them without the need of any additional acquisitions.

Introduction

Field maps are an essential tool in many areas of magnetic resonance, from susceptibility based imaging, to echo planar imaging, to active B0 shimming. Field maps are often acquired using a gradient multi-echo sequence. However, such sequences intrinsically contain geometric distortions along the readout axes, producing a distorted output, $$$B_0^{(dist)}(\mathbf{r})$$$. It is possible to correct such distortions given an accurate, undistorted field map [1] - the very thing being sought.

Here, we set out to: (1) Estimate the detrimental effect of geometric B0 distortions on the mapping of shim coils and, consequently, high order B0 shimming of the human brain [2]; (2) Propose an iterative algorithm which completely corrects them; (3) Validate the algorithm's performance using in-vivo data and computer simulations.

Methods

Data acquisition: : T1-weighted MPRAGE (1 mm isotropic resolution) and whole brain field maps using a double echo field‑mapping sequence (TR/TE1/TE2=740/4.71/7.17) were acquired from 75 healthy volunteers.

Data processing: Nine anatomical regions of interest (ROIs) were delineated in each subject: the whole brain (WB), brainstem (BS), cerebellum (Cereb), intracalcarine cortex (ICC), left hippocampus (LHipp), right hippocampus (RHipp), left thalamus (LThal), right thalamus (RThal), posterior cingulate gyrus (PCG). The whole brain masks were delineated using FSL Bet [3]. The other ROIs were selected using the Harvard Subcortical and Cortical atlases, after registering each subject’s MPRAGE to the common MNI space using the FSL software package [4,5].

Mapping of shim coils (in simulations): A 4th order spherical harmonic basis set was assumed, consisting of 25 independent ideal coils, $$$B_k^{(true)}(\mathbf{r})$$$ . Bloch equations simulations were used to produce distorted maps of each of the shim coils, $$$B_k^{(dist)}(\mathbf{r})$$$. The mapping was carried out by a double-echo gradient field mapping sequence with TR/TE1/TE2=740/4.71/7.17, 2 mm isotropic voxel size and a bandwidth of 200 Hz per pixel, inside a 12-cm radius spherical ROI (corresponding to a large spherical phantom, as often used for mapping shim coils). Two field maps were acquired at two shim coil currents: I=0 and I=I_max, where I_max was chosen to generate a maximal B₀ field of 500 Hz within a spherical ROI – i.e., approximately the maximum possible without leading to phase wrapping in the resulting maps.

Assessment of The Effect of Distortions: Active shimming was carried out in each of the anatomical ROIs, in each subject, using his or her specific B₀(r) map. A least squares algorithm was used to calculate the optimal (unrestricted) shim currents, using the distorted maps $$$B_k^{(dist)}(\mathbf{r})$$$ for each coil:

$$min_{I_k} ||B_0(\mathbf{r}) - \Sigma_{k=1}^N I_k B_k^{(dist)}(\mathbf{r})||$$

The optimal currents $$$I_k^{(opt)}$$$ were then applied using the true coils, $$$B_k^{(true)}(\mathbf{r})$$$ , and the standard deviation of in each region was calculated: $$$ ||B_0(\mathbf{r}) - \Sigma_{k=1}^N I_k^{(opt)} B_k^{(true)}(\mathbf{r})||$$$.

Iterative Correction Algorithm: Geometric distortion occurs only along the readout axis and hence the problem is one dimensional. One has $$$B_{meas}(x) = B_{true}\left(x + \frac{B_{true}(x)}{\gamma G}\right) \equiv B_{true} \left(x+ \delta_f(x) \right)$$$, where $$$B_{true}(x)$$$ is the true B₀ field, $$$B_{meas}(x)$$$ is the measured field and $$$\delta_f(x) \equiv \frac{\B_{true}(x)}{\gamma G}$$$ is the forward distortion field. One can also define an inverse distortion field, $$$\delta_b (x)$$$, for which $$$B_{meas}(x-\delta_b(x)) = B_{true}(x)$$$. Combining both definitions, one has:

$$\delta_f(x) = \delta_b(x + \delta_f(x))$$ (Eq. 1)

Eq. 1 is the basis of our iterative algorithm: one initializes $$$\delta_f(x) = \frac{ B_{meas}(x)}{\gamma G}$$$ and applies Eq. 1 repeatedly until convergence.

Validation of Algorithm: Each distorted shim map $$$B_k^{(dist)}(\mathbf{r})$$$ was corrected using the proposed iterative algorithm to yield a corrected $$$B_k^{(corr)}(\mathbf{r})$$$ (with 5 iterations), and active shimming was carried out again in each of the anatomical ROIs as detailed above.

Results

Discussion & Conclusions

Acknowledgements

References

[1] Jezzard, Peter, and Robert S. Balaban. "Correction for geometric distortion in echo planar images from B0 field variations." Magnetic resonance in medicine 34.1 (1995): 65-73.

[2] Pan, Jullie W., Kai‐Ming Lo, and Hoby P. Hetherington. "Role of very high order and degree B0 shimming for spectroscopic imaging of the human brain at 7 tesla." Magnetic resonance in medicine 68.4 (2012): 1007-1017.

[3] Smith, Stephen M. "Fast robust automated brain extraction." Human brain mapping 17.3 (2002): 143-155.

[4] Woolrich, Mark W., et al. "Bayesian analysis of neuroimaging data in FSL." Neuroimage 45.1 (2009): S173-S186.

[5] Diedrichsen, Jörn, et al. "A probabilistic MR atlas of the human cerebellum." Neuroimage 46.1 (2009): 39-46.