Synopsis

Eddy current characterizations are needed for pre-emphasis implementations in dynamic shimming applications. However, since a high spatio-temporal sampling of the eddy current fields is required, this a challenging task. Image-based approaches are well-suited for this purpose, but require substantial acquisition times.

This work presents a 2D image-based sampling scheme, which is fast compared to existing 3D techniques and still provides sufficient information for an unambiguous pre-emphasis parameter reconstruction. Moreover, a model-based fit is proposed, which jointly applies the spatial and temporal eddy current model to the acquired data. It is shown, that this approach is well-suited for reducing fitting noise.

Introduction

Methods

The proposed EC characterization is based on measuring phase offsets that are induced in 2D images, which are acquired at different time-points during a shim-pulse induced EC decay. It exploits the fact that an off-center placement of three orthogonal slices introduces sufficient spatial dependence into the data to unambiguously determine their SH components. The sequence and the sampling scheme are illustrated in Fig. 1. An additional reference scan is acquired without the application of a shim pulse and subtracted from the EC data to eliminate phase offsets from sources other than the shim ECs.

Given the acquired EC data, $$$\Delta\text{B}_{0}$$$, and with $$$\text{f}_{n,m}(\cdot)$$$ denoting a SH of order n and degree m and $$$a_{n,m}^{(i)}$$$ and $$$\tau_{n,m}^{(i)}$$$ being the amplitude and time constant of a modulating exponential function with index i, EC fields are modelled at position r and time t following

$$\Delta\text{B}_{0}\left(\mathbf{r},t\right) = \underbrace{\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\text{f}_{n,m}\left(\mathbf{r}\right)}_{\text{Spatial Term}}\cdot\underbrace{\sum_{i=1}^{\infty}a_{n,m}^{\left(i\right)}\cdot\exp\left(-\frac{t}{\tau_{n,m}^{\left(i\right)}}\right)}_{\text{Temporal Evolution}}.$$

Conventional EC processing routines regard the spatial and temporal components independently, by first performing a SH decomposition at each time-point and then fitting the temporal model to the SH coefficients. However, because the SH fit is an ill-posed, inverse problem, this can introduce strong fitting noise.

When analyzing all data points simultaneously, through joint application of the spatio-temporal model, this noise can be reduced. Let $$$\bar{\mathbf{A}}$$$ be a block diagonal matrix of copies of the SH system matrix $$$\mathbf{A}$$$, $$$\mathbf{x}$$$ be a vector of all SH EC amplitudes at each time-point and $$$\mathbf{b}$$$ be a vector of the measured EC field at each time-point. Applying a linear operator, $$$\mathcal{H}(\cdot)$$$, the time-course of the amplitudes of the EC terms can be rewritten as a stack of Hankel matrices. Minimizing the sum of the nuclear norms of these matrices enforces the time-course of the data to be approximated with as few damped exponentials as possible. Data consistency can be established to a desired accuracy, $$$\epsilon$$$, leading to the optimization problem

$$\underset{x}{\text{min}} \left\| \mathcal{H}(\mathbf(x)) \right\|_{*,1} \quad \text{s.t.} \quad \left\| \bar{\mathbf{A}}\mathbf{x}-\mathbf{b} \right\|^{2}_{2} \leq \epsilon^{2}.$$

The optimization was implemented using Bregman iterations with ADMM updates. Simulations were performed and EC data was acquired on a 3T TRIO (Siemens) equipped with a dynamically-driven high-order SH shim insert (RRI). Using a matrix size of 32x32 at 3 mm isotropic resolution and sampling 3000 ms of the EC decay, the total scan time was 9:36 min per shim.

Results

To test the applicability of the proposed sampling scheme, the pre-emphasis module of a C3-shim was deliberately misadjusted to generate artificial ECs with a known time-course. Fig. 2 shows the acquired data and illustrates the slice-based sampling of the high-order SH ECs.

The performance of the proposed model-based EC fit was compared in simulations to the conventional approach using realistic EC parameters⁶. Fig. 3 illustrates simulated data for a low-amplitude EC decay generated by a C2-shim, confounded by strong additive Gaussian noise (σ=2.5 Hz). Based on this data, Fig. 4 compares the SH decomposition performed with the proposed model-based approach and the conventional approach. The model-based approach is suitable to reconstruct the input parameters with an average accuracy of 99% as compared to 84% for the conventional approach.

Fig. 5 shows an experimentally acquired EC decay induced by the Z3-shim and demonstrates that the individual EC components are robustly reconstructed

Discussion and conclusions

Image-based EC measurements need no additional hardware, but data acquisition is time-consuming. Compared to a 3D approach, the presented sampling scheme substantially reduces acquisition times while still providing sufficient information for a pre-emphasis parameter reconstruction. Considering equal, isotropic resolutions, the acquisition time is reduced by a factor of NPE/3 (NPE ≙ number of phase encoding steps).

The proposed model-based EC processing has shown to substantially reduce fitting noise, thus to be applicable to even recover low-amplitude EC terms in the presence of high noise-levels. Being independent of the data acquisition scheme, it can also be applied to non-image-base EC measurement techniques.

Acknowledgements

References

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