Yuan Zheng^{1}, Yu Ding^{1}, Yongquan Ye^{1}, and Weiguo Zhang^{1}

Multispectral acquisition is an important technique for MRI near metal. It is critical to estimate the field map and correct for displacements among bin images before bin combination in order to eliminate blurring. However, current field-estimation methods are either susceptible to noise or are computationally intensive, limiting their clinical applications. We propose a robust and efficient algorithm for calculating the field map from multispectral datasets based on a previous matched-filter field estimation technique. The proposed technique was tested on a digital phantom and generated accurate field maps and high quality images with a very short calculation time.

The previous MF method has two steps. It first roughly estimates the field by matching the signal intensity across bins to the RF profile (Fig.1), and then accounts for displacements among bin images by a fine search of the field that maximizes the goodness-of-fit, which is defined as:

$$g(x,y,z,f)=1-\frac{|\overrightarrow{S}(x,y,z,f) - \alpha \overrightarrow{RF}(f)|_2}{\alpha |\overrightarrow{S}(x,y,z)|_2|\overrightarrow{RF}(f)|_2} \, ,\: \textrm{with} \: \overrightarrow{S}(x,y,z,f) = [s_1(x - \frac{f-F_1}{BW}) \quad s_2(x - \frac{f-F_2}{BW}) \quad \cdots \quad s_n(x - \frac{f-F_n}{BW})] \, ,$$

where $$$\alpha$$$ is a scalar chosen to match the magnitudes, $$$F_i$$$ and $$$s_i$$$ are bin frequencies and images. Since image interpolation is needed for updating $$$\overrightarrow{S}$$$ in every iteration step for all frequency bins, this method is computationally intensive.

We propose a variant of the MF approach that avoids interpolation and is computationally efficient (MF-Fast). Workflow of MF-Fast is shown in Fig.2. Firstly, original bin images are properly shifted along the RO direction so that they are all aligned (but distorted), which is equivalent to demodulating different acquisitions at the same frequency ($$$F_0$$$). Secondly, a field map (still distorted) is efficiently calculated using the aligned bin images by finding the maximum correlation between the RF profile and signal intensity across bins. Thirdly, field maps corresponding to demodulation at different bin frequencies are generated by shifting the previous one demodulated at $$$F_0$$$ along the RO direction and adding global offsets according to $$$f_b(x,y,z)=f_0(x-\frac{F_b - F_0}{BW},y,z) + (F_b-F_0)$$$, where $$$f_b$$$ and $$$f_0$$$ are voxel frequencies corresponding to demodulation frequencies $$$F_b$$$ and $$$F_0$$$, and RO is along the x direction. Finally, we search for the frequency closest to 0 across bins for each voxel, and add the corresponding bin frequency to generate the final field map:

$$b = \underset{b}{\operatorname{argmin}} \,abs(f_b (x,y,z)) \, ,\textrm{and} \: f(x,y,z) = f_b(x,y,z) + F_b \, ,$$

where b indicates the frequency bin closest to the frequency of voxel (x, y, z) , $$$F_b$$$ is the bin frequency and $$$f_b$$$ is the voxel frequency relative to $$$F_b$$$. The final field map is distortion free because each voxel is effectively demodulated at the closest bin frequency.

A digital phantom containing a titanium inclusion (χ=182ppm^{4}) was used to demonstrate MF-Fast at 3T. The true
field map was calculated using the dipole model^{5}. Effects of the B_{0}
field perturbation were considered in simulation of the MRI signal.
30 Spectra bins from -14KHz to 15KHz with a step size of 1KHz were collected
with BW=1KHz/pixel, and the matrix size=384x192. A Gaussian profile was
assumed for the RF with a FWHM of 2KHz. Gaussian noise with an SNR of 50 was
added to the bin images. Both CM and MF-Fast were used to calculate the field
maps, which were subsequently used to correct for displacements among bin
images^{2}. The final images were generated using RF-weighted spectral
bin combination^{3} (RF Comb). We also reconstructed the same dataset
using direct root-sum-of-squares (RSOS) bin combination for comparison.

- Koch KM et. al., A multispectral three-dimensional acquisition technique for imaging near metal implants, Magn Reson Med. 2009 Feb;61(2):381-90.
- Koch KM et. al., Imaging near metal with a MAVRIC-SEMAC hybrid, Magn Reson Med. 2011 Jan;65(1):71-82.
- Quist B et. al., Improved field-mapping and artifact correction in multispectral imaging, Magn Reson Med. 2017 Nov;78(5):2022-2034, Magn Reson Med. 2017 Nov;78(5):2022-2034.
- Schenck JF, The role of magnetic susceptibility in magnetic resonance imaging: MRI magnetic compatibility of the first and second kinds, Med Phys. 1996 Jun;23(6):815-50.
- J.P. Marques et. al., Application of a Fourierābased method for rapid calculation of field inhomogeneity due to spatial variation of magnetic susceptibility, Concepts Magn Reson 2005:25B:65-78.

As pointed out by Quist
et. al., the CM method suffers from frequency biases for noisy data, while filter
matching can more precisely locate the resonance frequency. However, since the
original bin images have relative displacements due to different demodulation
frequencies, a subsequent computationally intensive step of maximizing the
goodness-of-fit while considering the displacements is needed to improve the frequency
accuracy^{3}.

Workflow of the proposed
MF-Fast method.
Core computations are
carried out using images with no relative displacements. Therefore numerous image
interpolation is avoided and this algorithm is computationally efficient.

a), Field maps of the
same slice demodulated at different bin frequencies. The
field map shifts are exaggerated for display. b), Field of each voxel is
determined by searching for its frequency closest to 0 across all bins and then adding
the corresponding bin frequency. For instance, the closest-to-zero frequency of the voxel
indicated by the red circle is -0.2 KHz, found at the demodulation frequency
of 6 KHz. Therefore its frequency is -0.2 KHz + 6 KHz = 5.8 KHz.

a), Field maps
of the same slice calculated using the dipole model (true field), the CM method,
and the MF-Fast method. Error maps of the CM and MF-Fast field with respect to
the true field are shown in b). The CM map shows frequency-dependent biases,
while the MF-Fast map only has visible errors very close to the metal.

a)-c), Images
reconstructed by direct RSOS, CM field map with RF Comb, and
MF-Fast field map with RF Comb of the bin images. d)-f) show zoomed-in portions indicated by the red
boxes. Blurring was observed in the RSOS image. Signal voids and distortions
were observed in the CM field with RF Comb image. The MF-Fast field with RF
Comb reconstruction showed the best overall image quality.