Ultrafast 3D Bloch-Siegert B 1 + -mapping using variational modeling

Abstract

Purpose: Highly accelerated B ${}_1^{+}$ -mapping based on the Bloch-Siegert shift to allow 3D acquisitions even within a brief period of a single breath-hold.

Theory and methods: The B ${}_1^{+}$ dependent Bloch-Siegert phase shift is measured within a highly subsampled 3D-volume and reconstructed using a two-step variational approach, exploiting the different spatial distribution of morphology and B ${}_1^{+}$ -field. By appropriate variable substitution the basic non-convex optimization problem is transformed in a sequential solution of two convex optimization problems with a total generalized variation (TGV) regularization for the morphology part and a smoothness constraint for the B ${}_1^{+}$ -field. The method is evaluated on 3D in vivo data with retro- and prospective subsampling. The reconstructed B ${}_1^{+}$ -maps are compared to a zero-padded low resolution reconstruction and a fully sampled reference.

Results: The reconstructed B ${}_1^{+}$ -field maps are in high accordance to the reference for all measurements with a mean error below 1% and a maximum of about 4% for acceleration factors up to 100. The minimal error for different sampling patterns was achieved by sampling a dense region in k-space center with acquisition times of around 10-12 s for 3D-acquistions.

Conclusions: The proposed variational approach enables highly accelerated 3D acquisitions of Bloch-Siegert data and thus full liver coverage in a single breath hold.

Keywords: B 1 + -Bloch-Siegert shift; B 1 + -mapping; fast imaging; single breath hold acquisition; total generalized variation.

Figure 1

Retrospectively subsampled: B${}_1^{+}$‐map in μT for fully sampled reference, low resolution estimate and the result of the proposed two‐step reconstruction method for a retrospectively subsampled dataset in the brain of a healthy volunteer for a block size of 4 × 4, 10 × 6 and 12 × 4 encodings in the k‐space center. The right part of each column shows the error map for the corresponding result as normalized error in percent of the desired B₁ peak‐magnitude. The MAE is given as the mean of the error map over the whole 3D‐brain inside the cranial bone structure for each case

Figure 2

Retrospectively subsampled: Error histogram for the retrospectively subsampled dataset compared to the fully sampled reference in percent of the desired B₁ peak‐magnitude for block sizes of 4 × 4, 10 × 6 and 12 × 4 encodings in the k‐space center. The error histograms are shown for zero padded low resolution estimate and the result of our proposed two‐step reconstruction method

Figure 3

MAE inside the described ROI as a function of both regularization parameters λ and μ for different block sizes (8 × 8, 10 × 4, 12 × 4 and 12 × 12 encodings in k‐space center) in percent of the desired B₁ peak‐magnitude. For this evaluation the retrospective subsampled brain dataset shown in Figure 1 was used

Figure 4

Retrospectively subsampled: First row: Reconstruction results of the proposed two‐step reconstruction method in μT using different subsampling patterns. Second row: Corresponding error maps as normalized error in percent of the desired B₁ peak‐magnitude (Reference see Figure 1). Third row: Combinations of subsampling patterns for the first step P ₊ (TGV, +ω _BS, left pattern) and the second step P ₋ (H1, −ω _BS, right pattern) were investigated as follows: Case 1 and 2: 12 × 4 block‐pattern in k‐space center in H1‐part and a variable density pattern out of41 in TGV‐part with p = 14.4 and p = 25 respectively. Case 3: Different instances of this pattern with p = 14.4 in both parts. Case 4 and 5: The same instances of this pattern with p = 14.4 and p = 25 respectively. Cases 6 and 7: Pattern with Gaussian density function with σ _y = 5 and σ _z = 2, in case 7 with a higher acceleration factor. For each case the achieved acceleration R, the MAE, the medAE and the q _99% quantile inside the described ROI are given in percent of the desired B₁ peak‐magnitude as mean over 10 trails with different realizations out of the described probability distribution

Figure 5

Prospectively subsampled: B${}_1^{+}$‐map in μT for fully sampled reference and the result of our proposed two‐step reconstruction method with 10 × 6 and 12 × 4 encodings in the k‐space center and the corresponding error map in percent of the desired B₁ peak‐magnitude from prospectively subsampled data. The B${}_1^{+}$‐maps are shown for a brain and knee dataset from two different healthy volunteers. All results are shown in a transverse, coronal and sagittal orientation

Figure 6

Prospectively subsampled: B${}_1^{+}$‐map in μT as result of our proposed two‐step reconstruction method with 10 × 6, 12 × 4 and 12 × 6 encodings in the k‐space center in the liver of a healthy volunteer. These results are compared to those gained with zero padding using the same amount of data. The datasets were measured prospectively subsampled and acquired in a single breath hold. Due to the lack of a reference in the liver dataset, the reconstructed B${}_1^{+}$‐map (10 × 6 encodings) is also shown as an overlay to a morphological scan to show the underlying morphological structure. All results are shown in a transverse, coronal and sagittal orientation