Synopsis

Geometric distortions caused by off-resonant spins are a major issue in EPI based functional and diffusion imaging. We present a novel approach to the problem of geometric distortions. An extension of the model-based, algebraic hybrid-SENSE reconstruction method in combination with a known fieldmap, calculated from blip up/down scans, allows for correction of off-resonance effects during the reconstruction. This enables a joint blip up/down reconstruction that significantly reduces g-factor penalty if the blip-down trajectory is chosen to fill in missing k-space samples from the blip-up scan. The resulting high SNR images are automatically eddy-current corrected and geometric distortions are minimized.

Introduction

Geometric distortions caused by off-resonant spins are a major issue in EPI-based functional and diffusion imaging. Typically, correction methods use an off-resonance map and operate as a post-processing step on the magnitude images. A robust method to measure off-resonance maps is the blip up/down approach, where the polarity of the 2^nd readout is inverted (2). Brain regions that are compressed in the blip-up acquisitions are stretched in the blip-down case and vice versa. From that, FSL’s topup implementation calculates the off-resonance map.

We present a novel two-stage approach to correct for geometric distortions that makes use of the blip up/down concept. First, we propose an extension of the model-based, algebraic hybrid-SENSE reconstruction method (1) that allows correction for off-resonance effects. For accelerated scans (SMS and/or in-plane) we then employ hybrid-SENSE for a joint blip up/down reconstruction where we choose the sampling pattern of the blip-down trajectory (Figure 4) such that it partly fills in missing k-space samples. A joint reconstruction not only benefits from signal averaging but also from reduced g-factor penalty because the effective acceleration factor is reduced.

Theory

Ignoring off-resonance effects along the readout, and performing an FT along the $k_x$ domain, transforms the k-space data $s_0(k_x,k_y,k_z)$ to an hybrid space $s(x,k_y,k_z)$. For every position $x=x_n$ we can write the signal equation (forward model) in matrix notation as $${\bf s=Fm} \hspace{10mm} [1]$$ with the unknown magnetization ${\bf m} \in \mathbb{C}^{N_yN_x \times 1}$ and the signal vector ${\bf s} \in \mathbb{C}^{N_sN_c \times 1}$ that consists of all $N_s$ acquired k_y-k_z k-space samples times the number of receiver coils $N_c$. The effective encoding matrix $\bf F= C \circ E \circ W \circ R$ with ${ \bf C,E,W,R} \in \mathbb{C}^{N_s N_c \times N_y N_z}$ is the Hadamard product of the individual matrices that model the encoding process where: $\bf C$=coil sensitivity variation, $\bf E$=gradient encoding from$k_y$ and $k_z$-space, $\bf W$=off-resonance effects, and $\bf R$=intensity variations. In order to invert the rectangular matrix $\bf F$, we use a truncated singular value decomposition approach (TSVD) (3) where eigenvalues below a certain threshold $\lambda$ are completely suppressed (Figures 1,2). The solution is then given as $${\bf m = F}_{pinv}^{TSVD} {\bf s} \hspace{10mm} [2]$$

For the joint blip up/down reconstruction we rewrite the signal model as $${\bf F_{\uparrow \downarrow}}=\left(\begin{array}{c}{\bf F_{\uparrow}}\\ {\bf F_{\downarrow}}\end{array}\right) \hspace{10mm} [3]$$ where $\bf F_{\uparrow}$ and $\bf F_{\downarrow}$ are the effective encoding matrices for the blip up$(\uparrow)$ and down$(\downarrow)$ trajectories. The g-factor map is calculated as $$g(x_n,y,z)=diag(\sqrt{\frac{{\bf F}_{pinv}^{TSVD}({\bf F}_{pinv}^{TSVD})^T}{{\bf C C}^T} })/\sqrt{R_{tot}} \hspace{10mm} [4]$$

where ${\bf C C}^T$ corresponds to a SENSE reconstruction with R=1.

Methods

Results

Discussion

Acknowledgements

References