Synopsis

Echo planar imaging (EPI) is widely used clinically for its speed, but is known to be sensitive to non-idealities like B0 field inhomogeneity, eddy currents, and gradient nonlinearity. Such non-idealities are not typically managed during image reconstruction, resulting in geometrically distorted images. Post-processing corrections (e.g., image-based interpolation) usually tend to degrade resolution. In this work, a comprehensive model-based reconstruction framework that prospectively and simultaneously accounts for non-idealities in accelerated single-shot EPI acquisitions is proposed. Sparsity regularization is also incorporated to mitigate noise amplification. The proposed algorithm is demonstrated on brain MRI data acquired on a compact 3T MRI system.

Overview

Echo-planar-imaging$$$^1$$$ (EPI) is widely used clinically for its speed, but is known to be sensitive to hardware non-idealities including B0 field-inhomogeneity$$$^2$$$, eddy-currents$$$^3$$$, and gradient nonlinearity$$$^{4}$$$ (GNL). Such non-idealities are not typically managed during image reconstruction, resulting in geometrically distorted images. To correct such distortion, image-based post-processing methods$$$^{5-6}$$$ (e.g., interpolation) are commonly applied, but these tend to degrade resolution and provide only partial corrections. Hybrid reconstruction models that leverage image-based corrections within an iterative framework have also been proposed, with some success$$$^{7-10}$$$. In this work, a comprehensive model-based iterative reconstruction (MBIR) framework that prospectively and simultaneously accounts for multiple non-idealities in accelerated single-shot-EPI acquisitions is proposed. Sparsity regularization is also incorporated to mitigate noise amplification. Following technical description of the framework setup, associated calibration procedures, and a nonlinear optimization strategy for solving it, the practical benefits of the proposed algorithm are demonstrated on brain MRI data acquired on a high-performance, compact 3T-MRI system. Comparisons against conventional reconstruction and correction pipelines are also presented.

Methods

Comprehensive EPI Signal Model: Let $$$TE$$$, $$$\Delta t$$$, and $$$T$$$ denote the echo, dwell, and echo-spacing times of a single-shot EPI acquisition. Neglecting $$$T_2^*$$$ and presuming pre-correction of eddy-current and gradient delay effects$$$^3$$$, the signal measured during readout $$$m \in[0,M)$$$ of phase-encoding line $$$n\in[0,N)$$$ can be modeled as:$$g_c[m,n]=\int_{\Omega _x}\int_{\Omega_y}s_c(x,y)f(x,y)e^{-j\Delta\omega_0(x,y)\tau[m,n]}e^{-j((-1)^nk_x[m]\eta_{x}(x,y)+k_y[n]\eta_{y}(x,y))}dxdy+\epsilon[m,n]\qquad(1)$$where $$$f$$$ is the target signal, $$$k_x$$$ and $$$k_y$$$ are sampling maps, $$$s_c$$$ is the sensitivity profile for coil $$$c\in[0,C)$$$, $$$\Delta\omega_0$$$ is the B0-field map, $$$\tau[m,n]=TE+(m-\frac{M-1}{2})\Delta{t}+(n-\frac{N-1}{2})T$$$ denotes sampling-time, $$$\eta_{x}(x,y)$$$ and $$$\eta_{y}(x,y)$$$ are GNL maps$$$^{11}$$$, and $$$\varepsilon$$$ is Gaussian noise. Note that Eq.1 accommodates ramp-sampling, parallel imaging, and partial Fourier acceleration.

As EPI uses high readout-bandwidth, $$$\Delta{t}<<T$$$ and off-resonance primarily manifest along its phase encoded (i.e., blipped) direction -- thus $$$\Delta{t}\approx{0}$$$ can be assumed. Letting $$$u(x,y)=f(x,y)e^{i\Delta\omega_0(x,y)TE}$$$, discretizing$$$^{12}$$$ and time-segmenting$$$^{13}$$$ (Eq.1) yields:$$g_c[m,n]=\sum\limits_{l=0}^L\Phi_l(n)\left\{\sum\limits_{p=0}^{P-1}\sum\limits_{q=0}^{Q-1}s_c[p,q]u[p,q]e^{-j\Delta\omega_0[p,q][\gamma_{l}-\frac{N-1}{2}]T}e^{-j((-1)^nk_x[m]\eta_{p}[p,q])+k_y[n]\eta_{q}(p,q))}\right\}+\epsilon[m,n]\qquad(2)$$where $$$\Phi_l$$$ is a spectral window, $$$\gamma_{l}$$$ denotes the center of segment $$$l$$$, and $$$p\in[0,P)$$$ and $$$q\in[0,Q)$$$ are pixel indices. Defining $$$W_l=diag\left\{e^{-i\Delta\omega_0[p,q][\gamma_l-\frac{N-1}{2}]T}\right\}$$$ and $$$S=[diag\left\{s_0\right\}\cdots{diag}\left\{s_{C-1}\right\}]^{T}$$$, Eq.2 abstracts to:$$g=(I\otimes\sum\limits_{l=0}^L\Phi_lFW_l)Su+\epsilon=Au+\epsilon\qquad(3)$$where $$$F$$$ is a Fourier transform. If a phase reference ($$$\theta$$$) for $$$u$$$ is available, the complex target reduces to $$$u=diag\left\{e^{j\theta}\right\}v=P(\theta)v$$$, where $$$v\in\mathbb{R}$$$$$$^{12}$$$.

EPI Image Reconstruction: The target image $$$u$$$ in (Eq.2) can be estimated via regularized least squares estimation:$$\mathop{\min}_{v\in\mathbb{R}}\left\{J(v)\triangleq\beta{R(v)}+\|AP(\theta)v-g\|_2^2\right\}\qquad(4)$$where $$$\beta>0$$$ is a regularization parameter. Here, we adopt joint redundant wavelet sparsity regularization:$$R(v)=\left\|\sum\limits_{k\in\xi}(\delta^{T}_k\otimes\Psi\Gamma{_k}v)\right\|_{1,2}\qquad(5)$$where $$$\delta$$$and $$$\otimes$$$ are Kronecker's delta and product, $$$\Psi$$$ is the Daubechies-4-wavelet, $$$\Gamma{_k}$$$ is a shift operator over neighborhood $$$\xi$$$, and $$$\|\cdot\|_{1,2}$$$ is the matrix-1,2 norm. As Eq.4 is convex, it can be solved using FISTA$$$^{14}$$$ with composite splitting$$$^{15}$$$. For efficiency, $$$F$$$ is implemented as a type-III non-uniform fast Fourier transform (NUFFT)$$$^{16}$$$. Vendor-provided coil sensitivity profiles, and ramp-sampling and $$$GNL$$$ information were incorporated. B0 field maps were estimated from a dual-echo GRE calibration scan using a graph cut-based procedure$$$^{16-17}$$$. The phase reference, $$$\theta$$$, was obtained by minimizing Eq.4 without phase constraints.

Data Acquisition and Processing: Two healthy volunteers were imaged on a compact 3T MRI system: (G$$$_{max}$$$=80mT/m, SR=700T/m/s) under an IRB-approved protocol, with $$$C=32$$$ or $$$C=8$$$ head-only receiver-only RF head coils$$$^{18-21}$$$. Following field map, coil sensitivity, and eddy current calibration, single-shot EPI acquisitions were performed with the parameters in Table 1. Prior to MBIR, eddy current correction was performed in k-space using vendor-provided tools. MBIR was performed in Matlab, using a 2X-oversampled NUFFT with width=5 Kaiser-Bessel kernel, L=N time segments, and a $$$7\times{7}$$$ shift window. FISTA was executed for 30 iterations. The regularization parameter was chosen manually ($$$\beta=0.02$$$). For comparison, vendor reconstructions were saved and post-processed using standard intensity-corrected image interpolation for distortion correction$$$^{22}$$$.

Results

Discussion

Acknowledgements

References

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