Synopsis

Prostate diffusion MRI is recognized as a potential biomarker for tumour detection but currently it is unusable in some patients due to significant distortions. We proposed a novel model based joint image and B₀ reconstruction framework that can correct these distortions by using data acquired from opposite phase encoding gradient directions. Using sampling time shift between the two acquisitions, the proposed method is robust against any dynamic changes in the off resonance effects in the prostate-rectal air region.

Introduction

Method

The proposed framework (Fig.1) acquires two EPI data sets (blip-up and blip-down) in opposite phase encoding gradient directions with a constant time shift (T_diff) between acquisition of k-samples in the two scans.Starting from an initial B₀ field estimated from reconstructed images in the two scans,a joint estimation framework is proposed that can estimate both corrected EPI image and corrected field maps.Let Y₁ and Y₂ be the k-space for blip-up and blip-down EPI scans, respectively. Mathematically, we can write^1,2:

\[\mathbf{Y_1} (k,l)=∑_{n=0}^{N-1}∑_{m=0}^{M-1}\mathbf{x}(m,n) e^{-i2\pi(mk/M+nl/N)} e^{-i2\pi(\mathbf{ΔB_0} (m,n).t_1 (k,l))} \]

\[\mathbf{Y_2} (k,l)=∑_{n=0}^{N-1}∑_{m=0}^{M-1}\mathbf{x}(m,n) e^{-i2\pi(mk/M+nl/N)} e^{-i2\pi(\mathbf{ΔB_0} (m,n).t_2 (k,l))} \]

\[\quad \quad \quad \quad \quad \quad \qquad \qquad \it{Eq (1)} \]

where m,n are image coordinate indices, M,N are image dimensions, k,l are k-space coordinate indices, ΔB₀ is the B₀ field in Hz; t₁(k,l) and t₂(k,l) are the acquisition times of location (k,l) in k-space for blip-up and blip-down scans, respectively such that t₂(k,l) = -t₁(k,l) +T_diff, where T_diff is a constant time shift, the value is set in range of 1-3 msec. For spin echo sequence, time shift T_diff can be achieved by shifting the timing of the refocussing pulse in the sequence development. Eq (1) can be summarized as Y₁=E₁(ΔB₀)x and Y₂=E₂(ΔB₀)x, where E₁ and E₂ summarize the encoding operators for blip-up and blip-down scans, respectively. The data from both phase encoding directions can be combined into a single formulation in Eq(2) by setting Y=[Y₁ Y₂]^T and E=[E₁ E₂]^T in Eq (1).

\[ \mathbf{Y=E(ΔB_0)x} \quad \quad \it{Eq (2)} \]

The proposed joint image and B₀ field estimation framework can be summarized as:

\[ arg \it{min}_{\bf{x,ΔB_0}} \bf{Ψ(x, ΔB_0)} \quad \it{Eq (3)} \] where \[ \bf{Ψ(x, ΔB_0)} = \parallel\bf{Y-E(ΔB_0)x}\parallel^2+ β_1\it{R}(\bf{ x}) + β_2\it{R}(\bf{ΔB_0} ) \]where R(x)and R(ΔB₀) are quadratic regularization terms ||Dx||² and ||DΔB₀||² respectively, D being the first order finite difference operator; β₁,β₂ are regularization weights.

The above formulation is solved using alternating minimization scheme^3,4.The image update in the k^th iteration is estimated using a previous field map estimate ΔB₀^k-1:

\[ {\bf{x}}^{k}=arg \it{min}_{\bf{x}}\parallel\bf{Y}-\bf{E}({\bf{ΔB_0}}^{k-1})\bf{x}\parallel^2+ β_1\it{R}(\bf{x}) \quad \it{Eq(4)} \]

Using estimate x^k ,the updated field map in the k^th iteration is estimated as:

\[ {\bf{ΔB_0}}^{k}=arg \it{min}_{\bf{ΔB_0}}\parallel\bf{Y}-\bf{E}({\bf{ΔB_0}}){\bf{x}}^{k}\parallel^2+ β_2\it{R}(\bf{ΔB_0}) \quad \it{Eq(5)} \]The above two step iterative process is repeated until the convergence is achieved.

Experiments

Results and Discussion

Acknowledgements

References

[1] Munger et al, TMI 2000

[2] Usman et al, MRM 2018

[3] Matakos et al, ISBI 2010

[4] Fessler et al, "Michigan Image Reconstruction Toolbox," available at https://web.eecs.umich.edu/~fessler/code/index.html

[5] Funai et al, TMI 2008