Introduction

Balanced SSFP (bSSFP) imaging is highly SNR-efficient but suffers from ‘bands’ of reduced signal intensity across the image. The inclusion of gradient spoiling in FISP (or PSIF) eliminates these bands from the reconstructed images but substantially reduces the SNR. Although banding patterns are not observable in FISP images, they still occur at a sub-voxel scale and cause aliasing in k-space. This is a source of confounding but can also be used to give additional information^1-2.

Here we describe a strategy for improving the SNR efficiency of FISP by exploiting inherent k-space aliasing. We call this kaFISP (k-space aliased FISP).

Theory

k-space aliasing in FISP
In FISP, the characteristic off-resonance banding pattern of bSSFP occurs as a repeating signal modulation along the direction of the unbalanced gradient (Figure 1a, columns i & iii). Typically, the time integral of this gradient is chosen so that the periodicity of the pattern is smaller than the voxel spacing, thus ensuring that undesirable bands are no longer visible. In k-space, this is equivalent to ensuring that the centre of the aliased signals occur outside of the sampled region (Figure 1b).

At high flip angles (α≥40° for grey matter/white matter/CSF at 3T), the banding pattern resembles a sinusoid, with two configuration states (F₀ and F_-1) contributing approximately equally to the measured signal (Figure 1a, columns ii & iv). Their relative phases can be described in terms of the local ΔB₀ and chosen linear RF phase cycling increment (Δφ), which together determine the spatial shifting of the sinusoidal pattern within the voxel. Furthermore, the position of the aliased signal in k-space can be modified by changing the polarity of the unbalanced gradient (Figure 1c).

4-shot kaFISP experiment
This enables, for example, a 4-shot experiment. We acquire 4 acquisitions where we vary Δφ (0, π) and unbalanced gradient polarity (+ve, -ve; Figure 2a). In each acquisition we only acquire the central 1/3 of k-space, containing the aliased signals. We then combine these data to generate the full alias-free k-space (Figure 2b).

This is more SNR-efficient than traditional FISP imaging due to the repeated sampling of the centre of k-space (Figure 3). When the F₀ and F_-1 signals are approximately equal (which holds for T2>>TR at high α), this provides a potential 34% improvement in SNR efficiency compared to a traditional FISP acquisition.

Accounting for off-resonance: direct reconstruction
The key challenge to the core idea is that off-resonance (ΔB₀) introduces non-linear phase terms which must also be included in the forward model. Using a similar approach to Hennel³, we can apply separate ΔB₀ phase terms to the F₀ and F_-1 signals in image space, Fourier transform them, and then shift them to give their k-space representations ($$$S_0$$$ and $$$S_{-1}$$$ ­respectively):
$$S_0(k)=FT(\beta_0Xe^{2πi∆B_0(x)TE})(k)$$ $$S_{-1}(k)=FT(\beta_{-1}Xe^{-2πi∆B_0(x)TE}e^{-i(∆φ+π)})(k±k')$$ where $$$X$$$ is the image, TE is the echo time, $$$k'$$$ is the k-space shift from the unbalanced gradient, and $$$β$$$ is a scalar signal weighting. At high flip angles (α≥40° in brain at 3T), $$$\beta_0 \approx \beta_{-1}$$$. Therefore, we can reconstruct the full k-space provided we have a separate measurement of ΔB₀.

Accounting for off-resonance: iterative reconstruction
Alternatively, we can jointly reconstruct $$$X$$$ and ΔB₀ from the data itself, without the need for a separate ∆B₀ measurement. The forward operator $$$\mathbb{F}_X$$$ for each aliased dataset $$$d$$$ is:
$$d=\mathbb{F}_XX=SDM\mathscr{F}ΨX$$ where $$$Ψ$$$ combines the phase terms due to ΔB₀ (non-linear), Δφ (0 or π) and $$$k'$$$ (+ve or -ve ramp); $$$\mathscr{F}$$$ is the Fourier operator; $$$M$$$ mixes the aliased signals; $$$D$$$ is a down-sampling operator; and $$$S$$$ is the coil sensitivity matrix. Next, we linearise this forward operator in ΔB₀ (giving $$$\mathbb{F}_{B}$$$) using Taylor expansion⁴. Finally, we formulate the joint reconstruction as an alternating minimisation problem, calculating the following at the $$$n$$$th iteration:
$$X^n=argmin_X(\Vert\mathbb{F}_{X}(∆{B_0}^{n-1})X-d\Vert_2^2+\lambda_1\Vert X\Vert_2^2)$$$${∆B_0}^n=argmin_{∆B_0}(\Vert\mathbb{F}_{B}(X^n){∆B_0}-d\Vert_2^2+\lambda_2\Vert {ΔB_0}\Vert_2^2)$$ where $$$λ_1$$$ and $$$λ_2$$$ are regularisation weights, and the initial estimate $$$Δ{B_0}^0$$$ is an array of zeros.

Methods

We performed an experiment on a healthy volunteer on a 3T Magnetom Prisma (Siemens Healthineers, Erlangen, Germany) with a 64ch head coil. We acquired a 3D kaFISP (TR/TE=8.4ms/4.2ms, FA=60°, FOV=250x250x160mm³, four acquisitions with a resolution of 1.0x3.0x1.0mm³ reconstructed to 1.0x1.0x1.0mm³, BW=1kHz/pix, TA=8:11 [2:03 per acquisition]) and a matched 3D FISP (TR/TE=8.4ms/4.2ms, FA=60°, FOV=250x250x160mm³, 1.0x1.0x1.0mm³, BW=1kHz/pix, TA=5:45). Multicoil data were combined using ESPIRiT⁵. A low-resolution ΔB₀ map was acquired from a dual-echo GRE (TR/TE₁/TE₂=476ms/4.92ms/7.38ms, FA=60°, 2.0x2.0x3.0mm³).

Due to the reduced gradient requirements in kaFISP, its readout bandwidth could have also been reduced by a factor of nearly 3 to provide an additional SNR benefit over FISP, or the TR reduced to shorten the scan time to 6:43, but this was not done to ensure images were comparable.

Results & Discussion

kaFISP brain images are shown in Figure 4, alongside the low-resolution acquired data and reference FISP. Direct (Figure 5c) and iterative reconstructions (Figure 5d) are also shown. The former shows residual ringing artifacts from phase inconsistencies between adjacent k-space bands. This is improved via joint reconstruction but could further be improved with a reduced unbalanced gradient area, ensuring overlap between bands³.

Conclusions

k-space aliasing is inherent in FISP and should not be ignored, but it can be exploited to give an SNR-efficient acquisition strategy, while also providing a map of B₀ inhomogeneity.

Acknowledgements

We wish to thank Dr Iulius Dragonu (Siemens Healthineers) for providing advice and support for sequence development. PJL acknowledges generous funding from The Wellcome Trust (220473/Z/20/Z), The Edmond J Safra Foundation, UK Dementia Research Institute, and National Institutes of Health (R01EB002524).

References

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