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Disentangling diffusion-weighted SSFP: ADC estimates in terms of an effective diffusion time

Benjamin C Tendler¹, Saad Jbabdi¹, Sean Foxley², Menuka Pallebage-Gamarallage³, Moises Hernandez-Fernandes⁴, Martin R Turner³, Olaf Ansorge³, and Karla Miller¹

¹Wellcome Centre for Integrative Neuroimaging, FMRIB, Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, United Kingdom, ²Department of Radiology, University of Chicago, Chicago, IL, United States, ³Clinical Neurology, Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, United Kingdom, ⁴Biomedical Image Computing and Analytics, University of Pennsylvania, Philadelphia, PA, United States

Synopsis

Diffusion-weighted SSFP (dwSSFP) is a high-SNR-efficiency diffusion imaging method. Unlike conventional diffusion measurements, the dwSSFP signal reflects a range of diffusion times because the signal is recycled over multiple excitations. This complicates interpretation and leads to an ill-defined b-value. We present a framework to describe dwSSFP-derived ADC estimates in terms of an effective diffusion time. To achieve this, we require dwSSFP measurements at two flip-angles. Experimental results are presented in a whole, postmortem brain at 7T. This enables us to simultaneously addresses flip-angle inhomogeneity at 7T and provide ADC estimates that are more comparable to conventional diffusion MRI.

Introduction

Diffusion-weighted steady-state free precession (dwSSFP) is a highly efficient diffusion-weighted imaging method. This efficiency results from a steady-state signal that recycles magnetisation over multiple excitations (i.e. there are multiple signal-forming coherence pathways). As a result of this signal-forming mechanism, the signal reflects a range of diffusion times, each with its own b-value. Moreover, each echo pathway has distinct T₁ and T₂ weighting, and the relative contribution of these pathways depends on flip-angle and TR, leading to a complicated signal dependence¹ (Figure 1). dwSSFP has demonstrated higher SNR efficiency than spin echoes for imaging fixed, post-mortem tissue with very short T₂². Further SNR benefits have been gained at 7T³, but require addressing flip-angle (B₁) inhomogeneity. Previous work at 7T acquired dwSSFP images at two flip-angles, demonstrating improved homogeneity of principal diffusion direction (PDD) estimates⁴. However, that work encountered the additional challenge that the weighting of diffusion times is also flip-angle dependent⁵. Here, we turn this challenge into an opportunity: this dependence means that multiple flip-angle dwSSFP measurements can in theory be extrapolated to a single effective diffusion time, $\Delta_{eff}$ , making dwSSFP measurements more comparable to conventional diffusion measurements. We present a framework to obtain ADC estimates as a function of $\Delta_{eff}$ from multiple flip-angle dwSSFP data. Experimental validation is performed in a whole, postmortem brain.

Theory

The “two transverse approximation” of dwSSFP signal considers only coherence pathways where the magnetisation is in the transverse plane for two TR periods^1,6, as a summation of pathways corresponding to one spin- and many stimulated-echoes: $M_{\text{ADC}}=\frac{M_{0}(1-E_{1})E_{1}E_{2}^{2}\sin\alpha}{2\left(1-E_{1}\cos\alpha\right)}\left(\frac{1-\cos\alpha}{E_{1}}\cdot A_{\text{ADC}}+\sin^{2}\alpha\sum_{n=1}^{\infty} \left[(E_{1}\cos\alpha)^{n-1}\cdot A_{\text{ADC}}^{n+1}\right]\right)\;\;\;[1]$ where $E_{1}=e^{-TR/T_{1}}$ , $E_{2}=e^{-TR/T2}$ , $A_{ADC}=e^{-q^{2}\cdot ADC\cdot TR}$ , $q$ is the area under the diffusion gradient, $\alpha$ is the flip-angle and $n$ is the number of TRs between the transverse periods for a given stimulated-echo⁶.

The diffusion time of the signal is well defined for each pathway (spin-echo: $\Delta=TR$ , stimulated-echo: $\Delta=(n+1)\cdot TR$ ). We define $\Delta_{eff}$ as the weighted-mean of these diffusion times, with each echo weighted by its relative signal contribution, obtaining (Figure 2): $\Delta_{\text{eff}}=\frac{(1+2E_{1}-E_{1}^2\cos\alpha)}{(1+E_{1})(1-E_{1}\cos\alpha)}\cdot\text{TR}\;\;\;[2]$ The dependence of calculated diffusivity on diffusion time reflects non-gaussian diffusion, which can be captured with a Gamma distribution of diffusion coefficients⁷. For a given distribution, we calculate the dwSSFP signal and translate this into a single ADC. For a Gamma pdf of mean $\mu$ and standard deviation $\sigma$ , this gives (Figure 3): $\text{ADC}=-\frac{1}{q^{2}\text{TR}}\cdot\ln{\left[\frac{-(S\cdot E_{1}\cos\alpha+1)+[(S\cdot E_{1}\cos\alpha+1)^{2}+4E_{1}\cdot S]^{\frac{1}{2}}}{2E_{1}}\right]}\;\;\;[3]$ where: $S=\left(\frac{\mu}{\mu+q^{2}\cdot\text{TR}\cdot\sigma^2}\right)^{\frac{\mu^{2}}{\sigma^{2}}}+ (1+\cos\alpha)\cdot E_{1}\cdot\left(\frac{\mu}{q^{2}\cdot\text{TR}\cdot\sigma^2}\right)^{\frac{\mu^{2}}{\sigma^{2}}}\cdot\Phi\left(E_{1}\cos\alpha,\frac{\mu^{2}}{\sigma^{2}},2+\frac{\mu}{q^{2}\cdot\text{TR}\cdot\sigma^2}\right)\;\;\;[4]$

and $\Phi$ is the Lerch transcendent. As Eqs. [2] and [3] both depend on flip-angle, we can use measurements at multiple flip-angles to calculate the ADC for a target $\Delta_{eff}$ .

Method

A fixed postmortem brain was scanned at 7T using a protocol described previously⁸. Briefly, the dwSSFP acquisition used flip-angles 24^o and 94^o, 120 directions per flip-angle, q=300/cm, resolution=0.85mm³, along with T₁, T₂ and B₁ mapping. Diffusion-tensor (DT) estimates were calculated using the full dwSSFP signal model^1,9, defining unique eigenvalues ( $L_{1,2,3}$ ) at each flip-angle, but shared eigenvectors ( $\overrightarrow{V}_{1,2,3}$ ). $L_{1,2,3}$ estimates at each flip-angle were fit to Eq. [3] to determine $\mu$ and $\sigma$ : $\min_{\mu,\sigma}||ADC_{\alpha_{low},\alpha_{high}}(\mu,\sigma)-L_{\alpha_{low},\alpha_{high}}||_2^2+||\mu-L_{\alpha_{high}}||_2^2\;\;\;[5]$ where regularisation ensured that $\mu$ remains on the order of $L_{1,2,3}$ .

The $\mu$ and $\sigma$ maps were subsequently input into Eq. [3], where we substituted $\cos\alpha$ with an expression in terms of $\Delta_{eff}$ (rearranging Eq. [2]) to obtain $L_{1,2,3}$ estimates for $\Delta_{eff}$ =200ms over the entire brain. $\Delta_{eff}$ =200ms was selected due to the low diffusivity of postmortem tissue. Furthermore, it corresponds to $\approx$ 26^o in our sample, an SNR-efficient flip-angle in dwSSFP.

Results and discussion

Figure 4 reveals how $L_{1,2,3}$ varies with B₁ in the postmortem brain. The reconstructed $\Delta_{eff}$ =200ms maps (corresponding to $\approx$ 26^o flip-angle) give good agreement to the 24^o dataset at high B₁ whilst maintaining a flatter distribution, highlighting the removal of B₁-induced ADC variability in our reconstruction. At 94^o, we observe higher $L_{1,2,3}$ estimates in B₁ matched areas, in agreement with simulation (Figure 3). However, little B₁-influence is observed at 94^o, suggesting a reduced ADC-sensitivity at high flip-angles (Figure 3).

The dwSSFP signal is SNR-efficient at low flip-angles¹ (Figure 1). Figure 5 reveals that by extrapolating our dual flip-angle datasets to $\Delta_{eff}$ =200ms, we are able to simultaneously preserve the SNR of the low flip-angle dataset whilst removing the influence of B₁, generating results in terms of a single effective diffusion time over the entire brain.

Conclusion

The framework presented here generates high-SNR dwSSFP ADC estimates in terms of a single effective diffusion time, removing the influence of B₁. This methodology could be extended to other diffusion models, distributions, or to derive ADC estimates in terms of

$\Delta_{eff}$ for a single-orientation, multiple flip-angle dwSSFP dataset, making dwSSFP more comparable to conventional diffusion measurements.

Acknowledgements

No acknowledgement found.

References

¹Buxton, Richard B. "The diffusion sensitivity of fast steady‐state free precession imaging." Magnetic resonance in medicine 29.2 (1993): 235-243.

²Miller, Karla L., et al. "Diffusion tractography of post-mortem human brains: optimization and comparison of spin echo and steady-state free precession techniques." Neuroimage 59.3 (2012): 2284-2297.

³Foxley, Sean, et al. "Improving diffusion-weighted imaging of post-mortem human brains: SSFP at 7 T." Neuroimage 102 (2014): 579-589.

⁴Foxley, Sean, et al. “Correcting for B1 inhomogeneities in post-mortem DWSSFP human brain data at 7T using multiple flip-angles.” 22nd Proc. Intl. Soc. Mag. Reson. Med., 2014: #4438.

⁵Jbabdi, Saad, et al. “Modelling multiple flip-angle diffusion weighted SSFP data.” 23rd Proc. Intl. Soc. Mag. Reson. Med., 2015: #2928.

⁶McNab, Jennifer A., and Karla L. Miller. "Sensitivity of diffusion weighted steady state free precession to anisotropic diffusion." Magnetic Resonance in Medicine 60.2 (2008): 405-413.

⁷Jbabdi, Saad, et al. "Model‐based analysis of multishell diffusion MR data for tractography: How to get over fitting problems." Magnetic Resonance in Medicine 68.6 (2012): 1846-1855.

⁸Pallebage-Gamarallage, Menuka, et al. "Dissecting the pathobiology of altered MRI signal in amyotrophic lateral sclerosis: A post mortem whole brain sampling strategy for the integration of ultra-high-field MRI and quantitative neuropathology." BMC neuroscience 19.1 (2018): 11.

⁹Hernandez-Fernandez Moises, et al. Using GPUs to accelerate computational diffusion MRI: From microstructure estimation to tractography and connectomes. BioRxiv. 2018

¹⁰McNab, Jennifer A., and Karla L. Miller. "Steady‐state diffusion‐weighted imaging: theory, acquisition and analysis." NMR in Biomedicine 23.7 (2010): 781-793.

¹¹Jenkinson, Mark, et al. "Fsl." Neuroimage 62.2 (2012): 782-790.

Figures

Figure 1: The diffusion-weighting scheme for dwSSFP (a) consists of a single diffusion gradient. The short TR (typically TR<T2) prevents the transverse magnetisation from decaying before the next RF pulse. Over many TRs, the signal (

$M_{diff}$ ) will be decomposed into multiple coherence pathways, which will be sensitised to the diffusion gradient during transverse periods. (b) displays a spin-echo, a short- and a long-diffusion time stimulated-echo pathway. Echoes are formed when the diffusion-gradient rephases the transverse magnetisation. Relative contribution of each pathway depends on the input flip-angle (c), leading to flip-angle dependent diffusion and signal weighting. (a) and (b) adapted from¹⁰.

Figure 2: How

$\Delta_{eff}$ varies as a function of flip-angle. Lower flip-angles correspond to an increased

$\Delta_{eff}$ , with a decreased rate of change at higher flip-angles. Excellent agreement is shown comparing the analytical form of

$\Delta_{eff}$ (Eq. [2]) vs a summed solution over echo number,

$n$ . The summed solution was computed from Eq. [1] over 1000 TRs, substituting

$A_{ADC}^{n+1}\rightarrow\Delta$ . Simulation generated defining TR = 28ms and T₁ = 600ms.

Figure 3: How the ADC varies as a function of flip-angle (left) when considering a Gamma distribution of diffusion coefficients (right), with

$\mu=3.5\cdot 10^{-4}$ mm²/s and

$\sigma=3\cdot 10^{-4}$ mm²/s. The measured ADC decreases with reduced flip-angle, with a decreased rate of change at higher flip-angles. Excellent agreement is observed when comparing the analytical form of ADC (Eq. [3], substituting Eq. [4] for

$S$ ) to the ADCs estimated integrating Eq. [1] over a Gamma PDF with 10⁴ samples. Simulation generated defining

$q$ =300/cm, TR = 28ms, T₁ = 600ms and T₂= 25ms.

How

$L_{1,2,3}$ varies as a function of B₁ in grey and white matter for the 24^o, 94^o and the reconstructed

$\Delta_{eff}$ =200ms datasets (details of processing described in the Method section). Grey/white matter segmentation was performed with FAST¹¹ and white/grey matter masks were defined over the entire brain, covering a range of B₁ values. Here we observe that the

$\Delta_{eff}$ =200ms (corresponding to

$\approx$ 26^o flip-angle in dwSSFP) shows good agreement with the 24^o dataset at high B₁, whilst maintaining a flat

$L_{1,2,3}$ at reduced B₁ values, highlighting that the procesing has removed the influence of B₁ on

$L_{1,2,3}$

$L_{1,3}$ and fractional anisotropy (FA) maps of the postmortem data at 24^o, 94^o and the reconstructed

$\Delta_{eff}$ =200ms (details of processing described in the Method section). The

$\Delta_{eff}$ =200ms dataset (corresponding to

$\approx$ 26^o flip-angle in dwSSFP), is able to preserve the high-SNR of the low flip-angle dataset, whist removing the B₁ dependency, yielding high-SNR diffusivity estimates at a single effective diffusion time.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

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