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

Acknowledgements

References

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