Introduction

Non-vanishing diffusion kurtosis and time-dependent diffusion are both hallmarks of nongaussian diffusion in biological tissues. Here we use the 199 fast kurtosis acquisition protocol [1-3] to estimate radial, axial and mean kurtosis and diffusivity with reduced data requirements, allowing us to cover an extended range of diffusion times (6-350 ms ) using the same PFG based sequence. Combined with biophysical modelling, we use this data from spinal cord white matter to provide microstructurally specific information, i.e. axonal water fraction, intra-axonal diffusivity and extracellular diffusivity in radial and axial directions as functions of diffusion time, and compare with effective medium theory. The time dependence of axonal diffusivities help resolve an inherent ambiguity in diffusion modelling related to the relative magnitude of intra- and extra-cellular diffusivity.

Methods

Fixed porcine spinal cord was imaged with a stimulated echo diffusion imaging sequence (EPI) on an ultrahigh-field 16.4T Bruker Aeon Ascend magnet (700MHz) equipped with a micro5 probe capable of producing up to 3000mT/m in all directions. Six b=0 images were acquired, as well as six b-values from 500 to 3000 µm2/ms, with nine directions for the 199 DKI scheme each[1]. Other imaging parameters were: 4 averages, TE=16 ms, TR=7.5s, in-plane resolution 140 µm x140 µm. Gradient pulse width was kept constant at 1.15ms, and diffusion times ranged from 6 ms to 350 ms in 57 steps. The raw images were corrected for Gibbs ringing[4] and denoised[5] before the subsequent analysis. Axial, radial, and mean diffusion and kurtosis were estimated with axially symmetric DKI fitting[3], and analysed in seven white matter ROIs: A=dCST – dorsal corticospinal tract, B=FG – Fasiculus Gracilis, C=FC – Fasiculus Cuneatis, D=ReST – Reticulospinal tract, E=RST – Rubrospinal tract, F=STT – Spinothalamic tract, G=VST – Vestibulospinal tract. In these regions of highly aligned axons, biexponential modelling (WMTI[6]) is appropriate at least for sufficiently large diffusion times, and allows kurtosis and diffusion tensors to be expressed in terms of compartmental biophysical parameters: $$$D_{a}$$$ (intra-axonal diffusivity), $$$D_{e,||}$$$ (axial extra axonal diffusivity), $$$D_{e,\bot}$$$ (radial extra axonal diffusivity), and $$$f$$$ (axonal water fraction). These relationships can be analytically inverted in axially symmetric regions, resulting in $$f = (1 + 3/K_{\bot})^{-1}$$ $$D_{e,\bot} = D_{\bot}(1 + K_{\bot}/3)$$ $$D_{e,||} = D_{||}(1 - \eta\sqrt{K_{||}K_{\bot}}/3)$$ $$D_{a} = D_{||} (1 + \eta \sqrt{K_{||}/K_{\bot}})$$ where $$$\eta = \pm 1$$$. The unknown sign $$$\eta$$$ prevents an unambiguous solution for the diffusion coefficients[7, 8]. The long time behavior of the diffusivities has been predicted using effective medium theory[9, 10] $$ D_{a}(t) \sim (D_{a}(\infty) + \frac{c_1}{\sqrt{t}})$$ $$ D_{e,||}(t) \sim (D_{e,||}(\infty) + \frac{c_2}{\sqrt{t}})$$ $$ D_{\bot}(t)\approx (1-f) D_{e,\bot}(t) \sim (1-f)(D_{e,\bot}(\infty) + \frac{c_3}{t}\ln(t/t_c))$$

Results

Figure 1 shows the ROI-averaged diffusion (top) and kurtosis metrics (bottom). A pronounced, smoothly decaying time dependence was observed in axial and radial diffusivity, and remarkably, neither seem to have plateaued at 350 ms. Axial kurtosis displays two types of behaviours. In regions D,G a clear nonmonotonic behavior is seen, with a maximum around ~80 ms. Such nonmonotonic behavior is expected, as diffusion is approximately Gaussian at very short and very long diffusion times. (For confined diffusion, $$$K(t)\sim \sqrt{t}$$$ for $$$t\to 0$$$, $$$K(t)\propto 1-\textrm{constant}\cdot\exp(-\lambda_1 t)$$$ for $$$t\to \infty$$$. The remaining ROIs show only decreasing behavior, perhaps because of a smaller characteristic length scale. Radial kurtosis is noisier, but appears to increase over the observable time range. Finally, mean kurtosis increases sharply initially, plateauing above $$$t\sim 100$$$ ms. Figure 2 shows the two branches of intra-axonal diffusivity as function of time. The + branch shows the expected smooth decay of diffusivity for increasing diffusion times, whereas the - branch shows an increasing behavior as a function of diffusion time, although noisier. This indicates that the negative branch is unphysical, and it is therefore discarded in the following. Figure 3 shows the remaining WMTI parameters, f (a), $$$D_{e,||} $$$ (b) and $$$D_{e,\bot}$$$. Although noisy, the axonal water fraction is relatively stable for most ROIs at diffusion times above 100 ms. For lower diffusion times, there is a systematic decrease, which may be due to non-negligible intra-compartmental kurtosis. Extracellular axial diffusivity smoothly decays with time over the entire range, fulfilling $$$D_{e,||}< D_a$$$. Radial diffusivity also decays with time, but with a much lower rate at large diffusion times. Finally, figure 4 demonstrates fits to EMT diffusivity predictions with reasonable agreement. Fit parameters are available from the table.

Conclusions

We demonstrated strong time dependence of directional kurtosis and diffusion metrics over an extended range in fixed pig spinal cord. Biophysical modelling revealed intra-axonal diffusivity to typically exceed extracellular axial diffusivity, and effective medium theory provided reasonable predictions.

Acknowledgements

Danish Ministry of Science, Technology and Innovation’s University Investment Grant (MINDLab, Grant no. 0601-01354B), and Lundbeck Foundation R83-A7548 .

References

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