Synopsis

White matter diffusion MRI enables non-invasive estimation of the axon diameter distribution, which is of interest as it modulates communication speed and delays between brain regions, and changes during development and pathology. Distribution mapping is challenging: current methods simplify it by either estimating the mean diameter, imposing parametric distributions, or combining non-parametric approaches with Double Diffusion Encoding. We present a non-parametric framework based on a PGSE protocol. Simulations show robust reconstruction of unimodal and bimodal distributions. The method is sensitive to population specific changes within bimodal distributions, as long as the underlying populations are separated by a minimum distance.

Introduction

Methods

Reconstruction framework:

We assume the dMRI signal can be expressed as follows:³ $$y=f_{ic}\sum_{i=1}^{N_r}\Psi(d_i)S_{cyl}(d_i,d_\parallel,\Omega^\star)+(1-f_{ic})S_{tensor}(d_\perp,d_\parallel,\Omega^\star)$$ where $f_{ic}$ is the Intra-axonal Compartment (IC) volume fraction, $\Psi(d)$ is the ADD, $S_{cyl}$ is the dMRI signal for a cylinder⁵ of diameter $d_i$, parallel diffusivity $d_\parallel$ and PGSE protocol settings $\Omega^\star$, and $S_{tensor}$ is the dMRI signal for a zeppelin⁶ with perpendicular diffusivity $d_\perp$. The signal is acquired in 60 directions. We focus on ex-vivo imaging and IC signal only, and therefore fixed $d_\parallel=0.6\times10m.s^{-1}$ and $f_{ic}=1$. Equation 1 can be expressed as a linear formulation $y=Ax$, where x can be recovered from the convex inverse problem:^3,7,8 $$argmin_{x\geq 0}||Ax-y||^2_2+\lambda||\Gamma x_{ic}||_2^2$$ Ill-posedness can be reduced by decreasing the mutual coherence of $A$ (using DDE⁹ for example), by using regularization $||\Gamma x||_2^2$),⁷ and/or by extracting the mean diameter index $a'$ instead of $\Psi$.^4,8 We propose to combine Laplacian regularization $\Gamma=L$,⁷ with a PGSE protocol designed to maximize sensitivity to a range of diameters according to our biophysical model.

Protocol design and ADD reconstructions:

The PGSE parameter space is $\Omega={G,\Delta,\delta}$. We performed a grid-search on $\Omega$¹⁰ in order to find a set of 20 shells $\Omega^\star$ that maximized the sensitivity $S'(d_i)$ to a set of 20 diameters: $S'(d_i)=||S_{cyl}(d_i+\epsilon,\Omega_i)-S_{cyl}(d_i-\epsilon,\Omega_i)||_2$. We bounded $\Omega$ to $G_{max}=550mT/m$, 2.0<$\delta\leq$70ms and $\Delta\geq\delta+6ms$.

We computed the IC signal for cylinders with diameters sampled from different gamma distributions from histological samples,⁴ and reconstructed $\Psi$ from 100 noisy realisations (Rician noise, SNR=30, $\lambda=0.2$).

We then created bimodal distributions containing two gaussians: a first fixed population $N_1$ ($\mu_1$=4.0$\mu$m) and a second moving population $N_2$ with increasing mean $\mu_2$. We extracted two estimated means $\hat{\mu_1}$ and $\hat{\mu_2}$ by fitting a gaussian mixture model to the estimated $\Psi$. A second experiment tested if the method is sensitive to a reduction of 50% in the amplitude of $N_2$.

Results and discussion

The 20 selected shells maximize the sensitivity to diameters between 4.5$\mu$m and 10.0$\mu$m (Figure 1). By maximizing sensitivity to a set of diameters, differences between columns of $A$ are increased, reducing its mutual coherence.³

Unimodal distributions

Unimodal distributions are reliably reconstructed for all simulations (Figure 2). The mean diameter index⁴ $a'$ is robustly estimated for values down to almost 1.0$\mu$m, although the expected lower bound has previously been shown to be around 2$\mu$m.¹¹ The smaller lower bound could be due to prediction or extrapolation properties of the regularization.

Resolving two axonal populations

As shown in Figure 3, $N_1$ and $N_2$ should be separated by at least 4.0$\mu$m for their respective means to be recovered ($\mu_1$ and $\mu_2$ are not within the 1st and 3rd quartiles of $\hat{\mu_1}$ and $\hat{\mu_2}$ for smaller separations). This minimal separation depends on amplitude and variance of the populations. When the two populations are separated enough, $\mu_1$ and $\mu_2$ are well estimated and population specific changes can be detected (Figure 4). Interestingly, preliminary results indicate that removing the last 10 shells of $\Omega^\star$ compromised bimodal reconstructions while preserving robustness for unimodal distributions, showing that $\Omega^\star_{10-20}$ provides information for bigger diameters.

Conclusion

Acknowledgements

References

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