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Insights for in vivo MR axon radius mapping from simulations based on large-scale histology

Laurin Mordhorst¹, Luke J. Edwards², Maria Morozova^2,3, Carsten Jäger^2,3, Henriette Rusch³, Nikolaus Weiskopf^2,4,5, Markus Morawski³, and Siawoosh Mohammadi^1,2,6
¹Institute of Systems Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg, Germany, ²Department of Neurophysics, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany, ³Paul Flechsig Institute - Center of Neuropathology and Brain Research, Medical Faculty University of Leipzig, Leipzig, Germany, ⁴Felix Bloch Institute for Solid State Physics, Faculty of Physics and Earth Sciences, Leipzig University, Leipzig, Germany, ⁵Wellcome Centre for Human Neuroimaging, Institute of Neurology, University College London, London, United Kingdom, ⁶Max Planck Research Group MR Physics, Max Planck Institute for Human Development, Berlin, Germany

Synopsis

Keywords: Microstructure, White Matter, Histology, Axon Radius Distribution, Intra-axonal signal, Rician bias

Motivation: To understand deviations between axon radii from in vivo MR experiments and histology.

Goal(s): To assess the sensitivity of the intra-axonal MR signal to the axon radius; to assess the impact of confounders (extra-axonal signal and Rician noise bias); to discuss deviations between in vivo MR experiments and simulations.

Approach: We simulated MR signals for axon radii distributions from large-scale histology with and without confounders; we compared radii fitted to in vivo MR experiments and simulations.

Results: Large MR radii are inherently underestimated; confounders are expected to further bias MR radii and can potentially explain deviations between in vivo MR experiments and simulations.

Impact: We reveal an inherent bias in the MR axon radius model for in vivo measurements. Furthermore, we identified two main confounders that can significantly narrow the dynamic range of MR radius measurements and reduce sensitivity to small-axon radii regions.

Introduction

Diffusion-weighted MRI (dMRI) can estimate the tail-weighted effective MR axon radius^1,2
$$r_{\text{eff}} = \sqrt[4]{\frac{\langle r^6 \rangle}{\langle r^2 \rangle}}.~\quad~(\text{Eq.1})$$
However, current validation^2-4 of $$$r_{\text{eff}}$$$ in the human brian relies on small-scale histology data^5-7, which insufficiently samples the tail of the axon radii⁸. Furthermore, MR signal simulations for validation² or MR protocol optimization⁹ neglect the axon radii distribution and use single axon approximations instead. Using MR simulations based on axon radii distributions of the human corpus callosum (CC) from large-scale histology¹⁰, we assessed the sensitivity of the intra-axonal MR signal to $$$r_{\text{eff}}$$$ and characterized the impact of potential confounders, specifically, residual extra-axonal signal and Rician noise bias. Furthermore, we assessed and discussed deviations of MR simulations to in vivo MR experiments.

Theory

The magnitude diffusion-weighted MR signal normalized to S(b=0) in white matter can be approximated as^11-13 $$S(h(r),\vec{g},b)~=~f_{\text{a}}\underbrace{\frac{\langle~r^2e^{-bD_{\text{a}}^{\perp}(r)}\int~P(\vec{n})e^{-b(D_{\text{a}}^{\parallel}-D_{\text{a}}^{\perp}(r))(\vec{g}^T\vec{n})} d \vec{n} \rangle}{\langle~r^2\rangle}}_{S_{\text{a}}(h(r),\vec{g},b)}+(1-f_{\text{a}})\cdot\underbrace{e^{-bD_{\text{e}}^{\perp}}\int~P(\vec{n})e^{-b(D_{\text{e}}^{\parallel}-D_{\text{e}}^{\perp})(\vec{g}^T\vec{n})}d \vec{n}}_{S_{\text{e}}(\vec{g},b)}~\quad~(\text{Eq.2})$$ with

$$$\vec{g}$$$: diffusion gradient direction
$$$\vec{n}$$$: main fiber direction
$$$\text{P}(\vec{n})$$$: fiber orientation distribution function
$$$f_{\text{a}}$$$: $$$\text{T}_2$$$-weighted axonal water fraction
$$$\text{S}_{\text{e}}(\vec{g},b)$$$: extra-axonal signal
$$$\text{S}_{\text{a}}(h(r),\vec{g},b)$$$: intra-axonal signal
$$$\text{D}_{\text{e}}^{\perp}$$$ and $$$\text{D}_{\text{e}}^{\parallel}$$$: extra-axonal perpendicular and parallel diffusivity
$$$\text{D}_{\text{a}}^{\perp}(r)$$$ and $$$\text{D}_{\text{a}}^{\parallel}$$$: intra-axonal perpendicular and parallel diffusivity
$$$r$$$: axon radius
h(r): axon radii distribution
b: b-value

At sufficiently high diffusion weighting (estimated as $$$b \geq 6 \text{ms}/\text{μm}^2$$$ for in vivo data^2,14), $$$\text{S}_{\text{e}}(\vec{g},b)$$$ becomes negligible, and so $$$r_{\text{eff}}$$$ can be fit to the spherically-averaged signal of $$$\text{S}_{\text{a}}(h(r),\vec{g},b)$$$^2,15.

Materials and Methods

Histology Image Acquisition: We acquired large-scale light microscopy images for several ROIs of three CC specimens. See Table 1A for details.

Histology Image Processing: We derived empirical h(r) and $$$r_{\text{eff}}$$$ per ROI as in¹⁰.

MR Image Acquisition: We acquired in vivo, dMRI images of five healthy adults following the protocol in⁹ (see Table 1B).

MR Image Processing: We preprocessed dMRI images^16-19 and computed neurite dispersion (κ)²⁰. For $$$b \geq 6 \text{ms}/\text{μm}^2$$$, we fitted spherically-averaged signals using a Rician ML estimator²¹ with estimated noise level $$$\hat{𝜎}$$$ obtained^19,22-23 from raw dMR images.

MR Simulations: For histology-based h(r) per ROI, we simulated in vivo MR signals for $$$b = [6, 30]~\text{ms}/\text{μm}^2$$$ using Eq.2 (see Table 1B-C for parameters) to study:

the sensitivity of $$$\text{S}_{\text{a}}(h(r),\vec{g},b)$$$ to $$$r_{\text{eff}}$$$: Eq.2 with $$$\text{S}_{\text{e}}(\vec{g},b) = 0$$$ and SNR = inf; we repeated the analysis for single axons of radius $$$r \in [0, 5]~\text{μm}$$$ instead of histology-based h(r),
the impact of residual $$$\text{S}_{\text{e}}(\vec{g},b)$$$: Eq.2 with $$$\text{D}_{\text{e,sim}}^{\parallel} \in [\text{D}_{\text{e}}^{\parallel} \pm$$$ 25%], $$$\text{D}_{\text{e,sim}}^{\perp} \in [\text{D}_{\text{e}}^{\perp} \pm$$$ 25%] and SNR = inf,
the impact of imperfect Rician bias correction: Eq.2 with $$$\text{S}_{\text{e}}(\vec{g},b) = 0$$$, added Rician noise with noise level σ, and biased estimated Rician noise level $$$\hat{\sigma} \in [\sigma \pm$$$ 25%] for spherical averaging,
differences between in vivo MR experiments and simulations: we combined major confounders identified in (2)-(3), i.e., $$$\text{D}_{\text{e,sim}}^{\perp}$$$ and $$$\hat{\sigma}$$$.

To compensate for tissue shrinkage in histology, we repeated simulations after rescaling radii (see Table 1C) in Eq.1 and Eq.2 by η, i.e., $$r’ = r \cdot η~\quad~(\text{Eq.3})$$.

Results

Figure 1 shows that in vivo $$$r_{\text{eff}}$$$ of histology-based axon radii distributions differ significantly from single-axon $$$r_{\text{eff}}$$$, revealing a bias for larger $$$r_{\text{eff}}$$$.When including extra-axonal signal in the simulations (see Figure 2), small $$$r_{\text{eff}}$$$ are overestimated mainly as a function of $$$\text{D}_{\text{e}}^{\perp}$$$. Imperfect Rician bias correction (see Figure 3) can cause either under- or overestimation of $$$r_{\text{eff}}$$$, which is inherited from the bias in noise level estimation ($$$\hat{\sigma}$$$). When $$$\hat{\sigma}$$$ is underestimated, $$$r_{\text{eff}}$$$ fitting is likely to fail for small $$$r_{\text{eff}}$$$. Compared to $$$r_{\text{eff}}$$$ from in vivo MR experiments (see Figure 4), simulated $$$r_{\text{eff}}$$$ distributions are wider and have smaller modes; their appearance depends mostly on $$$\hat{\sigma}$$$. In the presence of strong extra-axonal signal (low $$$\text{D}_{\text{e}}^{\perp}$$$) and overestimated $$$\hat{\sigma}$$$, $$$r_{\text{eff}}$$$ distributions of MR experiments and simulations resemble each other, implying that the in vivo maps may be biased.

Discussion

We demonstrated through MR simulations that $$$r_{\text{eff}}$$$ of realistic human axon radii distributions are biased for large $$$r_{\text{eff}}$$$. Combined with additional confounders, specifically, residual extra-axonal signal and Rician noise bias, the narrow $$$r_{\text{eff}}$$$ range of in vivo MR experiments can be reproduced. However, there remain several confounders not discussed here, e.g., glial cells²⁴ or along-axon radius variation^25-26. Even though we corrected for tissue shrinkage with values outside (from no shrinkage to radii twice as large) the reported range^5,27-29, our results show that confounders can make the influence of tissue shrinkage negligible.

Conclusion

To improve sensitivity to $$$r_{\text{eff}}$$$, MR protocols should be optimized for realistic axon radii distributions instead of single axon approximations and the need for Rician bias correction should be eliminated by, e.g., getting Gaussian-distributed real data from complex MR images³⁰.

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 616905.This work was supported by the German Research Foundation (DFG Priority Program 2041 "Computational Connectomics”, [MO 2397/5-1; MO 2397/5-2], by the Emmy Noether Stipend: MO 2397/4-1; MO 2397/4-2) and by the BMBF (01EW1711A and B) in the framework of ERA-NET NEURON and the Forschungszentrums Medizintechnik Hamburg (fmthh; grant 01fmthh2017).

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Figures

Table 1: (A) Histology: case information, tissue processing and acquisition details. (B) MRI subject and acquisition details. (C) Parameters used for MR simulations.

Fig. 1: Sensitivity of the intra-axonal signal to $$$r_{\text{eff}}$$$. The plot compares reff between histology and simulated intra-axonal MR signal $$$\text{S}_{\text{a}}(h(r),\vec{g},b)$$$ (Eq.2). MR simulations are based on empirical axon radii distributions from histology (reds) and single axons with $$$r = r_{\text{eff}}$$$ (green). Markers show median and 95 % interval over 1000 random fiber directions $$$\vec{n}$$$. $$$\eta$$$ is a radius scaling factor (Eq.3). The bias at small $$$r_{\text{eff}}$$$ is due to the finite number of gradient directions³².

Fig. 2: Impact of residual extra-axonal signal on r_eff estimation. Each plot compares r_eff between histology and MR simulation of Eq.2 with (blues) and without (reds) extra-axonal signal S_e($$$\vec{g}$$$,b) for varying D_e,sim^⟂ (rows) and D_e,sim^∥ (columns). D_e,sim^⟂ and D_e,sim^∥ cover literature values D_e^⟂ and D_e^∥ $$$\pm$$$ 25 % (see Table 1C). Markers show median and 95 % interval over 1000 random fiber directions n. η is a radius scaling factor used to compensate for tissue shrinkage.

Fig. 3: Impact of imperfect Rician bias correction on r_eff estimation. Each plot compares r_eff between histology and MR simulation of Eq.2 with (blues) and without (reds) Rician noise (SNR = 30). The estimated noise level $$$\hat{σ}$$$ (columns) was used for spherical averaging with Rician ML. Markers show median and 95 % interval over 1000 noise realizations with random fiber directions $$$\vec{n}$$$. The gray area shows the fraction of successful fits of r_eff. η is a radius scaling factor (Eq.3).

Fig. 4: Comparison of r_eff between in vivo MR experiments (magenta) and simulations (blues). The comparison is shown for varying combinations of D_e,sim^⟂ (rows) and estimated noise level $$$\hat{σ}$$$ (columns) used to fit spherically-averaged signals with Rician ML. For each histogram, we pooled over all subjects (MR experiments) and histology ROIs (MR simulations) as well as 1000 Rician noise (SNR = 30) realizations with random fiber directions $$$\vec{n}$$$. η is a radius scaling factor (Eq.3).

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)

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DOI: https://doi.org/10.58530/2024/3663