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Quantifying Cell Size and Membrane Permeability with Microstructure Fingerprinting
Khoi Minh Huynh1,2, Ye Wu2, and Pew-Thian Yap1,2
1Biomedical Engineering, UNC Chapel Hill, Chapel Hill, NC, United States, 2Department of Radiology and Biomedical Research Imaging Center (BRIC), UNC Chapel Hill, Chapel Hill, NC, United States

Synopsis

In diffusion MRI, biophysical models offer a non-invasive means of probing the tissue micro-architecture of the human brain. However, most models rely on closed-form formulas derived with simplifying assumptions such as short gradi- ent pulse, Gaussian phase distribution, and the absence of compartmental ex- change. We present a fast microstructure fingerprinting framework for accurate estimation of axon/soma radii and membrane permeability without relying on these assumptions.

Introduction

Closed-form microstructure models typically rely on simplifying assumptions such as short gradient pulse, Gaussian phase distribution, and the absence of inter-compartmental exchange. Deviations from these assumptions diminish the reliability and interpretability of these models [1]. Microstructure fingerprinting (MF) [2,3] uses simulated signals from accurate and physically interpretable geometries for accurate tissue quantification. However, existing methods [2,3] typically require an inordinate amount of time to generate a sufficiently large dictionary of fingerprints. In this work, we introduce a computationally feasible MF method to (i) quantify cell size and membrane permeability, and (ii) elimi- nate confounding factors, such as extra-axonal water and fiber dispersion, in the estimation of axon/soma radii.

Methods

Microstructure fingerprinting
The normalized diffusion signal $$$\frac{S}{S(0)}$$$ can be seen as a combination of signal fingerprints $$$\{\frac{S_i}{S_i(0)}\}_{i}$$$ from multiple microenvironments:
$$\frac{S}{S(0)}=\sum_{i} v[i] \frac{S_i}{S_i(0)}.$$
MF determines the contribution of each fingerprint $$$v[i]$$$ to the voxel signal. We generate a dictionary of realistic fingerprints, covering four diffusion patterns:
  1. Intra-axonal diffusion using cylinders with radii $$$r$$$’s from 0 to 6 μm and permeabilities $$$\kappa$$$’s from 0 (impermeable) to 50 × 10−6 μm μs−1 [4,5,3,6,7,8,9].
  2. Extra-axonal diffusion using a tensor model with $$$\frac{\lambda_{\parallel}}{\lambda_{\bot}} < \tau ^2$$$ as in [10].
  3. Intra-soma diffusion using impermeable spheres with radii from 0 to 20 μm [11].
  4. Free-water diffusion using a tensor model with $$$\lambda_{\parallel}=\lambda_\perp \gt 1.0 \times 10^{-3}$$$ mm2 s−1 [12].
Unlike other methods using computationally expensive Monte Carlo simulations [3,2], we generate the fingerprints by solving the Bloch-Torrey partial differential equation (BT-PDE) using SpinDoctor [13]. The reason is two-fold:
  1. SpinDoctor numerically solves the BT-PDE and is typically 50 times faster than Monte Carlo simulations.
  2. SpinDoctor has been released as a MATLAB toolbox and is widely accessible, allowing the user to customize shape, permeability, pulse sequence, pulse duration, etc. It does not require GPU and can be parallelized.
We solve for the volume fractions $$$\{v[i]\}_{i}$$$ using the spherical mean spectrumimaging (SMSI) approach described in [10] to remove confounding factors such as extra-cellular water and fiber dispersion to improve estimation of axon/soma radii. Examples of SpinDoctor configurations are shown in Fig. 1.

Tackling estimation bias
The average axon radius for each voxel can be calculated by averaging the radii of the respective fingerprints, weighted by the volume fractions. However, since the volume fractions $$$v$$$ are estimated from the normalized signal model [14], they are actually ‘signal fractions’. Weighted averaging using signal fractions results in a radius estimate called axon radius index [8], which is biased towards axons with larger B0 signals [15]. We correct for this bias by deriving a weight accounting for the B0 signal of each microenvironment:
$$w[i] =\frac{\frac{v[i]}{S_i(0)}}{\sum_{i}\frac{v[i]}{S _i(0)}}.$$

Results

Quantifying volume fraction
We applied our method, MF-SMSI, to synthetic data with known intra-axonal (IA), extra-cellular (EC), intra-soma (IS), and free-water (FW) volume fractions as in [12]. Fig. 2 indicates that our method estimates accurately the volume fraction of each compartment.

Validating unbiased radii and permeability
To investigate the efficacy of our bias correction procedure on the quantification of radius and permeability, we performed three experiments:
  1. Axonal radius – We simulated signals of two axons with radii 2 and 4 μm and equal volume fractions. In Fig. 3 (left panel), the axon radius is biased toward the larger axon with higher B0 signal, similar to [14]. Our unbiased estimate of the radius is markedly closer to the ground truth.
  2. Membrane permeability – We simulated signals of two axons with perme- abilities 4 and 6 (×10−6 μmμs−1), common radius 3 μm, and equal volume fraction. In Fig. 3 (middle panel), the biased permeability estimate is biased toward the axon with lower permeability. Our method estimates the correct permeability value matching the ground truth.
  3. Soma radius – We simulated signals of two somas with radii 4 and 6 μm and equal volume fraction. From Fig. 3 (right panel), the bias toward the soma with higher radius (higher B0) is not severe. Our method yields results closer to the ground truth.
In-Vivo Data
We applied our method on data from the Human Connectome Project (HCP) [16]. Fig. 4 presents the indices for Subject ‘105923’. Indices are in great agreement with previous studies [12,10,11,17] with higher $$$v_{\text{IA}}$$$, MAI, and OCI in deep white matter, high $$$v_{\text{FW}}$$$ in CSF, and higher $$$v_{\text{IS}}$$$ in the cortical ribbon. Axonal radius and permeability are lower in the deep white matter, especially at the body of the corpus callosum and the forceps major, where axons are myelinated and densely packed [10]. Axonal radii are slightly lower in the anterior than the posterior part of the brain, similar to [3,4]. The unbiased estimates $$$r_{\text{axon}}^\dagger$$$ and $$$r_{\text{axon}}^\dagger$$$ reveal the correct patterns, avoiding the bias toward axons with higher radius and lower permeability, as can be observed in Fig. 3. The soma radii in the cortical ribbon have a mean value of 11 μm, similar to [11].

Conclusion

We have demonstrated that microstructure fingerprinting provides accurate and unbiased estimates of tissue properties. Our method takes five hours for a one- time precomputation of the fingerprint dictionary for the HCP data. This represents a significant speedup over methods relying on Monte Carlo simulations and therefore improves the practicality of microstructure fingerprinting.

Acknowledgements

This work was supported in part by NIH grant NS093842.

References

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Figures

Fig. 1. Examples of SpinDoctor geometric configurations. From left to right: a sphere representing a soma with impermeable membrane a cylinder representing an axon with impermeable membrane, a cylinder tightly wrapped with extra-cellular space, and a cylinder in a boxed extra-cellular space representing an axon with perme- able membrane allowing water exchange with extra-cellular space.

Fig. 2. Volume Fraction. Estimation of the volume fractions of various tissue com- partments. The dashed lines represent the ground-truth values. The solid lines indicate mean values of 1000 instances across noise levels. The shaded regions indicate standard deviations.

Fig. 3. Cell Size and Membrane Permeability. Estimation bias associated with axonal radius, membrane permeability, and soma radius. The dashed lines represent the ground truth. The solid lines indicate mean values of 1000 instances across noise levels. The shaded regions indicate standard deviations.

Fig. 4. MF-SMSI Indices. $$$v_{\text{IA}}$$$, $$$v_{\text{EC}}$$$, $$$v_{\text{FW}}$$$, $$$v_{\text{IS}}$$$, MAI, OCI, axonal radius $$$r_{\text{axon}}$$$ (μm), unbiased axonal radius $$$r_{\text{axon}}^\dagger$$$ (μm), axonaxonal permeability $$$\kappa_{\text{axon}}$$$ (10−6 μm μs−1), unbiased membrane permeability $$$\kappa_{\text{axon}}^\dagger$$$ (10−6 μm μs−1), soma radius $$$r_{\text{soma}}$$$ (μm), and unbiased soma radius $$$r_{\text{soma}}^\dagger$$$ (μm). Radius and permeability maps are overlaid on the T1w image.


Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)
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