Background

While White Matter models have the Standard Model of Diffusion¹, Gray Matter (GM) models lack such a unifying framework. Two of the most extensively studied GM models, NEXI², proposed in parallel as SMEX³, and SANDI⁴, correspond to extreme cases where soma and neurite membranes are either considered fully and partially permeable, respectively (NEXI), or impermeable (SANDI). In this work, we propose a means of quantifying exchange in simple structures, such as spheres or cylinders. This enables us to define a new three-compartment model, the Generalized Exchange Model (GEM), with soma and neurites, each exchanging with the extracellular space (ECS). In such a framework, most GM diffusion models can be understood as special cases of GEM. SANDIX³ corresponds to the case where t_ex,s is infinite, SANDI to the case where both exchange times are infinite, NEXI to the case where the fraction f_s is 0, with two implementations: NEXI_NPA in the case of the Narrow Pulse Approximation² (δ →0) and NEXI_AWP for Actual Wide Pulses³ . We aimed to test the performance of GEM in the human cortex in vivo on a clinical MRI scanner, using an acquisition protocol originally designed for the NEXI model⁵. We then compared the GEM parameter estimates to the other GM models and to literature values.

Methods

Theory: GEM parameters and equations are laid out in Fig.1. In the first order, the signal from a compartment exchanging with other compartments follows the equation: $$\frac{dS}{dt}=\frac{dS_{imper}}{dt}\frac{S}{S_{imper}}-rS+\sum_{i}{\alpha_i}r_{ext,i}S_{ext,i}$$ where S(t) and S_imper(t) correspond to the signal in the permeable and impermeable cases, r to the mean life rate of a particle in the compartment and $$$\sum_{i}{\alpha_i}r_{ext,i}S_{ext,i}$$$ to the external contributions of the other compartments. Obtaining the signal in the presence of exchange is a matter of expressing the signal S_imper(t). In the case of a sphere, we can derive it from the spherical mean displacement formula⁶ (Fig.2). Simulated data: Numerical PGSE diffusion signal was simulated inside a single impermeable sphere of radius R_s=10µm and bulk diffusivity D_s=2.5µm²/ms using the MC/DC simulator⁷ with 50’000 walkers and parameters b=3ms/µm², Δ=50ms, δ=16.5ms . This result was used to validate the analytical expression (Fig.2). Clinical data: Six healthy volunteers (3M, 26.5±1.1 years old) were scanned on a 3T system (MAGNETOM Prisma, Siemens Healthcare, Erlangen, Germany) and rescanned on a different day. Acquisition: An MPRAGE was acquired for anatomical reference (1-mm isotropic resolution). Diffusion-weighted images were acquired using a PGSE-EPI research sequence with the following parameters: b-values=1.00, 2.00, 3.20, 4.44 and 5.00ms/µm², 20 directions per shell, diffusion times Δ=28.3, 36.0, 45.0, 55.0 and 65.0ms, δ=16.5ms, 4 b=0 images, spatial resolution=2x2x2 mm³, total scan time: 35min. Processing: Multi-shell multi-diffusion time data were preprocessed jointly using a standard pipeline⁸. The GM region of interests (ROIs) from the Desikan-Killiany-Tourville atlas⁹ were segmented on the MPRAGE image using FastSurfer¹⁰ and projected onto the diffusion native space using linear registration¹¹. Parametric maps of NEXI_NPA, NEXI_AWP, SANDI, SANDIX and GEM, all corrected for Rician Mean¹², were estimated from the powder-average signals using non-linear least-squares. The spatial distribution of NEXI features was also examined using inflated brain surfaces obtained using Connectome Workbench¹³.

Results and discussion

The GEM signal calculation can be accelerated by using the analytical solution in the form of matrix exponential between times δ and Δ. For the other intervals, the ODE solver solve_ivp from scipy.integrate¹⁴ was used. Once applied on the clinical data and despite the potential degeneracy of the model, the ROI medians of all subjects were well centered for each parameter (Fig.3). Compared to other GM models, GEM estimates are very close to those of SANDIX, while giving a plausible t_ex,s, of the same order of magnitude as t_ex,n (Fig.4). The GEM intra-cellular fractions seem very low, but neurite diffusivities are more biologically plausible^15,16 than those obtained with SANDI or NEXI_NPA. The soma radii appear to be more moderate than those obtained with SANDI, even though they remain high compared to previous studies and histology^3,17. The cortical surface maps finally show that this model is sensitive to certain recognizable patterns (Fig.5) such as the longer neurite exchange time on the sensorimotor cortex or on the temporal lobe, matching higher myelination and previously reported in humans using NEXI_NPA¹².

Conclusion

GEM was derived and successfully estimated in the human cortex on data acquired on a clinical scanner. Estimated parameters are similar to those obtained with other GM models and literature values. The optimal acquisition for estimating this model has yet to be established. One possible approach would be to vary δ. Future work will also focus on estimating its accuracy and precision using synthetic data.

Acknowledgements

The authors thank Thorsten Feiweier for providing access to the PGSE-EPI research sequence. This work was supported by SNSF Eccellenza grant PCEFP2_194260 and ERC Starting Grant ‘FIREPATH’, SERI no. MB22.00032.

References

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