Validating the Standard Model of diffusion MRI in white matter with Numerical Substrates

The non-invasive estimation of intra- and extracellular microstructural parameters using biophysical models has been a major focus in brain microstructure imaging with MRI. The Standard Model (SM) of diffusion in white matter (WM) provides a unifying framework for various modelling approaches, representing axons as impermeable narrow cylinders embedded within a locally anisotropic extra-axonal space. However, the SM relies on simplifying assumptions that may not hold in realistic WM tissue, as they do not take into account axonal undulations, beading, the presence of glial cells, or membrane permeability. In this work, we investigate how SM-derived estimates behave when the model is applied to realistic numerical WM substrates generated by the CATERPillar tool. Specifically, we vary (i) axonal morphological features such as beading and undulations, (ii) axonal packing density, (iii) orientation dispersion, (iv) membrane permeability of axons and astrocytes separately, (v) myelin volume fraction, and (vi) diffusion time. In each part of the analysis, different noise levels are introduced. Overall, according to our results, the relative changes in SM estimates show that the intra-axonal volume fraction f increased with stronger beading, higher packing density, and greater myelin volume, and was strongly influenced by axonal and astrocytic permeability. The orientation dispersion index p2 was affected by undulation, but was substantially biased at low packing densities, with stronger beading and when astrocytes were impermeable. The effective intra-axonal diffusivity Da decreased with stronger beading and undulation and tended to be overestimated in most scenarios. The parallel extra-axonal diffusivity De|| was strongly influenced by axonal permeability, as well as packing density, dispersion, and undulation, and was the most noise-sensitive parameter, showing systematic overestimation at low SNR. Finally, the effective perpendicular extra-axonal diffusivity De{perp} was the most stable parameter relative to the effective ground truth across the tested conditions, while remaining sensitive to packing density, axonal permeability, myelin volume fraction, and undulation. These findings enable users to identify potential biases introduced by varying conditions and to adjust their interpretations accordingly.