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Diffusion time dependence of microstructural parameters in fixed spinal cord

Sune Nørhøj Jespersen, Jonas Lynge Olesen, Brian Hansen, Noam Shemesh

arXiv:1705.03630v2physics.med-phphysics.bio-ph

TL;DR

Biophysical modeling can relate diffusion MRI measurements to tissue microstructure, but diffusion and kurtosis tensors do not uniquely determine all standard-model parameters. This study models axonal dispersion and uses axisymmetric DKI across diffusion times; in fixed porcine spinal cord, the results support larger intra- than extra-axonal diffusivity and show substantial time dependence.

  • Problem

    Biophysical modeling is needed to infer specific microstructure, but diffusion and kurtosis tensors do not uniquely determine all standard-model parameters.

    Source: Introduction · p. 3

  • Method

    The study models axonal dispersion and combines a spinal-cord biophysical model with fast axisymmetric DKI to examine diffusion-time dependence.

    Source: Introduction · pp. 5, 9, 11

  • Results

    In fixed porcine spinal cord white matter, intra-axonal diffusivities were substantially larger than extra-axonal diffusivity, both varied substantially with diffusion time, and diffusivity time dependence reasonably matched effective medium theory.

    Source: Summary · p. 17, Conclusion · p. 23

  • Takeaways & Limitations

    The findings provide evidence relevant to microstructural modeling and interpretation of diffusion MRI in fixed spinal cord.

    Source: Conclusion · p. 23

    Estimates may be biased where gradients are perpendicular to the largest axons, such as in the reticulospinal tract.

    Source: Branch choice · p. 21

Abstract

from arXiv · show

Biophysical modelling of diffusion MRI is necessary to provide specific microstructural tissue properties. However, estimating model parameters from data with limited diffusion gradient strength, such as clinical scanners, has proven unreliable due to a shallow optimization landscape. On the other hand, estimation of diffusion kurtosis (DKI) parameters is more robust as the clinical acquisitions typically probe a regime in which the associated 4th order cumulant expansion is adequate; however, its parameters are not microstructurally specific a priori. Given an appropriate biophysical model, its parameters may be connected to DKI parameters, but it was previously shown that at the DKI level, it still does not provide sufficient information to uniquely determine all model parameters. Earlier work has shown that by neglecting axonal dispersion, this parameter degeneracy reduces to the question of whether intra-axonal diffusivity is larger than or smaller than extra-axonal diffusivity. Here we develop a model of diffusion in spinal cord white matter including axonal dispersion and demonstrate stable estimation of all model parameters from DKI. By employing the recently developed fast axisymmetric DKI, we use stimulated echo acquisition mode to collect data over an unprecedented diffusion time range with very narrow diffusion gradient pulses, enabling finely resolved measurements of diffusion time dependence of both net diffusion and kurtosis metrics, as well as model intra- and extra-axonal diffusivities, and axonal dispersion. Our results demonstrate substantial time dependence of all parameters except volume fractions, and the additional time dimension provides support for intra-axonal diffusivity to be larger than extra-axonal diffusivity in spinal cord white matter, although not unambiguously. We compare our findings to predictions from effective medium theory.

Introduction

The introduction motivates time-dependent measurements and explicit axonal-dispersion modeling as a route to microstructural inference from DKI, whose parameters can be degenerate. It outlines axisymmetric DKI, spinal-cord measurements across diffusion times, and comparison of fitted diffusivities with effective-medium predictions, noting that time dependence favors—but does not unambiguously establish—one parameter branch.

Introduction

  • Although DKI is broadly applicable at clinical diffusion weightings, its tensors cannot generally determine all standard-model parameters; in aligned fibers, two solutions differ in whether intra-axonal diffusivity exceeds extra-axonal diffusivity.Diffusion-time variation may provide the additional information needed to distinguish the branches; kurtosis time dependence has been less studied, in part because it requires more data and SNR than DTI.Source: Introduction · pp. 4–5
  • Measurements across a broad diffusion-time range provide more evidence for the branch with extra-axonal diffusivity higher than intra-axonal diffusivity, and the compartmental diffusivities are compared with effective-medium predictions.The evidence for the branch choice is based on expected physical behavior.Source: Introduction · p. 5
  • The spinal-cord model uses non-exchanging Gaussian axonal and extra-axonal compartments, represents axons as zero-radius sticks, and assumes an axially symmetric fiber orientation distribution.The Gaussian-compartment assumption may be limiting because net kurtosis can include intracompartmental contributions; the model also omits other tissue complexities, so its applicability requires further validation.Source: Introduction · pp. 5–7
  • Direct nonlinear fitting of the model signal expansion and a shell-by-shell coefficient-fitting procedure were unstable, motivating the use of fast axisymmetric DKI.Analytical expressions relate diffusion and kurtosis tensors to model parameters, supporting inference through the DKI representation.Source: Introduction · p. 7
  • Explicitly modeling dispersion with a one-parameter family of axially symmetric orientation distributions lets the system be solved for five independent microstructural parameters.The two orientation-distribution parameters share one degree of freedom.Source: Introduction · pp. 8–9

Effective medium theory

  • Effective medium theory predicts how diffusion approaches its longtime Gaussian limit according to the type and dimensionality of structural disorder.The described functional forms correspond to Poissonian disorder in one or two dimensions.Source: Introduction · p. 10
  • The effective-medium fit starts at a cutoff time chosen where normalized error rises steeply as the cutoff decreases; the cutoff itself has no obvious physical interpretation.Because the correlation time is unknown beforehand, fitting is monitored from successive cutoff times to the largest diffusion time.Source: Introduction · p. 11

Axisymmetric DKI

  • Axisymmetric DKI estimates all DKI parameters from nine diffusion directions at two nonzero b-values plus a b=0 image, assuming axially symmetric fiber ODFs (Hansen et al., 2016c).Its reduced parameter count (8 versus 22 for general DKI) makes the 199 protocol sufficient for nonlinear least-squares estimation.Source: Introduction · pp. 11–12

Acquisition

  • Measurements in fixed porcine spinal cord covered 57 diffusion times from 6 to 350 ms, with four averages per dataset and gradient pulse width δ = 1.15 ms.Six b=0 images and six shells from 0.5 to 3 ms/µm2 were acquired using nine directions per shell.Source: Introduction · p. 12

Results

Across 6–350 ms, diffusion and kurtosis measures and inferred compartmental parameters varied with time, whereas axonal water fraction was largely stable. Plus-branch intra-axonal estimates decreased smoothly, and EMT fit quality differed by compartment.

Results

  • Diffusion and kurtosis metrics and most model parameters varied substantially over 6–350 ms, while axonal water fraction was largely independent of diffusion time.Some net metrics had not fully plateaued at 350 ms.Source: DKI Metrics · pp. 13, 15
  • Net diffusivities decreased monotonically, while kurtosis was noisier and sometimes rose initially before decreasing at longer diffusion times.Radial kurtosis initially increased and possibly plateaued in some ROIs; axial and mean kurtosis decreased at sufficiently long times.Source: DKI Metrics · p. 13
  • Measured and computed signals agreed closely, supporting DKI adequacy; EMT fits matched intra-axonal diffusivity across ROIs but varied more for extra-axonal diffusivities.Histological axonal fractions aligned more with the plus branch, and fitted correlation lengths were on the cellular scale of a few micrometers.Source: DKI Metrics · p. 14, Comparison to EMT · p. 16
  • 30–40%: plus-branch intra-axonal diffusivity decreased smoothly by this amount, whereas minus-branch estimates were noisier and sometimes increased.The minus-branch increase was more pronounced when fibers were assumed fully aligned.Source: DKI Metrics · p. 14
  • Plus-branch extra-axonal axial diffusivity plateaued beyond about 200 ms, while minus-branch estimates continued to decay; radial extra-axonal diffusivity declined in both branches.At the longest times, minus-branch radial diffusivities reached 40%–50% of initial values, versus 34%–43% for the plus branch.Source: DKI Metrics · pp. 14–15
  • Axonal water fraction was largely time-independent; plus-branch dispersion fell from about 29° to 15°–19°, while minus-branch estimates hit the imposed maximum.Raising the minus-branch dispersion bound to 100 had minimal effect on the other estimated parameters.Source: DKI Metrics · p. 15

Discussion

Time-dependent measurements from fixed spinal cord support the plus branch, in which intra-axonal diffusivity is generally larger than extra-axonal diffusivity, though branch selection is not fully resolved. The analysis also finds time-dependent compartmental diffusivities and compares them with effective medium theory.

Discussion

  • The 57 diffusion times from 6 to 350 ms, acquired with very narrow pulses, provided densely sampled time-dependent diffusion and kurtosis measurements in fixed pig spinal cord.The measurements used multi-shell data and covered seven white-matter regions of interest.Source: Summary · pp. 16–17
  • Mapping axisymmetric DKI measurements to the spinal-cord biophysical model yielded two a priori equally acceptable solutions, which the added diffusion-time dimension allowed the authors to compare.The model-derived microstructural parameters included axonal volume fraction, dispersion, and intra- and extra-axonal diffusivities.Source: Summary · p. 17
  • The observed diffusivity and dispersion time dependence supports the plus branch, with intra-axonal diffusivity generally larger than extra-axonal diffusivity, but does not resolve branch choice unambiguously.The plus branch showed smoother intra-axonal diffusivity decay; the minus branch was noisier and increased in some ROIs, which the authors describe as unphysical.Source: Summary · p. 17, Branch choice · p. 18
  • The authors caution that branch preference may depend on tissue, including gray versus white matter, ex vivo versus in vivo conditions, and species.The study also cautions against generalizing fixed-tissue results to in vivo tissue because death, fixation, and temperature may influence microstructure.Source: Branch choice · pp. 19, 22
  • In the plus branch, extra-axonal axial diffusivity was generally less than half the intra-axonal value, and both varied strongly with diffusion time.The estimated intra-axonal water fraction was stable at longer times and ranged from 0.45 to 0.65, closer to rat spinal-cord histology than the minus-branch range of 0.2 to 0.4.Source: Branch choice · p. 20
  • Ignoring compartmental kurtosis may be a minor approximation beyond 20 ms, but could explain overestimated intra-axonal diffusivity at shorter times.The model assumes Gaussian diffusion within compartments despite the observed time-dependent diffusivities.Source: Branch choice · p. 21
  • The two-compartment model omits myelin water and exchange; the omission may affect estimated parameters, with myelin-water signal potentially more important at lower field strengths.Exchange may help explain some observed time dependence, mainly at the longest diffusion times.Source: Branch choice · p. 22
  • The inferred intra-axonal diffusivity time dependence was consistent with effective medium theory predictions for Poissonian disorder, while extra-axonal diffusivities showed more variable agreement.The authors identify the biological nature of the putative obstacles as unknown.Source: Branch choice · pp. 22–23

Conclusion

In fixed spinal cord, diffusion and kurtosis varied strongly across an extensive range of diffusion times. Biophysical modeling related this dependence to microstructural parameters, with intra-axonal diffusivities substantially larger than extra-axonal diffusivity and reasonable correspondence to effective medium theory.

Conclusion

  • Fixed porcine spinal cord white matter showed substantial time dependence in intra-axonal diffusivities, which were larger than extra-axonal diffusivity.The inferred time-dependent diffusivities showed reasonable correspondence to effective medium theory.Source: Conclusion · p. 23
  • Diffusion and kurtosis varied strongly across an extensive diffusion-time range in well-defined anatomical white-matter tracts.The measurements used a 16.4T microimaging system.Source: Conclusion · p. 23
  • Biophysical modeling related the measurements to axonal water fraction, dispersion, and intra- and extra-axonal diffusivities.Source: Conclusion · p. 23

Figures/Tables

The figures map the spinal-cord ROIs and show diffusion, kurtosis, and compartmental measures across diffusion time, alongside signal-fit and effective-medium-theory comparisons. Table 1 summarizes effective-medium-theory fitting parameters and cutoff times for the investigated white-matter ROIs.

Figure 1

  • Figure 1 identifies seven spinal-cord white-matter tract ROIs chosen to represent a range of axonal radii and densities.Source: Figure 1 · p. 24

Figure 2

  • Figure 2 shows diffusion-time dependence of net diffusion tensor metrics in the top row and kurtosis metrics in the bottom row across seven ROIs.Source: Figure 2 · p. 24

Figure 3

  • Figure 3 compares acquired and model-predicted signal, with each point representing a voxel from one diffusion-weighted image.Source: Figure 3 · p. 25

Figure 4

  • Figure 4 plots intra-axonal, extra-axonal axial, and extra-axonal radial diffusivities over diffusion time for both solution branches, with free-water diffusivity marked.Source: Figure 4 · p. 26

Figure 5

  • Figure 5 plots axonal volume fraction for both solutions, which should remain constant over diffusion time assuming no membrane exchange or T1 effects.Source: Figure 5 · p. 26

Figures 7–8 and Table 1

  • Figure 7 illustrates effective-medium-theory fitting across variable lower diffusion-time cutoffs, while Figure 8 compares plus-branch predictions across diffusivities and ROIs.Figure 7 includes cutoff selection based on the rate of change in normalized fitting error; Table 1 lists fitting parameters and cutoff times for the investigated ROIs.Source: Figure 7 · p. 27, Figures/Tables · p. 27, Figure 8 · p. 29
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