Jelle Veraart^{1}, Els Fieremans^{1}, Umesh Rudrapatna^{2}, Derek K Jones^{2}, and Dmitry S Novikov^{1}

The use of term “neurites” implies that many biophysical models of diffusion in the white matter can be applied in the gray matter as well. However, the validity of representing dendrites as a collection of zero-radius impermeably sticks, a widely adopted representation of myelinated axons, has not been evaluated yet. By evaluating the diffusion-weighted signal decay as a function of the $$$b$$$-value up to $$$b=25000\,\mathrm{ms/\mu\,m^2}$$$ in the living human brain, we show that a more accurate representation of the diffusion in neuronal processes in the gray matter must account for a fast proton exchange between the intra- and extra-cellular compartments.

Introduction

The gray matter is mainly composed of cell bodies, glial cells, and synapses, which are embedded in an intricately organized network of cellular processes, that are projections of the cell body that can be either dendrites or myelinated as well as unmyelinated axons$$$^1$$$. A clear differentiation between dendrites and axons is sometimes lacking, and an overarching term "neurite" is then used to refer to either of the projections.

We observe an increasing use of the term neurite instead of
(myelinated) axon in the context of biophysical modeling of diffusion in the
brain, thereby implying the model's applicability in both the white and gray
matter$$$^{2,3}$$$. However, the overarching assumption behind compartmental
models of diffusion in neuronal tissue is that the intra-cellular compartment
is well represented by narrow* impermeable *channels ("sticks") in
which the radial diffusivity is negligible$$$^{2,4}$$$. The proton exchange between intra- and
extra-cellular spaces must be sufficiently slow to be considered impermeable on
the time scale of the experiment.

This assumption has been validated for myelinated axons on a
state-of-the art clinical scanner by studying exchange$$$^{5,6}$$$ or by retrieving a nontrivial power law scaling
of the diffusion-weigthed signal as function of the $$$b$$$-value$$$^{7,8}$$$. Here we validate the “stick” assumptions
in the cortical gray matter while exploiting the strong gradients of the
Siemens Connectom 3T scanner to boost the SNR.
We interpret observed deviations from the power law scaling in terms of
finite neurite radii and exchange times to provide a comprehensive picture of
the challenges of translating our white matter models to the gray matter.

Four volunteers were scanned on a Connectom scanner with $$$\Delta/\delta\,=\,30/13\,\mathrm{ms}$$$
and maximal gradient amplitude $$$G=289\,\mathrm{mT/m}$$$. Diffusion-weighting
was applied along 60 gradient directions for a spectrum of $$$b$$$-values up to
$$$25\,\mathrm{ms/\mu\,m^2}$$$.
Furthermore, $$$\mathrm{TR/TE}\,=3500/62\,\mathrm{ms}$$$ and
resolution
$$$3\,\times\,3\times\,3\,\mathrm{mm}^3$$$. Data was denoised^{9}, gibbs^{10}, eddy current^{11},
Rician bias corrected^{12}, and isotropically-averaged per
$$$b$$$-shell to cancel the orientation distribution function.

In many biophysical models, the intra-cellular compartment is represented by
an array of zero-radius, impermeable “sticks” inside which diffusion is locally
one-dimensional, i.e. radial intra-cellular diffusivity
$$$D_c^\perp\,\equiv\,0$$$^{2,3}.
The stick model
yields an intra-cellular signal decay $$$\tilde{S}$$$ that asymptotically scales according
to a power law with exponent
$$$\alpha=\,1/2$$$: $$ \tilde{S}(b)\,\simeq\,\beta\,b^{-\alpha
}\,+\,\gamma\quad\mathrm{for}\quad\,bD_c^\parallel\gg1,\quad\,[1]$$ with $$$\gamma$$$ a immobile water fraction^{ }and $$$\beta$$$ a scaling factor that depends on the intra-cellular signal fraction^{7,8}. This decay only holds in
the absence of any exponentially fast decaying extra-axonal signal. Therefore, we restrict our analysis to
$$$b>6\,\mathrm{ms/\mu\,m^2}$$$.
Sensitivity of MR to either finite neurite radius or a notable exchange
rate between different compartments would break the $$$b^{-1/2}$$$-scaling:

** 1. Finite neurite radius**: A finite $$$D_c^\perp > 0$$$
results in a truncated power law: $$\tilde{S}(b)\,\simeq\,\beta\,e^{-bD_c^\perp}b^{-1/2}\,+\,\gamma.\quad\,[2]$$

** 2. Exchange:** The isotropic averaging of a two-compartment

We evaluate the signal decay as function of the
$$$b$$$-value and study the impact of accounting for non-zero radii, exchange
rates, and/or immobile water fraction qualitatively. We also assess the goodness-of-fit in comparison to the
power law that reflects the ideal "stick" model using the corrected
Akaike Information Criterion^{14} (AICc) in all voxels of the cortical gray matter
that were segmented using FreeSurfer^{15}.

**Fig. 1** shows the signal decay, averaged over all cortical GM
voxels, as a function of $$$b$$$. The
non-linear scaling of the signal as a function of $$$1/\sqrt(b)$$$ indicates deviations from the "stick" model.

The exchange model ($$$\gamma \equiv 0$$$) fits the data
best in all subjects, qualitatively (**Fig. 2**) and statistically (**Fig 3**). The average residence times, $$$1/r$$$ vary from approximately 10-15ms or 20-30ms if we
assume $$$D_e^\perp\,=\,1\,\mathrm{\mu\,m^2/ms}$$$ or
$$$D_e^\perp\,=\,0.5\,\mathrm{\mu\,m^2/ms}$$$, respectively.

In **Fig. 4**, we show the average residence times, computed within the
different cortical areas, and map them on the three-dimensional rendering of a
single brain for anatomical reference. All subjects demonstrate spatial
variability of the residence times, but overall -considering the precision of
the measurement- they show good mutual correspondence.

By observing significant deviations from the power law scaling, we uncovered deviations from the stick model in the gray matter. Biophysical models that are building upon that assumption may need to be interpreted with care when applied in the gray matter.

Evaluation of models of exchange and
finite radii suggests that neurites are in notable exchange with another
compartment in the gray matter. The measured exchange might result from actual membrane
permeability. However the significantly lower residence times in comparison with previous studies^{16,17} might reflect *apparent exchange* due to protons temporarily residing in the multitude of dendritic spines^{18}. Monte-Carlo simulations and model
refinements will address this pending question.

1. H. Gray. Gray's Anatomy, 30th edition. Philadelphia: Lippincott Williams & Wilkins (1985).

2. S.N. Jespersen, C.D. Kroenke, L. Østergaard, J.J.H. Ackerman, D.A. Yablonskiy, Modeling dendrite density from magnetic resonance diffusion measurements, NeuroImage 34(4), 1473-1486 (2007).

3. H. Zhang, T. Schneider, C.A. Wheeler-Kingshott, and D.C. Alexander, NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain, Neuroimage, 61(4), 1000-1016 (2012).

4. C. D. Kroenke, J. J. Ackerman, and D. A. Yablonskiy, On the nature of the NAA diffusion attenuated MR signal in the central nervous system, Magnetic resonance in medicine 52, 1052–1059 (2004).

5. M. Nilsson, J. Latt, D. van Westen, S. Brockstedt, S. Lasic, F. Stahlberg, and D. Topgaard. Noninvasive mapping of water diffusional exchange in the human brain using filter-exchange imaging, Magnetic Resonance in Medicine 69,1573–1581 (2013).

6. M. Nilsson, J. Latt, E. Nordh, R. Wirestam, F. Stahlberg, and S. Brockstedt. On the effects of a varied diffusion time in vivo: is the diffusion in white matter restricted? Magnetic Resonance Imaging 27:176–187 (2009).

7. J. Veraart, E. Fieremans, D.S. Novikov , Universal power-law scaling of water diffusion in human brain defines what we see with MRI. arXiv:1609.09145 (2016).

8. E. T. McKinnon, J. H. Jensen, G. R. Glenn, and J. A. Helpern, “Dependence on b-value of the direction-averaged diffusion-weighted imaging signal in brain,” Magnetic resonance imaging 36, 121–127 (2017).

9. J. Veraart, D.S. Novikov, D. Christiaens, B. Ades-Aron, J. Sijbers, and E. Fieremans, Denoising of diffusion MRI using random matrix theory. NeuroImage, 142, 394-406 (2016).

10. E. Kellner, B. Dhital, V. G. Kiselev, and M. Reisert, “Gibbs-ringing artifact removal based on local subvoxel-shifts,” Magnetic resonance in medicine 76, 1574–1581 (2016).

11. J. L. Andersson and S. N. Sotiropoulos, “An integrated approach to correction for off-resonance effects and subject movement in diffu-sion MR imaging,” Neuroimage 125, 1063–1078 (2016).

12. C. G. Koay and P. J. Basser, “Analytically exact correction scheme for signal extraction from noisy magnitude MR signals,” Journal of Magnetic Resonance 179, 317–322 (2006).

13. J. Kärger, NMR self-diffusion studies in heterogeneous systems. Adv. Colloid. Interfac,23:129–148 (1985).

14. K. P. Burnham and D. R. Anderson, “Information and Likelihood Theory: A Basis for Model Selection and Inference,” in Model Selection and Multimodel Inference (Springer New York, 2002) pp. 49–97.

15. A.M. Dale, B. Fischl, M.I. Sereno, Cortical surface-based analysis. I. Segmentation and surface reconstruction. Neuroimage 9:179-194 (1999).

16. J.D. Quirk, G.L.Bretthorst, T.Q. Duong, A.Z. Snyder, C.S. Springer, J.J.H. Ackerman, and J.J. Neil, Equilibrium water exchange between the intra and extracellular spaces of mammalian brain. Magnetic Resonance in Medicine 50:493–499 (2003).

17. D.M. Yang, J.E. Huettner, G.L. Bretthorst, J.J. Neil, J.R. Garbow, and J.J.H. Ackerman, Intracellular water preexchange lifetime in neurons and astrocytes. Magnetic Resonance in Medicine, doi:10.1002/mrm.26781 (2017).

16. E.G. Gray, "Electron microscopy of synaptic contacts on dendrite spines of the cerebral cortex", Nature 183: 1592–1593. (1959).

For all four subjects, we show the isotropically-averaged diffusion-weighted signal decay as function of $$$1/\sqrt{b}$$$. The signal decays are very consistent across the subjects (3 male, 1 female and the age range is 22 to 45 years). In all subjects, the exchange model with $$$\gamma\equiv\,0$$$ provides the best fit to the data.

The curvature of signal decay as a function of
$$$1/\sqrt{b}$$$ is a characteristic
signature of evaluated models: stick (straight), finite radius (concave), and
exchange (convex) within the $$$b$$$-range that was explored within this
experiment. The exchange model here introduced and evaluated is the solution of an isotropically-averaged anisotropic Kärger model, which is asymptotically well approximated by Eq. [3]. The experimental data shows
the convex signal decay as predicted by the exact and approximated exchange model.

Fit quality of the listed models was evaluated for the decay
of diffion-weighted signal with $$$b\geq\,7\,\mathrm{ms/\mu\,m^2}$$$. AICc
analysis shows that an exchange model that excludes the immobile water
compartment $$$\gamma$$$ should be
preferred over the others. Red crosses indicates the models with an AICc that
is not significantly larger than the
smallest one, i.e. $$$\Delta \mathrm{AICc}<2$$$.

Within each cortical area, as segmented using FreeSurfer, we estimated the residence time.
The estimates are used to color the three-dimensional rendering of a single
brain for anatomical reference (one value per comical area). Overall, the four
subjects show good consistency in the observed spatially varying pattern.