Noninvasive quantification of axon radii using diffusion MRI

1Axons facilitate connectivity between distant neurons. Along with myelination, the axon radius determines the conduction velocity, thereby shaping the timing of neuronal computations and communication. Using a model of action potential neurophysiology, it has been shown that the axon radius explains the largest proportion of variance in conduction velocity. Histological studies demonstrated that axon sizes vary widely within the human brain, ranging from 0.1 to more than 3, and across species. Moreover, axon radii have been shown to be altered in various disease processes. For example, direct axon counting in post-mortem tissue has suggested that smaller axons may be preferentially susceptible to axonal injury in multiple sclerosis due to inflammation. Electron microscopy has revealed a higher percentage of small-radius axons and a lower percentage of large-radius axons in several anatomically and functionally distinct segments of the corpus callosum in autistic subjects compared to healthy controls. From the animal literature, morphometric analysis of adult rat brains showed reduced axonal radii without axonal loss after chronic alcohol feeding. Such studies indicate that non-invasive metrics capable of reporting on features of the axon radius distribution could provide important neuroimaging biomarkers for basic research and clinical applications.

2A particularly relevant neuroimaging modality attuned to the microarchitecture of living brain tissue is diffusion-weighted MRI (dMRI). dMRI is sensitive to the thermal motion of water molecules and their interference with microscopic boundaries, such as imparted by cells and subcellular structures in the brain. Applications of dMRI specialize in revealing macroscopic brain connections and in the interpretation of contrast differences in diffusion-weighted images. However, reproducible and specific biomarkers for studying disease onset and progression non-invasively and quantitatively in the entire brain, in particular vis-a-vis axonal properties, would confer clear advantages. Several studies have used various methods to report on axon radius parameters; still, despite many attempts, axon radius mapping using dMRI remains highly contested. Discrepancies between histology and dMRI-derived axon radii uncovered various confounding factors, for example orientation dispersion, time-dependent extra-axonal diffusion overshadowing the intra-axonal signal at low diffusion weighting, weak signal attenuation for typically very narrow axons, especially in the realistic experimental regime of long diffusion gradient duration, and/or putative shrinkage during tissue preparation.

3Recent advances in biophysical modeling and hardware prompted a revival of MR axon radius mapping. First, several of the most crucial confounding factors have been removed using powder-averaging concepts. Averaging diffusion-weighted signals that are isotropically distributed on a sphere with constant diffusion-weighting strength has been shown to factor out the orientation dispersion, thereby eliminating one of the most important confounding factors in axon radius mapping. Second, gradient systems capable of producing relatively strong gradient pulses have been introduced in human scanners. Third, it has been shown that dMRI can be made specific to a particular water population restricted into long, yet micrometer-thin cylindrical objects by imparting high diffusion-weighting regimes. Often, an axon is too narrow to yield a measurable diffusion-weighted MR signal decay, hence the popular use of ‘sticks’ when referring to axons (and possibly glial cell processes) within the context of biophysical modeling of white matter using dMRI.

4The intuition behind promoting specificity to intra-axonal water comes from Callaghan’s model of diffusion inside infinitely narrow one-dimensional randomly-oriented cylinders, as applied to intra-neurite diffusion by. The spatial Fourier transform of the diffusion propagator (with respect to the diffusion wave vector ) for a single stick as measured with MRI, averaged over the orientations of the sticks, yields the asymptotic scale-invariant power law as a function of the diffusion weighting parameter, with the scaling exponent. Evidently, this power law scaling should be only approximate, for, where is the cylinder radius. Its observation in the range is a manifestation of the insensitivity of dMRI to the transverse axonal dimensions on clinical scanners. However, for sufficiently strong diffusion weighting, the power law scaling eventually breaks down, and the dMRI measurement becomes sensitive to the axonal diameter.

5Technically, this work addresses the detection and the interpretation of the deviation from the radius-insensitive power law signal behavior at the largest possible (by varying at fixed diffusion time ), in rat and human white matter. Indeed, either sensitivity of MR to a finite axonal radius, or a notable exchange rate between intra- and extra-axonal water at the clinical dMRI time scales ∼100, would alter the very particular power law scaling.

6Following theoretical considerations, we demonstrate the breaking of the power law scaling at very high -values in ex vivo rodent brains, from which metrics associated with the axon radius distribution can be mapped quantitatively. Confocal microscopy of the rat corpus callosum (CC) validated that (i) the signal arises mainly from the intra-axonal space, and (ii) the MR-derived axon radius metrics are in good quantitative agreement with those derived from histology. We then observe the same signal signatures in living human brain on the Connectom 3T scanner, that is, a high performance research scanner with a maximal gradient amplitude of 300 mT/m – a fourfold increase compared to state-of-the art clinical scanners. Our findings both validate the mechanism with which axon radii are weighted in dMRI, and demonstrate the accuracy of which properties of the radius distributions can be estimated. After validating and evaluating our methodology in rat and human brain, we further discuss the impact of axon radius measurements in health and disease.

7In most biophysical models for diffusion in white matter, axons (and possibly glial cell processes) are represented by zero-radius impermeable ‘sticks’, characterized by locally one-dimensional diffusion, that is radial intra-axonal diffusivity. The stick model then yields an asymptotic intra-axonal orientationally averaged signal decay,with an intercept (discussed below), the power law exponent, and the coefficient where is the -weighted axonal water fraction and the parallel intra-axonal diffusivity. This particular signal decay only holds in the absence of extra-axonal signal, which is assumed to decay exponentially fast and, as such, to be fully suppressed at sufficiently high -values. Therefore, we restrict our in vivo analysis to. Our lower bound on the -value is significantly higher than previous predictions from Monte Carlo simulations, thereby minimizing the likelihood of residual extra-axonal signal contributions. For the ex vivo analysis, we increase this lower bound to to compensate for the reduced diffusivities in fixed tissue.

8The following computations always assume that the signal is normalized to. Sensitivity of MR to either finite axon radius or notable exchange rate between intra- and extra-cellular water would break the -scaling at large as follows:

9Finite axon radius: A finite results in a truncated power law:

10with the signal fraction of a fully restricted immobile water compartment (the so-called ‘dot’ compartment). If, the power law offset (found from extrapolating the signal to ), should give the fraction of the dot compartment,. However, for nonzero and finite, the Taylor expansion of,

11around any finite point predicts the intercept, Figure 1. The always negative difference depends on,, and; its maximal magnitude is achieved at the curve’s inflection point. Hence, the lower bound for the intercept may be negative. A negative is biophysically implausible if the stick model holds,; however, becomes a natural consequence of a finite (and hence, of a finite axonal diameter), Figure 1, in the case when the extrapolated negative intercept overcomes the positive immobile fraction. Recently, was shown to be negligible in healthy human white matter. Therefore, a negative intercept is a novel hallmark of MR sensitivity to the inner axon diameter, even if the signal scaling might appear linear as a function of for -ranges accessible on human MR scanners. Importantly, the finite axon radius model, Equation 3, is poorly conditioned as a result of which the simultaneous estimation of and is practically impossible, especially for human MR experiments, see Figure 1. An accurate and precise measurement of depends on the prior knowledge of and requires a dedicated measurement.

12Exchange: The spherical integration of the two-compartment ‘Kärger’ model with a finite exchange rate yields approximately the following signal decay:

13with the radial diffusivity in the interstitial space, and, the echo time, during which exchange can happen. Importantly, Equation 4 is convex as a function of.

14The relative fit quality of the models (i.e., Equations 1, 2, and 4) to the dMRI signal decays can be evaluated qualitatively (convex versus concave shape) or statistically by means of the corrected Akaike information criterion (AICc).

15The radial signal attenuation inside the cylinder of radius in the Gaussian phase approximation:with and defines the connection between the intra-axonal radial diffusivity and the radius. Here, is the diffusivity of the axoplasm, the gradient of the Larmor frequency, is the th root of, where is the Bessel function of the first kind, and and are the gradient duration and separation, respectively.

16In the long-pulse limit, that is when, the dependence on drops out, and Equation 5 approaches the Neuman’s limit

17This limit practically applies to the majority of axons. Importantly, the attenuation is proportional to the fourth power of the radius and, as such, it is very weak for narrow axons. Hence the low sensitivity of dMRI to the inner axon diameter.

18For an unknown distribution of axons with radii, the total intra-axonal signal attenuation becomes a volume-average over the histogram bins:such that the signal contribution of an axon scales quadratically with its radius. The Taylor expansion of the net signal attenuation demonstrates the sensitivity of the dMRI signal to the distribution’s higher order moments:

19where the effective axon radius:captures the contribution from the whole distribution in a single metric. The ability to represent the whole distribution by the ratio of its 6th and 2nd moments relies on almost all axons falling into the Neuman’s limit, Equation 6. Representing, we can calculatethe MRI estimate of after estimating from the orientation-averaged signal using Equation 2.

20Note that the effective radius, Equation 9, is heavily weighted by the tail of. Physically, this happens due to the combination of the weak NMR signal attenuation by small radii,, in the diffusion-narrowing (Neuman’s) regime, and of the subsequent volume-weighting that emphasizes the thickest axons by an extra factor of. The error associated with these modeling assumptions is discussed in the Results section.

21First, we evaluate the accuracy of axon radius mapping as a function of for axon radius distributions extracted from histology; Figure 2 (left and middle panels). We used a simulation framework based on the matrix formalism for diffusion signal attenuation within fully restricted cylinders, as implemented in the MISST toolbox, while mimicking the entire experimental setup, for both the human and preclinical experiments.

22In the case of diffusion restricted in a single cylinder with radius, the error in the estimated radius increases with. Indeed, the missing higher-order corrections to Equations (5)-(6) set an upper bound on the achievable accuracy for large axons, as estimated recently.

23The combined error in the estimation of associated with the approximations made in Equation 6 and Equation 8 is only 5% for the human set-up when considering the axon radius distribution provided by; the distributions of result in a subpercent error. Additionally, we show the errors in the estimation of for the axon caliber distributions that were observed in our different histological sections while considering the scan parameters from our fixed tissue experiments. The shorter diffusion timings increase the approximation errors, leading to an underestimation up to 9%.

24Figure 2 (right panel) shows a theoretical lower bound on the 95% confidence interval in the voxelwise estimation of from Equation 2, as predicted using a Cramér-Rao lower bound analysis. Using the dependence, Equation 8, that approximately identifies with the single cylinder radius in van Gelderen’s model, can be used to translate the lower bound on to that on.

25Notably, it follows from Figure 2, that an estimate of exceeds zero with a statistical threshold of, if the corresponding and, when mimicking the diffusion acquisition and SNR on the Siemens Connectom and Bruker Aeon MR scanners, respectively. In comparison, for a typical acquisition on a modern clinical scanner with, this lower bound is 3.2.

26Because has been reported to be significant in fixed tissue by, we first have to estimate the signal fraction of the immobile water compartment in the three fixed brain samples from a dedicated MRI acquisition (see Materials and methods). The estimate is in the range of 8 to 17% with a median value of 13%. The range is defined here by the 5th and 95th percentile of the distribution of the estimated dot fractions in all CC voxels, across the three samples.

27Figure 3 shows the signal decay, averaged across all CC voxels, based on diffusion measurements in the three rat brain samples with up to 100. We notice that an extrapolation to infinite, that is, yields a small but significant negative offset, of the order of a few per cent of the non-attenuated signal, in all three samples after subtracting the from the diffusion measurements.

28We re-evaluate the validity of a perfect stick assumption in the high -regimes using a AIC analysis. To study fit robustness with respect to the number of degrees of freedom by considering the full, nested, and extended models to Equation 1. Specifically, we evaluated the following models:

29Our analysis shows that a truncated power law (vii), which explicitly accounts for (and hence does not require a negative intercept parameter), yet sets, fits the experimental data significantly better than pure power law forms (models (i), (ii), (v), and (vi)) (the difference in ), with or without an offset, if the immobile (dot) compartment is corrected for, that is when using. Without dot-correcting the ex vivo data, the power law form (ii) with an intercept outperforms the other models. In that case, the intercept is negative, while is defined to be positive. Hence, the intercept encodes both the still water fraction and a negative offset to the intercept associated with the sensitivity to the axon diameter, such that the overall.

30Axon radii were estimated from the diffusion MRI data for the different CC ROIs (Figure 4) along with the axon radius distributions extracted histologically, Figure 4. The errors between the associated tail-weighted effective radii and MR-derived vary between 5 and 21% in the different ROIs. Notably, a consistent residual overestimation was observed, whereas the previous simulations (Figure 2) predicted an underestimation.

31To further examine the correlations between the MR-derived parameters and underlying microstructure, we analyzed 16 patches with the same size as an imaging voxel, that is 100 × 100 within the genu of the CC of the second sample, CC#2. The confocal light microscopy images of two of those patches are shown for the various stainings, Figure 5. Two notable biological components other than axons were highlighted, namely, astrocytic processes and (neuronal or glial) cell bodies, which were found to have volume fractions of about 5%. Although the radius of the astrocytic processes cannot be measured due to their random orientations w.r.t. the image plane, it is clear that some of the processes have a large diameter, for example up to 7 in the first patch. The average cell body radius is 2.6, with an effective radius of 4.3. It is worth highlighting that the -weighted signal fraction of both cell types remains unknown since the corresponding relaxation times are unknown. This unknown difference between compartmental volume (histology) and signal (MRI) fractions remains the Achilles’s heel of comparisons between MRI measurements and histological evaluations of tissue microstructure.

32Within each of the 16 patches, we extracted the axon radii distribution and derived the average and effective radius. The box plots of those metrics are shown in Figure 5. The median and median effective radius across all patches are 0.61 and 1.06, respectively. In comparison, the median, derived from dMRI in 16 voxels within the genu of the CC, is 1.16, 1.10, and 1.19 for the three rat samples, respectively. The median MR-derived effective axon radius is between 81 and 97% larger than the median, whereas the error to the median, as derived from histology, is only 3.4 to 12.8%.

33ROI measurements provided robust estimation, but a remaining question is whether dMRI could be used to map the effective MR radius in a voxelwise manner. Figure 6 shows the maps of the MR-derived effective axon radii for all three rat CC’s. The maps appear smooth with very few outlier voxels, suggesting that the estimation is robust even when voxelwise data is used. Furthermore, the qualitative trends are in good agreement with previously reported observations of larger axons in the body of the CC in comparison to the genu and splenium. Inter-subject variability is not very large and can be attributed to slightly different slice positions. When computing the effective radius of the CC-averaged signal, the intersubject variability nearly nullifies. Indeed, we estimate and 1.25 µm in the three samples, respectively.

34To assess the applicability of this approach in more realistic settings available for human imaging, experiments were performed in human subjects on the Connectom scanner, which is capable of producing 300 mT/m gradient amplitudes. The dMRI signal decay curves, averaged across all WM voxels, with -values up to 25 ms/µm2 for the four human subjects are shown in Figure 3. Importantly, we find that — in excellent correspondence with the previous preclinical data — the linear extrapolation of the signal decay as a function of to produces a significant negative offset in all subjects.

35Note that the dot compartment was not measured directly, because previous dedicated studies revealed a negligible dot compartment, that is, in the healthy white matter; see Discussion.

36The AICc analysis of various models demonstrated that also for the human white matter, the truncated power law (vii) with and negligible dot compartment fits the data significantly better than pure power law forms, with or without intercept. However, this statistical analysis cannot be interpreted as a data-driven justification for because of the degeneracy of Equation 3, as highlighted in Figure 1.

37Since direct histological evaluation in volunteers is unfeasible, we turn to validate the MR-derived metrics in humans with previously reported literature of human corpus callosum microstructure. In Figure 7, the MR-derived metrics were directly compared with axon radius distributions of multiple histological studies. Various reports and histological studies show a good correspondence for the bulk of the distributions, represented by the average radius, that is the average radius only ranges between 0.54 and 0.69 µm. In histological samples, the corresponding effective radius dominated by large axons, shows strong variability. Indeed, compared to, varies more than three-fold, from 0.91 to 2.9.

38The four human subjects show good correspondence in terms of. In Figure 7, we show the individual and combined distributions describing for all voxels in the midbody of the CC for all four subjects. It is apparent that the combined distribution falls almost entirely within the range spanned by -values as predicted from histology – even without introducing a putative axonal shrinkage factor (maximally 35%, and typically within 10%).

39In Figure 8, the distribution and map of for WM voxels in all human subjects, estimated using the ODF-independent model, Equation 2 with, are shown. Considering the statistical bound from Figure 2, it is to be expected that the estimated is biophysically meaningful for the vast majority of WM voxels for the Connectom scanner (Figure 8 shows a representative map of and associated for a single subject of the Connectom cohort), whereas similar measurements on a modern clinical scanner result in a biophysically implausible negative in approximately 35% of all WM voxels. Note that the data from a clinical scanner (Siemens Prisma with 80 mT/m gradients) are adopted from our recent work. This suggests that estimating and the associated effective axonal radius is only possible on MR systems with ultra-strong gradients. The spatial variability as well as the observed asymmetry between the hemispheres in, for example, the occipital lobes was noted for all subjects. However, our cohort is too small and not sufficiently characterized to study the whole brain characterization or the role of lateralization in large axons of human brain.

40It is worth examining the power law scaling also in areas outside the white mater. Therefore, Figure 9 shows the diffusion-weighted signal decay, averaged over all cortical gray matter (GM) voxels as a function of in the human subjects. The signal scaling in the WM is shown for qualitative comparison. The non-linear scaling of the isotropically-averaged signal as a function of of all human subjects indicates strong deviations from the ‘stick’ model in the cortical GM,. Accounting for a finite neurite radius, Equation 2, does not describe the data well either. Instead, the convex signal decay as a function of at high -values is in good agreement with the anisotropic exchange model that we derived from the expansion of the anisotropic Kärger model in the powers of inverse, Equation 4. Both the finite radius and exchange model predictions are shown in the absence of an immobile water fraction. The exchange model fits the data better than all other evaluated models in all subjects according to an AIC analysis (data not shown). The residence time within the neurites varies from approximately 10 to 15 or 20 to 30 if we assume or, respectively. Dedicated experiments are required for a more precise measurement of the exchange rate.

41Noninvasively estimating metrics associated with axon radius distributions is a formidable task, yet it could have a strong impact for numerous areas of research including neuroscience, biomedicine and even for clinical research and applications. Histological studies have extensively reported axon diameters to be in the range 0.5 − 2 µm for human WM, with only 1% of all axons having a diameter larger than 3 µm. A vigorous debate has emerged in the MRI and neuroanatomy communities as in vivo, MRI-derived axon diameters are reported to fall within the range 3.5 −15 µm. On the MRI side, the discrepancy has been attributed to the long diffusion pulses that strongly reduce the signal attenuation of protons restricted in a narrow cylinder. Therefore, the time-dependence of extra-axonal diffusion, and the undulation or along-axon caliber variation, potentially overshadow the relatively small.

42On the other hand, shrinkage during tissue fixation has been suggested as a potential shortcoming of histology, implying that in vivo axons are thicker than their histologically reported values.

43This study aimed at investigating what the dMRI signal can measure in terms of axon radius, as well as provide insight into the aforementioned debate. Our wide range of diffusion weightings in both human and preclinical dMRI enables a suppression of the extra-axonal contribution (that otherwise biases the radii; ), thereby allowing us to shed light on this controversy. We claim that the effective MR radii measured in this study quantitatively agree with those derived from histology — to the extent that histology correctly captures the tail of the axonal radii distribution. That is, obtained from dMRI appears to be a self-averaging quantity in each imaging voxel, as large MRI voxels ensure that the moments of sampled from a voxel represent well the ‘true’ underlying in that WM region, so that the spatial variations in stem mainly from genuine biological variations of the tails of axon distributions across the brain.

44Histology-derived are prone to mesoscopic fluctuations due to small sampling sizes, Figure 7. Despite a good correspondence of the bulk of axon radii distributions obtained from different histological studies and samples, the tail of the distribution is typically coarsely sampled, with only a few spikes representing the occasional observation of large axons within the relatively small histological sections, Figure 7. It is precisely for the detectable large, that the relative fluctuations for the bin counts are observed for bin values of (Table 2 of and Figure 5), according to the Poissonian statistics governing each. Not surprisingly, derived from discrete histological histograms exhibits strong fluctuations, as depicted by dotted vertical lines in Figure 7a and the error bars in Figure 5.

45Although the average radius, as reported in human literature, only ranges between 0.54 and 0.69, the corresponding varies from 0.91 to 2.9. In comparison, the average dMRI-derived estimated from the four Connectom data sets within the same region-of-interest, the midbody of the CC, only varied from 2.48 to 2.82 (Figure 7b).

46The average radius, as measured in this study, varies between 0.54 and 0.68 across the 16 patches of the genu of the CC, while the associated varies from 0.81 to 1.30. The MR-derived effective axon radii vary similarly, that is 0.94 to 1.4 across several voxels within the genu of the CC for all three scanned samples.

47For dMRI, the variability in the estimation of is determined by thermal MRI noise, and genuine anatomical – inter-voxel and inter-subject – differences. For human MRI, the mesoscopic fluctuations are much weaker, due to the large MRI voxels in comparison to the histological samples. Indeed, the variance in the estimation of is expected to decay inversely with the number of axons within a field of view. However, for rodent MRI, in which the MRI voxels have the same surface area as the histological patches, the precision in the estimation of the effective radius is similar for both modalities.

48Overall, dMRI provides a precise measurement of the largest axons, which are captured within an MRI voxel. In contrast, histology, so far, mainly probes the bulk that consists of smaller axons with high precision. Therefore, both modalities are complementary, especially in human MRI for which the voxels are significantly larger than a typical histological sample.

49As the signal attenuation inside axons, Equations (5)-(6), scales as, getting to two-times smaller would require another four-fold increase in gradients. However, even with stronger gradient systems, the main bottleneck might be the missing prior knowledge about the shape of the expected axon radius distribution. Even when assuming a particular functional form of, one is limited to estimating a single parameter to describe the axon radius distribution, whereas realistic distributions such as the generalized extreme value distribution are parameterized by at least two variables. Hence, the reconstruction of from only is technically ill-posed and, as such, prone to mis– or over–interpretation due to biases towards user-defined distribution shapes and parameters, even more so if confounding factors such as dispersion or fixed diffusivities are ignored.

50With unknown, only a single metric representing the entire distribution, that is, for which the strength of tail-weighting is determined by the gradient pulse width, can be estimated reliably. In the best case, that is the narrow-pulse limit, see 'From to effective MR radius’, will depend on the fourth rather than the sixth order moment of, that is, thereby reducing, but not eliminating the difference between and. Other methods, such as oscillating gradient diffusion weighting or double diffusion encoding, may provide other sources of contrast encoding different aspects of the size distribution, although the low-frequency limit of the oscillating-gradient attenuation has been shown to be equivalent to the Neuman’s limit, not providing any extra information. It can be hypothesized that the combination of methods could perhaps recover more accurate information on the underlying.

51The presence of isotropic immobile water has been conjectured by as water possibly restricted inside the soma of various cell types, such as neurons or oligodendrocytes. Several previous studies, for example,, and, demonstrated with various diffusion encoding strategies that in vivo dMRI is practically insensitive, that is < 0.2%, to such signal contributions in the healthy white matter of the living human brain, excluding the cerebellum. The lack of sensitivity of dMRI to immobile water might be explained by a small volume fraction, a short T2 relaxation time, and/or a fast water exchange rate on the scale of our diffusion time for treating them as coming from separate compartments. In contrast, the dot compartment has been observed in fixed brain samples in various studies. The origin of this signal contribution is not well understood yet, but the still water compartment needs to be considered when validating or studying biophysical models in fixed tissue.

52In this work, for the human experiments, we build upon the previously reported observations to fix in the healthy white matter to avoid fitting degeneracies that are associated with the poor conditioning of model (iii). However, any underestimation of the dot compartment, for example due to fixing, leads directly to an underestimation of the effective MR radius, see Figure 1. Therefore, in future studies, especially those that focus on the developing, aging, or pathological brain, we encourage the independent measurement of the dot compartment to complement the axon radius acquisitions. The fast measurement of the dot compartment is promoted by the availability of spherical diffusion-encoding, as demonstrated by, and.

53In our ex vivo experiments, the measurement of the dot compartment is based on the diffusion-weighing in the direction parallel to the average fiber direction in the CC at the maximal -value of 100 ms/µm2. The measured signal fraction of such a still water compartment in our fixed brain samples was in the range of 8 to 17%, in line with. Applying a radial or planar diffusion-weighting filter prior to this measurement would suppress any contribution of anisotropic signal compartments, such as crossing or dispersed axons, to the isotropically restricted dot compartment. Although we aimed to minimize this confounding factor by using a very high -value, the dot compartment fraction, and as such the effective MR radius, might be slightly overestimated because of various complex fiber configurations. Additional confounding factors are listed in the following section.

54The apparent discrepancy between histology and dMRI, when confounding factors such as extra-axonal water and orientational dispersion are addressed, is mainly a result of the difference between and – that is between the bulk and the tail of axonal distribution. This already provides an important insight into the discussion on why the radii reported in the literature vary so much between the methods. When comparing apples-to-apples, despite the excellent agreement observed in this study between and its histological counterpart, in our own histological validation we still observed a small, yet consistent overestimation of between 5 and 20% in axon radius using dMRI. Aside from the previously discussed dot compartment, various other factors might contribute to this discrepancy.

55First, an underlying assumption of all studies targeting the measurement of the axon radius is specificity: that the signal observed at these high -values could be attributed exclusively to the intra-axonal space. However, this assumption is not established nor in our opinion is it justified given that water resides in all cellular compartments of the central nervous system. We cannot exclude that water trapped in other ‘stick’-like features such as the radiating processes of astrocytes contribute observable signals; it has been previously reported, but also observed in our histological sample, that such glial processes can have large diameters, up to 7 in our sample. In the future, this contribution could be investigated using the increased cellular specificity of (diffusion-weighted) spectroscopy.

56Second, and, more recently, demonstrated that at shorter diffusion times, the signal contribution from cell bodies might be characterized by a specific -value dependent signature ( and ) that might enable the extraction of MR effective cell body sizes in both the white and gray matter. In this study, the potential -value dependent signal contribution of cell bodies was unaccounted for, and, given our and other (e.g. ) observations of a finite cell body volume fraction, the axon radius measurements could be biased. However, deviations to the power law scaling due to the presence of cell bodies are more likely to be expected in the GM because of larger volume density of large somas in comparison to the WM. In our histological sample, we observed a significant volume fraction of cell bodies in the genu of the CC, that is 5%, but due to unknown compartmental relaxation times, the associated, yet more important, signal fraction is unknown. Regardless, a biophysical model parameterized by various volume fractions, axon radii, soma sizes, and compartmental diffusivities may be poorly conditioned and degenerate.

57Third, along-axon undulations and curvature might result in an increased apparent radial diffusivity and, as such, contribute to an overestimation of the axon radius using dMRI, especially for increasingly long diffusion times.

58Finally, the estimation of the MR effective axon radius depends on the unknown intrinsic diffusivity of the axoplasm. In ex vivo samples of a well-aligned WM bundle, one could estimate directly by exploring the time dependence of the apparent diffusivity at very short diffusion times,. In this study, we were not able to achieve a reproducible and precise estimate of and opted to use the longitudinal diffusivity as a proxy for, with. Therefore, we might actually underestimate the positive bias in the estimation of (Equation 10). However, the propagation of the error in the estimation of to is strongly reduced by the fourth root relation between both metrics.

59In our study, the effective MR radius in humans was significantly higher than in rats when comparing similar regions of interest, for example the midbody of the CC. This difference is in agreement with several studies that compared the callosal fiber composition as a function of the brain size of various mammals and concluded that large brains have more large axons and an increased maximal radius, whereas the bulk of axons is not altered. Since the effective MR radius is predominantly sensitive to the larger axons, observed differences between humans and rats will be amplified when comparing effective MR radii. Overall, this observation favors future application of MR axon radius mapping in species with relatively large brain sizes.

60Although this work mainly focuses on the WM, we do report significantly different signal scaling for the cortical GM. We suggest that the proton exchange between dendrites and interstitial water might explain this scaling behavior, in particular due to the convex scaling with. However, the abundance of cell bodies in the gray matter might confound this analysis. Moreover, the study of the cortical GM is challenged by its low SNR and susceptibility to partial voluming. Nonetheless, we conclude that the stick assumption does not hold in the cortical GM and that biophysical models building upon that assumption must be interpreted with caution if applied to tissue regions outside of WM.

61In summary, we provide a realistic perspective on MR axon radius mapping by showing MR-derived effective radii that have good quantitative agreement with histology. First, we compared the MR-derived axon radii directly to confocal microscopy of the same rat brain samples. Second, the distribution of dMRI-derived of the living human brain falls almost entirely within the range spanned by histology-derived that has been reported in the literature — even without introducing a putative axonal shrinkage factor. This estimation is inherently bound to a single scalar that encodes moments of the axon distribution, which is – by virtue of the signal encoding – dominated by the largest axons. Therefore, the average axon radius and can be practically considered as two complementary metrics probing the underlying axon caliber distribution: histology, so far, mainly probes its bulk, that is, while dMRI probes, that is its tail. Due to the intrinsic bias of MR-derived axon radii to larger axons, clinical applications should focus on pathologies that specifically target those larger axons, until other methods are developed that probe the smaller axon diameter.

62Animals used in this study were handled in agreement with the European FELASA guide-lines and all procedures were approved by the Champalimaud Animal Welfare Body and by the national authorities, Direção Geral de Alimentação e Veterinária, Lisbon, Portugal, under the approved protocol number 0421/000/000/2016. All animal care procedures were conducted in agreement with the European Directive 2010/63, at the vivarium of the Champalimaud Foundation, a research facility part of CONGENTO, project number Lisboa-01–0145-FEDER-022170.

63Three Long Evans rats (Female, 12-weeks-old) were transcardially perfused using 4% paraformaldehyde. The extracted brains were kept for 24 hr in 4% paraformaldehyde and washed using PBS over two days (changed daily). Given our focus on the CC of the rat brains, we will refer to the samples as CC#1, CC#2, and CC#3.

64Multi-shell dMRI data: The three samples were scanned on an 16.4T MR scanner (Bruker BioSpin) at room temperature with interfaced with an AVANCE IIIHD console and a micro2.5 imaging probe with maximal gradient amplitude. Diffusion-weighting was applied using a RARE sequence in the midsagittal plane along 60 gradient directions for a densely sampled spectrum of 18 different -values up to 100 ms/µm2. Furthermore, and the spatial resolution was 100 × 100 × 850µm3.

65‘dot fraction’: we acquired 60 repeated measurements of diffusion-weighing applied in the direction parallel to the average fiber direction in the corpus callosum (CC) at the maximal -value of 100 ms/µm2. The average fiber orientation was defined as the first eigenvector of the dyadic tensor that was computed from the voxelwise first eigenvectors of the diffusion tensors that were estimated by fitting the DTI model to the lowest -values, that is, of the multi-shell data in each voxel within the manually segmented CC.

66The average SNR for was 195 and some examples of the acquired images at various low and high -values are shown in Figure 10, where the quality of the raw data can be evaluated. Notably, since images are spherically averaged over many directions, the signal is characterized by high SNR even at high -values.

67From the multi-shell data, the spherically-averaged signals are estimated per -value as the zeroth order spherical harmonic using a Rician maximum likelihood estimator of the even order spherical harmonic coefficients up to the 6th order.

68The spatially localized dot fraction is computed as the signal estimated from the repeated measurements in the direction parallel to the principal fibre direction using a Rician maximum likelihood estimator with pre-computed noise level, normalized by the respective non-diffusion weighted signal. We compute the dot-free signal in each voxel as follows. In the remainder of the work, analyses were done on both ‘dot contaminated’ and ‘dot free’ signals.

69The intra-axonal radial diffusivity is estimated voxelwise by fitting Equation 2 with to using a nonlinear least squares estimator (code is available for download on GitHub [; https://github.com/NYU-DiffusionMRI/AxonRadiusMapping; copy archived at https://github.com/elifesciences-publications/AxonRadiusMapping]).

70Thereafter, the estimated effective axon radius is derived from using Equation 6. The alternative approach, that is the simultaneous estimation of and from is very poorly conditioned. Hence, disentangling both parameters from only the linearly-encoded multi-shell data is impossible, even at unrealistically high SNR.

71Full details of the immunohistochemistry for sample preparation, confocal microscopy, and image analysis are provided in. Study-specific elements are described below.

72After MRI scanning, free-floating horizontal sections 50 -thick were collected from the medial lateral center of two rat brains, CC#1 and CC#2, corresponding to the MR imaged volume. For CC#2, we used antibodies against neurofilaments 160/200 (axonal marker; Sigma-Aldrich, cat.# N2912) and GFAP (astrocytes marker; ThermoFisher Scientific, cat.# PA1-10019), as well as a staining for cell bodies using DAPI (Sigma-Adrich, cat.# D9542). For CC#1, the staining was limited to the neurofilaments to focus on the axon radius count.

73A Zeiss LSM 710 laser scanning confocal microscope was used for immunohistochemistry image acquisition. A tile scan using a 10× objective (EC Plan Neofluar, numerical aperture = 0.3, Zeiss, Germany) was used to cover the entire CC. Various ROIs were imaged using a 63× immersion objective (Plan Apochromat, numerical aperture = 1.4, Zeiss, Germany) in confocal mode, with pixel resolution of 65 × 65 × 150 nm3 and field-of-view of 135 × 135 µm2 (Figure 11). The placement of the ROIs is shown in Figures 4 and 5 for CC#1. and CC#2, respectively.

74Images were processed using the ImageJ software. Noise suppression of the confocal single frames was done using a subsequent application of a 2D anisotropic diffusion filter and bandpass filtering in the frequency domain,. Thereafter, axons were identified as particles with a minimum area size of 0.2 µm2 and a circularity larger than 0.4 in the confocal images that were stained for neurofilaments. The number of extracted axons varied from about 500 to 2000, depending on the placement of the ROI. The long axes of fitted ellipsoids served as proxies for the respective axon radii. For each ROI, we obtain a distribution of axon radii from which we compute the associate effective radius using Equation 9.

75Data were acquired after obtaining written informed consent and consent to publish. The project was approved by the Cardiff University School of Psychology Ethics Committee (approval number EC.06.05.02.891).

76Four healthy volunteers (3 males and 1 female between 22 and 45 years) were recruited for this study. We will refer to the human subjects as 22M, 25F, 32M, and 45M to encode both the age and gender. The data were collected under the approval of the Cardiff University School of Psychology Ethics Committee.

77All four subjects were scanned on a Siemens Connectom 3T MR scanner using a 32-channel receiver coil. The 300 mT/m gradient system was used to achieve -values up to 25. The diffusion gradients were characterized by and maximal gradient amplitude of 289 mT/m. Diffusion weighting was applied along 60 isotropically distributed gradient directions for 1, 3, 5, 7, 9, 11, 12.1, 13.5, 15, 16.9, 19.1, 21.7, and 25, with TR/TE: 3500/62, matrix: 74 × 74, and 42 slices with a spatial resolution of 3 × 3 × 3. The average SNR for was 52. See Figure 11 for the image quality and contrast at various -values.

78The dot compartment was not measured directly (see and ).

79Image processing was done according to the DESIGNER pipeline using the FSL and MRtrix software packages. In particular, MPPCA noise estimation and denoising were used for estimating noise maps by exploiting the inherent redundancy in dMRI data. The positive signal bias, inherent to low-SNR magnitude MR data, was removed by using the method of moments, where the denoised signal was used as a proxy for the Rician expectation value. Denoised and Rice-floor-corrected images were subsequently corrected for Gibbs ringing, geometric eddy current distortions and subject motion. The pipeline is available on https://github.com/NYU-DiffusionMRI/DESIGNER. We used tract-density imaging based on whole-brain probabilistic fiber-tracking of the -shell for identifying all WM voxels. To avoid voxels affected by partial voluming with the gray matter (GM), an additional, more conservative, segmentation was obtained by omitting all voxels with a fractional anisotropy smaller than 0.6. In addition, the cortical GM was segmented using FreeSurfer.