Intra-axonal diffusivity in brain white matter

NeuroImage 189 (2019) 543–550

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Intra-axonal diffusivity in brain white matter Bibek Dhital *, Marco Reisert, Elias Kellner, Valerij G. Kiselev Department of Diagnostic Radiology, University Medical Center, University of Freiburg, Freiburg, 79106, Germany

A R T I C L E I N F O

A B S T R A C T

Keywords: White matter Diffusion MRI Diffusion encoding Planar encoding Micro-structure imaging Intra-axonal diffusivity Orientation dispersion

Biophysical modeling lies at the core of evaluating tissue cellular structure using diffusion-weighted MRI, albeit with shortcomings. The challenges lie not only in the complexity of the diffusion phenomenon, but also in the need to know the diffusion-specific properties of diverse cellular compartments in vivo. The likelihood function obtained from the commonly acquired Stejskal-Tanner diffusion-weighted MRI data is degenerate with different parameter constellations explaining the signal equally well, thereby hindering an unambiguous parameter estimation. The aim of this study is to measure the intra-axonal water diffusivity which is one of the central parameters of white matter models. Estimating intra-axonal diffusivity is complicated by (i) the presence of other compartments, and (ii) the orientation dispersion of axons. Our measurement involves an efficient signal suppression of water in extra-axonal space and all cellular processes oriented outside a narrow cone around the principal fiber direction. This is achieved using a planar water mobility filter that suppresses signal from all molecules that are mobile in the plane transverse to the fiber bundle. After the planar filter, the diffusivity of the remaining intra-axonal signal is measured using linear and spherical diffusion encoding. We find the average intra-axonal diffusivity D0 ¼ 2:25  0:03 μm2 =ms for the timing of the applied gradients, ð∞Þ

which gives D0  2:0 μm2 =ms when extrapolated to infinite diffusion time. The result imposes a strong limitation on the parameter selection for biophysical modeling of diffusion-weighted MRI.

1. Introduction Diffusion-weighted magnetic resonance imaging (dMRI) in brain white matter (WM) has been used to detect tissue anomalies (Moseley et al., 1990; Werring et al., 1999; Kono et al., 2001) and to reconstruct axonal tracts in vivo (Basser et al., 2000; Tuch, 2004; Tournier et al., 2004; Reisert et al., 2011). Recent years have seen a significant growth in research interest to evaluate living tissue microstructure using dMRI (Reisert et al., 2017; Novikov et al., 2018a). Histology, the gold standard for tissue microstructure evaluations, cannot access parameters that are central to dMRI such as diffusivities inside different cell species. Furthermore, as diffusivities change dramatically upon cell death, they must be measured in vivo – their physical relevance does not leave much room for alternative (non-MRI) measurement techniques. The problem is exacerbated by the typically featureless shape of commonly acquired dMRI signal. This renders estimating parameters through model fitting ill-posed: different parameter sets can explain the measured data equally well (Jelescu et al., 2016; Novikov et al., 2018b).

In this study, we pursue a direct estimation of intra-axonal water diffusivity in the normal human brain relying on minimal model assumptions. Towards this endeavour, we implemented a dedicated dMRI measurement technique to suppress the signal from extra-axonal space and measure diffusivity of the remaining intra-axonal signal. 2. Methods 2.1. Theory Diffusion MRI signal is sensitive to the loss of coherence between individual spins due to random motion during the application of magnetic field gradients. In the commonly employed Stejskal-Tanner method (Stejskal and Tanner, 1965), each acquisition sensitizes the signal to diffusion in one particular direction. Herein, we refer to this as linear encoding. Varying gradient direction within a single acquisition can sensitize the signal to motion in all three spatial direction, spherical encoding, also known as isotropic encoding, to motion in a selected plane, planar encoding, or in general, any scalene ellipsoid (Westin et al., 2014,

* Corresponding author. E-mail address: [email protected] (B. Dhital). https://doi.org/10.1016/j.neuroimage.2019.01.015 Received 27 June 2018; Received in revised form 2 January 2019; Accepted 7 January 2019 Available online 16 January 2019 1053-8119/© 2019 Elsevier Inc. All rights reserved.

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2016). We characterize the diffusion weighting in the traditional terms of bmatrix. The b-matrix defines the signal attenuation for Gaussian diffusion in the form S ¼ eTr ðbDÞ , where D is the arbitrary diffusion tensor, both b and D are real, symmetric matrices. Each b-matrix can be decomposed into linear bL , planar bP and spherical bS components (De Almeida Martins and Topgaard, 2016) in its eigenvector basis, 2

0 b ¼ bL 4 0 0

0 0 0

3 2 1 0 0 bP 4 5 0 1 0 þ 2 0 0 1

3 2 0 1 bS 4 5 0 þ 0 3 0 0

3 0 0 1 05 : 0 1

(1)

The diffusion weighting gradient waveforms can be designed to obtain the desired b-matrix. Fig. 2. Schematic of the sequence used for our measurements. All the three gradient directions are orthogonal to each other and the gradient waveforms do not have any cross terms in the b-matrix. The top two gradients provide planar diffusion filtering, the bottom gradient provides an additional linear weighting. Each of these gradient lobes has ramp-up time of 1.5 ms and flat-top time of 4.3 ms. Slice selection gradients, spoilers and gradients for imaging are not shown.

2.2. Measurement technique matches the target geometry Our approach is to match the diffusion weighting geometry to the white matter tissue architecture (Fig. 1). We rely on the smallness of axonal diameter in the human brain (Aboitiz et al., 1992; Caminiti et al., 2013). In other words, diffusion in the axons is effectively one-dimensional with the radial diffusivity close to zero. First, we use the planar filter – planar encoding as a filter to suppress the signal from water molecules mobile in the plane orthogonal to the principal fiber direction. The planar encoding can be ensured by the condition bL ¼ bS ¼ 0. The strength of this filter is characterized by its b-value, bP , which can be understood as the signal suppression by the factor ebP D in a homogeneous medium with diffusivity D (the filter b-matrix has the eigenvalues ½bP =2 ; bP =2 ; 0). The precise nature of diffusion in the extra-axonal compartment is not relevant for the present study, since the signal is strongly suppressed by the planar filter. Along with suppressing signal from other compartments, the planar filter also narrows the native axonal orientation distribution around the principal fiber direction (Fig. 1). As shown in Fig. 2, the planar filter can be obtained by applying two orthogonal gradient waveforms G1 and G2 . The resulting b-matrix contains two degenerate eigenvalues that span the plane we intend to filter. To measure the axial diffusivity, an additional weak gradient is applied

resulting in a small non-zero bL . The third gradient G3 is orthogonal to both G1 and G2 – the mutual orthogonality of all three gradients insures absence of any cross-terms in the b-matrix. It should be noted that when the planar filter is combined with additional linear gradients the resulting b-matrix is neither planar nor linear but a combination of two as described in Eq. (1). The ratio between two measurements-one with and another without G3 allow us to measure diffusivity of the remaining signal as a function of the planar filter strength, Dk ðbP Þ. When the planar filter is perpendicular to the fiber bundle orientation, Dk ðbP Þ is a function of the true intra-axonal diffusivity D0 and the width of the axonal orientation distribution function PðsinθÞ. For axononly monocompartment, it is straightforward to quantify this measurement as  Dk ðbP Þ ¼ D0 cos2 θiP ;

Fig. 1. The principle of the measurement. A: Neuronal axons are represented by cylinders positioned at a common origin to show their orientation distribution (which results in the massive overlapping as a side effect). The in-plane cylinders represent axons incoherent with the principal direction and glial processes in extra-axonal space. The second row shows the top views. B: Application of a linear water mobility filter suppresses extra-axonal space and a part of axons, that are shown with the darker color. Diffusivity measured along the fiber bundle underestimates the true value due to the contribution of tilted axons. C: The planar water mobility filter, which is used in this study, suppresses signal from all compartments, but those that are uni-directional and close to the principal fiber direction. Diffusivity in this direction approaches the ground truth. Isotropic diffusion measurement is applied to eliminate the residual effect of the orientation dispersion.

(2)

Fig. 3. Axial diffusion coefficient, Dk , and trace of diffusion tensor, Tr D, in single bundle voxels in all subjects as functions of the strength of the planar water mobility filter, bP . With increasing bP , Diso ¼ Tr D (circles) rapidly approaches the intra-axonal Tr Da , which is dominated by the intra-axonal diffusivity, D0 . Dk (triangles) is measured in the principal fiber bundle direction and underestimates D0 due to the residual axonal orientation dispersion (Fig. 1). The slow approach to the asymptote contains information about the width of this distribution as discussed in the text. The black lines and the text show the results of fitting the corresponding models to the pooled data for all subjects. 544

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where θ is the angle an axon makes with the principal fiber direction and the averaging is taken over the filter-reshaped orientation distribution. At strong filter strength, the additional linear gradients act only on the remaining signal from the axon-only monocompartment. Overlooking hardware and signal limitations, a very strong planar filter leaves signal from a cone very narrow such that hcos2 θiP is unity thereby revealing the intrinsic intra-axonal diffusivity, D0 . With realistically strong planar filter, Dk ðbP Þ asymptotically approaches D0 but is underestimated due to the residual axonal dispersion. This trend is seen in Fig. 3 that presents our main result. The right-hand side of Eq. (2) can be estimated by assuming orientation distribution to be axially symmetric so that Pðsin θÞ can be approximated by accounting only for the curvature near the maximum,   sin2 θ Pðsin θÞ  exp  ; 2 2σ

(3)

which is justified for distributions effectively narrowed by the planar filter, Fig. 1. This distribution is accompanied by the integration measure sin θ dθ. The variance of θ with this distribution is hθ2 i  2σ 2 for small θ. The parameter σ defines the distribution width affected by the filter application, 1

σ2

¼

1

σ 20

þ bP D0 ;

Fig. 4. Schematic of the workflow employed in this study. The left side of the figure shows the data acquisition and the right side shows the analysis scheme. We first employ a DTI measurement that is analyzed to obtain the three eigenvectors. This information is used to select single bundle WM voxels and their primary eigenvector direction is stored. When the planar filter is applied perpendicular to the primary eigenvector we use this particular filter direction for further analysis. The Dk is simply calculated by taking the ratio between two data points one with the added linear encoding and one without.

(4)

where σ 0 defines the width of the native fiber orientation distribution in the absence of filter. Accurate calculation of hcos2 θiP in Eq. (2) using this width gives "

#  1=2 2 ez 1 1 bP D0  2 ; z¼ þ ; 2σ 20 2 π z erfi ðzÞ 2z

Dk ¼ D0 pffiffiffi

axon-only mono-compartment, spherical encoding thus gives an estimate of the intra-axonal diffusivity insensitive to the axonal orientation distribution. The spherical encoding estimates the trace of the compartmentaveraged diffusion tensor, Tr D,

(5)

where the error function of imaginary argument is defined as Z z 2 2 erfi ðzÞ ¼ pffiffiffi dt eþt :

π

Tr D ¼ vP Tr Da þ ð1  vP ÞTr De ;

where Da and De are the intra- and extra-axonal diffusion tensors, respectively, Tr Da  D0 as discussed above and vP is the relative fraction of axonal signal after the application of the planar filter,

(6)

0

With increasing bP , the asymptotic approach of Dk to D0 can be described as " Dk  D0 1 

vP ¼

#

1 2

ð1=σ 20 þ bP D0 Þ

;

(8)

(7)

va ; va þ ð1  va ÞebP D?

(9)

where va is the true water fraction of axons (more precisely, of all effectively one-dimensional processes) and D? is the diffusivity in extraaxonal space in the transverse direction. The value of vP approaches unity for very strong filter, bP D? ≫ 1. To avoid proliferation of unknown parameters, for large bP ) a constant Tr Da (which is the asymptotic form of Eq. (8)) can be fitted to the data obtained from spherical encoding. In principle, measuring the trace of the remaining signal requires an isotropic (spherical) weighting in addition to the planar filter. However, since all the three gradients are orthogonal to each other, a measurement 0 with a high bP and a small bL can be re-interpreted as one with bP ¼ bP  2bL and bS ¼ 3bL . In other words, a planar filter of 2.8 ms=μm2 plus a linear weighting of 0.45 ms=μm2 can also be considered as a planar filter of 1.9 ms=μm2 plus an isotropic weighting of 1.35ms=μm2 (see Eq. (1)). In this way, estimating the trace for a planar filter of 1.9 ms=μm2 , can be estimated from two data points – one obtained from a planar filter of 1.9 ms=μm2 and another obtained from a planar filter of 2:8 ms=μm2 with an additional linear weighting of 0.45ms=μm2 . Practically, this enables us to reinterpret the same data set towards estimating Dk and Tr Da – a single workflow allows us to obtain both. The workflow of our experiments is illustrated in Fig. 4. For axons that are not quite colinear with the principal fiber direction, using Gaussian diffusion needs a justification. The Gaussian form of intraaxonal axial diffusion should hold for significant signal attenuation

which is eventually fitted to the experimental data. We note that suppression of signal in one direction and measurement in the other can also be performed with Diffusion-Diffusion correlation experiment (Callaghan et al., 2003), also known as double PFG method (Cory et al., 1990; Nilsson et al., 2013; Skinner et al., 2015). The filtered-dPFG (Skinner et al., 2015) method uses two pairs of gradient pulses perpendicular to each other (Callaghan and Komlosh, 2002) and has been used to show local anisotropy in macroscopically isotropic material including gray matter (Komlosh et al., 2007). A lower suppression efficiency of filtered-PFG for dispersed axons is illustrated in Fig. 1. Hence, even in the absence of extra-axonal compartment, parallel diffusivity obtained with filtered-PFG method produces a greater downward bias of the true intra-axonal diffusivity than that obtained with the planar filter. The issue of underestimation can be circumvented if, instead of a linear encoding, a spherical encoding is added to the planar filter. Again, ratio between two signals obtained with planar filter only and with planar filter plus spherical encoding measures the diffusion trace of the remaining signal (Lasi
c et al., 2014; Szczepankiewicz et al., 2015; Dhital et al., 2017; Vellmer et al., 2017a, b), which is primarily intra-axonal. For long diffusion times, the eigenvalues of the intra-axonal diffusion tensor, Tr Da , can be approximated by ½0 ; 0 ; D0 , so that Tr Da  D0 . For 545

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for 30 directions of G3 and the bP values were uniformly distributed in the same interval.

before the kurtosis and other higher-order terms start manifesting. This follows from the long diffusion length in our experiments relative to the correlation length of intra-axonal disorder, which is estimated to be 3–7 μm (Fieremans et al., 2016). After the application of planar filter, we are only interested in measuring intra-axonal diffusivity that requires weak gradients; the Gaussian signal approximation thus remains valid (Kiselev, 2017; Novikov et al., 2016, 2018a).

2.4. Data analysis The data was analyzed using in-house written code in MatlabⓇ. All images were corrected for Gibbs-ringing by interpolating the image based on subvoxel-shifts that samples the ringing pattern at the zerocrossings of the oscillating sinc-function (Kellner et al., 2016). For images obtained with the partial Fourier acquisition, the Gibbs ringing correction works only in the readout direction. From the diffusion tensor (Basser et al., 1994) single bundle voxels were selected using the fractional anisotropy (FA) along with measures of linearity ðcl ¼ ðλ1  λ2 Þ=λ1 Þ, planarity ðcp ¼ ðλ2  λ3 Þ=λ1 Þ and sphericity ðcs ¼ λ3 =λ1 Þ, where λ1 , λ2 , and λ3 are the first, second, and third eigenvalues of the diffusion tensor estimated at that voxel. These coefficients describe proximity of the tensor to a line, plane and sphere (Westin et al., 2002). The single bundle voxels had to fulfill the limit on the fractional anisotropy, FA > 0:5 and cl  0:4, cp  0:2, cs  0:35 (Fieremans et al., 2011). The rest of the analysis focused on single bundle voxels in which the direction of the primary eigenvector reflects the fiber orientation. The

2.3. dMRI measurements In vivo measurements were performed on four informed volunteers in a 3 T human scanner (Siemens PRISMA, max gradient strength 80 mT/m, 32 channel receive coil). The procedure was approved by the ethics board of University Medical Center, Freiburg. Written consents were obtained from all volunteers. For all volunteers,14 slices were acquired with field of view (FOV) ¼ 25.6 cm, 4 mm isotropic resolution, echo time (TE) ¼ 140 ms, repetition time (TR) ¼ 2500 ms, bandwidth ¼ 2005 Hz/ pixel and partial Fourier factor of 0.75. The 14 slices were selected from the bottom of the corpus callosum to 6.4 cm upwards. All imaging parameters, FOV, TE, bandwidth, resolution and number of slices, were kept the same for all measurements. A simple dMRI sequence comprising of 30 single direction measurements was measured for two b-values of 0.05 and 1 ms=μm2 to estimate the diffusion tensor (Basser et al., 1994). Planar filter was applied in many pre-defined planes with varying strength of the planar filter. The maximum gradient amplitude used for the planar filter was 60 mT/m. For each planar filter, a measurement with additional linear encoding was also performed. Effectively, each planar filter was applied once without the linear gradient, G3 , and once with an additional linear encoding with a b-value of 0.45 ms=μm2 . As mentioned above, the direction of the linear weighting was always kept normal to the filter plane. We note here that there are concomitant fields induced by G1 and G2 but not by G3 . However, since diffusivity is measured at each planar filter strength, these fields do not affect diffusivity measurements. The concomitant fields are quadratic in the inducing field – the field induced by, say, G1 has a constant sign. Since G3 is balanced on the other side of the refocusing pulse – it does not contribute to the concomitant field. Therefore, concomitant fields affect both measurements, with G3 ¼ 0 and G3>0, in the same way, thus leaving the estimated diffusivity unaffected. In two subjects (S1 and S2), planar filter were employed for 18 directions of G3 , with strength of the planar filter 0.1, 1.0, 1.8, 2.7, 3.6, 4.5 and 5.4 ms=μm2 . To account for signal-to-noise ratio (SNR) loss, data at higher b-values were acquired with more (up to six) repetitions. In two other subjects, instead of repetitions, the measurements were performed

Fig. 6. Signal remaining after the planar filter for selected single bundle voxels. On the y-axis is the remaining signal normalized to the non-diffusion weighted value and the x-axis is the angle made between the normal to the planar filter and the primary eigenvector of the diffusion tensor. It is seen that signal is maximum when the normal to the planar filter is aligned with the primary eigenvector i.e, when the planar filter is perpendicular to the main fiber direction.

Fig. 5. (A) Voxels selected only by FA > 0:5 condition (green) overlaid on a S0 image. (B) Selected voxels after applying additional criteria cl  0:4, cp  0:2, and cs  0:35 . (C) Maximum intensity projection of all planar filters applied at the planar filter strength of b ¼ 5.5 ms=μm2 . It can be seen that the planar weighting suppresses all of CSF and gray matter while leaving intact a significant signal fraction from the white matter regions. For the selected single bundle voxels, the maximum intensity is obtained when the planar weighting in perpendicular to the main fiber orientation.

546

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selected single bundle voxels were mostly from corpus callosum and cerebrospinal tract and a few from the internal capsule. The DTI data was used to obtain the relative angle between the primary eigenvector and the measured directions of the gradient waveform G3 . Only those directions and voxels were selected for further analysis where the measurement direction (G3) and the principle direction of single bundle voxels were less than 15∘ apart. For each planar filter, Dk was estimated by ordinary least squares method where the natural logarithm of the signal were fitted against the two b-values of 0 and 0.45 ms=μm2 . Tr D was obtained by a similar fitting, but in this case signal from each planar filter with no linear weighting was fitted with signal from an increased planar filter and a linear weighting of 0.45 ms=μm2 . The resulting Dk and Tr D as functions of the planar filter strength (Fig. 3) were interpreted in the context of the model presented in Fig. 1. This allowed us to estimate the orientation distribution, θ0 , of the axons. Estimating θ0 requires some model assumptions, but irrespective of the interpretation, the measured Dk and Tr Da can be considered as lower and upper-limit for intra-axonal diffusivity respectively.

Fig. 7. Dependence of the fitted intra-axonal diffusivity on the fitting interval, upwards. Systematic deviation for small values of bmin are due to the from bmin P P extra-axonal contributions not accounted for in the models. The large error for large bmin follow from too short fitting intervals. The final selection is shown P with the horizontal lines. Note that any other reasonable selection affects the result values within their error bars.

3. Results We find that in addition to FA, using the indices of linearity, planarity and sphericity to select single bundle voxels makes a substantial difference in the number of voxels selected. Fig. 5, on the left shows a representative non-diffusion weighted data, the green overlay shows voxels that would be selected if only FA threshold was used. On the middle figure, the overlay shows the actual voxels that were selected. The SNR of the data used was quite high – at the highest planar filter of 5.5 ms=μm2 , when the planar filter was perpendicular to the principal fiber direction the mean signal was about 9.5 times the Rayleigh noise estimated from the background. On the right side of Fig. 5 is the maximum signal intensity projection of all directions obtained after maximum planar filter of b ¼ 5:5 ms=μm2 . For single bundle voxels, the signal is maximum when the filter is applied perpendicular to the mean fiber orientation. In these single bundle voxels, Fig. 6 shows the normalized remaining signal at the maximum filter strength as a function of the angle between the primary eigenvector and the filter plane. Fig. 3 shows the tissue-averaged Dk and Tr D as functions of the planar filter strength for all subjects. Results in each subject were obtained from the signal averaged over all selected single bundle voxels. All results are presented in Table 1. We observe that Dk gradually increases with increasing filter strengths. For weak filter, Dk is the weighted mean of intra- and extra-axonal compartments. Increasing the filter strength reduces the weight of the extra-axonal contribution, resulting in an increased Dk . For even stronger filter, further increase in Dk is associated with the narrowing the axonal orientation distribution (Fig. 1) (as discussed after Eq. (2)). In contrast to Dk , Tr D rapidly levels off at a moderate filter strength. This is expected if the extra-axonal signal is exponentially suppressed and the remaining signal is only from the axon-

only monocompartment. Furthermore, Tr D without planar filter is only slightly above its asymptotic value after the strongest planar filter (Fig. 3) implying that Tr De is only slightly larger than Tr Da , in agreement with our recent conclusion (Dhital et al., 2017). At the strongest filter, Tr D and Dk approach closer but seem to level-off leaving a small gap in between. We fitted a constant Tr Da to the data obtained from spherical encoding and estimated D0 and σ 0 (Eq. (4)), by fitting Eq. (5) to corresponding data obtained from linear encoding (Fig. 3). Since neither equation account for the extra-axonal signal, it poses an intrinsic dependence on the data fitting interval, bP > bmin P . Whereas, too low a bmin results in a biased estimate, too large bmin admits a shorter fitting P P interval and reduces precision (see Fig. 7). The choice was made by visual inspection of Fig. 7. Nevertheless, the arbitrariness of this action is akin to the selection of the significance level for statistics. The fitting intervals ¼ 1:2 ms=μm2 and bmin ¼ 3 ms=μm2 for the singlewere selected as bmin P P direction and isotropic measurements, respectively. The difference between these values is interpreted in Discussion. From this choice, pooled all-subject data set yields mean values of D0 and Tr Da for linear and spherical encoding, respectively (Fig. 3): D0 ¼ 2:25  0:03 μm2 =ms and Tr Da ¼ 2:39  0:02 μm2 =ms. Fitting Eq. (5) to data obtained for linear encoding also provided an estimate of the width of the axonal orientation distribution (more precisely, it is the curvature of the distribution near the peak). From the fit, we obtain σ 0 ¼ 0:45  0:03; the corresponding width hsin2 θi0 ¼ 0:42  0:05 and the angle θ0 ¼ arcsinðhsin2 θi0 Þ ¼ 40  3∘ . However, these quantities overestimate the genuine width of the axonal orientation distribution due to the additional orientation dispersion introduced by 1=2

Table 1 Results for all subjects. Subject

voxels

Tr Da ðμm2 =msÞ

D0 ðμm2 =msÞ

σ 0 uncorrected

hsin2 θi0 uncorrected

hsin2 θi0 corrected

θ0 (degree) corrected

S1 S2 S3 S4 group mean pooled data

75 90 241 202 152  71 608

2:39  0:01 2:37  0:06 2:46  0:03 2:31  0:02 2:38  0:07 2:39  0:02

2:24  0:005 2:26  0:02 2:36  0:03 2:15  0:02 2:25  0:09 2:25  0:03

0:49  0:006 0:44  0:02 0:48  0:03 0:41  0:02 0:46  0:04 0:45  0:03

0:46  0:01 0:41  0:04 0:45  0:06 0:38  0:03 0:43  0:06 0:42  0:05

0:43  0:01 0:38  0:04 0:42  0:06 0:35  0:03 0:39  0:06 0:39  0:05

41  0:7 38  2 40  4 36  2 39  3 39  3

The errors show one standard deviation. The error in the group mean consists of variance within the group and the variance-averaged fit errors for individual subjects. The bottom row shows results of first pooling all data together, then fitting (Fig. 3). The axonal orientation distribution is approximated by Eq. (3). The corresponding 1=2

angle is calculated as arcsinðhsin2 θi0 Þ. In the two right most columns, σ 0 is corrected by subtracting the variance of the principal fiber direction in the selected voxels as explained in the text. 547

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pffiffiffiffiffiffiffiffiffiffi signal in proportion to 1= bP D0 . However, the planar filter suppresses the signal from such processes exponentially, as ebP D0 =2 , similar to the relation between the linear and the three-dimensional (isotropic) diffusion weighting (Lasi
c et al., 2014; Szczepankiewicz et al., 2015; Dhital et al., 2017; Vellmer et al., 2017a, b). The strongest effect on the present results could only come from the cells in which water is effectively immobilized in all three dimensions. The signal from such restricted compartment would not be suppressed by the filter contribution. The signal then would take the form

the voxel selection. The effect can be rectified assuming uniform distribution of voxels within the aperture of 30∘ and independence of microscopic parameters from the fiber orientation. The corrected values are hsin2 θi0 ¼ 0:39  0:05 with the corresponding angle, θ0 ¼ 39  3∘ (Table 1). 4. Discussion We find the intra-axonal diffusivity D0 ¼ 2:25  0:03 μm2 =ms is almost three-fourth of the free water diffusion coefficient at the body temperature which is comparable to previous direct measurement of intra-axonal diffusivity. In live excised lamprey spinal cord, Takahashi et al. (2002) obtained intra-axonal diffusivity of around 1 μm2 =ms at an ambient tem perature of 6 C. Similarly, at 20 C Beaulieu and Allen (1994) estimated parallel diffusivity along the long axis of giant squid axon as 1.6 μm2 =ms. Considering activation energy of water diffusion in white matter is 14 kJ/mol (Dhital et al., 2016), the two measurements would be equivalent to 1.82 μm2 =ms and 2.20 μm2 =ms at 37 C. A rather high values of intra-axonal diffusivity found here, rules out a large domain of the parameter space available for dMRI signal interpretation. For instance, intra-axonal diffusivity of more that 2μm2 =ms resolves the bi-modality of parameter estimation arising from multiple single-direction measurements (Jelescu et al., 2016; Novikov et al., 2018b). Our measurement is subject to a compromise between the measurement precision and sensitivity to possible regional variations of the investigated parameter. Massive averaging over many voxels allows for a precise determination of D0 , but hides possible differences between fiber bundles. The high precision of our result should not be over-interpreted, since it applies to a specific signal averaging; investigation of regional variations was beyond the scope of this study. In contrast to the majority of the present theoretical approaches, our measurement technique does not require assuming a three-dimensional Gaussian water mobility in the extra-axonal compartment. It copes with possible complex composition of the extra-axonal compartment by including one-dimensional cellular processes, restrictions to planes and connected three-dimensional space. The present method has a minimal assumptions about the white matter microstructure. Its robustness with respect to deviations from these assumptions and minor corrections to the obtained values are discussed below.

S ¼ ð1  vo ÞebDtrue þ vo ;

(10)

where vo is the fraction of immobilized water and Dtrue is the genuine target value, either Dk or Tr Da . The experimental value, Dapp , is calculated as

∂

Dapp ¼  jb¼0 ln S ¼ ð1  vo ÞDtrue ; ∂b

(11)

Using isotropic weighting, it has been shown that the presence of restricted water was limited to below 2% (Dhital et al., 2017) and also by pffiffiffi analyzing the 1= b signal behavior at very high b (Veraart et al., 2019). Using the above estimate with vo slightly below 2% results in an underestimation of the true diffusivity by 0:04 μm2 =ms, which is close to the measurement error. 4.2. Partial volume effect We chose 4 mm isotropic resolution to increase the SNR, which is crucial when an essential fraction signal is suppressed by application of strong planar filter. However, the question still retains anatomical relevance only when we can clearly resolve not just white and gray matter but also find single bundle WM voxels where the fibers are coherently aligned. Owing to high diffusivity in the cerebrospinal fluid, moderate planar water mobility filter efficiently suppresses any partial volumning due to this compartment. Possibility of having some admixture of gray matter in the selected voxels is strongly controlled by the selection criteria of high anisotropy. Inclusion of more than one fiber bundle in the selected voxels is also limited by the selection of voxels with high anisotropy. Furthermore, the planar filter also suppresses signal from possible sub-dominant fiber bundles. With this picture in mind, we maintain that the partial volume effect results in a broader orientation distribution of axons, without a significant effect on the measured intra-axonal diffusivity. Nevertheless, the use of large voxel size is also a likely reason for our estimation of the orientation dispersion to be greater than that found in the literature as discussed below. Implementation of de-noising procedure (Veraart et al., 2016) could further push the resolution limit.

4.1. Realistic white matter architecture The values of Tr Da is larger than D0 by 0:14  0:04 μm2 =ms, which remains to be explained. We speculate that this difference can be attributed either to the deviations of axon geometry from that of thin, ideally straight cylinders or to strongly non-Gaussian diffusion of the extra-axonal compartment. Non-Gaussian diffusion in the extra-axonal compartment would appear as a gradually decreasing Tr Da which is not seen in our result. Deviations from the idealized axonal geometry include all irregularities of their shape effective over the water diffusion length, in particular, in the form of axonal undulation (Nilsson et al., 2012). A simple estimate shows that curvature of axons with the typical radius 80 μm would explain the difference. Assigning such a curvature to all axons does not sound realistic, but a large contribution from a relatively small sub-population cannot be excluded. In this context, the value D0 ¼ 2:25  0:03 μm2 =ms obtained with the linear weighting is interpreted as the diffusivity along the axons, while the isotropic weighting adds about 0:07 μm2 =ms for each transverse direction due to deviations of axons from the ideal cylindrical form. Incomplete extra-axonal signal suppression by the planar filter would most likely occur if the suppression was performed with linear gradient orthogonal to the measurement direction (illustrated in the middle column in Fig. 1). For example, assuming a higher population of cellular processes in the equatorial plane, the linear gradient would only suppress

4.3. Correction for finite diffusion time Due to finite duration of the applied magnetic field gradients, the values of diffusivities found in this study slightly overestimate the genuine long-time values. The overestimation results from the finite width of the gradient power spectrum in the following signal form for weak diffusion weighting, Z 2 dω  ln S ¼  qðωÞ D ðωÞ ; 2π

(12)

where qðωÞ is the Fourier transform of the time-integrated gradient, gðtÞ ¼ γGðtÞ ¼ dqðtÞ=dt and 2D ðωÞ is the autocorrelation function of molecular velocity (Novikov and Kiselev, 2011; Kiselev, 2017; Novikov et al., 2016). The magnitude of this effect depends on the form of D ðωÞ for small ω. We use the theoretical result supported by experimental evidences that 548

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4.5. Other filter techniques and white matter microstructure

ð∞Þ

this dependence takes the form D ðωÞ  D0 þ Const ω1=2 (Novikov et al., 2014; Fieremans et al., 2016). The calculated correction for the

Although not subject to direct comparison, data obtained outside the human brain support the present result. Skinner et al. (2017) obtained the axial diffusivity of 2:16  0:22 μm2 =ms in the normal rat spinal cord using the linear water mobility filter (double PFG) in the direction orthogonal to the spinal cord. Their method is akin to the filter-dPFG method (Cory et al., 1990; Nilsson et al., 2013; Skinner et al., 2015). A lower suppression efficiency of filtered-PFG for dispersed axons is illustrated in Fig. 1. Therefore, similarity between our result and from Skinner et al. might reflect a higher degree of fiber orientation coherence in the spinal cord. The majority of white matter models treat extra-axonal space as hindered by axons, but otherwise structure-less. If this were true, both the filtered-PFG method and the planar filter would have equivalent suppression efficiency for the extra-axonal compartment. In reality, white matter is ‘structurally crowded’ being comprised of oligodendrocytes, astrocytes, microglia and vasculature. Both astrocytes and oligodendrocytes possess processes. The astrocyte processes are smooth, thin and have relatively less branching but extend more than 100 μm. The oligodendrocyte processes wrap around the axons to form the myelin layers. It is quite plausible that glial processes have anisotropic diffusion properties. The assumption of Gaussian extra-axonal space is based on the effective coarse-graining of structural features for long diffusion times (Novikov et al., 2014, 2016). The course-graining of the whole extra-axonal space within the experimental diffusion time could be effectively hindered by the low permeability of cell membranes (Nilsson et al., 2013; Yang et al., 2017). In such a scenario, the advantage of the present planar filter over the linear filter (Fig. 1) becomes crucial. Present study focused on providing an accurate estimate of the intraaxonal water diffusion coefficient. Here, we have only selected WM voxels where the major fiber orientation could be easily obtained using DTI analysis. In principle, our technique could also be used for voxels containing multiple fiber orientation if the orientations of each fiber bundle is known a priori. In that case, the planar filter would have to be selected perpendicular to each fiber bundle and the associated diffusivity could be subsequently measured. We would like to emphasize that the measurements like the one presented here are geared towards understanding a particular quantity of interest which is complimentary to deconvolving underlying parameters using post-processing techniques. While creation of solid ground for modeling efforts is the main focus of this study, testing the sensitivity of intra-axonal diffusivity in pathology or variation in intra-axonal diffusivity between different WM bundles is an open area for future research.

ð∞Þ

gradients used in our experiments is about D0  D0 ¼ 0:25 μm2 =ms. Note that this correction is only noticeable due to the dependence of D ðωÞ on the square root of frequency. A linear dependence would result ð∞Þ

in a negligible difference between D0 and D0 . 4.4. Comparison with other results The present result for D0 does not agree with the branch selection made for WMTI (Fieremans et al., 2011) and the fixed values assumed by NODDI (Zhang et al., 2012). The obtained width of the fiber orientation distribution of θ0 ¼ 39  3∘ is higher than the values around 17∘ obtained from histology (Leergaard et al., 2010; Ronen et al., 2014; Lee et al., 2018) and MRI-based studies (Leergaard et al., 2010; Ronen et al., 2014; Veraart et al., 2019; Kunz et al., 2018). For proper comparison note that histology is performed using optical microscopy in slices where the fiber images are projected on the focal plane. For small angles, this gives a pffiffiffi pffiffiffi correction factor 2 for the three-dimensional fiber tilt angle, 17∘ 2 ¼ 24∘ , which agrees with 22∘ found by Lee et al. (2018). The residual discrepancy can be a consequence of the large voxel size as discussed above. Our result agrees reasonably well with the interval ½1:9; 2:2 μm2 =ms pffiffiffi found from Da obtained using the 1= b signal scaling at very high b (Veraart et al., 2019). Similar values although with large regional variations were found by studying the combined echo- and diffusion time dependencies (Veraart et al., 2017) and using the rotational invariants of signal weighted in multiples single directions and b-values (Novikov et al., 2018b). The result also agrees with observations fiber ball imaging that resulted in intra-axonal diffusivity of 2.2 – 2:5 μm2 =ms (McKinnon et al., 2018). Kunz et al. (2018) analyzed the effect of Gd injection in the rat brain and concluded about the intra-axonal diffusivity being larger than the axial extra-axonal one, D0 > Dke . The suggested value of D0 was large, close to the absolute limit of water diffusion coefficient at the body temperature, 3μm2 =ms. Jespersen et al. (2017) investigated the time dependence of both the diffusivity and kurtosis metrics in fixed pig spinal cord and found the long-time intra-axonal diffusivity to be close to 1/2 of the free water diffusivity but larger than the extra-axonal axial diffusivity. Application of the same reduction factor to in vivo data results in the value 1:5 μm2 = ms in place of 2:25  0:03 μm2 =ms. The reduced diffusivity is attributable to the axonal damage in ex-vivo samples. Present results can also be compared with data obtained using intraneurite reporter molecule (Ackerman and Neil, 2010). To enable comparison between different molecular species, the intra-axonal diffusivity should be normalized on its value in aqueous solutions of the same molecule. Without correcting for axonal dispersion, in rat brain the corresponding reduction factor for N-acetyl-L-aspartate (NAA) was reported to be 0.46 (Kroenke et al., 2004) This should be compared with the factor D0 hcos2 θi0 =Dw ¼ 0:45  0:04, where Dw ¼ 3:08 μm2 =ms is the water diffusivity at 37∘ C and hcos2 θi0 enters according to Eq. (2) with the numerical value after the correction for the voxel selection hcos2 θi0 ¼ 0:61  0:05 according to Table 1. Accounting for axonal orientation dispersion, diffusion measurements of N-acetylaspartate þ N-acetyl aspartyl glutamate (tNAA) in the human brain corpus callosum found a reduction factor of 0:65  0:08 (Ronen et al., 2014), in a reasonable agreement with the present value D0 =Dw ¼ 0:73  0:01. The nearly equal diffusivity reduction for different molecules suggests the purely geometric (not chemical) mechanism of this effect due to impermeable membranes and other intracellular obstacles as it was discussed using data for small molecules obtained in Xenopus oocyte (Sehy et al., 2002).

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