Towards MRI axon radius mapping in clinical settings: insights from MRI-scale histology and experimental validation

Abstract

The MRI-visible axon radius is a potential clinical biomarker for, e.g., neurological disorders. However, its clinical potential remains untapped, as in-vivo MRI-based estimation lacks validation in humans and currently requires specialized research scanners. Here, we assess state-of-the-art MRI methods for axon radius estimation against a new, open-access histological gold standard of two densely sampled human corpora callosa, enabling validation via quantitative spatial correlations. Our findings show a significant correlation between estimates from histology and in-vivo dMRI acquired with a research scanner. Critically, our simulations suggest that these findings can be translated from research to clinical scanners, enabling clinical adoption. We propose specific clinical scanner protocols and illustrate their potential in a hypothetical application distinguishing individuals with autism spectrum disorder from healthy controls. Overall, our study provides promising evidence for the validity of the MRI-visible axon radius and outlines a pathway to its clinical application, while critically discussing remaining challenges.

Full text

Towards MRI axon radius mapping in clinical
settings: insights from MRI-scale histology and
experimental validation
Laurin Mordhorst
University of Luebeck
Luke Edwards
Max Planck Institute for Human Cognitive and Brain Sciences https://orcid.org/0000-0002-8320-7298
Maria Morozova
Max Planck Institute for Human Cognitive and Brain Sciences
Mohammad Ashtarayeh
University Medical Center Hamburg-Eppendorf
Tobias Streubel
University Medical Center Hamburg-Eppendorf
Björn Fricke
University of Luebeck
Francisco Fritz
University Medical Center Hamburg-Eppendorf
Henriette Rusch
Paul Flechsig Institute - Centre of Neuropathology and Brain Research https://orcid.org/0000-0001-
5989-5310
Carsten Jäger
Max Planck Institute for Human Cognitive and Brain Sciences https://orcid.org/0000-0001-6512-9903
Ludger Starker
Max Delbrück Center for Molecular Medicine in the Helmholtz Association
Thomas Gladytz
Max Delbrück Center for Molecular Medicine in the Helmholtz Association
Ehsan Tasbihi
Max Delbrück Center for Molecular Medicine in the Helmholtz Association
Joao Periquito
Max Delbrück Center for Molecular Medicine in the Helmholtz Association https://orcid.org/0000-
0003-3702-9264
Andreas Pohlmann

Max Delbrück Center for Molecular Medicine in the Helmholtz Association
Thoralf Niendorf
Max Delbrück Center for Molecular Medicine in the Helmholtz Association
Nikolaus Weiskopf
Max Planck Institute for Human Cognitive and Brain Sciences https://orcid.org/0000-0001-5239-1881
Markus Morawski
Paul Flechsig Institute - Centre of Neuropathology and Brain Research https://orcid.org/0000-0002-
3817-5186
Siawoosh Mohammadi
University of Luebeck
Article
Keywords: diffusion-weighted MRI, axon radius, biomarker, neurological disorders
Posted Date: March 4th, 2025
DOI: https://doi.org/10.21203/rs.3.rs-6114978/v1
License:   This work is licensed under a Creative Commons Attribution 4.0 International License. 
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Additional Declarations: There is NO Competing Interest.

Towards MRI axon radius mapping in clinical
settings: insights from MRI-scale histology and
experimental validation
Laurin Mordhorst1,2*, Luke J. Edwards3, Maria Morozova3,4,
Mohammad Ashtarayeh2, Tobias Streubel2, Björn Fricke1,2,
Francisco J. Fritz2, Henriette Rusch4, Carsten Jäger3,4,
Ludger Starke5, Thomas Gladytz5, Ehsan Tasbihi5,6,
Joao S. Periquito5, Andreas Pohlmann5, Thoralf Niendorf5,6,
Nikolaus Weiskopf3,7,8, Markus Morawski3,4,
Siawoosh Mohammadi1,2,9*
1Department, University Hospital of Luebeck, Luebeck, Germany.
2Institute of Systems Neuroscience, University Medical Center
Hamburg-Eppendorf, Hamburg, Germany.
3Department of Neurophysics, Max Planck Institute for Human
Cognitive and Brain Sciences, Leipzig, Germany.
4Paul Flechsig Institute - Centre of Neuropathology and Brain Research,
Medical Faculty, Leipzig University, Leipzig, Germany.
5Berlin Ultrahigh Field Facility (B.U.F.F.), Max Delbrück Center for
Molecular Medicine in the Helmholtz Association, Berlin, Germany.
6Charité – Universitätsmedizin Berlin, Berlin, Germany.
7Felix Bloch Institute for Solid State Physics, Leipzig University,
Leipzig, Germany.
8Wellcome Centre for Human Neuroimaging, Institute of Neurology,
University College London, London, United Kingdom.
9Max Planck Research Group MRI Physics, Max Planck Institute for
Human Development, Berlin, Germany.
Contributing authors: laurin.mordhorst@uni-luebeck.de;
siawoosh.mohammadi@uni-luebeck.de;
1

Abstract
The MRI-visible axon radius is a potential clinical biomarker for, e.g., neuro-
logical disorders. However, its clinical potential remains untapped, as in-vivo
MRI-based estimation lacks validation in humans and currently requires special-
ized research scanners. Here, we assess state-of-the-art MRI methods for axon
radius estimation against a new, open-access histological gold standard of two
densely sampled human corpora callosa, enabling validation via quantitative spa-
tial correlations. Our findings show a significant correlation between estimates
from histology and in-vivo dMRI acquired with a research scanner. Critically,
our simulations suggest that these findings can be translated from research to
clinical scanners, enabling clinical adoption. We propose specific clinical scanner
protocols and illustrate their potential in a hypothetical application distinguish-
ing individuals with autism spectrum disorder from healthy controls. Overall,
our study provides promising evidence for the validity of the MRI-visible axon
radius and outlines a pathway to its clinical application, while critically discussing
remaining challenges.
Keywords: diffusion-weighted MRI, axon radius, biomarker, neurological disorders
1 Introduction
Axons are critical to neural communication, with their radii inŕuencing communication
speed [1,2]. Axon radii vary spatially along and across white matter őber bundles [3ś
5] but also change temporally over the lifespan. While typical changes occur during
development and aging, others can indicate neurodevelopmental disorders [6ś8] or
neurodegenerative diseases [8,9], positioning the axon radius as a potential clinical
biomarker.
This biomarker might be measurable via water diffusion-weighted magnetic res-
onance imaging (dMRI). In the dMRI-based signal models relevant here, axons are
represented as cylindrical structures, with diffusion perpendicular to the cylinder axes
reŕecting the axon radius [10ś14]. Since axon radii are micrometer-sized ś much smaller
than millimeter-scale dMRI voxels ś the dMRI signal reŕects a combined contribution
from the individual axons within a voxel. This results in an ensemble average radius,
referred to as the effective radius (reff) [14,15]. reff is heavily inŕuenced by the largest
axons, representing the tail of the axon radius distribution.
While reff estimates from dMRI have been attempted for decades, it is only recently
that biologically plausible values could be obtained in-vivo [5,14,16ś18]. This mile-
stone has been difficult to achieve because the sensitivity of dMRI to reff is inherently
weak, requiring powerful water diffusion sensitizing magnetic őeld gradients to reli-
ably detect reff-related signal attenuation amidst noise [14,16,19]. In fact, diffusion
gradient requirements appear to be so high that promising in-vivo measurements in
humans have so far only been demonstrated at a few research sites worldwide, equipped
with specialized scanners capable of achieving gradient amplitudes up to 300 mT/m.
While these amplitudes remain far beyond the capabilities of state-of-the-art clinical
scanners with gradient amplitudes up to 80 mT/m, next-generation clinical scanners
2

with gradient amplitudes up to 200 mT/mcould enable reff measurements in clinical
environments.
While making dMRI sensitive to reff is challenging, there are also several confound-
ing factors that reduce speciőcity, i.e., the ability to correctly attribute signal changes
to variations in reff. Potential confounding factors include the contribution of other
signal compartments [14,20,21], orientation dispersion [22,23], and Rician noise bias
[24]. State-of-the art methods [5,14] promise to account for these factors through
experimental design [5,14] or advanced processing [25ś30].
To ensure that dMRI-based reff truly reŕects microstructure, the method must be
validated against a histological gold standard derived from ex-vivo tissue. However,
this validation lags behind, as the current histological gold standard [3,31,32] for
reff in the human brain [12,14,16,17,33] suffers from inadequate sampling along
several dimensions. First, the small number of donors is inhibitive if not prohibitive to
capture inter-donor variability. Second, the spatial sampling within tissue samples is
sparse, allowing only for coarse tests of biological plausibility rather than a quantitative
assessment of spatial correlations. Third, the size of individual regions of interest
(ROIs) is too small (edge length: ∼100 µm) to be representative of much larger dMRI
voxels (edge length: ∼1 mm or more). Such small ROIs likely fail to capture the tail of
the axon radius distribution ś a critical component in determining reff. While Veraart
et al. [14] used more comprehensive sampling to validate reff in rats than currently
available for humans, the authors did not quantitatively assess spatial correlation
between reff from dMRI and histology. Furthermore, translating validation in rats
to humans is uncertain due to species differences in axon radius distributions [34,
35]. Besides these sampling limitations, tissue shrinkage in histological samples is a
confounding factor for validation, introducing at least a scaling between reff from
histology and dMRI [36].
In this study, we examine the validity of dMRI-based reff mapping in experi-
mental dMRI data with histology of two densely sampled human corpora callosa,
enabling quantitative assessment of spatial correlations. First, we establish our light
microscopy-based histology dataset as a new gold standard for reff in the human
brain by demonstrating improved accuracy and precision over current histology data
[3,31,32]. Next, we show that reff from experimental in-vivo dMRI [5,14] data cor-
relates with our histological gold standard, albeit strong dynamic range reduction of
dMRI-based reff. In simulations, we show that this dynamic range reduction is due
to a limitation of the signal model. Finally, we assess clinical feasibility, and show
that our experimental results from an advanced research scanner could be translated
to next-generation clinical scanners. We identify speciőc protocols and illustrate their
potential with a hypothetical study differentiating healthy individuals from those with
autism spectrum disorder within realistic group size constraints.
2 Results
2.1 A dMRI-scale histological gold standard for reff
We obtained toluidine-stained light microscopy data of two human corpus callo-
sum tissue samples including 35 ROIs (see Fig. 1a). For each ROI, we acquired one
3

light microscopy image and extracted empirical axon radius distributions using deep
learning-based segmentation [37] (see Fig. 1b-c).
Light microscopy enables improved spatial sampling
Fig. 1d-f compares sampling statistics of our dataset against existing histology datasets
of the human corpus callosum [3,31,32]. While all datasets include only few donors
(see Fig. 1d), our dataset improves spatial sampling both by including a greater num-
ber of ROIs per donor (see Fig. 1e) and by capturing more axons per ROI, spanning
cross-sections of in-vivo dMRI voxels used in our study (see Fig. 1f).
Larger ROIs improve accuracy and precision of reff through enhanced tail
sampling
Fig. 1g-h illustrates that light microscopy ROI sizes enable smoother sampling of the
tail of the axon radius distribution than ROI sizes used by Aboitiz et al. [31], which
would result in occasional spikes in the tail and deviations in reff. This effect of ROI
size on reff is further explored in Fig. 1i, which shows sampling distributions of reff
computed from repeated sampling at different ROI sizes. Smaller ROIs underestimate
reff but increase the likelihood of overestimated outliers (see example in Fig. 1h),
indicating lower accuracy and precision. The low accuracy and precision of smaller
ROIs is quantiőed across all ROIs in Fig. 1j-k. As ROI size increases, both accuracy
(bias) and precision (coefficient of variation) improve, with accuracy improving more
rapidly. For ROI sizes of existing histology data [3,31,32], the expected bias would
be 4to 12 %, whereas the expected coefficient of variation would be 14 to 21 %.
2.2 Comparison of reff across modalities
To validate reff experimentally against our histological gold standard, we acquired
in-vivo and ex-vivo dMRI data. In-vivo, we acquired magnitude dMRI data of őve
healthy human subjects on a Siemens Connectom 3 T scanner with maximum gradient
amplitude of 300 mT/m, following the protocol suggested by Veraart et al. [5]. Ex-vivo,
we acquired magnitude dMRI data of tissue sample CC-01 (see Fig. 1a) on a Bruker
Biospin 9.4 T preclinical scanner with maximum gradient amplitude of 1500 mT/m
using a protocol akin to that suggested by Veraart et al. [14]. As a bridge between
histology and dMRI, we conducted dMRI simulations under both idealized (SNR =
∞) and experiment-like conditions with added Rician noise (in-vivo: SNR = 32; ex-
vivo: SNR ranging from 17 to 51, depending on the diffusion-weighting b). For a fair
comparison between in-vivo dMRI experiments/simulations and histology, we scaled
axon radii from histological distributions by 1.3 to account for tissue shrinkage [31,38].
Histological samples agree across broader regions but vary locally
Fig. 2a-b show the spatial patterns of reff in histology across the mid-sagittal section
of the corpus callosum. Both samples exhibit similar inter-region trends, with an alter-
nating low-high pattern across the anterior midbody, midbody, posterior midbody,
and splenium (see also Section SI1). However, there’s strong intra-region variability
within the splenium, inconsistent across tissue samples. In other regions, intra-region
variability cannot be conclusively assessed due to sparser sampling.
4

bc
CC-01
a
j
ik
CC-02
df
Fig. 1 A dMRI-scale histological gold standard for reff. (a) Human corpus callosum tissue
samples with annotated regions of interest (ROIs) scanned via light microscopy. (b) Light microscopy
image of an example ROI. (c) Magnified view of (b) showing myelin sheath (black) and axonal
body (white) segmentation. (d-f ) Sampling statistics of human corpus callosum histology datasets
[3,31,32]: (d) donors per dataset, (e) ROIs per donor (mean across donors) and (f ) mean sample area
and axon count per ROI (double-logarithmic scale). The dashed line represents the cross-sectional area
of in-vivo dMRI voxels used in our study. (g-h) Axon radius distribution for (g) a light microscopy
ROI and (h) a random subsample of the distribution in (g) including 103axons, mimicking a ROI as
presented by Aboitiz et al. [31]. Vertical dotted lines denote reff ; insets highlight tails of axon radius
distributions. (i) Sampling distribution of reff as a function of ROI size (axon count) for the ROI in
(g). The blue marker and dashed line represent reff computed from all axons within the ROI, while
boxplots show simulated sampling distributions for smaller ROI sizes, indicating the median (line),
interquartile range (IQR, box), whiskers (1.5 IQR), and outliers (dots). Box colors reflect datasets,
categorized by ROI size (see legend over d-g). Note that we do not explicitly mark the dataset of
Caminiti et al. [3] as the ROI size roughly coincides with the ROI size used by Aboitiz et al. [31].
(j-k) Bias and coefficient of variation as a function of the ROI size based on sampling distributions
as shown in (i). Markers showing mean ±standard deviation across ROIs. Color encoding follows
definitions in (i).
5

In-vivo dMRI captures coarse spatial reff pattern at reduced dynamic range
Fig. 2c-j compare spatial reff patterns across the corpus callosum between histology,
dMRI experiments and simulations, both for the ex-vivo (see Fig. 2c-f) and in-vivo
scenario (see Fig. 2g-j). Ex-vivo dMRI-based reff underestimate histological values
(see Fig. 2d), aligning with simulations (see Fig. 2e-f). While in-vivo simulations also
predict an underestimation of reff (see Fig. 2i-j), experimental reff overestimate his-
tological values (see Fig. 2h), indicating effects not captured by simulations. Overall,
both ex-vivo and in-vivo reff patterns exhibit a reduced dynamic range, suggesting low
sensitivity to microstructure (see Fig. 2d-f,h-j). This low sensitivity complicates the
capture of spatial patterns ex-vivo (see Fig. 2d-f), but the group-average pattern of
in-vivo dMRI experiments (see Fig. 2h) shows some resemblance to histology, hinting
at a similar alternating low-high pattern across anterior midbody, midbody, poste-
rior midbody, and splenium. However, the high values in the genu, relative to other
regions, do not align with histological patterns. Additionally, partial volume effects
may inŕuence the in-vivo dMRI pattern, as suggested by extreme values near border
regions.
In-vivo dMRI-based reff correlate with histology
The resemblance of the group-average spatial reff pattern from in-vivo dMRI exper-
iments with histology is reŕected in a signiőcant correlation (see Fig. 2k). However,
this analysis exhibited some variability due to the non-deterministic nature of our
in-vivo dMRI processing (Fig. 2k shows a representative iteration; over 10 iterations
we yielded: R= 0.414 ±0.03, all p < 0.05; see Section SI2). The signiőcant corre-
lation of in-vivo dMRI-based reff with histological values was not predicted by our
simulations (see Fig. 2l), which, however, assume a single subject rather than a group-
average. Section SI3 provides a more comparable scenario to simulations by showing
per-subject correlations, revealing no signiőcant correlation with histology for most
individual subjects as predicted by simulations. For ex-vivo dMRI experiments (see
Fig. 2k), there’s no signiőcant correlation with histology. This is likely due to reduced
precision compared to simulations (see Fig. 2l) and the need to estimate an additional
parameter, fim, which can confound reff estimation (see Section SI4).
A model-inherent bias reduces the dynamic range
The idealized dMRI simulations (see Fig. 2m) reveal a primary cause of the reduced
dynamic range of dMRI-based reff: a proportional bias for larger reff, which we refer
to as "model-inherent bias". This bias affects absolute agreement, as measured by the
normalized root mean square error (NRMSE), by shifting values from the unity line.
Additionally, it reduces Rby limiting the dynamic range on the upper end of reff
values, thereby obscuring correlations under noisy conditions (see Fig. 2k-l). Notably,
the model-inherent bias is stronger ex-vivo than in-vivo (see slopes in Fig. 2m). In
Sections SI5 and SI6, we characterize the model-inherent bias across broader range
of reff and identify driving factors, revealing that the model-inherent bias scales non-
linearly with reff and becomes more pronounced for high gradient amplitudes as used
in our ex-vivo dMRI protocol.
6

Low sensitivity to small reff additionally reduces the dynamic range
In-vivo experiment-like simulations (see Fig. 2l) show a mild, noise-induced overesti-
mation of smaller reff values, affecting sensitivity at the lower end of reff values. This
reduced sensitivity to small reff hints at the practical resolution limit, below which reff
values may no longer be reliably distinguished from noise [19].
2.3 Optimal in-vivo dMRI protocols for reff mapping
We optimized in-vivo dMRI protocols for next-generation 3 T clinical scanners with
maximum gradient amplitude (gmax) up to 200 mT/m. To this end, we conducted
a grid search for optimal protocol parameters and evaluated protocol candidates by
simulating their reff estimates for our corpus callosum dataset and maximizing the
correlation (R) with histological reff. We accounted for SNR variations due to protocol
parameter choices, but also considered increased baseline SNR levels, independent of
protocol parameters, to explore potential gains achievable through technical or acquisi-
tion improvements. In contrast to our dMRI experiments, we assumed Gaussian- rather
than Rician distributed signals, which can be achieved with advanced preprocessing
techniques [39ś41].
Next-generation clinical scanners could narrow gap to research scanners
Fig. 3a-b show Rand NRMSE as a function of gmax, contextualizing the achievable
performance of next-generation clinical scanners in comparison to state-of-the-art 3 T
clinical scanners, state-of-the-art research scanners as used in our dMRI experiments,
and next-generation research scanners, assuming 90 % of nominal gmax values. For any
baseline SNR level, Rconverges to a maximum value at a certain minimum gmax, where
NRMSE is also optimal or close to optimal. While state-of-the-art clinical scanners
consistently perform well below optimal Rand NRMSE values, next-generation clinical
scanners achieve Rvalues much closer to those of research scanners and can reach near-
optimal NRMSE at higher SNR baseline levels. The protocol parameters and further
analyses for all optimal protocols referenced in Fig. 3a-b are presented in Section SI7.
Next-generation clinical scanners may reveal correlation at increased SNR
Fig. 3c-e show simulated reff for optimal next-generation clinical scanner protocol at
each baseline SNR level (corresponding to colored markers in Fig. 3a-b). At SNR ≈
27, reŕecting the expected SNR of the protocol candidate under our experimental
conditions, next-generation clinical scanners would not reveal a signiőcant correlation
with our histology data (R= 0.29,p= 0.13, see Fig. 3c). This aligns roughly with
simulations of our experimental protocol on a state-of-the-art research scanner in
Fig. 2m, although these simulations assume Rician- rather than Gaussian-distributed
signals (see Section SI8 for impact of noise distribution on reff estimaton). However,
our simulations suggest that a signiőcant correlation could be revealed at SNR ≈48
(R= 0.49,p= 3.7e−3, see Fig. 3d) and a stronger correlation at SNR ≈68 (R= 0.63,
p < 0.05, see Fig. 3e).
7

in-vivo
g
group-average
across donors
group-average
across subjects
hij
group-average
across donors
group-average
across donors
ex-vivo
cdef
idealized dMRI
simulations
experiment-like
dMRI simulations
dMRI experiments
histology
group-average
across donors
group-average
across donors
group-average
across donors
CC-01
histology
a
splenium
posterior
midbody midbody
anterior
midbody
CC-01
bposterior
midbody midbody
anterior
midbody
CC-02
genu
splenium
genu
Fig. 2 Comparison of reff across modalities. (a-b) Histological spatial patterns of reff across
the corpus callosum, shown in mid-sagittal MNI slice with subregions indicated (dashed lines). (c-f)
Ex-vivo comparison of spatial patterns: (c) histology, (d) dMRI experiments, (e) experiment-like
dMRI simulations (experimental SNR), and (f) idealized dMRI simulations (SNR = ∞). Patterns in
(c,e-f) show the group-average across donors, whereas the pattern in (d) covers the 15 ROIs of CC-01
scanned with ex-vivo dMRI (void area indicates ROI not scanned with ex-vivo dMRI; see Fig. 5b,e).
For experiment-like simulations in (e), the pattern reflects the median across 1000 noise realizations.
(h-k) In-vivo comparison of spatial patterns analogously to (c-f) with the following exceptions: spa-
tial patterns in (h,j-k) are based on histological axon radii scaled by 1.3 to compensate for tissue
shrinkage [31,38] and pattern in (i) reflects the group-average across in-vivo subjects (see Section SI3
for per-subject patterns). (l-n) Quantitative comparisons of reff from dMRI experiments/simulation
scenarios in (d-f,i-k) against histology. Markers represent histological ROIs in Fig. 1a, with colors
encoding experimental conditions (in-vivo vs. ex-vivo, see legend). For dMRI experiments in (l), ex-
vivo markers include the 15 ROIs of CC-01 scanned with ex-vivo dMRI (see Fig. 5b,e), whereas
in-vivo markers denote group-average reff values (see Section SI3 for per-subject analyses). Note that
reff from in-vivo dMRI experiments exhibited some variability due to non-deterministic processing
(across 10 iterations: R= 0.414 ±0.03, all p < 0.05; see Section SI2). The simulations in (m-n) use
all histological ROIs and assume a single subject/donor scanned with dMRI. The 95 % confidence
intervals (shaded areas in (m)) were computed across 1000 noise realizations. The dashed lines illus-
trate theoretical perfect agreement. The legends provide metrics computed over all ROIs, including
Pearson’s correlation coefficient (R) and the corresponding p-value, the normalized root-mean-square
error (NRMSE), and the fitting success rate (S) (see Section 4.7 for metric definitions).
8

The model-inherent bias is a relevant factor in procotol design
While Rremains stable after reaching the optimum at some gmax (see Fig. 3a), NRMSE
decreases thereafter (see Fig. 3b). We attribute this loss of absolute agreement to the
increasing inŕuence of the model-inherent bias, making it a relevant factor in protocol
design for scanners with very high gmax, such as next-generation research scanners.
2.4 Simulation of clinical application for reff mapping
We investigated whether reff could distinguish between healthy individuals and those
with autism spectrum disorder (ASD) as a potential clinical application of the optimal
next-generation scanner protocols identiőed in Fig. 3c-e. Based on a reported 28.6 %
reduction of axon radii in the splenium under ASD conditions [7], we simulated ASD
conditions by adjusting our histological axon radius distributions accordingly, while
leaving distributions untouched for healthy controls (see Fig. 4a). In a Monte Carlo
simulation, we estimated the statistical power to detect group differences in mean reff
in the splenium when using our optimized dMRI protocols for next-generation clinical
scanners (see Fig. 4b-c).
Fig. 4d shows the statistical power as a function of the group size, assuming equal
group sizes for subjects with ASD and healthy controls. A typical statistical power
target of 0.9could be reached for group sizes of 12 subjects at SNR ≈27, reŕecting the
conditions of our experimental protocol. However, increased SNR ≥48 would strongly
reduce the group size requirements to 6 or fewer subjects per group.
9

ab
cde
Fig. 3 Optimal in-vivo dMRI protocols for reff mapping.(a-b) Optimal Pearson’s correlation
coefficient (R) and normalized root mean square error (NRMSE) as a function of maximum gradient
amplitude (gmax). Markers encode gmax of existing clinical scanners and research scanners (assuming
90 % of the nominal gmax). Colored markers highlight optimal protocols for next-generation clinical
scanners. Line styles indicate different SNR baseline levels. While the reference SNR baseline level
reflects our experimental conditions, increased SNR baseline levels assume an SNR increase through
potential technical or acquisition advances. In addition, we accounted for SNR variation due to
protocol parameter differences (see Eq. (13)). For our experimental protocol, baseline SNR levels
would correspond to SNR values of 32 (reference), 56 (75 % increased) and 80 (150 % increased).
Note that we optimized protocols by minimizing R, whereas NRMSE is an auxiliary metric. (c-
e) Comparison of estimated reff with histological gold standard for optimal next-generation clinical
scanner protocols across baseline SNR levels (color coding matches the highlighted protocols in (a-b)).
SNR values of protocols are annotated above plots. Markers represent histological ROIs in Fig. 1a. The
95 % confidence intervals (shaded areas) were computed across 1000 noise realizations. The dashed
lines illustrate theoretical perfect agreement. The legends provide metrics computed over all ROIs,
including Pearson’s correlation coefficient (R) and the corresponding p-value, the normalized root-
mean-square error (NRMSE), and the fitting success rate (S) (see Section 4.7 for metric definitions).
10

b1.5 3
ROI
average:
cd
Fig. 4 Simulation of clinical application for reff mapping.(a) Exemplary axon radius dis-
tributions in the splenium for subjects with autism spectrum disorder (ASD) and healthy controls,
based on distributions from a splenium ROI in Fig. 1a. ASD conditions were simulated by scaling
down radii by 28.6 % [7], whereas distributions for healthy controls were used as is. (b) Illustration of
subject-level mean reff in the splenium, denoted as ¯reff . Values were computed over 11 voxels, reflect-
ing the average number of splenium voxels in the mid-sagittal slice in our in-vivo dMRI subjects.
(c) Illustration of one Monte Carlo iteration for statistical power estimation. We assessed group dif-
ferences in ¯reff between healthy and ASD subjects with a t-test using a significance level α= 0.05,
with the null hypothesis being that the group means are equal. We then approximated the statisti-
cal power over 5000 Monte Carlo iterations as the fraction of tests that rejected the null hypothesis.
(d) Statistical power as a function of the group size. Colors indicate optimal next-generation clinical
scanner protocols for different SNR baseline levels determined in Fig. 3c-e. The dashed line indicates
a typical statistical power target value of 0.9. We assumed equal group sizes.
3 Discussion
We examined the validity of water diffusion-weighted MRI (dMRI) methods to esti-
mate the effective axon radius (reff) in humans by assessing spatial correlations with a
new, light microscopy-based histological gold standard of two densely sampled corpora
callosa. In contrast to ex-vivo dMRI experiments, in-vivo dMRI-based reff showed a
signiőcant correlation with histology at the group level, providing promising but pre-
liminary evidence for its validity. Notably, this correlation emerges despite a newly
discovered limitation of the signal model, which weakens the relationship between
dMRI-based reff and histological values. While our in-vivo acquisitions were performed
on an advanced research scanner, our simulations suggest that the method can be
translated to more widely available clinical scanners with realistic SNR advances. We
11

propose speciőc clinical scanner protocols and illustrate their potential in a hypothet-
ical application distinguishing subjects with autism spectrum disorder from healthy
controls within realistic group size constraints.
While recent studies have assessed whether dMRI-based estimates of reff fall within
a biologically plausible value range [5,14,16,17], our study quantitatively compares
spatial trends of reff across the human corpus callosum with histological values. We
do not őnd absolute agreement between reff from in-vivo dMRI and histology, but
reveal a signiőcant spatial correlation across these modalities on the group level. Visual
evaluation suggests that this correlation largely reŕects a coarse low-high pattern
across the anterior midbody, midbody, posterior midbody, and splenium. This pattern
is well documented, both in humans [5,17,31,33,40] and macaques [3,42]. Beyond
this coarse pattern on the group level, patterns of individual subjects/donors exhibit
notable variability (see Fig. 2a-b and Section SI3), raising the question of whether
őner common spatial patterns exist at all. Moreover, we see hints of partial volume
effects, potentially inŕuencing the observed pattern. While validation via ex-vivo dMRI
promises to mitigate the aforementioned challenges, it seems to introduce new ones.
For our experimental protocols, ex-vivo dMRI is more sensitive to the "model-inherent
bias" than in-vivo dMRI, leading to strong underestimation and a weak relationship
with histology. Additionally, the presence of the immobile water compartment may
further confound reff estimation (see Section SI4). Although our quantitative spatial
correlation analysis provides the most realistic perspective on the validity of MRI-
based reff measurements to date, the next step should be to conőrm their validity on
an independent dataset.
Even though we revealed a correlation of in-vivo dMRI-based reff with histology
on an advanced research scanner, our simulations suggest that in-vivo reff-mapping
is well within reach of clinical environments. This is possible because emerging next-
generation clinical scanners őll the gap in strong gradient amplitudes essential for
reff mapping, bridging the range between current clinical scanners (80 mT/m) and
research scanners (≥300 mT/m) with gradient strengths of up to 200 mT/m. In fact,
the performance of next-generation clinical scanners (as assessed by Pearson’s Rand
NRMSE) could be remarkably close to that of research scanners, enabling reliable
detection of reff at increased SNR. At SNR ≈27, reŕecting our experimental conditions
after adjusting for protocol differences, next-generation clinical scanners would not yet
reveal a correlation of dMRI-based reff from a single subject with our histology dataset.
However, assuming SNR increases through technical advances, a signiőcant correlation
(R= 0.49,p= 3.7e−3) could be reached at 75 % increased SNR ≈48, and a stronger
correlation (R= 0.63,p < 1e−3) at 150 % increased SNR ≈68, although the latter
may be unrealistic. Nonetheless, meaningful SNR advances seem possible, as recently
demonstrated through advanced acquisition techniques with a 30 % gain using the
same protocol as in our study [18], corresponding to SNR ≈35. On the processing side,
adapting gaussian-distributed dMRI [39,40] promises to enable effective denoising
[41], likely relaxing SNR requirements. Looking further ahead, ongoing advancements
in coil design and array conőgurations hold promise for further boosting SNR [43,44].
12

Moving from feasibility to potential clinical impact, SNR advances could also be
critical for translating reff into actual applications. In a hypothetical clinical appli-
cation, we explored the group size requirements to differentiate subjects with autism
spectrum disorder from healthy controls, based on an assumed reduction of axon radii
by 28.6 % in the mid-sagittal area of the splenium [7]. While our simulations suggest
that group sizes of 12 subjects suffice to achieve a statistical power target of 0.9 at
SNR ≈27, group sizes of only 6 or less subjects would be needed at SNR ≥48.
These modest group size requirements are facilitated by spatial averaging across sple-
nium voxels, which effectively increases the SNR ś a strategy previously employed to
characterize reff along white matter őber tracts [5]. However, we suggest that these
results rather emphasize the strong positive impact of potential SNR gains rather than
providing accurate estimates for the required group sizes.
Expanding beyond clinical protocols, we provide broader insights into protocol
design for in-vivo reff measurements, exploring gradient amplitudes ranging from state-
of-the-art clinical scanners (gmax = 80 mT/m) to next-generation research systems
(gmax = 500 mT/m). Consistent with previous őndings [14,19], we conőrm that high
gmax values are generally beneőcial across realistic SNR regimes for optimizing both R
and NRMSE. However, at high SNR and large reff , next-generation research scanners
offer only marginal improvements in Rover next-generation clinical scanners while
yielding suboptimal NRMSE due to increasing model-inherent bias. These őndings
suggest that the model-inherent bias is an important consideration in protocol design
and that improved modeling would be necessary to fully leverage the potential of
next-generation research scanners.
Our histology dataset enables not only the assessment of quantitative spatial corre-
lations but also provides broader insights and validation resources for reff. Critically, it
facilitates realistic dMRI simulations that capture the combined contributions of giant
axon ensembles representative of MRI voxels. Furthermore, the giant axon ensembles
improve both precision of reff by up to 21 % and bias by up to 12 % compared to
commonly cited histology datasets [3,31,32]. This enhanced precision mitigates the
risk of extreme outliers, a potential problem in current dMRI-histology comparisons
[14,16,17]. While the bias can be corrected in such comparisons, it also provides an
intriguing insight: reff may be up to 12 % larger than previously estimated, facilitat-
ing the resolution of small reff values with dMRI [19]. Overall, our dataset establishes
a new histological gold standard for reff and dMRI-scale axon radius distributions
through improved spatial sampling, offering a valuable resource for future validation
studies.
We acknowledge several limitations in our study. First, we applied spatial smooth-
ing to in-vivo dMRI-based reff maps. While smoothing reduces sensitivity to local
variation, our results in Section SI9 suggest that it is essential for mitigating noise and
alignment inaccuracies inherent to comparison of voxel-sized ROIs across subjects and
modalities. In addition, it seems questionable whether őner spatial structures could be
successfully evaluated at all between modalities, given both the notable intra-cohort
variabilities and potential inter-cohort variability due to age differences (mean ages:
61 years for histology, 31 years for in-vivo dMRI). Second, we assumed uniform scaling
13

of axon radius distributions to account for tissue shrinkage. While potentially over-
simpliőed [36], this assumption does not affect the observed correlation with in-vivo
dMRI-based reff, nor does it alter key insights on the model-inherent bias. As shown
in Section SI6, the bias and its severity are driven by the magnitude of reff-values,
a property unlikely to be strongly altered by reőned tissue shrinkage models. Third,
our dMRI simulations rely on őxed literature values to model tissue properties, poten-
tially calibrating outcomes to these őxed values. Fourth, we assumed Gaussian rather
than Rician noise for protocol optimization and clinical application, as Rician noise
more strongly obscures correlation (see Section SI8) but can, in principle, be mitigated
through advanced preprocessing techniques [16,39,41].
Our research identiőes areas for future work. First, validating the correlation
between dMRI-based reff and histology needs to be conőrmed on an independent
dataset. Second, reőning the signal model seems an immediate avenue for improve-
ment, either by incorporating higher-order terms in Eq. (A9) or by exploring the
apparent linear decay of S◦(b, H(r)) with reff (see Section SI6). Third, our simulations
could be extended to account for prominent confounding factor candidates, such as,
e.g., residual extra-axonal signal [14,20]. This could potentially explain the remain-
ing differences between dMRI and histology, particularly the strong overestimation of
in-vivo dMRI-based reff (consistent with [5,18]).
In conclusion, our őndings highlight both the promise and challenges of reff as
a biomarker in the human brain. We provide promising evidence that reff reŕects
microstructure and can be measured in clinical settings, encouraging further research
into its potential applications for diagnosing and monitoring disorders where axon
morphology plays a critical role. At the same time, we identify critical constraints
of the state-of-the-art reff estimation method and discuss limitations of our valida-
tion approach, emphasizing the need for further methodological improvements and
independent validation. To support this effort, we provide an open-access dataset as
a resource for future research. Together, these contributions offer a roadmap to har-
ness the full potential of MRI-based reff mapping and pave the way for its clinical
application.
4 Materials and Methods
4.1 The dMRI-visible effective axon radius reff
Here, we provide a brief introduction to the MRI-visible effective axon radius reff and
the dMRI signal model used for its estimation. For a more detailed description, see
Section A. The dMRI-visible effective axon radius
reff =4
s⟨r6⟩
⟨r2⟩.(1)
is a scalar, tail-weighted statistic of the axon radius distribution within a dMRI voxel
[14,15]. reff can be estimated in a regime of strong diffusion weighting (b), suggested
to be around b≥6 ms/µm2for in-vivo dMRI [14,20] and b≥20 ms/µm2for ex-vivo
14

Table 1 Tissue sample information for the ex-vivo dataset. Histology ROIs and Ex-vivo
dMRI ROIs indicate the number of regions of interest (ROIs) analyzed in each modality. See
Fig. 1a for locations of all histology ROIs and Fig. 5a-b,e for common ROIs between histology
and ex-vivo dMRI.
Sample Sex Age
[years]
Postmortem delay
[hours]
Cause of death Histology ROIs Ex-vivo dMRI ROIs
CC-01 m 61 20 myocardial infection 16 15
CC-02 f 60 24 multi organ failure 19 0
35 15
dMRI [14]. In this regime, one can approximate the powder-averaged dMRI signal as
S◦(b)≈β
√b·e−bD⊥
a+fim,(2)
where βis a signal scaling factor, D⊥
ais the intra-axonal perpendicular diffusivity, and
fim is the signal of the immobile water compartment [21]. reff is directly linked to D⊥
a
through
reff =4
r48
7δ(∆ −δ
3)D⊥
aD0,(3)
where δis the diffusion gradient duration, ∆is the diffusion gradient separation, and
D0is the diffusivity of the axoplasm [14]. Using Eqs. (2) and (3) and an estimate for
D0, one can jointly estimate reff and β, e.g., via non-linear őtting [45].
4.2 Histology
Tissue samples
We used two human corpus callosum tissue samples, CC-01 and CC-02. See Table 1
for sample information.
Tissue preparation
We immersion-őxed whole brains in 3 % paraformaldehyde and 1 % glutaraldehyde
in phosphate-buffered saline (pH 7.4). We then extracted the corpora callosa and
bisected them along the mid-sagittal plane, yielding hemispheric sections (see Fig. 5e).
We prepared one hemispheric section from each donor for histology and preserved
the other hemisphere for ex-vivo dMRI. In the histology hemisphere, we cut a slice
of the tissue sample orthogonal to the mid-sagittal plane and extracted ROIs (see
Fig. 5a,e). The ROI segments were then contrasted with osmium tetroxide and uranyl
acetate, dehydrated in graded acetones, and embedded in Durcupan resin. For imaging
with light microscopy, we cut semi-thin sections (≈500 nm thickness) parallel to the
mid-sagittal plane on a Reichert Ultractut II. The sections were mounted on Thermo
Scientiőc SuperFrost Plus glass slides, stained with 1 % toluidine blue, air dried and
coverslipped with Sigma-Aldrich Entellan toluene.
Image Acquisition
We acquired light microscopy images (one per ROI) using a Zeiss AxioScan Z1
(objective: 40×, numerical aperture: 0.95, resolution: 0.1112 µm/pixel; resolution limit:
292 nm). An example image is shown in Fig. 1b-c.
15

d
c
CC-01
Histology
Histology ex-vivo dMRI Histology
MNI
MNI
MNI
in-vivo dMRI
in-vivo dMRI
CC-02
non-linear
registration (2D)
non-linear
registration
T1w + FA
template (MNI)
T1w + FA
image (subject)
corpus callosum
tissue mask
(histology)
histology
hemisphere thresholded FA
template (MNI)
L
a
efg
A
A
A
I
I
SS
S
S
S
A
A
AA
A
P
P
P
P
P
PP
P
II
I
I
S
ex-vivo dMRI
b
R
ex-vivo dMRI
hemisphere
CC-01
A
I
S
P
IS
A
P
Fig. 5 Regions of interest in different spaces and their registration. (a-d) Regions of
interest (ROIs) shown in different spaces: (a) histology, (b) ex-vivo dMRI, (c) MNI space (overlaid
on T1-weighted template), (d) in-vivo dMRI (overlaid on T1-weighted image). Polygons and circles
indicate ROI boundaries and centroids, with colors representing tissue sample CC-01 (magenta) or
CC-02 (green). (e) Registration between histology and ex-vivo dMRI. We bisected the brain along
the mid-sagittal plane, indicated by the red line, yielding hemispheric sections for histology (left)
and ex-vivo dMRI (right). We first defined histological ROIs near the mid-sagittal plane; then, we
manually defined corresponding ROIs in ex-vivo dMRI. Magnified views illustrate an example of
matching ROIs in histology and ex-vivo dMRI (extracted tissue area in histology and magenta area
in ex-vivo dMRI). Note that we scanned only part of the genu with ex-vivo dMRI. (f ) Registration
between histology and MNI space. We manually created two-dimensional tissue masks (left image)
for the images in (a) and registered these masks with the mid-sagittal slice of a fractional anisotropy
(FA) atlas (the FSL HCP-1065 FA atlas [46], thresholded at FA ≥0.3) in MNI space (see red area in
right image). (g) Registration between MNI space and in-vivo dMRI. We simultaneously registered
T1-weighted image and FA map in native space to their corresponding templates in MNI space (the
FSL HCP-1065 FA atlas and the FSL MNI152 T1-weighted template [46]).
Parameter estimation
We segmented axons using a deep-learning-based method [37] (see Fig. 1c) and derived
empirical axon radius distributions by calculating the radius of circles with equivalent
areas for each axon. For comparison with in-vivo dMRI, we compensated for tissue
shrinkage by scaling each axon radius (scaled radii: r′= 1.3r), where the factor 1.3
was estimated as the mean of previously reported values [31,38]. Finally, we computed
reff values from the empirical axon radius distributions using Eq. (1).
4.3 Ex-vivo dMRI
Tissue samples
We used the remaining hemisphere of tissue sample CC-01 for ex-vivo dMRI (see
Section 4.2 and Fig. 5e for the bisection into hemispheres). Sample information is
provided in Table 1.
16

Tissue preparation
We cut the tissue sample into őve segments along the anterior-posterior axis (see
Fig. 5b) using a Reichert Ultracut II, and embedded the segments in 1.5 % agarose in
phosphate buffer saline in a custom-made container.
Image Acquisition
We acquired magnitude dMRI data using a Bruker Biospin 9.4T scanner with a
single-channel transceiver volume coil and a gradient insert coil with a maximum
gradient amplitude of 1500 mT/mat the Berlin Ultrahigh Field Facility in Berlin,
Germany. We followed a protocol similar to that suggested by Veraart et al. [14]
for ex-vivo reff mapping in rats. Brieŕy, we applied diffusion-weighting for 65 gradi-
ent directions using a segmented EPI sequence with four segments and the following
őxed parameters: δ= 7 ms,∆ = 20.1 ms, echo time TE= 34.7 ms, repetition
time TR= [15000,25000] ms (segment-dependent), and isotropic voxel edge length
of 0.35 mm. For different tissue segments, the őeld-of-view was adjusted between
22 ×28 ×9 mm3and 25 ×30 ×10.5 mm3. We varied bbetween 2.5and 100 ms/µm2,
and gradient amplitude (g) between 200 and 1278 mT/m, as detailed in Table 2. To
enhance SNR, we averaged repeated measurements prior to image reconstruction for
higher b-values, as shown in the Repetitions column of Table 2.
Table 2 Ex-vivo dMRI acquisition
parameters.Repetitions denotes the
number of repeated measurements per
diffusion gradient direction, which were
averaged in k-space prior to image
reconstruction.
b
[ms/µm2]
g
[mT/m] Repetitions
2.5 200 1
5.0 283 1
7.5 347 1
10.0 401 1
20.0 567 2
30.0 695 2
40.0 802 3
50.0 896 3
60.0 982 4
70.0 1061 5
80.0 1134 6
90.0 1203 7
100.0 1278 8
Preprocessing
We corrected for Gibbs ringing artifacts [47,48]. To account for signal drift across b-
shells, we normalized images within each b-shell to an S(b= 0) image acquired at the
start of acquisition for that shell.
17

Parameter estimation
Per b, we estimated the noise level ˆσusing Marchenko-Pastur principal component
analysis [48ś50] prior to preprocessing. Then, we estimated S◦(b)as the zeroth order
spherical harmonic using an estimator of the even order spherical harmonic coeffi-
cients up to the sixth order. Speciőcally, we determined the spherical harmonics basis
functions [51ś54] and estimated the coefficients using a Rician maximum likelihood
estimator [30], which relied on the b-dependent ˆσmaps.
For b≤10 ms/µm2, we estimated the main őber direction µ using NODDI [55,56],
using the ˆσmap for b= 2.5 ms/µm2.
For b≥20 ms/µm2, we estimated reff from S◦(b)using Eqs. (2) and (3) via non-
linear őtting [45], assuming D0= 0.35 µm2
/ms [57]. We estimated fim (see Eq. (2))
from strongly decayed directional signals. To this end, we selected signals from the
highest b-shell with high alignment between g and µ (angle ≤20◦), őtted a Rician
distribution, and approximated fim as its expected value.
4.4 In-vivo dMRI
Subjects
We recruited őve healthy adult subjects (age: 31 ±3years, representing mean ±
standard deviation; sex: 2 male, 3 female).
Image Acquisition
We acquired magnitude dMRI data using a 32-channel receive coil and 300 mT/m
gradient coils on a Siemens Connectom 3T scanner at the Max Planck Institute for
Human Cognitive and Brain Sciences in Leipzig, Germany. We followed the dMRI
protocol described by Veraart et al. [5]. Brieŕy, we used a single-shot multi-band echo-
planar imaging (EPI) sequence with blipped-CAIPI (multi-band factor: 2) and in-plane
GRAPPA acceleration (acceleration factor: 2). We applied diffusion-weighting with
the following őxed parameters: δ= 15 ms,∆ = 30 ms,TE= 66 ms,TR= 3500 ms,
matrix size of 88 ×88 with 54 slices, and isotropic voxel edge length of 2.5 mm. We
varied b={0.5,1,2.5,6,30.45}ms/µm2for {30,30,30,60,120}gradient directions and
used variable gradient amplitude g={36,51,80,124,279}mT/m.
For geometric susceptibility correction, we acquired 23 non-diffusion-weighted
images with the same and 10 images with reverse phase encoding. Additionally, we
acquired T1-weighted MP-RAGE images [58].
Preprocessing
We corrected for Gibbs ringing artifacts [47,48], eddy current and motion artifacts
[48,59,60], and gradient non-linearity distortions [61,62].
Parameter estimation
For b≤2.5 ms/µm2, we estimated the noise level ˆσusing Marchenko-Pastur principal
component analysis [48ś50] prior to preprocessing. After preprocessing, we estimated
the apparent diffusion tensor [48,63,64] and mapped fractional anisotropy (FA) [48,
64].
18

For b≥6 ms/µm2, we estimated S◦(b)as the zeroth order spherical harmonic
using an estimator of the even order spherical harmonic coefficients up to the sixth
order. Speciőcally, we determined the spherical harmonics basis functions [51ś54] and
estimated the coefficients using a Rician maximum likelihood estimator [30], which
relied on the ˆσmaps. Finally, we estimated reff using Eqs. (3) and (A14), assuming
D0= 2.07 µm2
/ms [65] and fim = 0 [66].
4.5 dMRI simulations
We conducted simulations to replicate dMRI signal generation and reff-estimation
under ex-vivo and in-vivo conditions. The simulations are described in greater detail
in Section B. In brief, we generated dMRI signals for each diffusion gradient direction
by computing volume-weighted average signals [67] over our empirical axon radius
distributions. For in-vivo simulations, we used axon radius distributions adjusted for
tissue shrinkage as described in Section 4.2. For signal simulation, we modeled three
compartments: intra-axonal, extra-axonal and immobile water compartment with T2-
weighted volume fractions fa,feand fim. We simulated intra-axonal signal using the
matrix method [68] to capture effects beyond the Gaussian phase approximation [69].
We assumed fully decayed extra-axonal signal (but fe>0) and used őxed fim. To
estimate reff from simulated signals, we followed the procedure for experimental ex-
vivo and in-vivo dMRI data described in Sections 4.3 and 4.4, assuming known fim.
We considered both an idealized scenario (SNR = ∞), as well as an "experiment-like"
scenario mimicking experimental Rician noise conditions (in-vivo: SNR = 32; ex-vivo:
b-dependent SNR ranging from 17 to 51), for which we repeated simulations 1000
times.
4.6 Quantification of improvements through dMRI-scale
histology
For each histology ROI, we empirically determined the sampling distribution of reff
for different subsample sizes between 102and 105axons, reŕecting smaller ROI sizes
typical for existing histology data [3,31,32]. Per subsample size, we assessed accuracy
using the normalized mean bias error
NMBE = PM
m=1(ˆreff ,m −reff )
reff
(4)
and precision using the coefficient of variation
CV = std({ˆreff,m|m∈M})
reff
,(5)
where M= 1000,ˆreff ,mis a subsample estimate, reff is the reference value computed
from the full empirical axon radius distribution, and std({ˆreff ,m|m∈M})denotes the
standard deviation across all subsample estimates.
19

4.7 Comparison of reff across modalities
Qualitative comparison of spatial reff patterns
To compare the spatial patterns of reff in the mid-sagittal section of the corpus callo-
sum across modalities, we mapped reff values from all modalities onto the mid-sagittal
slice in MNI space [70]. Speciőcally, we proceeded as follows:
•For histology and histology-based dMRI simulations, we őrst registered two-
dimensional tissue masks to the mid-sagittal slice of an FA atlas in MNI space (the
FSL HCP-1065 FA atlas [46], see Fig. 5a,c,f). Then, we transformed histological ROI
centroids to MNI space, assigned histological/simulated reff values, and generated
continuous spatial patterns using nearest-neighbor interpolation.
•Since each ex-vivo dMRI ROI covered multiple voxels, we őrst averaged reff values
within each ROI in native space. We then mapped these averages to MNI space
using the same approach as for histology, given that ROIs between these modalities
were registered per our study design (see Fig. 5e).
•For in-vivo dMRI, we used both T1-weighted image and FA map for registration
to corresponding templates in MNI space (the FSL HCP-1065 FA atlas and the
FSL MNI152 T1-weighted template [46]; see Fig. 5g). Using this registration, we
transformed per-subject reff maps to MNI space. To select voxels inside the corpus
callosum, we applied a coarse mask derived from the JHU ICBM-DTI-81 white
matter atlas [46,71], along with FA and reff thresholds (FA ≥0.65 and reff ≥
0.1µm).
Quantitative comparison of spatial reff patterns
To quantitatively compare reff from dMRI experiments to histological values, we
determined corresponding reff values in the respective native spaces.
•For ex-vivo dMRI, we computed the mean reff across all voxels per ROI, with ROIs
registered to histological ROIs per our study design (see Fig. 5e).
•For in-vivo dMRI, we determined histological ROI coordinates in native dMRI space
by őrst transforming them to MNI space and then mapping them to the nearest
voxel in dMRI native space (see Fig. 5a,c-d). We applied the same corpus callo-
sum voxel selection criteria as in the qualitative analysis (see last paragraph). For
comparison with histological values, we used the group-average reff, computed from
spatially smoothed per-subject reff maps (FWHM = 3.75 mm), ensuring that spatial
smoothing was restricted to corpus callosum voxels [72]. We assessed the impact of
spatial smoothing on quantitative analyses in Section SI9.
Error metrics
Between histological values (reff ) and estimated values (ˆreff ) from dMRI experi-
ments, we computed a linear regression slope to assess the scaling behavior of ˆreff .
Additionally, we determined the őtting success rate
S=1
N
N
X
i=1
I(ˆreff,i >0.1µm),(6)
20

as the proportion of ˆreff values that exceeded 0.1µm, where N= 35 is the number of
ROIs and I(·)is the indicator function, which equals 1 if the condition inside is true
and 0 otherwise.
To quantify absolute agreement, we computed the normalized root-mean-square
error:
NRMSE = qPi= 1N(ˆreff,i −reff,i )2
PN
i=1 reff,i
,(7)
using ˆreff ,i = 0 µmfor unsuccessfully őtted values.
To quantify the ability to capture linear relationships, we computed Pearson’s
correlation coefficient:
R=PN
i=1(ˆreff ,i − ⟨ˆreff ⟩)(reff,i − ⟨reff ⟩)
qPN
i=1(ˆreff ,i − ⟨ˆreff ⟩)2qPN
i=1(reff ,i − ⟨reff ⟩)2
,(8)
where ⟨reff ⟩and ⟨ˆreff ⟩denote the mean histological and estimated reff values across
ROIs. To assess statistical signiőcance, we performed a Monte Carlo permutation test
under the null hypothesis that ˆreff and reff are uncorrelated (R= 0). We computed
the associated p-value as:
p=1
K
K
X
i=1
I(|R′i| ≥ |R|),(9)
where R′iwere computed from shuffled reff and őxed ˆreff to approximate the null
distribution, using K= 106permutations.
For dMRI simulations with M= 1000 repetitions, we pooled over all M×Nvalues
to compute the linear regression, S,Rand NRMSE. Accordingly, we computed pover
M×Kiterations with K= 1000 so that M×K= 106.
4.8 Optimal in-vivo dMRI protocols for reff mapping
Goals and setup
We optimized in-vivo dMRI protocols for a range of maximum gradient strengths
(gmax = [40,600] mT/m), covering the capabilities of existing clinical and research
3 T scanners. The optimization aimed to maximize Rbetween histological reff and
simulated reff for protocol candidates, where the simulations were conducted analo-
gously to those for our experimental protocols described in Section 4.5. In contrast
to Section 4.5, we assumed Gaussian rather than Rician noise, as Rician noise more
strongly obscures correlation (see Section SI8) but can, in principle, be mitigated
through advanced preprocessing techniques [16,39,41].
Definition of the optimization problem
To streamline the parameter search, we formulated the optimization problem using
the constraints of our experimental in-vivo dMRI protocol. Speciőcally, we focused on
two-shell protocols with őxed diffusion timing parameters (δ,∆) and őxed minimum
21

b(bmin = 6 µm2
/ms) to suppress extra-axonal signal, while allowing the minimum g
(gmin) to vary. Thus, we modeled the optimization problem as
θ∗|gmax = arg max θR(θ)|gmax , θ ={δ, ∆, gmin}.(10)
The search grid for θand additional parameters are detailed in Table B1. Additionally,
we enforced ∆≥δ+ 4 ms.
Modeling echo time, intra-axonal signal fraction and SNR
We modeled the effect of the protocol-dependent TEon T2-weighted intra-axonal water
fraction (fa) and SNR. The echo time of a protocol candidates was estimated as
TE(θ) = δ+ ∆ + C,(11)
where the constant C= 21 ms, derived from our experimental protocol, accounts
for additional contributions, such as the RF pulse and readout gradients. Assuming
fim = 0 [66], we computed faas
fa(TE) = f0·e−TE/T2,a
f0·e−TE/T2,a+ (1 −f0)·e−TE/T2,e,(12)
where f0= 0.41 is the non-T2-weighted intra-axonal water fraction, T2,a= 82 ms is
the intra-axonal T2-value and T2,e= 44 ms is the extra-axonal T2-value, as reported
by Veraart et al. [65]. Using SNRref = 32 and TE,ref = 66 ms, derived from our
experimental protocol as reference values, we extrapolated
SNR(TE) = SNRref ·f0·e−TE/T2,a+ (1 −f0)·e−TE/T2,e
f0·e−TE,ref /T2,a+ (1 −f0)·e−TE,ref /T2,e.(13)
To evaluate a potential elevation of the baseline SNR level through technical or
acquisition improvements, we repeated the protocol optimization analyses for SNRref
increased by 75 % and 150 %, yielding SNR values of 56 and 80 for our experimental
protocol.
4.9 Simulation of clinical application for reff mapping
Motivated by reports of a 28.6 % reduction in axon radii in the splenium of individuals
with autism spectrum disorder (ASD), we conducted a statistical power analysis to
evaluate the feasibility of using reff to distinguish individuals with ASD from healthy
controls. The analysis was conducted using the optimized protocols for next-generation
clinical scanners, as determined in Section 4.8. We estimated statistical power using a
Monte Carlo simulation over M= 5000 iterations. We assumed L= 11 voxels in the
splenium, reŕecting the number of voxels in the mid-sagittal slice of the splenium in
our in-vivo dMRI data. For various group sizes (N), we proceeded as follows:
•We randomly sampled L×Naxon radius distributions of splenium ROIs from the
histological dataset in Fig. 1a per group (healthy and ASD), with replacement.
22

•For the ASD group, we scaled down each sampled axon radius by 28.6 %.
•For each sampled axon radius distribution, we simulated dMRI signals and estimated
reff.
•We computed the mean reff across the Lsplenium voxels to obtain subject-level
mean reff.
•We performed a two-sample t-test to assess group differences in subject-level mean
reff with signiőcance level α= 0.05.
Finally, based on the t-test results, we estimated the statistical power as the proportion
of iterations that rejected the null hypothesis, which states that there is no difference
between group means.
Supplementary information. Supplementary information is available for this
paper.
Data Availability Statement. The histology and ex-vivo dMRI will be made pub-
lic upon publication of this article. The in-vivo dMRI data are available from the
corresponding author upon reasonable request.
Code Availability Statement. The dMRI processing and simulation code is pub-
licly available at https://github.com/quantitative-mri-and-in-vivo-histology/mri_
radius_validation. Our code made use of other publicly available packages, such as
MRtrix3 [48], FSL [46], the Standard Model Imaging (SMI) toolbox [51ś54], the
Microstructure Imaging Sequence Simulation ToolBox (MISST) [73,74], and the
Advanced Normalization Tools (ANTs) [75].
Ethics. In-vivo dMRI: The subjects were scanned under the approval of the Ethics
Commission of the Medical Faculty of Leipzig University (reference number 293/18-
ek). The participants gave written informed consent before participation in the study.
Ex-vivo dMRI and histology: The entire procedure of case recruitment, acquisition
of the patients personal data, the protocols and the informed consent forms, performing
the autopsy and handling the autopsy material have been approved by the responsible
authorities (Approval #205/17-ek and WF-74/16).
Acknowledgements. The research leading to these results has received funding
from the European Research Council under the European Union’s Seventh Framework
Programme (FP7/2007-2013) / ERC grant agreement number 616905.
This work was supported by the German Research Foundation (DFG Priority
Program 2041 "Computational Connectomics", [MO 2397/5-1, MO 2397/5-2, MO
2249/3-1, MO 2249/3-2], by the Emmy Noether Stipend: MO 2397/4-1; MO 2397/4-
2) and by the BMBF (01EW1711A and B) in the framework of ERA-NET NEURON
and the Forschungszentrums Medizintechnik Hamburg (fmthh; grant 01fmthh2017).
We thank Jan Malte Oeschger for insightful discussions.
Author contributions. L.M.: conceptualization, methodology, software, data cura-
tion, investigation, formal analysis, visualization, writing of the original draft. L.J.E.:
conceptualization, methodology, data acquisition and curation, review and editing.
M.M.: data acquisition and curation, conceptualization, investigation, review and edit-
ing. M.A.: data curation, review and editing. T.S.: data acquisition and curation,
23

review and editing. B.F.: data acquisition and curation, review and editing. F.J.F.
data acquisition and curation, review and editing. H.R.: data acquisition and cura-
tion, review and editing. C.J.: data acquisition and curation, review and editing. L.J.:
data acquisition and curation, review and editing. T.G.: data acquisition and curation,
review and editing. E.T.: data acquisition and curation, review and editing. J.S.P.:
data acquisition and curation, review and editing. A.P.: data acquisition and curation,
review and editing. T.N.: resources, review and editing. N.W.: resources, review and
editing. M.M.: conceptualization, resources, funding acquisition, review and editing.
S.M.: conceptualization, data acquisition and curation, resources, funding acquisition,
review and editing.
Appendix A Theory
A.1 Multi-compartment signal model
In white matter, consisting of densely packed, cylindrical axons with radius distribu-
tion H(r), the dMRI signal for one diffusion gradient direction g = [gx, gy, gz]Tcan be
written as a volume-weighted average signal [67]
S(b, g, H (r)) = R∞
r=0 H(r)r2S(b, g, r)dr
R∞
r=0 H(r)r2dr ,(A1)
where πand the common cylinder length cancel out. When normalized to the signal
at b= 0,S(b, g, r)can be modeled as a multi-compartment signal [20]
S(b, g, r) =fa·Z|n|=1
P(n)e−bD∥
aξ2
|{z }
S∥
a(b,ξ(g,n),D∥
a)
e−bD⊥
a(r)(1−ξ2)+O(b2)
|{z }
S⊥
a(b,ξ(g,n),r)
dn
|{z }
Sa(b,g ,r)
+fe·Z|n|=1
P(n)e−bD∥
eξ2
|{z }
S∥
e(b,ξ(g,n),D∥
e)
e−bD⊥
e(1−ξ2)
|{z }
S⊥
e(b,ξ(g,n),D⊥
e)
dn
|{z }
Se(b,g )
+fim,(A2)
with T2-weighted intra- and extra-axonal compartment signal fractions (fa,fe) and
the signal of immobile water compartment fim [20,21,66,76] summing up to 1.
Sa(b, g, r)and Se(b, g)are the intra- and extra-axonal compartment signals, which can
be modeled as a convolution between the őber orientation function P(n)normalized to
RP(n)dn = 1 and the signal of a őber pointing in direction n [20,54]; the latter signal
can be represented as the product of parallel (S∥
a(b, ξ(g, n), D ∥
a)and S∥
e(b, ξ(g, n), D ∥
e))
and perpendicular signals (S⊥
a(b, ξ(g, n), D ⊥
a(r)) and S⊥
e(b, ξ(g, n))) under the assump-
tion of axisymmetric diffusion, where ξ(g, n)denotes the scalar product between g and
n.S∥
a(b, ξ(g, n), D ∥
a),S∥
e(b, ξ(g, n)) and S⊥
e(b, ξ(g, n)) are determined by the scalar dif-
fusivities (D∥
a,D∥
eand D⊥
e) and the diffusion-weighting b=g2γ2δ2(∆−δ/3), where γis
24

the gyromagnetic ratio, gis the diffusion gradient amplitude, δis the diffusion gradient
duration and the ∆is the diffusion gradient separation [77,78]. S⊥
a(b, ξ(g, n), r )can be
modeled as a function of r; we introduce several approximations for S⊥
a(b, ξ(g, n), r )
in the next section.
A.2 Perpendicular signal attenuation inside a cylinder
There is no known analytical solution for the O(b2)terms in
ln S⊥
a(b, ξ(g, n), r ) = −(1 −ξ2)bD⊥
a(r) + O(b2)(A3)
for őnite δ. In the Gaussian phase approximation (GPA), the O(b2)terms are neglected
and the equation for a cylindrical geometry can be written as [69]
ln S⊥
a,GPA(b, ξ(g, n), r)≈ − (1 −ξ2)bD⊥
a(r)(A4)
≈ − (1 −ξ2)2g2γ2r4
D0
∞
X
m=1
tc
α6
m(α2
m−1)
·[2α2
m
δ
tc−2 + 2e−α2
mδ
tc+ 2e−α2
m∆
tc−e−2α2
m
∆−δ
tc−e−2α2
m
∆+δ
tc],
(A5)
where the b-dependence is captured implicitly through g,δand ∆;tc=r2/D0is the
correlation time, D0is the diffusivity of the axoplasm and αmis the m-th root of
dJ1(α)/dα= 0, where J1(α)is the Bessel function of the őrst kind. In the wide-pulse
approximation (WPA), ∆> δ ≫tc[79], the dependency of Eq. (A5) on ∆can be
neglected so that
ln S⊥
a,WPA(b, ξ(g, n), r)≈ −(1 −ξ2)κr4, κ =7
48
g2γ2δ
D0
.(A6)
A.3 Axon radius estimation via powder-averaged signals
The orientation dependence and interference of Se(b,g)in Eq. (A2) complicate estab-
lishing a robust connection to H(r). To mitigate interference of Se(b, g), one can study
signals in the high b-regime, where its contribution has been reported to be negligi-
ble (b≥6 ms/µm2for in-vivo dMRI [14,20] and b≥20 ms/µm2for ex-vivo dMRI
[14]). To mitigate orientation dependency, powder-averaging can be applied [25ś27].
Combining these measures, the powder-averaged signal at sufficiently high bcan be
approximated as
S◦(b, H(r)) ≈β
√b·e−bD⊥
a
|{z }
S⊥,◦
a(b,H(r))
+fim, β =rπ
4D∥
a
fa.(A7)
25

The perpendicular component of this signal, S⊥,◦
a(b, H(r)), can be further simpliőed
through a series of approximations [14,15]
S⊥,◦
a(b, H(r)) ≈R∞
r=0 H(r)r2S⊥,◦
a,WPA(b, r)dr
R∞
r=0 H(r)r2dr , S⊥,◦
a,WPA(b, r) = S⊥
a,WPA(b, ξ = 0, r)
(A8)
=R∞
r=0 H(r)r2e−κr4dr
R∞
r=0 H(r)r2dr
=⟨r2(1 −κr4+O(r8))⟩
⟨r2⟩
≈1−κ⟨r6⟩
⟨r2⟩(A9)
≈e−κr4
eff (A10)
=S⊥,◦
a,WPA(b, reff)(A11)
including the WPA in Eq. (A8), a Taylor series approximation in Eq. (A9), and an
exponential approximation in Eq. (A10). Here,
reff =4
s⟨r6⟩
⟨r2⟩.(A12)
is a scalar, tail-weighted statistic of H(r). Alternatively, reff can be expressed in terms
of dMRI parameters and diffusivities from the relation e−bD⊥
a≈e−κr4
eff (see Eqs. (A7)
and (A10)), yielding
reff ≈4
r48
7δ(∆ −δ
3)D⊥
aD0.(A13)
Using Eqs. (A7) and (A13), one can jointly estimate reff and β, e.g., via non-linear
őtting [45]. If only two different b(bmin and bmax) are used, one can directly determine
D⊥
a=
log(S◦(bmin ,H(r))
S◦(bmax,H (r)) qbmin
bmax )
bmax −bmin
(A14)
and subsequently determine reff [17].
Appendix B dMRI signal simulations
We simulated dMRI signals S(b, g, H (r)) normalized to S(b,g, H (r))|b=0 according to
Eq. (A1). Below, we outline the key parameter and implementation choices:
•We assumed Se(b, g) = 0 for both in-vivo and ex-vivo dMRI. Although Se(b,g) = 0,
the extra-axonal compartment still inŕuences S(b, g, H(r)) as fe>0.
•We assumed fim = 0 for in-vivo dMRI, assuming negligible signal contribution [66].
26

•We assumed fim = 0.27 for ex-vivo dMRI, reŕecting the mean value in our
experimental dMRI data (see Section 4.3 for estimation approach).
•For S⊥
a(b, ξ(g, n), r ), we used the matrix method [68]. In particular, we used the
implementation of the MISST toolbox [73,74] to determine the relevant matrices
and computed S⊥
a(b, ξ(g, n), r )using an equation for rectangular waveform (see Eq.
(26) in [68]). The latter implies the assumption of inőnite slew rate.
•For S∥
a(b, ξ(g, n)), we used őxed values from the literature [14,65] (see Table B1)
•We assumed a single őber bundle with őxed mean orientation µ = [0,0,1]Tand
Watson-distributed őbers with orientation n [80].
•We numerically approximated the integral over n using Lebedev quadrature of
degree 590 [81,82].
•We discretized rusing anisotropic binning with edges ∈
{0,0.1, ..., 5,5.2, ..., 10,10.5, ...20}.
The full set of simulation parameters is listed in Table B1.
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27

Table B1 In-vivo and ex-vivo simulation parameters. Annotations denote: ∗the default value was used; 1estimated from our experimental data; 2we did
not add noise; 3estimated as the mean of previously reported values [31,38]; 4parameters of our experimental protocols; 5reported by West et al. [57]; 6
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scanner [65] to 9.4 T by scaling with a conversion factor T2(9.4 T)/T2(3 T) ≈30 ms/83.8 ms ≈0.358 using literature values [83,84].
Ex-vivo simulations
z}| {
In-vivo simulations
z }| {
Parameter Symbol Unit
Default value
Comparison across
modalities (experiment-like)
Comparison across
modalities (idealized)
Default value
Comparison of reff
across modalities (experiment-like)
Comparison of reff
across modalities (idealized)
Protocol optimization
signal-to-noise ratio SNR - - [17,51] 1∞2-32 1∞2Eq. (13)
noise distribution - - - Rician - - Rician - Gaussian
powder-average estimator - - - Rician ML Gaussian ML - Rician ML Gaussian ML Gaussian ML
radius scaling factor - - 1.0* * 1.33* * *
neurite dispersion - - 8.21* * 8.21* * *
diffusion shells - - 94* * 24* * *
gradient directions per shell - - 65 4* * {60,120}4* * *
axoplasmic diffusivity D0µm2
/ms 0.35 5* * 2.07 6* * *
parallel intra-axonal diffusivity D∥
aµm2
/ms 0.35 5* * 2.07 6* * *
minimum b bmin ms/µm220 4* * 64* * *
maximum b bmax ms/µm2100 4* * 30.45 4* * f(δ, ∆, gmax)7
minimum g gmin mT/m200 4* * 124 4* * [40,600]
maximum g gmax mT/m1278 4* * 279 4* * f(δ, ∆, bmin)7
diffusion gradient time δms 74* * 15 4* * [2,60]
diffusion gradient separation ∆ ms 20.14* * 29.25 4* * [6,80]
intra-axonal water fraction f00.41 6* * 0.41 6* * *
immobile water fraction fim -0.27 1* * 08* * *
T2-weighted intra-axonal water fraction fa-0.58 9* * 0.58 9* * Eq. (12)
T2-weighted extra-axonal water fraction fe-0.15 10 * * 0.42 10 * * f(fa, fim)10
extra-axonal signal Se-0* * 0* * *
intra-axonal transverse relaxation time T2,a ms 29.411 * * 82 6* * *
extra-axonal transverse relaxation time T2,e ms 15.811 * * 44 6* * *
echo time TEms 34.74* * 66 4* * Eq. (11)
28

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