Calibration datasets in adults.

To calibrate T1_c and T2_c of each compartment c, we acquired data on 3 healthy adult volunteers (1 female, 2 males, mean age: 23.2±2 years) using long protocols, based on large numbers (N = 30–60) of TI (sampled between 100ms and 3100ms) and TE (sampled between 30ms and 340ms) leading to acquisition times between 35 and 47min (S1 Table and S1 Fig). The TI and TE samplings were denser at the beginning of the sampling range to ensure good parameters estimation for the fast-relaxing myelin-related compartment with low T1_my and T2_my.

Reduced datasets in adults and infants.

The strategy was tested in two groups using short acquisition protocols: an adult group of 13 healthy subjects (6 females, 7 males, mean age: 22.4±1.6 years), and a group of 18 healthy infants born at term (8 girls, 10 boys), with a maturational age (i.e. chronological age corrected for gestational age at birth) between 3 and 21 weeks for 17 infants, plus one infant of 34 weeks old. A reduced number of measurements was acquired. For T1 relaxometry, 8 different TIs (from 250 to 1500ms every 250ms, 2000 and 2500ms) were used, leading to a scan duration of 3min03s/2min11s (adults/infants). For T2 relaxometry, 8 linearly spaced TE values were used between TE = 50 and 260ms, leading to a scan duration of 4min/2min51s. To identify different white matter bundles used as regions-of-interest for f_my quantification, a non-diffusion weighted image (b = 0 s/mm2) and 30 diffusion-weighted (DW) images (b = 700 s/mm2) were acquired using a DW-SE-EPI sequence (TE = 72ms, TR = 14s/10s, parallel imaging GRAPPA factor 2, partial Fourier sampling factor 6/8) within an acquisition time of 7min56s/5min40s.

The study protocol was approved by the regional ethical committee for biomedical research from Kremlin-Bicêtre (for adult experiments: CPP #2008-A00241-54; for infant experiments: INSERM 123 011, CPP #05 14). All parents and adult subjects gave written informed consents. The infants were spontaneously asleep during MR imaging (the protocol length was sufficiently short to be acquired during the nap). Particular precautions were taken to minimize noise exposure, by using customized headphones and covering the magnet tunnel with special noise protection foam (“plastison”, Serenata, http://www.serenata.tm.fr/product.php?id_product=17). We also restrained the slew rate of MR gradients in EPI acquisitions. To reduce motion, the infants’ heads were gently restricted by the headphones and MRI foam pads.

Data preparation.

All data were pre- and post-processed using PTK toolkit and Connectomist software both developed in-house at NeuroSpin [30,31]. For T1 and T2 relaxometry images, the signal-to-noise ratio (SNR) was computed as the ratio between the mean signal from the brain (without ventricles) to the standard deviation of the noise from the image background. For all TIs and TEs, the SNR of T1 and T2 relaxation images was above 5.3 and 6.2 respectively. These noise levels allowed us to approximate the non-centered chi-noise present in GRAPPA-reconstructed images by a Gaussian noise, and thus to use NNLS estimators.

In each subject, T1 and T2 relaxometry images were co-registered with anatomical images using affine transformations maximizing their mutual information. A brain mask was computed from the T2-weighted images with the highest SNR (corresponding to the lowest TE), based on thresholding and mathematical morphology tools (combination of opening and closing). All images were masked with this brain mask. Quantitative T1 and T2 maps were further computed.

Computation of reference f_my maps.

In the calibration datasets (3 adults), we first performed a voxel-wise fitting of the 3-compartment model (Eq 1–4), to estimate f_c, T1_c and T2_c for each compartment c in each brain voxel. We aimed to both ensure that there was no particular regional dependence of T1_c and T2_c, and to compute reference f_my maps. The upper boundary for f_my was set to 40%. An original combination of a region contraction approach [19,32] and a standard NNLS algorithm [28] was used (Fig 1). This procedure was programmed in Python using NumPy and SciPy libraries. The initial search ranges for the individual compartment T1_c and T2_c were set based on literature evidence [9,19,33,34] and to ensure their continuity:

• for myelin-related compartment: T1_my ∈ [300; 570] ms, T2_my ∈ [1; 40] ms
• for intra/extra-cellular water compartment: T1_ie ∈ [570; 1600] ms, T2_ie ∈ [40; 200] ms
• for CSF compartment: T1_csf ∈ [1600; 4000] ms, T2_csf ∈ [200; 2000] ms.

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Fig 1. Schematic representation of the fitting algorithm.

The fitting procedure was performed either for each voxel (voxel-wise approach) or for each slice (calibration approach) independently. The fitting steps were repeated until both the fitting error and the differences between the upper and lower borders of the search ranges became less than 1%.

https://doi.org/10.1371/journal.pone.0163143.g001

In each voxel, random T1_c and T2_c values were first selected from the corresponding search ranges using uniform random distributions. Then the simplified model (with corresponding T1_c and T2_c) was fitted using a NNLS algorithm. The fitting error was calculated by increasing the weight of Eq 3 (describing T1 relaxation) proportionally to the number of TI points: this was performed to balance the contributions of T1 (only one equation in the model) and T2 (TE equations) in the fitting error computation. Similar to [19], this step was repeated 10000 times in each voxel, and the best 100 selections of T1_c and T2_c values (i.e. providing the smallest fitting errors) were used to contract the search ranges. After contraction, the whole procedure was re-started: 90% of random T1_c and T2_c values were chosen from the new search ranges, while the remaining 10% were selected from the initial ranges to avoid falling into a local minimum. This fitting procedure was executed until both the fitting errors and the difference between the ranges boundaries became less than 1%. For each voxel, this procedure took around 70-80sec on Intel I3 machine using a single core CPU (the algorithm convergence is illustrated in S2 Fig), leading to around 120 hours per subject for the central slices of the brain (covering the ventricles, corpus callosum and basal ganglia, i.e. containing portions of all three compartments).

At the same time, f_my maps were computed using Eq 1–4 by fixing T1_c and T2_c based on the literature: either according to [9] (T1_my = 350ms, T1_ie = 850ms, T1_csf = 2800ms, T2_my = 10ms, T2_ie = 40ms, T2_csf = 130ms) or [19] (T1_my = 465ms, T1_ie = 9665ms, T1_csf = 3500ms, T2_my = 12ms, T2_ie = 90ms, T2_csf = 250ms). The resulting f_my maps were compared with maps obtained with voxel-wise fitting, with the aim to demonstrate that T1_c and T2_c values could not be simply taken from the literature but required specific calibration for our acquisition protocol.

Calibration of the 3-compartment relaxation model.

T1_c and T2_c were then calibrated on the same datasets, using the same fitting procedure as for the voxel-wise estimation, except that it was performed at the slice level and not at the voxel level for two reasons: to save computation time, and to increase the fitting accuracy (as detailed below, calculation errors decrease with the number of voxels used for fitting). In each of the 3 subjects, we considered the same 10 central slices as they contained portions of all three compartments. The total fitting error was calculated over all voxels within each slice independently (averaged number per slice: 5718±406). Once the calibration was performed for each slice, T1_c and T2_c were set to the weighted averages over all slices of all subjects (10 slices x 3 subjects, weighted with inversed fitting errors). These calibrated values were compared with T1_c and T2_c distributions obtained with the voxel-wise approach. The resulting f_my maps were also compared with the reference f_my maps. This calibration stage took approximately 24 hours per subject.

Selection of f_my upper boundary.

We explored the effect of f_my upper boundary at the calibration stage, since it was shown to have a considerable impact on f_my values derived from the relaxometry model based on mcDESPOT protocol [35]. It has been shown that the quality of model fitting (sum-of-square of residuals) using stochastic region contraction approach is the best when the upper boundary is between 0.3 and 0.5 and is insensitive to changes within this range [35]. Thus the search interval for f_my was initially set to [0; 0.4], and we also considered two other upper boundaries (0.3 and 0.5) at the calibration stage, to compare the estimated T1_c and T2_c, and f_my maps. Given the results, the upper boundary was fixed to 0.4 in the following tests.

Impact of TI/TE numbers on the calibration of T1_c and T2_c.

We investigated the impact of TI and TE numbers on the estimation of T1_c and T2_c, to make sure that each calibration dataset had enough sampling points for reliable calibration. From the initial dataset with 60 TI/TE sampling points (subject #3, S1 Fig), we progressively removed data points in a regular manner from 60 down to 10 points (S1 Fig), and we performed independent calibrations for each resulting dataset.

Evaluating the fitting accuracy using simulated data.

We also checked using simulated data whether the proposed calibration strategy provided reliable computation of f_my values. T1 and T2 relaxometry signals were simulated using Eq 2–4 at all TI and TE sampling points. We fixed T1_c and T2_c values either equal to those identified at the calibration stage or as in previous studies [9,19,34,35] (T1_my = 465ms, T1_ie = 965ms, T1_csf = 3500ms, T2_my = 12ms, T2_ie = 90ms, T2_csf = 250ms). Different compartment fractions were considered, as possibly observed in the white matter: f_my = [5,10,15,20,25,30,40]%, f_csf = [0,1,2,3,4,5]% and f_ie = (100 − f_my − f_csf)%. Simulated signals were additionally corrupted with a Gaussian noise with a dispersion varying from 0 to 20% relative to signal values. Then the calibration procedure was applied to simulated signals to identify the compartment fractions as well as T1_c and T2_c. The error in f_my calculation was estimated as the averaged absolute percent difference between identified and simulated values. We also investigated how the number of voxels used for the model fitting impacted f_my errors. In theory, calculation errors are proportional to the inversed square root of the voxel number [36]. Thus, simulations were performed for voxel numbers equal to [1², 2², …, 75² = 5625], and errors were fitted with the inverse square root function. Finally, we compared f_my calculation errors obtained in simulations using both T1 and T2 relaxometry signals together, vs. only T2 relaxometry signal.

Selection of f_my upper boundary.

We investigated how the initial upper search boundary for f_my impacted the estimated f_my, T1_c and T2_c values at the calibration stage. With a [0, 0.4] interval, quite high f_my values were observed in the adult white matter (Fig 2b), in comparison with previous studies [7,9]. When changing the upper boundary to 0.3 and 0.5 for the 3 calibration datasets, we observed a strong impact on f_my maps (S3 Fig): higher upper boundary resulted in higher f_my values in all 3 subjects. However, the estimated T1_c and T2_c were very close (S2 Table). Of interest, when the upper boundary was set to 0.4, distributions of f_my values across the 3 subjects were the most similar (i.e. they had the biggest overlap, S3 Fig), and standard deviations for T1_c and T2_c estimations were the lowest (S2 Table). These results suggested that fixing the upper boundary for f_my to 0.4 was valid although it might provide high f_my values.

Accuracy of the calibration strategy.

We further assessed whether our calibration strategy was sensitive to different acquisition parameters (sampling points, noise level). First, reducing the number of calibration TI and TE points until N = 25 for both T1 and T2 relaxometry signals, had no significant effect on T1_c and T2_c calibration in subject #3 (ad-hoc paired t-test between T1_c and T2_c values from individual slices: p>0.06, S4 Fig). This suggested that our calibration stage was performed in a stable manner across all 3 subjects despite the different numbers of TI and TE points.

Application of the calibration strategy to simulated data showed that as the noise level increased, reliable calculation of f_my values required increasing the number of voxels used to estimate the compartment relaxation times (Fig 4a). In agreement with previously conducted simulations [19], the average estimation errors were less than 10% even for a noise level of 20% of the signal values, on condition that more than 2000 voxels were used simultaneously (Fig 4a). In simulations with a 20% noise level and a number of voxels higher than 5000, we observed that estimation errors decreased with increasing f_my values, and that maximal errors did not exceed 15% for f_my values of 5% (Fig 4b). This suggested that the number of voxels considered in our calibration stage (~5718±406 voxels for each slice) was sufficient for reliable f_my estimation.

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Fig 4. f_my calculation accuracy at the calibration stage assessed in numerical simulations.

(a) The f_my calculation errors were calculated for simulated f_my = [5,10,15,20,25,30,40]% and f_csf = [0,1,2,3,4,5]%, and the resulting average values were represented as a function of the number of voxels used for the model fitting [1², 2², …, 75² = 5625], for different noise levels (from 0 to 20%). As expected, relationship between the f_my calculation errors and the number of voxels could be described by inverse square root relationship (R² > 0.75). (b) For a noise level of 20%, calculation errors for different simulated f_my values were averaged for voxel numbers higher than 5000 ([71², …, 75²]) and f_csf = [0,1,2,3,4,5]%, considering T1_c and T2_c values fixed according to our calibration stage (set1) or to previous studies (set2 [9,19,34,35]). (c, d) Examples of the theoretical and estimated T2 (c) and T1 (d) relaxation signals for data simulated with f_my = 20%, f_ie = 78%, f_csf = 2% and T1_c and T2_c fixed according to our calibration stage.

https://doi.org/10.1371/journal.pone.0163143.g004

Besides, in identical conditions (same noise level, number of voxels, compartment fractions, T1_c and T2_c), removing T1 relaxation from the model (i.e. removing Eq 3) significantly reduced the accuracy of f_my calculations as compared to the full model (ad-hoc paired t-test, p < 10⁻⁶). This demonstrated that modeling T1 and T2 relaxometry signals together was more appropriate than considering T2 relaxation alone, this was probably because we could not sample T2 signals for very short TE as compared with the very low T2_my value as discussed below.

Fixing T1_c and T2_c with calibrated values.

In the calibration datasets, fixing T1_c and T2_c with calibrated values to fit the 3-compartment model had little effect on f_my maps, whether the number of data points was kept complete (Fig 2e) or reduced to 8 TI and 8 TE values matching those of the reduced datasets (Fig 2f). The average absolute differences between the reference f_my maps and f_my maps computed with fixed calibrated T1_c and T2_c were very low (-0.3±1.8%; Fig 2e and 2f).

f_my maps in the reduced datasets.

In adults, f_my maps from the reduced datasets were very similar to the maps from the calibration datasets, both in terms of visual appearance (Figs 2 vs 5) and distribution of f_my values (ad-hoc χ² test: p>0.95). In infants, f_my maps showed the myelination progression described by post-mortem studies [43]. Within white matter, the youngest infants had high f_my values only in the posterior limb of the internal capsule, and f_my values progressively increased with age in a caudo-rostral direction, from the center of the brain to the periphery (Fig 5).

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Fig 5. T1, T2 and fraction maps computed in reduced datasets.

Quantitative T1 (a) and T2 (b) maps are presented for infants at different ages and for an adult subject, with corresponding anatomical images (T2w for infants / T1w for the adult). Axial slices are displayed in radiological convention. T1 and T2 maps show the expected decreases with age. Fraction maps for the compartments of myelin-related water f_my (c), intra/extra-cellular water f_ie (d) and unrestricted water f_csf (e) obtained at the testing stage are also presented. In agreement with the known age-related decrease in water content, f_csf decreased with age in most voxels except in the CSF, while f_my increased in white matter regions with myelination. Interestingly, the early increase in f_ie during infancy was followed by a decrease from 34 weeks on in locations where f_my showed concomitant sharp increase (i.e. posterior limb of the internal capsule and optic radiations).

https://doi.org/10.1371/journal.pone.0163143.g005

f_my in the adult bundles.

In adults, f_my values were highly variable across bundles (Fig 6a and 6b): from 0.14 in the spino-thalamic tract up to 0.36 in the optic radiations. In the spino-thalamic tract, surprisingly low values and high inter-individual variability were probably due to subtle misalignment between f_my maps and DW images in the brainstem of some adults. High variability in f_my values was also observed in the genu of corpus callosum, possibly due to partial volume effects with the ventricle CSF. In all bundles, f_my values were strongly correlated with T1 values across adult subjects (R² > 0.95), but not with T2 values.

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Fig 6. f_my quantification across white matter bundles in infants and adults.

(a) White matter bundles reconstructed with tractography (adapted from [42]). (b) f_my values across bundles in infants (light) and adults (dark). (c) Age-related changes in infants’ normalized f_my values (in % of the mean adult values) for the same bundles. Lines show significant linear regressions with age (R² > 0.58, p<0.015), for all bundles except the spino-thalamic tract. Abbreviations: Projection bundles: cortico-spinal tract CST (inf—inferior, mid—middle, sup—superior portions), spino-thalamic tract STT, optic radiations OR, anterior limb of the internal capsule ALIC; Association bundles: external capsule EC, arcuate fasciculus AF, superior SLF and inferior ILF longitudinal fascicles, uncinate fasciculus UF, fronto-occipital fasciculus FOF; Limbic and Commissural bundles: fornix FX, inferior CGinf and superior CGsup parts of the cingulum; corpus callosum: genu CCg, body CCb and splenium CCs.

https://doi.org/10.1371/journal.pone.0163143.g006

f_my in the infant bundles.

In infants, f_my values were smaller than in adults and below 0.05 for most bundles (on average over the group) (Fig 6b). The highest f_my values were observed in the cortico-spinal and spino-thalamic tracts and in the optic radiations, which are known to myelinate early on. According to the 3-compartment model, age-related changes in T1 and T2 relaxation times (measured at the voxel level) over the infant group could be explained by age-related changes in the compartment fractions, notably by changes in f_my values. For instance, in the middle cortico-spinal tract, we observed that f_my changed from 5% at 3 weeks of age to 17% at 21 weeks, in relation with changes in f_ie from 89% to 80%, and in f_csf from 6% to 3%. According to Eq 3 and calibrated T1_c and T2_c, T1 was expected to change from 1321 to 979ms, in agreement with experimental observations (the measured T1 changed from 1322 to 978ms on average over the tract). These results were coherent with our initial hypothesis that age-related changes in relaxometry signals and T1 / T2 relaxation times (measured at the voxel level), could be modeled exclusively by age-related changes in the compartment fractions.

f_my values normalized by the mean values from the adult group increased with age (from 3 to 21 weeks) in all bundles but in an asynchronous manner (Fig 6c): in certain bundles (e.g. spino-thalamic tract, cortico-spinal tract, external capsule, fornix, optic radiations), normalized f_my values increased earlier than in other bundles (e.g. arcuate and uncinate fasciculi, anterior limb of the internal capsule, corpus callosum). In all bundles except the spino-thalamic tract, linear regressions could describe these age-related increases over this short developmental period. However, since certain bundles had zero f_my values in the youngest infants, non-linear fitting may be more pertinent for describing the earliest subtle changes in bundle myelination [29].

Checking the measures of f_my, T1_c and T2_c.

One may first wonder whether our main hypothesis (i.e. fixing the compartment relaxation times to identical values across all brain regions and tissues) is valid in the adult brain. Indeed, when we applied a voxel-wise fitting to the calibrating datasets, T1_c and T2_c did not show any regional dependence across voxels, and their distributions showed mean values close to those identified during calibration. Furthermore differences in f_my values across methods were rather low (<3% Fig 2a, 2b and 2e). Thus local variations in T1_c and T2_c were not expected to significantly vitiate our results in adults. However, these magnetic properties likely depend on acquisition protocol, and fixing their values based on the literature (Fig 2c and 2d) was less reliable than considering calibrated values. Similarly, in future studies with different acquisition settings, calibrating T1_c and T2_c values would be required rather than applying the values reported here.

At the calibration stage, the initial search ranges for T1_c and T2_c were selected based on literature evidence [9,19,34,35], and identified values were in agreement with reported values [6,9,33,44]. Before fixing T1_c and T2_c to compute f_my maps in the reduced datasets, we checked that their calibration was stable by 1) showing that all calibration datasets had enough sampling points for robust estimation of the compartment magnetic properties (S4 Fig); 2) verifying using simulated data that the proposed calibration procedure resulted in reliable estimations of f_my values even in the presence of noise in relaxometry signals (Fig 4a and 4b); 3) proving that fixing these relaxation times and down-sampling the calibration datasets did not change the generated f_my maps (Fig 2e and 2f). Experiments with simulated data further suggested that in the presence of noise, calculation of f_my values might not be reliable on a voxel-wise level, and that increasing the number of voxels used to calibrate T1_c and T2_c was then required (Fig 4a).

Comparison with other methods.

With our approach, the calibration stage naturally required long acquisition (>35min) and post-processing times (up to 24 hours, without parallel computing). However, it must be performed only once for a given acquisition protocol (magnetic field, acquisition sequences, spatial resolution, etc.), and could be drastically accelerated using parallel computing. The main advantage of this approach is that, once calibration is performed, f_my values can be computed over the whole brain within much shorter acquisition (~5/7min for 50/70 slices with 1.8mm isotropic resolution, in infants/adults correspondingly) and post-processing times (<5min) than in previous studies. Indeed, in conventional multi-compartment relaxometry imaging studies [11], acquisition time takes up to 15 min for 8–12 contiguous slices because estimation of the T2 spectrum requires acquisition of the relaxometry data with a large number of TE. Although alternative approaches using mcDESPOT protocol [8] have enabled to shorten the acquisition time (between 18 and 25 min for a 1.8mm isotropic whole brain coverage on a 3T scanner, including B₀ and B₁ corrections [21], or around 10–12 min without these corrections [7]), these times remain quite long for unsedated infants, and rather long post-processing times (i.e. ~14 hours per subject [19]) are still required. Thus, using the proposed strategy can save at least 10 min of acquisition time and ~14 hours of post-processing time.

Our computed f_my values were higher than in most previous studies of infants [7,9] and adults [45], despite some exceptions [26,28]. Simulations suggested that our strategy was reliable even in the presence of noise in relaxometry signals. Nevertheless, local variations in the biophysical tissue properties and hence, in T1_c and T2_c across voxels might have introduced some additional “physiological” noise, leading to slight overestimation of f_my values (indeed, f_my values from the calibration stage were slightly higher than voxel-wise f_my values). It should also be noted that our model did not directly take into account the exchange between the different compartments: it only assumed that this exchange was fast relatively to T1 relaxation and slow relatively to T2 relaxation but it did not explicitly use exchange rate constants. Although these assumptions are generally true [9], ignoring exchange between compartments may also result in systematic bias [19,35]. However, including exchange through additional free parameters (exchange rate constants), would prevent the model to be linear, and our post-processing strategy would lose in reliability and speed. Similarly, our model neglected susceptibility differences across compartments, which might interfere particularly at high magnetic field, when these differences may impact the compartments relaxation. Correcting this would require to model a local susceptibility field gradient with a sinc function, but at the expense of an additional free parameter (i.e. the gradient amplitude). With fast-gradient echo sequences and in mcDESPOT protocols [21], not correcting for B0 and B1 field inhomogeneities might lead to errors in f_my calculations (e.g. overestimated f_my values [46]) since the signal is spatially dependent on the accuracy of the net flip angle. However, our protocol to measure T1 and T2 magnetic properties was based on spin echo EPI sequences, with repetition times much longer than T1 relaxation times, and MR signal was sampled for different TIs and TEs respectively. Thus inhomogeneities in B0 and B1 fields would impact images in the same way at all different TIs and TEs, and might corrupt proton density maps but without any impact on T1 and T2 estimations.

Eventually, our potential bias in high f_my values may arise from the stochastic fitting strategy used during calibration. Actually, in white matter structures, voxel-wise region contraction approach tend to assign f_my values to the upper part of the initial search interval, and the corresponding T2_my values to the lower part of the initial search interval [35]. Thus the initially selected search interval for f_my may have a considerable impact on f_my values. Indeed, we observed a strong impact on f_my maps when the upper boundary was changed from 0.4 to 0.3 or 0.5: higher boundary resulted in higher f_my values, and the 0.4 boundary provided the highest similarity in f_my distributions across the 3 subjects while the compartment relaxation times were stable. These observations point out to a main limitation of the present approach and possibly of other approaches using stochastic model fitting, meaning that f_my values strongly depend on the model fitting parameters. f_my quantification is also very sensitive to acquisition settings, including field strength, phase rewinding, sampling schemes of the inversion- and echo-times, voxel size, etc. [12,14,33,46]. This also justified the requirement to calibrate T1_c and T2_c instead of considering values from the literature. As a consequence, quantitative comparisons of f_my values across studies remain difficult.

Modeling both T1 and T2 relaxometry signals to measure f_my.

As myelin-related compartment has rather fast relaxation properties (e.g. T2_my~20ms), robust estimation of its fraction from T2 spectrum requires to start TE sampling with small values [10–12]. In this study, single-shot EPI sequences were used to keep the acquisition time as short as possible; and such short TEs could not be achieved, mainly due to the further lines of the partial k-space located before the echo time [47]. This means that even at the shortest TE, T2 relaxometry signal from myelin-related water was already strongly degraded. Consequently, estimating f_my values from T2 spectral analysis only was not possible, and reliably fitting the 3-compartment model required adding information from T1 relaxation in order to reveal the contribution of myelin-water compartment. However, information from T2 relaxometry signal did help to separate intra-/extracellular water from unrestricted water compartments. Thus in such experimental conditions, considering both T1 and T2 relaxometry signals was necessary. In our study the SNRs were high and similar for both T1 and T2 relaxation images; thus, to balance T1 and T2 contributions to the model fitting error, the weight of Eq 3, describing T1 relaxation, was increased proportionally to the number of TI points. However, it may be necessary to adjust the relative weights of the model equations if SNRs differ between T1 and T2 relaxation signals. These different issues might explain at least in part why f_my and T1 values were strongly correlated in all adult bundles, similarly to previous studies using mcDESPOT strategy [45], in addition to the dramatic influence on T1 values of water compartmentalization and of macromolecules (large lipids and proteins) in the myelin sheath.

Estimation of f_my in the infant brain from T1_c and T2_c calibration in the adult brain.

In addition to the verified hypothesis on compartmental property stability across adult brain tissues and regions, we also assumed stability throughout development, enabling us to compute f_my maps in infants using calibrated T1_c and T2_c in adults. Actually the hypothesis that age-related changes in T1 and T2 relaxation signals measured at the voxel level could be explained exclusively by changes in the compartment fractions within voxels, was verified in our infant group.

This assumption of compartment properties stability is often implicitly used in f_my studies [9,48] as well as in multi-compartment diffusion imaging studies [49]. Nevertheless, it might be questioned as compartments might change their biophysical properties during development, thus possibly affecting their relaxation times. For example, does the increase in myelin compactness with maturation shorten T1_my and T2_my values of the myelin-related compartment? Interestingly, the chemical composition of myelin in the infant brain was shown to be similar to that of the adult brain [50], meaning that T1_my and T2_my should be identical across different ages. Similarly, do T1_ie and T2_ie values of the intra-/extra-cellular compartment change with the development of intra-cellular cytoskeleton and microtubules?

Anyhow, investigating possible age-related changes in the compartments relaxation times was beyond the scope of this study for two main reasons: healthy infants cannot withstand long acquisition protocols without sedation, and maturation is not homogeneous across brain regions [43,51,52]. First, to be coherent with the issue raised previously, the estimation of T1_c and T2_c at the calibration stage should be performed independently for regions with different maturational levels. This would require voxel-wise (and not slice-wise) fitting at the individual level, which is hardly achievable because of acquisition time. Estimating these magnetic properties is not realistic at the group level either, i.e. using data from different infants: even at the same age, inter-individual differences in brain development are observed, and it would be a major challenge to register tissues with strictly similar microstructure and maturation. Second, even if calibration of maturation-specific T1_c and T2_c was realistic, it would not have much sense at the testing stage. To fit the model within a single infant brain, different T1_c and T2_c would be needed for different regions with different maturations. Such an approach would require the maturation level to be known a priori in each voxel, creating a vicious circle. To summarize, only a voxel-wise estimation of f_my, T1_c and T2_c would be entirely free from model hypotheses, and considering exchange between compartments would be helpful. Nevertheless these aspects might be hardly achievable in healthy infants. Despite possible limitations related to the assumption of T1_c and T2_c stability across brain regions and throughout development, our approach based on calibration and testing stages remains a reliable and realistic strategy in such populations.

As detailed in S1 Supporting Information, we observed a strong correlation between the age-matched infant f_my maps that we obtained and those computed by another group [8,29]. These latter studies used an alternative approach based on age-range optimized mcDESPOT protocols and the voxel-wise 3-component fitting of relaxometry signals. Some differences in f_my values were observed across these and our studies. They might rely either on the use of adult T1_c and T2_c to fit our model equations in infants, or on the fact that mcDESPOT f_my estimates might represent not only the magnetization associated with the water pool trapped between the myelin sheaths, but also the magnetization transfer from non-aqueous myelin protons [35]. Thus comparing our results with previous studies across all ages remains difficult [8]. Furthermore, the authors did not indicate whether T1_c and T2_c were stable across the life span. Some correlations between f_my values, T1 and T2 (computed at the voxel level) were observed in infants. As both relaxation times do not exclusively reflect white matter myelination and as they change according to different maturational processes [5], their correlations with f_my values are likely to be age-dependent [8].

Modeling 3 compartments to measure f_my.

In our model, the number of model compartments was set to 3 because it has been shown to be adequate under a wide range of different conditions, including in the developing white matter [9,19,29,53,54]. Although it is possible to quantify f_my based on 2 compartments, 3-compartment models are thought to be more reliable, especially in regions with partial volume [19]. Furthermore, consistently with a 3-compartment model, T2 spectrum in both in vitro and in vivo experiments has shown three major peaks [22,39,55–57]. If necessary, our model could be easily extended with additional compartments, at the expense of increased model fitting complexity due to additional free-parameters. However, theoretical considerations suggest that there is no scope for deriving more than 2 or 3 components in healthy tissues [58].

The most reliable approach to estimate f_my without assumptions on the number of compartments and on their magnetic properties, might be to calculate T2 spectrum [10,11]. However in this case reliable estimation would require acquisition of relaxometry signals for a large number of TEs (making acquisition time unacceptably long for infants), including very short TEs that were not achievable with EPI sequences. Recent acquisition sequences with ultra-short TE (UTE) might be helpful in the future to sample the T2-weighted signal over a wider range of TEs while keeping a short acquisition time. Simulations further suggested that in the presence of noise, using a 3-compartment model with fixed T1_c and T2_c could significantly improve accuracy and reproducibility in determining compartments fractions, as compared to T2 spectrum [22].

Potential applications under pathological conditions.

Although 3-compartment models have been previously applied to investigate pathologies [19,24,53,54,59,60], such as multiple sclerosis [60], autism [53] and partial deletions of chromosome 18q [54], one should be careful when trying to apply the suggested approach to diseases. Indeed it might require including additional compartments (e.g., microvascular, tumor, inflammatory cells, gliosis, etc.) to account for pathological tissues [33]. For example, in demyelinating diseases, gliotic tissue (tissue matrix that replaces myelin, being filled with water, sparse cells and macromolecules) might be modeled as a separate compartment; if not, it would likely contribute to the intra-/extracellular compartment, which has the most similar properties to gliotic tissue among the 3 compartments. Consequently, applying the suggested approach to pathological conditions would require taking into account possible additional compartments specific to the pathology, and re-calibrating the compartment relaxation times in patients.

In conclusion, our f_my quantification strategy overcame existing difficulties (long acquisition/post-processing times) that might limit its practical application in infants, children and clinical patients. Although reliable comparison with previous studies [7,8] was not achievable, our infant f_my maps were able to capture myelin-related changes across early development, suggesting that the proposed approach was relevant at least for evaluation of normal maturation.