Improved precision in CHARMED assessment of white matter through sampling scheme optimisation and model parsimony testing

Abstract

The composite hindered and restricted model of diffusion was proposed a few years ago to characterise anisotropic water diffusion in white matter. The model provides various micro-structural indices, e.g., the axonal density, which can potentially be more specific in characterising white matter structure as compared to metrics from diffusion tensor imaging. Nevertheless, the increased biological specificity comes at the cost of an increased acquisition time, which is challenging for clinical applications, and an elaborate post-processing step, which involves the estimation of several parameters whose confidence intervals can be large. Here an optimisation of the experimental acquisition scheme and data processing pipeline was developed. An optimized multi-shell sampling scheme is proposed based on the electrostatic repulsion algorithm, combined with optimised ordering. Parsimonious model selection criteria, based on Bayesian information, are used to choose among up to three restricted compartments, allowing discrimination between regions of high directional coherence among fibres and regions with more complex geometries. Marked improvements in data quality are demonstrated both through Monte-Carlo simulations and in vivo data. An optimised version of the CHARMED pipeline is developed, that balances scan duration with accuracy/precision on the estimated parameters, needing only a 12 minutes acquisition for whole-brain coverage.

Introduction

The conventional Diffusion Tensor MRI (DT-MRI) model [1,2] provides estimates of mean diffusivity (MD) and fractional anisotropy (FA), which have been demonstrated to be indicators of major micro-structural changes resulting from, for example, development [3–6] or pathology [7–10]. Despite their large sensitivity to brain tissue alterations, these indices are known to be non-specific, since in cerebral white matter (WM) very different configurations of axon density, size and myelination may generate the same measured MD and FA.

The composite hindered and restricted model of diffusion (CHARMED) was proposed a few years ago to characterise anisotropic water diffusion in brain white matter [11,12]. In this model, the diffusion-weighted signal is expressed as the contribution from two different pools of water: a hindered extra-axonal compartment, whose properties are characterised by an effective diffusion tensor, and one (or more) intra-axonal compartments, whose properties are characterised by a model of restricted diffusion within impermeable cylinders.

A CHARMED protocol consists of diffusion-weighted images acquired over a wide range of b-values (up to 10000 s/mm²) along different gradient orientations. The model is able to provide various micro-structural parameters, such as the nerve fibre orientation(s), the T2-weighted extra- and intra-axonal volume fractions, and the principal diffusivities. These parameters can potentially be more specific in characterising white matter structure as compared to conventional DT-MRI. As an example, it has been shown in simulations that the determination of the restricted compartment’s orientation provides improved angular resolution as compared to the principal eigenvector estimated via DT-MRI [11].

More recently, the CHARMED framework was extended to provide estimates of axon diameter distribution within a nerve bundle [12–16]. This framework, named AxCaliber, achieves such a goal by acquiring diffusion-weighted data at different degrees of diffusion weighting and different diffusion times. The CHARMED model has also been used to explain the biological substrates underpinning contrast in Diffusional Kurtosis Imaging [17], a diffusion-based technique that has recently gained much attention thanks to the increased sensitivity to aging-related changes in brain microstructure [18–19].

Notwithstanding the potential improvement that these approaches might offer for white matter characterisation, no clinical studies have been recorded so far. The most likely explanation for the slow uptake is the fact that a deeper characterisation of diffusion is reached at the cost of an increased acquisition time, which is challenging for clinical applications, and an elaborate post-processing step, which involves the estimation of several parameters whose confidence intervals can be large. For this reason, it is essential to improve precision in the measured parameters, either by designing the experiment or in processing the acquired data, such that the acquisition time for a given data quality can be shortened. Furthermore, to adapt the method to clinical applications, it is important to identify a protocol that strikes a good balance between scan duration, which increases with the number of applied gradient orientations, and precision/accuracy on the extracted indices, which decreases as the total number of measurements is reduced.

Alexander proposed a general framework for experimental design optimisation in diffusion-weighted imaging experiments where the tuneable variables are the gradient amplitude g, the diffusion time Δ and the diffusion pulse width δ [20]. These variables are chosen so as to minimise the sum of the coefficients of variation of the model parameters. The diffusion gradient orientations for each combination of g,☐Δ and δ are fixed and distributed evenly on the unit hemisphere, so that the number of orientations employed at high and low b-values is the same. This framework is particularly suitable for small dimensionality of the optimisation and is implemented for a simplified version of the CHARMED protocol [20, 21].

The aims of the present work are threefold: 1) to optimise the experimental acquisition scheme of CHARMED; 2) to propose an optimal protocol for CHARMED acquisitions; and 3) to improve the data processing pipeline including a model selection.

To optimise the acquisition scheme, we propose an approach based on electrostatic repulsion [22], combined with optimised temporal ordering of the measurements [23–24] and extended to different b-value shells. The b-value sampling scheme and the maximum b-value amplitude are chosen on the basis of the results of Monte-Carlo simulations. The optimised ordering approach allowed us to determine the optimal protocol to be applied subject to a given time constraint, using as few measurements as possible (and thus reducing the scan time) without compromising the accuracy/precision with which the biomarkers are estimated. Reducing the scan duration is expected to have a significant impact on the clinical feasibility of the CHARMED analysis and facilitate its widespread adoption.

We demonstrate that the CHARMED model is able to resolve up to three different orientations per voxel. In addition, instead of choosing a priori the number of restricted compartments (RC’s) in the model, parsimonious model selection criteria [25], based on Bayesian information, are used to choose voxel-wise the highest number of RC’s that the data can support without over-fitting. Marked improvements in precision are demonstrated using both Monte-Carlo simulations and a bootstrap approach based on in-vivo data. Specifically, the parsimonious model selection procedure provides increased precision on extracted parameters such as the intra-axonal volume fractions and the restricted fibre orientations.

Methods

Optimisation approach

The CHARMED protocol comprises diffusion-weighted data obtained at different b-values. The original implementation [12] suggested that higher b-values need higher angular sampling resolution while more recent publications [20] employed the same gradient orientations in each shell. In this section, we develop, optimise and compare three different schemes:

• 8 shells in which the number of gradient orientations in each shell increases with the increasing b-value (i.e., similar to Assaf and Basser’s original protocol [11,12]), where both the angular coverage of each single shell and the total angular coverage (i.e., once the gradient directions in all the shells are projected onto a single shell) are maximised, referred to as ‘UNEVEN’ throughout the paper

• 8 shells with the same number of gradient orientations in each shell, where both the angular coverage of each single shell and the total angular coverage are maximised, referred to as ‘EVEN’ throughout the paper

• 8 shells with the same gradient orientations (i.e., similar to Alexander et al.’s protocol [20]), where only the angular coverage of the single shell is maximised, referred to as ‘EVENSAME’ throughout the paper The total number of measurements is set to 200, which allows the total acquisition time for whole brain coverage to be kept within an hour. Fig. 1 a illustrates the differential gradient arrangements in the three schemes.

Figure 1.

Graphic example showing how the gradient orientations are cast in sets to obtain the b-value order for the three schemes UNEVEN (left), EVEN (centre) and EVENSAME (right), shown in panels a). Each set, shown in Fig. 1b, is composed of different gradient orientations, sampling all the 8 b-value shells. For illustration purposes, each shell is represented as a square, whose color goes from yellow (lowest b-value) to light blue (highest b-value). Fig. 1 c shows how the 200 measurements are divided into sets. Fig. 1d shows the b-value order, which is obtained as a continuous sequence of all the sets.

Gradient Scheme Optimisation

As previously described [22], the gradient sampling vectors should be distributed in space as uniformly as possible so as to aim for an average signal-to-noise ratio (SNR) that is as uniform as possible, irrespective of the fibre orientation. Moreover, the order in which the gradient orientations are sampled also needs to be optimized. These goals are achieved through three different simulated annealing algorithms (written in Matlab R2007a, The Mathworks), based on the minimisation of the electrostatic forces between each orientation [22,26]. The algorithm used to arrange the gradient vectors uniformly in 3-dimensional space adjusts the orientations of the gradient vectors until the sum of the repulsive forces between every possible pair of charges, i.e.:

$$F=\underset{ij}{{{\displaystyle \sum }}^{\text{}}}\frac{{\overrightarrow{r}}_{ij}}{{\left(\left|{\overrightarrow{r}}_{ij}\right|\right)}^3}$$ (eq.1)

is minimised, resulting in uniform coverage on the sphere [22]. This algorithm is used for calculating the gradient orientation in each shell for all three schemes (UNEVEN, EVEN and EVENSAME).

For the EVENSAME scheme, the same orientations are repeated across all the shells, but for the UNEVEN and EVEN schemes, a further simulated annealing algorithm is run to ensure that when the gradients of all the shells are considered together the same orientation is never sampled more than once (this may happen, for example, if any of the points belonging to two different shells co-align). This optimisation is performed by projecting all the orientations onto the unit sphere and maximising the coverage of the sphere by means of a second annealing schedule. This algorithm is similar to the first, except that, rather than minimising the electrostatic repulsion of eq.1 between individual charges, all points on a given shell are fixed, and the sum of the electrostatic forces between each set of points is minimised by means of rotation of one shell at a time with respect to the others to maintain an optimal coverage in each shell and maximum overall coverage at the same time.

The optimal acquisition order is obtained, for each scheme, by parsing the measurements into subsets of measurements, each sampling all the b-value shells. Each set comprised gradient orientations belonging to every shell with a different proportion that depends on the scheme, as shown in Fig. 1b. The b-value order is thus obtained as a contiguous sequence of all the sets in order, as shown in Fig 1c. Having low and high b-values interleaved during the scan facilitates our motion correction procedure, interpolating the corrections for high b-values images (generally characterised by a poor signal to noise ratios) on the basis of the correction obtained for the low b-value images (having higher signal to noise ratios).

Initially, the choice of the specific gradient orientations belonging to each set is random. Given the b-value order of Fig.1c, a third simulated annealing is run to optimise the order in which the gradient orientations are sampled, following [23,24]. If the scan is corrupted or terminated before completion, due to subject non-compliance, for example, one aims at maintaining the most uniform coverage of the sphere. In our case, the importance of performing such an optimisation is twofold: it provides the highest possible data quality obtainable for a given number of measurements and, in turn, it allows a posteriori an empirical determination of the smallest possible number of acquisitions needed to have an acceptable data quality.

The method proposed in [24] optimises N-6 nested subsets of the first P directions, with 6<P<N, searching for the ordering that minimises the normalised sum of the electrostatic energy of the subsets. Here, this idea is extended to multiple b-values shells calculating, for each choice of P, the force F^P_TOT as a summation over all the different b-value shells:

$$\begin{array}{l}{F}_{TOT}^P=\underset{SHELL}{{{\displaystyle \sum }}^{\text{}}}{F}_{SHELL}\hfill \\ F_{\text{SHELL }}=\underset{ij,SHELL}{{{\displaystyle \sum }}^{\text{}}}\frac{{\overrightarrow{r}}_{ij}}{{\left(\left|{\overrightarrow{r}}_{ij}\right|\right)}^3}\hfill \end{array}$$ (eq.2)

The initial order is changed randomly swapping the gradient orientations in pairs belonging to the same shell and the best configuration is chosen as the one that minimised $F_{TOT}=\underset{P}{{{\displaystyle \sum }}^{\text{}}}{F}_{TOT}^P$. As a final remark, the convergence of the three simulated annealings has been verified running each algorithm ten times (data not shown). The final energy configuration has a coefficient of variation across repetitions that is always smaller that 3.5%.

Simulations

The three schemes are used to generate synthetic data for comparing different protocols. For each scheme, 8 different maximum b-values are tested, ranging from b_max=2500 s/mm² to 11250 s/mm². Monte-Carlo simulations are used to model the population of spins as random walkers in a 3D environment, which contains a static configuration of barriers, undergoing a diffusion-weighted sequence. Each spin has a magnetization that evolves depending on position and magnetic field gradient at each time step, and the spins accumulate a phase change that depends on the sequence parameters. Synthetic signals are then generated by summing the contributions from all spins at a given echo time. All Monte-Carlo simulations are performed using the CAMINO toolkit [27] using the following parameters: 50000 walkers, 20000 time steps, restricted volume fraction of 0.42, one and two orthogonal fibre populations with a radii distribution drawn from a gamma distribution of parameters a= 9.1477 and b=4.977 mm, longitudinal diffusivity equal to 10^-3 mm²/s. For each scheme, the set of Δ/δ that minimizes the TE is chosen. To provide the most realistic model of noise possible, accounting for the fact that most contemporary studies employ multi-channel coils, non central chi distributed noise is added in each simulation [28] for three different SNRs (64, 43 and 23), that represent the mean SNR of WM in the b=0 s/mm² image, the mean SNR minus one standard deviation and the mean SNR minus two standard deviations. The simulations are repeated 30 times for each scheme using different starting conditions. The synthetic signals are then fitted to the CHARMED model using a custom script written in Matlab R2007a, based on a Levenberg-Marquardt fit of the signal decay to the original CHARMED expression [11]:

$$S=\sqrt{{\left\{S_0\left[\underset{N}{{{\displaystyle \sum }}^{\text{}}}R{F}_N\cdot S_N^{RES}+(1-RF)\cdot S^{HIN}\right]\right\}}^2+{\eta }^2}$$ (eq.3)

where S₀ is the un-weighted image intensity, RF_N is the intra-axonal volume fraction or restricted fraction for the N-th fibre population, S^RES is the restricted signal decay based on Neuman’s model of diffusion within cylinders [29], S^HIN is the hindered signal based on conventional diffusion tensor framework [1,2] and η is the background noise level. The standard deviation of the parameters is calculated using a residual bootstrap approach with n=500 bootstraps. To identify the shortest possible protocol for each scheme and for each maximum b-value, we truncate the ordered datasets by removing one measurement at a time from the end of the acquisition, and run the CHARMED analysis.

The mean and standard deviations of the CHARMED parameters are used to evaluate the performance of the schemes. Specifically, accuracy and precision are calculated, respectively, as the difference between the nominal and measured mean values and as the relative error. The uncertainty in fibre orientation is evaluated calculating the 95% confidence in the principal fibre orientation (i.e., that associated with the biggest RF) using the percentile method [30]. The best scheme is then established by finding, amongst all the combinations of maximum b-value and number of measurements, the shortest possible protocol that maintains the bias in accuracy and the coefficient of variation (CoV) for RF and fibre orientation below 5% in the single fibre configuration and below 10% in the crossing fibre configurations. This is done separately for the three schemes, UNEVEN, EVEN and EVENSAME. The three outcomes are then compared to pick the scheme that provides the same level of accuracy/precision with the least measurements, and thus the least time. As shown in the results section, the UNEVEN scheme with maximum b-value of 8750 s/mm² outperforms the others and results in a reduced protocol comprising 40 measurements, with accuracy/precision still within the chosen threshold, in a scan time of only 12 minutes This reduced optimal scheme will be referred to as ‘UNEVEN SHORT’ throughout the paper.

Data Acquisition

Diffusion-weighted data were acquired from a healthy subject (age = 37 years), using a CHARMED protocol (Assaf and Basser, 2005) on a 3T HDx Signa (General Electric) MR system. Diffusion-weighted data were collected by means of a diffusion-weighted spin-echo EPI sequence with the following parameters: TR/TE = 6000/122 ms, resolution 1.8x1.8x2.4 mm, matrix size 128x128x9, diffusion pulses separation/duration Δ/δ=50/43ms. Diffusion-weighted measurements were obtained using the three gradient schemes described in the previous paragraphs. The total acquisition time was 60 minutes. A b=0 s/mm² image was interleaved every 20 diffusion-weighted images, so that the resulting total number of measurements acquired was 210 for each scheme. The acquisition was repeated three times in three different scan sessions to assess the reproducibility of CHARMED parameters. The reproducibility index was calculated voxel-wise as:

$$R=100\cdot \left(1-\frac{.5(\max -\min )}{mean}\right)$$ (eq.4)

, so that perfectly reproducible voxels have R=100 and voxels whose variability across scans is large compared to the mean have R approaching 0.

Data Analysis and Model Parsimony

Diffusion-weighted data are initially co-registered to the first scan of the first session, then corrected for eddy current/motion combining the approaches proposed in ref. [31–32], which are specifically designed for correcting high b-value data. The data are then analysed with a custom script based on a Levenberg-Marquardt fitting routine written in Matlab R2007a.

The first step was to estimate how many restricted compartments the CHARMED model is able to resolve. The CHARMED model proposed by Assaf et al. [11–12] considers one hindered and up to two restricted components for each voxel (i.e., 16 parameters to be estimated). However, Monte-Carlo simulations reported in the results section demonstrate that the CHARMED model is able to resolve three fibres for each voxel (i.e., 20 parameters to be estimated) with SNR=43, both using the full UNEVEN protocol and its reduced UNEVEN SHORT version.

It is clear that assigning cylindrical-restricted components in CSF makes no logical sense, but the original implementation of the CHARMED approach did not prohibit this. Moreover, there is heterogeneity of fibre configurations within WM, with some voxels containing complex crossing fibre configurations, while others contain only a single fibre population [33]. While in the former, as many as three restricted components may be required [34], in the latter, one component is often sufficient. In a second analysis, the number of restricted compartments was thus chosen voxel-wise on the basis of a Bayesian Information Criterion (BIC) [35]. The BIC is intended to provide a measure of the weight of evidence favouring one model among several competing models. The BIC value:

$$BIC=-2\ln (L)+k\ln (N)$$ (eq.5)

depends on two terms: the first is proportional to the negative log-likelihood L of the model under consideration and gets larger as the model gets farther from the data, while the second term is a penalty that increases as more parameters k are included in the model (characterised by N measurements). The best (most parsimonious) model is thus the one that strikes a balance between fitting the data well and using only a few parameters. Here, a BIC value is separately calculated for each model under consideration, i.e., a CHARMED model characterised by zero, one, two and three restricted compartments, and the model with the smallest BIC value chosen as the best model.

Results

Monte-Carlo simulations

Figure 2 shows the best configurations for each of the three schemes: UNEVEN, EVEN and EVENSAME, obtained by imposing a threshold on accuracy/precision for fibre orientation (left) and restricted fraction (right). The scheme that outperforms the others is the UNEVEN scheme, as it allows better estimates of both RF and fibre orientations with smaller b-values and fewer measurements (i.e., shorter acquisition time), for all the SNR levels tested (with the exception of the plot reported in the upper left corner, where the number of gradient orientations needed is comparable with that needed by the EVENSAME scheme, but the b-value is smaller).

Figure 2.

Best configurations, in terms of maximum b-value and number of gradient orientations, for each of the three schemes UNEVEN (blue), EVEN (red) and EVENSAME (yellow), obtained imposing a threshold on the bias in accuracy and on the CoV for fibre orientation (left) and restricted fraction (right). For further details on the criteria for determining the best configuration, please refer to the methods section. The results are repeated for three different noise levels (SNR=23,43 and 64).

Figure 3 reports the number of gradient orientations/scan time required for measuring RF as a function of the SNR of the non-diffusion weighted scan for the UNEVEN gradient scheme. Clearly, the lower the SNR, the longer the scan time needed to obtain acceptable estimates of RF. We decided to employ the best scheme for SNR=43, since for our scanner the mean SNR was 64 and the standard deviation was 21, i.e., 84.2% of the pixels in WM had SNR larger than 43. However, this plot provides a general indication for when designing diffusion experiments.

Figure 3. Number of gradient orientations/scan time required for measuring RF as a function of the SNR of the non diffusion-weighted scan for the UNEVEN gradient scheme.

In vivo validation

Figure 4 reports the results of the in-vivo validation of the results of the simulation. The first two lines show the RF maps for the three schemes UNEVEN, EVEN and EVENSAME obtained using the full (upper line) protocol and the reduced (lower line) protocols. The UNEVEN SHORT scheme provides estimates of RF that are closer to those obtained using the full protocol; this is demonstrated by the maps of the difference between full and reduced acquisitions, reported in the third line. The bottom line reports the reproducibility of the maps across three scans for the short protocols, with the UNEVEN scheme showing the highest reproducibility.

Figure 4.

RF for the full protocols, RF for the reduced protocols, difference between the RF calculated using the full and the reduced protocol and reproducibility index across three repetitions, calculated for the three schemes UNEVEN (left), EVEN (centre) and EVENSAME (right).

Effect of the Model Parsimony

Figure 5 reports the results of Monte-Carlo simulations for a geometry composed by three orthogonal fibres. The plot shows that the CHARMED model succeeded in resolving three fibres for each voxel when using both the complete UNEVEN scheme and the UNEVEN SHORT scheme.

Figure 5.

CHARMED parameters calculated fitting the CHARMED model to the synthetic data generated using three fibre populations, oriented along the x, y and z axes: angle with the true fibre orientation (left) and restricted fraction (right). The black markers correspond to the mean values across 30 repetitions. The errors bars represent the 95% percentile for the fibre orientation and standard deviation for the restricted fraction. The dashed line is the ground truth. The results are displayed for both the full UNEVEN scheme and the UNEVEN SHORT scheme.

In Fig. 6, Monte-Carlo simulations results show that using a BIC approach leads to higher accuracy and precision in estimates of both RF and the fibre orientation(s), as opposed to using a fixed number of restricted compartments to fit the data. This is true for both the complete UNEVEN scheme and the reduced UNEVEN SHORT one.

Figure 6.

CHARMED parameters calculated fitting the CHARMED model to the synthetic data with and without using a model parsimony approach. The angle with the true fibre direction (left) and the value of RF (right), calculated using or not the model selection, are shown in the same plots for a geometry comprising a single fibre population (upper panels) and two fibre populations (lower panels). The results are shown for both the complete UNEVEN protocol and the reduced UNEVEN SHORT. The markers correspond to the mean values across 30 repetitions. The errors bars represent the 95% percentile for the fibre orientation and standard deviation for the restricted fraction. The dashed line is the ground truth.

In Fig. 7, maps of the difference between the standard deviation associated with RF and fibre(s) orientation are reported. Importantly, the variation associated with the values estimated using model parsimony is always lower (i.e., blue) as compared to that obtained with the fixed RC’s approach, indicating that the precision on the estimates is higher, hence confirming the simulation results.

Figure 7. Maps of the difference between the standard deviation associated with RF and fibre(s) orientation, such that blue-like voxels corresponds to a larger precision using model selection while red-like voxels corresponds to large precision without using model selection.

Discussion

In this work, an approach to optimising the CHARMED protocol is proposed, with the final aim of increasing the confidence in the estimated parameters per unit time and therefore allowing for reduced acquisition times. Although the approach is developed for the CHARMED model, due to the compatibility of CHARMED protocol with other advanced diffusion techniques, the proposed optimisation pipeline which is used to compare different performances, is a general approach to be considered when using high b-value diffusion acquisitions to extract information on the WM.

The gradient order optimisation ensures the highest possible data quality, even if the scan is interrupted due to patient non-compliance. For the purposes of this paper, it also allows us to readily identify the minimum number of gradient orientations that needs to be used to have the shortest acquisition time while maintaining acceptable data quality. We compared three different acquisition schemes and proved that the original implementation of Assaf and Basser [11] with a maximum b-value of 8750 s/mm² and 40 unique gradient orientations is the gradient scheme that produces the best results (in terms of accuracy, precision and reproducibility) with the shortest experimental time. This was confirmed using both Monte-Carlo simulations and in-vivo data.

Our finding that the best value of those tested was bmax=8750 s/mm² is compatible with other results reported in the literature [37]. In addition, in this work, we matched the number of b-value shells used to that of the original implementation of Assaf and Basser [11]. Supplementary Monte-Carlo simulations demonstrate that there is no appreciable difference between the 8 b-value gradient scheme and a scheme in which each measurement has a unique b-value, incremented linearly from the minimum to the maximum (data not shown). Reducing the total number of shells does not affect the experimental time in our approach, since the experimental time is driven by the total number of measurements. Our results allow us to define the best protocol for measuring the restricted fraction and the best protocol for measuring fibre orientations in a given scan time. Given that high-resolution single-shell approaches are already used to estimate the fibre orientation [36], we believe the most interesting parameter the CHARMED analysis produces is the restricted volume fraction (often interpreted as ‘axonal density’). Therefore, we propose the UNEVEN SHORT protocol as the optimal protocol to run CHARMED analysis, given the SNR of our scanner. If we reduce the number of measurements from 200 to 40, we reduce the experimental time by 80% (from 60 to 12 minutes for whole brain coverage) and decrease accuracy/precision by less than 5% for voxels characterised by a single fibre population and less than 10% for voxels characterised by crossing fibers. The present paper also provides indications about the best protocol when the SNR is lower or higher, serving as a general guideline for choosing CHARMED acquisition parameters.

We propose the protocol reported in Table 1 as the best compromise between the scanning duration and the quality of the estimated parameters for estimating RF. In Table 2, the gradient orientations are listed. This optimised gradient scheme is calculated from scratch following the pipeline of par 2.2 with 40 gradient orientations (plus 5 gradient orientations in the inner shell, that are needed to calculate conventional DTI maps that are used as a reference in the registration [31–32]). Monte-Carlo simulations (data not shown) demonstrate that there is no appreciable difference between the performance of the optimised scheme and the performance of the original scheme with the same number of gradient orientations.

In the second part of this work, we demonstrated for the first time via simulations that the CHARMED model is able to resolve up to three different fibre orientations within a voxel. However, introducing a BIC means that a higher number of RC’s, and thus a higher number of fitting parameters, is invoked only when the fit quality is sufficiently improved. This implies that, for example, one can obtain a better confidence in the estimated fibre orientation when fitting parsimoniously to the data, as shown in Fig. 6 on synthetic data and in Fig. 7 in vivo. As reported in the introduction, it has been shown by simulation [11] that the orientation of the restricted compartment provides improved angular resolution as compared to the main eigenvector obtained with DT-MRI technique, when one RC is fitted, which would confer an advantage in tractography studies. With the proposed approach, we can thus obtain improved angular resolution as compared to DT-MRI for all voxels characterised by one RC. At the same time, we can resolve the fiber crossing issue in all voxels characterised by two or three RCs, implying a further benefit for tractography. Specifically, incorporating both the fiber orientations and the RC’s number information into tractography algorithms can improve the reliability on the derived fiber tracts, and is the matter of a future work.

Using a parsimonious fitting approach, lower standard deviations are also obtained in estimated scalar parameters as the population fraction of the restricted components. By means of q-space techniques, it has been shown that characterization of the restricted diffusion can extend the diagnostic ability of diffusion imaging [37–38]; nevertheless, q-space imaging is even more data intensive than CHARMED and thus less suitable to clinical application. We believe instead that the protocol proposed here will make the CHARMED approach more accessible in investigations into white matter disorders. Specifically, we believe this protocol to be the starting point for applying CHARMED analysis to WM diseases such as multiple sclerosis, where RF can offer complementary information to DTI measures [39–41].

A possible limitation of the model parsimony approach is the fact that the uncertainty in parameters is a function of orientation of the structure with respect to the encoding gradients, as shown by Jones [42]. As statistical rotational invariance is compromised, a problem with the parsimony testing would become increasingly apparent, i.e. for some orientations, the variance in parameters would be larger than in others. Thus, by pure virtue of fibre orientation(s), we might be forced by the BIC to infer a less complex fibre architecture, comprising fewer unique compartments. Ensuring as uniform a coverage of the sphere as possible (par. 2.2) will serve to mitigate against such occurrences.

Conclusions

We have developed a comprehensive optimised CHARMED pipeline, comprising an optimised data acquisition scheme and an analysis approach that incorporates a measure of model parsimony. This approach clearly improves the data quality and results in increased confidence in the estimated parameters. Moreover, we proposed an optimised protocol that reaches a sensible compromise between the scanning duration and the quality of the estimated parameters.