Investigating apparent differences between standard DKI and axisymmetric DKI and its consequences for biophysical parameter estimates

The purpose of the study is to identify differences between axisymmetric diffusion kurtosis imaging (DKI) and standard DKI, their consequences for biophysical parameter estimates, and the protocol choice influence on parameter estimation.


INTRODUCTION
Diffusion kurtosis imaging (DKI) has increasingly been used to study the neuronal tissue microstructure and derive biophysical parameters relevant for understanding brain function and impact of disease [1][2][3] during the past ten years.DKI is a more complex extension of the diffusion tensor imaging (DTI) framework and provides diffusion kurtosis metrics that can provide complementary information [4][5][6] to DTI.However, the increased complexity comes with an increase in acquisition time.Since time is a limited resource in scientific and especially clinical settings, needing more time poses a major hurdle for a more extensive implementation and application of DKI.
8][9] Axisymmetric DKI reduces the parameter space by imposing additional symmetry assumptions, that is, axisymmetrically organized fibers in the imaged tissue structure and axisymmetric DKI (eight parameters) can be fitted with less data than is required by standard DKI (22 parameters).One time-efficient acquisition scheme that leverages axisymmetric DKI's data demand advantages is the fast "199" scheme 7,10 that relies on a total of 19 images (18 diffusion weighted images and one nondiffusion weighted image), only.However, parameter estimation from fewer diffusion weighted images is more susceptible to noise and it can be expected that the signal-to-noise ratio (SNR) requirements for accurate parameter estimates are higher when fitting data acquired with the fast "199" protocol compared to a standard diffusion MRI (dMRI) protocol.To mitigate the influence of noise, denoising algorithms can be used. 11,12n important parameter subset existing in both axisymmetric DKI and standard DKI are the five axisymmetric DKI tensor metrics (AxTM), the parallel and perpendicular diffusivity (D || and D ⊥ ) and kurtosis (W || and W ⊥ ) and mean of the kurtosis tensor (W).Since the AxTM are attainable with both DKI models, they are perfectly suited to be used for a model comparison.The five AxTM are also directly related to the five parameters of the biophysical standard model, 13 the axon water fraction AWF, axon dispersion , parallel and perpendicular extra-axonal diffusivities D e,|| and D e,⊥ and intra axonal diffusivity D a , here estimated with the WMTI-Watson model. 14,15t has been suggested that the symmetry assumptions made in axisymmetric DKI are likely a reasonable approximation to diffusion in major white matter fiber bundles, for example, occurring in white matter. 7Violation of these additional symmetry assumptions, for example, in white matter voxels containing crossing fibers, might lead to a deviation of axisymmetric DKI fit results from their standard DKI reference counterpart.
In this work, we simulate synthetic white matter data using standard DKI as forward model to explicitly include complex, non-axisymmetric fiber configurations.We hypothesize that any observed deviation between both DKI model variants is due to an error in axisymmetric DKI rooted in an underlying complex and thus non-axisymmetric fiber configuration.To test this hypothesis, voxel selection masks are used to filter out voxels with high fiber complexity that likely break axisymmetric DKI's symmetry assumptions.Furthermore, we establish the "baseline difference" between axisymmetric DKI and standard DKI AxTM estimates from noise-free data.The baseline differences are inherent to axisymmetric DKI and will always be there with respect to standard DKI.Another topic is the influence of noise on the standard deviation (SD) of axisymmetric DKI fit results.Here the question at which SNR the baseline difference between both DKI models becomes bigger than axisymmetric DKI's SD is investigated.This will establish an SNR regime where the standard deviation caused by noise dominates parameter estimation and at what SNR the baseline difference between both DKI models becomes visible which is here referred to as the "tipping point".To investigate this, first a noise simulation study is performed where the SNR space is densely sampled to precisely establish the tipping point without using denoising methods.Then, we use Multi-shell Position-Orientation Adaptive Smoothing (msPOAS), an adaptive denoising algorithm that is preserving tissue boundaries without introducing blurring 11,16,17 to establish if the tipping point can be reached for realistic SNRs in voxels where the axisymmetric conditions were best fulfilled.In both noise studies, the influence of the acquisition protocol (fast "199" protocol vs. a standard dMRI protocol) is investigated as an additional variable.Finally, the median bias based on the two acquisition protocols is quantified in five well known fiber tracts for a typical SNR found in dMRI datasets. 18Throughout the work, the noise-free estimates of the five AxTM or the biophysical parameters based on standard DKI are used as a ground truth reference and axisymmetric DKI fit results are compared to them on a voxel-wise basis, hence the baseline difference between axisymmetric DKI and standard DKI is also referred to as "bias".

METHODS
A detailed description of the standard DKI model and the axisymmetric DKI model is provided in the Sections S1.1 and S1.2 but can also be found in References 7 and 19.

Acquisition
Multishell, in vivo dMRI data with 153 diffusion gradient directions and b-values of 0 Image reconstruction at the scanner from the multichannel information was done using a sum of squares algorithm where all channels had the same weights.The following pre-processing was applied to the dMRI data using the ACID toolbox 20 (in that order): eddy current and motion artifact correction using the ACID "ECMO-CO" module, susceptibility artifact correction using the ACID "HySCO" module and Rician bias correction using the ACID "Rician bias correction" module that uses a second moment approach and acts on the signals of the individual diffusion weighted images as described in Reference 21.The noise level  was calculated using the ACID "Noise estimation" module with the "repeated measures" method where  is estimated as the averaged SD over the diffusion measurements of the highest diffusion shell in the ventricles, see Reference 20.Additionally, we acquired multiparameter mapping data 22 on the same subject and calculated the R 1 map using the hMRI toolbox, 23 for anatomical visualization.
The in vivo dMRI data used for this study was acquired with the help of a human research participant.The participant provided written informed consent.The local ethics committees at University Medical Center Hamburg-Eppendorf approved the study (PV5141).

Generation of synthetic data
The acquired multishell, in vivo dMRI data was fitted with standard DKI to obtain the 22 standard DKI tensor metrics per voxel which were then used for generation of noise-free, synthetic dMRI data using standard DKI as a forward model with the same diffusion shells and gradient directions that were used to acquire the data.

Biophysical parameters
The framework presented in Reference 13,31, here referred to as "WMTI-Watson," was used to establish an analytical connection between the five AxTM Ω = {W, W || , W ⊥ , D || , D ⊥ } and the biophysical parameters  = {AWF, , D e,⊥ , D e,|| , D a }.A detailed description of the framework can be found in Section S1. 3.
We neglected a potential model error of the WMTI-Watson model with respect to the biological tissue ground truth and assumed the biophysical parameters based on AxTM estimates from standard DKI to be the ground truth.In this study, we focused on quantifying the propagation of the error in the AxTM estimates into the respective WMTI-Watson parameter estimates introduced by axisymmetric DKI.

Computation of difference between both DKI models and substantially differing voxels
The estimated parameters, either the set of AxTM Ω or the set of biophysical parameters  were estimated based on standard DKI and axisymmetric DKI and compared using the voxel-wise absolute percentage error (A-PE): Here  is an element of either Ω or  and the subscript indicates whether the parameter was estimated based on standard DKI or axisymmetric DKI.If A-PE > 5%, the corresponding voxel was classified as a "substantially differing voxel" (SDV).The study focused on white matter only, to obtain the white matter mask, we segmented the R 1 map into tissue probability maps (TPM) and thresholded the white matter TPM (TPM > 0.9), see green contour in Figure 2. To summarize the results we estimated a) the number of SDV in white matter in percent and b) the median A-PE in the population of SDV.We implemented the condition  ∈ Ω ≥ 0 and  ∈  ≥ 0 because negative diffusivities are non-physical and AWF and  are ≥ 0 by definition.Furthermore, kurtosis estimates in the healthy brain have been found 24 well above 0.

Influence of voxel selection masks on number of substantially differing voxels
To identify voxels in white matter that likely brake the axisymmetric tissue symmetry assumption, 25,26 two different masks were used: a fractional anisotropy (FA) mask and a so-called "Westin mask."To generate the FA mask, white matter voxels with FA ≥ 0.55 were selected based on the FA of an unidirectional phantom, see Reference 27.This voxel selection is referred to as "FA mask."To generate the Westin mask, white matter voxels that fulfill the conditions imposed by a threshold for the Westin indices computed via the diffusion tensor eigenvalues , C L = 26 were investigated, this voxel selection is referred to as "Westin mask."For comparison, a third mask containing the entire white matter was used.

Simulation of noisy dMRI data and fast "199" protocol
The noise-free, in vivo human brain data Snoise−free were noise contaminated according to S cont = | Snoise−free +  + i|, where ,  ∈  (0, ) were drawn from a zero mean Gaussian with SD .A special property of axisymmetric DKI is that due to it's reduced parameter space, it needs fewer measurements and-at a minimum-all five AxTM can be estimated from a fast two shell setup with nine distinct gradient directions n (i+∕−)7,10 and one additional b = 0 measurement.Here, the fast protocol was simulated for b-values of b = 1100 s mm 2 and b = 2500 s mm 2 .

Single voxel analysis
Here we asked: at what SNR is the baseline difference between standard DKI and axisymmetric DKI bigger than the SD of axisymmetric DKI?The SNR at which this happens is referred to as "tipping point."To investigate this, we simulated 12 single voxels (see Section S1.7) selected from the white matter of the in vivo human brain data set for densely sampled SNRs = [1, 2, 3, ...140] according to the procedure reported above for 2500 noise samples for the standard MRI protocol described in Section 2.1 and additionally for the fast "199" protocol. 7,10Then, axisymmetric DKI was fitted to the simulated data.To evaluate the simulation and make it comparable to the noise-free study regarding the SDV, the differences between standard DKI and axisymmetric DKI were normalized to the corresponding average AxTM values in white matter and expressed as a percentage thereof.

White matter Westin mask analysis with adaptive denoising
To investigate whether the tipping point can be reached for typical SNRs of published DKI protocols, we performed another noise simulation study that used the adaptive denoising "msPOAS" module of the ACID toolbox with two set-ups for the adaptation parameter ,  = 10 and  = 100 on noisy, whole brain simulation data.The adaptation bandwidth  controls the adaptivity of msPOAS, 11 ranging from  = 0 (complete adaptation) where the original image is unchanged, to  = ∞ (non-adaptive denoising) similar to, for example, Gaussian smoothing.Smoothing was performed in the white matter mask because msPOAS needs coherent regions to work properly, while parameter estimation was done in the Westin mask.For the simulated SNRs, values of published protocols were chosen: SNR = [5, 15, 30, 52] (SNR: 52, see Reference 28, SNRs: 30, 15, 5, see Reference 29) and SNR 100.That data was then fitted with axisymmetric DKI in the Westin mask (see Section 2.3), since the Westin mask turned out to be most effective in reducing the number of SDV (see Section 3.2).To evaluate this analysis, n = 100 noise samples were simulated, the SD per voxel was calculated and the median was computed.Second, the percentage of voxels reaching the tipping point was calculated.

Quantification of bias in fiber tracts
To quantify the difference between both DKI models based on the two acquisition protocols under realistic conditions, the whole brain human dMRI data was simulated and fitted with the fast "199" and the standard acquisition protocol for SNR = 39, which was the SNR reported in one of the original fast "199" protocol studies. 18Data were smoothed using msPOAS and  = 100."Bias" in this context refers to the difference of the noisy, voxel-wise fit results to the ground truth (noise-free, standard DKI fit results), again computed as the A-PE, see Equation (1).Results were evaluated with a specific focus on five well-known fiber tracts: corpus callosum (cc), superior corona radiata (scr), external capsule (exc), superior longitudinal fasciculus (slf), and posterior corona radiata (pcr).The fiber tracts were identified with the JHU-ICBM-DTI-81 white matter atlas 30 for which it was nonlinearly registered to the subject space of the in vivo dMRI data using the spatial normalization tool in SPM12, see Figure 1.

3.1.1
Differences between AxTM across the white matter using the two DKI models Figure 2 shows the spatial distribution of SDV (see Section 2.3) as red dotted voxels in a slice of the AxTM and biophysical parameters, Figure 3 summarizes the The five regions of interest corpus callosum (cc), superior corona radiata (scr), external capsule (exc), superior longitudinal fasciculus (slf) and posterior corona radiata (pcr) identified with the JHU-ICBM-DTI-81 white matter atlas indicated in the coregistered R 1 image of the human brain in vivo dataset used in this study.

F I G U R E 2
Examples of the axisymmetric DKI tensor metrics (top) and biophysical parameters (bottom) in a slice of the human brain data used in this study.The green contour outlines the white matter, the red dots indicate voxels where the A-PE ≥ 5% ("substantially differing voxels").The red barplots of the top row in Figure 3 quantify the percentage of substantially differing voxels in the whole white matter.number of SDV and the median A-PE in that population using barplots.
The number of SDV in the white matter mask is highly parameter dependent (Figure 3A.1), for example, only 1% of D || , D ⊥ and 2% of the mean of the kurtosis tensor W. The W || (22%) and W ⊥ (51%) were affected much more, see also spatial distribution of SDV.However, the median difference in the SDV across the AxTM was more similar and ranged between 7% (W) and 11% (W ⊥ ).

Differences between biophysical parameters across the white matter based on the two DKI models
Both the number of SDV and the median A-PE in the SDV population (11%-18%) was higher for the biophysical parameters than the AxTM.Again, the percentage of SDV was parameter dependent and spanned from 28% For neither the AxTM nor the biophysical parameters Figure 2 revealed a spatial distribution pattern of the SDV in the depicted white matter slice.

Influence of voxel selection mask on number of SDV and median bias
Confining the analysis to white matter voxels in the Westin mask reduces the number of SDV more than confining the analysis to white matter voxels in the FA mask.For the AxTM, the median bias remained approximately the same for all three voxel masks, except for D || in the Westin mask analysis that was heavily affected by very few outliers with high median bias (64%), while in the case of the biophysical parameters the Westin mask and FA mask could reduce the median bias.

AxTM
Both, the FA mask and the Westin mask, reduced the number of SDV for all AxTM, compare Figure 3A.2 and A.3.They were most effective for W || and least effective for W ⊥ .For example, for W ⊥ the number of SDV could be reduced from 51% in the whole white matter to 46% in the Westin mask while for W || the number of SDV could be reduced from 22% to 3%.In general, the Westin mask was more effective in reducing the SDV than the FA mask.Interestingly, the spread of the median A-PE remained similar between ≈7% to ≈11% when applying both masks, see Figure 3A.2, 3A.3, except for D || in the Westin mask that was affected by outliers.

Biophysical parameters
The same trend as for the AxTM was also observed for the biophysical parameters with regards to the number of SDV: both masks reduced the number of SDV in all parameters significantly and the Westin mask was more effective than the FA mask.But, as opposed to the AxTM, the spread of the median A-PE was also reduced from (11%-18%) to (8%-10%) for the Westin mask and (8%-11%) for the FA mask, see Figure 3B.2,B.3.

Single voxel analysis
Figure 4 shows the SNR of the tipping point for each of the 12 simulated voxels.The term "bias" here refers to the normalized difference between standard DKI and axisymmetric DKI in %.

F I G U R E 4
Signal-to-noise ratio (SNR) at which the tipping point (bias > std) is reached, computed for n = 2500 noise realizations for the 12 simulated voxels.Top shows the results for the standard diffusion kurtosis imaging (DKI) protocol, the bottom shows the results for the fast "199" protocol.The connecting gray line's purpose is to indicate the trend that for higher biases the SNR required to reach the tipping point tends to be lower.

Standard protocol
The likelihood of finding an SNR threshold ≤140 increases for increasing axisymmetric DKI biases, that is, from left to right on the x-axis.W ⊥ and W had the highest number of voxels for which a tipping point below the simulation limit (SNR 140) could be found, where the lowest tipping point was found at SNR = 13 for the W ⊥ voxel with the highest bias (voxel 12).For the diffusion parameters D ⊥ and D || , most voxels did not reach the tipping point up to SNRs = 140.

Fast "199" protocol
For the fast "199" protocol, too, the likelihood of finding an SNR tipping point ≤140 increases for increasing biases.However, much fewer voxels reached the tipping point below the simulation limit compared to the standard protocol.Again the lowest tipping point was found for the W ⊥ voxel with the highest bias (voxel 12).

White matter Westin mask analysis with adaptive denoising
Figure 5 shows the median SD in the Westin mask (blue and cyan markers), the median difference between standard DKI and axisymmetric DKI in the Westin mask (dashed black line) and the percentage of voxels crossing the voxel-wise tipping point (red and magenta markers, right y-axis).

Standard protocol
The voxel-wise tipping point was reached for realistic, higher SNRs up to 52 predominantly for the kurtosis parameters but the percentage depended on the adaptation parameter .For example, for W at SNR = 52, ≈ 37% of voxels reached the tipping point for  = 100 while it was only ≈ 14% for  = 10.A similar pattern was observed for W ⊥ where for SNR = 52, ≈ 28% of voxels reached the tipping point for  = 100 and only ≈ 19% in case of  = 10.
For W || at SNR = 52 the number of voxels reaching the tipping point was ≈ 7% for  = 100 and below 5% for  = 10.
For the diffusion parameters the number of voxels was close to zero for all realistic SNRs up to 52.

Fast "199" protocol
The number of voxels reaching the tipping point was lower when the fast protocol was used compared to the standard protocol.For SNR= 52 and  = 100 the tipping point was Median SD in subset of white matter voxels in Westin mask, computed for n = 100 noise samples (blue and cyan datapoints, left y-axis), median difference between standard diffusion kurtosis imaging (DKI) and axisymmetric DKI in that subsample of voxels (dashed black line) and percentage of voxels crossing the voxel-wise tipping point (red and magenta data, right y-axis).Shown are the results when fitting the standard MRI protocol (top) and the fast "199" protocol (bottom) data, see Sections 2.1 and 2.4.
reached by ≈ 13% of voxels in case of W and ≈ 14% in case of W ⊥ .For SNRs up to 52 and in case of  = 10 the number of voxels reaching the tipping point was always close to zero for all parameters.

3.3.7
Median SD Independent of the protocol, the median SD always became smaller for increasing SNRs and was also smaller for a higher adaptation parameter .

Quantification of bias in fiber tracts
Figure 6 shows the median bias computed in the five fiber tracts corpus callosum (cc), superior corona radiata (scr), external capsule (exc), superior longitudinal fasciculus (slf), and posterior corona radiata (pcr).The median bias of standard DKI and axisymmetric DKI is always very similar when a standard MRI protocol is used.For W the combination of the fast protocol and axisymmetric DKI performs very similarly to when a standard protocol is used while for the other parameters it performs worse.

DISCUSSION
In this work, we found that depending on the parameter, there is a significant number of voxels where axisymmetric DKI and standard DKI inherently differ more than 5%.For D || , D ⊥ and W we found little differences between the two DKI models, while W ⊥ and W || showed larger differences.All five axisymmetric DKI based biophysical parameters were strongly different from their standard DKI based counterparts.Introduction of two voxel selection masks that reduced the number of voxels containing more complex fiber configurations mitigated the observed differences between both DKI models (especially for W || and the biophysical parameters) suggesting that fiber complexity is one cause of the observed baseline differences between both DKI models.The noise simulations revealed the challenges of the fast "199" protocol.For a higher but still realistic SNR of 52, the single voxel noise simulation without denoising showed that the baseline differences between axisymmetric DKI and standard DKI are often invisible when a standard MRI acquisition protocol is used (the most tipping points were reached for W: 5/12 voxels, that is, ≈ 42% and W ⊥ : 2/12 voxels, that is, ≈ 17%) and that they are almost always invisible when the fast "199" protocol is used (only 1/12 voxels for W ⊥ reached the Median bias for noisy human brain data at a signal-to-noise ratio (SNR) = 39, smoothed with multi-shell position-orientation adaptive Smoothing (msPOAS) using  = 100 in the five white matter fiber tracts corpus callosum (cc), superior corona radiata (scr), external capsule (exc), superior longitudinal fasciculus (slf) and posterior corona radiata (pcr).The median bias was computed for a simulation of the standard protocol that was fitted with standard diffusion kurtosis imaging (DKI) (blue data points) and axisymmetric DKI (red data points) and for a simulation of the fast "199" protocol that was fitted with axisymmetric DKI (pink data points).
tipping point, that is, ≈ 8%).We also found that the SNR of the tipping point depended on the absolute value of the bias.However, in another noise simulation, adaptive denoising improved axisymmetric DKI parameter estimation from the fast protocol to a point where ≈ 13% of W voxels and ≈ 14% W ⊥ voxels reached the tipping point for realistic SNRs while it did not substantially improve estimation from the standard protocol.Furthermore, our fiber tract analysis revealed that under realistic experimental conditions (realistic SNR and one repeated measurement only), the combination of the fast protocol with adaptive denoising performed similar to standard protocol estimates for W.

Noise-free simulation: baseline differences between AxTM across the white matter using the two DKI models
The AxTM capture different properties of diffusion in tissue and it is not surprising that the observed differences between both DKI models is AxTM dependent.We used the number of SDV to quantify the baseline differences between axisymmetric DKI and standard DKI.We found that the diffusion parameters D || and D ⊥ and the mean of the kurtosis tensor W have very few SDV compared to W || and W ⊥ .
With a median A-PE of 7%-11% in white matter, the error made in axisymmetric DKI might be acceptable depending on the application.Purely judging from the number of SDV, the diffusion parameters were "safest" where only 1% of white matter voxels were SDV, followed by the mean of the kurtosis tensor W with 2%.It can generally be expected that the kurtosis parameters are more sensitive to a model error since they are quadratic in the b-value b compared to the linear diffusion parameter counterparts.The propagation of error when fitting the axisymmetric DKI model to dMRI data therefore will be more severe for the kurtosis parameters.Interestingly, the number of SDV increases tenfold from W to W || and roughly doubles from W || to W ⊥ .The reason for this trend still needs to be explored.

Noise-free simulation: differences between biophysical parameters across the white matter based on the two DKI models
The number of SDV were significantly enhanced in the biophysical parameters (in case of the whole white matter mask up to 54%, see Figure 3) compared to the AxTM from which they were computed.A reason for the enhancement might be that all five AxTM are required to estimate the biophysical parameters, see Section S1.3, and that the connection is complex and nonlinear.The observed AxTM differences could therefore be amplified due to nonlinear effects but also synergistically enhance the number of SDV in the biophysical parameters.However, the found median A-PE of 11%-18% might be acceptable depending on the study.Also, similar to Reference 31, here the "−" branch tended to yield physically unfeasible, constant high (50)  values where the objective function did not have a well defined minimum for .Furthermore, for this branch D e,|| > D a which in healthy white matter was found to be the biologically invalid solution by most studies. 32We therefore did not report the results of the − branch.

Noise-free simulation: influence of voxel selection mask on number of SDV
The difference between standard DKI and axisymmetric DKI is likely linked to fiber complexity that brakes the symmetry assumptions of axisymmetric DKI.Reducing the number of voxels with complex fiber structures using voxel selection masks reduced the number of SDV.
The Westin mask reduced the number of SDV in both the AxTM and the biophysical parameters, supporting the hypothesis that differences between both DKI models are linked to the underlying fiber complexity.However, the Westin mask effectiveness was parameter dependent and, for example, worked particularly well for W || where it more than quartered the number of SDV while it had much smaller effects on W ⊥ .The FA mask performed very similarly but reduced the number of SDV slightly less than the Westin mask.In particular, both masks struggled with reducing the number of SDV of W ⊥ .
These findings indicate that next to fiber complexity other factors may play a role in determining the difference between axisymmetric DKI and standard DKI estimates.Other factors that could play a role are differences glia-cell density, 33 the axonal diameters distribution or the fiber dispersion.With regards to the FA, for example, in vivo tissue FA could also be influenced by factors like the degree of myelination or axon density.This means that the conclusion "if FA ≥ 0.55 then the voxel has a unidirectional fiber configuration" is not necessarily strictly true and voxels in the FA mask might still have a complex fiber structure.The smaller effectiveness of both the FA mask and the Westin mask on W ⊥ could originate from axisymmetric DKI generally oversimplifying estimation of W ⊥ since it is directly estimated as one model parameter instead of calculated from three separate tensor metrics as in standard DKI, see, for example, Reference 34.

Inter-dependence of A-PE and difference in main fiber orientation
It was shown that axisymmetric DKI produces the same results as standard DKI if two requirements 35 are met, see Section S1.6.Fulfillment of condition b) was not explicitly checked in Reference 35 where differences between axisymmetric DKI and standard DKI were reported.The degree to which the main fiber orientations, estimated with both DKI models, differ can be quantified with the angle  between them.The majority of angles  were between ≈ 1 and 5 degrees in white matter (Figure S3) demonstrating that condition b) is not fulfilled in most cases.Investigating the dependency of the A-PE on angle  using density scatter plots showed an inter-dependency predominantly for W ⊥ and W || , Figure S3.For these parameters these findings indicate that at least to some extent, there is a causal relationship between  and the A-PE.
It could be ruled out that the observed differences between both DKI models in this study is only due to violating condition a) of Reference 35 by implementing a log-of-signals fit demonstrating that this fit implementation still produced different fit results for both DKI models, see Section S1.5.

Single voxel analysis
Simulating 12 single voxels for n = 2500 noise realizations revealed that the tipping point at which the axisymmetric DKI inherent bias is larger than the SD of the axisymmetric DKI parameter estimates was only reached for high, unrealistic SNRs ≥ 52 in most cases.Therefore, the differences between both DKI models will likely not be visible under realistic experimental conditions if techniques like denoising are not used.We found that the tipping point depended on the axisymmetric DKI bias with the strong tendency that the higher the bias, the earlier the tipping point is reached.Furthermore and in line with the main findings that were dealing with the number of SDV per AxTM, it was harder to find the tipping point for the diffusion parameters than the kurtosis parameters, most likely because the diffusion parameters were less biased than the kurtosis parameters and therefore the requirements for the SD to reach the tipping point were too high.

White matter Westin mask analysis with adaptive denoising
Intuitively one would expect axisymmetric DKI to have better precision than standard DKI because of it's reduced parameter space compared to standard DKI.However, in an earlier study performed on simulations of noisy dMRI data 19 we have found that axisymmetric DKI does not improve precision compared to standard DKI.Furthermore, in the "Single voxel analysis" we have found that without denoising, the tipping point at which the difference between axisymmetric DKI and standard DKI becomes visible typically requires very high SNRs, often above 140.We therefore were interested to see if the tipping point could be reached for realistic SNRs (up to 52) in combination with additional adaptive denoising (msPOAS) with two settings of adaptation parameter . 11The influence of  was significant for low SNRs while its effect was decreasing the higher the SNR was.We found that for a realistic but high SNR 52, the tipping point was only reached for W and W ⊥ in roughly 1∕3 of voxels in the Westin mask if a standard MRI protocol is used and for a high adaptation parameter  = 100.In all other cases the number of voxels reaching the tipping point was significantly lower and especially for the diffusion parameters it was close to 0 for realistic SNRs.This indicates that the bias introduced by the symmetry assumption made in axisymmetric DKI is not going to be visible under realistic, experimental conditions in most of the voxels.

Quantification of bias in fiber tracts
The noise simulations are more realistic than the noise-free simulation but introduce additional factors that could cause a difference between standard DKI and axisymmetric DKI, for example, model dependent noise susceptibility and the Rician bias.Noisy data was simulated at SNR = 39 and in an earlier study 19 this SNR was found to be sufficient to reduce the difference between both DKI models below 5% in in vivo white matter, see Fig. 6 in Reference 19.
If the noise induced Rician noise bias can be considered negligible if it is below 5% at SNR = 39, the observed difference between axisymmetric DKI and standard DKI estimated for the standard protocol is most likely the axisymmetric DKI inherent bias.Consistent with the findings concerned with the median bias found in Figure 3, the fiber tract analysis, Figure 6, showed that the median bias of the kurtosis AxTM per fiber tract was highest for W ⊥ , followed by W || and lowest for W. Also, both DKI models based on the standard protocol performed very similarly (except for W ⊥ ) which again demonstrates that the tipping point is not reached in most cases under realistic conditions, that is, noise is dominating the bias and not the model differences.
However, some of the findings in the fiber tract analysis were counter-intuitive.In an other study, the corpus callosum was found to predominantly have voxels with a single fiber orientation 36 and could therefore be considered to have less of a complex structure than other tracts making it more likely to fulfill the assumptions made in axisymmetric DKI.However, the corpus callosum had the highest median bias for 2 of 5 AxTM.Furthermore, in Reference 36 the slf tract was reported to host many voxels with two fiber orientations, that is, fiber crossings which would make it prone to failing the assumption of axisymmetric DKI but the slf was one of the tracts with the lowest differences between both DKI models.This finding, again, points to other factors than fiber complexity, here in the form of fiber crossings, playing a role in determining the difference between standard DKI and axisymmetric DKI, as already discussed in Section 4.3.

4.5.4
Fast "199" protocol versus standard protocol Axisymmetric DKI's major advantage is its reduced data demand which can save acquisition time if it is used with a fast acquisition protocol.Combination with additional, well known image acceleration techniques like multislice and GRAPPA could speed up acquisition even more.Moreover, the shorter sampling scheme was reported to have better contrast-to-noise ratio compared to standard DKI in rat brains with induced stroke 37 and it is less susceptible to motion artifacts which is even more important in clinical settings than for basic research where the subjects are more used to scanning and thus less probable to move during the acquisition.We set out to investigate potential problems going along with axisymmetric DKI and its assumptions and the fast "199" image acquisition scheme.
For the single voxel analysis, investigating the influence of the MRI protocol on the tipping point revealed that the fast "199" protocol had even higher SNR requirements to reach the tipping point than the standard protocol.This is plausible since reducing the number of diffusion gradients in the MRI protocol makes parameter estimation more prone to the influence of noise because fitting is done with fewer data points.Furthermore, the effective SNR, SNR eff = SNR ⋅ √ number of b = 0 images is higher for the standard acquisition protocol, since here more and optimally distributed b = 0 images are acquired compared to the fast "199" protocol where only one b = 0 image is acquired.In the "white matter Westin mask analysis with adaptive denoising" analysis we found that when switching to the fast "199" measurement protocol, the number of individual voxels that reached the tipping point in total was smaller compared to the standard protocol which is plausible as described above.It is also important to consider that fewer acquired diffusion gradients can go along with disadvantages, for example, FSL's eddy current correction ("eddy") as a preprocessing step will be challenging since it ideally requires to densely sample the whole sphere with the diffusion gradients and a minimum of approximately 10-15 directions for a b = 1500 shell with increasing demands for higher shells, see https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/eddy.Note that these requirements do not apply to the eddy current and motion correction algorithm implemented in the ACID toolbox that was used to pre-process the measured dMRI data used for synthetic data generation since here the eddy-current field is estimated independent of the diffusion gradient directions. 38Furthermore, Rician bias correction depends on an accurate estimate of the noise level  and the chosen "repeated measures" method (see Section 2.1) works better with more acquired diffusion gradients since it  is estimated as the SD over the highest diffusion shell.All in all, parameter estimation with the fast "199" protocol is challenged by a higher bias and the baseline difference between both DKI models is likely not visible for realistic SNRs.

CONCLUSION
Axisymmetric DKI offers advantages like a reduced data demand that is relevant in scientific and clinical practice.We asked the question whether these advantages are counteracted by an error related to the intrinsic simplification of the model.We found that in the noise-free case and when using a standard MRI acquisition protocol D ⊥ , D || , and W could be estimated with few SDV with respect to their standard DKI counterpart.For all other parameters, that is, W ⊥ , W || and the biophysical parameters, the number of SDV was high.However, the number of SDV can be reduced if a Westin mask is used, suggesting that fiber complexity is one main driver for the differences between both DKI models.Under realistic conditions with noise in the acquired data, the model inherent baseline difference between both DKI variants requires very high SNRs to become visible since it is hidden in the SD otherwise.Furthermore, our results showed that the fast "199" protocol is particularly vulnerable to the effects of noise in the dMRI data but also that adaptive denosing can help counteract these effects.Here, we only contrasted two extreme acquisition protocols, one suited for standard and one optimized for fast axisymmetric DKI.To find the "sweet spot" between noise susceptibility and time reduction in the acquisition protocol a follow-up study is required.
(D a ) to 54% (D e,|| ), see Figure 3B.1.D e,|| , D e,⊥ and AWF had most SDV while D a and  had the least.Figure S1 in Section S1.4 documents the underlying A-PE histogram distributions.
Barplots summarizing the number of substantially differing voxels (SDV) of the axisymmetric DKI tensor metrics (AxTM) (top) and biophysical parameters (bottom).Shown is the rounded number of SDV (red barplots) and the median absolute percentage error (A-PE) or "bias" in those voxels (blue barplots).The subplot's titles indicate the subset of voxels that were analyzed, that is, (A.1, B.1) "WM mask" = white matter voxels were analyzed, (A.2, B.2) "WM mask & FA mask" = white matter voxels in the FA mask were analyzed and (A.3, B.3) "WM mask & Westin mask" = white matter voxels in the Westin mask were analyzed.Number of voxels in the white matter mask: 101 521, number of voxels in the WM mask & FA mask analysis: 28 741, number of voxels in the WM mask & Westin mask analysis: 20 527.