The importance of b-values selection and the precision of diffusion kurtosis estimation by the conventional schemes
Andrey Chuhutin1, Noam Shemesh2, Brian Hansen1, and Sune Nørhøj Jespersen1,3

1CFIN, Aarhus University, Aarhus, Denmark, 2Champalimaud Neuroscience Programme, Champalimaud Centre for the Unknown, Lisbon, Portugal, 3Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

### Synopsis

In DKI imaging studies, a wide range of different gradient strengths is used, which is known to affect the estimated kurtosis. Being driven by a need to assess the validity of the DKI expression and the accuracy of the estimated parameters as a function of b-value we both evaluated the variability of the kurtosis parameter for the in vivo and ex vivo data for different fitting techniques and studied the error in a mean kurtosis parameter with respect to the ground truth. The results and conclusions suggest circumspection while preferring a specific technique.

### Purpose and Introduction

Diffusion Kurtosis Imaging (DKI)[1] aims to approximate the diffusion weighted signal in a more precise manner by accounting for the leading non-Gaussian diffusion effects. Although the kurtosis expression originates from Taylor expansion of the log diffusion signal around $b=0$[2], normally b-values up to $b ∼ 2-3 \mathrm{ms /μ m}^2$ are used. Practically, it is necessary to ensure sufficient signal attenuation to quantify the second order term, while still remaining within the appropriate regime of the cumulant expansion without hitting the noise floor. In DKI imaging studies, a wide range of gradient strengths is used, which affects the estimated diffusivity and kurtosis parameters[2,3].

DKI is being increasingly used both as a biomarker, and as a starting point for microstructural modelling[4,5,6]. Hence, there is a need to assess the range of validity of the kurtosis expansion and how the accuracy of the estimated parameters depends on b-values and fitting strategy. A high level of discrepancy between the analytically evaluated kurtosis and the fit results was previously established[7], but here we evaluate the variability and error of the kurtosis parameter for different fitting techniques using diverse data.

### Theory

The standard DKI approximation reads$$\log\left(\frac{S\left(b,\mathbf{\hat{n}}\right)}{S_{0}}\right)=-b\mathbf{\hat{n}}^{\top}\mathbf{D}\mathbf{\hat{n}}+\frac{b^{2}\bar{D}^{2}}{6}\mathbf{\hat{n}}\mathbf{\hat{n}}^{\top}\mathbf{W}\mathbf{\hat{n}}\mathbf{\hat{n}}^{\top}$$and the mean of the kurtosis tensor may be calculated through[8,9]$$\bar{W}=\frac{1}{5}\left(W_{xxxx}+W_{yyyy}+W_{zzzz}+2W_{xxyy}+2W_{xxzz}+2W_{yyzz}\right)=\frac{1}{5}\mathrm{Tr}\left(W\right)$$To assess the accuracy of mean kurtosis estimation, a microstructural model for which kurtosis can be calculated is necessary. Here the biexponential and a biophysical model[10] were used to provide the ground truths. The latter model is described by $$S=\left(1-\nu\right)S_{h}+\nu\cdot S_{c}$$ where $S_{h}\left(b,\mathbf{\hat{n}}\right)=\exp\left(-b\mathbf{\hat{n}}^{\top}\mathbf{D}\mathbf{\hat{n}}\right)$ and $S_{c}$ is the expansion of the signal from a population of cylindrically symmetric diffusion tensors in spherical harmonics. Neurite densities derived from this model were found to be highly correlated with histology findings[11]. In this study we implemented fitting routines that have their data requirements met by shorter scanning protocols[3,12]: weighted linear least squares fit (LLSQ), non-linear least squares fit (NLSQ), linear and non-linear constrained objective function minimisation (CLLSQ,CNLSQ), directionwise fit (signal along each direction fit linearly, and then diffusion and kurtosis tensors are constructed) where higher terms of the cumulant expansion are absent (DW) or present ($b^{3}$:DW+, $b^{3}$,$b^{4}$:DW++).

### Methods

Human data was acquired in normal volunteers on a Siemens Trio 3T with 32 channel head coil and a double spin echo DW EPI sequence. Data consisted of $b=0$ image and 14 shells with 33 directions at $b=0.2−3.0\mathrm{ms /μm}^2$, 2.5mm isotropic, TR/TE=4300/103ms.

Fixed rat brain data acquired as in [7], was fit with the biophysical[10] and biexponential models using NLLS to provide ground truth $\bar{W}_\mathrm{th}$ computed from the estimated model parameters. These parameters were then used to generate synthetic signals at 12 directions and 14 equispaced b-values so that $0<b_{i}\leq b_{\mbox{max}},\,i=1\ldots14$($b_{\mathrm{max}},\mbox{SNR}$ are varied) and were subsequently fit with Eq1. The fit results were then compared to $\bar{W}_\mathrm{th}$ as a function of $b_{\mathrm{max}}$, reported in terms of the normalised absolute difference between the experimental $\bar{W}_\mathrm{exp}$ and $\bar{W}_\mathrm{th}$, henceforth referred as an error.

### Results

The fit to clinical data revealed that kurtosis varies substantially as the maximum b-value changes (Figures 1-2).

We observe that the degree of the variation depends significantly on the fitting techniques. Our findings are confirmed with analysis of ex-vivo spinal cord (data will be presented). These results are in agreement with [7] and show that the results of kurtosis evaluation depend on $b_{\mbox{max}}$ and algorithm choice.

We also examined a number of fitting techniques using simulated data (Figure 3). The fit with one higher term of cumulant expansion performs best for the grey matter tissue for 'clinical' $b_{\mbox{max}}\sim2-2.5\mbox{ms}/\mbox{μm}^{2}$. However we should take into consideration that the standard deviation of the error is considerably higher for this type of fit. A high error in evaluation of kurtosis was also observed with the biexponential model as a ground truth, for grey and especially white matter. In this case there is no obvious advantage of any specific fitting technique for $b_{\mbox{max}}\sim2-2.5\,\mbox{ms}/\mbox{μm}^{2}$.

### Discussion

We demonstrated that estimation of kurtosis parameters is very sensitive to the choice of fitting technique and maximum b-value. Even the most successful strategy could not enhance the estimation accuracy above the 20% for the biophysical model in grey matter or biexponential in white matter. The measurement bias for each one of the techniques evidently depends on the microstructural model generating the signal and tissue types, and in particular influences the choice of the maximum b-value. Our results suggest caution when using kurtosis to draw conclusions about tissue structure or estimating microstructural parameters[4,5] and even comparing studies that employ different methods. This does not subtract from the value of the kurtosis model in providing sensitive biomarkers.

### Acknowledgements

Lundbeck Foundation grant R83–A7548 and Simon Fougner Hartmanns Familiefond. AC and BH acknowledge support from NIH 1R01EB012874-01. The authors wish to thank Lippert’s Foundation and Korning’s Foundation for financial support. The 9.4T lab was made possible by funding from the Danish Research Counsil's Infrastructure program, the Velux Foundations, and the Department of Clinical Medicine, AU.

### References

[1] Jens H Jensen, Joseph A Helpern, Anita Ramani, Hanzhang Lu, Kyle Kaczynski: “Diffusional kurtosis imaging: The quantification of non-gaussian water diffusion by means of magnetic resonance imaging”, Magnetic Resonance in Medicine, pp. 1432—1440, 2005.

[2] Valerij G Kiselev: “The cumulant expansion: an overarching mathematical framework for understanding diffusion NMR”, Diffusion MRI: Theory, Methods, and Applications, pp. 152—168, 2011.

[3] Jelle Veraart, Dirk HJ Poot, Wim Van Hecke, Ines Blockx, Annemie Van der Linden, Marleen Verhoye, Jan Sijbers: “More accurate estimation of diffusion tensor parameters using diffusion Kurtosis imaging”, Magn. Reson. Med., pp. 138—145, 2011.

[4] Edward S Hui, G Russell Glenn, Joseph A Helpern, Jens H Jensen: “Kurtosis analysis of neural diffusion organization”, NeuroImage, pp. 391—403, 2015.

[5] Els Fieremans, Jens H Jensen, Joseph A Helpern: “White matter characterization with diffusional kurtosis imaging”, Neuroimage, pp. 177—188, 2011.

[6] Dmitry S. Novikov, Ileana O. Jelescu, Els Fieremans: “From diffusion signal moments to neurite diffusivities, volume fraction and orientation distribution: An exact solution”, ISMRM , 2015.

[7] Andrey Chuhutin, Ahmad Raza Khan, Brian Hansen, Sune Nørhøj Jespersen: “The Mean Kurtosis evaluation measurements show a considerable disparity from the analytically evaluated ones for a clinically used range of b-values”, ISMRM, 2015.

[8] Brian Hansen, Torben E Lund, Ryan Sangill, Sune Nørhøj Jespersen: “Experimentally and computationally fast method for estimation of a mean kurtosis”, Magnetic Resonance in Medicine, pp. 1754—1760, 2013.

[10] Jens H Jensen, Joseph A Helpern: “MRI quantification of non-Gaussian water diffusion by kurtosis analysis”, NMR Biomed, pp. 698—710, 2010.

[11] Sune N Jespersen, Carsten R Bjarkam, Jens R Nyengaard, M Mallar Chakravarty, Brian Hansen, Thomas Vosegaard, Leif Østergaard, Dmitriy Yablonskiy, Niels Chr Nielsen, Peter Vestergaard-Poulsen: “Neurite density from magnetic resonance diffusion measurements at ultrahigh field: comparison with light microscopy and electron microscopy”, Neuroimage, pp. 205—216, 2010.

[12] Jelle Veraart, Jeny Rajan, Ronald R Peeters, Alexander Leemans, Stefan Sunaert, Jan Sijbers: “Comprehensive framework for accurate diffusion MRI parameter estimation”, Magn. Reson. Med., pp. 972—984, 2013.

### Figures

Changes in the evaluation of mean kurtosis tensor a human subject as a function of $b_{\mbox{max}}$ and fit technique, rows are for $b_{\mbox{max}}=1.2,\,2,\,3$, columns(left to right): NLSQ, DW, DW+, CLLSQ

Variability of evaluated kurtosis values for a human diffusion weighted data (average of 300 voxels in each case) in grey matter (a) and white matter (b) as a function of the maximum b-value (values are relative to the NLSQ-evaluated value at $b_{\mbox{max}}=2$)

Error(left column) and the standard deviation of noise in estimation(right column) in evaluation of $\bar{W}$ averaged for 30 randomly chosen voxels along 30 noise realisations (SNR=100) in grey matter, biophysical ground truth (a) grey matter, biexponential ground truth (b) and white matter, biexponential ground truth(c)

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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