Diffusion kurtosis imaging (DKI) is becoming increasingly popular in diffusion weighted imaging due to its higher sensitivity to tissue microstructure compared to conventional DTI¹, while remaining within a clinically feasible scan time. However, the kurtosis tensor (KT) model is also more sensitive to noise and other data artifacts. To improve robustness and accuracy, several techniques have been proposed that address various components of the fitting process. However, there is no study to this date that compared the accuracy and robustness of these techniques against some gold standard. This work compares three KT algorithms: sparsity constraint² (6KT), iterative outlier removal³ (REKINDLE), and directional weighting and regularization⁴ (DWAR). The data set to be used as the ground truth is derived from a subset of the HCP data using a novel approach.

For least-squares algorithms, the mean square error (MSE) can be decomposed into the squared bias (bias²), variance, and an irreducible error term. This is known as bias-plus-variance decomposition⁵. Bias² error results from simplifying assumptions built into the method. Variance error is caused by the model fitting noise. By minimizing both of these terms, there is a balance between accuracy and robustness to noise. Here, our goal is to evaluate the effectiveness of the kurtosis tensor fitting algorithms with an in-vivo-based “ground truth” (GT) dataset and bias-plus-variance decomposition.

Preprocessed diffusion-weighted and T1-weighted images from 10 subjects (6 female, 22-30yrs) were retrieved from the WU-Minn HCP database⁶. Diffusion weighted images (DWI) were processed through software developed in-house to estimate DKI tensors based on the algorithm by Jensen et al⁷ to calculate 22 tensor elements. Then, the tensor images were registered to standard space in FSL⁸ and averaged across subjects to create GT tensor images. From these images, first the GT metric images (mean diffusivity (MD), fractional anisotropy (FA), and mean kurtosis (MK)) were generated (Figure 1). Then, a set of DWI images was generated using the same q-space sampling scheme as the HCP. A white matter (WM) mask was applied to reduce processing time.

Using the synthetically generated DWI dataset as a baseline, Rician noise was added to achieve SNR levels of 20 and 40 to simulate typical acquisitions. For both SNR levels, 100 noise iterations were performed. With singular value decomposition (SVD) as the control, the synthetic DWI signals were processed by the KT algorithms. Outputs were compared to the GT metric images at each voxel by MSE, bias², and variance.

MSE-Bias²-Variance analysis illuminates the differences between algorithm outputs and GT images. For MD (Figure 2), there is minimal contribution from algorithm bias. Noise has some effect on variance at low SNR, but the magnitude of the error is negligible (~1%). For FA (Figure 3), 6KT exhibits a bias in the white matter of about 5%; its variance is lower than the other algorithms but not enough to overcome the bias. The other three algorithms perform similarly. For MK (Figure 4), 6KT has bias and, to a lesser extent, variance error. In major WM tracts, its performance is comparable to SVD. REKINDLE has the lowest MSE of the algorithms, especially in the peripheral WM. The variance is the source of its error. DWAR performed similarly to SVD in nearly all regions.

Based on the analysis, there is no clear winner for optimal KT fitting algorithm. The MSE was within about 5% of the actual metric value indicating good algorithm performance throughout the WM. 6KT has a slight advantage in robustness to noise, but the model simplicity underfits the data resulting in greater bias. With higher SNR, the robustness of 6KT no longer matters, yet the bias persists. REKINDLE is an improvement over SVD in terms of accuracy. Conversely, it is an iterative algorithm, and its processing time was approximately a 10-fold increase over SVD. DWAR performed similarly to SVD for all metrics in nearly all regions.

Although the GT metric images appear to be representative of typical in vivo values from human subjects, there could be errors from misregistration between subjects or partial volume effects. This might affect the GT tensor values at the edges of major fiber tracts and peripheral WM regions. These effects are mitigated to some extent by synthesizing the GT signal and metric images from the tensor images but should be considered when interpreting results. This GT template and analysis approach can be extended to arbitrary sampling schemes and other diffusion models.