The negative residuals of diffusion kurtosis model: a study on rat brain imaging
Wei-Cheng Lee1, Sheng-Min Huang 1, Cheng-He Li1, Kung-Chu Ho2, and Fu-Nien Wang1

1Department of Biomedical Engineering and Environmental Sciences, National Tsing Hua University, Hsinchu, Taiwan, 2Nuclear Medicine, Chang Gung Memorial Hospital, Taoyuan, Taiwan

### Synopsis

The well-behaved residual of a successful fitting model should present as an independent random variable in its histogram, otherwise it will contain structure that is not accounted for in this model. In the study, we found that there are two groups of fitting residuals. And the spatial mapping of these smaller negative pixels reveals that the overestimation of kurtosis model is related to the contribution of vessel signal. Therefore, we proposed a semi-kurtosis model to calculate the restricted diffusion adequately and obtain correct DKI information in these regions with vascular interference.

### Introduction:

Diffusion MRI is a noninvasive MR imaging technique that allows in vivo characterization and quantification of the molecular water diffusion in tissues. Diffusion kurtosis imaging (DKI) model has been used to describe the restricted diffusion condition in tissue, which showed elevated intensities from the estimation of conventional monoexponential diffusion model in high b-value [1]. Theoretically, the well-behaved residual of a successful fitting model should present as an independent random variable in its histogram, otherwise it will contain structure that is not accounted for in this model. In the study, we conducted DKI experiments on rat brains and aimed to investigate the characteristic of some specific pixels with extraordinarily large fitting residual from the DKI model.

### Materials and Methods:

An adult Sprague-Dawley rat was scanned under 1.6% isoflurane anesthesia on 7T Bruker Clinscan scanner. The multi-directional diffusion weighting sequence was used with 64 gradient directions and six b-values (0, 500, 1000, 1500, 2000, 2500 s/mm2). Imaging parameters were: TR/TE=3000/30ms, matrix size=92*92, FOV=30*30 mm, phase partial=7/8, GRAPPA=2, 11 slices with thickness=1mm, averages=4. The following equation was used as the diffusion kurtosis model [1]:$$\frac{S(b)}{S(b=0)}=e^{-bD+\frac{1}{6}b^{2}D^{2}K}$$ , where S(b) is the signal intensity as a function of the b-value, D is the diffusion coefficient, K is the diffusion kurtosis. Residuals were calculated by the difference between measured signal and estimated value which was calculated from each b-value and gradient direction. Histogram analysis was performed on the pixels in brain to evaluate the normality of its distribution. Furthermore, pixels with residual value smaller than an observed watershed of the main group in histogram were mapping to its anatomical locations. A monoexponential:$$\frac{S(b)}{S(b=0)}=e^{-bD}$$, and a proposed semi-kurtosis:$$\frac{S(b)}{S(b=0)}=f\times e^{-bD*}+(1-f)\times e^{-bD+\frac{1}{6}b^{2}D^{2}K}$$ , were used on fitting these pixels, where D* is the fast diffusion coefficient, f is the fraction of the fast component, D is the slow diffusion coefficient.[3]

### Result:

Fig 1 shows a representative residual histogram from b=500s/mm2 at one of the gradient direction (#40). A small group of pixels with large negative residuals was found on the left side of the main group. Identical phenomenon was observed in all of the b values and gradient directions. The spatial locations of these small groups were shown in Fig 2(a), (b), (c). According to anatomic atlas [2], these pixels are closely matching to blood vessels. Fig 3 illustrates the fitting curves of a typical pixel in Fig 2 (white arrow) using three different diffusion models. Among these three models, the semi-kurtosis model performs best on describing the data. Furthermore, after subtracting the fast diffusion component, the residual fitted curve shown in Fig 4 is well depicted by kurtosis model. Fig 5 depicts the residual contribution of five b-values along 64 gradient directions in three specific pixels: Vessel-included pixel, gray matter, and white matter. Generally, fitting the diffusion kurtosis model at the vessel-included pixel results in large overestimation than other tissues.

### Discussion:

A small group with large negative fitting residual implies inherent source of nonrandom effect. The spatial mapping of these pixels reveals that the overestimation of kurtosis model is related to the contribution of vessel signal. Interestingly, the consistent negative residuals of these pixels only exist in diffusion kurtosis model, but not conventional monoexponential model (i.e. not exist in DTI). It is anticipated that the signal drop between b values=0 and 500 s/mm2 is dominant by the elimination of vascular signal [4]. The observed flat signal changes in higher b-values mainly due to the restricted microenvironment in the partial volume of these pixels. Therefore, the proposed semi-kurtosis model may be useful to calculate the restricted diffusion adequately and obtain correct DKI information in these regions with vascular interference. Further studies could be carried out in optimization of the b-values to improve the fitting accuracy of semi-kurtosis model under vascular effect. The vascular fraction in the semi-kurtosis model may have potential clinical applications and worthy for future study.

### Acknowledgements

The Ministry of Science and Technology provided the grant support of this work. (MOST 103-2221-E-007-008-, 104-2221-E-007-063-)

### References

[1] Jens H. Jensen et al. MRM 53:1432–1440 (2005)

[2] Oscar U. Scremin 3rd edn. Elsevier, Amsterdam, Boston, pp 1167–1202(2004)

[3] Denis Le Bihan et al. Phys. Med. Biol. 52 R57–R90(2007)

[4] R Attariwala et al. JMRI 38:253–268 (2013)

### Figures

Fig 1. The histogram of residual value of DKI. Note that a small separated group showed large negative residuals (<-0.2).

Fig. 2. Spatial mapping of the pixels with large negative residuals at different slice (a) Bregma -5.2mm (b) Bregma -3.2mm (c) Bregma 0.2mm. The location of these pixels matched with blood vessels.

Fig. 3. The measured signal at (Fig 2(a) white arrow) pixel in each b value (red dot). Fitted by mono-exponential model ( blue line), kurtosis model( red line), semi-kurtosis model( green line). Note that the semi-kurtosis model fitted curve performs much better than others.

Fig. 4. After subtracting the fast diffusion component, all residual data can be well fitted by the kurtosis model.

Fig. 5. Analysis of residual contribution from different b-values and gradient directions (a) vessel-included pixel (b) gray matter (c) white matte. Note the consistent negative residuals from almost all b-value and gradient directions.

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