Wei-Cheng Lee^{1}, Sheng-Min Huang ^{1}, Cheng-He Li^{1}, Kung-Chu Ho^{2}, and Fu-Nien Wang^{1}

^{1}Department of Biomedical Engineering and Environmental Sciences, National Tsing Hua University, Hsinchu, Taiwan, ^{2}Nuclear 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)