Hajime Tamura^{1}, Hideki Ota^{2}, Tatsuo Nagasaka^{2}, Naoko Mori^{2}, and Shunji Mugikura^{2}

To know how much the intravoxel
incoherent motion (IVIM) parameters deduced by a
bi-exponential model are affected by neglecting non-Gaussian
diffusion restriction effects, we performed Monte-Carlo simulations: fitting the bi-exponential model to simulated data containing the diffusion
restriction effects. The
results showed that non-Gaussian diffusion restriction effects may considerably affect estimation of IVIM parameters
even when data acquired with low *b*-values (*b*≤1000
s/mm^{2}) are used. This should be taken into account when interpreting
the results of IVIM analyses based on the bi-exponential model.

Non-Gaussian diffusion restriction
effects are not clearly visible when low b-values
(*b*≤1000
s/mm^{2}) are used in clinical MRI. Therefore, the restriction effects
are usually neglected for intravoxel incoherent motion (IVIM) imaging with *b*-values less than 1000 s/mm^{2}
and a simple bi-exponential signal decay model is used^{1}:$$
S(b)=S(0)\left\{fe^{-bD^{*}}+(1-f)e^{-bD}\right\}, \quad
(Equation\quad 1)$$
where *S* denotes the signal intensity; *b*,
the *b*-value; *f*, the flowing blood (perfusion) fraction; *D*^{*}, the pseudo-diffusion coefficient of vascular
compartment; *D*, the apparent
diffusion coefficient of non-vascular compartment. However, the accuracy of parameters
(*f*, *D*, *D*^{*})
estimated by fitting the bi-exponential model to IVIM data that contain non-Gaussian
diffusion restriction effects has
not been investigated systematically. The purpose of this study is to know how much
the estimated parameters are affected by the non-Gaussian diffusion restriction effects.

We numerically simulated IVIM data
containing non-Gaussian diffusion restriction
effects. We adopted the diffusion
kurtosis as an indicator of the restriction
effects. Theoretical IVIM signals were generated using the diffusion kurtosis
model^{2}:$$
S(b)=S(0)\left\{fe^{-bD^{*}}+(1-f)e^{(-bD+Kb^{2}D^{2}/6)}\right\}, \quad (Equation\quad 2)$$
where *K* denotes the diffusion kurtosis. The signal intensities were
calculated for 10 *b*-values (*b*≤500-1000 s/mm^{2}, Table 1) with
ranges of parameters: *K* = 0-1.5, *f* = 0.03-0.1,
*D*^{*} = 6-20 µm^{2}/ms,
and *D* = 0.8-1.5
µm^{2}/ms.
*S*(*b*)'s were calculated 1024 times for each
set of parameters with adding Rician noise randomly so that the signal-to-noise
ratio was 100 for the signal at *b* = 0.

*D*
was estimated by non-linear least squares using the simulated data of *b≥*400
s/mm^{2} where the first term in *Equation *1 is assumed to be negligibly
small. Then *f* and *D*^{*} were calculated using the
data of b≤80
s/mm^{2} and the estimated *D*
above. *f* and *D*^{*} were constrained as
0 < *f* < 0.5, and 3<*D*^{*}<50 µm^{2}/ms. We also calculated the apparent diffusion coefficient for a
single-exponential decay model (ADC) as$$ADC = \log\left[S(0)/S(b_{max})\right]/b, \quad (Equation\quad 3)$$
where *b*_{max} is the maximum *b*
in the set of 10 *b*-values.

The estimated parameters were compared with ground-truth values using box and whisker plots.

1. Federau C. Intravoxel incoherent motion MRI as a means to measure in vivo perfusion: A review of the evidence. NMR in Biomed. 2017;e3780.

2. Iima M and Le Bihan D. Clinical Intravoxel Incoherent Motion and Diffusion MR Imaging: Past, Present, and Future. Radiology 2016;278(1):13-32.

3. Lätt J, Nilsson M, Wirestam R, et al. Regional values of diffusional kurtosis estimates in the healthy brain. J. Magn. Reson. Imaging 2013;37(3):610–618.

4. Wurnig M, Kenkel D, Filli L, et al. A Standardized parameter-free algorithm for combined intravoxel incoherent motion and diffusion kurtosis analysis of diffusion imaging data. Invest Radiol. 2016;51(3):203-210.

5. Cauter S, Veraart J, Sijbers J, et al. Gliomas: Diffusion kurtosis MR imaging in grading. Radiology 2012;263(2):492-501.

6. Iima M, Kataoka M, Kanao S, et al. Intravoxel incoherent motion and quantitative non-Gaussian diffusion MR imaging: evaluation of the diagnostic and prognostic value of several markers of malignant and benign breast lesions.Radiology 2017:162853. https://doi.org/10.1148/radiol.2017162853.

Table 1.
Sets of 10 *b*-values adopted in this
study varying the maximum *b*-values
500 to 1000 s/mm^{2}.

Figure 1. Box and whisker plots of
estimated *D*’s using the data of *D* = 0.8
µm^{2}/ms,
*D*^{*} = 10 µm^{2}/ms, and *f* = 0.05 with various Kurtosis (*K*)
and sets of *b*-values. The each set
has 10 *b*-values with the maximum *b*-value (*b*_{max}) of 500-1000 s/mm^{2} (Table 1). The horizontal line indicates the ground truth. The estimated *D’s*
decrease successively with both *K*
and *b*_{max} as low as 0.73 of
the ground truth at *K* = 1.5 and *b*_{max} = 1000 s/mm^{2}.

Figure 2. Box and
whisker plots of estimated *D*^{*}’s from the data of *D*/*D*^{*}=0.8/10 µm^{2}/ms,
and *f* = 0.05 with various Kurtosis (*K*) and sets of *b*-values. The each set has 10 *b*-values
data with the maximum *b*-value (*b*_{max}) of 500-1000 s/mm^{2}
(Table 1). The horizontal line
indicates the ground truth. The estimated *D*^{*}’s tend to decrease successively with *K* or *b*_{max}.

Figure 3. Box and whisker plots of
estimated *f*’s from the data of *D*/*D**=0.8/10 µm^{2}/ms, and *f* = 0.05 with various Kurtosis (*K*)
and sets of *b*-values. The each set
has 10 *b*-values data with the maximum
*b*-value (*b*_{max}) of 500-1000 s/mm^{2} (Table 1). The horizontal line indicates the ground truth. The estimated *f’s*
increase successively with both *K*
and *b*_{max} as large as 2.2
of the ground truth at *K* = 1.5 and *b*_{max} = 1000 s/mm^{2}.

Figure 4. Box-and-whisker plots of ADC’s calculated
by Equation 3 in the text using the data of *D*/*D**=0.8/10
µm^{2}/ms,
and *f* = 0.05 with various Kurtosis (*K*)
and pairs of *b*-values: *b* = 0 and the maximum *b*-value (*b*_{max}) of 500-1000 s/mm^{2}. The horizontal line indicates the ground truth. The estimated ADC’s
decrease successively with *K* or *b*_{max}. The ADC’s are overestimated at *K* = 0 and underestimated at *K*≥1.0
and *b*_{max}≥800 s/mm^{2}. The
overestimation is caused by “perfusion effect” (the first term of the
bi-exponential model: Equation 1 or 2 in the text).