Data Modeling

The following four mathematical models were applied to the DWI signal obtained using low and high b-values:

1. Mono-exponential model:

where b is the b-value, S(0) is the signal intensity at b-value of 0 s/mm², and ADC_m is the apparent diffusion coefficient calculated using the mono-exponential model.

2. Stretched exponential model also known as Kohlrausch-Williams-Watts model (20):

where ADC_s is the apparent diffusion coefficient calculated using the stretched exponential model, and α is the heterogeneity index. The dimensionless α parameter varies from 0 to 1. During the fitting procedure, α parameter was constrained to be in the range of 0 to 1.

3. Kurtosis model:

where ADC_k is the apparent diffusion coefficient calculated using the kurtosis model, and K is the kurtosis. Jensen at al. (21) originally developed the kurtosis model to fit deviation of diffusion tensor signal from the mono-exponential function. The dimensionless positive K parameter characterizes the deviation from the mono-exponential signal decay.

4a. Bi-exponential model for low b-values:

where f_p is the “pseudodiffusion” fraction, D_f is the fast diffusion coefficient, and D_p is the “pseudodiffusion” coefficient. The intravoxel incoherent motion (IVIM) theory is an advanced method to separate diffusion and perfusion effects using DWI (6) at low b-values. According to the IVIM theory, the blood flow in the capillaries causes a dephasing of the magnetization when motion-encoding gradients are applied. This means that the motion of water molecules due to microcirculation of blood in the capillaries has a similar effect on the resulting DWI signal as their motion due to molecular diffusion.

4b. Bi-exponential model for high b-values:

where f_f is the fraction of fast diffusion, D_f is the fast diffusion coefficient, and D_s is the slow diffusion coefficient.

The DWI signal decay of each individual voxel has been fitted using four mathematical models, as described above, to generate parametric maps of the parameters. The fitting procedure has been performed using in-house written C++ code utilizing Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (22) in dlib library (23).

Following multiple initializations values were used to prevent local minima in the fitting procedure in order to avoid local minima in the fitting procedure (initializations values for high b-values data are in brackets):

1. Mono-exponential:

ADCm– from 0.001 (0.001) to 0.003 (0.003) with the step size of 0.0005 (0.0005)

2. Stretched exponential:

ADCs– from 0.001 (0.001) to 0.003 (0.003) with the step size of 0.0005 (0.0005)

α–from 0.1 (0.1) to 1.0 (1.0) with the step size of 0.15 (0.15)

3. Kurtosis:

ADCk– from 0.001 to 0.003 with the step size of 0.0003

K–from 0.0001 (0.0001) to 4.0 (2.0) with the step size of 0.05 (0.02)

4a (b). Biexponential:

fp(ff)–from 0.0 (0.5) to 1.0 (1.0) with the step size of 0.1 (0.1)

Dp(Df)–from 0.0001 (0.0001) to 0.04 (0.003) with the step size of 0.0005 (0.0003)

Df(Ds)–from 0.0001 (0.00002) to 0.003 (0.001) with the step size of 0.0003 (0.00005)