When examining brain diffusion weighted MRI (DW-MRI) images acquired at multiple b-values spanning from 0 to ~5000 s/mm², it has become apparent that the DW-MRI measurements can not be completely characterized by a functional corresponding to a single component or process. The intravoxel incoherent motion IVIM model¹ was proposed to explain the perfusion effects on the low to moderately diffusion weighted data. The stretch exponential model² was proposed as an empirical characterization of the moderately (<3000 s/mm²) weighted diffusion data. This model was later supported by theoretical analysis and named the fractional order calculus (FROC) model³. With the advent of clinical MR scanner that can acquire acceptable SNR diffusion weighted images within the 3000 to 5000 s/mm² range, it appears that stretch exponential model may not fully explain these images collected at these high b-values, suggesting an additional diffusion component may exist. Here we propose a trimodal model to fit the full spectrum of multi-b DW-MRI (from 0 to 5000 s/mm²) by incorporating the perfusion component, the stretch exponential component (or the FROC component), and a 3^rd distinct diffusion component.

Existing Models:

IVIM: $$$ S_b/S_0 = fe^{-D^*b} + (1-f)e^{Db} $$$, where f is the perfusion fraction, D* is the pseudo-diffusion coefficient, and D is the Gaussian diffusion coefficient.

Stretch exponential / FROC: $$$ S_b/S_0 = e^{-(DDC b)^\beta} $$$, where DDC is the distributed diffusion coefficient, and β is the heterogeneity index.

Proposed Model:

Trimodal: $$$ S_b/S_0 = f_fe^{-D^*b}+ f_se^{-(D_s b)^\beta}+ f_re^{-D_r b} $$$, where S_b is the diffusion weighted signal at b-value, f_f and D*are the IVIM fraction and pseudo-diffusion coefficient – respectively, f_s, D_s and β_s are the stretch exponential fraction, diffusion coefficient and heterogeneity index - respectively, and, f_r and D_r are the remaining high b-value fraction and diffusion coefficient - respectively.

MR imaging: Diffusion weighted brain images of three healthy adults were acquired using a 3T MRI scanner (MR750, GE Healthcare, Milwaukee, WI) with a 8-element phased-array coil and a customized single-shot spin-echo EPI sequence (TR/TE = 3000/91 ms, field of view = 240x240 mm², matrix = 256x256, slice thickness = 5 mm, number of slices = 24, diffusion b-value = 0, 10, 50, 100, 300, 400, 500, 700, 1000, 1250, 1800, 2000, 2500, 3000, 3300, 3500, 3700, 3800, 4000, 5000 s/mm²).

Model fitting: Per voxel fitting was performed in two steps. Step 1: with D_r fixed at 0.005x10^-3mm²/s, iteratively performed a nonlinear fit to the stretch exponential to the b=1000-3000 s/mm² range and then estimated the f_r that produced the least fitting error over the b=3000-5000s/mm² range. Step 2: with the f_r and D_r fixed and D* constrained above 3.5x10^-3mm²/s, as suggested by IVIM, a nonlinear fit was performed using the whole trimodal equation and complete range of b-values. A per voxel fit to the stretch exponential only model was also performed for comparison.

Fitting the trimodal model produces six maps as shown in Figure 1. The perfusion fraction f_f (figure 1a) is low and sparse as expected, and the pseudo-diffusion D* (figure 1b), as has been generally reported, is highly variable. The stretch fraction f_s (Figure 1c) indicates that the diffusion data is predominantly explained by the stretch exponential portion of the trimodal model, where D_s and β_s are presented in figure 1 d and e, respectively. The 3^rd fraction f_r is presented in figure 1f.

The stretch exponential parameters show the similar relative contrast in both models, for comparison see figure 2. An ROI within the superior corona radiata reveals trimodal (D_s = 0.62±0.10 x10^-3mm²/s, β_s=0.76±0.05) and stretch only (DDC=0.70±0.03 x10^-3mm²/s, β=0.60±0.04).

To appreciate the white matter information contained in the f_r map, figure 3 contains the 3^rd fraction f_r; stretch exponential only model χ² error map, where large fitting errors were observed from white matter regions, an anatomical T₁ image, and the trimodal χ² error.

The proposed trimodal model was developed to extend the stretch exponential model to incorporate both the perfusion information predominant around low b-values and the additional information available at b-values between 3000 and 5000 s/mm². The trimodal χ² error map as compared to the stretch exponential only χ² error map suggests that the trimodal model provided a better fit to the acquired data, especially in white matter regions, which suggests there was remaining information in white matter regions. The f_r map compared to the T₁ weighted MR image seems to support this claim. Since f_r seems to explain one aspect of the diffusion inhomogeneity within the voxel, a rise in β_s, i.e., more homogeneous, in the trimodal model would be expected.