A Trimodal non-Gaussian Diffusion Model for the Full Spectrum of Multi-b Diffusion MRI in Brain Tissue

Nicholas W. Damen^{1,2}, Kejia Cai^{2,3,4}, Yi Sui^{2,4}, Muge Karaman^{2}, and Frederick C. Damen^{2,3}

When examining brain diffusion weighted MRI (DW-MRI) images acquired at multiple b-values spanning from 0 to ~5000 s/mm^{2}, 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 model^{1} was proposed to explain the perfusion effects on the low to moderately diffusion weighted data. The stretch exponential model^{2} was proposed as an empirical characterization of the moderately (<3000 s/mm^{2}) weighted diffusion data. This model was later supported by theoretical analysis and named the fractional order calculus (FROC) model^{3}. With the advent of clinical MR scanner that can acquire acceptable SNR diffusion weighted images within the 3000 to 5000 s/mm^{2} 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/mm^{2}) 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 mm^{2}, 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/mm^{2}).

*Model fitting*: Per voxel fitting was performed in two steps. Step 1: with D_{r} fixed at 0.005x10^{-3}mm^{2}/s, iteratively performed a nonlinear fit to the stretch exponential to the b=1000-3000 s/mm^{2} range and then estimated the f_{r} that produced the least fitting error over the b=3000-5000s/mm^{2} range. Step 2: with the f_{r} and D_{r} fixed and D* constrained above 3.5x10^{-3}mm^{2}/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^{-3}mm^{2}/s, β_{s}=0.76±0.05) and stretch only (DDC=0.70±0.03 x10^{-3}mm^{2}/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 χ^{2} error map, where large fitting errors were observed from white matter regions, an anatomical T_{1} image, and the trimodal χ^{2} 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/mm^{2}. The trimodal χ^{2} error map as compared to the stretch exponential only χ^{2} 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 T_{1} 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.

We would like to thank Dr X.J.Zhou for valuable discussion on this topic.

The MRI facility is supported by a grant from the NIH (1S10RR028898).

1) Le Bihan, D. Radiology 1986; 161:401-407.

2) Bennett, KM. MRM 2003; 50:727-734.

3) Zhou, XJ. MRM 2010; 63:562-9.

Figure 1: Trimodal maps, IVIM a) fraction and b) pseudo-diffusion D*, stretch exponential c) fraction, d) diffusion coefficient D_{s}, e) heterogeneity index β_{s}, remaining f) fraction f_{r}.

Figure 2: Stretch parameters. Top row, stretch exponential only model, a) distributed diffusion coefficient DDC (x10^{-3}mm^{2}/s), and b) homogeneity index β. Bottom row, Trimodal model, c) diffusion coefficient D_{s} (x10^{-3}mm^{2}/s), and, d) homogeneity index β_{s}.

Figure 3: a) Trimodal remaining fraction f_{r}, b) Stretch exponential only model χ^{2} error map, c) T_{1} weighed image, d) trimodal χ^{2} error map.

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

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