MRI PLAYGROUND
Beyond the b-value
How far does water explore?
During the diffusion time, a water molecule typically explores a neighborhood only a few micrometres across. A millimetre-wide voxel collects signal from many molecules, each sampling its own local environment. These environments can differ across the voxel.
If water mixes within each compartment and exchange between compartments is slow, we can describe each water population by its own diffusion tensor. One voxel then contains a distribution of diffusion tensors. Compare the dMRI signals from two such distributions below.
For more details, see RICE paper.
VOXEL A
Dispersed orientations
Each compartment has the same elongated diffusion tensor, rotated in a different direction.
VOXEL B
Mixed diffusivities
Each compartment diffuses equally in all directions. Larger spheres represent higher diffusivity.
Why do A and B agree with linear encoding?
LTE measures the apparent diffusivity along one direction. In A it varies with compartment orientation. In B it varies with the size of the spherical tensor. At low b, the first two terms of the log signal measure the mean and variance of that apparent diffusivity.
| LTE term | Voxel A | Voxel B |
|---|---|---|
| O(b) · Mean diffusivity (µm²/ms) | 0.80 | 0.80 |
| O(b²) · Diffusivity variance ((µm²/ms)²) | 0.20 | 0.20 |
Both log signals start with the same slope and curvature: ln(S/S₀) = −bμ + b²σ²/2 + O(b³), with b in ms/µm². Here the entire LTE diffusivity distribution matches, so the curves overlap even beyond second order. Try planar or spherical encoding to separate them.
Shape the encoding
Spherical mean signal vs. b-value
Each curve averages the normalized signal over encoding orientations. The selected encoding is highlighted. All three curves coincide for voxel B.
Other voxel A curves in gray: LTE solid · PTE dashed · STE dotted
How the two voxels are matched
Both voxels are mixtures of non-exchanging Gaussian compartments. In A, the principal diffusivities are 1.8, 0.3 and 0.3 µm²/ms with uniformly distributed orientations. B samples the distribution of apparent diffusivities that LTE sees in A. This matches the entire LTE curve, including the mean diffusivity of 0.8 µm²/ms.
The signal is the average of exp(−B:D) across compartments. At fixed b, the encoding tensor assigns its weight to one axis (LTE), equally to two axes (PTE), or equally to three axes (STE). The drawings show representative diffusion tensors. Read about multidimensional encoding and RICE.