Analytical derivation of the b matrix of a time-efficient isotropic diffusion weighting gradient waveform

Diffusion-weighted magnetic resonance imaging of (3)He provides information about lung structure. If rotationally invariant measures of diffusion are desired, an equal diffusion weighting in all three spatial directions is necessary to obtain. In order to achieve such isotropic diffusion weighting, gradients have to be applied in these three spatial directions, which can be time consuming. Therefore, the purpose of this study was the analytic derivation of a time-efficient isotropic diffusion weighting scheme.

The complete b matrix of a preselected gradient waveform was derived analytically. The effect of ramp times and the contribution of the imaging gradients were included in the calculation. The time-efficient waveform was compared to a standard isotropic diffusion weighting scheme by determining the mean diffusivity of hyperpolarized (3)He in human lungs.

An analytically derived expression of the b matrix for a time-efficient gradient scheme allowing isotropic diffusion weighting was derived. Additionally, the b matrix of a common set of imaging gradients was calculated. Diffusion measurements of hyperpolarized (3)He in human lungs using the derived optimized gradient scheme and a standard gradient waveform used for isotropic diffusion weighting, respectively, gave results for the mean diffusivity which did not show any statistical difference. However, the echo time using the optimized scheme was reduced by 2.5 ms in comparison with the standard scheme which leads to a theoretical signal increase of 30%.

The analytically derived b matrix allows for the straightforward determination of time-efficient isotropic diffusion weighting schemes. By using those schemes, a substantial gain in signal can be achieved whereas the resulting values for the mean diffusivity did not show any statistical difference to the values obtained when using standard waveforms for isotropic diffusion weighting.