Efficient and precise calculation of b-matrix elements in diffusion-weighted imaging pulse sequences

Precise NMR diffusion measurements require detailed knowledge of the cumulative dephasing effect caused by the numerous gradient pulses present in most NMR pulse sequences. This effect, which ultimately manifests itself as the diffusion-related NMR signal attenuation, is usually described by the b-value or the b-matrix in the case of multidirectional diffusion weighting, the latter being common in diffusion-weighted NMR imaging. Neglecting some of the gradient pulses introduces an error in the calculated diffusion coefficient reaching in some cases 100% of the expected value. Therefore, ensuring the b-matrix calculation includes all the known gradient pulses leads to significant error reduction. Calculation of the b-matrix for simple gradient waveforms is rather straightforward, yet it grows cumbersome when complexly shaped and/or numerous gradient pulses are introduced. Making three broad assumptions about the gradient pulse arrangement in a sequence results in an efficient framework for calculation of b-matrices as well providing some insight into optimal gradient pulse placement. The framework allows accounting for the diffusion-sensitising effect of complexly shaped gradient waveforms with modest computational time and power. This is achieved by using the b-matrix elements of the simple unmodified pulse sequence and minimising the integration of the complexly shaped gradient waveform in the modified sequence. Such re-evaluation of the b-matrix elements retains all the analytical relevance of the straightforward approach, yet at least halves the amount of symbolic integration required. The application of the framework is demonstrated with the evaluation of the expression describing the diffusion-sensitizing effect, caused by different bipolar gradient pulse modules.