A General Framework for Automated Accurate Calculation of b-Matrix (Auto-b) in Diffusion MRI Pulse Sequences

To derive accurate diffusion metrics, both imaging and diffusion-sensitizing gradient pulses should be accounted for when calculating the diffusion-weighted b-matrix. However, it is complex to derive analytical solutions due to complicated interactions between gradient pulses, including orthogonal directions. This study proposes a general framework to calculate the b-matrix automatically (dubbed as Auto-b). Based on the divide-and-conquer approach, the b-matrix calculation is appropriately segmented, and the symbolic mathematical library is applied to handle integration operations for each interval. If the specifications of all gradient pulses are provided to Auto-b, an accurate b-matrix can be obtained. Three examples are explored to validate the accuracy of Auto-b and to detect b-value errors when using approximate calculations. (1) In the conventional spin-echo example, Auto-b exhibits high accuracy, as indicated by the maximum relative deviation of 1.68 parts per thousand between its calculated b-matrices and those obtained from analytical expressions. (2) Auto-b is applied to investigate the contribution of imaging gradients to the b-matrix in an optimized spin-echo echo planar imaging sequence at submillimeter resolution. Specifically, ignoring the contribution of imaging gradients results in a b-value error of 12.16 s/mm(2) at the 0.8 x 0.8 x 0.8 mm(3) resolution and 22.47 s/mm(2) at the 0.6 x 0.6 x 0.8 mm(3) resolution, respectively, when nominal b = 0. (3) Auto-b is also utilized to analyze the influence of approximate calculations in the spatiotemporally encoded sequence. The results showed that neglecting the contribution of phase-encoding blips causes large b-value errors up to 11.02 s/mm(2). In addition, the rectangularization of trapezoidal waveforms led to a high relative b-value error of 39.91%. This study validates the high accuracy of Auto-b and underscores the importance of accurate b-value calculations in both submillimeter imaging and spatiotemporally encoded sequences. Attributed to its automation, accuracy, and broad applicability, Auto-b is helpful for developers of diffusion sequences.