Analytical Calculation of the Matrix in Diffusion Imaging

The clinical ability to measure molecular diffusion non­ invasively and in vivo using magnetic resonance imag­ ing (MRI) has generated great interest (1). In tissues, the effective diffusivity depends upon tissue anatomy, microstructure, and gradient pulse parameters. Diffu­ sion MR spectroscopy (2-4), 2DFT spin-echo imaging (5-7), and echo-planar imaging (EPI) (8-11), of in vivo isotropic media, to estimate the apparent diffusion con­ stant (ADC) one must first calculate a bfactor from the gradient pulse sequences (see Chapters 1 and 2). The scalar effective diffusivity also depends strongly on the direction of diffusion-sensitizing gradients with respect to the grain or fiber direction of the sample. · This dependence has been observed in both diffusion NMR spectroscopy and imaging studies of skeletal musCle (12) and brain white matter (13). In these het­ erogeneous, macroscopically anisotropic media, d sive transport is characterized by an effective diffusiop, tensor, Derr, not an effective scalar diffusivity (14-18). Analogously, in diffusion tensor imaging (DTI), to estiinate the diffusion tensor in each voxel, one must first calculate the nine elements of its coefficient matrix (the b matrix) from the gradient pulse sequences (19,20). In anisotropic media, imaging and diffusion gra­ dients appear to interact with one another, producing "cross terms" that are contained in the b matrix 1 ,22). If these cross-terms are unaccounted for, then the estimate of the diffusion tensor will be corrupted '(see Chapter 5, Part I). In isotropic media, in prindgle, these cross-terms only exist between magnetic field gradient pulses applied along the same direction, how­ ever, in anisotropic media (17,18), additional cross-· terms arise between magnetic field gradient pulses ap­ plied in orthogonal directions. We cannot assume a priori that off-diagonal elements of the effective diffu­ sion tensor vanish (19). Using imaging gradient parameters derived from pulse sequences, one can calculate numerically or ana­ lytically all the elements of the b matrix which contain the cross-terms between gradient pulses. Using these b matrix values, one can estimate the effective diffusion tensors and construct diffusion ellipsoid images as de­ s cribed by B a sser et al. (19,20,23). We have previously presented data using these method to assess the quality of the DTI protocol for isotropic media (22,24). Here we present the analytical expressions to calculate tpe b'inatrix in 2DFT and EPI pulse sequences, as weH as the Mathematica computer code which allows one to calculate the b matrix.