Effect of impermeable boundaries on diffusion-attenuated MR signal

The nonlinear dependence between the logarithm of the diffusion weighted signal, ln S, and the b-value, b, has often been interpreted as a manifestation of two physically distinct compartments, resulting in a biexponential form of the signal. This model fits to experimental data, however, has failed to yield realistic compartment sizes, severely jeopardizing the use of DWI to infer structural information on a cellular level. It has been hypothesized that the biexponential behavior can be attributed to the effect of confining boundaries that restrict diffusion in individual physical compartments. This interpretation is based on the analysis of diffusion in the presence of impermeable interfaces for short diffusion times such that the layer in which diffusion is affected by the boundary is thin as compared with the dimensions of the whole compartment. This model system is analyzed from the point of view of the cumulant expansion of the diffusion-weighted signal that results in a Taylor expansion of ln S in powers of b. Termination of this expansion to a polynomial form provides an excellent accuracy for small b-factors, but the series diverges for large b. The convergence of the series is studied, yielding a large range of b-values in which the absolute error of terminating the series at the second term remains smaller than 1% relative to the signal magnitude without diffusion weighting. With this accuracy, the signal in the studied model can be described as ln S ≈ −A · bD + B · (bD)2, where the parameters A and B can be expressed in terms of correlation functions of molecular velocity. Fitting of these parameters to the exact signal is more stable than for the three parameters of the biexponential function. This description fails for large b, for which the cumulant expansion diverges. The signal at even larger b-values is proportional to 1/b, 1/b3/2, and 1/b2 in one-, two-, and three-dimensional systems, respectively.