The effect of impermeable boundaries of arbitrary geometry on the apparent diffusion coefficient

The apparent diffusion coefficient (ADC) obtained from NMR measurements is modelled for diffusion in a compartment restricted by an impermeable boundary. For a given pulse sequence, the ADC can be determined from the connected velocity autocorrelation function (the second-order velocity cumulant), which we show can be expressed as a double surface integral over the boundary, involving the probability for molecules to diffuse from one boundary point to another. There is no restriction on the geometry of the boundary. This result allows a fast calculation of the ADC for an arbitrary time course of the diffusion-sensitizing gradient. Explicit examples are given for diffusion within three basic geometries for different pulse sequences. The ADCs measured with the Stejskal-Tanner pulse sequence and a more realistic pulse sequence with slice selection gradient and eddy current compensation are found to yield almost identical results. The application of the results are discussed in relation to determination of the microscopic structure of brain white matter.