Notes on the inverse Gaussian distribution and choice of boundary conditions for the dispersion model in the analysis of local pharmacokinetics.

The dispersion model has been widely used to analyze local pharmacokinetics in the organs and the tissues since the 1980's. However, an ambiguity still remains in selecting the boundary conditions which are necessary to solve the basic equation of the model. In this note, theoretical considerations are given to this problem and we present here some deficiencies of the mixed boundary conditions. It seems that theoretical confusion exists in the literature for the mixed boundary conditions. It is well-known that the solution of the dispersion model with a bolus input is the inverse Gaussian distribution for the mixed boundary conditions. However, it is rarely recognized that the inverse Gaussian distribution requires an open boundary at either the inlet or the outlet. For the analysis of local pharmacokinetics, the use of the classical Danckwerts (or closed) boundary conditions is recommended.