Calculating the hybrid (macro) rate constants of a three-compartment mamillary pharmacokinetic model from known micro-rate constants

While there are published equations for calculating the hybrid (macro) rates constants (lambda1 and lambda2) of a two-compartment mamillary pharmacokinetic model from its micro-rate constants (e.g., k12, k21 etc.), there appears to be no report of an analogous method for a three-compartment model. The hybrid rate constants are the exponents of the multi-exponential equation describing the time-course of the predicted blood concentrations.

Using the method of Wagner, the differential equations of a three-compartment model were solved by transformation into the Laplace domain then matrix manipulation. The inversion of the result back into the time domain requires finding the roots of a cubic polynomial. The equations of a convenient method for doing so are reported. This "analytical" method for finding the hybrid rate constants was compared with an alternative "simulation and fitting" method. For this, a model with known micro-rate constants was used to predict a time-course of blood concentrations for a bolus dose, which was then fitted to a tri-exponential equation to find the hybrid rate constants.

The hybrid rate constants for the two methods were identical to at least four significant figures, confirming the validity of the analytical equations.

The equations presented here fill a gap in the pharmacokinetic literature, which may be useful in some applications considering the widespread use of the three-compartment mamillary pharmacokinetic model.