Derivation and presentation of formulas for drug concentrations in two-, three- and four-compartment pharmacokinetic models

Although compartment models are frequently used in pharmacokinetics, it is difficult to find complete analytical formulas describing the behaviour of drugs in universal simpler compartment models in the accessible literature. The paper presents derivations of formulas for general two- and three-compartment models, including the possibilities of original non-zero quantity in all compartments and elimination from all compartments. Formulas for four-compartment models are also derived with the restriction that original quantity is non-zero in only one compartment. Derivation uses Laplace transformation but does not require prior knowledge of the technique. The derived analytical formulas are verified numerically. These formulas can be easily simplified to less complex cases.