Validity of model approximations for receptor-ligand kinetics in nuclear medicine

An appropriate mathematical model is required for quantitative analysis of high affinity radioligands as direct or surrogate probes to measure receptor distribution, affinity, concentration, binding potential, and endogenous or exogenous ligand occupancy levels. For studies with positron emission tomography (PET) or single photon emission computed tomography (SPECT), the receptor-ligand compartment model has been well established and widely used. This pharmacokinetic model is represented mathematically by a set of nonlinear ordinary differential equations. Variations of models for PET and SPECT account for radioactive decay differently. These are not equivalent and entail assumptions or approximations that may be not appreciated. In this study, a general form of the model is presented and compared with others with various approximations, which are valid only under specific conditions. The various approximate formulations were analytically compared to the exact model to identify the terms that were neglected in the approximate formulations. The extent to which the approximations impact the model solutions was assessed by computer simulations based on numerical solutions to each set of equations. Specifically, each model formulation was tested using three simulated injection protocols representing a typical PET experiment, a typical SPECT experiment, and an extreme experiment where both the injected activity and the specific activity were very high. No significant differences were found among the output of the three model formulations when the PET and SPECT injection protocols were tested. The only conditions that produced significant differences occurred when the specific activity and the administered activity were simultaneously very high. These conditions, however, have little practical relevance to experimentally achievable conditions due to radiation dose and specific activity of radiopharmaceuticals.