Some Mathematical Models for Concentration-Response Relationships

This paper discusses several mathematical models for concentration-response relationships in pharmacology. The stress is on models which have a mechanistical interpretation of receptor-mediated effects. Their differentiation from purely descriptive approaches is one main goal of this paper. Starting with a simple equation based on the Law of Mass Action, a model allowing for a receptor reserve will be introduced. A further model is obtained by including a scaling factor for the concentration axis. This model can be used as a touchstone, if a receptor reserve exists. The mathematical properties of the so-called operational model of receptor agonism, which is sometimes used in pharmacology, are discussed. Finally, a model is developed, which allows to analyze experimental data on the disinhibition of transmitter release: Different tissues are stimulated by different numbers of pulses at different frequencies. The models are discussed mathematically and are shown graphically, and examples are presented. The value of model parameters certainly depends on whether or not they may have a biological meaning. The development of a model for the evaluation of concentration-response data should be based, therefore, as much as possible on the physico-chemical grounds of pharmacology.