Kinetic interpretation of log-logistic dose-time response curves

A Hill-type time-response curve was derived using a single-step chemical kinetics approximation. The rate expression for the transformation is a differential equation that provides an interpolation formula between the logistic growth curve and second order kinetics. The solution is equivalent to the log-logistic cumulative distribution function with the time constant expressed in terms of a kinetic rate constant. This expression was extended to a full dose-time-response equation by postulating a concentration dependence for the rate constant. This was achieved by invoking a modified form of Haber’s law that connects an observed toxic effect with the concentration of the active agent and the elapsed exposure time. Analysis showed that the concept of Concentration Addition corresponds to a special case where the rate constant for the overall transformation rate is proportional to the sum of the rate constants that apply when the agents act individually. Biodiesel “survival” curves were measured and used to test the applicability of the empirical model to describe the effects of inhibitor dosage and binary inhibitor mixtures. Positive results suggest that the proposed dose-response relationship for the toxicity of agents to organisms can be extended to inanimate systems especially in cases where accurate mechanistic models are lacking.