Understanding the effect of measurement time on drug characterization

In order to determine correct dosage of chemotherapy drugs, the effect of the drug must be properly quantified. There are two important values that characterize the effect of the drug: ε(max) is the maximum possible effect of a drug, and IC(50) is the drug concentration where the effect diminishes by half. There is currently a problem with the way these values are measured because they are time-dependent measurements. We use mathematical models to determine how the ε(max) and IC(50) values depend on measurement time and model choice. Seven ordinary differential equation models (ODE) are used for the mathematical analysis; the exponential, Mendelsohn, logistic, linear, surface, Bertalanffy, and Gompertz models. We use the models to simulate tumor growth in the presence and absence of treatment with a known IC(50) and ε(max). Using traditional methods, we then calculate the IC(50) and ε(max) values over fifty days to show the time-dependence of these values for all seven mathematical models. The general trend found is that the measured IC(50) value decreases and the measured ε(max) increases with increasing measurement day for most mathematical models. Unfortunately, the measured values of IC(50) and ε(max) rarely matched the values used to generate the data. Our results show that there is no optimal measurement time since models predict that IC(50) estimates become more accurate at later measurement times while ε(max) is more accurate at early measurement times.