Parabolic Quantitative Structure‐Activity Relationships and Photodynamic Therapy: Application of a Three‐Compartment Model with Clearance to the In Vivo Quantitative Structure‐Activity Relationships of a Congeneric Series of Pyropheophorbide Derivatives Used as Photosensitizers for Photodynamic Therapy

Abstract

An open three-compartment pharmacokinetic model was applied to the in vivo quantitative structure-activity relationship (QSAR) data of a homologous series of pyropheophorbide photosensitizers for photodynamic therapy (PDT). The physical model was a lipid compartment sandwiched between two identical aqueous compartments. The first compartment was assumed to clear irreversibly at a rate K0. The measured octanol-water partition coefficients, P(i) (where i is the number of carbons in the alkyl chain) and the clearance rate K0 determined the clearance kinetics of the drugs. Solving the coupled differential equations of the three-compartment model produced clearance kinetics for each of the sensitizers in each of the compartments. The third compartment was found to contain the target of PDT. This series of compounds is quite lipophilic. Therefore these drugs are found mainly in the second compartment. The drug level in the third compartment represents a small fraction of the tissue level and is thus not accessible to direct measurement by extraction. The second compartment of the model accurately predicted the clearance from the serum of mice of the hexyl ether of pyropheophorbide a, one member of this series of compounds. The diffusion and clearance rate constants were those found by fitting the pharmacokinetics of the third compartment to the QSAR data. This result validated the magnitude and mechanistic significance of the rate constants used to model the QSAR data. The PDT response to dose theory was applied to the kinetic behavior of the target compartment drug concentration. This produced a pharmacokinetic-based function connecting PDT response to dose as a function of time postinjection. This mechanistic dose-response function was fitted to published, single time point QSAR data for the pheophorbides. As a result, the PDT target threshold dose together with the predicted QSAR as a function of time postinjection was found.