Mathematical Modeling of Oxygen Transport, Cell Killing and Cell Decision Making in Photodynamic Therapy of Cancer

Abstract

In this study we present a model of in vitro cell killing through type II Photodynamic Therapy (PDT) for simulation of the molecular interactions leading to cell death in time domain in the presence of oxygen transport within a spherical cell. By coupling the molecular kinetics to cell killing, we develop a modeling method of PDT cytotoxicity caused by singlet oxygen and obtain the cell survival ratio as a function of light fluence or initial photosensitizer concentration with different photon density or irradiance of incident light and other parameters of oxygen transport. A systems biology model is developed to account for the detailed molecular pathways induced by PDT treatment leading to cell killing. We derive a mathematical model of cell decision making through a binary cell fate decision scheme on cell death or survival, during and after PDT treatment, and we employ a rate distortion theory as the logical design for this decision making proccess to understand the biochemical processing of information by a cell. Rate distortion theory is also used to design a time dependent Blahut-Arimoto algorithm of three variables where the input is a stimulus vector composed of the time dependent concentrations of three PDT induced signaling molecules and the output reflects a cell fate decision. The concentrations of molecules involved in the biochemical processes are determined by a group of rate equations which produce the probability of cell survival or death as the output of cell decision. The modeling of the cell decision strategy allows quantitative assessment of the cell survival probability, as a function of multiple parameters and coefficients used in the model, which can be modified to account for heterogeneous cell response to PDT or other killing or therapeutic agents. The numerical results show that the present model of type II PDT yields a powerful tool to quantify various processes underlying PDT at the molecular and cellular levels and to interpret experimental results of in vitro cell studies. Finally, following an alternative approach, the cell survival probability is modeled as a predator - prey equation where predators are cell death signaling molecules and prey is the cell survival. The two models can be compared to each other as well as directly to the experimental results of measured molecular concentrations and cell survival ratios for optimization of models, to gain insights on in vitro cell studies of PDT.