A comprehensive mathematical model of microscopic dose deposition in photodynamic therapy

Abstract

We have developed a comprehensive theoretical model for rigorously describing the spatial and temporal dynamics of oxygen (3O2) consumption and transport and microscopic photodynamic dose deposition during photodynamic therapy (PDT) in vivo. Previously published models have been improved by considering perfused vessels as a time-dependent 3O2 source and linking the 3O2 concentration in the vessel to that within the tissue through the Hill equation. The time-dependent photochemical 3O2 consumption rate incorporates sensitizer photobleaching effects and an experimentally determined initially nonuniform photosensitizer distribution. The axial transport of 3O2 is provided for in the capillaries and in the surrounding tissue. A self-sensitized singlet oxygen (1O2)-mediated bleaching mechanism and the measured, initially nonuniform distribution of mesotetrahydroxyphenyl chlorin at 3 h after intravascular administration were used to demonstrate the capabilities of the model. Time-evolved distributions of 3O2 concentration were obtained by numerically solving two-dimensional diffusion-with-reaction equations both in the capillary and the adjacent tissue. Using experimentally established physiological and photophysical parameters, the mathematical model allows computation of the dynamic variation of hemoglobin-3O2 saturation (SO2) within the vessels, irreversible sensitizer degradation due to photobleaching, and the microscopic distributions of 3O2, sensitizer concentration, and 1O2 dose deposition under various irradiation conditions. The simulations reveal severe axial gradients in 3O2 and in photodynamic dose deposition in response to a wide range of clinically relevant treatment parameters. Thus, unlike former Krogh cylinder-based models, which assume a constant 3O2 concentration at the vessel, this new model identifies conditions in which 3O2 depletion and minimal deposition of reacting 1O2 exist near the end of axial segments of vessels and shows that treatment-limiting 3O2 depletion is induced at fluence rates as low as 10 mW cm(-2). These calculations also demonstrate that intercapillary heterogeneity of photosensitizer contributes significantly to the distribution of photodynamic dose. This more rigorous mathematical model enables comparison with experimentally observable, volume-averaged quantities such as SO2 and the loss of sensitizer fluorescence through bleaching that have not been included in previous analyses. Further, it establishes some of the intrinsic limitations of such measurements. Specifically, our simulations demonstrate that tissue measurements of SO2 and of photobleaching are necessarily insensitive to microscopic heterogeneity of photodynamic dose deposition and are sensitive to intercapillary spacing. Because prior knowledge of intercapillary distances in tumors is generally unavailable, these measurements must be interpreted with caution. We anticipate that this model will make useful dosimetry predictions that should inform optimal treatment conditions and improve current clinical protocols.