Mathematical Modelling of Radiobiological Effect of Oxygen

In this paper we shall concentrate on designing a mathematical model of the recombination and diffusion of the radicals formed when a radioactive particle passes through the cell. This process is described by a system of partial differential equation in three space dimension. We shall demonstrate how to transform this problem to one space variable and propose a method how to solve it. Introduction In the radiotherapy the main aim is to destroy the tumor cells with avoiding the damage of healthy tissue. It has been observed already at the beginning of the twentieth century (as noted in [5]) that the response of the irradiated tissue to the ionizing radiation is lower if its contact with oxygen is restricted. This so-called “oxygen effect” has still not been entirely understood. In this paper we shall try to provide a tool for examination of the concentration of oxygen to the one specific part of the radiobiological mechanism. To provide a better comprehension of the biological, physical and chemical background of the problem we shall briefly introduce the processes taking place in the cell after the impact of the ionizing particle: It has been noted that the main reason of the cell’s inactivation is the damage of the DNA. This does not mean that a cell with a cracked DNA has to die immediately, but it can for example cause the cell’s inability to reproduce. Therefore we should mainly study how the irradiation can cause DNA harm. The processes in the cell after the impact of ionizing radiation can be divided into the following 4 phases (more detailed description can be found in [3]): • Phase 1 Physical (about 10−11 seconds) After the impact of the ionizing particle to the cellular (i.e. in fact aqueous) environment the particle transfers its kinetic energy to the surrounding matter which causes the creation of the aggressive radicals. The ionizing particle itself can also hit the DNA in which case it always causes its damage (so-called direct effect), however the probability of this kind of interaction is very low. • Phase 2 Physiochemical (about 10−5 seconds) In this phase the radicals and ions generated in the previous phase diffuse and recombine of and interact with DNA which with a certain probability can harm it (so called indirect effect). Compared to the direct effect this kind of interaction causes more than 99% of all reactions in which DNA is destroyed. • Phase 3 Cellular biological (minutes to hours) When the phase 2 is over the cells start a sequence of enzymatic reactions in order to try to repair the DNA. If this is not possible the cell turns on the inactivation mechanisms. • Phase 4 Tissular biological (days to years) In this phase the reaction of the tissue or the organism to the damage can be observed. In the rest of this paper we shall show how to describe the second phase of the radiobiological process by means of differential equations and propose a technique how to solve the model. WDS'05 Proceedings of Contributed Papers, Part I, 185–189, 2005. ISBN 80-86732-59-2 © MATFYZPRESS