Mathematical Modeling of Physio-chemical Phase of the Radiobiological process J. Barilla, J. Felcman, S. Kucková Department of Numerical Mathematics, Charles University in Prague Faculty of Mathematics and Physics Department of Computer Science, J. E. Purkyne University in Usti nad Labem, Institute of Science Acknowledgement: The work is a part of the research project MSM financed by MSMT.

Outline Radiobiological process Mathematical model of the physio-chemical phase Spherical symmetry of the solution and transformation to one dimension Initial condition and a an idea of a numerical method

Radiobiological process Radiobiology deals with study of the influence of ionizing radiation to living organisms. This can be used e.g. in the treatment of cancer, or, on the other hand, in the fields where we need to protect people against radiation. To be effective in this, we would like to be able to determine a probability of the death of an irradiated cell. The key reason for the cell’s death caused by irradiation is damage of DNA. From this point of view the processes taking place in the cell after the impact of a radioactive particle can be divided in the following four phases

Physical phase (  sec) transfer of the energy from the particle to the cellular environment and creation of radicals Physio-chemical phase (  sec) diffusion and recombination of the radicals Biological cellular phase (minutes – hours) reparation of the damaged DNA or beginning of the inactivation mechanism Biological tissular phase (days - years) reaction of the tissue or organism to the consequences of irradiation

Notations c i (x,t) concentration of i-th species in the point (x,t) D i diffusion coeffitients k ij reaction rates for the reaction of c i and c j N i initial number of the radicals of i-th species

The diffusion and recombination process can be described by the following system of equations

To solve the system numerically we need to handle the following two difficulties 1. The problem is formulated in three space dimensions  we show that the (unknown) solution is spherically symmetric 2. The initial condition is singular  we use the advantage of the fact that we can solve the diffusion part analyticaly

Spherical symmetry A clasical way for showing the spherical symmetry of the solution would be to transform to the standard spherical coordinates (r,ξ,φ) and to show that the first derivatives with respect to ξ and φ are equal to zero. However, we do not know anything about the formula describing the solution, so we cannot use this approach. What to do?

Let u(x 1,x 2,x 3 ) be a spherically symmetric function and U(y 1,y 2,y 3 ) = u(x 1,x 2,x 3 ) be a transform of u, where the coordinate system y 1,y 2,y 3 is given by an arbitrary rotation of the original coordinate system x 1,x 2,x 3. Then U(z 1,z 2,z 3 ) = u(z 1,z 2,z 3 )  z.  It is enough to show that our system of equations is the same before and after the rotation of coordinates. Spherical symmetry can be also defined as follows

An arbitrary rotation of coordinates can be represented by a composition of rotations by an arbitrary angles α,β and γ around the axes x 1, x 2, and x 3 respectively. E. g. rotation around x 3 can be written as y 1 = x 1 sin γ + x 2 cos γ y 2 = x 1 sin γ – x 2 cos γ y 3 = x 3 …and it is easy to show that this transform of coordinates does not change the form of our system (it only causes the change of notation of variables)

Using the standard spherical coordinates and the fact that the solution is spherically symmetric, we get

The initial condition We shall assume, that for a very short time period t 0 at the beginning of the physio-chemical phase, there are no reactions and the distribution of radicals is only given by the diffusion. The solution of the three dimensional diffusion equation with “our” singular initial condition is

The solution of the diffusion equation

Using the solution of the diffusion equation at time t 0 as an initial condition we come to the following (final) model of the physio-chemical phase:

Idea of the method for the solution assume that the diffusion and recombination do not proceed simultaneously but in turns (diffusion – recombination – diffusion …) assume that the increase or decrease of radical during the recombination respects the Gaussian distribution from the diffusion step note that the diffusion equation can be solved exactly and the recombination can be solved numerically (e.g. S. K. Dey)

Algorithm Compute one time step of diffusion (analytically) Compute one time step of recombination (numerically) using the previous result as an initial condition Count the numbers of radicals in the system Multiply the result from diffusion by the number of radicals in previous step/number of radicals in this step Repeat until the total number of radicals in the system is less or equal one

Next time perhaps… Numerical results Existence of the solution

Thank you for your attention Questions?