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 0021620839 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 processRadiobiology 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 (10-11sec)

transfer of the energy from the particle to the cellular

environment and creation of radicals • Physio-chemical phase (10-5sec)

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

ci(x,t) concentration of i-th species in the point (x,t)

Di diffusion coeffitients

kij reaction rates for the reaction of ci and cj

Ni initial number of the radicals of i-th species

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

3

,

,0,..1

),(),(

),(),(),(),(

Rxtniwhere

txcktxc

txcktxctxcDtxt

c

ikjkjkj

ijjijiixi

i

i

R

ii Ndxxcxxcand 3

)0,(,00)0,(

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(x1,x2,x3) be a spherically symmetric function and U(y1,y2,y3) = u(x1,x2,x3) be a transform of u, where the coordinate system y1,y2,y3 is given by an arbitrary rotation of the original coordinate system x1,x2,x3 .

Then U(z1,z2,z3) = u(z1,z2,z3) 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 x1, x2, and x3 respectively.

E. g. rotation around x3 can be written as

y1= x1 sin γ + x2 cos γ

y2 = x1 sin γ – x2 cos γ

y3 = x3

…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)

ikjkjkj

ijjiji

ii

i

trcktrc

trcktrctrr

cr

rrDtr

t

c

,

22

),(),(

),(),(),(1

),(

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

iii Ndxrrcrrcand

2

0

4)0,(,00)0,(

0,0,..1 rtniwhere

The initial conditionWe shall assume, that for a very short time period t0

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

tD

r

i

ii

ietD

Ntrc 4

3

2

4),(

The solution of the diffusion equation

ikjkjkj

ijjiji

ii

i

trcktrc

trcktrctrr

cr

rrDtr

t

c

,

22

),(),(

),(),(),(1

),(

Using the solution of the diffusion equation at time t0 as an initial condition we come to the following (final) model of the physio-chemical phase:

0,0,..1 rtniwhere

0

2

4

304

)0,( tD

r

i

ii

ietD

Nrcand

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 attentionQuestions?