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 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 (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
ci
( x, t )  Di  x ci ( x, t )   ci ( x, t )kij c j ( x, t ) 
t
j i

 c ( x, t ) k
j ,k i
where
and
j
c ( x, t )
jk k
i  1..n, t  0, x  R 3
ci ( x,0)  0 x  0,  ci ( x,0) dx  Ni
R3
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?
Spherical symmetry can be also
defined as follows
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.
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 = 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
 1   2 ci
ci

(r , t )  Di  2
(r , t )    ci (r , t )kij c j (r , t )
r
t
 j i
 r r  r

 c (r , t )k
j ,k i
where
and
j
c (r , t )
jk k
i  1..n, t  0, r  0

ci (r ,0)  0 r  0,  ci (r ,0) 4r 2 dx  Ni
0
The initial condition
We 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
ci (r , t ) 
Ni
4Dit 
3
e
r2

4 Di t
The solution of the diffusion equation
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:
 1   2 ci
ci

(r , t )  Di  2
(r , t )    ci (r , t )kij c j (r , t )
r
t
 j i
 r r  r

 c (r , t )k
j ,k i
where
and
j
c (r , t )
jk k
i  1..n, t  0, r  0
r2

Ni
ci (r ,0) 
e 4 Dit0
4Dit0 3
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?