Plutonium Biokinetics in Human Body A. Luciani - Kit-Bibliothek - FZK

On the basis of the matrix form, the system can be solved by calculating the
eigenvalues of the R-λ RI matrix. The relating eigenvectors v i will provide the solution of the
system of differential equations [56]:
where:
c i is the i-th coefficient;
λ i is the i-th eigenvalue;
v i is the i-th eigenvector.
n
q(t) c i
i 1
84
v ie λ i t
equation 3.1.3
It should be pointed out that although the previous methodology is a very elegant way
for solving a system of differential equations, the solution functions q i(t) are given by a sum
of exponential terms, each with its own exponent and coefficient, generally equal to the
number of compartments constituting the model. For a model describing the metabolism of
Plutonium isotopes after inhalation almost 50 compartments would be necessary, or even 100
if the metabolism of daughter isotopes too (as 241 Am from 241 Pu) have to be modelled. The use
of functions formed by a summation of 100 terms is not easy.
The previous methodology for solving a system of differential equations is appropriate
when the transfer rates among the model compartments are constant, e.g. they are not
depending on time. In ICRP models time dependent transfer rates are normally approximated
with chains of compartments with different constant rates. Therefore the complexity of the
model is increased, as a larger number of compartments is necessary, but all the transfer rates
are constant, allowing to solve the model as previously described.
In the most general case, in which a compartmental model is based on time dependent
transfer rates, the system of differential equations can be efficiently solved by means of
numerical methods. The solution of the system is therefore given for a discrete and finite
number of time points. These discrete solutions can be successively fitted to obtain an
approximated function that allows calculating the solution at any time. It should be noted that
the numerical functions approximating the solution are generally of a more simple and handy
form than the summation of exponential terms obtained with the eigenvalues method.
In the present work the numerical methodology for solving systems of differential
equations was adopted in order to consider general biokinetic models with time dependent
transfer rates. For this purpose commercially available software [152] for dealing with general
mathematical problems was used. Functions representing the amount of activity in the organs
and tissues of the biokinetic models at any time after intake were calculated. The amount of
activity calculated for the urinary bladder and the lower large intestine were multiplied by the
respective clearance transfer rates to evaluate the instantaneous urinary and fecal excretion,
respectively. In the practical situation the urine and fecal bioassays obviously do not refer to
instantaneous rates but to a cumulative excretion over 24 hours. This is important for the first
days after an initial single intake, while for times greater than about 20 days the difference
among the instantaneous excretion and the 24 hours integrated function is negligible.
Therefore to calculate realistic excretion values at a short time, comparable with the
experimental data, the instantaneous excretion rates were integrated over a period of 24 hours
for the first 20 days after exposure. The blood content of Plutonium was estimated by