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

commonly in urine. The methodology and the results of the present analysis were in part
already published in a recent publication [164].
Many of the methods available for conducting sensitivity analysis have been described
and compared in the general context of models development [165]. One of the most common
methods is the differential sensitivity analysis, based on partial differentiation of the equations
mathematically describing the model. The sensitivity coefficient for an independent variable
is calculated from the partial derivative of the dependent variable with respect to the
independent variable. In the specific case of the present work the dependent variable is the
excreted Plutonium activity and the independent variables are the rates of transfer of activity
among the compartments. As a rule the derivatives are multiplied by the ratio of the transfer
rate value to the model predictions of excreted activity for the base-case scenario in order to
normalize the results and avoiding the effects of the units.
Concerning the interpretation of the sensitivity coefficients as defined above, an
independent variable to which a dependent variable is particular sensitive has a sensitivity
coefficient that tends to be unity or larger; on the contrary with low sensitivity the coefficient
will tend to zero. For example a sensitivity coefficient of + 0.1 (- 0.1) indicates that an
increase of +1% in the independent variable results in a + 0.1% (- 0.1%) variation in the
dependent variable.
In the sensitivity studies for the first order biokinetic models the differential analysis
has been often used because the differential equations that describe the model can be easily
recasted to provide a set of equations by which to calculate the sensitivity coefficients [166].
This method can be efficiently applied when exact solutions of the system of differential
equations are available. As time-dependent transfer rates are assumed in the optimized model
(Model-b), only a numerical solution can be obtained. Therefore the sensitivity coefficients
were calculated by numerical partial differentiation. The sensitivity coefficient for the daily
urinary excretion of Plutonium activity (e u) with respect to the i-th transfer rate (λ i) is defined
as:
S i = Δe u
Δ i
122
equation 3.1.9
Similar expressions were used for the sensitivity coefficients relating to fecal excretion
and blood content of Plutonium. The ratio Δe u/Δλ i was evaluated by changing the rate
constant λ i only by 1% and calculating the resulting increment Δe u of the daily excretion in
urine. Some preliminary calculations of S i with respect to some transfer rates for different
increments (0.1%, 0.5%, 1%, 5% and 10%) were carried out. It was found that the increment
of 1% is small enough to represent a valid approximation of the derivative and, at the same
time, is large enough to avoid rounding problems [164].
The sensitivity coefficients so determined are time dependent through both the
function e u, which depends on the time post intake, and the transfer rates of the skeletal
system, for which the new model has introduced an age-related dependency. In order to
consider the sensitivity over a large period, the coefficients S i were calculated at 10, 100,
1,000 and 10,000 days post intake.
The sensitivity analysis was carried out for the Model-b parameters both for a direct
uptake into blood (i.e. injection) and for ingestion or inhalation. In the latter two cases a
sensitivity analysis for the parameters that describes the non-systemic phase of the
contamination was carried out as well. As in previous analyses, the systemic model (Model-b)
was connected to the ICRP models for the gastrointestinal tract [58] and for the respiratory
i
e u