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

calculating the amount of Plutonium activity in the blood compartment (often named transfer
compartment too).
3.1.1.2 Target function for data vs. model comparison
The comparison of model predictions for a certain variable (e.g; activity in an organ or
excreted in bioassays) with available data or empirical curves can be carried out qualitatively,
as a first attempt, by plotting them. For a quantitative comparison it is necessary to select a
target function that quantifies how good a certain model fits a given data set best.
An example of target function is given by Jones [111]. It is based on the assumption
that bioassay data, specifically for urinary excretion, are distributed according to a lognormal
distribution [153]. It is defined as:
where:
d(t i) is the experimental datum at time t i;
x(t i) is the value predicted by the model at time t i;
n is the number of empirical data.
[ ] 2
n
∑
L = Logd(t i)− Logx(t i)
i=1
85
equation 3.1.4
Other functions can be designed for such purpose, considering that the more a target
function points out a disagreement between data and a curve, the more this function is
appropriate. A target function was already used in a preliminary study to the present work
[154]:
F =
n
∑
i=1
⎡
⎣
⎢
d(t i)−x(t i )
x(t i )
equation 3.1.5
where the different elements have the same meaning as in Jones’s target function. This target
function has a similar structure to the χ 2 function. It is based on the deviation of each
experimental datum from the relative value predicted by the model. The terms are then
squared in order to eliminate the sign effect of the deviation. For such purpose the absolute
value operation, instead of term squaring, could be also adopted and the target function would
be defined as:
A =
n
∑
i=1
d(t i)−x(t i)
x(t i )
equation 3.1.6
By comparing the target function F and A in the equation 3.1.5 and equation 3.1.6, it
can be seen that the F function will give more weight to large deviations. In fact if the
experimental datum exceeds by two times the model’s prediction, the squared deviation is
greater than the absolute value of the deviation. On the contrary if the experimental datum
⎤
⎦
⎥
2
⎥