1 General Purpose
Plasma concentration time curves are
the basis of pharmacokinetics. If traditional nonlinear models do
not fit the data well, spline and loess (locally weighted scatter
plot smoothing) modeling will provide a possible solution.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
Does loess and spline modeling produce
a better fit model for the plasma concentration – time
relationships of medicines than traditional curvilinear estimations
(Chap. 25).
4 Data Example
In 90 patient a plasma concentration
time curve study of intravenous administration of zoledronic acid
(ng/ml) was performed.
|
Conc
|
Time
|
|
1,10
|
1,00
|
|
,90
|
1,00
|
|
,80
|
1,00
|
|
,78
|
2,00
|
|
,55
|
2,00
|
|
,65
|
3,00
|
|
,48
|
4,00
|
|
,45
|
4,00
|
|
,32
|
4,00
|
|
,30
|
5,00
|
5 Some Background Information
Usually, the relationship between
plasma concentration and time of a drug is described in the form of
an exponential model. This is convenient, because it enables to
calculate pharmacokinetic parameters like plasma half-life and
equations for clearance. Using the Non-Mem program of the
University of San Francisco a non linear mixed effect model of the
data is produced (= multi-exponential model). The underneath figure
of the data shows the exponential model. There is a wide spread in
the data, and, so, the pharmacokinetic parameters derived from the
model do not mean too much.
6 Spline Modeling
If the traditional models do not fit
your data very well, you may use a method called spline modeling.
The term spline stems from thin flexible wooden splines formerly
used by shipbuilders and cardesigners to produce smooth shapes. A
spline model consists of 4, 5 or more intervals with different
cubic curves (= third order polynomes, like y = a + bx3,
see also Chap. 25) that have the same y-value,
slope, and curvature at the junctions.
Command:
-
Graphs….Chart Builder….click Scatter/Dot….click in Simple Scatter and drag to Chart Preview…. click plasma concentration and drag to the Y-Axis….click time and drag to the X-Axis….OK…..double-click in GGraph ….Chart Editor comes up….click Elements….click Interpolation….dialog box Properties….mark Spline….click Apply….click Edit….click Copy Chart.
The underneath figure shows the best
fit spline model of the above data.
7 Loess (Locally Weighted Scatter Plot Smoothing) Modeling
Also loess modeling works with cubic
curves (third order polynomes), but unlike spline modeling it does
not work with junctions, but, instead, it chooses the best fit
cubic curves for each value with outlier data given less weight.
Command:
-
Graphs….Chart Builder….click Scatter/Dot….click in Simple Scatter and drag to Chart Preview…. click plasma concentration and drag to the Y-Axis….click time and drag to the X-Axis….OK…..double-click in GGraph ….Chart Editor comes up….click Elements….Fit Line at Total….in dialog box Properties….mark: Loess….click: Apply…. click Edit….click Copy Chart.
The underneath figure shows the best
fit Loess model of the above data.
8 Conclusion
Both spline and loess modeling are
computationally very intensive methods that do not produce simple
regression equations like the ones given in the Chap. 25 on curvilinear regression. They
also require fairly large, densely sampled data sets in order to
produce good models. For making predictions from such models direct
interpolations / extrapolations from the graphs can be made, and,
given the mathematical refinement of these methods, these
predictions should, generally, give excellent precision. We
conclude.
1.
Both spline and loess modeling are
computationally intensive models that are adequate, if the data
plot leaves you with no idea about the relationship between the y-
and x-values.
2.
They do not produce simple regression
equations like the ones given in Chap. 25 on curvilinear regression.
3.
For making predictions from such
models direct interpolations / extrapolations from the graphs can
be made, and, given the mathematical refinement of these methods,
these predictions generally give excellent precision.
4.
Maybe, the best fit for many types of
nonlinear data is offered by loess.
9 Note
More background, theoretical, and
mathematical information of loess and spline modeling is given in
Statistics applied to clinical studies 5th edition, Chap. 24,
Springer Heidelberg Germany, 2012, from the same authors.