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pdftitle = {Examples for kinetic evaluations using mkin},
pdfsubject = {Manuscript},
pdfauthor = {Johannes Ranke},
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\begin{document}
\title{Examples for kinetic evaluations using mkin}
\author{\textbf{Johannes Ranke} \\[0.5cm]
%EndAName
Eurofins Regulatory AG\\
Weidenweg 15, CH--4310 Rheinfelden, Switzerland\\[0.5cm]
and\\[0.5cm]
University of Bremen\\
}
\maketitle
%\begin{abstract}
%\end{abstract}
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\clearpage
\tableofcontents
\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation
\section{Kinetic evaluations for parent compounds}
\label{intro}
These examples are also evaluated in a parallel vignette of the
\Rpackage{kinfit} package \citep{pkg:kinfit}. The datasets are from Appendix 3,
of the FOCUS kinetics report \citep{FOCUS2006, FOCUSkinetics2011}.
\subsection{Laboratory Data L1}
The following code defines example dataset L1 from the FOCUS kinetics
report, p. 284
\begin{Schunk}
\begin{Sinput}
R> library("mkin")
R> FOCUS_2006_L1 = data.frame(
+ t = rep(c(0, 1, 2, 3, 5, 7, 14, 21, 30), each = 2),
+ parent = c(88.3, 91.4, 85.6, 84.5, 78.9, 77.6,
+ 72.0, 71.9, 50.3, 59.4, 47.0, 45.1,
+ 27.7, 27.3, 10.0, 10.4, 2.9, 4.0))
R> FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
\end{Sinput}
\end{Schunk}
The next step is to set up the models used for the kinetic analysis. Note that
the model definitions contain the names of the observed variables in the data.
In this case, there is only one variable called \Robject{parent}.
\begin{Schunk}
\begin{Sinput}
R> SFO <- mkinmod(parent = list(type = "SFO"))
R> FOMC <- mkinmod(parent = list(type = "FOMC"))
R> DFOP <- mkinmod(parent = list(type = "DFOP"))
\end{Sinput}
\end{Schunk}
The three models cover the first assumption of simple first order (SFO),
the case of declining rate constant over time (FOMC) and the case of two
different phases of the kinetics (DFOP). For a more detailed discussion
of the models, please see the FOCUS kinetics report.
The following two lines fit the model and produce the summary report
of the model fit. This covers the numerical analysis given in the
FOCUS report.
\begin{Schunk}
\begin{Sinput}
R> m.L1.SFO <- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet=TRUE)
R> summary(m.L1.SFO)
\end{Sinput}
\begin{Soutput}
mkin version: 0.9.10
R version: 2.15.2
Date of fit: Sat Feb 16 21:38:15 2013
Date of summary: Sat Feb 16 21:38:15 2013
Equations:
[1] d_parent = - k_parent_sink * parent
Starting values for optimised parameters:
initial type transformed
parent_0 100.0 state 100.000000
k_parent_sink 0.1 deparm -2.302585
Fixed parameter values:
None
Optimised, transformed parameters:
Estimate Std. Error
parent_0 92.471 1.368
k_parent_sink -2.347 0.041
Backtransformed parameters:
Estimate
parent_0 92.471
k_parent_sink 0.096
Residual standard error: 2.948 on 16 degrees of freedom
Chi2 error levels in percent:
err.min n.optim df
All data 3.424 2 7
parent 3.424 2 7
Estimated disappearance times:
DT50 DT90
parent 7.249 24.08
Estimated formation fractions:
ff
parent_sink 1
Parameter correlation:
parent_0 k_parent_sink
parent_0 1.0000 0.6248
k_parent_sink 0.6248 1.0000
Data:
time variable observed predicted residual
0 parent 88.3 92.471 -4.1710
0 parent 91.4 92.471 -1.0710
1 parent 85.6 84.039 1.5610
1 parent 84.5 84.039 0.4610
2 parent 78.9 76.376 2.5241
2 parent 77.6 76.376 1.2241
3 parent 72.0 69.412 2.5884
3 parent 71.9 69.412 2.4884
5 parent 50.3 57.330 -7.0301
5 parent 59.4 57.330 2.0699
7 parent 47.0 47.352 -0.3515
7 parent 45.1 47.352 -2.2515
14 parent 27.7 24.247 3.4527
14 parent 27.3 24.247 3.0527
21 parent 10.0 12.416 -2.4163
21 parent 10.4 12.416 -2.0163
30 parent 2.9 5.251 -2.3513
30 parent 4.0 5.251 -1.2513
\end{Soutput}
\end{Schunk}
A plot of the fit is obtained with the plot function for mkinfit objects.
\begin{Schunk}
\begin{Sinput}
R> plot(m.L1.SFO)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L1_SFO_plot}
The residual plot can be obtained using the information contained in the
mkinfit object, which is in fact a derivative of an modFit object defined by
the \Rpackage{FME} package.
\begin{Schunk}
\begin{Sinput}
R> plot(m.L1.SFO$data$time, m.L1.SFO$data$residual,
+ xlab = "Time", ylab = "Residual", ylim = c(-8, 8))
R> abline(h = 0, lty = 2)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L1_SFO_residuals}
For comparison, the FOMC model is fitted as well, and the $\chi^2$ error level
is checked.
\begin{Schunk}
\begin{Sinput}
R> m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet=TRUE)
R> s.m.L1.FOMC <- summary(m.L1.FOMC)
R> s.m.L1.FOMC$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.03618911 3 6
parent 0.03618911 3 6
\end{Soutput}
\end{Schunk}
Due to the higher number of parameters, and the lower number of degrees of freedom
of the fit, the $\chi^2$ error level is actually higher for the FOMC model (3.6\%) than
for the SFO model (3.4\%).
\subsection{Laboratory Data L2}
The following code defines example dataset L2 from the FOCUS kinetics
report, p. 287
\begin{Schunk}
\begin{Sinput}
R> library("mkin")
R> FOCUS_2006_L2 = data.frame(
+ t = rep(c(0, 1, 3, 7, 14, 28), each = 2),
+ parent = c(96.1, 91.8, 41.4, 38.7,
+ 19.3, 22.3, 4.6, 4.6,
+ 2.6, 1.2, 0.3, 0.6))
R> FOCUS_2006_L2_mkin <- mkin_wide_to_long(FOCUS_2006_L2)
\end{Sinput}
\end{Schunk}
Again, the SFO model is fitted and a summary is obtained.
\begin{Schunk}
\begin{Sinput}
R> m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet=TRUE)
R> summary(m.L2.SFO)
\end{Sinput}
\begin{Soutput}
mkin version: 0.9.10
R version: 2.15.2
Date of fit: Sat Feb 16 21:38:15 2013
Date of summary: Sat Feb 16 21:38:15 2013
Equations:
[1] d_parent = - k_parent_sink * parent
Starting values for optimised parameters:
initial type transformed
parent_0 100.0 state 100.000000
k_parent_sink 0.1 deparm -2.302585
Fixed parameter values:
None
Optimised, transformed parameters:
Estimate Std. Error
parent_0 91.4656 3.807
k_parent_sink -0.4112 0.107
Backtransformed parameters:
Estimate
parent_0 91.466
k_parent_sink 0.663
Residual standard error: 5.51 on 10 degrees of freedom
Chi2 error levels in percent:
err.min n.optim df
All data 14.38 2 4
parent 14.38 2 4
Estimated disappearance times:
DT50 DT90
parent 1.046 3.474
Estimated formation fractions:
ff
parent_sink 1
Parameter correlation:
parent_0 k_parent_sink
parent_0 1.0000 0.4295
k_parent_sink 0.4295 1.0000
Data:
time variable observed predicted residual
0 parent 96.1 91.4656079103 4.6344
0 parent 91.8 91.4656079103 0.3344
1 parent 41.4 47.1395280371 -5.7395
1 parent 38.7 47.1395280371 -8.4395
3 parent 19.3 12.5210295280 6.7790
3 parent 22.3 12.5210295280 9.7790
7 parent 4.6 0.8833842647 3.7166
7 parent 4.6 0.8833842647 3.7166
14 parent 2.6 0.0085318162 2.5915
14 parent 1.2 0.0085318162 1.1915
28 parent 0.3 0.0000007958 0.3000
28 parent 0.6 0.0000007958 0.6000
\end{Soutput}
\end{Schunk}
The $\chi^2$ error level of 14\% suggests that the model does not fit very well.
This is also obvious from the plots of the fit and the residuals.
\begin{Schunk}
\begin{Sinput}
R> plot(m.L2.SFO)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L2_SFO_plot}
In the FOCUS kinetics report, it is stated that there is no apparent systematic
error observed from the residual plot up to the measured DT90 (approximately at
day 5), and there is an underestimation beyond that point.
\begin{Schunk}
\begin{Sinput}
R> plot(m.L2.SFO$data$time, m.L2.SFO$data$residual,
+ xlab = "Time", ylab = "Residual", ylim = c(-10, 10))
R> abline(h = 0, lty = 2)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L2_SFO_residuals}
We may add that it is difficult to judge the random nature of the residuals just
from the three samplings at days 0, 1 and 3. Also, it is not clear why a
consistent underestimation after the approximate DT90 should be irrelevant.
For comparison, the FOMC model is fitted as well, and the $\chi^2$ error level
is checked.
\begin{Schunk}
\begin{Sinput}
R> m.L2.FOMC <- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet=TRUE)
R> plot(m.L2.FOMC)
R> s.m.L2.FOMC <- summary(m.L2.FOMC)
R> s.m.L2.FOMC$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.06204245 3 3
parent 0.06204245 3 3
\end{Soutput}
\end{Schunk}
\includegraphics{examples-L2_FOMC}
The error level at which the $\chi^2$ test passes is much lower in this case.
Therefore, the FOMC model provides a better description of the data, as less
experimental error has to be assumed in order to explain the data.
Fitting the four parameter DFOP model does not further reduce the
$\chi^2$ error level.
\begin{Schunk}
\begin{Sinput}
R> m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet=TRUE)
R> plot(m.L2.DFOP)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L2_DFOP}
Here, the default starting parameters for the DFOP model obviously do not lead
to a reasonable solution. Therefore the fit is repeated with different starting
parameters.
\begin{Schunk}
\begin{Sinput}
R> m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin,
+ parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),
+ quiet=TRUE)
R> plot(m.L2.DFOP)
R> summary(m.L2.DFOP)
\end{Sinput}
\begin{Soutput}
mkin version: 0.9.10
R version: 2.15.2
Date of fit: Sat Feb 16 21:38:16 2013
Date of summary: Sat Feb 16 21:38:16 2013
Equations:
[1] d_parent = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time))) * parent
Starting values for optimised parameters:
initial type transformed
parent_0 1e+02 state 100.0000000
k1 1e+00 deparm 0.0000000
k2 1e-02 deparm -4.6051702
g 8e-01 deparm 0.9802581
Fixed parameter values:
None
Optimised, transformed parameters:
Estimate Std. Error
parent_0 93.9500 NA
k1 4.9589 NA
k2 -1.0880 NA
g -0.2821 NA
Backtransformed parameters:
Estimate
parent_0 93.950
k1 142.434
k2 0.337
g 0.402
Residual standard error: 1.732 on 8 degrees of freedom
Chi2 error levels in percent:
err.min n.optim df
All data 2.529 4 2
parent 2.529 4 2
Estimated disappearance times:
DT50 DT90
parent NA NA
Estimated formation fractions:
[1] ff
<0 rows> (or 0-length row.names)
Data:
time variable observed predicted residual
0 parent 96.1 93.950000 2.1500
0 parent 91.8 93.950000 -2.1500
1 parent 41.4 40.143423 1.2566
1 parent 38.7 40.143423 -1.4434
3 parent 19.3 20.464500 -1.1645
3 parent 22.3 20.464500 1.8355
7 parent 4.6 5.318322 -0.7183
7 parent 4.6 5.318322 -0.7183
14 parent 2.6 0.503070 2.0969
14 parent 1.2 0.503070 0.6969
28 parent 0.3 0.004501 0.2955
28 parent 0.6 0.004501 0.5955
\end{Soutput}
\begin{Sinput}
R> s.m.L2.DFOP <- summary(m.L2.DFOP)
R> s.m.L2.DFOP$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.02528763 4 2
parent 0.02528763 4 2
\end{Soutput}
\end{Schunk}
\includegraphics{examples-L2_DFOP_2}
Therefore, the FOMC model is clearly the best-fit model based on the
$\chi^2$ error level criterion.
\subsection{Laboratory Data L3}
The following code defines example dataset L3 from the FOCUS kinetics
report, p. 290
\begin{Schunk}
\begin{Sinput}
R> library("mkin")
R> FOCUS_2006_L3 = data.frame(
+ t = c(0, 3, 7, 14, 30, 60, 91, 120),
+ parent = c(97.8, 60, 51, 43, 35, 22, 15, 12))
R> FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)
\end{Sinput}
\end{Schunk}
SFO model, summary and plot:
\begin{Schunk}
\begin{Sinput}
R> m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet=TRUE)
R> summary(m.L3.SFO)
\end{Sinput}
\begin{Soutput}
mkin version: 0.9.10
R version: 2.15.2
Date of fit: Sat Feb 16 21:38:16 2013
Date of summary: Sat Feb 16 21:38:16 2013
Equations:
[1] d_parent = - k_parent_sink * parent
Starting values for optimised parameters:
initial type transformed
parent_0 100.0 state 100.000000
k_parent_sink 0.1 deparm -2.302585
Fixed parameter values:
None
Optimised, transformed parameters:
Estimate Std. Error
parent_0 74.873 8.458
k_parent_sink -3.678 0.326
Backtransformed parameters:
Estimate
parent_0 74.873
k_parent_sink 0.025
Residual standard error: 12.91 on 6 degrees of freedom
Chi2 error levels in percent:
err.min n.optim df
All data 21.24 2 6
parent 21.24 2 6
Estimated disappearance times:
DT50 DT90
parent 27.43 91.12
Estimated formation fractions:
ff
parent_sink 1
Parameter correlation:
parent_0 k_parent_sink
parent_0 1.0000 0.5484
k_parent_sink 0.5484 1.0000
Data:
time variable observed predicted residual
0 parent 97.8 74.873 22.92734
3 parent 60.0 69.407 -9.40654
7 parent 51.0 62.734 -11.73403
14 parent 43.0 52.563 -9.56336
30 parent 35.0 35.083 -0.08281
60 parent 22.0 16.439 5.56137
91 parent 15.0 7.510 7.48961
120 parent 12.0 3.609 8.39083
\end{Soutput}
\begin{Sinput}
R> plot(m.L3.SFO)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L3_SFO}
The $\chi^2$ error level of 22\% as well as the plot suggest that the model
does not fit very well.
The FOMC model performs better:
\begin{Schunk}
\begin{Sinput}
R> m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet=TRUE)
R> plot(m.L3.FOMC)
R> s.m.L3.FOMC <- summary(m.L3.FOMC)
R> s.m.L3.FOMC$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.07321867 3 5
parent 0.07321867 3 5
\end{Soutput}
\begin{Sinput}
R> endpoints(m.L3.FOMC)
\end{Sinput}
\begin{Soutput}
$distimes
DT50 DT90
parent 7.729478 431.2428
$ff
logical(0)
$SFORB
logical(0)
\end{Soutput}
\end{Schunk}
\includegraphics{examples-L3_FOMC}
The error level at which the $\chi^2$ test passes is 7\% in this case.
Fitting the four parameter DFOP model further reduces the $\chi^2$ error level
considerably:
\begin{Schunk}
\begin{Sinput}
R> m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet=TRUE)
R> plot(m.L3.DFOP)
R> s.m.L3.DFOP <- summary(m.L3.DFOP)
R> s.m.L3.DFOP$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.02223992 4 4
parent 0.02223992 4 4
\end{Soutput}
\end{Schunk}
\includegraphics{examples-L3_DFOP}
Therefore, the DFOP model is the best-fit model based on the $\chi^2$ error
level criterion for laboratory data L3.
\subsection{Laboratory Data L4}
The following code defines example dataset L4 from the FOCUS kinetics
report, p. 293
\begin{Schunk}
\begin{Sinput}
R> library("mkin")
R> FOCUS_2006_L4 = data.frame(
+ t = c(0, 3, 7, 14, 30, 60, 91, 120),
+ parent = c(96.6, 96.3, 94.3, 88.8, 74.9, 59.9, 53.5, 49.0))
R> FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)
\end{Sinput}
\end{Schunk}
SFO model, summary and plot:
\begin{Schunk}
\begin{Sinput}
R> m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet=TRUE)
R> summary(m.L4.SFO)
\end{Sinput}
\begin{Soutput}
mkin version: 0.9.10
R version: 2.15.2
Date of fit: Sat Feb 16 21:38:17 2013
Date of summary: Sat Feb 16 21:38:17 2013
Equations:
[1] d_parent = - k_parent_sink * parent
Starting values for optimised parameters:
initial type transformed
parent_0 100.0 state 100.000000
k_parent_sink 0.1 deparm -2.302585
Fixed parameter values:
None
Optimised, transformed parameters:
Estimate Std. Error
parent_0 96.44 1.949
k_parent_sink -5.03 0.080
Backtransformed parameters:
Estimate
parent_0 96.442
k_parent_sink 0.007
Residual standard error: 3.651 on 6 degrees of freedom
Chi2 error levels in percent:
err.min n.optim df
All data 3.288 2 6
parent 3.288 2 6
Estimated disappearance times:
DT50 DT90
parent 106 352
Estimated formation fractions:
ff
parent_sink 1
Parameter correlation:
parent_0 k_parent_sink
parent_0 1.0000 0.5865
k_parent_sink 0.5865 1.0000
Data:
time variable observed predicted residual
0 parent 96.6 96.44 0.1585
3 parent 96.3 94.57 1.7324
7 parent 94.3 92.13 2.1744
14 parent 88.8 88.00 0.7972
30 parent 74.9 79.26 -4.3589
60 parent 59.9 65.14 -5.2376
91 parent 53.5 53.18 0.3167
120 parent 49.0 43.99 5.0054
\end{Soutput}
\begin{Sinput}
R> plot(m.L4.SFO)
\end{Sinput}
\end{Schunk}
\includegraphics{examples-L4_SFO}
The $\chi^2$ error level of 3.3\% as well as the plot suggest that the model
fits very well.
The FOMC model for comparison
\begin{Schunk}
\begin{Sinput}
R> m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet=TRUE)
R> plot(m.L4.FOMC)
R> s.m.L4.FOMC <- summary(m.L4.FOMC)
R> s.m.L4.FOMC$errmin
\end{Sinput}
\begin{Soutput}
err.min n.optim df
All data 0.02027643 3 5
parent 0.02027643 3 5
\end{Soutput}
\end{Schunk}
\includegraphics{examples-L4_FOMC}
The error level at which the $\chi^2$ test passes is slightly lower for the FOMC
model. However, the difference appears negligible.
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
% vim: set foldmethod=syntax:
