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diff --git a/docs/articles/FOCUS_Z.Rnw b/docs/articles/FOCUS_Z.Rnw new file mode 100644 index 00000000..5a66db24 --- /dev/null +++ b/docs/articles/FOCUS_Z.Rnw @@ -0,0 +1,274 @@ +%\VignetteIndexEntry{Example evaluation of FOCUS dataset Z} +%\VignetteEngine{knitr::knitr} +\documentclass[12pt,a4paper]{article} +\usepackage{a4wide} +\input{header} +\hypersetup{ + pdftitle = {Example evaluation of FOCUS dataset Z}, + pdfsubject = {Manuscript}, + pdfauthor = {Johannes Ranke}, + colorlinks = {true}, + linkcolor = {blue}, + citecolor = {blue}, + urlcolor = {red}, + hyperindex = {true}, + linktocpage = {true}, +} + +\begin{document} + +<<include=FALSE>>= +require(knitr) +opts_chunk$set(engine='R', tidy = FALSE, cache = TRUE) +options(width=70) +@ + +\title{Example evaluation of FOCUS dataset Z} +\author{\textbf{Johannes Ranke} \\[0.5cm] +%EndAName +Wissenschaftlicher Berater\\ +Kronacher Str. 8, 79639 Grenzach-Wyhlen, Germany\\[0.5cm] +and\\[0.5cm] +University of Bremen\\ +} +\maketitle + +\thispagestyle{empty} \setcounter{page}{0} + +\clearpage + +\tableofcontents + +\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation + +\section{The data} + +The following code defines the example dataset from Appendix 7 to the FOCUS kinetics +report \citep{FOCUSkinetics2011}, p.350. + +<<FOCUS_2006_Z_data, echo=TRUE, eval=TRUE>>= +require(mkin) +LOD = 0.5 +FOCUS_2006_Z = data.frame( + t = c(0, 0.04, 0.125, 0.29, 0.54, 1, 2, 3, 4, 7, 10, 14, 21, + 42, 61, 96, 124), + Z0 = c(100, 81.7, 70.4, 51.1, 41.2, 6.6, 4.6, 3.9, 4.6, 4.3, 6.8, + 2.9, 3.5, 5.3, 4.4, 1.2, 0.7), + Z1 = c(0, 18.3, 29.6, 46.3, 55.1, 65.7, 39.1, 36, 15.3, 5.6, 1.1, + 1.6, 0.6, 0.5 * LOD, NA, NA, NA), + Z2 = c(0, NA, 0.5 * LOD, 2.6, 3.8, 15.3, 37.2, 31.7, 35.6, 14.5, + 0.8, 2.1, 1.9, 0.5 * LOD, NA, NA, NA), + Z3 = c(0, NA, NA, NA, NA, 0.5 * LOD, 9.2, 13.1, 22.3, 28.4, 32.5, + 25.2, 17.2, 4.8, 4.5, 2.8, 4.4)) + +FOCUS_2006_Z_mkin <- mkin_wide_to_long(FOCUS_2006_Z) +@ + +\section{Parent compound and one metabolite} + +The next step is to set up the models used for the kinetic analysis. As the +simultaneous fit of parent and the first metabolite is usually straightforward, +Step 1 (SFO for parent only) is skipped here. We start with the model 2a, +with formation and decline of metabolite Z1 and the pathway from parent +directly to sink included (default in mkin). + +<<FOCUS_2006_Z_fits_1, echo=TRUE, fig.height=6>>= +Z.2a <- mkinmod(Z0 = mkinsub("SFO", "Z1"), + Z1 = mkinsub("SFO")) +m.Z.2a <- mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.2a) +summary(m.Z.2a, data = FALSE)$bpar +@ + +As obvious from the parameter summary (the \texttt{bpar} component of the +summary), the kinetic rate constant from parent compound Z to sink +is negligible. Accordingly, the exact magnitude of the fitted parameter +\texttt{log k\_Z0\_sink} is ill-defined and the covariance matrix is not +returned (not shown, would be visible in the complete summary). +This suggests, in agreement with the analysis in the FOCUS kinetics report, to +simplify the model by removing the pathway to sink. + +A similar result can be obtained when formation fractions are used in the model +formulation: + +<<FOCUS_2006_Z_fits_2, echo=TRUE, fig.height=6>>= +Z.2a.ff <- mkinmod(Z0 = mkinsub("SFO", "Z1"), + Z1 = mkinsub("SFO"), + use_of_ff = "max") + +m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.2a.ff) +summary(m.Z.2a.ff, data = FALSE)$bpar +@ + +Here, the ilr transformed formation fraction fitted in the model takes a very +large value, and the backtransformed formation fraction from parent Z to Z1 is +practically unity. Again, the covariance matrix is not returned as the model is +overparameterised. + +The simplified model is obtained by setting the list component \texttt{sink} to +\texttt{FALSE}.\footnote{If the model formulation without formation fractions +is used, the same effect can be obtained by fixing the parameter \texttt{k\_Z\_sink} +to a value of zero.} + +In the following, we use the parameterisation with formation fractions in order +to be able to compare with the results in the FOCUS guidance, and as it +makes it easier to use parameters obtained in a previous fit when adding a further +metabolite. + +<<FOCUS_2006_Z_fits_3, echo=TRUE, fig.height=6>>= +Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE), + Z1 = mkinsub("SFO"), use_of_ff = "max") +m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.3) +summary(m.Z.3, data = FALSE)$bpar +@ + +As there is only one transformation product for Z0 and no pathway +to sink, the formation fraction is internally fixed to unity. + +\section{Including metabolites Z2 and Z3} + +As suggested in the FOCUS report, the pathway to sink was removed for metabolite Z1 as +well in the next step. While this step appears questionable on the basis of the above results, it +is followed here for the purpose of comparison. Also, in the FOCUS report, it is +assumed that there is additional empirical evidence that Z1 quickly and exclusively +hydrolyses to Z2. + +<<FOCUS_2006_Z_fits_5, echo=TRUE, fig.height=7>>= +Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO"), use_of_ff = "max") +m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.5) +@ + +Finally, metabolite Z3 is added to the model. We use the optimised +differential equation parameter values from the previous fit in order to +accelerate the optimization. + +<<FOCUS_2006_Z_fits_6, echo=TRUE, fig.height=8>>= +Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO", "Z3"), + Z3 = mkinsub("SFO"), + use_of_ff = "max") +m.Z.FOCUS <- mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, + parms.ini = m.Z.5$bparms.ode, + quiet = TRUE) +plot_sep(m.Z.FOCUS) +summary(m.Z.FOCUS, data = FALSE)$bpar +endpoints(m.Z.FOCUS) +@ + +This fit corresponds to the final result chosen in Appendix 7 of the FOCUS +report. Confidence intervals returned by mkin are based on internally +transformed parameters, however. + +\section{Using the SFORB model for parent and metabolites} + +As the FOCUS report states, there is a certain tailing of the time course of metabolite +Z3. Also, the time course of the parent compound is not fitted very well using the +SFO model, as residues at a certain low level remain. + +Therefore, an additional model is offered here, using the single first-order +reversible binding (SFORB) model for metabolite Z3. As expected, the $\chi^2$ +error level is lower for metabolite Z3 using this model and the graphical +fit for Z3 is improved. However, the covariance matrix is not returned. + +<<FOCUS_2006_Z_fits_7, echo=TRUE, fig.height=8>>= +Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO", "Z3"), + Z3 = mkinsub("SFORB")) +m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.mkin.1) +summary(m.Z.mkin.1, data = FALSE)$cov.unscaled +@ + +Therefore, a further stepwise model building is performed starting from the +stage of parent and two metabolites, starting from the assumption that the model +fit for the parent compound can be improved by using the SFORB model. + +<<FOCUS_2006_Z_fits_9, echo=TRUE, fig.height=8>>= +Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO")) +m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE) +plot_sep(m.Z.mkin.3) +@ + +This results in a much better representation of the behaviour of the parent +compound Z0. + +Finally, Z3 is added as well. These models appear overparameterised (no +covariance matrix returned) if the sink for Z1 is left in the models. + +<<FOCUS_2006_Z_fits_10, echo=TRUE, fig.height=8>>= +Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO", "Z3"), + Z3 = mkinsub("SFO")) +m.Z.mkin.4 <- mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, + parms.ini = m.Z.mkin.3$bparms.ode, + quiet = TRUE) +plot_sep(m.Z.mkin.4) +@ + +The error level of the fit, but especially of metabolite Z3, can be improved if +the SFORB model is chosen for this metabolite, as this model is capable of +representing the tailing of the metabolite decline phase. + +<<FOCUS_2006_Z_fits_11, echo=TRUE, fig.height=8>>= +Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE), + Z1 = mkinsub("SFO", "Z2", sink = FALSE), + Z2 = mkinsub("SFO", "Z3"), + Z3 = mkinsub("SFORB")) +m.Z.mkin.5 <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, + parms.ini = m.Z.mkin.4$bparms.ode[1:4], + quiet = TRUE) +plot_sep(m.Z.mkin.5) +@ + +The summary view of the backtransformed parameters shows that we get no +confidence intervals due to overparameterisation. As the optimized +\texttt{k\_Z3\_bound\_free} is excessively small, it seems reasonable to fix it to +zero. + +<<FOCUS_2006_Z_fits_11a, echo=TRUE>>= +m.Z.mkin.5a <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, + parms.ini = c(m.Z.mkin.5$bparms.ode[1:7], + k_Z3_bound_free = 0), + fixed_parms = "k_Z3_bound_free", + quiet = TRUE) +plot_sep(m.Z.mkin.5a) +@ + +As expected, the residual plots for Z0 and Z3 are more random than in the case of the +all SFO model for which they were shown above. In conclusion, the model +\texttt{Z.mkin.5a} is proposed as the best-fit model for the dataset from +Appendix 7 of the FOCUS report. + +A graphical representation of the confidence intervals can finally be obtained. + +<<FOCUS_2006_Z_fits_11b, echo=TRUE>>= +mkinparplot(m.Z.mkin.5a) +@ + +The endpoints obtained with this model are + +<<FOCUS_2006_Z_fits_11b_endpoints, echo=TRUE>>= +endpoints(m.Z.mkin.5a) +@ + +It is clear the degradation rate of Z3 towards the end of the experiment +is very low as DT50\_Z3\_b2 (the second Eigenvalue of the system of two differential +equations representing the SFORB system for Z3, corresponding to the slower rate +constant of the DFOP model) is reported to be infinity. However, this appears +to be a feature of the data. + +\bibliographystyle{plainnat} +\bibliography{references} + +\end{document} +% vim: set foldmethod=syntax: |