From 8f38147e145c66ebd27f5d9fcc6f00172c25872f Mon Sep 17 00:00:00 2001
From: Johannes Ranke <jranke@uni-bremen.de>
Date: Wed, 26 Oct 2016 23:18:20 +0200
Subject: Static documentation rebuilt by pkgdown::build_site()

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+%\VignetteIndexEntry{Example evaluation of FOCUS dataset Z}
+%\VignetteEngine{knitr::knitr}
+\documentclass[12pt,a4paper]{article}\usepackage[]{graphicx}\usepackage[]{color}
+%% maxwidth is the original width if it is less than linewidth
+%% otherwise use linewidth (to make sure the graphics do not exceed the margin)
+\makeatletter
+\def\maxwidth{ %
+  \ifdim\Gin@nat@width>\linewidth
+    \linewidth
+  \else
+    \Gin@nat@width
+  \fi
+}
+\makeatother
+
+\definecolor{fgcolor}{rgb}{0.345, 0.345, 0.345}
+\newcommand{\hlnum}[1]{\textcolor[rgb]{0.686,0.059,0.569}{#1}}%
+\newcommand{\hlstr}[1]{\textcolor[rgb]{0.192,0.494,0.8}{#1}}%
+\newcommand{\hlcom}[1]{\textcolor[rgb]{0.678,0.584,0.686}{\textit{#1}}}%
+\newcommand{\hlopt}[1]{\textcolor[rgb]{0,0,0}{#1}}%
+\newcommand{\hlstd}[1]{\textcolor[rgb]{0.345,0.345,0.345}{#1}}%
+\newcommand{\hlkwa}[1]{\textcolor[rgb]{0.161,0.373,0.58}{\textbf{#1}}}%
+\newcommand{\hlkwb}[1]{\textcolor[rgb]{0.69,0.353,0.396}{#1}}%
+\newcommand{\hlkwc}[1]{\textcolor[rgb]{0.333,0.667,0.333}{#1}}%
+\newcommand{\hlkwd}[1]{\textcolor[rgb]{0.737,0.353,0.396}{\textbf{#1}}}%
+\let\hlipl\hlkwb
+
+\usepackage{framed}
+\makeatletter
+\newenvironment{kframe}{%
+ \def\at@end@of@kframe{}%
+ \ifinner\ifhmode%
+  \def\at@end@of@kframe{\end{minipage}}%
+  \begin{minipage}{\columnwidth}%
+ \fi\fi%
+ \def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep
+ \colorbox{shadecolor}{##1}\hskip-\fboxsep
+     % There is no \\@totalrightmargin, so:
+     \hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}%
+ \MakeFramed {\advance\hsize-\width
+   \@totalleftmargin\z@ \linewidth\hsize
+   \@setminipage}}%
+ {\par\unskip\endMakeFramed%
+ \at@end@of@kframe}
+\makeatother
+
+\definecolor{shadecolor}{rgb}{.97, .97, .97}
+\definecolor{messagecolor}{rgb}{0, 0, 0}
+\definecolor{warningcolor}{rgb}{1, 0, 1}
+\definecolor{errorcolor}{rgb}{1, 0, 0}
+\newenvironment{knitrout}{}{} % an empty environment to be redefined in TeX
+
+\usepackage{alltt}
+\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},
+}
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\begin{document}
+
+
+
+\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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{require}\hlstd{(mkin)}
+\end{alltt}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: mkin}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: minpack.lm}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: rootSolve}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: inline}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: methods}}
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Loading required package: parallel}}\begin{alltt}
+\hlstd{LOD} \hlkwb{=} \hlnum{0.5}
+\hlstd{FOCUS_2006_Z} \hlkwb{=} \hlkwd{data.frame}\hlstd{(}
+  \hlkwc{t} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{0.04}\hlstd{,} \hlnum{0.125}\hlstd{,} \hlnum{0.29}\hlstd{,} \hlnum{0.54}\hlstd{,} \hlnum{1}\hlstd{,} \hlnum{2}\hlstd{,} \hlnum{3}\hlstd{,} \hlnum{4}\hlstd{,} \hlnum{7}\hlstd{,} \hlnum{10}\hlstd{,} \hlnum{14}\hlstd{,} \hlnum{21}\hlstd{,}
+        \hlnum{42}\hlstd{,} \hlnum{61}\hlstd{,} \hlnum{96}\hlstd{,} \hlnum{124}\hlstd{),}
+  \hlkwc{Z0} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{100}\hlstd{,} \hlnum{81.7}\hlstd{,} \hlnum{70.4}\hlstd{,} \hlnum{51.1}\hlstd{,} \hlnum{41.2}\hlstd{,} \hlnum{6.6}\hlstd{,} \hlnum{4.6}\hlstd{,} \hlnum{3.9}\hlstd{,} \hlnum{4.6}\hlstd{,} \hlnum{4.3}\hlstd{,} \hlnum{6.8}\hlstd{,}
+         \hlnum{2.9}\hlstd{,} \hlnum{3.5}\hlstd{,} \hlnum{5.3}\hlstd{,} \hlnum{4.4}\hlstd{,} \hlnum{1.2}\hlstd{,} \hlnum{0.7}\hlstd{),}
+  \hlkwc{Z1} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{18.3}\hlstd{,} \hlnum{29.6}\hlstd{,} \hlnum{46.3}\hlstd{,} \hlnum{55.1}\hlstd{,} \hlnum{65.7}\hlstd{,} \hlnum{39.1}\hlstd{,} \hlnum{36}\hlstd{,} \hlnum{15.3}\hlstd{,} \hlnum{5.6}\hlstd{,} \hlnum{1.1}\hlstd{,}
+         \hlnum{1.6}\hlstd{,} \hlnum{0.6}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{),}
+  \hlkwc{Z2} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{2.6}\hlstd{,} \hlnum{3.8}\hlstd{,} \hlnum{15.3}\hlstd{,} \hlnum{37.2}\hlstd{,} \hlnum{31.7}\hlstd{,} \hlnum{35.6}\hlstd{,} \hlnum{14.5}\hlstd{,}
+         \hlnum{0.8}\hlstd{,} \hlnum{2.1}\hlstd{,} \hlnum{1.9}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{),}
+  \hlkwc{Z3} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{0}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{NA}\hlstd{,} \hlnum{0.5} \hlopt{*} \hlstd{LOD,} \hlnum{9.2}\hlstd{,} \hlnum{13.1}\hlstd{,} \hlnum{22.3}\hlstd{,} \hlnum{28.4}\hlstd{,} \hlnum{32.5}\hlstd{,}
+         \hlnum{25.2}\hlstd{,} \hlnum{17.2}\hlstd{,} \hlnum{4.8}\hlstd{,} \hlnum{4.5}\hlstd{,} \hlnum{2.8}\hlstd{,} \hlnum{4.4}\hlstd{))}
+
+\hlstd{FOCUS_2006_Z_mkin} \hlkwb{<-} \hlkwd{mkin_wide_to_long}\hlstd{(FOCUS_2006_Z)}
+\end{alltt}
+\end{kframe}
+\end{knitrout}
+
+\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).
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.2a} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{),}
+                \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.2a} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.2a, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.2a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_1-1} 
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.2a,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+##               Estimate se_notrans      t value       Pr(>t)
+## Z0_0      9.701488e+01 3.55313691 2.730401e+01 1.679214e-21
+## k_Z0_sink 1.281376e-11 0.22689470 5.647447e-11 5.000000e-01
+## k_Z0_Z1   2.236006e+00 0.16507604 1.354531e+01 7.396594e-14
+## k_Z1_sink 4.821248e-01 0.06585369 7.321150e+00 3.552015e-08
+##                Lower       Upper
+## Z0_0      91.4058170 102.6239462
+## k_Z0_sink  0.0000000         Inf
+## k_Z0_Z1    1.8419826   2.7143172
+## k_Z1_sink  0.4005856   0.5802613
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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:
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.2a.ff} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{),}
+                   \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),}
+                   \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.2a.ff} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.2a.ff, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.2a.ff)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_2-1} 
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.2a.ff,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+##              Estimate se_notrans   t value       Pr(>t)      Lower
+## Z0_0       97.0148812 3.55314944 27.303912 1.679369e-21 91.3287912
+## k_Z0        2.2360064 0.21684747 10.311425 3.661846e-11  1.8052909
+## k_Z1        0.4821248 0.06585372  7.321147 3.552045e-08  0.3996826
+## f_Z0_to_Z1  1.0000000 0.10147342  9.854798 9.707056e-11  0.0000000
+##                  Upper
+## Z0_0       102.7009713
+## k_Z0         2.7694843
+## k_Z1         0.5815722
+## f_Z0_to_Z1   1.0000000
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.3} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+               \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),} \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.3} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.3, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.3)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_3-1} 
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.3,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+##        Estimate se_notrans  t value       Pr(>t)      Lower
+## Z0_0 97.0148815 2.68177093 36.17568 2.363587e-25 91.5215237
+## k_Z0  2.2360064 0.14686110 15.22531 2.246514e-15  1.9545310
+## k_Z1  0.4821248 0.04268712 11.29439 3.068581e-12  0.4021552
+##            Upper
+## Z0_0 102.5082392
+## k_Z0   2.5580177
+## k_Z1   0.5779966
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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. 
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.5} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+               \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+               \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),} \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.5} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.5, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.5)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_5-1} 
+
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.FOCUS} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                   \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                   \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+                   \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{),}
+                   \hlkwc{use_of_ff} \hlstd{=} \hlstr{"max"}\hlstd{)}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.FOCUS} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.FOCUS, FOCUS_2006_Z_mkin,}
+                     \hlkwc{parms.ini} \hlstd{= m.Z.5}\hlopt{$}\hlstd{bparms.ode,}
+                     \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.FOCUS)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_6-1} 
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.FOCUS,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{bpar}
+\end{alltt}
+\begin{verbatim}
+##               Estimate se_notrans   t value       Pr(>t)       Lower
+## Z0_0       96.83849566 2.05886133 47.034977 5.583742e-44 92.70517559
+## k_Z0        2.21540897 0.11813046 18.753919 7.744034e-25  1.99051134
+## k_Z1        0.47829896 0.02928763 16.331089 3.333376e-22  0.42297603
+## k_Z2        0.45161653 0.04421268 10.214638 3.110936e-14  0.37103495
+## k_Z3        0.05869343 0.01429485  4.105914 7.285393e-05  0.03599566
+## f_Z2_to_Z3  0.47150952 0.05705233  8.264510 2.809889e-11  0.36038193
+##                   Upper
+## Z0_0       100.97181573
+## k_Z0         2.46571661
+## k_Z1         0.54085782
+## k_Z2         0.54969886
+## k_Z3         0.09570371
+## f_Z2_to_Z3   0.58553462
+\end{verbatim}
+\begin{alltt}
+\hlkwd{endpoints}\hlstd{(m.Z.FOCUS)}
+\end{alltt}
+\begin{verbatim}
+## $ff
+##     Z2_Z3   Z2_sink 
+## 0.4715095 0.5284905 
+## 
+## $SFORB
+## logical(0)
+## 
+## $distimes
+##          DT50      DT90
+## Z0  0.3128755  1.039350
+## Z1  1.4491923  4.814113
+## Z2  1.5348136  5.098540
+## Z3 11.8096217 39.230714
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.1} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+                    \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.1} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.1, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.1)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_7-1} 
+\begin{kframe}\begin{alltt}
+\hlkwd{summary}\hlstd{(m.Z.mkin.1,} \hlkwc{data} \hlstd{=} \hlnum{FALSE}\hlstd{)}\hlopt{$}\hlstd{cov.unscaled}
+\end{alltt}
+\begin{verbatim}
+## NULL
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.3} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.3} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.3, FOCUS_2006_Z_mkin,} \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.3)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_9-1} 
+
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.4} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+                    \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.4} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.4, FOCUS_2006_Z_mkin,}
+                      \hlkwc{parms.ini} \hlstd{= m.Z.mkin.3}\hlopt{$}\hlstd{bparms.ode,}
+                      \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.4)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_10-1} 
+
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{Z.mkin.5} \hlkwb{<-} \hlkwd{mkinmod}\hlstd{(}\hlkwc{Z0} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{,} \hlstr{"Z1"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z1} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z2"}\hlstd{,} \hlkwc{sink} \hlstd{=} \hlnum{FALSE}\hlstd{),}
+                    \hlkwc{Z2} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFO"}\hlstd{,} \hlstr{"Z3"}\hlstd{),}
+                    \hlkwc{Z3} \hlstd{=} \hlkwd{mkinsub}\hlstd{(}\hlstr{"SFORB"}\hlstd{))}
+\end{alltt}
+\begin{verbatim}
+## make[1]: warning: jobserver unavailable: using -j1.  Add '+' to parent make rule.
+\end{verbatim}
+
+
+{\ttfamily\noindent\itshape\color{messagecolor}{\#\# Successfully compiled differential equation model from auto-generated C code.}}\begin{alltt}
+\hlstd{m.Z.mkin.5} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.5, FOCUS_2006_Z_mkin,}
+                      \hlkwc{parms.ini} \hlstd{= m.Z.mkin.4}\hlopt{$}\hlstd{bparms.ode[}\hlnum{1}\hlopt{:}\hlnum{4}\hlstd{],}
+                      \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.5)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11-1} 
+
+\end{knitrout}
+
+The summary view of the backtransformed parameters shows that we get no
+confidence intervals due to overparameterisation. As the optimized
+\texttt{k\_Z3\_bound\_free} is excessively small, it seems reasonable to fix it to
+zero. 
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlstd{m.Z.mkin.5a} \hlkwb{<-} \hlkwd{mkinfit}\hlstd{(Z.mkin.5, FOCUS_2006_Z_mkin,}
+                       \hlkwc{parms.ini} \hlstd{=} \hlkwd{c}\hlstd{(m.Z.mkin.5}\hlopt{$}\hlstd{bparms.ode[}\hlnum{1}\hlopt{:}\hlnum{7}\hlstd{],}
+                                     \hlkwc{k_Z3_bound_free} \hlstd{=} \hlnum{0}\hlstd{),}
+                       \hlkwc{fixed_parms} \hlstd{=} \hlstr{"k_Z3_bound_free"}\hlstd{,}
+                       \hlkwc{quiet} \hlstd{=} \hlnum{TRUE}\hlstd{)}
+\hlkwd{plot_sep}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11a-1} 
+
+\end{knitrout}
+
+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.
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{mkinparplot}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\end{kframe}
+\includegraphics[width=\maxwidth]{figure/FOCUS_2006_Z_fits_11b-1} 
+
+\end{knitrout}
+
+The endpoints obtained with this model are
+
+\begin{knitrout}
+\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe}
+\begin{alltt}
+\hlkwd{endpoints}\hlstd{(m.Z.mkin.5a)}
+\end{alltt}
+\begin{verbatim}
+## $ff
+##   Z0_free_Z1        Z1_Z2      Z2_sink   Z2_Z3_free Z3_free_sink 
+##    1.0000000    1.0000000    0.4634431    0.5365569    1.0000000 
+## 
+## $SFORB
+##       Z0_b1       Z0_b2       Z3_b1       Z3_b2 
+## 2.447138640 0.007512589 0.080007099 0.000000000 
+## 
+## $distimes
+##         DT50     DT90 DT50_Z0_b1 DT50_Z0_b2 DT50_Z3_b1 DT50_Z3_b2
+## Z0 0.3042972 1.184811   0.283248   92.26476         NA         NA
+## Z1 1.5147787 5.031986         NA         NA         NA         NA
+## Z2 1.6413857 5.452565         NA         NA         NA         NA
+## Z3        NA       NA         NA         NA   8.663571        Inf
+\end{verbatim}
+\end{kframe}
+\end{knitrout}
+
+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:
-- 
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