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<h1 data-toc-skip>Short introduction to mkin</h1>
<h4 data-toc-skip class="author">Johannes
Ranke</h4>
<h4 data-toc-skip class="date">Last change 18 May 2023
(rebuilt 2023-05-19)</h4>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/mkin.rmd" class="external-link"><code>vignettes/mkin.rmd</code></a></small>
<div class="hidden name"><code>mkin.rmd</code></div>
</div>
<p><a href="https://www.jrwb.de" class="external-link">Wissenschaftlicher Berater, Kronacher
Str. 12, 79639 Grenzach-Wyhlen, Germany</a><br> Privatdozent at the
University of Freiburg</p>
<div class="section level2">
<h2 id="abstract">Abstract<a class="anchor" aria-label="anchor" href="#abstract"></a>
</h2>
<p>In the regulatory evaluation of chemical substances like plant
protection products (pesticides), biocides and other chemicals,
degradation data play an important role. For the evaluation of pesticide
degradation experiments, detailed guidance has been developed, based on
nonlinear optimisation. The <code>R</code> add-on package
<code>mkin</code> implements fitting some of the models recommended in
this guidance from within R and calculates some statistical measures for
data series within one or more compartments, for parent and
metabolites.</p>
<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="st"><a href="https://pkgdown.jrwb.de/mkin/">"mkin"</a></span>, quietly <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span><span class="co"># Define the kinetic model</span></span>
<span><span class="va">m_SFO_SFO_SFO</span> <span class="op"><-</span> <span class="fu"><a href="../reference/mkinmod.html">mkinmod</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M1"</span><span class="op">)</span>,</span>
<span> M1 <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"M2"</span><span class="op">)</span>,</span>
<span> M2 <span class="op">=</span> <span class="fu"><a href="../reference/mkinmod.html">mkinsub</a></span><span class="op">(</span><span class="st">"SFO"</span><span class="op">)</span>,</span>
<span> use_of_ff <span class="op">=</span> <span class="st">"max"</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span></span>
<span></span>
<span><span class="co"># Produce model predictions using some arbitrary parameters</span></span>
<span><span class="va">sampling_times</span> <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">3</span>, <span class="fl">7</span>, <span class="fl">14</span>, <span class="fl">28</span>, <span class="fl">60</span>, <span class="fl">90</span>, <span class="fl">120</span><span class="op">)</span></span>
<span><span class="va">d_SFO_SFO_SFO</span> <span class="op"><-</span> <span class="fu"><a href="../reference/mkinpredict.html">mkinpredict</a></span><span class="op">(</span><span class="va">m_SFO_SFO_SFO</span>,</span>
<span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>k_parent <span class="op">=</span> <span class="fl">0.03</span>,</span>
<span> f_parent_to_M1 <span class="op">=</span> <span class="fl">0.5</span>, k_M1 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html" class="external-link">log</a></span><span class="op">(</span><span class="fl">2</span><span class="op">)</span><span class="op">/</span><span class="fl">100</span>,</span>
<span> f_M1_to_M2 <span class="op">=</span> <span class="fl">0.9</span>, k_M2 <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html" class="external-link">log</a></span><span class="op">(</span><span class="fl">2</span><span class="op">)</span><span class="op">/</span><span class="fl">50</span><span class="op">)</span>,</span>
<span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span>parent <span class="op">=</span> <span class="fl">100</span>, M1 <span class="op">=</span> <span class="fl">0</span>, M2 <span class="op">=</span> <span class="fl">0</span><span class="op">)</span>,</span>
<span> <span class="va">sampling_times</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Generate a dataset by adding normally distributed errors with</span></span>
<span><span class="co"># standard deviation 3, for two replicates at each sampling time</span></span>
<span><span class="va">d_SFO_SFO_SFO_err</span> <span class="op"><-</span> <span class="fu"><a href="../reference/add_err.html">add_err</a></span><span class="op">(</span><span class="va">d_SFO_SFO_SFO</span>, reps <span class="op">=</span> <span class="fl">2</span>,</span>
<span> sdfunc <span class="op">=</span> <span class="kw">function</span><span class="op">(</span><span class="va">x</span><span class="op">)</span> <span class="fl">3</span>,</span>
<span> n <span class="op">=</span> <span class="fl">1</span>, seed <span class="op">=</span> <span class="fl">123456789</span> <span class="op">)</span></span>
<span></span>
<span><span class="co"># Fit the model to the dataset</span></span>
<span><span class="va">f_SFO_SFO_SFO</span> <span class="op"><-</span> <span class="fu"><a href="../reference/mkinfit.html">mkinfit</a></span><span class="op">(</span><span class="va">m_SFO_SFO_SFO</span>, <span class="va">d_SFO_SFO_SFO_err</span><span class="op">[[</span><span class="fl">1</span><span class="op">]</span><span class="op">]</span>, quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Plot the results separately for parent and metabolites</span></span>
<span><span class="fu"><a href="../reference/plot.mkinfit.html">plot_sep</a></span><span class="op">(</span><span class="va">f_SFO_SFO_SFO</span>, lpos <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"topright"</span>, <span class="st">"bottomright"</span>, <span class="st">"bottomright"</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<p><img src="mkin_files/figure-html/unnamed-chunk-2-1.png" width="768"></p>
</div>
<div class="section level2">
<h2 id="background">Background<a class="anchor" aria-label="anchor" href="#background"></a>
</h2>
<p>The <code>mkin</code> package <span class="citation">(J. Ranke
2021)</span> implements the approach to degradation kinetics recommended
in the kinetics report provided by the FOrum for Co-ordination of
pesticide fate models and their USe <span class="citation">(FOCUS Work
Group on Degradation Kinetics 2006, 2014)</span>. It covers data series
describing the decline of one compound, data series with transformation
products (commonly termed metabolites) and data series for more than one
compartment. It is possible to include back reactions. Therefore,
equilibrium reactions and equilibrium partitioning can be specified,
although this often leads to an overparameterisation of the model.</p>
<p>When the first <code>mkin</code> code was published in 2010, the most
commonly used tools for fitting more complex kinetic degradation models
to experimental data were KinGUI <span class="citation">(Schäfer et al.
2007)</span>, a MATLAB based tool with a graphical user interface that
was specifically tailored to the task and included some output as
proposed by the FOCUS Kinetics Workgroup, and ModelMaker, a general
purpose compartment based tool providing infrastructure for fitting
dynamic simulation models based on differential equations to data.</p>
<p>The ‘mkin’ code was first uploaded to the BerliOS development
platform. When this was taken down, the version control history was
imported into the R-Forge site (see <em>e.g.</em> <a href="https://cgit.jrwb.de/mkin/commit/?id=30cbb4092f6d2d3beff5800603374a0d009ad770" class="external-link">the
initial commit on 11 May 2010</a>), where the code is still being
updated.</p>
<p>At that time, the R package <code>FME</code> (Flexible Modelling
Environment) <span class="citation">(Soetaert and Petzoldt 2010)</span>
was already available, and provided a good basis for developing a
package specifically tailored to the task. The remaining challenge was
to make it as easy as possible for the users (including the author of
this vignette) to specify the system of differential equations and to
include the output requested by the FOCUS guidance, such as the <span class="math inline">\(\chi^2\)</span> error level as defined in this
guidance.</p>
<p>Also, <code>mkin</code> introduced using analytical solutions for
parent only kinetics for improved optimization speed. Later, Eigenvalue
based solutions were introduced to <code>mkin</code> for the case of
linear differential equations (<em>i.e.</em> where the FOMC or DFOP
models were not used for the parent compound), greatly improving the
optimization speed for these cases. This, has become somehow obsolete,
as the use of compiled code described below gives even faster execution
times.</p>
<p>The possibility to specify back-reactions and a biphasic model
(SFORB) for metabolites were present in <code>mkin</code> from the very
beginning.</p>
<div class="section level3">
<h3 id="derived-software-tools">Derived software tools<a class="anchor" aria-label="anchor" href="#derived-software-tools"></a>
</h3>
<p>Soon after the publication of <code>mkin</code>, two derived tools
were published, namely KinGUII (developed at Bayer Crop Science) and
CAKE (commissioned to Tessella by Syngenta), which added a graphical
user interface (GUI), and added fitting by iteratively reweighted least
squares (IRLS) and characterisation of likely parameter distributions by
Markov Chain Monte Carlo (MCMC) sampling.</p>
<p>CAKE focuses on a smooth use experience, sacrificing some flexibility
in the model definition, originally allowing only two primary
metabolites in parallel. The current version 3.4 of CAKE released in May
2020 uses a scheme for up to six metabolites in a flexible arrangement
and supports biphasic modelling of metabolites, but does not support
back-reactions (non-instantaneous equilibria).</p>
<p>KinGUI offers an even more flexible widget for specifying complex
kinetic models. Back-reactions (non-instantaneous equilibria) were
supported early on, but until 2014, only simple first-order models could
be specified for transformation products. Starting with KinGUII version
2.1, biphasic modelling of metabolites was also available in
KinGUII.</p>
<p>A further graphical user interface (GUI) that has recently been
brought to a decent degree of maturity is the browser based GUI named
<code>gmkin</code>. Please see its <a href="https://pkgdown.jrwb.de/gmkin/" class="external-link">documentation page</a> and <a href="https://pkgdown.jrwb.de/gmkin/articles/gmkin_manual.html" class="external-link">manual</a>
for further information.</p>
<p>A comparison of scope, usability and numerical results obtained with
these tools has been recently been published by <span class="citation">Johannes Ranke, Wöltjen, and Meinecke
(2018)</span>.</p>
</div>
</div>
<div class="section level2">
<h2 id="unique-features">Unique features<a class="anchor" aria-label="anchor" href="#unique-features"></a>
</h2>
<p>Currently, the main unique features available in <code>mkin</code>
are</p>
<ul>
<li>the <a href="https://pkgdown.jrwb.de/mkin/articles/web_only/compiled_models.html">speed
increase</a> by using compiled code when a compiler is present,</li>
<li>parallel model fitting on multicore machines using the <a href="https://pkgdown.jrwb.de/mkin/reference/mmkin.html"><code>mmkin</code>
function</a>,</li>
<li>the estimation of parameter confidence intervals based on
transformed parameters (see below) and</li>
<li>the possibility to use the <a href="https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html">two-component
error model</a>
</li>
</ul>
<p>The iteratively reweighted least squares fitting of different
variances for each variable as introduced by <span class="citation">Gao
et al. (2011)</span> has been available in mkin since <a href="https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-22-2013-10-26">version
0.9-22</a>. With <a href="https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-49-5-2019-07-04">release
0.9.49.5</a>, the IRLS algorithm has been complemented by direct or
step-wise maximisation of the likelihood function, which makes it
possible not only to fit the variance by variable error model but also a
<a href="https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html">two-component
error model</a> inspired by error models developed in analytical
chemistry <span class="citation">(Johannes Ranke and Meinecke
2019)</span>.</p>
</div>
<div class="section level2">
<h2 id="internal-parameter-transformations">Internal parameter transformations<a class="anchor" aria-label="anchor" href="#internal-parameter-transformations"></a>
</h2>
<p>For rate constants, the log transformation is used, as proposed by
Bates and Watts <span class="citation">(1988, 77, 149)</span>.
Approximate intervals are constructed for the transformed rate constants
<span class="citation">(compare Bates and Watts 1988, 135)</span>,
<em>i.e.</em> for their logarithms. Confidence intervals for the rate
constants are then obtained using the appropriate backtransformation
using the exponential function.</p>
<p>In the first version of <code>mkin</code> allowing for specifying
models using formation fractions, a home-made reparameterisation was
used in order to ensure that the sum of formation fractions would not
exceed unity.</p>
<p>This method is still used in the current version of KinGUII (v2.1
from April 2014), with a modification that allows for fixing the pathway
to sink to zero. CAKE uses penalties in the objective function in order
to enforce this constraint.</p>
<p>In 2012, an alternative reparameterisation of the formation fractions
was proposed together with René Lehmann <span class="citation">(J. Ranke
and Lehmann 2012)</span>, based on isometric logratio transformation
(ILR). The aim was to improve the validity of the linear approximation
of the objective function during the parameter estimation procedure as
well as in the subsequent calculation of parameter confidence intervals.
In the current version of mkin, a logit transformation is used for
parameters that are bound between 0 and 1, such as the g parameter of
the DFOP model.</p>
<div class="section level3">
<h3 id="confidence-intervals-based-on-transformed-parameters">Confidence intervals based on transformed parameters<a class="anchor" aria-label="anchor" href="#confidence-intervals-based-on-transformed-parameters"></a>
</h3>
<p>In the first attempt at providing improved parameter confidence
intervals introduced to <code>mkin</code> in 2013, confidence intervals
obtained from FME on the transformed parameters were simply all
backtransformed one by one to yield asymmetric confidence intervals for
the backtransformed parameters.</p>
<p>However, while there is a 1:1 relation between the rate constants in
the model and the transformed parameters fitted in the model, the
parameters obtained by the isometric logratio transformation are
calculated from the set of formation fractions that quantify the paths
to each of the compounds formed from a specific parent compound, and no
such 1:1 relation exists.</p>
<p>Therefore, parameter confidence intervals for formation fractions
obtained with this method only appear valid for the case of a single
transformation product, where currently the logit transformation is used
for the formation fraction.</p>
<p>The confidence intervals obtained by backtransformation for the cases
where a 1:1 relation between transformed and original parameter exist
are considered by the author of this vignette to be more accurate than
those obtained using a re-estimation of the Hessian matrix after
backtransformation, as implemented in the FME package.</p>
</div>
<div class="section level3">
<h3 id="parameter-t-test-based-on-untransformed-parameters">Parameter t-test based on untransformed parameters<a class="anchor" aria-label="anchor" href="#parameter-t-test-based-on-untransformed-parameters"></a>
</h3>
<p>The standard output of many nonlinear regression software packages
includes the results from a test for significant difference from zero
for all parameters. Such a test is also recommended to check the
validity of rate constants in the FOCUS guidance <span class="citation">(FOCUS Work Group on Degradation Kinetics 2014,
96ff)</span>.</p>
<p>It has been argued that the precondition for this test, <em>i.e.</em>
normal distribution of the estimator for the parameters, is not
fulfilled in the case of nonlinear regression <span class="citation">(J.
Ranke and Lehmann 2015)</span>. However, this test is commonly used by
industry, consultants and national authorities in order to decide on the
reliability of parameter estimates, based on the FOCUS guidance
mentioned above. Therefore, the results of this one-sided t-test are
included in the summary output from <code>mkin</code>.</p>
<p>As it is not reasonable to test for significant difference of the
transformed parameters (<em>e.g.</em> <span class="math inline">\(log(k)\)</span>) from zero, the t-test is
calculated based on the model definition before parameter
transformation, <em>i.e.</em> in a similar way as in packages that do
not apply such an internal parameter transformation. A note is included
in the <code>mkin</code> output, pointing to the fact that the t-test is
based on the unjustified assumption of normal distribution of the
parameter estimators.</p>
</div>
</div>
<div class="section level2">
<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
</h2>
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Ranke, J., and R. Lehmann. 2012. <span>“Parameter Reliability in Kinetic
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</div>
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