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<body>
<h1 class="title toc-ignore">Calculation of time weighted average
concentrations with mkin</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">Last change 18 September 2019 (rebuilt 2023-01-05)</h4>
<p>Since version 0.9.45.1 of the ‘mkin’ package, a function for
calculating time weighted average concentrations for decline kinetics
(<em>i.e.</em> only for the compound applied in the experiment) is
included. Strictly speaking, they are maximum moving window time
weighted average concentrations, <em>i.e.</em> the maximum time weighted
average concentration that can be found when moving a time window of a
specified width over the decline curve.</p>
<p>Time weighted average concentrations for the SFO, FOMC and the DFOP
model are calculated using the formulas given in the FOCUS kinetics
guidance <span class="citation">(FOCUS Work Group on Degradation
Kinetics 2014, 251)</span>:</p>
<p>SFO:</p>
<p><span class="math display">\[c_\textrm{twa} = c_0 \frac{\left( 1 -
e^{- k t} \right)}{ k t} \]</span></p>
<p>FOMC:</p>
<p><span class="math display">\[c_\textrm{twa} = c_0 \frac{\beta}{t (1 -
\alpha)}
\left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha}
- 1 \right) \]</span></p>
<p>DFOP:</p>
<p><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left(
\frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
\frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p>
<p>HS for <span class="math inline">\(t > t_b\)</span>:</p>
<p><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left(
\frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) +
\frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)}
\right) \right) \]</span></p>
<p>Often, the ratio between the time weighted average concentration
<span class="math inline">\(c_\textrm{twa}\)</span> and the initial
concentration <span class="math inline">\(c_0\)</span></p>
<p><span class="math display">\[f_\textrm{twa} =
\frac{c_\textrm{twa}}{c_0}\]</span></p>
<p>is needed. This can be calculated from the fitted initial
concentration <span class="math inline">\(c_0\)</span> and the time
weighted average concentration <span class="math inline">\(c_\textrm{twa}\)</span>, or directly from the
model parameters using the following formulas:</p>
<p>SFO:</p>
<p><span class="math display">\[f_\textrm{twa} = \frac{\left( 1 - e^{- k
t} \right)}{k t} \]</span></p>
<p>FOMC:</p>
<p><span class="math display">\[f_\textrm{twa} = \frac{\beta}{t (1 -
\alpha)}
\left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha}
- 1 \right) \]</span></p>
<p>DFOP:</p>
<p><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left(
\frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) +
\frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p>
<p>HS for <span class="math inline">\(t > t_b\)</span>:</p>
<p><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left(
\frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) +
\frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)}
\right) \right) \]</span></p>
<p>Note that a method for calculating maximum moving window time
weighted average concentrations for a model fitted by ‘mkinfit’ or from
parent decline model parameters is included in the
<code>max_twa_parent()</code> function. If the same is needed for
metabolites, the function <code>pfm::max_twa()</code> from the ‘pfm’
package can be used.</p>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-FOCUSkinetics2014" class="csl-entry">
FOCUS Work Group on Degradation Kinetics. 2014. <em>Generic Guidance for
Estimating Persistence and Degradation Kinetics from Environmental Fate
Studies on Pesticides in EU Registration</em>. 1.1 ed. <a href="http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
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