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<h1>Calculation of time weighted average concentrations with mkin</h1>
<h4 data-toc-skip class="author">Johannes
Ranke</h4>
<h4 data-toc-skip class="date">Last change 18 September 2019
(rebuilt 2025-02-13)</h4>
<small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/twa.rmd" class="external-link"><code>vignettes/twa.rmd</code></a></small>
<div class="d-none name"><code>twa.rmd</code></div>
</div>
<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><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>k</mi><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mi>k</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = c_0 \frac{\left( 1 - e^{- k t} \right)}{ k t} </annotation></semantics></math></p>
<p>FOMC:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mfrac><mi>β</mi><mrow><mi>t</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><msup><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>t</mi><mi>β</mi></mfrac><mo>+</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>−</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = c_0 \frac{\beta}{t (1 - \alpha)}
\left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) </annotation></semantics></math></p>
<p>DFOP:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">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) </annotation></semantics></math></p>
<p>HS for
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>></mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t > t_b</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">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) </annotation></semantics></math></p>
<p>Often, the ratio between the time weighted average concentration
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><annotation encoding="application/x-tex">c_\textrm{twa}</annotation></semantics></math>
and the initial concentration
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mn>0</mn></msub><annotation encoding="application/x-tex">c_0</annotation></semantics></math></p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><msub><mi>c</mi><mn>0</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{c_\textrm{twa}}{c_0}</annotation></semantics></math></p>
<p>is needed. This can be calculated from the fitted initial
concentration
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mn>0</mn></msub><annotation encoding="application/x-tex">c_0</annotation></semantics></math>
and the time weighted average concentration
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><annotation encoding="application/x-tex">c_\textrm{twa}</annotation></semantics></math>,
or directly from the model parameters using the following formulas:</p>
<p>SFO:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>k</mi><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mi>k</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{\left( 1 - e^{- k t} \right)}{k t} </annotation></semantics></math></p>
<p>FOMC:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mi>β</mi><mrow><mi>t</mi><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="true" form="postfix">)</mo></mrow></mrow></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><msup><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>t</mi><mi>β</mi></mfrac><mo>+</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>−</mo><mn>1</mn><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{\beta}{t (1 - \alpha)}
\left( \left(\frac{t}{\beta} + 1 \right)^{1 - \alpha} - 1 \right) </annotation></semantics></math></p>
<p>DFOP:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">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) </annotation></semantics></math></p>
<p>HS for
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>></mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t > t_b</annotation></semantics></math>:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">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) </annotation></semantics></math></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><a href="../reference/max_twa_parent.html">max_twa_parent()</a></code> function. If the same is needed for
metabolites, the function <code><a href="https://pkgdown.jrwb.de/pfm/reference/max_twa.html" class="external-link">pfm::max_twa()</a></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" class="external-link">http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a>.
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