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<img src="" class="logo" alt=""><h1>Calculation of time weighted average concentrations with mkin</h1> + + <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 2023-11-16)</h4> +(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> @@ -118,48 +104,44 @@ 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><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><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><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><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left( +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = \frac{c_0}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + - \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p> -<p>HS for <span class="math inline">\(t > t_b\)</span>:</p> -<p><span class="math display">\[c_\textrm{twa} = \frac{c_0}{t} \left( + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) </annotation></semantics></math></p> +<p>HS for +<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>></mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t > t_b</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><msub><mi>c</mi><mn>0</mn></msub><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">c_\textrm{twa} = \frac{c_0}{t} \left( \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) + - \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} -\right) \right) \]</span></p> + \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 -<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> +<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 <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> +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><span class="math display">\[f_\textrm{twa} = \frac{\left( 1 - e^{- k -t} \right)}{k t} \]</span></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><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><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><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left( +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mi>g</mi><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>g</mi></mrow><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{1}{t} \left( \frac{g}{k_1} \left( 1 - e^{- k_1 t} \right) + - \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) \]</span></p> -<p>HS for <span class="math inline">\(t > t_b\)</span>:</p> -<p><span class="math display">\[f_\textrm{twa} = \frac{1}{t} \left( + \frac{1-g}{k_2} \left( 1 - e^{- k_2 t} \right) \right) </annotation></semantics></math></p> +<p>HS for +<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>></mo><msub><mi>t</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">t > t_b</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mtext mathvariant="normal">twa</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mfrac><mn>1</mn><msub><mi>k</mi><mn>1</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo>+</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><msub><mi>t</mi><mi>b</mi></msub></mrow></msup><msub><mi>k</mi><mn>2</mn></msub></mfrac><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mrow><mo stretchy="true" form="prefix">(</mo><mi>t</mi><mo>−</mo><msub><mi>t</mi><mi>b</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow></mrow></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="true" form="postfix">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_\textrm{twa} = \frac{1}{t} \left( \frac{1}{k_1} \left( 1 - e^{- k_1 t_b} \right) + - \frac{e^{- k_1 t_b}}{k_2} \left( 1 - e^{- k_2 (t - t_b)} -\right) \right) \]</span></p> + \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 @@ -173,28 +155,26 @@ 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>. </div> </div> - </main><aside class="col-md-3"><nav id="toc"><h2>On this page</h2> + </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2> </nav></aside> </div> <footer><div class="pkgdown-footer-left"> - <p></p> -<p>Developed by Johannes Ranke.</p> + <p>Developed by Johannes Ranke.</p> </div> <div class="pkgdown-footer-right"> - 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