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<body>




<h1 class="title toc-ignore">Library of models</h1>


<div id="TOC">
<ul>
<li><a href="#pharmacokinetic-models" id="toc-pharmacokinetic-models">Pharmacokinetic models</a>
<ul>
<li><a href="#compartmental-models-and-parameters" id="toc-compartmental-models-and-parameters">Compartmental models and
parameters</a></li>
<li><a href="#one-compartment-models" id="toc-one-compartment-models">One-compartment models</a></li>
<li><a href="#models-with-linear-elimination" id="toc-models-with-linear-elimination">Models with linear
elimination</a>
<ul>
<li><a href="#one-compartment-models-1" id="toc-one-compartment-models-1">One-compartment models</a></li>
<li><a href="#two-compartment-models" id="toc-two-compartment-models">Two-compartment models</a></li>
</ul></li>
<li><a href="#models-with-michaelis-menten-elimination" id="toc-models-with-michaelis-menten-elimination">Models with
Michaelis-Menten elimination</a>
<ul>
<li><a href="#one-compartment-models-2" id="toc-one-compartment-models-2">One-compartment models</a></li>
<li><a href="#two-compartment-models-1" id="toc-two-compartment-models-1">Two-compartment models</a></li>
</ul></li>
</ul></li>
<li><a href="#pharmacodynamic-models" id="toc-pharmacodynamic-models">Pharmacodynamic models</a>
<ul>
<li><a href="#immediate-response-models" id="toc-immediate-response-models">Immediate response models</a>
<ul>
<li><a href="#drugactionmodels" id="toc-drugactionmodels">Drug action
models</a></li>
<li><a href="#baselinediseasemodel" id="toc-baselinediseasemodel">Baseline/disease models</a></li>
</ul></li>
<li><a href="#turnover-response-models" id="toc-turnover-response-models">Turnover response models</a>
<ul>
<li><a href="#models-with-impact-on-the-input-r_in" id="toc-models-with-impact-on-the-input-r_in">Models with impact on the
input <span class="math inline">\((R_{in})\)</span></a></li>
<li><a href="#models-with-impact-on-the-output-k_out" id="toc-models-with-impact-on-the-output-k_out">Models with impact on
the output <span class="math inline">\((k_{out})\)</span></a></li>
</ul></li>
</ul></li>
</ul>
</div>

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<div id="pharmacokinetic-models" class="section level1">
<h1>Pharmacokinetic models</h1>
<div id="compartmental-models-and-parameters" class="section level2">
<h2>Compartmental models and parameters</h2>
<!-- In the ensuing section, the mammillary models with one, two or three compartments are presented with the associated parameters and the different parameterisations.  -->
<p>Six parameters are common to one, two or three compartment
models:</p>
<ul>
<li><span class="math inline">\(V\)</span> or <span class="math inline">\(V_1\)</span>, the volume of distribution in the
central compartment</li>
<li><span class="math inline">\(k\)</span>, the elimination rate
constant</li>
<li><span class="math inline">\(CL\)</span>, the clearance of
elimination</li>
<li><span class="math inline">\(V_m\)</span>, the maximum elimination
rate for Michaelis-Menten elimination</li>
<li><span class="math inline">\(K_m\)</span>, the Michaelis-Menten
constant</li>
<li><span class="math inline">\(k_a\)</span>, the absorption rate
constant for oral administration</li>
</ul>
</div>
<div id="one-compartment-models" class="section level2">
<h2>One-compartment models</h2>
<p>There are two parameterisations implemented in PFIM for
one-compartment models, <span class="math inline">\(\left(V\text{ and
}k\right)\)</span> or <span class="math inline">\(\left(V\text{ and
}CL\right)\)</span>. The equations are given for the first
parameterisation <span class="math inline">\(\left(V, k\right)\)</span>.
For extra-vascular administration, <span class="math inline">\(V\)</span> and <span class="math inline">\(CL\)</span> are apparent volume and clearance. The
equations for the second parameterisation <span class="math inline">\(\left(V, CL\right)\)</span> are derived using
<span class="math inline">\(k={\frac{CL}{V}}\)</span>.</p>
</div>
<div id="models-with-linear-elimination" class="section level2">
<h2>Models with linear elimination</h2>
<div id="one-compartment-models-1" class="section level3">
<h3>One-compartment models</h3>
<div id="intravenous-bolus" class="section level4">
<h4>Intravenous bolus</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[\begin {equation}
\begin{aligned}
C\left(t\right)=\frac{D}{V}e^{-k\left(t-t_{D}\right)}
\end{aligned}   
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
\begin{aligned}
&amp;
C\left(t\right)=\sum^{n}_{i=1}\frac{D_{i}}{V}e^{-k\left(t-t_{D_{i}}\right)}\\
&amp;
\end{aligned}   
\end {equation}\]</span></p>
<ul>
<li>steady state</li>
</ul>
<p><span class="math display">\[\begin {equation}
C(t)=\frac{D}{V}\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}\\        
\end {equation}\]</span></p>
</div>
<div id="infusion" class="section level4">
<h4>Infusion</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[\begin{equation}
C\left(t\right)=
\begin{cases}
{\frac{D}{Tinf}\frac{1}{kV}\left(1-e^{-k\left(t-t_{D}\right)}\right)}
&amp; \text{if $t-t_{D}\leq Tinf$,}\\[0.5cm]
{\frac{D}{Tinf}\frac{1}{kV}\left(1-e^{-kTinf}\right)e^{-k\left(t-t_{D}-Tinf\right)}}
&amp; \text{if not.}\\
\end{cases}\\    
\end{equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin{equation}
C\left(t\right)=
\begin{cases}
\begin{aligned}
\sum^{n-1}_{i=1}\frac{D_{i}}{Tinf_{i}}
\frac{1}{kV}
&amp;\left(1-e^{-kTinf_{i}}\right)
e^{-k\left(t-t_{D_{i}}-Tinf_i\right)}\\
&amp;+\frac{D_{n}}{Tinf_{n}}
\frac{1}{kV}
\left(1-e^{-k\left(t-t_{D_{n}}\right)}\right)
\end{aligned}
&amp; \text{if $t-t_{D_{n}} \leq Tinf_{n}$,}\\[1cm]
{\displaystyle\sum^{n}_{i=1}\frac{D_{i}}{Tinf_{i}}
\frac{1}{kV}}
\left(1-e^{-kTinf_{i}}\right)
e^{-k\left(t-t_{D_{i}}-Tinf_i\right)}                                                   &amp;
\text{if not.}\\
\end{cases}
\end{equation}
\]</span></p>
<ul>
<li>steady state</li>
</ul>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
&amp;
C\left(t\right)=    
\begin{cases}
{\frac{D}{Tinf}
\frac{1}{kV}}
\left[
\left(1-e^{-k(t-t_D)}\right)
+e^{-k\tau}
{\frac{\left(1-e^{-kTinf}\right)e^{-k\left(t-t_D-Tinf\right)}}{1-e^{-k\tau}}}
\right]                                                                                                                                                             &amp;\text{if
$(t-t_D)\leq Tinf$,}\\[0.6cm]
{\frac{D}{Tinf}
\frac{1}{kV}
\frac{\left(1-e^{-kTinf}\right)e^{-k\left(t-t_D-Tinf\right)}}{1-e^{-k\tau}}}                            &amp;\text{if
not.}\\
\end{cases}\\
&amp;
\end{aligned}
\end{equation}\]</span></p>
</div>
<div id="first-order-absorption" class="section level4">
<h4>First order absorption</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=\frac{D}{V}
\frac{k_{a}}{k_{a}-k}
\left(e^{-k\left(t-t_{D}\right)}-e^{-k_{a}\left(t-t_{D}\right)}\right)
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=\sum^{n}_{i=1}\frac{D_{i}}{V}
\frac{k_{a}}{k_{a}-k}
\left(e^{-k\left(t-t_{D_{i}}\right)}-e^{-k_{a}\left(t-t_{D_{i}}\right)}\right)
\end {equation} \]</span></p>
<ul>
<li>steady state</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=\frac{D}{V}
\frac{k_{a}}{k_{a}-k}
\left(\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}-\frac{e^{-k_{a}(t-t_D)}}{1-e^{-k_a\tau}}\right)
\end {equation}\]</span></p>
</div>
</div>
<div id="two-compartment-models" class="section level3">
<h3>Two-compartment models</h3>
<p>For two-compartment model equations, <span class="math inline">\(C(t)=C_1(t)\)</span> represent the drug
concentration in the first compartment and <span class="math inline">\(C_2(t)\)</span> represents the drug concentration
in the second compartment.</p>
<p>As well as the previously described PK parameters, the following PK
parameters are used for the two-compartment models:</p>
<ul>
<li><span class="math inline">\(V_2\)</span>, the volume of distribution
of second compartment</li>
<li><span class="math inline">\(k_{12}\)</span>, the distribution rate
constant from compartment 1 to compartment 2</li>
<li><span class="math inline">\(k_{21}\)</span>, the distribution rate
constant from compartment 2 to compartment 1</li>
<li><span class="math inline">\(Q\)</span>, the inter-compartmental
clearance</li>
<li><span class="math inline">\(\alpha\)</span>, the first rate
constant</li>
<li><span class="math inline">\(\beta\)</span>, the second rate
constant</li>
<li><span class="math inline">\(A\)</span>, the first
macro-constant</li>
<li><span class="math inline">\(B\)</span>, the second
macro-constant</li>
</ul>
<p>There are two parameterisations implemented in PFIM for
two-compartment models: <span class="math inline">\(\left(V\text{,
}k\text{, }k_{12}\text{ and }k_{21}\right)\)</span>, or <span class="math inline">\(\left(CL\text{, }V_1\text{, }Q\text{ and
}V_2\right)\)</span>. For extra-vascular administration, <span class="math inline">\(V_1\)</span> (<span class="math inline">\(V\)</span>), <span class="math inline">\(V_2\)</span>, <span class="math inline">\(CL\)</span>, and <span class="math inline">\(Q\)</span> are apparent volumes and
clearances.</p>
<p>The second parameterisation terms are derived using:</p>
<ul>
<li><span class="math inline">\(V_1=V\)</span></li>
<li><span class="math inline">\(CL=k \times V_1\)</span></li>
<li><span class="math inline">\(Q=k_{12} \times V_1\)</span></li>
<li><span class="math inline">\(V_2= {\frac{k_{12}}{k_{21}}}\times
V_1\)</span></li>
</ul>
<p>For readability, the equations for two-compartment models with linear
elimination are given using the variables <span class="math inline">\(\alpha\text{, }\beta\text{, }A\text{ and
}B\)</span> defined by the following expressions:</p>
<p><span class="math display">\[\alpha = {\frac{k_{21}k}{\beta}} =
{\frac{{\frac{Q}{V_2}}{\frac{CL}{V_1}}}{\beta}}\]</span></p>
<p><span class="math display">\[\beta=
\begin{cases}
{\frac{1}{2}\left[k_{12}+k_{21}+k-\sqrt{\left(k_{12}+k_{21}+k\right)^2-4k_{21}k}\right]}\\[0.4cm]
{
\frac{1}{2}
\left[
\frac{Q}{V_1}+\frac{Q}{V_2}+\frac{CL}{V_1}-\sqrt{\left(\frac{Q}{V_1}+\frac{Q}{V_2}+\frac{CL}{V_1}\right)^2-4\frac{Q}{V_2}\frac{CL}{V_1}}
\right]
}
\end{cases}\]</span></p>
<p>The link between A and B, and the PK parameters of the first and
second parameterisations depends on the input and are given in each
subsection.</p>
<div id="intravenous-bolus-1" class="section level4">
<h4>Intravenous bolus</h4>
<p>For intravenous bolus, the link between <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span>, and the parameters (<span class="math inline">\(V\)</span>, <span class="math inline">\(k\)</span>, <span class="math inline">\(k_{12}\)</span> and <span class="math inline">\(k_{21}\)</span>), or (<span class="math inline">\(CL\)</span>, <span class="math inline">\(V_1\)</span>, <span class="math inline">\(Q\)</span> and <span class="math inline">\(V_2\)</span>) is defined as follows:</p>
<p><span class="math display">\[A={\frac{1}{V}\frac{\alpha-k_{21}}{\alpha-\beta}}
={\frac{1}{V_1}\frac{\alpha-{\frac{Q}{V_2}}}{\alpha-\beta}}\]</span></p>
<p><span class="math display">\[B={\frac{1}{V}\frac{\beta-k_{21}}{\beta-\alpha}}
={\frac{1}{V_1}\frac{\beta-{\frac{Q}{V_2}}}{\beta-\alpha}}\]</span></p>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=D\left(Ae^{-\alpha \left(t-t_D\right)}+Be^{-\beta
\left(t-t_D\right)}\right)
\end {equation}\]</span></p>
<ul>
<li>multiples doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=\sum^{n}_{i=1}D_{i}\left(Ae^{-\alpha
\left(t-t_{D_{i}}\right)}+Be^{-\beta \left(t-t_{D_{i}}\right)}\right)
\end {equation} \]</span></p>
<ul>
<li>steady state</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=D\left(\frac{Ae^{-\alpha t}}{1-e^{-\alpha
\tau}}+\frac{Be^{-\beta t}}{1-e^{-\beta \tau}}\right)
\end{equation}\]</span></p>
</div>
<div id="infusion-1" class="section level4">
<h4>Infusion</h4>
<p>For infusion, the link between <span class="math inline">\(A\)</span>
and <span class="math inline">\(B\)</span>, and the parameters (<span class="math inline">\(V\)</span>, <span class="math inline">\(k\)</span>, <span class="math inline">\(k_{12}\)</span> and <span class="math inline">\(k_{21}\)</span>), or (<span class="math inline">\(CL\)</span>, <span class="math inline">\(V_1\)</span>, <span class="math inline">\(Q\)</span> and <span class="math inline">\(V_2\)</span>) is defined as follows:</p>
<p><span class="math display">\[A={\frac{1}{V}\frac{\alpha-k_{21}}{\alpha-\beta}}
={\frac{1}{V_1}\frac{\alpha-{\frac{Q}{V_2}}}{\alpha-\beta}}\]</span></p>
<p><span class="math display">\[B={\frac{1}{V}\frac{\beta-k_{21}}{\beta-\alpha}}
={\frac{1}{V_1}\frac{\beta-{\frac{Q}{V_2}}}{\beta-\alpha}}\]</span></p>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[  \begin {equation}
C\left(t\right)=
\begin{cases}
{\frac{D}{Tinf}}\left[
\begin{aligned}
\frac{A}{\alpha}\left(1-e^{-\alpha \left(t-t_D\right)}\right)\\[0.1cm]
+ \frac{B}{\beta}\left(1-e^{-\beta \left(t-t_D\right)}\right)
\end{aligned}               
\right] &amp; \text{if $t-t_D\leq Tinf$,}\\[1cm]
{\frac{D}{Tinf}}\left[
\begin{aligned}
\frac{A}{\alpha}\left(1-e^{-\alpha Tinf}\right) e^{-\alpha
\left(t-t_D-Tinf\right)}\\[0.1cm]
+ \frac{B}{\beta}\left(1-e^{-\beta Tinf}\right) e^{-\beta
\left(t-t_D-Tinf\right)}
\end{aligned}   
\right] &amp; \text{if not.}\\
\end{cases}
\end {equation} \]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=
\begin{cases}
\begin{aligned}
\sum^{n-1}_{i=1}&amp;\frac{D_i}{Tinf_i}
\left[
\begin{aligned}
\frac{A}{\alpha}\left(1-e^{-\alpha Tinf_i}\right) e^{-\alpha
\left(t-t_{D_{i}}-Tinf_i\right)}\\[0.1cm]
+ \frac{B}{\beta}\left(1-e^{-\beta Tinf_i}\right) e^{-\beta
\left(t-t_{D_{i}}-Tinf_i\right)}
\end{aligned}               
\right]\\[0.2cm]
&amp;+\frac{D}{Tinf_n}
\left[
\begin{aligned}
\frac{A}{\alpha}\left(1-e^{-\alpha
\left(t-t_{D_{n}}\right)}\right)\\[0.1cm]
+ \frac{B}{\beta}\left(1-e^{-\beta \left(t-t_{D_{n}}\right)}\right)
\end{aligned}               
\right]                                                              
\end{aligned}                                                           &amp;
\text{if $t-t_{D_{n}}\leq
Tinf$,}\\                                                                                           
{\displaystyle \sum^{n}_{i=1}\frac{D_i}{Tinf_i}}
\left[
\begin{aligned}
\frac{A}{\alpha}\left(1-e^{-\alpha Tinf_i}\right) e^{-\alpha
\left(t-t_{D_{i}}-Tinf_i\right)}\\[0.1cm]
+ \frac{B}{\beta}\left(1-e^{-\beta Tinf_i}\right) e^{-\beta
\left(t-t_{D_{i}}-Tinf_i\right)}
\end{aligned}   
\right]                                                                 &amp;
\text{if not.}
\end{cases}                             
\end {equation} \]</span></p>
<ul>
<li>steady state</li>
</ul>
<p>$$$$</p>
</div>
<div id="first-order-absorption-1" class="section level4">
<h4>First-order absorption</h4>
<p>For first order absorption, the link between <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span>, and the parameters (<span class="math inline">\(k_a\)</span>, <span class="math inline">\(V\)</span>, <span class="math inline">\(k\)</span>, <span class="math inline">\(k_{12}\)</span> and <span class="math inline">\(k_{21}\)</span>), or <span class="math inline">\(\left(k_a\text{, } CL\text{, }V_1\text{, }Q\text{
and }V_2\right)\)</span> is defined as follows:</p>
<p><span class="math display">\[A={\frac{k_a}{V}\frac{k_{21}-\alpha}{\left(k_a-\alpha\right)\left(\beta-\alpha\right)}}
={\frac{k_a}{V_1}\frac{{\frac{Q}{V_2}}-\alpha}{\left(k_a-\alpha\right)\left(\beta-\alpha\right)}}\]</span></p>
<p><span class="math display">\[B={\frac{k_a}{V}\frac{k_{21}-\beta}{\left(k_a-\beta\right)\left(\alpha-\beta\right)}}
={\frac{k_a}{V_1}\frac{{\frac{Q}{V_2}}-\beta}{\left(k_a-\beta\right)\left(\alpha-\beta\right)}}\]</span></p>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[          \begin {equation}
C\left(t\right)=D
\left(
Ae^{-\alpha \left(t-t_D\right)}+Be^{-\beta
\left(t-t_D\right)}-(A+B)e^{-k_a \left(t-t_D\right)}             
\right)
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=\sum^{n}_{i=1}D_{i}
\left(
Ae^{-\alpha \left(t-t_{D_{i}}\right)}+Be^{-\beta
\left(t-t_{D_{i}}\right)}-(A+B)e^{-k_a \left(t-t_{D_{i}}\right)}       
\right)
\end {equation}\]</span></p>
<ul>
<li>steady state</li>
</ul>
<p><span class="math display">\[\begin {equation}
C\left(t\right)=D
\left(
\frac{Ae^{-\alpha (t-t_D)}}{1-e^{-\alpha \tau}}
+\frac{Be^{-\beta (t-t_D)}}{1-e^{-\beta \tau}}
-\frac{(A+B)e^{-k_a (t-t_D)}}{1-e^{-k_a \tau}}              
\right)         
\end {equation}\]</span></p>
</div>
</div>
</div>
<div id="models-with-michaelis-menten-elimination" class="section level2">
<h2>Models with Michaelis-Menten elimination</h2>
<p>The list of PK models with Michaelis-Menten elimination implemented
in PFIM are summarised in Appendix I.2. Presently, there is no
implementation for multiple dosing with IV bolus administration in the
PFIM software. For infusion and oral administration, the implementation
in PFIM does not allow designs with different groups of doses as the
dose is included in the model.</p>
<div id="one-compartment-models-2" class="section level3">
<h3>One-compartment models</h3>
<div id="intravenous-bolus-2" class="section level4">
<h4>Intravenous bolus</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[\begin{equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C\left(t\right)&amp;= 0 \text{  for $t&lt;t_D$}\\[0.05cm]
C\left(t_{D}\right)&amp;= {\frac{D}{V}}\\
\end{cases}\\[0.2cm]
&amp;\frac{dC}{dt}= -\frac{{V_m}\times C}{K_m+C}\\
\end{aligned}
\end {equation}\]</span></p>
</div>
<div id="infusion-2" class="section level4">
<h4>Infusion</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[  \begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }C\left(t\right)=0 \text{  for
$t&lt;t_D$}\\[0.05cm]
&amp;\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+input\\[0.2cm]
&amp;input\left(t\right)=
\begin{cases}
{\frac{D}{Tinf}\frac{1}{V}} &amp;\text{if $0\leq t-t_{D}\leq
Tinf$}\\[0.05cm]
0 &amp;\text{if not.}
\end{cases}
\end{aligned}
\label{infusion1mmsd}
\end {equation} \]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }C\left(t\right)=0 \text{  for
$t&lt;t_{D_{1}}$}\\[0.05cm]
&amp;\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+input\\[0.2cm]
&amp;input\left(t\right)=
\begin{cases}
{\frac{D_{i}}{Tinf_{i}}\frac{1}{V}} &amp;\text{if $0\leq t-t_{D_{i}}\leq
Tinf_{i}$,}\\[0.05cm]
0 &amp;\text{if not.}
\end{cases}
\end{aligned}\label{infusion1mmss}  
\end {equation}\]</span></p>
</div>
<div id="first-order-absorption-2" class="section level4">
<h4>First order absorption</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[
\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }C\left(t\right)=0\text{   for
$t&lt; t_D$}\\[0.05cm]
&amp;\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+ input\\[0.2cm]
&amp;input\left(t\right)=\frac{D}{V}k_ae^{-k_a\left(t-t_D\right)}
\end{aligned}
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[          \begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }C\left(t\right)=0\text{   for
$t&lt; t_{D_{1}}$}\\[0.05cm]
&amp;\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+ input\\[0.2cm]
&amp;input\left(t\right)=\sum^{n}_{i=1}\frac{D_i}{V}k_ae^{-k_a\left(t-t_{D_{i}}\right)}
\end{aligned}\label{oral11mmss}
\end {equation}\]</span></p>
</div>
</div>
<div id="two-compartment-models-1" class="section level3">
<h3>Two-compartment models</h3>
<div id="intravenous-bolus-3" class="section level4">
<h4>Intravenous bolus</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[
\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C_1\left(t\right)= &amp;0 \text{    for $t&lt;t_D$}\\[0.05cm]
C_2\left(t\right)= &amp;0 \text{    for $t\leq t_D$}\\[0.05cm]
C_1\left(t_{D}\right)=&amp;{\frac{D}{V}}\\[0.05cm]
\end{cases}\\[0.15cm]
&amp;\frac{dC_{1}}{dt}=-\frac{{V_m}\times C_1}{K_m+C_1}-k_{12}C_1+\frac{
k_{21 }V_2}{V}C_2\\[0.2cm]
&amp;\frac{dC_{2}}{dt}=\frac{ k_{12 }V}{V_2}C_1-k_{21}C_2   \\
\end{aligned}
\end {equation}\]</span></p>
</div>
<div id="infusion-3" class="section level4">
<h4>Infusion</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[  \begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C_1\left(t\right)=&amp;0 \text{ for $t&lt;t_D$}\\[0.05cm]
C_2\left(t\right)=&amp;0 \text{ for $t\leq t_D$}\\[0.05cm]
\end{cases}\\[0.15cm]
&amp;\frac{dC_{1}}{dt}=-\frac{{V_m}\times C_1}{K_m+C_1}-k_{12}C_1+\frac{
k_{21 }V_2}{V}C_2+input\\[0.2cm]
&amp;\frac{dC_{2}}{dt}=\frac{ k_{12 }V}{V_2}C_1-k_{21}C_2\\[0.2cm]
&amp;input\left(t\right)=\begin{cases}
{\frac{D}{Tinf}\frac{1}{V}} &amp;\text{if $0\leq t-t_{D}\leq
Tinf$}\\[0.05cm]
0 &amp;\text{if not.}
\end{cases}
\end{aligned}
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C_1\left(t\right)=&amp;0 \text{ for $t&lt;t_{D_{1}}$}\\[0.05cm]
C_2\left(t\right)=&amp;0 \text{ for $t\leq t_{D_{1}}$}\\[0.05cm]
\end{cases}\\[0.15cm]
&amp;\frac{dC_{1}}{dt}=-\frac{{V_m}\times C_1}{K_m+C_1}-k_{12}C_1+\frac{
k_{21 }V_2}{V}C_2 + input\\[0.2cm]
&amp;\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\\[0.2cm]
&amp;input\left(t\right)=
\begin{cases}
{\frac{D_{i}}{Tinf_{i}}\frac{1}{V}} &amp;\text{if $0\leq t-t_{D_{i}}\leq
Tinf_{i}$,}\\[0.05cm]
0 &amp;\text{if not.}
\end{cases}
\end{aligned}
\end {equation}\]</span></p>
</div>
<div id="first-order-absorption-3" class="section level4">
<h4>First order absorption</h4>
<ul>
<li>single dose</li>
</ul>
<p><span class="math display">\[
\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C_1\left(t\right)=&amp;0 \text{ for $t&lt; t_D$}\\[0.05cm]
C_2\left(t\right)=&amp;0 \text{ for $t\leq t_D$}\\
\end{cases}\\[0.15cm]
&amp;\frac{dC_{1}}{dt}=-\frac{{V_m}\times C_1}{K_m+C_1}-k_{12}C_1+\frac{
k_{21}V_2}{V}C_2+input\\[0.2cm]
&amp;\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\\[0.2cm]
&amp;input\left(t\right)=\frac{D}{V}k_ae^{-k_a\left(t-t_D\right)}
\end{aligned}
\end {equation}\]</span></p>
<ul>
<li>multiple doses</li>
</ul>
<p><span class="math display">\[\begin {equation}
\begin{aligned}
\text{Initial }&amp;\text{conditions:    }\begin{cases}
C_1\left(t\right)=&amp;0 \text{ for $t&lt; t_{D_{1}}$}\\[0.05cm]
C_2\left(t\right)=&amp;0 \text{ for $t\leq t_{D_{1}}$}\\
\end{cases}\\[0.15cm]
&amp;\frac{dC_{1}}{dt}=-\frac{{V_m}\times C_1}{K_m+C_1}-k_{12}C_1+\frac{
k_{21}V_2}{V}C_2+input\\[0.2cm]
&amp;\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\\[0.2cm]
&amp;input\left(t\right)=\sum^{n}_{i=1}\frac{D_i}{V}k_ae^{-k_a\left(t-t_{D_{i}}\right)}
\end{aligned}
\end {equation} \]</span></p>
</div>
</div>
</div>
</div>
<div id="pharmacodynamic-models" class="section level1">
<h1>Pharmacodynamic models</h1>
<div id="immediate-response-models" class="section level2">
<h2>Immediate response models</h2>
<p>For these response models, the effect <span class="math inline">\(E\left(t\right)\)</span> is expressed as:</p>
<p><span class="math display">\[  \begin{equation}
E\left(t\right)=A\left(t\right)+S\left(t\right)
\end{equation}\]</span></p>
<p>where <span class="math inline">\(A\left(t\right)\)</span> represents
the model of drug action and <span class="math inline">\(S\left(t\right)\)</span> corresponds to the
baseline/disease model. <span class="math inline">\(A\left(t\right)\)</span> is a function of the
concentration <span class="math inline">\(C\left(t\right)\)</span> in
the central compartment.</p>
<p>The drug action models are presented in section <a href="#drugactionmodels">Drug action models</a> for <span class="math inline">\(C(t)\)</span>. The baseline/disease models are
presented in section <a href="#baselinediseasemodel">Baseline/disease
models</a>. Any combination of those two models is available in the PFIM
library.</p>
<p>Parameters</p>
<ul>
<li><span class="math inline">\(A_{lin}\)</span>: constant associated to
<span class="math inline">\(C\left(t\right)\)</span></li>
<li><span class="math inline">\(A_{quad}\)</span>: constant associated
to the square of <span class="math inline">\(C\left(t\right)\)</span></li>
<li><span class="math inline">\(A_{log}\)</span>: constant associated to
the logarithm of <span class="math inline">\(C\left(t\right)\)</span></li>
<li><span class="math inline">\(E_{max}\)</span>: maximal agonistic
response</li>
<li><span class="math inline">\(I_{max}\)</span>: maximal antagonistic
response<br />
</li>
<li><span class="math inline">\(C_{50}\)</span>: concentration to get
half of the maximal response ( drug potency)</li>
<li><span class="math inline">\(\gamma\)</span>: sigmoidicity
factor</li>
<li><span class="math inline">\(S_0\)</span>: baseline value of the
studied effect</li>
<li><span class="math inline">\(k_{prog}\)</span>: rate constant of
disease progression</li>
</ul>
<p>NB: <span class="math inline">\(V_m\)</span> is in concentration per
time unit and <span class="math inline">\(K_m\)</span> is in
concentration unit.</p>
<div id="drugactionmodels" class="section level3">
<h3>Drug action models</h3>
<ul>
<li><p>linear model <span class="math display">\[\begin{equation}
A\left(t\right)=A_{lin}C\left(t\right)
\end{equation}\]</span></p></li>
<li><p>quadratic model <span class="math display">\[\begin{equation}
A\left(t\right)=A_{lin}C\left(t\right)+A_{quad}C\left(t\right)^{2}
\end{equation}\]</span></p></li>
<li><p>logarithmic model <span class="math display">\[\begin{equation}
A\left(t\right)=A_{log}log(C\left(t\right))
\end{equation}\]</span></p></li>
<li><p><span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
A\left(t\right)=\frac{E_{max}C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
A\left(t\right)=\frac{E_{max}C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}}
\end{equation}\]</span></p></li>
<li><p><span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
A\left(t\right)=1-\frac{I_{max}C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
A\left(t\right)=1-\frac{I_{max}C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}}
\end{equation}\]</span></p></li>
<li><p>full <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
A\left(t\right)=-\frac{C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}\]</span></p></li>
<li><p>sigmoïd full <span class="math inline">\(I_{max}\)</span> model
<span class="math display">\[\begin{equation}
A\left(t\right)=-\frac{C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}}
\end{equation}\]</span></p></li>
</ul>
</div>
<div id="baselinediseasemodel" class="section level3">
<h3>Baseline/disease models</h3>
<ul>
<li>null baseline</li>
</ul>
<p><span class="math display">\[\begin{equation}
S\left(t\right)=0
\end{equation}\]</span></p>
<ul>
<li>constant baseline with no disease progression</li>
</ul>
<p><span class="math display">\[\begin{equation}
S\left(t\right)=S_{0}
\end{equation}\]</span></p>
<ul>
<li>linear disease progression</li>
</ul>
<p><span class="math display">\[\begin{equation}
S\left(t\right)=S_{0}+k_{prog}t
\end{equation}\]</span></p>
<ul>
<li>exponential disease increase</li>
</ul>
<p><span class="math display">\[\begin{equation}
S\left(t\right)=S_{0}e^{-k_{prog}t}
\end{equation}\]</span></p>
<ul>
<li>exponential disease decrease</li>
</ul>
<p><span class="math display">\[\begin{equation}
S\left(t\right)=S_{0}\left(1-e^{-k_{prog}t}\right)
\end{equation}\]</span></p>
</div>
</div>
<div id="turnover-response-models" class="section level2">
<h2>Turnover response models</h2>
<p>In these models, the drug is not acting on the effect <span class="math inline">\(E\)</span> directly but rather on <span class="math inline">\(R_{in}\)</span> or <span class="math inline">\(k_{out}\)</span>.</p>
<p>Thus the system is described with differential equations, given <span class="math inline">\({\frac{dE}{dt}}\)</span> as a function of <span class="math inline">\(R_{in}\)</span>, <span class="math inline">\(k_{out}\)</span> and <span class="math inline">\(C\left(t\right)\)</span> the drug concentration at
time t.</p>
<p>The initial condition is: while <span class="math inline">\(C\left(t\right)=0\)</span>, <span class="math inline">\(E\left(t\right)=
{\frac{R_{in}}{k_{out}}}\)</span>.</p>
<p>Parameters</p>
<ul>
<li><span class="math inline">\(E_{max}\)</span>: maximal agonistic
response</li>
<li><span class="math inline">\(I_{max}\)</span>: maximal antagonistic
response<br />
</li>
<li><span class="math inline">\(C_{50}\)</span>: concentration to get
half of the maximal response (=drug potency)</li>
<li><span class="math inline">\(\gamma\)</span>: sigmoidicity
factor</li>
<li><span class="math inline">\(R_{in}\)</span>: input (synthesis)
rate</li>
<li><span class="math inline">\(k_{out}\)</span>: output (elimination)
rate constant</li>
</ul>
<div id="models-with-impact-on-the-input-r_in" class="section level3">
<h3>Models with impact on the input <span class="math inline">\((R_{in})\)</span></h3>
<ul>
<li><p><span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1+\frac{E_{max}C}{C+C_{50}}\right)-k_{out}E
\label{indirect_emax}   
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1+\frac{E_{max}C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)-k_{out}E    
\label{indirect_sig_emax}               
\end{equation}\]</span></p></li>
<li><p><span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{I_{max}C}{C+C_{50}}\right)-k_{out}E   
\label{indirect_imax}
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{I_{max}C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)-k_{out}E    
\label{indirect_simax}
\end{equation}\]</span></p></li>
<li><p>full <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{C}{C+C_{50}}\right)-k_{out}E  
\label{indirect_fimax}
\end{equation}\]</span></p></li>
<li><p>sigmoïd full <span class="math inline">\(I_{max}\)</span> model
<span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)-k_{out}E   
\label{indirect_sfimax}             
\end{equation}\]</span></p></li>
</ul>
</div>
<div id="models-with-impact-on-the-output-k_out" class="section level3">
<h3>Models with impact on the output <span class="math inline">\((k_{out})\)</span></h3>
<ul>
<li><p><span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1+\frac{E_{max}C}{C+C_{50}}\right)E   
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(E_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1+\frac{E_{max}C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)E    
\end{equation}\]</span></p></li>
<li><p><span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{I_{max}C}{C+C_{50}}\right)E   
\end{equation}\]</span></p></li>
<li><p>sigmoïd <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{I_{max}C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)E    
\end{equation}\]</span></p></li>
<li><p>full <span class="math inline">\(I_{max}\)</span> model <span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{C}{C+C_{50}}\right)E  
\end{equation}\]</span></p></li>
<li><p>sigmoïd full <span class="math inline">\(I_{max}\)</span> model
<span class="math display">\[\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{C^{\gamma}}{C^{\gamma}+C_{50}^{\gamma}}\right)E   
\label{indirect_osfimax}
\end{equation}\]</span></p></li>
</ul>
</div>
</div>
</div>



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