Library of models"

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Pharmacokinetic models

Compartmental models and parameters

Six parameters are common to one, two or three compartment models:

One-compartment models

There are two parameterisations implemented in PFIM for one-compartment models, $\left(V\text{ and }k\right)$ or $\left(V\text{ and }CL\right)$. The equations are given for the first parameterisation $\left(V, k\right)$. For extra-vascular administration, $V$ and $CL$ are apparent volume and clearance. The equations for the second parameterisation $\left(V, CL\right)$ are derived using $k={\frac{CL}{V}}$.

Models with linear elimination

One-compartment models

Intravenous bolus

$$\begin {equation} \begin{aligned} C\left(t\right)=\frac{D}{V}e^{-k\left(t-t_{D}\right)} \end{aligned}
\end {equation}$$

$$\begin {equation} \begin{aligned} & C\left(t\right)=\sum^{n}{i=1}\frac{D{i}}{V}e^{-k\left(t-t_{D_{i}}\right)}\ & \end{aligned}
\end {equation}$$

$$\begin {equation} C(t)=\frac{D}{V}\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}\
\end {equation}$$

Infusion

$$\begin{equation} C\left(t\right)= \begin{cases} {\frac{D}{Tinf}\frac{1}{kV}\left(1-e^{-k\left(t-t_{D}\right)}\right)} & \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)}} & \text{if not.}\ \end{cases}\
\end{equation}$$

$$\begin{equation} C\left(t\right)= \begin{cases} \begin{aligned} \sum^{n-1}{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)}\ &+\frac{D_{n}}{Tinf_{n}} \frac{1}{kV} \left(1-e^{-k\left(t-t_{D_{n}}\right)}\right) \end{aligned} & \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)} & \text{if not.}\ \end{cases} \end{equation} $$

$$\begin{equation} \begin{aligned} & 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] &\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}}} &\text{if not.}\ \end{cases}\ & \end{aligned} \end{equation}$$

First order absorption

$$\begin {equation} C\left(t\right)=\frac{D}{V} \frac{k_{a}}{k_{a}-k} \left(e^{-k\left(t-t_{D}\right)}-e^{-k_{a}\left(t-t_{D}\right)}\right) \end {equation}$$

$$\begin {equation} C\left(t\right)=\sum^{n}{i=1}\frac{D{i}}{V} \frac{k_{a}}{k_{a}-k} \left(e^{-k\left(t-t_{D_{i}}\right)}-e^{-k_{a}\left(t-t_{D_{i}}\right)}\right) \end {equation} $$

$$\begin {equation} C\left(t\right)=\frac{D}{V} \frac{k_{a}}{k_{a}-k} \left(\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}-\frac{e^{-k_{a}(t-t_D)}}{1-e^{-k_a\tau}}\right) \end {equation}$$

Two-compartment models

For two-compartment model equations, $C(t)=C_1(t)$ represent the drug concentration in the first compartment and $C_2(t)$ represents the drug concentration in the second compartment.

As well as the previously described PK parameters, the following PK parameters are used for the two-compartment models:

There are two parameterisations implemented in PFIM for two-compartment models: $\left(V\text{, }k\text{, }k_{12}\text{ and }k_{21}\right)$, or $\left(CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$. For extra-vascular administration, $V_1$ ($V$), $V_2$, $CL$, and $Q$ are apparent volumes and clearances.

The second parameterisation terms are derived using:

For readability, the equations for two-compartment models with linear elimination are given using the variables $\alpha\text{, }\beta\text{, }A\text{ and }B$ defined by the following expressions:

$$\alpha = {\frac{k_{21}k}{\beta}} = {\frac{{\frac{Q}{V_2}}{\frac{CL}{V_1}}}{\beta}}$$

$$\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}$$

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.

Intravenous bolus

For intravenous bolus, the link between $A$ and $B$, and the parameters ($V$, $k$, $k_{12}$ and $k_{21}$), or ($CL$, $V_1$, $Q$ and $V_2$) is defined as follows:

$$A={\frac{1}{V}\frac{\alpha-k_{21}}{\alpha-\beta}} ={\frac{1}{V_1}\frac{\alpha-{\frac{Q}{V_2}}}{\alpha-\beta}}$$

$$B={\frac{1}{V}\frac{\beta-k_{21}}{\beta-\alpha}} ={\frac{1}{V_1}\frac{\beta-{\frac{Q}{V_2}}}{\beta-\alpha}}$$

$$\begin {equation} C\left(t\right)=D\left(Ae^{-\alpha \left(t-t_D\right)}+Be^{-\beta \left(t-t_D\right)}\right) \end {equation}$$

$$\begin {equation} C\left(t\right)=\sum^{n}{i=1}D{i}\left(Ae^{-\alpha \left(t-t_{D_{i}}\right)}+Be^{-\beta \left(t-t_{D_{i}}\right)}\right) \end {equation} $$

$$\begin {equation} C\left(t\right)=D\left(\frac{Ae^{-\alpha t}}{1-e^{-\alpha \tau}}+\frac{Be^{-\beta t}}{1-e^{-\beta \tau}}\right) \end{equation}$$

Infusion

For infusion, the link between $A$ and $B$, and the parameters ($V$, $k$, $k_{12}$ and $k_{21}$), or ($CL$, $V_1$, $Q$ and $V_2$) is defined as follows:

$$A={\frac{1}{V}\frac{\alpha-k_{21}}{\alpha-\beta}} ={\frac{1}{V_1}\frac{\alpha-{\frac{Q}{V_2}}}{\alpha-\beta}}$$

$$B={\frac{1}{V}\frac{\beta-k_{21}}{\beta-\alpha}} ={\frac{1}{V_1}\frac{\beta-{\frac{Q}{V_2}}}{\beta-\alpha}}$$

$$ \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] & \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] & \text{if not.}\ \end{cases} \end {equation} $$

$$\begin {equation} C\left(t\right)= \begin{cases} \begin{aligned} \sum^{n-1}{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]\[0.2cm] &+\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} & \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] & \text{if not.} \end{cases}
\end {equation} $$

$$\begin {equation} \hspace{-0.5cm} C\left(t\right)=\begin{cases} {\frac{D}{Tinf}} \left[ \begin{aligned} &\frac{A}{\alpha} \left( \begin{aligned} &\left(1-e^{-\alpha (t-t_D)}\right)\ &+ e^{-\alpha \tau} \frac{ \left(1-e^{-\alpha Tinf}\right) e^{-\alpha \left(t-t_D - Tinf\right)}} {1-e^{-\alpha \tau}} \end{aligned} \right)\[0.1cm] &+ \frac{B}{\beta} \left( \begin{aligned} &\left(1-e^{-\beta (t-t_D)}\right)\ &+ e^{-\beta \tau} \frac{ \left(1-e^{-\beta Tinf}\right) e^{-\beta \left(t-t_D - Tinf\right)}} {1-e^{-\beta \tau}} \end{aligned} \right) \end{aligned}
\right] &!!!!!\text{if $t-t_D\leq Tinf$,}\vspace*{0.5cm}\

{\frac{D}{Tinf}} \left[ \begin{aligned} &\frac{A}{\alpha} \left( \frac{ \left(1-e^{-\alpha Tinf}\right) e^{-\alpha \left(t-t_D - Tinf\right)}} {1-e^{-\alpha \tau}} \right)\[0.1cm] &+ \frac{B}{\beta} \left( \frac{ \left(1-e^{-\beta Tinf}\right) e^{-\beta \left(t-t_D - Tinf\right)}} {1-e^{-\beta \tau}} \right) \end{aligned}
\right] &!!!!!\text{if not.}
\end{cases} \label{infusion2lss}
\end {equation}$$

First-order absorption

For first order absorption, the link between $A$ and $B$, and the parameters ($k_a$, $V$, $k$, $k_{12}$ and $k_{21}$), or $\left(k_a\text{, } CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$ is defined as follows:

$$A={\frac{k_a}{V}\frac{k_{21}-\alpha}{\left(k_a-\alpha\right)\left(\beta-\alpha\right)}} ={\frac{k_a}{V_1}\frac{{\frac{Q}{V_2}}-\alpha}{\left(k_a-\alpha\right)\left(\beta-\alpha\right)}}$$

$$B={\frac{k_a}{V}\frac{k_{21}-\beta}{\left(k_a-\beta\right)\left(\alpha-\beta\right)}} ={\frac{k_a}{V_1}\frac{{\frac{Q}{V_2}}-\beta}{\left(k_a-\beta\right)\left(\alpha-\beta\right)}}$$

$$ \begin {equation} C\left(t\right)=D \left( Ae^{-\alpha \left(t-t_D\right)}+Be^{-\beta \left(t-t_D\right)}-(A+B)e^{-k_a \left(t-t_D\right)}
\right) \end {equation}$$

$$\begin {equation} C\left(t\right)=\sum^{n}{i=1}D{i} \left( Ae^{-\alpha \left(t-t_{D_{i}}\right)}+Be^{-\beta \left(t-t_{D_{i}}\right)}-(A+B)e^{-k_a \left(t-t_{D_{i}}\right)}
\right) \end {equation}$$

$$\begin {equation} C\left(t\right)=D \left( \frac{Ae^{-\alpha (t-t_D)}}{1-e^{-\alpha \tau}} +\frac{Be^{-\beta (t-t_D)}}{1-e^{-\beta \tau}} -\frac{(A+B)e^{-k_a (t-t_D)}}{1-e^{-k_a \tau}}
\right)
\end {equation}$$

Models with Michaelis-Menten elimination

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.

One-compartment models

Intravenous bolus

$$\begin{equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C\left(t\right)&= 0 \text{ for $t<t_D$}\[0.05cm] C\left(t_{D}\right)&= {\frac{D}{V}}\ \end{cases}\[0.2cm] &\frac{dC}{dt}= -\frac{{V_m}\times C}{K_m+C}\ \end{aligned} \end {equation}$$

Infusion

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }C\left(t\right)=0 \text{ for $t<t_D$}\[0.05cm] &\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+input\[0.2cm] &input\left(t\right)= \begin{cases} {\frac{D}{Tinf}\frac{1}{V}} &\text{if $0\leq t-t_{D}\leq Tinf$}\[0.05cm] 0 &\text{if not.} \end{cases} \end{aligned} \label{infusion1mmsd} \end {equation} $$

$$\begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }C\left(t\right)=0 \text{ for $t<t_{D_{1}}$}\[0.05cm] &\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+input\[0.2cm] &input\left(t\right)= \begin{cases} {\frac{D_{i}}{Tinf_{i}}\frac{1}{V}} &\text{if $0\leq t-t_{D_{i}}\leq Tinf_{i}$,}\[0.05cm] 0 &\text{if not.} \end{cases} \end{aligned}\label{infusion1mmss}
\end {equation}$$

First order absorption

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }C\left(t\right)=0\text{ for $t< t_D$}\[0.05cm] &\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+ input\[0.2cm] &input\left(t\right)=\frac{D}{V}k_ae^{-k_a\left(t-t_D\right)} \end{aligned} \end {equation}$$

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }C\left(t\right)=0\text{ for $t< t_{D_{1}}$}\[0.05cm] &\frac{dC}{dt}=-\frac{{V_m}\times C}{K_m+C}+ input\[0.2cm] &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}$$

Two-compartment models

Intravenous bolus

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C_1\left(t\right)= &0 \text{ for $t<t_D$}\[0.05cm] C_2\left(t\right)= &0 \text{ for $t\leq t_D$}\[0.05cm] C_1\left(t_{D}\right)=&{\frac{D}{V}}\[0.05cm] \end{cases}\[0.15cm] &\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] &\frac{dC_{2}}{dt}=\frac{ k_{12 }V}{V_2}C_1-k_{21}C_2 \ \end{aligned} \end {equation}$$

Infusion

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C_1\left(t\right)=&0 \text{ for $t<t_D$}\[0.05cm] C_2\left(t\right)=&0 \text{ for $t\leq t_D$}\[0.05cm] \end{cases}\[0.15cm] &\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] &\frac{dC_{2}}{dt}=\frac{ k_{12 }V}{V_2}C_1-k_{21}C_2\[0.2cm] &input\left(t\right)=\begin{cases} {\frac{D}{Tinf}\frac{1}{V}} &\text{if $0\leq t-t_{D}\leq Tinf$}\[0.05cm] 0 &\text{if not.} \end{cases} \end{aligned} \end {equation}$$

$$\begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C_1\left(t\right)=&0 \text{ for $t<t_{D_{1}}$}\[0.05cm] C_2\left(t\right)=&0 \text{ for $t\leq t_{D_{1}}$}\[0.05cm] \end{cases}\[0.15cm] &\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] &\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\[0.2cm] &input\left(t\right)= \begin{cases} {\frac{D_{i}}{Tinf_{i}}\frac{1}{V}} &\text{if $0\leq t-t_{D_{i}}\leq Tinf_{i}$,}\[0.05cm] 0 &\text{if not.} \end{cases} \end{aligned} \end {equation}$$

First order absorption

$$ \begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C_1\left(t\right)=&0 \text{ for $t< t_D$}\[0.05cm] C_2\left(t\right)=&0 \text{ for $t\leq t_D$}\ \end{cases}\[0.15cm] &\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] &\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\[0.2cm] &input\left(t\right)=\frac{D}{V}k_ae^{-k_a\left(t-t_D\right)} \end{aligned} \end {equation}$$

$$\begin {equation} \begin{aligned} \text{Initial }&\text{conditions: }\begin{cases} C_1\left(t\right)=&0 \text{ for $t< t_{D_{1}}$}\[0.05cm] C_2\left(t\right)=&0 \text{ for $t\leq t_{D_{1}}$}\ \end{cases}\[0.15cm] &\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] &\frac{dC_{2}}{dt}=\frac{ k_{12}V}{V_2}C_1-k_{21}C_2\[0.2cm] &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} $$

Pharmacodynamic models

Immediate response models

For these response models, the effect $E\left(t\right)$ is expressed as:

$$ \begin{equation} E\left(t\right)=A\left(t\right)+S\left(t\right) \end{equation}$$

\noindent where $A\left(t\right)$ represents the model of drug action and $S\left(t\right)$ corresponds to the baseline/disease model. $A\left(t\right)$ is a function of the concentration $C\left(t\right)$ in the central compartment.

The drug action models are presented in section Drug action models for $C(t)$. The baseline/disease models are presented in section Baseline/disease models. Any combination of those two models is available in the PFIM library.

Parameters

NB: $V_m$ is in concentration per time unit and $K_m$ is in concentration unit.

Drug action models {#drugactionmodels}

Baseline/disease models {#baselinediseasemodel}

$$\begin{equation} S\left(t\right)=0 \end{equation}$$

$$\begin{equation} S\left(t\right)=S_{0} \end{equation}$$

$$\begin{equation} S\left(t\right)=S_{0}+k_{prog}t \end{equation}$$

$$\begin{equation} S\left(t\right)=S_{0}e^{-k_{prog}t} \end{equation}$$

$$\begin{equation} S\left(t\right)=S_{0}\left(1-e^{-k_{prog}t}\right) \end{equation}$$

Turnover response models

In these models, the drug is not acting on the effect $E$ directly but rather on $R_{in}$ or $k_{out}$.

Thus the system is described with differential equations, given ${\frac{dE}{dt}}$ as a function of $R_{in}$, $k_{out}$ and $C\left(t\right)$ the drug concentration at time t.

The initial condition is: while $C\left(t\right)=0$, $E\left(t\right)= {\frac{R_{in}}{k_{out}}}$.

Parameters

Models with impact on the input $(R_{in})$

Models with impact on the output $(k_{out})$



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PFIM documentation built on June 24, 2022, 9:06 a.m.