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Library of models"
In PFIM: Population Fisher Information Matrix
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<!-- The document presents the library of pharmacokinetic (PK) and pharmacodynamic (PD) models incorporated into the PFIM software. -->
# Pharmacokinetic models
## Compartmental models and parameters
Six parameters are common to one, two or three compartment models:
+ $V$ or $V_1$, the volume of distribution in the central compartment
+ $k$, the elimination rate constant
+ $CL$, the clearance of elimination
+ $V_m$, the maximum elimination rate for Michaelis-Menten elimination
+ $K_m$, the Michaelis-Menten constant
+ $k_a$, the absorption rate constant for oral administration
## 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
+ single dose
$$\begin {equation}
\begin{aligned}
C\left(t\right)=\frac{D}{V}e^{-k\left(t-t_{D}\right)}
\end{aligned}
\end {equation}$$
+ multiple doses
$$\begin {equation}
\begin{aligned}
&
C\left(t\right)=\sum^{n}_{i=1}\frac{D_{i}}{V}e^{-k\left(t-t_{D_{i}}\right)}\\
&
\end{aligned}
\end {equation}$$
+ Library of models
```r
Linear1BolusSingleDose_kV
Linear1BolusSingleDose_ClV
- steady state
$$\begin {equation}
C(t)=\frac{D}{V}\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}\
\end {equation}$$
Linear1BolusSteadyState_kVtau
Linear1BolusSteadyState_ClVtau
Infusion
- single dose
$$\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}$$
- multiple doses
$$\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}
$$
Linear1InfusionSingleDose_kV
Linear1InfusionSingleDose_ClV
- steady state
$$\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}$$
Linear1InfusionSteadyState_kVtau
Linear1InfusionSteadyState_ClVtau
First order absorption
- single dose
$$\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}$$
- multiple doses
$$\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} $$
Linear1FirstOrderSingleDose_kakV
Linear1FirstOrderSingleDose_kaClV
- steady state
$$\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}$$
Linear1FirstOrderSteadyState_kakVtau
Linear1FirstOrderSteadyState_kaClVtau
Two-compartment models
For two-compartment model equations, $C(t)=C_1(t)$ represent the drug concentration in the first compartment and $C_2(t)$ represents the drug concentration in the second compartment.
As well as the previously described PK parameters, the following PK parameters are used for the two-compartment models:
- $V_2$, the volume of distribution of second compartment
- $k_{12}$, the distribution rate constant from compartment 1 to compartment 2
- $k_{21}$, the distribution rate constant from compartment 2 to compartment 1
- $Q$, the inter-compartmental clearance
- $\alpha$, the first rate constant
- $\beta$, the second rate constant
- $A$, the first macro-constant
- $B$, the second macro-constant
There are two parameterisations implemented in PFIM for two-compartment models: $\left(V\text{, }k\text{, }k_{12}\text{ and }k_{21}\right)$, or $\left(CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$. For extra-vascular administration, $V_1$ ($V$), $V_2$, $CL$, and $Q$ are apparent volumes and clearances.
The second parameterisation terms are derived using:
- $V_1=V$
- $CL=k \times V_1$
- $Q=k_{12} \times V_1$
- $V_2= {\frac{k_{12}}{k_{21}}}\times V_1$
For readability, the equations for two-compartment models with linear elimination are given using the variables $\alpha\text{, }\beta\text{, }A\text{ and }B$ defined by the following expressions:
$$\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}}$$
- single dose
$$\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}$$
- multiples doses
$$\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} $$
Linear2BolusSingleDose_ClQV1V2
Linear2BolusSingleDose_kk12k21V
- steady state
$$\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}$$
Linear2BolusSteadyState_ClQV1V2tau
Linear2BolusSteadyState_kk12k21Vtau
Infusion
For infusion, the link between $A$ and $B$, and the parameters ($V$, $k$, $k_{12}$ and $k_{21}$), or ($CL$, $V_1$, $Q$ and $V_2$) is defined as follows:
$$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}}$$
- single dose
$$ \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} $$
- multiple doses
$$\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} $$
Linear2InfusionSingleDose_kk12k21V,
Linear2InfusionSingleDose_ClQV1V2,
- steady state
$$\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}$$
Linear2InfusionSteadyState_kk12k21Vtau
Linear2InfusionSteadyState_ClQV1V2tau
First-order absorption
For first order absorption, the link between $A$ and $B$, and the parameters ($k_a$, $V$, $k$, $k_{12}$ and $k_{21}$), or $\left(k_a\text{, } CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$ is defined as follows:
$$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)}}$$
- single dose
$$ \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}$$
- multiple doses
$$\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}$$
Linear2FirstOrderSingleDose_kaClQV1V2
Linear2FirstOrderSingleDose_kakk12k21V
- steady state
$$\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}$$
Linear2FirstOrderSteadyState_kaClQV1V2tau
Linear2FirstOrderSteadyState_kakk12k21Vtau
Models with Michaelis-Menten elimination
One-compartment models
Intravenous bolus
- single dose
$$\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}$$
MichaelisMenten1BolusSingleDose_VmKmV
Infusion
- single dose
$$ \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} $$
- multiple doses
$$\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}$$
??????
First order absorption
- single dose
$$
\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}$$
- multiple doses
$$ \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}$$
MichaelisMenten1FirstOrderSingleDose_kaVmKmV,
MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2
Two-compartment models
Intravenous bolus
- single dose
$$
\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}$$
MichaelisMenten2BolusSingleDose_VmKmk12k21V1V2
Infusion
- single dose
$$ \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}$$
- multiple doses
$$\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}$$
MichaelisMenten2InfusionSingleDose_VmKmk12k21V1V2
First order absorption
- single dose
$$
\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}$$
- multiple doses
$$\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} $$
MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2
MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2
Pharmacodynamic models
Immediate response models
For these response models, the effect $E\left(t\right)$ is expressed as:
$$ \begin{equation}
E\left(t\right)=A\left(t\right)+S\left(t\right)
\end{equation}$$
\noindent where $A\left(t\right)$ represents the model of drug action and $S\left(t\right)$ corresponds to the baseline/disease model. $A\left(t\right)$ is a function of the concentration $C\left(t\right)$ in the central compartment.
The drug action models are presented in section Drug action models for $C(t)$. The baseline/disease models are presented in section Baseline/disease models. Any combination of those two models is available in the PFIM library.
Parameters
- $A_{lin}$: constant associated to $C\left(t\right)$
- $A_{quad}$: constant associated to the square of $C\left(t\right)$
- $A_{log}$: constant associated to the logarithm of $C\left(t\right)$
- $E_{max}$: maximal agonistic response
- $I_{max}$: maximal antagonistic response
- $C_{50}$: concentration to get half of the maximal response (\textit{i.e.} drug potency)
- $\gamma$: sigmoidicity factor
- $S_0$: baseline value of the studied effect
- $k_{prog}$: rate constant of disease progression
NB: $V_m$ is in concentration per time unit and $K_m$ is in concentration unit.
Drug action models {#drugactionmodels}
- linear model
$$\begin{equation}
A\left(t\right)=A_{lin}C\left(t\right)
\end{equation}$$
ImmediateDrugLinear_S0Alin
- quadratic model
$$\begin{equation}
A\left(t\right)=A_{lin}C\left(t\right)+A_{quad}C\left(t\right)^{2}
\end{equation}$$
ImmediateDrugImaxQuadratic_S0AlinAquad
- logarithmic model
$$\begin{equation}
A\left(t\right)=A_{log}log(C\left(t\right))
\end{equation}$$
ImmediateDrugImaxLogarithmic_S0Alog
- $E_{max}$ model
$$\begin{equation}
A\left(t\right)=\frac{E_{max}C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}$$
ImmediateDrugEmax_S0EmaxC50
- sigmoïd $E_{max}$ model
$$\begin{equation}
A\left(t\right)=\frac{E_{max}C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}}
\end{equation}$$
ImmediateDrugSigmoidEmax_S0EmaxC50gamma
- $I_{max}$ model
$$\begin{equation}
A\left(t\right)=1-\frac{I_{max}C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}$$
ImmediateDrugImax_S0ImaxC50
- sigmoïd $I_{max}$ model
$$\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}$$
ImmediateDrugImax_S0ImaxC50_gamma
-
full $I_{max}$ model
$$\begin{equation}
A\left(t\right)=-\frac{C\left(t\right)}{C\left(t\right)+C_{50}}
\end{equation}$$
-
sigmoïd full $I_{max}$ model
$$\begin{equation}
A\left(t\right)=-\frac{C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}}
\end{equation}$$
ImmediateDrugImax_S0ImaxC50_gamma
Baseline/disease models {#baselinediseasemodel}
- null baseline
$$\begin{equation}
S\left(t\right)=0
\end{equation}$$
ImmediateBaselineConstant_S0
- constant baseline with no disease progression
$$\begin{equation}
S\left(t\right)=S_{0}
\end{equation}$$
ImmediateBaselineConstant_S0
- linear disease progression
$$\begin{equation}
S\left(t\right)=S_{0}+k_{prog}t
\end{equation}$$
ImmediateBaselineLinear_S0kprog
- exponential disease increase
$$\begin{equation}
S\left(t\right)=S_{0}e^{-k_{prog}t}
\end{equation}$$
ImmediateBaselineExponentialincrease_S0kprog
- exponential disease decrease
$$\begin{equation}
S\left(t\right)=S_{0}\left(1-e^{-k_{prog}t}\right)
\end{equation}$$
ImmediateBaselineExponentialdecrease_S0kprog
Turnover response models
In these models, the drug is not acting on the effect $E$ directly but rather on $R_{in}$ or $k_{out}$.
Thus the system is described with differential equations, given ${\frac{dE}{dt}}$ as a function of $R_{in}$, $k_{out}$ and $C\left(t\right)$ the drug concentration at time t.
The initial condition is: while $C\left(t\right)=0$, $E\left(t\right)= {\frac{R_{in}}{k_{out}}}$.
Parameters
- $E_{max}$: maximal agonistic response
- $I_{max}$: maximal antagonistic response
- $C_{50}$: concentration to get half of the maximal response (=drug potency)
- $\gamma$: sigmoidicity factor
- $R_{in}$: input (synthesis) rate
- $k_{out}$: output (elimination) rate constant
Models with impact on the input $(R_{in})$
- $E_{max}$ model
$$\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}$$
TurnoverRinEmax_RinEmaxCC50koutE
- sigmoïd $E_{max}$ model
$$\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}$$
TurnoverRinSigmoidEmax_RinEmaxCC50koutE
- $I_{max}$ model
$$\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}$$
TurnoverRinFullImax_RinCC50koutE
- sigmoïd $I_{max}$ model
$$\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}$$
TurnoverRinImax_RinImaxCC50koutE
- full $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{C}{C+C_{50}}\right)-k_{out}E
\label{indirect_fimax}
\end{equation}$$
TurnoverRinSigmoidImax_RinImaxCC50koutE
- sigmoïd full $I_{max}$ model
$$\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}$$
TurnoverRinFullImax_RinCC50koutE
Models with impact on the output $(k_{out})$
- $E_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1+\frac{E_{max}C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutEmax_RinEmaxCC50koutE
- sigmoïd $E_{max}$ model
$$\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}$$
TurnoverkoutSigmoidEmax_RinEmaxCC50koutEgamma
- $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{I_{max}C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutImax_RinImaxCC50koutE
- sigmoïd $I_{max}$ model
$$\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}$$
TurnoverkoutSigmoidImax_RinImaxCC50koutEgamma
- full $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutFullImax_RinCC50koutE
- sigmoïd full $I_{max}$ model
$$\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}$$
TurnoverkoutSigmoidFullImax_RinCC50koutE
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<!-- The document presents the library of pharmacokinetic (PK) and pharmacodynamic (PD) models incorporated into the PFIM software. -->
# Pharmacokinetic models
## Compartmental models and parameters
Six parameters are common to one, two or three compartment models:
+ $V$ or $V_1$, the volume of distribution in the central compartment
+ $k$, the elimination rate constant
+ $CL$, the clearance of elimination
+ $V_m$, the maximum elimination rate for Michaelis-Menten elimination
+ $K_m$, the Michaelis-Menten constant
+ $k_a$, the absorption rate constant for oral administration
## One-compartment models
There are two parameterisations implemented in PFIM for one-compartment models, $\left(V\text{ and }k\right)$ or $\left(V\text{ and }CL\right)$. The equations are given for the first parameterisation $\left(V, k\right)$. For extra-vascular administration, $V$ and $CL$ are apparent volume and clearance. The equations for the second parameterisation $\left(V, CL\right)$ are derived using $k={\frac{CL}{V}}$.
## Models with linear elimination
### One-compartment models
#### Intravenous bolus
+ single dose
$$\begin {equation}
\begin{aligned}
C\left(t\right)=\frac{D}{V}e^{-k\left(t-t_{D}\right)}
\end{aligned}
\end {equation}$$
+ multiple doses
$$\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}$$
+ Library of models
```r
Linear1BolusSingleDose_kV
Linear1BolusSingleDose_ClV
- steady state
$$\begin {equation}
C(t)=\frac{D}{V}\frac{e^{-k(t-t_D)}}{1-e^{-k\tau}}\
\end {equation}$$
Linear1BolusSteadyState_kVtau Linear1BolusSteadyState_ClVtau
Infusion
- single dose
$$\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}$$
- multiple doses
$$\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} $$
Linear1InfusionSingleDose_kV Linear1InfusionSingleDose_ClV
- steady state
$$\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}$$
Linear1InfusionSteadyState_kVtau Linear1InfusionSteadyState_ClVtau
First order absorption
- single dose
$$\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}$$
- multiple doses
$$\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} $$
Linear1FirstOrderSingleDose_kakV Linear1FirstOrderSingleDose_kaClV
- steady state
$$\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}$$
Linear1FirstOrderSteadyState_kakVtau Linear1FirstOrderSteadyState_kaClVtau
Two-compartment models
For two-compartment model equations, $C(t)=C_1(t)$ represent the drug concentration in the first compartment and $C_2(t)$ represents the drug concentration in the second compartment.
As well as the previously described PK parameters, the following PK parameters are used for the two-compartment models:
- $V_2$, the volume of distribution of second compartment
- $k_{12}$, the distribution rate constant from compartment 1 to compartment 2
- $k_{21}$, the distribution rate constant from compartment 2 to compartment 1
- $Q$, the inter-compartmental clearance
- $\alpha$, the first rate constant
- $\beta$, the second rate constant
- $A$, the first macro-constant
- $B$, the second macro-constant
There are two parameterisations implemented in PFIM for two-compartment models: $\left(V\text{, }k\text{, }k_{12}\text{ and }k_{21}\right)$, or $\left(CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$. For extra-vascular administration, $V_1$ ($V$), $V_2$, $CL$, and $Q$ are apparent volumes and clearances.
The second parameterisation terms are derived using:
- $V_1=V$
- $CL=k \times V_1$
- $Q=k_{12} \times V_1$
- $V_2= {\frac{k_{12}}{k_{21}}}\times V_1$
For readability, the equations for two-compartment models with linear elimination are given using the variables $\alpha\text{, }\beta\text{, }A\text{ and }B$ defined by the following expressions:
$$\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}}$$
- single dose
$$\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}$$
- multiples doses
$$\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} $$
Linear2BolusSingleDose_ClQV1V2 Linear2BolusSingleDose_kk12k21V
- steady state
$$\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}$$
Linear2BolusSteadyState_ClQV1V2tau Linear2BolusSteadyState_kk12k21Vtau
Infusion
For infusion, the link between $A$ and $B$, and the parameters ($V$, $k$, $k_{12}$ and $k_{21}$), or ($CL$, $V_1$, $Q$ and $V_2$) is defined as follows:
$$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}}$$
- single dose
$$ \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} $$
- multiple doses
$$\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} $$
Linear2InfusionSingleDose_kk12k21V, Linear2InfusionSingleDose_ClQV1V2,
- steady state
$$\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}$$
Linear2InfusionSteadyState_kk12k21Vtau Linear2InfusionSteadyState_ClQV1V2tau
First-order absorption
For first order absorption, the link between $A$ and $B$, and the parameters ($k_a$, $V$, $k$, $k_{12}$ and $k_{21}$), or $\left(k_a\text{, } CL\text{, }V_1\text{, }Q\text{ and }V_2\right)$ is defined as follows:
$$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)}}$$
- single dose
$$ \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}$$
- multiple doses
$$\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}$$
Linear2FirstOrderSingleDose_kaClQV1V2 Linear2FirstOrderSingleDose_kakk12k21V
- steady state
$$\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}$$
Linear2FirstOrderSteadyState_kaClQV1V2tau Linear2FirstOrderSteadyState_kakk12k21Vtau
Models with Michaelis-Menten elimination
One-compartment models
Intravenous bolus
- single dose
$$\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}$$
MichaelisMenten1BolusSingleDose_VmKmV
Infusion
- single dose
$$ \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} $$
- multiple doses
$$\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}$$
??????
First order absorption
- single dose
$$ \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}$$
- multiple doses
$$ \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}$$
MichaelisMenten1FirstOrderSingleDose_kaVmKmV, MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2
Two-compartment models
Intravenous bolus
- single dose
$$ \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}$$
MichaelisMenten2BolusSingleDose_VmKmk12k21V1V2
Infusion
- single dose
$$ \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}$$
- multiple doses
$$\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}$$
MichaelisMenten2InfusionSingleDose_VmKmk12k21V1V2
First order absorption
- single dose
$$ \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}$$
- multiple doses
$$\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} $$
MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2 MichaelisMenten2FirstOrderSingleDose_kaVmKmk12k21V1V2
Pharmacodynamic models
Immediate response models
For these response models, the effect $E\left(t\right)$ is expressed as:
$$ \begin{equation} E\left(t\right)=A\left(t\right)+S\left(t\right) \end{equation}$$
\noindent where $A\left(t\right)$ represents the model of drug action and $S\left(t\right)$ corresponds to the baseline/disease model. $A\left(t\right)$ is a function of the concentration $C\left(t\right)$ in the central compartment.
The drug action models are presented in section Drug action models for $C(t)$. The baseline/disease models are presented in section Baseline/disease models. Any combination of those two models is available in the PFIM library.
Parameters
- $A_{lin}$: constant associated to $C\left(t\right)$
- $A_{quad}$: constant associated to the square of $C\left(t\right)$
- $A_{log}$: constant associated to the logarithm of $C\left(t\right)$
- $E_{max}$: maximal agonistic response
- $I_{max}$: maximal antagonistic response
- $C_{50}$: concentration to get half of the maximal response (\textit{i.e.} drug potency)
- $\gamma$: sigmoidicity factor
- $S_0$: baseline value of the studied effect
- $k_{prog}$: rate constant of disease progression
NB: $V_m$ is in concentration per time unit and $K_m$ is in concentration unit.
Drug action models {#drugactionmodels}
- linear model $$\begin{equation} A\left(t\right)=A_{lin}C\left(t\right) \end{equation}$$
ImmediateDrugLinear_S0Alin
- quadratic model $$\begin{equation} A\left(t\right)=A_{lin}C\left(t\right)+A_{quad}C\left(t\right)^{2} \end{equation}$$
ImmediateDrugImaxQuadratic_S0AlinAquad
- logarithmic model $$\begin{equation} A\left(t\right)=A_{log}log(C\left(t\right)) \end{equation}$$
ImmediateDrugImaxLogarithmic_S0Alog
- $E_{max}$ model $$\begin{equation} A\left(t\right)=\frac{E_{max}C\left(t\right)}{C\left(t\right)+C_{50}} \end{equation}$$
ImmediateDrugEmax_S0EmaxC50
- sigmoïd $E_{max}$ model $$\begin{equation} A\left(t\right)=\frac{E_{max}C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}} \end{equation}$$
ImmediateDrugSigmoidEmax_S0EmaxC50gamma
- $I_{max}$ model $$\begin{equation} A\left(t\right)=1-\frac{I_{max}C\left(t\right)}{C\left(t\right)+C_{50}} \end{equation}$$
ImmediateDrugImax_S0ImaxC50
- sigmoïd $I_{max}$ model $$\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}$$
ImmediateDrugImax_S0ImaxC50_gamma
-
full $I_{max}$ model $$\begin{equation} A\left(t\right)=-\frac{C\left(t\right)}{C\left(t\right)+C_{50}} \end{equation}$$
-
sigmoïd full $I_{max}$ model $$\begin{equation} A\left(t\right)=-\frac{C\left(t\right)^{\gamma}}{C\left(t\right)^{\gamma}+C_{50}^{\gamma}} \end{equation}$$
ImmediateDrugImax_S0ImaxC50_gamma
Baseline/disease models {#baselinediseasemodel}
- null baseline
$$\begin{equation} S\left(t\right)=0 \end{equation}$$
ImmediateBaselineConstant_S0
- constant baseline with no disease progression
$$\begin{equation} S\left(t\right)=S_{0} \end{equation}$$
ImmediateBaselineConstant_S0
- linear disease progression
$$\begin{equation} S\left(t\right)=S_{0}+k_{prog}t \end{equation}$$
ImmediateBaselineLinear_S0kprog
- exponential disease increase
$$\begin{equation} S\left(t\right)=S_{0}e^{-k_{prog}t} \end{equation}$$
ImmediateBaselineExponentialincrease_S0kprog
- exponential disease decrease
$$\begin{equation} S\left(t\right)=S_{0}\left(1-e^{-k_{prog}t}\right) \end{equation}$$
ImmediateBaselineExponentialdecrease_S0kprog
Turnover response models
In these models, the drug is not acting on the effect $E$ directly but rather on $R_{in}$ or $k_{out}$.
Thus the system is described with differential equations, given ${\frac{dE}{dt}}$ as a function of $R_{in}$, $k_{out}$ and $C\left(t\right)$ the drug concentration at time t.
The initial condition is: while $C\left(t\right)=0$, $E\left(t\right)= {\frac{R_{in}}{k_{out}}}$.
Parameters
- $E_{max}$: maximal agonistic response
- $I_{max}$: maximal antagonistic response
- $C_{50}$: concentration to get half of the maximal response (=drug potency)
- $\gamma$: sigmoidicity factor
- $R_{in}$: input (synthesis) rate
- $k_{out}$: output (elimination) rate constant
Models with impact on the input $(R_{in})$
- $E_{max}$ model
$$\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}$$
TurnoverRinEmax_RinEmaxCC50koutE
- sigmoïd $E_{max}$ model
$$\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}$$
TurnoverRinSigmoidEmax_RinEmaxCC50koutE
- $I_{max}$ model
$$\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}$$
TurnoverRinFullImax_RinCC50koutE
- sigmoïd $I_{max}$ model
$$\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}$$
TurnoverRinImax_RinImaxCC50koutE
- full $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}\left(1-\frac{C}{C+C_{50}}\right)-k_{out}E
\label{indirect_fimax} \end{equation}$$
TurnoverRinSigmoidImax_RinImaxCC50koutE
- sigmoïd full $I_{max}$ model
$$\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}$$
TurnoverRinFullImax_RinCC50koutE
Models with impact on the output $(k_{out})$
- $E_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1+\frac{E_{max}C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutEmax_RinEmaxCC50koutE
- sigmoïd $E_{max}$ model
$$\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}$$
TurnoverkoutSigmoidEmax_RinEmaxCC50koutEgamma
- $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{I_{max}C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutImax_RinImaxCC50koutE
- sigmoïd $I_{max}$ model
$$\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}$$
TurnoverkoutSigmoidImax_RinImaxCC50koutEgamma
- full $I_{max}$ model
$$\begin{equation}
\frac{dE}{dt}=R_{in}-k_{out}\left(1-\frac{C}{C+C_{50}}\right)E
\end{equation}$$
TurnoverkoutFullImax_RinCC50koutE
- sigmoïd full $I_{max}$ model
$$\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}$$
TurnoverkoutSigmoidFullImax_RinCC50koutE
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