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

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

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:

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

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 ( 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

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

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

Library of models

Pharmacokinetic models

Compartmental models and parameters

One-compartment models

Models with linear elimination

One-compartment models

Intravenous bolus

Infusion

First order absorption

Two-compartment models

Intravenous bolus

Infusion

First-order absorption

Models with Michaelis-Menten elimination

One-compartment models

Intravenous bolus

Infusion

First order absorption

Two-compartment models

Intravenous bolus

Infusion

First order absorption

Pharmacodynamic models

Immediate response models

Drug action models

Baseline/disease models

Turnover response models

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

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

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

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