# Library of models

# 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