Library of models

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

  • 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

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

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