Tucuxi PK models

Linear elimination, 1 compartment, macro constants

Pk model Id : linear.1comp.macro

Bolus

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]

Equations:

\[\frac{dC}{dt} = - \frac{CL}{V} C\]

Initial conditions:

\[C(t_0) = C_{residual} + \frac{D}{V}\]

Infusion

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} - \frac{CL}{V} C , &\text{for $t\leq t_0 + Tinf$}\\ - \frac{CL}{V} C, &\text{for $t > t_0 + Tinf$} \end{cases} \end{align*}

Initial conditions:

\[C(t_0) = C_{residual}\]

Extra

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation in [0,1], [-]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} \cdot C1 \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]
Tlag	Lag time, [h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} \cdot C1 \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2(Tlag - \delta) + \frac{F \cdot D}{V}\end{split}\]

Linear elimination, 1 compartment, micro constants

Pk model Id : linear.1comp.micro

Bolus

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]

Equations:

\[\frac{dC}{dt} = - Ke \cdot C\]

Initial conditions:

\[C(t_0) = C_{residual} + \frac{D}{V}\]

Infusion

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} - Ke \cdot C , &\text{for $t\leq t_0 + Tinf$}\\ - Ke \cdot C, &\text{for $t > t_0 + Tinf$} \end{cases} \end{align*}

Initial conditions:

\[C(t_0) = C_{residual}\]

Extra

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]
Tlag	Lag time, [h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2(Tlag - \delta) + \frac{F \cdot D}{V}\end{split}\]

Linear elimination, 2 compartments, macro constants

Pk model Id : linear.2comp.macro

Bolus

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual}\end{split}\]

Infusion

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 , &\text{for $t\leq t_0 + Tinf$}\\ \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\ \end{align*}

Initial conditions:

\[\begin{split}C1(t_0) = C1_{residual} \\ C2(t_0) = C2_{residual} \\\end{split}\]

Extra

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]
Tlag	Lag time, [h]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} \cdot C2 - \frac{CL}{V1} \cdot C1 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= -Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Linear elimination, 2 compartments, micro constants

Pk model Id : linear.2comp.micro

Bolus

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual}\end{split}\]

Infusion

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 , &\text{for $t\leq t_0 + Tinf$}\\ K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \end{align*}

Initial conditions:

\[\begin{split}C1(t_0) = C1_{residual} \\ C2(t_0) = C2_{residual} \\\end{split}\]

Extra

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ka \cdot C3 + K21 \cdot C2 - Ke \cdot C1 - K12 \cdot C1 \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= -Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Linear elimination, 2 compartments, macro constants, with Q and V2 as ratios of CL and V1

Pk model Id : linear.2comp.macroRatios

Bolus

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, also used as the inter-compartmental clearance Q [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
RQCL	Ratio between Q and CL, where Q is the inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [-]
RV2V1	Ratio between V2 and V1, where V2 is the peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [-]

Equations:

\[\begin{split}Q &= RQCL \cdot CL \\ V2 &= RV2V1 \cdot V1 \\ \frac{dC1}{dt} &= \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual}\end{split}\]

Infusion

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, also used as the inter-compartmental clearance Q [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
RQCL	Ratio between Q and CL, where Q is the inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [-]
RV2V1	Ratio between V2 and V1, where V2 is the peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [-]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} Q &= RQCL \cdot CL \\ V2 &= RV2V1 \cdot V1 \\ \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 , &\text{for $t\leq t_0 + Tinf$}\\ \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\ \end{align*}

Initial conditions:

\[\begin{split}C1(t_0) = C1_{residual} \\ C2(t_0) = C2_{residual} \\\end{split}\]

Extra

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, also used as the inter-compartmental clearance Q [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
RQCL	Ratio between Q and CL, where Q is the inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [-]
RV2V1	Ratio between V2 and V1, where V2 is the peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [-]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]

Equations:

\[\begin{split}Q &= RQCL \cdot CL \\ V2 &= RV2V1 \cdot V1 \\ \frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} C2 - \frac{Q}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} C2 + \frac{Q}{V1} C1 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V}\end{split}\]

Linear, 3 compartments, macro constants

Pk model Id : linear.3comp.macro

Bolus

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
V3	Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with retardation, from and back to the central compartment, [L]
Q2	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the first peripheral compartments over its circulating concentration , [L/h]
Q3	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the second peripheral compartments over its circulating concentration , [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= \frac{Q3}{V3} C3 - \frac{Q3}{V1} C1 + \frac{Q2}{V2} C2 - \frac{Q2}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q2}{V2} C2 + \frac{Q2}{V1} C1 \\ \frac{dC3}{dt} &= - \frac{Q3}{V3} C3 + \frac{Q3}{V1} C1\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C2_{residual}\end{split}\]

Infusion

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
V3	Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with retardation, from and back to the central compartment, [L]
Q2	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the first peripheral compartments over its circulating concentration , [L/h]
Q3	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the second peripheral compartments over its circulating concentration , [L/h]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + \frac{Q3}{V3} C3 - \frac{Q3}{V1} C1 + \frac{Q2}{V2} C2 - \frac{Q2}{V1} C1 - \frac{CL}{V1} \cdot C1 , &\text{for $t\leq t_0 + Tinf$}\\ \frac{Q3}{V3} C3 - \frac{Q3}{V1} C1 + \frac{Q2}{V2} C2 - \frac{Q2}{V1} C1 - \frac{CL}{V1} C1, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - \frac{Q2}{V2} C2 + \frac{Q2}{V1} C1 \\ \frac{dC3}{dt} &= - \frac{Q3}{V3} C3 + \frac{Q3}{V1} C1 \end{align*}

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C2_{residual}\end{split}\]

Extra

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
V3	Peripheral volume of distribution, third compartment. Volume into which a drug is considered to distribute thirdly with retardation, from and back to the central compartment, [L]
Q2	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the first peripheral compartments over its circulating concentration , [L/h]
Q3	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the second peripheral compartments over its circulating concentration , [L/h]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C4 + \frac{Q3}{V3} C1 + \frac{Q2}{V2} C2 - \frac{Q3}{V1} C1 - \frac{Q2}{V1} C1 - \frac{CL}{V1} C1 \\ \frac{dC2}{dt} &= - \frac{Q2}{V2} C2 + \frac{Q2}{V1} C1 \\ \frac{dC3}{dt} &= - \frac{Q3}{V3} C3 + \frac{Q3}{V1} C1 \\ \frac{dC4}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C4(t_0) &= C4_{residual}+ \frac{F \cdot D}{V1}\end{split}\]

Linear, 3 compartments, micro constants

Pk model Id : linear.3comp.micro

Bolus

Parameters

Name	Description
K12	Elimination rate constant of drug from central compartment to peripheral compartment 2, [1/h]
K21	Elimination rate constant of drug from peripheral compartment 2 to central compartment, [1/h]
K13	Elimination rate constant of drug from central compartment to peripheral compartment 3, [1/h]
K31	Elimination rate constant of drug from peripheral compartment 3 to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= K31 \cdot C3 - K13 \cdot C1 + K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - K31 \cdot C3 + K13 \cdot C1\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C2_{residual}\end{split}\]

Infusion

Parameters

Name	Description
K12	Elimination rate constant of drug from central compartment to peripheral compartment 2, [1/h]
K21	Elimination rate constant of drug from peripheral compartment 2 to central compartment, [1/h]
K13	Elimination rate constant of drug from central compartment to peripheral compartment 3, [1/h]
K31	Elimination rate constant of drug from peripheral compartment 3 to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
Tinf	Infusion time (related to the dosage), [h]

Equations:

\begin{align*} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + K31 \cdot C3 - K13 \cdot C1 + K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 , &\text{for $t\leq t_0 + Tinf$}\\ K31 \cdot C3 - K13 \cdot C3 + K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - K31 \cdot C3 + K13 \cdot C1 \end{align*}

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C2_{residual}\end{split}\]

Extra

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
F	Bioavailability. Fraction of the drug’s administered dose that reaches unchanged the systemic circulation, in [0,1] [-]
K12	Elimination rate constant of drug from central compartment to peripheral compartment 2, [1/h]
K21	Elimination rate constant of drug from peripheral compartment 2 to central compartment, [1/h]
K13	Elimination rate constant of drug from central compartment to peripheral compartment 3, [1/h]
K31	Elimination rate constant of drug from peripheral compartment 3 to central compartment, [1/h]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C4 + K31 \cdot C3 + K21 \cdot C2 - K13 \cdot C1 - K12 \cdot C1 - Ke \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - K31 \cdot C3 + K13 \cdot C1 \\ \frac{dC4}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C4(t_0) &= C4_{residual}+ \frac{F \cdot D}{V1}\end{split}\]

Linear, 2 compartments, erlang absorption with 1 transit compartment, macro constants

Pk model Id : linear.2comp.erlang1.macro

Parameters :header: “Name”, “Description” :widths: 5, 40

CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C4 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 1 transit compartment, micro constants

Pk model Id : linear.2comp.erlang1.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C4 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 2 transit compartment, macro constants

Pk model Id : linear.2comp.erlang2.macro

Parameters :header: “Name”, “Description” :widths: 5, 40

CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ktr \cdot C5 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 2 transit compartment, micro constants

Pk model Id : linear.2comp.erlang2.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C5 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 3 transit compartment, macro constants

Pk model Id : linear.2comp.erlang3.macro

Parameters :header: “Name”, “Description” :widths: 5, 40

CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ktr \cdot C6 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6\end{split}\]

Extra

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 3 transit compartment, micro constants

Pk model Id : linear.2comp.erlang3.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C6 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 4 transit compartment, macro constants

Pk model Id : linear.2comp.erlang4.macro

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ktr \cdot C7 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7\end{split}\]

Extra

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 4 transit compartment, micro constants

Pk model Id : linear.2comp.erlang4.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C7 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 5 transit compartment, macro constants

Pk model Id : linear.2comp.erlang5.macro

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split} \frac{dC1}{dt} &= Ktr \cdot C8 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7 \\ \frac{dC8}{dt} &= Ktr \cdot C7 - Ktr \cdot C8\end{split}\]

Extra

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual} \\ C8(t_0) &= C8_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 5 transit compartment, micro constants

Pk model Id : linear.2comp.erlang5.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C8 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7 \\ \frac{dC8}{dt} &= Ktr \cdot C7 - Ktr \cdot C8\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual} \\ C8(t_0) &= C8_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 6 transit compartment, macro constants

Pk model Id : linear.2comp.erlang6.macro

Parameters

Name	Description
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C9 - \frac{CL}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Q}{V1} \cdot C1 \\ \frac{dC2}{dt} &= - \frac{Q}{V2} \cdot C2 + \frac{Q}{V1} \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7 \\ \frac{dC8}{dt} &= Ktr \cdot C7 - Ktr \cdot C8 \\ \frac{dC9}{dt} &= Ktr \cdot C8 - Ktr \cdot C9\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual} \\ C8(t_0) &= C8_{residual} \\ C9(t_0) &= C9_{residual}\end{split}\]

Linear, 2 compartments, erlang absorption with 6 transit compartment, micro constants

Pk model Id : linear.2comp.erlang6.micro

Parameters

Name	Description
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ktr \cdot C9 - Ke \cdot C1 + K21 \cdot C2 - K12 \cdot C1 \\ \frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\ \frac{dC3}{dt} &= - Ktr \cdot C3 \\ \frac{dC4}{dt} &= Ktr \cdot C3 - Ktr \cdot C4 \\ \frac{dC5}{dt} &= Ktr \cdot C4 - Ktr \cdot C5 \\ \frac{dC6}{dt} &= Ktr \cdot C5 - Ktr \cdot C6 \\ \frac{dC7}{dt} &= Ktr \cdot C6 - Ktr \cdot C7 \\ \frac{dC8}{dt} &= Ktr \cdot C7 - Ktr \cdot C8 \\ \frac{dC9}{dt} &= Ktr \cdot C8 - Ktr \cdot C9\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{D}{V1} \\ C4(t_0) &= C4_{residual} \\ C5(t_0) &= C5_{residual} \\ C6(t_0) &= C6_{residual} \\ C7(t_0) &= C7_{residual} \\ C8(t_0) &= C8_{residual} \\ C9(t_0) &= C9_{residual}\end{split}\]

Michaelis-Menten, 1 compartment

Pk model Id : michaelismenten.1comp

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [concentration/h]
Km	The Michaelis Menten constant, [concentration]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - \frac{Vmax\cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten, 1 compartment, with Vmax as an amount

Pk model Id : michaelismenten.1comp.vmaxamount

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [concentration/h]
Km	The Michaelis Menten constant, [concentration]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - \frac{\frac{Vmax}{V} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - \frac{\frac{Vmax}{V} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 1 compartment, macro constants

Pk model Id : michaelismentenlinear.1comp.macro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [concentration/h]
Km	The Michaelis Menten constant, [concentration]
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} C1 - \frac{Vmax\cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - \frac{CL}{V} C1 - \frac{Vmax\cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - \frac{CL}{V} C1 - \frac{Vmax\cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 1 compartment, micro constants

Pk model Id : michaelismentenlinear.1comp.micro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [concentration/h]
Km	The Michaelis Menten constant, [concentration]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 - \frac{Vmax\cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - Ke \cdot C1 - \frac{Vmax\cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - Ke \cdot C1 - \frac{Vmax\cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 1 compartment, with Vmax as an amount, macro constants

Pk model Id : michaelismentenlinear.1comp.vmaxamount.macro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - \frac{CL}{V} C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - \frac{CL}{V} C1 - \frac{\frac{Vmax}{V} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 1 compartment, micro constants

Pk model Id : michaelismentenlinear.1comp.vmaxamount.micro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
V	Volume of distribution. Ratio of the drug’s amount present in the body over its circulating concentration, [L]
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\ \frac{dC2}{dt} &= - Ka \cdot C2 \\\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\ C2(t_0) &= C2_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C2(Tlag) &= C2_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= - Ka \cdot C2\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual}\end{split}\]

Michaelis-Menten, 2 compartments, macro constants

Pk model Id : michaelismenten.2comp.macro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Vmax	Maximal elimination rate, [concentration/h]. Be careful, as the concentration unit should be the same as the active moiety and analyte group unit.
Km	The Michaelis Menten constant, [concentration]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 - \frac{Q}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

Equations:

\[\begin{split} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + Ka \cdot C3 - \frac{Q}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 - \frac{Q}{V1} \cdot C1 + \frac{Q}{V2} \cdot C2 - \frac{Vmax\cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten, 2 compartments, micro constants

Pk model Id : michaelismenten.2comp.micro

In this model, the concentration unit for Vmax and Km shall be the same as the unit of the active moiety and the analytes.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Vmax	Maximal elimination rate, [concentration/h]
Km	The Michaelis Menten constant, [concentration]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 - K12 \cdot C1 + K21 \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[ \begin{align}\begin{aligned}\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\\ .. _michaelismenten.2comp.micro.extra:\end{aligned}\end{align} \]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + Ka \cdot C3 - K12 \cdot C1 + K21 \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 - K12 \cdot C1 + K21 \cdot C2 - \frac{Vmax\cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten, 2 compartment, with Vmax as an amount, macro constants

Pk model Id : michaelismenten.2comp.vmaxamount.macro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten, 2 compartment, with Vmax as an amount, micro constants

Pk model Id : michaelismenten.2comp.vmaxamount.micro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - \frac{\frac{Vmax}{V1} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 2 compartment, macro constants

Pk model Id : _michaelismentenlinear.2comp.macro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Vmax	Maximal elimination rate, [concentration]
Km	The Michaelis Menten constant, [concentration]
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} \cdot C1 - \frac{Vmax \cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} \cdot C1 - \frac{Vmax \cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} - \frac{Vmax \cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 2 compartment, micro constants

Pk model Id : michaelismenten.2comp.micro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Vmax	Maximal elimination rate, [concentration]
Km	The Michaelis Menten constant, [concentration]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{Vmax \cdot C1}{Km + C1} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{Vmax \cdot C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{Vmax \cdot C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 2 compartment, with Vmax as an amount, macro constants

Pk model Id : _michaelismentenlinear.2comp.vmaxamount.macro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
V2	Peripheral volume of distribution, second compartment. Volume into which a drug is considered to distribute secondly with retardation, from and back to the central compartment, [L]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]
CL	Clearance. Ratio of the drug’s elimination rate from the body over its circulating concentration, [L/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} \cdot C1 - \frac{\frac{Vmax}{V1} C1}{Km + C1} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} \cdot C1 - \frac{\frac{Vmax}{V1} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + \frac{Q}{V2} \cdot C1 - \frac{Q}{V1} \cdot C2 - \frac{CL}{V1} - \frac{\frac{Vmax}{V1} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= \frac{Q}{V1} \cdot C1 - \frac{Q}{V2} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten and linear elimination, 2 compartment, with Vmax as an amount, micro constants

Pk model Id : michaelismenten.2comp.vmaxamount.micro

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
K12	Elimination rate constant of drug from central compartment to peripheral compartment, [1/h]
K21	Elimination rate constant of drug from peripheral compartment to central compartment, [1/h]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]
V1	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
Ke	Elimination rate constant. Relative rate constant of the drug’s elimination from the body, [1/h]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V1} C1}{Km + C1} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Infusion

Equations:

\[\begin{split}\frac{dC1}{dt} &= \begin{cases} \frac{D}{V\cdot Tinf} + Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V1} C1}{Km + C1} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 + K21 \cdot C1 - K12 \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V1} C1}{Km + C1}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= K12 \cdot C1 - K21 \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Michaelis-Menten, 1 compartment, with enzyme interaction

Pk model Id : michaelismenten.enzyme.1comp

This model has been specifically implemented for Rifampicin (Svensson model). It considers the concentrations to be in [mg/l]

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Vmax	Maximal elimination rate, [amount/h]
AllmCL	Allometric scaling on clearance, [-]
Km	The Michaelis Menten constant, [concentration]
MTT	Mean transit time, [h]
NN	Number of transits, [-]
Kenz	First-order rate constant for enzyme pool degradation, [1/h]
Emax	Maximal increase in enzyme production rate, [-]
ECmid	concentration at which half the Emax is reached, [mg/l]
Dmid	Half dose, [mg]
Ktr	Elimination rate constant of drug from each transit compartment to the next one, [1/h]
F	relative bioavailability in [0,1], [-]
Fmax	Maximal increase in relative bioavailability above dose, [-]
EDmid	Difference in dose from standard dose at which half the Fmax is reached, [mg]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C2 - \frac{Vmax \cdot C1 \cdot AllmCL \cdot C3}{Km + C1}\\ \frac{dC2}{dt} &= \begin{cases} - Ka \cdot C2 , &\text{for $MTT = 0$}\\ - Ka \cdot C2 + 2^{\log_2(e) \cdot \frac{NN^2 - 1}{MTT} + cumul} , &\text{for $MTT \ne 0$} \end{cases}\\ \frac{dC3}{dt} &= Kenz \cdot \left( 1 - C3 + \frac{C1 \cdot Emax}{ECmid + C1} \right)\end{split}\]

with:

\[cumul = \ln\left(bio \cdot D\right) + \ln\left(Ktr\right) - \left( 0.918935 + (NN + 0.5) \cdot \ln(NN) - NN + \ln\left(1 + \frac{1}{12 \cdot NN}\right) \right)\]

with:

\[\begin{split} bio = \begin{cases} F \cdot \frac{1 + Fmax \cdot \left(low\_bound - Dmid\right)}{EDmid + \left(low\_bound - Dmid\right)} , &\text{for $low\_bound < D < crit\_point$}\\ F \cdot \frac{1 + Fmax \cdot \left(high\_bound - Dmid\right)}{EDmid + \left(high\_bound - Dmid\right)} , &\text{for $crit\_point < D < high\_bound$}\\ F \cdot \frac{1 + Fmax \cdot \left(D - Dmid\right)}{EDmid + \left(D - Dmid\right)} , &\text{$otherwise$} \end{cases}\\\end{split}\]

with:

\[\begin{split}crit\_point &= Dmid - EDmid\\ low\_bound &= crit\_point - 23\\ high\_bound &= crit\_point + 37\end{split}\]

Infusion

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split} C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Michaelis-Menten, 1 compartment, 2 analytes, with Vmax as an amount, macro constants

Pk model Id : michaelismenten.1comp.2analytes.vmaxamount.macro

This model is meant to be used for 2 analytes. The first analyte is in C1, and the second in C2.

In this model, the concentration unit for Km shall be the same as the unit of the active moiety and the analytes.

In this model, Vmax is an amount, and not a concentration. Its unit has to be consistent with the unit of the active moiety and the analyte groups.

Parameters

Name	Description
Ka	Absorption rate constant. Relative rate constant of the drug’s absorption into the body, [1/h]
Q	Inter-compartmental clearance. Ratio of the drug’s distribution rate between the central compartment and the peripheral compartments over its circulating concentration , [L/h]
V	Central Volume of distribution. Volume into which the drug distributes initially without delay after its delivery into the circulation, [L]
Fm	Fraction of drug eliminated that transforms into the second analyte, [-]
Vmax	Maximal elimination rate, [amount/h]
Km	The Michaelis Menten constant, [concentration]

Equations:

\[\begin{split}\frac{dC1}{dt} &= Ka \cdot C3 - \frac{Vmax\cdot C1}{(Km + C1)V} \\ \frac{dC2}{dt} &= Fm \cdot \frac{Vmax\cdot C1}{(Km + C1)V} - \frac{CL}{V} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Bolus

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} + \frac{D}{V1} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]

Extra

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} + \frac{F \cdot D}{V1}\end{split}\]

Extra Lag

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual} \\ C3(Tlag) &= C3_{residual} + \frac{F \cdot D}{V}\end{split}\]

Infusion

Equations:

\[\begin{split} \frac{dC1}{dt} &= \begin{cases} \frac{D}{V1\cdot Tinf} + Ka \cdot C3 - \frac{Vmax\cdot C1}{(Km + C1)V} , &\text{for $t\leq t_0 + Tinf$}\\ Ka \cdot C3 - \frac{Vmax\cdot C1}{(Km + C1)V}, &\text{for $t > t_0 + Tinf$} \end{cases} \\ \frac{dC2}{dt} &= Fm \cdot \frac{Vmax\cdot C1}{(Km + C1)V} - \frac{CL}{V} \cdot C2 \\ \frac{dC3}{dt} &= - Ka \cdot C3\end{split}\]

Initial conditions:

\[\begin{split}C1(t_0) &= C1_{residual} \\ C2(t_0) &= C2_{residual} \\ C3(t_0) &= C3_{residual}\end{split}\]