.. _pkmodels:

****************
Tucuxi PK models
****************

.. contents:: Available PK models:
	:depth: 3

.. _linear.1comp.macro:

Linear elimination, 1 compartment, macro constants
==================================================

**Pk model Id : linear.1comp.macro**


.. _linear.1comp.macro.bolus:

Bolus
-----

.. csv-table:: 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]"
	V, "Volume of distribution. Ratio of the drug's amount present in the body over its circulating concentration, [L]"


Equations:

.. math::

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

Initial conditions:

.. math::

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


.. _linear.1comp.macro.infusion:

Infusion
--------


.. csv-table:: 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]"
	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:

.. math::
	:nowrap:

	\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:

.. math::

	C(t_0) = C_{residual}


.. _linear.1comp.macro.extra:

Extra
-----

.. csv-table:: 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]"
	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} \cdot C1 \\
	\frac{dC2}{dt} &= - Ka \cdot C2

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _linear.1comp.macro.extra.lag:

Extra Lag
---------

.. csv-table:: 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]"
	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} \cdot C1 \\
	\frac{dC2}{dt} &= - Ka \cdot C2

Initial conditions:

.. math::

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


.. _linear.1comp.micro:

Linear elimination, 1 compartment, micro constants
==================================================

**Pk model Id : linear.1comp.micro**

.. _linear.1comp.micro.bolus:

Bolus
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

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

Initial conditions:

.. math::

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


.. _linear.1comp.micro.infusion:

Infusion
--------

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::
	:nowrap:

	\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:

.. math::

	C(t_0) = C_{residual}

.. _linear.1comp.micro.extra:

Extra
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 \\
	\frac{dC2}{dt} &= - Ka \cdot C2

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}


.. _linear.1comp.micro.extra.lag:

Extra Lag
---------

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 \\
	\frac{dC2}{dt} &= - Ka \cdot C2

Initial conditions:

.. math::

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



.. _linear.2comp.macro:

Linear elimination, 2 compartments, macro constants
===================================================


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



.. _linear.2comp.macro.bolus:

Bolus
-----

.. csv-table:: 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]"
	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:

.. math::

	\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 \\

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual}


.. _linear.2comp.macro.infusion:

Infusion
--------


.. csv-table:: 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]"
	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:

.. math::
	:nowrap:

	\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:

.. math::

	C1(t_0) = C1_{residual} \\
	C2(t_0) = C2_{residual} \\


.. _linear.2comp.macro.extra:

Extra
-----

.. csv-table:: 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]"
	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:

.. math::

	\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

Initial conditions:

.. math::

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


.. _linear.2comp.macro.extra.lag:

Extra Lag
---------

.. csv-table:: 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]"
	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:

.. math::

	\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

Initial conditions:

.. math::

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


.. _linear.2comp.micro:

Linear elimination, 2 compartments, micro constants
===================================================

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


.. _linear.2comp.micro.bolus:

Bolus
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= K21 \cdot C2 - K12 \cdot C1 - Ke \cdot C1 \\
	\frac{dC2}{dt} &= - K21 \cdot C2 + K12 \cdot C1 \\

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual}


.. _linear.2comp.micro.infusion:

Infusion
--------


.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::
	:nowrap:

	\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:

.. math::

	C1(t_0) = C1_{residual} \\
	C2(t_0) = C2_{residual} \\


.. _linear.2comp.micro.extra:

Extra
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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

Initial conditions:

.. math::

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


.. _linear.2comp.micro.extra.lag:

Extra Lag
---------

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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

Initial conditions:

.. math::

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


.. _linear.2comp.macroSameCl:

.. _linear.2comp.macroRatios:

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

**Pk model Id : linear.2comp.macroRatios**



.. _linear.2comp.macro.bolusRatios:

Bolus
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	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 \\

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual}


.. _linear.2comp.macro.infusionRatios:

Infusion
--------


.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::
	:nowrap:

	\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:

.. math::

	C1(t_0) = C1_{residual} \\
	C2(t_0) = C2_{residual} \\


.. _linear.2comp.macro.extraRatios:

Extra
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	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

Initial conditions:

.. math::

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



.. _linear.3comp.macro:

Linear, 3 compartments, macro constants
=======================================

**Pk model Id : linear.3comp.macro**


.. _linear.3comp.macro.bolus:

Bolus
-----

.. csv-table:: 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]"
	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:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C2_{residual}


.. _linear.3comp.macro.infusion:

Infusion
--------

.. csv-table:: 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]"
	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:

.. math::
	:nowrap:

	\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:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C2_{residual}


.. _linear.3comp.macro.extra:

Extra
-----

.. csv-table:: 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]"
	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:

.. math::

	\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

Initial conditions:

.. math::

	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}

.. _linear.3comp.micro:

Linear, 3 compartments, micro constants
=======================================

**Pk model Id : linear.3comp.micro**

.. _linear.3comp.micro.bolus:

Bolus
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C2_{residual}


.. _linear.3comp.micro.infusion:

Infusion
--------

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::
	:nowrap:

	\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:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C2_{residual}


.. _linear.3comp.micro.extra:

Extra
-----

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang1.macro:

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

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

.. csv-table:: 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:

.. math::

	\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


.. _linear.2comp.erlang1.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _linear.2comp.erlang1.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang1.micro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _linear.2comp.erlang2.macro:

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

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

.. csv-table:: 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:

.. math::

	\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

.. _linear.2comp.erlang2.macro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang2.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang2.micro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang3.macro:

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

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

.. csv-table:: 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:

.. math::

	\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

.. _linear.2comp.erlang3.macro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang3.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang3.micro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang4.macro:

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

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

.. csv-table:: 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:

.. math::

	\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

.. _linear.2comp.erlang4.macro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang4.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang4.micro.extra:

Extra
-----

Initial conditions:

.. math::

	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}


.. _linear.2comp.erlang5.macro:

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

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

.. csv-table:: 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:

.. math::

	\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

.. _linear.2comp.erlang5.macro.extra:

Extra
-----

Initial conditions:

.. math::

	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}

.. _linear.2comp.erlang5.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang5.micro.extra:

Extra
-----

Initial conditions:

.. math::

	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}


.. _linear.2comp.erlang6.macro:

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

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

.. csv-table:: 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:

.. math::

	\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


.. _linear.2comp.erlang6.macro.extra:

Extra
-----

Initial conditions:

.. math::

	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}


.. _linear.2comp.erlang6.micro:

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

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

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _linear.2comp.erlang6.micro.extra:

Extra
-----

Initial conditions:

.. math::

	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}




.. _michaelismenten.1comp:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{Vmax\cdot C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismenten.1comp.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismenten.1comp.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismenten.1comp.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.1comp.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}



.. _michaelismenten.1comp.vmaxamount:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismenten.1comp.vmaxamount.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismenten.1comp.vmaxamount.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismenten.1comp.vmaxamount.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.1comp.vmaxamount.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}






.. _michaelismentenlinear.1comp.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} C1 - \frac{Vmax\cdot C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismentenlinear.1comp.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismentenlinear.1comp.macro.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismentenlinear.1comp.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.1comp.macro.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}



.. _michaelismentenlinear.1comp.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 - \frac{Vmax\cdot C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismentenlinear.1comp.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismentenlinear.1comp.micro.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismentenlinear.1comp.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.1comp.micro.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}















.. _michaelismentenlinear.1comp.vmaxamount.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - \frac{CL}{V} C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismentenlinear.1comp.vmaxamount.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismentenlinear.1comp.vmaxamount.macro.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismentenlinear.1comp.vmaxamount.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.1comp.vmaxamount.macro.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}



.. _michaelismentenlinear.1comp.vmaxamount.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\frac{dC1}{dt} &= Ka \cdot C2 - Ke \cdot C1 - \frac{\frac{Vmax}{V} C1}{Km + C1} \\
	\frac{dC2}{dt} &= - Ka \cdot C2 \\


.. _michaelismentenlinear.1comp.vmaxamount.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{F \cdot D}{V} \\
	C2(t_0) &= C2_{residual}


.. _michaelismentenlinear.1comp.vmaxamount.micro.extra:

Extra
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} + \frac{F \cdot D}{V}

.. _michaelismentenlinear.1comp.vmaxamount.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.1comp.vmaxamount.micro.infusion:

Infusion
--------

.. math::

	\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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual}























.. _michaelismenten.2comp.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismenten.2comp.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}

.. _michaelismenten.2comp.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.macro.infusion:

Infusion
--------


Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismenten.2comp.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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

.. _michaelismenten.2comp.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


	.. _michaelismenten.2comp.micro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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



.. _michaelismenten.2comp.micro.infusion:

Infusion
--------

Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismenten.2comp.vmaxamount.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismenten.2comp.vmaxamount.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}

.. _michaelismenten.2comp.vmaxamount.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.vmaxamount.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.vmaxamount.macro.infusion:

Infusion
--------

Equations:

.. math::

  \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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}



.. _michaelismenten.2comp.vmaxamount.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismenten.2comp.vmaxamount.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismenten.2comp.vmaxamount.micro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.vmaxamount.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.2comp.vmaxamount.micro.infusion:

Infusion
--------

Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}



.. _michaelismentenlinear.2comp.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismentenlinear.2comp.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}

.. _michaelismentenlinear.2comp.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.macro.infusion:

Infusion
--------

Equations:

.. math::

  \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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}



.. _michaelismentenlinear.2comp.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismentenlinear.2comp.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismentenlinear.2comp.micro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.micro.infusion:

Infusion
--------

Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismentenlinear.2comp.vmaxamount.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismentenlinear.2comp.vmaxamount.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}

.. _michaelismentenlinear.2comp.vmaxamount.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.vmaxamount.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.vmaxamount.macro.infusion:

Infusion
--------

Equations:

.. math::

  \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


Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}



.. _michaelismentenlinear.2comp.vmaxamount.micro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismentenlinear.2comp.vmaxamount.micro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismentenlinear.2comp.vmaxamount.micro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.vmaxamount.micro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismentenlinear.2comp.vmaxamount.micro.infusion:

Infusion
--------

Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}



.. _michaelismenten.enzyme.1comp:

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]

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

  \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)

with:

.. math::

  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:

.. math::

	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}\\

with:

.. math::

  crit\_point &= Dmid - EDmid\\
  low\_bound  &= crit\_point - 23\\
  high\_bound &= crit\_point + 37


.. _michaelismenten.enzyme.1comp.infusion:

Infusion
--------

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
  C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}


.. _michaelismenten.enzyme.1comp.extra:

Extra
-----

Initial conditions:

.. math::

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





.. _michaelismenten.1comp.2analytes.vmaxamount.macro:

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.

.. csv-table:: Parameters
	:header: "Name", "Description"
	:widths: 5, 40

	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:

.. math::

	\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


.. _michaelismenten.1comp.2analytes.vmaxamount.macro.bolus:

Bolus
-----

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} + \frac{D}{V1} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}

.. _michaelismenten.1comp.2analytes.vmaxamount.macro.extra:

Extra
-----

Initial conditions:

.. math::

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


.. _michaelismenten.1comp.2analytes.vmaxamount.macro.extra.lag:

Extra Lag
---------

Initial conditions:

.. math::

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


.. _michaelismenten.1comp.2analytes.vmaxamount.macro.infusion:

Infusion
--------


Equations:

.. math::

	\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

Initial conditions:

.. math::

	C1(t_0) &= C1_{residual} \\
	C2(t_0) &= C2_{residual} \\
	C3(t_0) &= C3_{residual}