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# Library of PK models with double absorption

Some drugs can display complex absorption kinetics. Common examples are mixed first-order and zero-order absorptions, either sequentially or simultaneously, and fast and slow parallel first-order absorptions. Multiple peaks in concentration-time curves can also be described with mixed absorptions, and can be explained by different physiological processes, such as enterohepatic recycling, delayed gastric emptying, or variability of absorption.

In most cases, a two mixed absorptions are sufficient to capture the main features of the absorption.

We have developed a library of double absorption models implemented in Monolix to take into account all the combinations of absorption types and delays. The absorptions can be specified as simultaneous or sequential, and with a pre-defined or independent order. This library simplifies the selection and testing of different types of absorptions and delays.

## Modeling choices

In this documentation we detail the filtering options in the interface of Monolix, shown on the figure below, and the corresponding parameters and modelling choices.

In all cases, a fraction F1 of the dose is absorbed via the first absorption and the remaining 1-F1 fraction of the dose is absorbed via the second absorption. As the parameter F1 must stay within [0,1], a logit distribution is automatically chosen, and the default initial value is 0.5.

The choices for the first absorption are:

• Type of absorption:
• zero order, modelled with the parameter Tk01
• first order, modelled with the parameter ka1
• Delay:
• no delay,
• a lag time, modelled with the parameter Tlag1
• transit compartments, modelled with the parameters Mtt1 (mean transit time) and Ktr1 (transfer rate)

The choices for the second absorption are:

• Type of absorption:
• zero order, modelled with the parameter Tk02
• first order, modelled with the parameter ka2
• Delay:
• no delay,
• a lag time, modelled with the parameter Tlag2
• transit compartments, modelled with the parameters Mtt2 (mean transit time) and Ktr2 (transfer rate)
• Order of absorption: this option is available if the first absorption is a zero-order type, otherwise absorptions are always simultaneous.
• Simultaneous: both absorptions are modelled with independent parameters. Depending on the values estimated for the parameters, the absorptions will be simultaneous or not.
• Sequential: the second absorption starts at the end of or after the zero-order process corresponding to the first absorption. So the second absorption is necessarily modelled with a delay, with two possible parameterizations:
• Tlag2 = Tk01 if “no delay” was chosen for both absorptions: the delay of the second absorption is equal to the duration of the first absorption.
• Tlag2 = Tlag1 + Tk01, if “lag time” was chosen for the first absorption and “no delay” was chosen for the second absorption.
• Tlag2 = Tk01 + diffTlag2, if “no delay” was chosen for the first absorption and “lag time” was chosen for the second absorption. In that case diffTlag2 is estimated instead of Tlag2, but the value of Tlag2 will be available in the outputs of Monolix thanks to the table statement. This parameterization ensures that the delay for the second absorption is greater than the duration of the first absorption.
• Tlag2 = Tlag1 + Tk01 + diffTlag2, if “lag time” was chosen for both absorptions. Similarly as before, diffTlag2 is estimated instead of Tlag2.
• Mtt2 = Tk01 + diffMtt2, if “no delay” was chosen for the first absorption and “transit compartments” was chosen for the second absorption. Similarly as before, diffMtt2 is estimated instead of Mtt2.
• Mtt2 = Tlag1 + Tk01 + diffMtt2, if “lag time” was chosen for the first absorption and “transit compartments” was chosen for the second absorption. Similarly as before, diffMtt2 is estimated instead of Mtt2.
• Force longer delay: this option is available if both absorptions where selected with a delay (lag time or transit compartments), and if the absorptions are not sequential.
• False: the parameters used to describe both delays are independent, so for the second absorption Tlag2 is estimated if “lag time” was chosen, and Mtt2 is estimated if “transit compartments” was chosen.
• True: similarly as above, the parameter for the delay of the second absorption is forced to be larger than the delay of the first absorption. For example, if “lag time” was chosen for both absorptions, Tlag2 is written as Tlag2 = Tlag1 + diffTlag2, and diffTlag2 is estimated instead of Tlag2.

Finally, just like for the PK library, the remaining choices are:

• Distribution:
• 1 compartment: the volume of the central compartment is V
• 2 compartments: the two compartments are modelled with the parameters V1 (volume of central compartment), Q (inter-compartment clearance) and V2 (volume of peripheral compartment), or with V (volume of central compartment), k12 and k21 (transfer rates between the central and peripheral compartments).
• 3 compartments: in the same way, the three compartments are modelled with V1, Q2, V2, Q3 and V3, or with V, k12, k21, k13 and k31.
• Elimination:
• Linear: modelled with Cl (clearance) or k (elimination rate)
• Michaelis-Menten: modelled with Vm and Km.

### File names

The PK double absorption library contains many models, because of the many combinations of absorption types and delays for both absorptions, constraint on the time of the second absorption, distribution and parameterizations. In total 418 model files are available. The file names follow the pattern below:

In this pattern, oral0 means zero-order absorption, oral1 means first-order absorption, seqAbs corresponds to the filtering option “Order of absorptions = sequential” (the second absorptions starts at the end of the first absorption), and seqDelay corresponds to the filtering option “Force longer delay = true” (the delay for the second absorption is necessarily longer than the delay for the first absorption). If no keyword “seqDelay” or “seqAbs” are present in the file name, this is equivalent to the filtering option “Order of absorptions = sequential” and it means that the parameters for both absorptions are independent.

## Examples

The models are written with PK macros.

For example, the model corresponding to a zero-order process with a lag time for the first absorption and a simulatenous (independent) first-order process with transit compartments for the second absorption, with a longer delay than the delay of the first absorption, a single compartment and parameterized with the clearance, is named oral0_oral1_seqDelay_1cpt_Tk01ka2F1Tlag1diffMtt2Ktr2VCl.txt and reads:

[LONGITUDINAL]
input = {Tk01, ka2, F1, Tlag1, diffMtt2, Ktr2, V, Cl}

EQUATION:
odeType=stiff

; Parameter transformations
k = Cl/V
Mtt2 = Tlag1 + diffMtt2

PK:
;PK model definition
compartment(cmt = 1, volume = V, concentration = Cc)
absorption(cmt = 1, Tk0 = Tk01, Tlag = Tlag1, p = F1)
absorption(cmt = 1, ka = ka2, Mtt = Mtt2, Ktr = Ktr2, p = 1-F1)
elimination(cmt = 1, k)

OUTPUT:
output = Cc
table = Mtt2