using Random using Pumas using PlottingUtilities using PumasPlots using CairoMakie Random.seed!(1234) # Set random seed for reproducibility

MersenneTwister(1234)

Absorption modeling is an integral part of pharmacokinetic modeling for drugs delivered by any route of administration other than intravenous (IV) (e.g. oral, subcutaneous, intramuscular, transdermal, nasal, etc.). In the simplest case, drugs administered orally may undergo first-order absorption, whereby the absorption from the depot compartment (i.e. gut) to the central compartment (i.e. plasma) is a first-order process. However, many more complex situations arise in practice, some of which will be covered in this tutorial.

In this tutorial, we will cover the following absorption models:

First-order

Zero-order

Parallel zero-order and first-order

Two parallel first-order processes

Weibull-type absorption

Absorption through a sequence of transit compartments (Erlang absorption)

We begin by simulating a population of 10 subjects to use for illustration purposes in this tutorial. For simplicity, we will use a 1-compartment distribution system with additional absorption compartment(s) as required. Note that in these examples we are simulating without residual error (it could be added, but isn't necessary here). Hence, we use the `@observed`

block that is designed to capture post dynamics outputs.

dose = DosageRegimen(100, time = 0) choose_covariates() = (wt = rand(55:80), dose = 100) subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end

Population Subjects: 10 Covariates: wt, dose Observations: conc

For a more in-depth discussion of simulating data with Pumas, please see the tutorial generating and simulation populations.

Here is the Pumas code for a 1-compartment model with first-order absorption, diagonal random-effects structure, and allometric scaling on clearance and volume:

foabs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvka ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(3) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) Ka = tvka*exp(η[3]) end @dynamics begin Depot' = -Ka*Depot Central' = Ka*Depot - (CL/Vc)*Central end @observed begin conc = @. Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvka, Ω Random effects: η Covariates: wt Dynamical variables: Depot, Central Derived: Observed: conc

Note that we have written the model using differential equations, but we could equally have specified an "analytical" (closed-form) solution using this `@dynamics`

block instead (see documentation on analytical solutions):

@dynamics Depots1Central1

Below are a set of parameter values for this model:

param = ( tvcl = 5, tvvc = 20, tvka = 1, Ω = Diagonal([0.04, 0.04, 0.04]))

(tvcl = 5, tvvc = 20, tvka = 1, Ω = [0.04 0.0 0.0; 0.0 0.04 0.0; 0.0 0.0 0. 04])

We can now use the data and model with parameters to simulate a first-order absorption profile:

sims = simobs(foabs, subj_with_covariates, param, obstimes = 0:.1:24)

And now, we plot the results:

sim_plot(foabs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "First-order absorption"))

Zero-order absorption is less common than first-order absorption. It is essentially like an IV infusion, but where the duration of the infusion is an estimated parameter rather than a known quantity. Here is the Pumas code for the same 1-compartment model but with zero-order absorption:

zoabs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvdur ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(3) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) end @dosecontrol begin duration = (Central = tvdur*exp(η[3]),) end @dynamics begin Central' = - (CL/Vc)*Central end @observed begin conc = @. Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvdur, Ω Random effects: η Covariates: wt Dynamical variables: Central Derived: Observed: conc

Notice that there is no depot compartment, and that instead the absorption takes place in the `@dosecontrol`

block by specifying a `duration`

for the zero-order process. In order for this to work, we need to set the `rate`

data item in `DosageRegimen`

to the value `-2`

; this is a clue to the Pumas engine that the `rate`

should be derived from the `duration`

and `amt`

.

Below are a set of parameter values for this model:

param = ( tvcl = 0.792, tvvc = 13.7, tvdur = 5.0, Ω = Diagonal([0.04, 0.04, 0.04]))

(tvcl = 0.792, tvvc = 13.7, tvdur = 5.0, Ω = [0.04 0.0 0.0; 0.0 0.04 0.0; 0 .0 0.0 0.04])

We can now use the data and model with parameters to simulate a zero-order absorption profile:

dose = DosageRegimen(100, time = 0, rate = -2) # Note: rate must be -2 for duration modeling subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end sims = simobs(zoabs, subj_with_covariates, param, obstimes = 0:0.1:48)

And now, we plot the results:

sim_plot(zoabs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "Zero-order absorption"))

This is a more complex absorption model, with both a zero-order process and a first-order process operating simultaneously. Essentially, the total dose is split into two parts, one of which undergoes first-order absorption and the other zero-order absorption (how this splitting is typically accomplished from a technical perspective is to treat each actual dose as two virtual doses with different bioavailable fractions, as in the example below). Furthermore, we will assume that the first-order process is subject to a lag (i.e., the absorption begins after a certain time delay, which is a model parameter).

zofo_paral_abs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvka ∈ RealDomain(lower=0) tvdur ∈ RealDomain(lower=0) tvbio ∈ RealDomain(lower=0) tvlag ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(2) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) Ka = tvka end @dosecontrol begin duration = (Central = tvdur,) bioav = (Depot = tvbio, Central = 1 - tvbio) lags = (Depot = tvlag,) end @dynamics begin Depot' = -Ka*Depot Central' = Ka*Depot - (CL/Vc)*Central end @observed begin conc = @. Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvka, tvdur, tvbio, tvlag, Ω Random effects: η Covariates: wt Dynamical variables: Depot, Central Derived: Observed: conc

Below are a set of parameter values for this model:

param = ( tvcl = 5, tvvc = 50, tvka = 1.2, tvdur = 2, tvbio = 0.5, tvlag = 1, Ω = Diagonal([0.04, 0.04]))

(tvcl = 5, tvvc = 50, tvka = 1.2, tvdur = 2, tvbio = 0.5, tvlag = 1, Ω = [0 .04 0.0; 0.0 0.04])

With these parameter values, the first-order process starts 1 hour after administration, while the zero-order process lasts 2 hours. Hence, there is one hour where they overlap.

We can now use the data and model with parameters to simulate a parallel first- and zero-order absorption profile:

dose_zo = DosageRegimen(100, time = 0, cmt = 2, rate = -2, evid = 1) # Note: rate must be -2 for duration modeling dose_fo = DosageRegimen(100, time = 0, cmt = 1, rate = 0, evid = 1) dose = DosageRegimen(dose_fo, dose_zo) # Actual dose is made up of 2 virtual doses subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end sims = simobs(zofo_paral_abs, subj_with_covariates, param, obstimes = 0:0.1:24)

And now, we plot the results:

sim_plot(zofo_paral_abs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "Zero- and first-order parallel absorption"))

This is similar to the example above, except that both parallel processes are first-order rather than one being zero-order. We will refer to the two processes as "immediate release" (IR) and "slow release" (SR); only the SR process is subject to a lag. The Pumas code is as follows:

two_parallel_foabs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvka1 ∈ RealDomain(lower=0) tvka2 ∈ RealDomain(lower=0) tvlag ∈ RealDomain(lower=0) tvbio ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(6) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) Ka1 = tvka1*exp(η[3]) Ka2 = tvka2*exp(η[4]) end @dosecontrol begin lags = (SR = tvlag*exp(η[5]),) bioav = (IR = tvbio*exp(η[6]), SR = (1 - tvbio)*exp(η[6])) end @dynamics begin IR' = -Ka1*IR SR' = -Ka2*SR Central' = Ka1*IR + Ka2*SR - Central*CL/Vc end @observed begin conc = @. Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvka1, tvka2, tvlag, tvbio, Ω Random effects: η Covariates: wt Dynamical variables: IR, SR, Central Derived: Observed: conc

Below are a set of parameter values for this model:

param = ( tvcl = 5, tvvc = 50, tvka1 = 0.8, tvka2 = 0.6, tvlag = 5, tvbio = 0.5, Ω = Diagonal([0.04, 0.04, 0.36, 0.36, 0.04, 0.04]))

(tvcl = 5, tvvc = 50, tvka1 = 0.8, tvka2 = 0.6, tvlag = 5, tvbio = 0.5, Ω = [0.04 0.0 … 0.0 0.0; 0.0 0.04 … 0.0 0.0; … ; 0.0 0.0 … 0.04 0.0; 0.0 0.0 … 0.0 0.04])

We can now use the data and model with parameters to simulate a profile with this type of absorption:

dose_fo1 = DosageRegimen(100, cmt = 1, time = 0) dose_fo2 = DosageRegimen(100, cmt = 2, time = 0) dose = DosageRegimen(dose_fo1, dose_fo2) # Actual dose is made up of 2 virtual doses subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end sims = simobs(two_parallel_foabs, subj_with_covariates, param, obstimes = 0:.1:48)

And now, we plot the results:

sim_plot(two_parallel_foabs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "Two Parallel first-order absorption"))

Yet other situations call for an absorption process that varies over time. One such model is this Weibull-type model, whereby the absorption is by a first-order process, but with a rate constant that is continuously changing over time according to a *Weibull function* (i.e., the CDF of a Weibull distribution). Here is the Pumas code for this model:

weibullabs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvka ∈ RealDomain(lower=0) tvγ ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(4) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) Ka∞ = tvka*exp(η[3]) # Maximum Ka as t → ∞ γ = tvγ*exp(η[4]) # Controls the steepness of the Ka curve Kaᵗ = 1 - exp(-(Ka∞*t)^γ) # Weibull function end @dynamics begin Depot' = -Kaᵗ*Depot Central' = Kaᵗ*Depot - (CL/Vc)*Central end @derived begin conc = Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvka, tvγ, Ω Random effects: η Covariates: wt Dynamical variables: Depot, Central Derived: conc Observed: conc

Below are a set of parameter values for this model:

param = ( tvcl = 5, tvvc = 50, tvka = 0.4, tvγ = 4, Ω = Diagonal([0.04, 0.04, 0.36, 0.04]))

(tvcl = 5, tvvc = 50, tvka = 0.4, tvγ = 4, Ω = [0.04 0.0 0.0 0.0; 0.0 0.04 0.0 0.0; 0.0 0.0 0.36 0.0; 0.0 0.0 0.0 0.04])

We can now use the data and model with parameters to simulate a Weibull-type absorption profile:

dose = DosageRegimen(100, cmt = 1, time = 0) subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end sims = simobs(weibullabs, subj_with_covariates, param, obstimes = 0:.1:24)

And now, we plot the results:

sim_plot(two_parallel_foabs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "Weibull absorption"))

The Erlang absorption model is based on the concept of transit compartments. The Erlang distribution is a special case of the Gamma distribution that arises as the sum of $k$ independent exponential distributions. This is exactly the distribution of the transit time taken by a particle passing through a sequence of $k$ compartments where the transition from one compartment to the next is governed by first-order processes with a common rate constant $K_{tr}$. In Pumas, this model can be written with the following code (we assume $N = 5$ transit compartments):

erlangabs = @model begin @param begin tvcl ∈ RealDomain(lower=0) tvvc ∈ RealDomain(lower=0) tvktr ∈ RealDomain(lower=0) Ω ∈ PSDDomain(3) end @random begin η ~ MvNormal(Ω) end @covariates wt @pre begin CL = tvcl*(wt/70)^0.75*exp(η[1]) Vc = tvvc*(wt/70)*exp(η[2]) Ktr = tvktr*exp(η[3]) end @dynamics begin Depot' = -Ktr*Depot Transit1' = Ktr*Depot - Ktr*Transit1 Transit2' = Ktr*Transit1 - Ktr*Transit2 Transit3' = Ktr*Transit2 - Ktr*Transit3 Transit4' = Ktr*Transit3 - Ktr*Transit4 Transit5' = Ktr*Transit4 - Ktr*Transit5 Central' = Ktr*Transit5 - (CL/Vc)*Central end @observed begin conc = @. Central/Vc end end

PumasModel Parameters: tvcl, tvvc, tvktr, Ω Random effects: η Covariates: wt Dynamical variables: Depot, Transit1, Transit2, Transit3, Transit4, Trans it5, Central Derived: Observed: conc

Below are a set of parameter values for this model:

param = ( tvcl = 7, tvvc = 32, tvktr = 2.6, Ω = Diagonal([0.09, 0.22, 0.10, 0.52]))

(tvcl = 7, tvvc = 32, tvktr = 2.6, Ω = [0.09 0.0 0.0 0.0; 0.0 0.22 0.0 0.0; 0.0 0.0 0.1 0.0; 0.0 0.0 0.0 0.52])

We can now use the data and model with parameters to simulate an Erlang absorption profile:

dose = DosageRegimen(100, cmt = 1, time = 0) subj_with_covariates = map(1:10) do i Subject(id = i, events = dose, covariates = choose_covariates(), observations = (conc = nothing,)) end sims = simobs(erlangabs, subj_with_covariates, param, obstimes = 0:1:24)

And now, we plot the results:

sim_plot(erlangabs, sims, observations =[:conc], figure = (fontsize = 18, ), axis = (xlabel = "Time (hr)", ylabel = "Predicted Concentration (ng/mL)", title = "Erlang absorption"))

We have seen in this tutorial some examples of how to implement various absorption models that are commonly employed in pharmacokinetic modeling using Pumas. The powerful Pumas modeling language framework makes it relatively straightforward to implement these and other complex PK and PKPD models for modeling and simulation purposes.

This concludes our tutorial on basic absorption models with Pumas. Thanks for reading, and please be sure to check out our other tutorials as well.