# PK08 - Two compartment distribution models

## 1 Background

• Structural model - Two compartment linear elimination with first order elimination
• Route of administration - IV bolus
• Dosage Regimen - 100 μg IV or 0.1 mg IV
• Number of Subjects - 1

## 2 Learning Outcome

This exercise demonstrates simulating single IV bolus dose kinetics from a two-compartment model.

## 3 Objectives

To build a two-compartment model, simulate the model for a single subject given a single IV bolus dose, and subsequently perform a simulation for a population.

## 4 Libraries

``````using PumasUtilities
using Random
using Pumas
using CairoMakie
using AlgebraOfGraphics
using CSV
using DataFramesMeta
using Dates``````

## 5 Model definition

Note the expression of the model parameters with helpful comments. The model is expressed with differential equations. Residual variability is a proportional error model.

``````pk_08_05 = @model begin
desc = "Two Compartment Model"
timeu = u"hr"
end

@param begin
"""
Clearance (L/hr)
"""
tvcl ∈ RealDomain(lower = 0)
"""
Volume of Distribution (L)
"""
tvvc ∈ RealDomain(lower = 0)
"""
Intercompartmental Clearance (L/hr)
"""
tvq ∈ RealDomain(lower = 0)
"""
Peripheral Volume of Distribution (L)
"""
tvvp ∈ RealDomain(lower = 0)
#Ω ∈ PDiagDomain(4)
Ω_cl ∈ RealDomain(lower = 0.0001)
Ω_vc ∈ RealDomain(lower = 0.0001)
Ω_q ∈ RealDomain(lower = 0.0001)
Ω_vp ∈ RealDomain(lower = 0.0001)
"""
Proportional RUV
"""
σ²_prop ∈ RealDomain(lower = 0)
end

@random begin
η_cl ~ Normal(0, sqrt(Ω_cl))
η_vc ~ Normal(0, sqrt(Ω_vc))
η_q ~ Normal(0, sqrt(Ω_q))
η_vp ~ Normal(0, sqrt(Ω_vp))
end

@pre begin
Cl = tvcl * exp(η_cl)
Vc = tvvc * exp(η_vc)
Vp = tvvp * exp(η_q)
Q = tvq * exp(η_vp)
end

@dynamics begin
Central' = -(Cl / Vc) * Central - (Q / Vc) * Central + (Q / Vp) * Peripheral
Peripheral' = (Q / Vc) * Central - (Q / Vp) * Peripheral
end

@derived begin
"""
PK08 Concentration (μg/L)
"""
cp = @. Central / Vc
"""
PK08 Concentration (μg/L)
"""
dv ~ @. Normal(cp, sqrt(cp^2 * σ²_prop))
end
end``````
``````PumasModel
Parameters: tvcl, tvvc, tvq, tvvp, Ω_cl, Ω_vc, Ω_q, Ω_vp, σ²_prop
Random effects: η_cl, η_vc, η_q, η_vp
Covariates:
Dynamical system variables: Central, Peripheral
Dynamical system type: Matrix exponential
Derived: cp, dv
Observed: cp, dv``````

## 6 Initial Estimates of Model Parameters

The model parameters for simulation are the following. Note that `tv` represents the typical value for parameters.

• `CL` - Clearance (L/hr),
• `Vc` - Volume of Central Compartment (L),
• `Vp` - Volume of Peripheral Compartment (L),
• `Q` - Inter-departmental clearance (L/hr),
• `Ω` - Between Subject Variability,
• `σ` - Residual error
``````param = (
tvcl = 6.6,
tvvc = 53.09,
tvvp = 57.22,
tvq = 51.5,
Ω_cl = 0.01,
Ω_vc = 0.01,
Ω_q = 0.01,
Ω_vp = 0.01,
σ²_prop = 0.047,
)``````

## 7 Dosage Regimen

Dosage Regimen - 100 μg or 0.1 mg of IV bolus.

``ev1 = DosageRegimen(100, time = 0, cmt = 1, evid = 1, addl = 0, ii = 0)``
1×10 DataFrame
Row time cmt amt evid ii addl rate duration ss route
Float64 Int64 Float64 Int8 Float64 Int64 Float64 Float64 Int8 NCA.Route
1 0.0 1 100.0 1 0.0 0 0.0 0.0 0 NullRoute

## 8 Single-individual that receives the defined dose

``sub1 = Subject(id = 1, events = ev1)``
``````Subject
ID: 1
Events: 1``````

## 9 Perform Single-Subject Simulation

Simulate the plasma concentration-time profile with the given observation time-points for a single subject.

Initialize the random number generator with a seed for reproducibility of the simulation.

``Random.seed!(123)``

Define the timepoints at which concentration values will be simulated.

``````sim_s1 = simobs(
pk_08_05,
sub1,
param,
obstimes = [
0.08,
0.25,
0.5,
0.75,
1,
1.33,
1.67,
2,
2.5,
3.07,
3.5,
4.03,
5,
7,
11,
23,
29,
35,
47.25,
],
)``````
``````SimulatedObservations
Simulated variables: cp, dv
Time: [0.08, 0.25, 0.5, 0.75, 1.0, 1.33, 1.67, 2.0, 2.5, 3.07, 3.5, 4.03, 5.0, 7.0, 11.0, 23.0, 29.0, 35.0, 47.25]``````

## 10 Visualize Results

``````@chain DataFrame(sim_s1) begin
dropmissing(:cp)
data(_) *
mapping(:time => "Time (hours)", :cp => "Concentration (μg/L)";) *
visual(Lines; linewidth = 4)
draw(;
figure = (; fontsize = 22),
axis = (;
yscale = log10,
xticks = 0:4:48,
yticks = map(i -> round(10^i, sigdigits = 1), -1:0.5:1),
),
)
end``````

## 11 Perform a Population Simulation

We perform a population simulation with 48 participants, and simulate concentration values for 72 hours following 6 doses administered every 8 hours.

This code demonstrates how to write the simulated concentrations to a comma separated file (`.csv`).

``````par = (
tvcl = 6.6,
tvvc = 53.09,
tvvp = 57.22,
tvq = 51.5,
Ω_cl = 0.04,
Ω_vc = 0.09,
Ω_q = 0.169,
Ω_vp = 0.0225,
σ²_prop = 0.0497,
)

ev1 = DosageRegimen(100, time = 0, cmt = 1, evid = 1, addl = 5, ii = 8)
pop = map(i -> Subject(id = i, events = ev1), 1:48)

Random.seed!(1234)
pop_sim = simobs(pk_08_05, pop, par, obstimes = 0:1:72)

pkdata_08_sim = DataFrame(pop_sim)

#CSV.write("pk_08_05_sim.csv", pkdata_08_sim)``````

## 12 Conclusion

This tutorial showed how to build a two compartment model and perform a single subject and population simulation.