Background

• Structural model - Two Compartment model with a Metabolite Compartment

• Route of administration - IV Bolus

• Dosage Regimen - 10μmol/kg, 50μmol/kg, 300μmol/kg

• Number of Subjects - 3

Learning Outcome

In this model, 3 different dose of the drug given as an IV Bolus to 3 different subjects, will help you estimate metabolite formation rate and elimination rate.

Objectives

In this tutorial, you will learn how to build two compartment model, a drug undergoing capacity limited metabolite kinetics.

Libraries

call the "necessary" libraries to get start.

using Pumas
using Plots
using CSV
using StatsPlots
using Random

Model

In this two compartment model, we administer 3 different doses in 3 different subjects of a drug that undergoes metabolite kinetics.

pk_19           = @model begin
  @param begin
    tvvc        ∈ RealDomain(lower=0)
    tvvp        ∈ RealDomain(lower=0)
    tvq         ∈ RealDomain(lower=0)
    tvvmax      ∈ RealDomain(lower=0)
    tvkm        ∈ RealDomain(lower=0)
    tvkme       ∈ RealDomain(lower=0)
    tvvme       ∈ RealDomain(lower=0)
    Ω           ∈ PDiagDomain(7)
    σ²_prop_cp  ∈ RealDomain(lower=0)
    σ²_prop_met ∈ RealDomain(lower=0)
  end

  @random begin
    η           ~ MvNormal(Ω)
  end

  @pre begin
    Vc          = tvvc * exp(η[1])
    Vp          = tvvp * exp(η[2])
    Q           = tvq * exp(η[3])
    Vmax        = tvvmax * exp(η[4])
    Km          = tvkm * exp(η[5])
    Kme         = tvkme * exp(η[6])
    Vme         = tvvme * exp(η[7])
  end

  @vars begin
    VMKM       := Vmax/(Km+(Central/Vc))
  end

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

  @derived begin
    cp          = @. Central/Vc
    dv_cp       ~ @. Normal(cp, sqrt(cp^2*σ²_prop_cp))
    met         = @. Metabolite/Vme
    dv_met      ~ @. Normal(met, sqrt(met^2*σ²_prop_met))
  end
end

PumasModel
  Parameters: tvvc, tvvp, tvq, tvvmax, tvkm, tvkme, tvvme, Ω, σ²_prop_cp, σ
²_prop_met
  Random effects: η
  Covariates: 
  Dynamical variables: Central, Peripheral, Metabolite
  Derived: cp, dv_cp, met, dv_met
  Observed: cp, dv_cp, met, dv_met

Parameters

The parameters are as given below. tv represents the typical value for parameters.

• Vc - Volume of Central Compartment (L/kg)

• Vp - Volume of Peripheral Compartment (L/kg)

• Q - Inter-Compartmental Clearance (L/min)

• Vmax - Maximum Velocity of Reaction (μmol/min/kg)

• Km - Michaelis-Menten constant (μmol/L)

• Kme - Rate of Elimination of Metabolite (min⁻¹)

• Vme - Volume of Metabolite Compartment (L/kg)

• Ω - Between Subject Variability

• σ - Residual error

param = ( tvvc        = 1.06405,
          tvvp        = 2.00748,
          tvq         = 0.128792,
          tvvmax      = 1.64429,
          tvkm        = 54.794,
          tvkme       = 0.145159,
          tvvme       = 0.290811,
          Ω           = Diagonal([0.0,0.0,0.0,0.0,0.0,0.0,0.0]),
          σ²_prop_cp  = 0.015,
          σ²_prop_met = 0.015)

(tvvc = 1.06405, tvvp = 2.00748, tvq = 0.128792, tvvmax = 1.64429, tvkm = 5
4.794, tvkme = 0.145159, tvvme = 0.290811, Ω = [0.0 0.0 … 0.0 0.0; 0.0 0.0 
… 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0], σ²_prop_cp = 0.015, σ
²_prop_met = 0.015)

Dosage Regimen

Three Subjects were adminitered with three different doses of 10μmol/kg, 50μmol/kg and 300μmol/kg.

ev1      = DosageRegimen(10, cmt=1, time=0)
sub1     = Subject(id=1, events=ev1)
ev2      = DosageRegimen(50, cmt=1, time=0)
sub2     = Subject(id=2, events=ev2)
ev3      = DosageRegimen(300, cmt=1, time=0)
sub3     = Subject(id=3, events=ev3)
pop3_sub = [sub1,sub2,sub3]

Population
  Subjects: 3
  Covariates:

Simulation

We will simulate the parent plasma concentration and metabolite plasma concentration.

Random.seed!(123)
sim_pop3_sub = simobs(pk_19, pop3_sub, param, obstimes=0:1:300)
df1          = DataFrame(sim_pop3_sub)

Dataframe & Plot

Use the datafreme for plotting

df1_dv = filter(x -> x.time in [0,5,10,20,30,60,90,120,180,300], df1)

@df df1 plot(:time,:cp, yaxis=:log,
              xlabel="Time (min)", ylabel="Concentration (umol/L)" , label="Pred - Parent",
              title = "Plasma concentration vs Time", linewidth=3,
              ylims=(0.1,1000), yticks=[0.1,1,10,100,1000], xticks=[0,50,100,150,200,250,300])
@df df1 plot!(:time,:met, yaxis=:log,
               label="Pred - Metabolite", linestyle=[:dot],linewidth=3)
@df df1_dv scatter!(:time, :dv_cp, yaxis=:log, label="Obs - Parent")
@df df1_dv scatter!(:time, :dv_met, yaxis=:log, label="Obs - Metabolite")

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