Background

• Structural Model - One compartment model with non-linear elimination.

• Route of administration - IV infusion

• Dosage Regimen - 310 μg, 520 μg, 780 μg

• Number of Subjects - 3

Learning Outcome

This is a one compartment model with capacity limited elimination. Concentration time profile was obtained for three subjects administered with three different dosage regimens.

Objectives

In this tutorial, you will learn how to build one compartment model with non-linear elimination.

Libraries

call the "necessary" libraries to get start.

using Pumas
using Plots
using CSV
using StatsPlots
using Random

Model

The following model describes the parameters and differential equation for a one-compartment model with capacity limited elimination

pk_41        = @model begin
  @param begin
    tvvmax   ∈ RealDomain(lower=0)
    tvkm     ∈ RealDomain(lower=0)
    tvvc     ∈ RealDomain(lower=0)
    Ω        ∈ PDiagDomain(3)
    σ²_prop  ∈ RealDomain(lower=0)
  end

  @random begin
    η        ~ MvNormal(Ω)
  end

  @pre begin
    Vmax     = tvvmax * exp(η[1])
    Km       = tvkm * exp(η[2])
    Vc       = tvvc  * exp(η[3])
  end

  @dynamics begin
    Central' = - (Vmax * (Central/Vc)/(Km + (Central/Vc)))
  end

  @derived begin
    cp       = @. Central/Vc
    dv       ~ @. Normal(cp, sqrt(cp^2*σ²_prop))
    nca     := @nca cp
    cl       = NCA.cl(nca)
  end
end

PumasModel
  Parameters: tvvmax, tvkm, tvvc, Ω, σ²_prop
  Random effects: η
  Covariates: 
  Dynamical variables: Central
  Derived: cp, dv, cl
  Observed: cp, dv, cl

Parameters

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

• Vmax - Maximum Metabolic Rate (μg/kg/hr)

• Km - Michaelis Menten Constant (μg/kg/L)

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

param = ( tvvmax  = 180.311,
          tvkm    = 79.8382,
          tvvc    = 1.80036,
          Ω       = Diagonal([0.0,0.0,0.0,0.0,0.0]),
          σ²_prop = 0.015)

(tvvmax = 180.311, tvkm = 79.8382, tvvc = 1.80036, 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 = 0.0
15)

Dosage Regimen

• Subject-1 receives a dose of 310 μg given as an IV Infusion over 5 hrs

• Subject-2 receives a dose of 520 μg given as an IV Infusion over 5 hrs

• Subject-3 receives a dose of 780 μg given as an IV Infusion over 5 hrs

ev1      = DosageRegimen(310, cmt=1, time=0, rate=62, route=NCA.IVInfusion)
sub1     = Subject(id=1, events=ev1)
ev2      = DosageRegimen(520, cmt=1, time=0, rate=104, route=NCA.IVInfusion)
sub2     = Subject(id=2, events=ev2)
ev3      = DosageRegimen(780, cmt=1, time=0, rate=156, route=NCA.IVInfusion)
sub3     = Subject(id=3, events=ev3)
pop3_sub = [sub1,sub2,sub3]

Population
  Subjects: 3
  Covariates:

Simulation

Simulate the plasma concentration of the drug for both the subjects

Random.seed!(123)
sim_pop3_sub = simobs(pk_41, pop3_sub, param, obstimes=0:0.01:10)
df1          = DataFrame(sim_pop3_sub)

Dataframe & Plot

Split the simulation to the necessary dataframe and use it for plotting

Regression Analysis - Concentration vs Time

df_id1 = filter(x -> x.id == "1", df1)
filter!(x -> x.time <= 6, df_id1)
df_id2 = filter(x -> x.id == "2", df1)
df_id3 = filter(x -> x.id == "3", df1)
df_dv_1 = filter(x -> x.time in [0.1,2,5,6], df_id1)
df_dv_2 = filter(x -> x.time in [0.1,2,5,6,8,10], df_id2)
df_dv_3 = filter(x -> x.time in [0.1,2,5,6,8,10], df_id3)

@df df_id1 plot(:time, :cp, yaxis=:log,
                title ="Concentration vs Time", xlabel="Time (hr)", ylabel="Concentration (ug/L)",
                label ="Pred - Conc ID 1", linewidth=3, legend=:bottomleft,
                xticks=[0,1,2,3,4,5,6,7,8,9,10], xlims=(-0.2,10.2), ylims=(0.3,300))
@df df_id2 plot!(:time, :cp, label ="Pred - Conc ID 2", linewidth=3)
@df df_id3 plot!(:time, :cp, label ="Pred - Conc ID 3", linewidth=3)
@df df_dv_1 scatter!(:time, :dv, label="Obs - Conc", color=[:purple])
@df df_dv_2 scatter!(:time, :dv, label=false, color=[:purple])
@df df_dv_3 scatter!(:time, :dv, label=false, color=[:purple])

NCA - Clearance vs Dose

df_cl = filter(x -> x.time == 0.1, df1)
df_cl[!, :dose] .= [310,520,780]
select!(df_cl, :dose, :cl)

@df df_cl plot(:dose, :cl,
                xlabel="Dose ug/kg", ylabel = "Cl (L/hr/kg)", label=false,
                title="Clearance vs Dose", linewidth=3,
                xticks=[300,400,500,600,700,800], xlims=(300,800), ylims=(0.8,1.8), yticks=[0.8,1.0,1.2,1.4,1.6,1.8])
@df df_cl scatter!(:dose, :cl, label=false)

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