Background

• Structural model - Two compartment disposition model with nonlinear elimination

• Route of administration - IV infusion

• Dosage Regimen - 0.4g/Kg (i.e.,28g for a 70Kg healthy individual), infused over a time span of 30 minutes

• Number of Subjects - 1

Learning Outcome

In this model, you will learn -

• To build a two compartment disposition model, the drug is given as an Intravenous Infusion which follows Michaelis-Menten Kinetics.

• To apply differential equation in the model as per the compartment model.

• To design the dosage regimen for the subjects and simulate the plot.

Objectives

In this tutorial, you will learn how to build a two Compartment disposition model with Non-linear elimination following Intravenous infusion and simulate the model for single subject and single dosage regimen.

Libraries

Call the "necessary" libraries to get started

using Random
using Pumas
using PumasUtilities
using CairoMakie

Model

In this two compartment model, we administer dose on Central compartment.

pk_18           = @model  begin
  @metadata begin
    desc = "Two Compartment Model - Nonlinear Elimination"
    timeu = u"hr"
  end

  @param begin
    "Maximum rate of elimination (mg/hr)"
    tvvmax      ∈ RealDomain(lower=0)
    "Michaelis-Menten rate constant (mg/L)"
    tvkm        ∈ RealDomain(lower=0)
    "Intercompartmental Clearance (L/hr)"
    tvQ         ∈ RealDomain(lower=0)
    "Volume of Central Compartment (L)"
    tvvc        ∈ RealDomain(lower=0)
    "Volume of Peripheral Compartment (L)"
    tvvp        ∈ RealDomain(lower=0)
    Ω           ∈ PDiagDomain(5)
    "Proportional RUV"
    σ_prop     ∈ RealDomain(lower=0)
  end

  @random begin
    η           ~ MvNormal(Ω)
  end

  @pre begin
    Vmax        = tvvmax * exp(η[1])
    Km          = tvkm * exp(η[2])
    Q           = tvQ * exp(η[3])
    Vc          = tvvc * exp(η[4])
    Vp          = tvvp * exp(η[5])
  end

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

  @derived begin
    cp         = @. Central/Vc
    """
    Observed Concentration (g/L)
    """
    dv         ~ @. Normal(cp, cp^2*σ_prop)
  end
end

PumasModel
  Parameters: tvvmax, tvkm, tvQ, tvvc, tvvp, Ω, σ_prop
  Random effects: η
  Covariates: 
  Dynamical variables: Central, Peripheral
  Derived: cp, dv
  Observed: cp, dv

Parameters

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

• $Vmax$ - Maximum rate of elimination (mg/hr)

• $Km$ - Michaelis-Menten rate constant (mg/L)

• $Q$ - Intercompartmental Clearance (L/hr)

• $Vc$ - Volume of Central Compartment (L)

• $Vp$ - Volume of Peripheral Compartment (L)

• $Ω$ - Between Subject Variability

• $σ$ - Residual error

param  = (tvvmax     = 0.0812189,
          tvkm       = 0.0125445,
          tvQ        = 1.29034,
          tvvc       = 8.93016,
          tvvp       = 31.1174,
          Ω          = Diagonal([0.0,0.0,0.0,0.0,0.0]),
          σ_prop    = 0.005)

(tvvmax = 0.0812189, tvkm = 0.0125445, tvQ = 1.29034, tvvc = 8.93016, tvvp 
= 31.1174, 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], σ_prop = 0.005)

Dosage Regimen

0.4g/Kg (i.e.,28g for a 70Kg healthy individual), infused over a time span of 30 minutes, given to a single subject.

ev1   = DosageRegimen(28, time = 0, cmt = 1, duration = 30)
sub1  = Subject(id = 1, events = ev1, observations = (cp = nothing,))

Subject
  ID: 1
  Events: 2
  Observations: cp: (nothing)

Simulation

Lets simulate for plasma concentration with the specific observation time points after Intravenous administration.

Random.seed!(1234)
sim_sub = simobs(pk_18,sub1, param, obstimes = 0.1:1:360)

Visualization

f, a, p = sim_plot(pk_18, [sim_sub], 
        observations = :cp, 
        color = :redsblues,
        linewidth = 4,
        axis = (xlabel = "Time (minute)", 
                ylabel = "PK18 Concentrations (g/L)",
                xticks = 0:50:400, yscale = log10))
axislegend(a) 
f

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