Background

• Structural model - Two compartment model with first order elimination

• Route of administration - IV-Bolus and IV-Infusion given simultaneously

• Dosage Regimen - 400 μg/kg IV-Bolus and 800 μg/kg IV-Infusion for 26 mins at time=0

• Number of Subjects - 1

Learning Outcome

• Write the differential equation for a two-compartment model in terms of Clearance and Volume

• Simulate data for both a bolus dose followed by a constant rate infusion regimen

• Administration of loading dose helps to achieve therapeutic concentrations faster

Objective

The objective of this exercise is to simulate data from a bolus followed by a constant rate infusion using differential equation model.

Libraries

Call the "necessary" libraries to get started

using Random
using Pumas
using PumasUtilities
using CairoMakie

Model

The given data follows a two compartment model in which the IV Bolus and IV-Infusion are administered at time=0

pk_13           = @model begin
  @metadata begin
    desc        = "Two Compartment Model"
    timeu       = u"minute"
  end

  @param begin
    "Clearance (L/min/kg)"
    tvcl        ∈ RealDomain(lower=0)
    "Volume of Central Compartment (L/kg)"
    tvvc        ∈ RealDomain(lower=0)
    "Inter-compartmental Clearance (L/min/kg)"
    tvq         ∈ RealDomain(lower=0)
    "Volume of Peripheral Compartment (L/kg)"
    tvvp        ∈ RealDomain(lower=0)
    Ω           ∈ PDiagDomain(4)
    "Proportional RUV"
    σ²_prop     ∈ RealDomain(lower=0)
    "Additive RUV"
    σ²_add      ∈ RealDomain(lower=0)
  end

  @random begin
    η           ~ MvNormal(Ω)
  end

  @pre begin
    Cl          = tvcl * exp(η[1])
    Vc          = tvvc * exp(η[2])
    Q           = tvq * exp(η[3])
    Vp          = tvvp * exp(η[4])
  end

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

  @derived begin
    cp          = @. Central/Vc
    """
    Observed Concentrations (ug/L)
    """
    dv          ~ @. Normal(cp, sqrt((cp*σ²_prop)^2 + σ²_add^2))
  end
end

PumasModel
  Parameters: tvcl, tvvc, tvq, tvvp, Ω, σ²_prop, σ²_add
  Random effects: η
  Covariates: 
  Dynamical variables: Central, Peripheral
  Derived: cp, dv
  Observed: cp, dv

Parameters

• $Cl$ - Clearance of central compartment (L/min/kg)

• $Vc$ - Volume of central compartment (L/kg)

• $Q$ - Inter-compartmental clearance (L/min/kg)

• $Vp$ - Volume of peripheral compartment (L/kg)

• $Ω$ - Between Subject Variability

• $σ$ - Residual Unexplained Variability

param = ( tvcl        = 0.344708,
          tvvc        = 2.8946,
          tvq         = 0.178392,
          tvvp        = 2.18368,
          Ω           = Diagonal([0.0, 0.0, 0.0, 0.0]),
          σ²_prop     = 0.0571079,
          σ²_add      = 0.1)

(tvcl = 0.344708, tvvc = 2.8946, tvq = 0.178392, tvvp = 2.18368, 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
.0571079, σ²_add = 0.1)

Dosage Regimen

• Single dose of 400 μg/kg given as IV-Bolus at time=0

• Single dose of 800 μg/kg given as an IV-Infusion for 26 mins at time=0

ev1  = DosageRegimen(400, time = 0, cmt = 1)
ev2  = DosageRegimen(800, time = 0, cmt = 1, rate = 30.769)
ev3  = DosageRegimen(ev1, ev2)
sub1 = Subject(id = 1, events = ev3)

Subject
  ID: 1
  Events: 3

Simulation

We will simulate the plasma concentration at the pre specified time points.

Random.seed!(123)
sim_sub1 = simobs(pk_13, sub1, param, 
                  obstimes=[2,5,10,15,20,25,30,33,35,37,40,45,50,60,70,90,110,120,150])

Visualization

sim_plot(pk_13, [sim_sub1],
                   observations = :cp, 
                   linewidth = 4,
                   color = :maroon,
                   axis = (xlabel = "Time (min)",
                           ylabel="Concentration (ug/L)", 
                           yscale = log10, xticks = 0:20:160))

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