Background

• Structural model - 1 compartment with first order absorption (without/with Lag time)

• Route of administration - Oral

• Dosage Regimen - 100 μg Oral

• Number of Subjects - 1

Learning Outcome

By the application of the present model, we will learn how to simulate model for first order input model with and without lag-time.

Objectives

In this exercise you will learn how to

• Simulate an Oral One Compartment (without/ with lag-time). Assuming oral bioavailability of 100%. The interpretation V includes bioavailability (i.e., it is really estimating V/F).

• Write a differential equation for a one-compartment model

Libraries

call the "necessary" libraries to get start.

using Random
using Pumas
using PumasUtilities
using CairoMakie

Model

In this one compartment model, we administer dose in Depot compartment at time= 0.

pk_02        = @model begin
  @metadata begin
    desc = "One Compartment Model with lag time"
    timeu = u"minute"
  end

  @param begin
    "Absorption rate constant (1/min)"
    tvka     ∈ RealDomain(lower=0)
    "Elimination rate constant (1/min)"
    tvkel    ∈ RealDomain(lower=0)
    "Volume (L)"
    tvvc     ∈ RealDomain(lower=0)
    "Lag Time (mins)"
    tvlag    ∈ RealDomain(lower=0)
    Ω        ∈ PDiagDomain(4)
    "Proportional RUV"
    σ²_prop  ∈ RealDomain(lower=0)
  end

  @random begin
    η        ~ MvNormal(Ω)
  end

  @pre begin
    Ka       = tvka * exp(η[1])
    Kel      = tvkel * exp(η[2])
    Vc       = tvvc * exp(η[3])
  end

  @dosecontrol begin
    lags     = (Depot=tvlag * exp(η[4]),)
  end

  @dynamics begin
    Depot'   = -Ka*Depot
    Central' =  Ka*Depot - Kel*Central
  end

  @derived begin
    """
    PK02 Concentration (ug/L)
    """
    cp       = @. Central/Vc
    """
    PK02 Concentration (ug/L)
    """
    dv       ~ @. Normal(cp, sqrt(cp^2*σ²_prop))
  end
end

PumasModel
  Parameters: tvka, tvkel, tvvc, tvlag, Ω, σ²_prop
  Random effects: η
  Covariates: 
  Dynamical variables: Depot, Central
  Derived: cp, dv
  Observed: cp, dv

Parameters

The compound follows a one compartment model, in which the various parameters are as mentioned below:

• $Ka$ - Absorption Rate Constant (min⁻¹)

• $Kel$ - Elimination Rate Constant(min⁻¹)

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

• $tlag$ - Lag-time (min)

• $Ω$ - Between Subject Variability

• $σ$ - Residual error

param = [ 
          (tvka    = 0.013,
          tvkel   = 0.013,
          tvvc    = 32,
          tvlag   = 0,
          Ω       = Diagonal([0.0,0.0,0.0,0.0]),
          σ²_prop = 0.015),

          (tvka    = 0.043,
          tvkel   = 0.0088,
          tvvc    = 32,
          tvlag   = 16,
          Ω       = Diagonal([0.0,0.0,0.0,0.0]),
          σ²_prop = 0.015)
        ]

Dosage Regimen

In this section the Dosage regimen is mentioned:

• Oral dosing of 100 μg at time=0 for a single subject

ev1  = DosageRegimen(100, time = 0, cmt = 1)
pop  = map(i -> Subject(id = i, events = ev1), ["1: No Lag", "2: With Lag"])

Population
  Subjects: 2
  Observations:

Simulation

When simulating a single subject, we can pass in an array of parameters to visualize multiple curves for the same subject.

Random.seed!(123)
sim = map(zip(pop, param)) do (subj, p)
  return simobs(pk_02, subj, p, obstimes = 0:1:400)
end

Simulated population (Vector{<:Subject})
  Simulated subjects: 2
  Simulated variables: cp, dv

Visualize results

From the plot below, we can clearly identify the subject with and without lag time.

f, a, p = sim_plot(pk_02, sim, 
        observations = :cp, 
        color = :redsblues, linewidth = 4,
        axis = (xlabel = "Time (minutes)", 
        ylabel = "PK02 Concentrations (ug/L)",
        xticks = 0:50:400,))
axislegend(a)
f

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