Background

• Structural model - One compartment first order elimination with zero order production of hormone

• Route of administration - Subcutaneous

• Dosage Regimen - 40 mcg/kg

• Number of Subjects - 1

Learning Outcome

With subcutaneous dose information, we learn to discriminate between the clearance and the rate of synthesis. Further simplification of the model is done by assuming bioavailability is 100% and concentration of the endogenous compound equals concentration at baseline (turnover/clearance)

Objectives

In this tutorial, you will learn how to build a one compartment PK turnover model, following first order elimination kinetics and zero order hormone production.

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 subcutaneously.

pk_30           = @model begin
  @metadata begin
    desc        = "Two Compartment Model"
    timeu       = u"hr"
  end

  @param begin
    "Absorption rate constant (hr⁻¹)"
    tvka        ∈ RealDomain(lower=0)
    "Clearance (L/kg/hr)"
    tvcl        ∈ RealDomain(lower=0)
    "Turnover Rate (hr⁻¹)"
    tvsynthesis ∈ RealDomain(lower=0)
    "Volume of Central Compartment (L/kg)"
    tvv         ∈ RealDomain(lower=0)
    Ω           ∈ PDiagDomain(4)
    "Additive RUV"
    σ_add       ∈ RealDomain(lower=0)
  end

  @random begin
    η           ~ MvNormal(Ω)
  end

  @pre begin
    Ka          = tvka * exp(η[1])
    Cl          = tvcl * exp(η[2])
    Synthesis   = tvsynthesis * exp(η[3])
    V           = tvv * exp(η[4])
  end

  @init begin
    Central     = Synthesis/(Cl/V) # Concentration at Baseline = Turnover Rate (0.78) / Cl of hormone (0.028)
  end

  @dynamics begin
    Depot'      = -Ka * Depot
    Central'    =  Ka * Depot + Synthesis - (Cl/V) * Central
  end

  @derived begin
    cp          = @. Central/V
    """
    Observed Concentration (mcg/L)
    """
    dv          ~ @. Normal(cp, σ_add)
  end
end

PumasModel
  Parameters: tvka, tvcl, tvsynthesis, tvv, Ω, σ_add
  Random effects: η
  Covariates: 
  Dynamical variables: Depot, Central
  Derived: cp, dv
  Observed: cp, dv

Parameters

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

• $Ka$ - Absorption rate constant (hr⁻¹)

• $Cl$ - Clearance (L/kg/hr)

• $Synthesis$ - Turnover Rate (hr⁻¹)

• $V$ - Volume of Central Compartment (L/kg)

• $Ω$ - Between Subject Variability

• $σ$ - Residual error

param =  ( tvka         = 0.539328,
           tvcl         = 0.0279888,
           tvsynthesis  = 0.781398,
           tvv          = 0.10244,
           Ω            = Diagonal([0.0,0.0,0.0,0.0]),
           σ_add        = 3.97427)

(tvka = 0.539328, tvcl = 0.0279888, tvsynthesis = 0.781398, tvv = 0.10244, 
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
_add = 3.97427)

Dosage Regimen

A dose of 40 mcg/kg is given subcutaneously to a Single subject.

ev1  = DosageRegimen(40,time = 0, cmt = 1)
sub1 = Subject(id = 1, events = ev1)

Subject
  ID: 1
  Events: 1

Simulation

To simulate plasma concentration with turnover rate after oral administration.

Random.seed!(1234)
sim_sub1 = simobs(pk_30, sub1, param, obstimes = 0:0.1:72)

Visualization

f, a, p = sim_plot(pk_30, [sim_sub1], 
        observations = :cp, 
        color = :redsblues,
        linewidth = 4,
        axis = (xlabel = "Time (hrs)", 
                ylabel = "PK30 Concentrations (μg/L)",
                xticks = 0:10:80, yscale = log10))
axislegend(a) 
f

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