Exercise PK34 - Reversible Metabolism

2021-09-06

Background

  • Structural model - Two compartment model

  • Route of administration - IV-infusion (with an infusion pump)

  • Dosage Regimen - 100 mg/m² of Cisplatin for 1-h at time=0 considering a patient with 1.7m²

  • Number of Subjects - 1

pk34

Learning Outcome

  • We will learn how to simulate kinetics of drug that exhibits reversible metabolism.

  • Simulate data for two IV-Infusion with two different rates of infusion regimen.

Objectives

In this exercise you will learn how to

  • Simulate an IV-infusion two compartment model and kinetic of reversible metabolism.

Certain assumptions to be considered:

  • A fraction of the dose (2.3%) is present as the monohydrated complex in the infusion solution,

  • That there is a reversible reactions between cisplatin (p) and its monohydrated complex (m),

  • The input rate can be split into cisplatin infusion rate (Inp) and monohydrate infusion rate (Inm).

Libraries

Call the "necessary" libraries to get start.

using Random
using Pumas
using PumasUtilities
using CairoMakie

Model - Microconstant Model

In this two compartment model, we administer the mentioned dose in the Central compartment as well as in the Metabolite compartment at 'time= 0'. Also, $K12$ and $K21$ are the rate constants for the conversion of cisplatin into monohydrated complex and the monohydrated complex into cisplatin, respectively.

pk_34             = @model begin
    @metadata begin
      desc        = "Microconstant Model"
      timeu       = u"minute"
    end

    @param begin
      "Volume of Central Compartment (L)"
      tvvc         RealDomain(lower=0)
      "Clearance of metabolite (L/min)"
      tvclm        RealDomain(lower=0)
      "Volume of Metabolite Compartment (μg/L)"
      tvvm         RealDomain(lower=0)
      "Clearance of parent (L/min)"
      tvclp        RealDomain(lower=0)
      tvk12        RealDomain(lower=0)
      tvk21        RealDomain(lower=0)
      Ω            PDiagDomain(6)
      "Proportional RUV"
      σ²_prop      RealDomain(lower=0)
    end

    @random begin
      η           ~ MvNormal(Ω)
    end

    @pre begin
      Vc          = tvvc * exp(η[1])
      CLm         = tvclm * exp(η[2])
      Vm          = tvvm * exp(η[3])
      CLp         = tvclp * exp(η[4])
      K12         = tvk12 * exp(η[5])
      K21         = tvk21 * exp(η[6])
    end

    @dynamics begin
      Central'    = -(CLp/Vc)*Central - K12*Central + K21*Metabolite*Vc/Vm
      Metabolite' = -(CLm/Vm)*Metabolite - K21*Metabolite + K12*Central*Vm/Vc
    end

    @derived begin
      cp          = @. Central/Vc
      """
      Observed Concentration - Cisplatin (ug/ml)
      """
      dv_cp       ~ @. Normal(cp, sqrt(cp^2*σ²_prop))
      met         = @. Metabolite/Vm
      """
      Observed Concentration - Metabolite (ug/ml)
      """
      dv_met      ~ @. Normal(met, sqrt(cp^2*σ²_prop))
    end
end
PumasModel
  Parameters: tvvc, tvclm, tvvm, tvclp, tvk12, tvk21, Ω, σ²_prop
  Random effects: η
  Covariates: 
  Dynamical variables: Central, Metabolite
  Derived: cp, dv_cp, met, dv_met
  Observed: cp, dv_cp, met, dv_met

Parameters

Parameters provided for simulation are as below. tv represents the typical value for parameters.

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

  • $CLm$ - Clearance of metabolite (L/min)

  • $Vm$ - Volume of metabolite compartment (μg/L)

  • $CLp$ - Clearance of parent (L/min)

  • $K12$ - Rate constant for the conversion of cisplatin into monohydrated complex (min⁻¹)

  • $K21$ - Rate constant for the conversion of monohydrated complex into cisplatin (min⁻¹)

  • $Ω$ - Between Subject Variability

  • $σ$ - Residual error

param = ( tvvc    = 14.1175,
          tvclm   = 0.00832616,
          tvvm    = 2.96699,
          tvclp   = 0.445716,
          tvk12   = 0.00021865,
          tvk21   = 0.021313,
          Ω       = Diagonal([0.0,0.0,0.0,0.0,0.0,0.0]),
          σ²_prop = 0.001)
(tvvc = 14.1175, tvclm = 0.00832616, tvvm = 2.96699, tvclp = 0.445716, tvk1
2 = 0.00021865, tvk21 = 0.021313, 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 = 0.001)

Dosage Regimen

In this section the Dosage regimen is mentioned for:

  • Cisplatin Infusion - A total dose of 170mg (100mg/m² * 1.7m²) split as Cisplatin 166.09 and Monohydrate 3.91.

  • Monohydrate Infusion - A total dose of 10 mg/L is given as Monohydrate

ev1  = DosageRegimen([166.09,3.91], time = 0, cmt = [1,2], duration = [60,60])
sub1 = Subject(id = "Cisplatin (Inf-Cisplatin)", events = ev1, time = 20:0.1:180)
ev2  = DosageRegimen(10, time = 0, cmt = 2, duration = 2)
sub2 = Subject(id = "Monohydrate (Inf-Cisplatin)", events = ev2, time = 5:0.1:180)
pop2_sub = [sub1, sub2]
Population
  Subjects: 2
  Observations:

Simulation

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

Random.seed!(123)
sim_sub1 = simobs(pk_34, pop2_sub, param)

Visualization

f1, a1, p1 = sim_plot(pk_34, sim_sub1, 
        observations = :cp, 
        color = :redsblues,
        linewidth = 4,
        axis = (xlabel = "Time (hr)", 
                ylabel = "PK34 Parent Concentrations (μg/mL)",
                xticks = 0:20:180, ))
axislegend(a1) 
f1
f2, a2, p2 = sim_plot(pk_34, sim_sub1, 
        observations = :met, 
        color = :redsblues,
        linewidth = 4,
        axis = (xlabel = "Time (hr)", 
                ylabel = "PK34 Metabolite Concentrations (μg/mL)",
                xticks = 0:20:180, ))
axislegend(a2) 
f2