Structural model - Two Compartment model with a Metabolite Compartment

Route of administration - IV Bolus

Dosage Regimen - 10μmol/kg, 50μmol/kg, 300μmol/kg

Number of Subjects - 3

In this model, 3 different dose of the drug given as an IV Bolus to 3 different subjects, will help you estimate metabolite formation rate and elimination rate.

In this tutorial, you will learn how to build two compartment model, a drug undergoing capacity limited metabolite kinetics.

Call the "necessary" libraries to get start.

using Random using Pumas using PumasUtilities using CairoMakie

In this two compartment model, we administer 3 different doses in 3 different subjects of a drug that undergoes metabolite kinetics.

pk_19 = @model begin @metadata begin desc = "Non-linear formation of Metabolite Model" timeu = u"hr" end @param begin "Volume of Central Compartment (L/kg)" tvvc ∈ RealDomain(lower=0) "Volume of Perpheral Compartment (L/kg)" tvvp ∈ RealDomain(lower=0) "Inter-Compartmental Clearance (L/min)" tvq ∈ RealDomain(lower=0) "Maximum Velocity of Reaction (μmol/min/kg)" tvvmax ∈ RealDomain(lower=0) "Michaelis-Menten constant (μmol/L)" tvkm ∈ RealDomain(lower=0) "Rate of Elimination of Metabolite (min⁻¹)" tvkme ∈ RealDomain(lower=0) "Volume of Metabolite Compartment (L/kg)" tvvme ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(7) "Proportional RUV - Plasma" σ_prop_cp ∈ RealDomain(lower=0) "Proportional RUV - Metabolite" σ_prop_met ∈ RealDomain(lower=0) end @random begin η ~ MvNormal(Ω) end @pre begin Vc = tvvc * exp(η[1]) Vp = tvvp * exp(η[2]) Q = tvq * exp(η[3]) Vmax = tvvmax * exp(η[4]) Km = tvkm * exp(η[5]) Kme = tvkme * exp(η[6]) Vme = tvvme * exp(η[7]) end @vars begin VMKM := Vmax/(Km+(Central/Vc)) end @dynamics begin Central' = -VMKM*(Central/Vc) - (Q/Vc)*Central + (Q/Vp)*Peripheral Peripheral' = (Q/Vc)*Central - (Q/Vp)*Peripheral Metabolite' = VMKM*(Central/Vc) - Kme*Metabolite end @derived begin cp = @. Central/Vc """ Observed Plasma Concentration (μmol/L) """ dv_cp ~ @. Normal(cp, cp*σ_prop_cp) met = @. Metabolite/Vme """ Observed Metabolite Concentration (μmol/L) """ dv_met ~ @. Normal(met, met*σ_prop_met) end end

PumasModel Parameters: tvvc, tvvp, tvq, tvvmax, tvkm, tvkme, tvvme, Ω, σ_prop_cp, σ_ prop_met Random effects: η Covariates: Dynamical variables: Central, Peripheral, Metabolite Derived: cp, dv_cp, met, dv_met Observed: cp, dv_cp, met, dv_met

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

$Vc$ - Volume of Central Compartment (L/kg)

$Vp$ - Volume of Peripheral Compartment (L/kg)

$Q$ - Inter-Compartmental Clearance (L/min)

$Vmax$ - Maximum Velocity of Reaction (μmol/min/kg)

$Km$ - Michaelis-Menten constant (μmol/L)

$Kme$ - Rate of Elimination of Metabolite (min⁻¹)

$Vme$ - Volume of Metabolite Compartment (L/kg)

$Ω$ - Between Subject Variability

$σ$ - Residual error

param = ( tvvc = 1.06405, tvvp = 2.00748, tvq = 0.128792, tvvmax = 1.64429, tvkm = 54.794, tvkme = 0.145159, tvvme = 0.290811, Ω = Diagonal([0.0,0.0,0.0,0.0,0.0,0.0,0.0]), σ_prop_cp = 0.12, σ_prop_met = 0.12)

(tvvc = 1.06405, tvvp = 2.00748, tvq = 0.128792, tvvmax = 1.64429, tvkm = 5 4.794, tvkme = 0.145159, tvvme = 0.290811, 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_cp = 0.12, σ_p rop_met = 0.12)

Three Subjects were adminitered with three different doses of **10μmol/kg, 50μmol/kg and 300μmol/kg**.

dose = [10, 50, 300] ids = ["ID:1 Dose 10", "ID:2 Dose 50", "ID:3 Dose 300"] ev(x) = DosageRegimen(dose[x], cmt = 1, time = 0) pop = map(zip(1:3, ids)) do (i, id) return Subject(id = id, events = ev(i), observations = (cp = nothing, met = nothing)) end

Population Subjects: 3 Observations: cp, met

We will simulate the parent plasma concentration and metabolite plasma concentration.

Random.seed!(123) sim_pop3_sub = simobs(pk_19, pop, param, obstimes = 0.1:1:300)

fig = Figure() ax, p1 = sim_plot(fig[1,1], pk_19, sim_pop3_sub, observations = :cp, color = :redsblues, linewidth = 4, axis = (xlabel = "Time (minute)", ylabel = "PK19 Parent Concentrations (μmol/L)", xticks = 0:50:300, yscale = log10)) axislegend(ax) fig

fig = Figure() ax, p1 = sim_plot(fig[1,1], pk_19, sim_pop3_sub, observations = :met, color = :redsblues, linewidth = 4, axis = (xlabel = "Time (minute)", ylabel = "PK19 Metabolite Concentrations (μmol/L)", xticks = 0:50:300, yscale = log10)) axislegend(ax) fig