Structural model - Target Mediated Drug Disposition Model (TMDD)

Route of administration - IV-Bolus

Dosage Regimen - 1.5, 5, 15, 45 mg/kg administered after complete washout

Number of Subjects - 4

To fit a full TMDD model with data from only ligand, ligand and target, target and ligand-target complex

Write a differential equation for a full TMDD model

The objective of this exercise is to simulate from a TMDD model

Call the "necessary" libraries to get started

using Random using Pumas using PumasUtilities using CairoMakie

pk_27 = @model begin @metadata begin desc = "Target Mediated Drug Disposition Model" timeu = u"hr" end @param begin "Clearance of central compartment (L/kg/hr)" tvcl ∈ RealDomain(lower=0) "Second oder on rate of ligand (L/mg/hr)" tvkon ∈ RealDomain(lower=0) "First order off rate of ligand (1/hr)" tvkoff ∈ RealDomain(lower=0) "Volume of Peripheral Compartment (L/kg)" tvvp ∈ RealDomain(lower=0) "Inter-compartmental clearance (L/kg/hr)" tvq ∈ RealDomain(lower=0) "Zero order receptor synthesis process (mg/L/hr)" tvkin ∈ RealDomain(lower=0) "First order receptor degenration process (1/hr)" tvkout ∈ RealDomain(lower=0) "First order elimination of complex (1/hr)" tvkerl ∈ RealDomain(lower=0) "Volume of Central Compartment (L/kg)" tvvc ∈ RealDomain(lower=0) Ω ∈ PDiagDomain(9) "Proportional RUV - Plasma" σ²_prop_cp ∈ RealDomain(lower=0) "Proportional RUV - Receptor" σ²_prop_rec ∈ RealDomain(lower=0) "Proportional RUV - Complex" σ²_prop_com ∈ RealDomain(lower=0) end @random begin η ~ MvNormal(Ω) end @pre begin Cl = tvcl * exp(η[1]) Kon = tvkon * exp(η[2]) Koff = tvkoff * exp(η[3]) Vp = tvvp * exp(η[4]) Q = tvq * exp(η[5]) Kin = tvkin * exp(η[6]) Kout = tvkout * exp(η[7]) Kerl = tvkerl * exp(η[8]) Vc = tvvc * exp(η[9]) end @dosecontrol begin Vc_ = tvvc * exp(η[9]) bioav = (Central = 1/Vc_,) end @init begin Receptor = Kin/Kout end @dynamics begin Central' = -(Cl/Vc)*Central - (Q/Vc)*Central + (Q/Vp)*Peripheral - Kon*Receptor*Central + Koff*Complex Peripheral' = (Q/Vc)*Central - (Q/Vp)*Peripheral Receptor' = Kin - Kout*Receptor - Kon*Receptor*Central + Koff*Complex Complex' = Kon*Receptor*Central - Koff*Complex - Kerl*Complex end @derived begin cp = @. Central """ Observed Concentration - Plasma (mg/L) """ dv_cp = @. Normal(cp, sqrt(cp^2*σ²_prop_cp)) rec = @. Receptor """ Observed Concentration - Receptor (mg/L) """ dv_rec = @. Normal(rec, sqrt(rec^2*σ²_prop_rec)) com = @. Complex """ Observed Concentration - Complex (mg/L) """ dv_com = @. Normal(com, sqrt(com^2*σ²_prop_com)) end end

PumasModel Parameters: tvcl, tvkon, tvkoff, tvvp, tvq, tvkin, tvkout, tvkerl, tvvc, Ω, σ²_prop_cp, σ²_prop_rec, σ²_prop_com Random effects: η Covariates: Dynamical variables: Central, Peripheral, Receptor, Complex Derived: cp, dv_cp, rec, dv_rec, com, dv_com Observed: cp, dv_cp, rec, dv_rec, com, dv_com

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

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

$Kon$ - Second oder on rate of ligand (L/mg/hr)

$Koff$ - First order off rate of ligand (1/hr)

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

$Q$ - Inter-compartmental clearance (L/kg/hr)

$Kin$ - Zero order receptor synthesis process (mg/L/hr)

$Kout$ - First order receptor degeneration process (1/hr)

$Kerl$ - First order elimination of complex (1/hr)

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

param = ( tvcl = 0.001, tvkon = 0.096, tvkoff = 0.001, tvvp = 0.100, tvq = 0.003, tvkin = 0.11, tvkout = 0.0089, tvkerl = 0.003, tvvc = 0.05, Ω = Diagonal([0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0]), σ²_prop_cp = 0.02, σ²_prop_rec = 0.012, σ²_prop_com = 0.015)

(tvcl = 0.001, tvkon = 0.096, tvkoff = 0.001, tvvp = 0.1, tvq = 0.003, tvki n = 0.11, tvkout = 0.0089, tvkerl = 0.003, tvvc = 0.05, 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_ cp = 0.02, σ²_prop_rec = 0.012, σ²_prop_com = 0.015)

Single dose of 1.5 mg/kg was administered as IV-Bolus at

`time=0`

Single dose of 5 mg/kg was administered as IV-Bolus at

`time=0`

Single dose of 15 mg/kg was administered as IV-Bolus at

`time=0`

Single dose of 45 mg/kg was administered as IV-Bolus at

`time=0`

dose = [1.5, 5, 15, 45] ids = ["1.5 mg/kg", "5 mg/kg", "15 mg/kg", "45 mg/kg"] dose_ind(x) = DosageRegimen(dose[x], time = 0, cmt = 1) pop4_sub = map(i -> Subject(id = ids[i], events = dose_ind(i), observations = (cp = nothing, rec = nothing, com = nothing)), 1:length(ids))

Population Subjects: 4 Observations: cp, rec, com

Random.seed!(123) sim_pop4 = simobs(pk_27, pop4_sub, param, obstimes = 0.1:1:500)

figure = Figure(; fontsize = 12) a1, p1 = sim_plot(figure[1,1], pk_27, sim_pop4, observations = :cp, color = :redsblues, linewidth = 4, axis = (title = "Ligand", xlabel = "Time (hr)", ylabel = "PK27 Concentrations (mg/L)", xticks = 0:100:500, yscale = log10)) a2, p2 = sim_plot(figure[1,2], pk_27, sim_pop4, observations = :rec, color = :redsblues, linewidth = 4, axis = (title = "Receptor", xlabel = "Time (hr)", ylabel = "PK27 Concentrations (mg/L)", xticks = 0:100:500, yscale = log10)) a3, p3 = sim_plot(figure[2,1], pk_27, sim_pop4, observations = :com, color = :redsblues, linewidth = 4, axis = (title = "Complex", xlabel = "Time (hr)", ylabel = "PK27 Concentrations (mg/L)", xticks = 0:100:500, yscale = log10)) Legend(figure[2, 2], a3, "Doses", tellwidth = false, framevisible = false) figure