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Tolerance Models

PD models for tolerance development through counter-regulation or receptor regulation mechanisms.

Counter-Regulation Tolerance

Usage

using NeoPKPD

# PK setup
pk_model = OneCompIVBolus()
pk_params = OneCompIVBolusParams(5.0, 50.0)
doses = [DoseEvent(i * 8.0, 50.0) for i in 0:20]  # TID for 7 days
pk_spec = ModelSpec(pk_model, "pk", pk_params, doses)

# Tolerance PD
pd_model = ToleranceCounterRegulation()
pd_params = ToleranceCounterRegulationParams(
    0.0,      # e0
    100.0,    # emax
    5.0,      # ec50
    1.0,      # gamma
    0.1,      # kin_mod
    0.05,     # kout_mod
    1.0       # alpha_feedback
)

pd_spec = PDSpec(pd_model, "tol", pd_params, :conc, :effect)

grid = SimGrid(0.0, 168.0, 0:0.5:168)
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10^7)

result = simulate_pkpd_coupled(pk_spec, pd_spec, grid, solver)

Parameters

Parameter Type Description
e0 Float64 Baseline effect
emax Float64 Maximum drug effect
ec50 Float64 EC50 for drug effect
gamma Float64 Hill coefficient
kin_mod Float64 Moderator production rate
kout_mod Float64 Moderator elimination rate
alpha_feedback Float64 Feedback strength

Model Equations

Drug effect: $\(E_{drug} = \frac{E_{max} \cdot C^\gamma}{EC_{50}^\gamma + C^\gamma}\)$

Moderator dynamics: $\(\frac{dM}{dt} = k_{in,mod} \cdot E_{drug} - k_{out,mod} \cdot M\)$

Net effect: $\(E_{net} = E_0 + E_{drug} - \alpha \cdot M\)$

Basic Example

using NeoPKPD

pk_model = OneCompIVBolus()
pk_params = OneCompIVBolusParams(5.0, 50.0)
doses = [DoseEvent(i * 8.0, 50.0) for i in 0:20]
pk_spec = ModelSpec(pk_model, "pk", pk_params, doses)

pd_model = ToleranceCounterRegulation()
pd_params = ToleranceCounterRegulationParams(0.0, 100.0, 5.0, 1.0, 0.1, 0.05, 1.0)
pd_spec = PDSpec(pd_model, "tol", pd_params, :conc, :effect)

grid = SimGrid(0.0, 168.0, 0:0.5:168)
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10^7)

result = simulate_pkpd_coupled(pk_spec, pd_spec, grid, solver)

effect = result.observations[:effect]

# Compare first vs last dose
first_peak = maximum(effect[1:16])
last_peak = maximum(effect[end-16:end])

println("First dose peak: $first_peak")
println("Last dose peak: $last_peak")
println("Tolerance: $((1 - last_peak/first_peak) * 100)%")

Receptor Regulation Model

Usage

using NeoPKPD

# PK setup
pk_model = OneCompIVBolus()
pk_params = OneCompIVBolusParams(5.0, 50.0)
doses = [DoseEvent(i * 8.0, 50.0) for i in 0:20]
pk_spec = ModelSpec(pk_model, "pk", pk_params, doses)

# Receptor regulation PD
pd_model = ReceptorRegulation()
pd_params = ReceptorRegulationParams(
    0.0,      # e0
    100.0,    # emax
    5.0,      # ec50
    1.0,      # gamma
    1.0,      # r_baseline
    0.05,     # kreg
    2.0,      # rmax
    0.02,     # kchange
    :down     # direction
)

pd_spec = PDSpec(pd_model, "receptor", pd_params, :conc, :effect)

grid = SimGrid(0.0, 168.0, 0:0.5:168)
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10^7)

result = simulate_pkpd_coupled(pk_spec, pd_spec, grid, solver)

Parameters

Parameter Type Description
r_baseline Float64 Baseline receptor density
kreg Float64 Receptor recovery rate
rmax Float64 Maximum receptor density
kchange Float64 Rate of receptor change
direction Symbol :down or :up

Receptor Equations

Down-regulation: $\(\frac{dR}{dt} = k_{reg} \cdot (R_0 - R) - k_{change} \cdot E_{drug} \cdot R\)$

Up-regulation: $\(\frac{dR}{dt} = k_{reg} \cdot (R_0 - R) + k_{change} \cdot E_{drug} \cdot (R_{max} - R)\)$

Net effect: $\(E_{net} = E_0 + R \cdot E_{drug}\)$

Down-Regulation Example

using NeoPKPD

pk_model = OneCompIVBolus()
pk_params = OneCompIVBolusParams(5.0, 50.0)
doses = [DoseEvent(i * 8.0, 50.0) for i in 0:20]
pk_spec = ModelSpec(pk_model, "pk", pk_params, doses)

pd_model = ReceptorRegulation()
pd_params = ReceptorRegulationParams(0.0, 100.0, 5.0, 1.0, 1.0, 0.05, 2.0, 0.02, :down)
pd_spec = PDSpec(pd_model, "beta_receptor", pd_params, :conc, :effect)

grid = SimGrid(0.0, 168.0, 0:0.5:168)
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10^7)

result = simulate_pkpd_coupled(pk_spec, pd_spec, grid, solver)

effect = result.observations[:effect]
receptor = result.states[:R]

println("Initial receptor: $(receptor[1])")
println("Day 7 receptor: $(receptor[end])")
println("First peak effect: $(maximum(effect[1:16]))")
println("Day 7 peak effect: $(maximum(effect[end-16:end]))")

Clinical Applications

Counter-Regulation

  • Opioid tolerance
  • Benzodiazepine tolerance
  • Beta-blocker tolerance

Receptor Regulation

  • Beta-receptor down-regulation
  • Opioid receptor adaptation
  • Hormone receptor changes

Equations Summary

Counter-Regulation

Quantity Formula
Drug effect \(E_{max} \cdot C^\gamma / (EC_{50}^\gamma + C^\gamma)\)
Moderator \(dM/dt = k_{in} \cdot E_{drug} - k_{out} \cdot M\)
Net effect \(E_0 + E_{drug} - \alpha \cdot M\)

Receptor Regulation

Direction Rate Equation
Down \(k_{reg}(R_0 - R) - k_{change} \cdot E \cdot R\)
Up \(k_{reg}(R_0 - R) + k_{change} \cdot E \cdot (R_{max} - R)\)

See Also