Michaelis-Menten Elimination

One-compartment model with saturable (capacity-limited) elimination kinetics, where clearance depends on concentration.

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Model Overview

graph LR
    D((Dose)) -->|Bolus| C[Central<br/>V]
    C -->|Vmax, Km| E((Saturable<br/>Elimination))

Clinical Applications

• High-dose therapeutics
• Phenytoin, ethanol, aspirin (high dose)
• Biologics with target-mediated disposition
• Enzyme saturation scenarios
• Dose-dependent kinetics

When to Use

Use When	Don't Use When
Nonlinear kinetics	Linear (first-order) elimination
Dose-dependent t½	Constant half-life
Disproportionate AUC ↑	Dose-proportional exposure
Saturable metabolism	Low concentration range

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Mathematical Formulation

Parameters

Parameter	Symbol	Units	Description	Constraints
Maximum rate	Vmax	mg/h	Maximum elimination rate	Vmax > 0
Michaelis constant	Km	mg/L	Concentration at 50% Vmax	Km > 0
Volume	V	L	Volume of distribution	V > 0

Differential Equation

In terms of amount:

\[\frac{dA}{dt} = -\frac{V_{max} \cdot A}{K_m \cdot V + A}\]

In terms of concentration:

\[\frac{dC}{dt} = -\frac{V_{max} \cdot C}{K_m + C}\]

Observation

\[C = \frac{A}{V}\]

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Kinetic Behavior

Concentration-Dependent Kinetics

Condition	Behavior	Effective CL
C << Km	First-order (linear)	CL ≈ Vmax/Km
C ≈ Km	Mixed-order	Variable
C >> Km	Zero-order (saturated)	Rate ≈ Vmax

Apparent Clearance

\[CL_{app} = \frac{V_{max}}{K_m + C}\]

At low concentrations: \(CL_{max} = V_{max}/K_m\)

Apparent Half-Life

Half-life is concentration-dependent:

\[t_{1/2} \approx \frac{0.693 \cdot V \cdot (K_m + C)}{V_{max}}\]

At low C: \(t_{1/2,min} = 0.693 \cdot V \cdot K_m / V_{max}\)

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Julia API

Type Definitions

# Model kind
struct MichaelisMentenElimination <: ModelKind end

# Parameters
struct MichaelisMentenEliminationParams <: AbstractParams
    Vmax::Float64  # Maximum elimination rate (mg/h)
    Km::Float64    # Michaelis constant (mg/L)
    V::Float64     # Volume of distribution (L)
end

Basic Simulation

using NeoPKPD

# Define parameters
# Vmax = 500 mg/h, Km = 10 mg/L, V = 50 L
params = MichaelisMentenEliminationParams(500.0, 10.0, 50.0)

# Single 1000 mg dose
doses = [DoseEvent(0.0, 1000.0)]

# Create specification
spec = ModelSpec(
    MichaelisMentenElimination(),
    "mm_example",
    params,
    doses
)

# Time grid
grid = SimGrid(0.0, 48.0, collect(0.0:0.25:48.0))

# Solver
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

# Run simulation
result = simulate(spec, grid, solver)

# Check concentration-dependent kinetics
conc = result.observations[:conc]
t = result.t

println("C0: ", round(conc[1], digits=2), " mg/L (C >> Km: saturated)")
println("C at 6h: ", round(conc[25], digits=2), " mg/L")
println("C at 24h: ", round(conc[97], digits=3), " mg/L (C < Km: linear)")

Dose-Dependent Half-Life

# Compare half-lives at different doses
doses_list = [100.0, 500.0, 1000.0, 2000.0]

for dose in doses_list
    doses = [DoseEvent(0.0, dose)]
    spec = ModelSpec(MichaelisMentenElimination(), "dose_$dose", params, doses)
    grid = SimGrid(0.0, 96.0, collect(0.0:0.5:96.0))

    result = simulate(spec, grid, solver)
    conc = result.observations[:conc]

    # Find t½ (time to reach half of initial)
    c0 = conc[1]
    target = c0 / 2
    idx = findfirst(x -> x <= target, conc)

    if idx !== nothing
        t_half = result.t[idx]
        println("Dose $(dose) mg: t½ = $(round(t_half, digits=2)) h")
    else
        println("Dose $(dose) mg: t½ > 96 h")
    end
end

Expected Results: - Higher doses → Longer t½ (saturation) - Low doses → Shorter t½ (linear kinetics)

---

Dose Proportionality

# AUC increases more than proportionally with dose
doses_list = [100.0, 200.0, 500.0, 1000.0]
auc_values = Float64[]

for dose in doses_list
    doses = [DoseEvent(0.0, dose)]
    spec = ModelSpec(MichaelisMentenElimination(), "dose_$dose", params, doses)
    grid = SimGrid(0.0, 120.0, collect(0.0:0.1:120.0))

    result = simulate(spec, grid, solver)
    conc = result.observations[:conc]

    # Approximate AUC (trapezoidal)
    auc = sum(conc) * 0.1
    push!(auc_values, auc)

    println("Dose $(dose) mg: AUC ≈ $(round(auc, digits=1)) mg·h/L")
end

# Check proportionality
println("\nDose ratio (10x): ", doses_list[4] / doses_list[1])
println("AUC ratio: ", round(auc_values[4] / auc_values[1], digits=2))
# AUC ratio > 10 indicates saturation

---

Clinical Example: Phenytoin

# Phenytoin typical parameters
# Vmax ≈ 7 mg/kg/day, Km ≈ 4-6 mg/L, V ≈ 0.65 L/kg
# For 70 kg patient:
params_phenytoin = MichaelisMentenEliminationParams(
    7.0 * 70 / 24,  # Vmax in mg/h (≈ 20 mg/h)
    5.0,            # Km (mg/L)
    0.65 * 70       # V (≈ 45.5 L)
)

# Loading dose followed by maintenance
loading = 15 * 70  # 15 mg/kg loading
maintenance = 300.0 / 24 * 24  # 300 mg/day as continuous equivalent

# Simulate loading
doses_loading = [DoseEvent(0.0, loading)]
spec_loading = ModelSpec(MichaelisMentenElimination(), "phenytoin_load",
                         params_phenytoin, doses_loading)
grid = SimGrid(0.0, 24.0, collect(0.0:0.5:24.0))

result = simulate(spec_loading, grid, solver)

# Therapeutic range: 10-20 mg/L
conc = result.observations[:conc]
println("Post-loading C: ", round(conc[1], digits=1), " mg/L")
println("24h C: ", round(conc[end], digits=1), " mg/L")

---

Steady-State Considerations

At steady state with constant infusion rate R:

\[C_{ss} = \frac{R \cdot K_m}{V_{max} - R}\]

This only applies when R < Vmax. If R ≥ Vmax, no steady state is achievable.

# Critical infusion rate
Vmax = params.Vmax
println("Maximum sustainable infusion rate: ", Vmax, " mg/h")

# At 80% of Vmax
R = 0.8 * Vmax
Css = (R * params.Km) / (Vmax - R)
println("At R = $(R) mg/h: Css = $(round(Css, digits=1)) mg/L")

---

Population Simulation

# Typical parameters
typical_params = MichaelisMentenEliminationParams(500.0, 10.0, 50.0)

# IIV: High variability on Vmax and Km
omega = OmegaMatrix([
    0.16 0.0  0.0;     # ω²_Vmax (40% CV)
    0.0  0.25 0.0;     # ω²_Km (50% CV)
    0.0  0.0  0.04     # ω²_V (20% CV)
])

doses = [DoseEvent(0.0, 1000.0)]
base_spec = ModelSpec(MichaelisMentenElimination(), "pop", typical_params, doses)

pop_spec = PopulationSpec(base_spec, 50, omega, 12345)

grid = SimGrid(0.0, 48.0, collect(0.0:0.5:48.0))
result = simulate_population(pop_spec, grid, solver)

# High variability in exposure due to nonlinear kinetics
# Small changes in Vmax/Km cause large changes in AUC at high doses

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Comparison with Linear Elimination

# Michaelis-Menten at low dose (approximately linear)
params_mm = MichaelisMentenEliminationParams(500.0, 10.0, 50.0)
# Equivalent linear: CL = Vmax/Km = 50 L/h

params_linear = OneCompIVBolusParams(50.0, 50.0)  # CL = 50 L/h, V = 50 L

# Low dose: 100 mg (C0 = 2 mg/L << Km = 10)
doses_low = [DoseEvent(0.0, 100.0)]

# High dose: 2000 mg (C0 = 40 mg/L >> Km = 10)
doses_high = [DoseEvent(0.0, 2000.0)]

grid = SimGrid(0.0, 24.0, collect(0.0:0.1:24.0))

# At low dose, MM ≈ linear
spec_mm_low = ModelSpec(MichaelisMentenElimination(), "mm_low", params_mm, doses_low)
spec_lin_low = ModelSpec(OneCompIVBolus(), "lin_low", params_linear, doses_low)

result_mm_low = simulate(spec_mm_low, grid, solver)
result_lin_low = simulate(spec_lin_low, grid, solver)

# Nearly identical at low dose
println("Low dose (100 mg):")
println("  MM at 4h: ", round(result_mm_low.observations[:conc][41], digits=3))
println("  Linear at 4h: ", round(result_lin_low.observations[:conc][41], digits=3))

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Equations Summary

Quantity	Formula
Elimination rate	\(-V_{max} \cdot C / (K_m + C)\)
CL_apparent	\(V_{max} / (K_m + C)\)
CL_max (C→0)	\(V_{max} / K_m\)
t½ (apparent)	\(0.693 \cdot V \cdot (K_m + C) / V_{max}\)
Css (infusion R)	\(R \cdot K_m / (V_{max} - R)\)

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See Also

• One-Compartment IV Bolus - Linear model
• Population Modeling - Adding variability
• Parameter Estimation - Fitting nonlinear models