Two-Compartment Oral¶

Two-compartment model with first-order oral absorption, combining absorption kinetics with distribution between central and peripheral compartments.

Model Overview¶

graph LR
    D((Dose)) -->|Oral| G[Gut<br/>Depot]
    G -->|Ka| C[Central<br/>V1]
    C <-->|Q| P[Peripheral<br/>V2]
    C -->|CL| E((Elimination))

Clinical Applications¶

Oral drugs with tissue distribution
Extended-release formulations
Lipophilic oral compounds
Drugs with enterohepatic recirculation (simplified)
Biologics with oral absorption

When to Use¶

Use When	Don't Use When
Oral with bi-exponential decline	Simple mono-exponential
Distribution phase evident	Rapid equilibration
Complex tissue binding	IV-only administration
Multi-phase elimination	Simple first-pass kinetics

Mathematical Formulation¶

Parameters¶

Parameter	Symbol	Units	Description	Constraints
Absorption rate	Ka	1/h	First-order absorption	Ka > 0
Clearance	CL	L/h	Apparent clearance (CL/F)	CL > 0
Central volume	V1	L	Central compartment (V1/F)	V1 > 0
Inter-compartmental clearance	Q	L/h	Distribution clearance (Q/F)	Q > 0
Peripheral volume	V2	L	Peripheral compartment (V2/F)	V2 > 0

State Variables¶

State	Symbol	Units	Description
Gut amount	A_gut	mg	Drug in absorption depot
Central amount	A1	mg	Drug in central compartment
Peripheral amount	A2	mg	Drug in peripheral compartment

Differential Equations¶

\[\frac{dA_{gut}}{dt} = -K_a \cdot A_{gut}\]

\[\frac{dA_1}{dt} = K_a \cdot A_{gut} - \frac{CL}{V_1} \cdot A_1 - \frac{Q}{V_1} \cdot A_1 + \frac{Q}{V_2} \cdot A_2\]

\[\frac{dA_2}{dt} = \frac{Q}{V_1} \cdot A_1 - \frac{Q}{V_2} \cdot A_2\]

Observation¶

\[C = \frac{A_1}{V_1}\]

Derived Parameters¶

Micro-Rate Constants¶

\[k_a = K_a\]

\[k_{10} = \frac{CL}{V_1}\]

\[k_{12} = \frac{Q}{V_1}\]

\[k_{21} = \frac{Q}{V_2}\]

Hybrid Constants (α, β)¶

Same as two-compartment IV:

\[\alpha, \beta = \frac{1}{2}\left[(k_{10} + k_{12} + k_{21}) \pm \sqrt{(k_{10} + k_{12} + k_{21})^2 - 4k_{10}k_{21}}\right]\]

Half-Lives¶

Absorption: \(t_{1/2,a} = 0.693/K_a\)
Distribution: \(t_{1/2,\alpha} = 0.693/\alpha\)
Terminal: \(t_{1/2,\beta} = 0.693/\beta\)

Tmax¶

No simple closed-form solution exists. Tmax is determined numerically when:

\[\frac{dC}{dt} = 0\]

Julia API¶

Type Definitions¶

# Model kind
struct TwoCompOral <: ModelKind end

# Parameters
struct TwoCompOralParams <: AbstractParams
    Ka::Float64    # Absorption rate constant (1/h)
    CL::Float64    # Clearance (L/h)
    V1::Float64    # Central volume (L)
    Q::Float64     # Inter-compartmental clearance (L/h)
    V2::Float64    # Peripheral volume (L)
end

Basic Simulation¶

using NeoPKPD

# Define parameters
# Ka = 1.2/h, CL = 8 L/h, V1 = 25 L, Q = 12 L/h, V2 = 60 L
params = TwoCompOralParams(1.2, 8.0, 25.0, 12.0, 60.0)

# Single 400 mg oral dose
doses = [DoseEvent(0.0, 400.0)]

# Create specification
spec = ModelSpec(
    TwoCompOral(),
    "twocomp_oral_example",
    params,
    doses
)

# Time grid
grid = SimGrid(0.0, 72.0, collect(0.0:0.25:72.0))

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

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

# Find Cmax and Tmax
conc = result.observations[:conc]
t = result.t
cmax, idx = findmax(conc)
tmax = t[idx]

println("Cmax: ", round(cmax, digits=2), " mg/L")
println("Tmax: ", round(tmax, digits=2), " h")

Multiple Dosing to Steady State¶

# 400 mg every 12 hours
doses = DoseEvent[]
for i in 0:13  # 7 days
    push!(doses, DoseEvent(i * 12.0, 400.0))
end

spec = ModelSpec(TwoCompOral(), "bid_7days", params, doses)
grid = SimGrid(0.0, 168.0, collect(0.0:0.5:168.0))

result = simulate(spec, grid, solver)

# Steady-state metrics (last dosing interval)
conc = result.observations[:conc]
t = result.t

# Find Cmax,ss and Cmin,ss in last interval (156-168 h)
ss_start = findfirst(x -> x >= 156.0, t)
ss_end = findfirst(x -> x >= 168.0, t)

cmax_ss = maximum(conc[ss_start:ss_end])
cmin_ss = minimum(conc[ss_start:ss_end])

println("Steady-state Cmax: ", round(cmax_ss, digits=2), " mg/L")
println("Steady-state Cmin: ", round(cmin_ss, digits=2), " mg/L")

Absorption Rate Effects¶

# Compare different absorption rates
ka_values = [0.5, 1.0, 2.0, 4.0]

for ka in ka_values
    params = TwoCompOralParams(ka, 8.0, 25.0, 12.0, 60.0)
    doses = [DoseEvent(0.0, 400.0)]
    spec = ModelSpec(TwoCompOral(), "ka_$ka", params, doses)
    grid = SimGrid(0.0, 24.0, collect(0.0:0.1:24.0))

    result = simulate(spec, grid, solver)
    conc = result.observations[:conc]
    cmax, idx = findmax(conc)
    tmax = result.t[idx]

    println("Ka = $ka: Cmax = $(round(cmax, digits=2)), Tmax = $(round(tmax, digits=2))")
end

Expected Results: - Higher Ka → Earlier Tmax, Higher Cmax - Lower Ka → Later Tmax, Lower Cmax - AUC remains constant (depends on CL, not Ka)

Population Simulation¶

# Typical parameters
typical_params = TwoCompOralParams(1.2, 8.0, 25.0, 12.0, 60.0)

# IIV: High variability on Ka (50% CV), moderate on others
omega = OmegaMatrix([
    0.25 0.0  0.0  0.0  0.0;   # ω²_Ka (50% CV)
    0.0  0.09 0.0  0.0  0.0;   # ω²_CL (30% CV)
    0.0  0.0  0.04 0.0  0.0;   # ω²_V1 (20% CV)
    0.0  0.0  0.0  0.09 0.0;   # ω²_Q  (30% CV)
    0.0  0.0  0.0  0.0  0.04   # ω²_V2 (20% CV)
])

doses = [DoseEvent(0.0, 400.0)]
base_spec = ModelSpec(TwoCompOral(), "pop", typical_params, doses)

pop_spec = PopulationSpec(base_spec, 100, omega, 12345)

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

# Summarize Cmax and Tmax variability
cmax_values = Float64[]
tmax_values = Float64[]

for ind in result.individuals
    conc = ind.observations[:conc]
    t = ind.t
    cmax, idx = findmax(conc)
    push!(cmax_values, cmax)
    push!(tmax_values, t[idx])
end

using Statistics
println("Cmax: ", round(mean(cmax_values), digits=2),
        " (CV: ", round(100*std(cmax_values)/mean(cmax_values), digits=1), "%)")
println("Tmax: ", round(median(tmax_values), digits=2),
        " h (range: ", round(minimum(tmax_values), digits=1), "-",
        round(maximum(tmax_values), digits=1), ")")

Food Effect Study Design¶

# Fasted state: Fast absorption
params_fasted = TwoCompOralParams(2.0, 8.0, 25.0, 12.0, 60.0)

# Fed state: Slower absorption, possibly enhanced bioavailability
# If F increases by 25%, apparent CL decreases
params_fed = TwoCompOralParams(0.8, 6.4, 25.0, 12.0, 60.0)

doses = [DoseEvent(0.0, 400.0)]
grid = SimGrid(0.0, 48.0, collect(0.0:0.25:48.0))

spec_fasted = ModelSpec(TwoCompOral(), "fasted", params_fasted, doses)
spec_fed = ModelSpec(TwoCompOral(), "fed", params_fed, doses)

result_fasted = simulate(spec_fasted, grid, solver)
result_fed = simulate(spec_fed, grid, solver)

# Compare exposure
auc_fasted = sum(result_fasted.observations[:conc]) * 0.25  # Approximate AUC
auc_fed = sum(result_fed.observations[:conc]) * 0.25

cmax_fasted = maximum(result_fasted.observations[:conc])
cmax_fed = maximum(result_fed.observations[:conc])

println("Fed/Fasted AUC ratio: ", round(auc_fed/auc_fasted, digits=2))
println("Fed/Fasted Cmax ratio: ", round(cmax_fed/cmax_fasted, digits=2))

Comparison with One-Compartment Oral¶

# Same total apparent volume and clearance
# Two-comp oral
params_2comp = TwoCompOralParams(1.2, 8.0, 25.0, 12.0, 60.0)

# One-comp oral (same AUC)
params_1comp = OneCompOralFirstOrderParams(1.2, 8.0, 85.0)  # V = V1 + V2

doses = [DoseEvent(0.0, 400.0)]
grid = SimGrid(0.0, 48.0, collect(0.0:0.1:48.0))

spec_2comp = ModelSpec(TwoCompOral(), "2comp", params_2comp, doses)
spec_1comp = ModelSpec(OneCompOralFirstOrder(), "1comp", params_1comp, doses)

result_2comp = simulate(spec_2comp, grid, solver)
result_1comp = simulate(spec_1comp, grid, solver)

# Key differences:
# - 2-comp has higher initial peak (smaller V1)
# - 2-comp has secondary distribution phase
# - Both have same terminal AUC

Equations Summary¶

Quantity	Formula
dA_gut/dt	\(-K_a \cdot A_{gut}\)
dA1/dt	\(K_a A_{gut} - (k_{10} + k_{12})A_1 + k_{21}A_2\)
dA2/dt	\(k_{12}A_1 - k_{21}A_2\)
C(t)	\(A_1(t)/V_1\)
AUC	\(D/CL\)
Absorption t½	\(0.693/K_a\)
Terminal t½	\(0.693/\beta\)