Two-Compartment IV Bolus¶

Two-compartment mammillary model with central and peripheral compartments, exhibiting bi-exponential concentration decline after IV bolus administration.

Model Overview¶

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

Clinical Applications¶

Drugs with distribution phase
Lipophilic compounds
Antibodies and biologics
Anesthetics
Drugs with tissue binding

When to Use¶

Use When	Don't Use When
Bi-exponential decline	Mono-exponential decline
Clear distribution phase	Rapid equilibration
Multi-tissue distribution	Single compartment kinetics
α and β phases visible	Tri-exponential needed

Mathematical Formulation¶

Parameters¶

Parameter	Symbol	Units	Description	Constraints
Clearance	CL	L/h	Systemic clearance	CL > 0
Central volume	V1	L	Volume of central compartment	V1 > 0
Inter-compartmental clearance	Q	L/h	Distribution clearance	Q > 0
Peripheral volume	V2	L	Volume of peripheral compartment	V2 > 0

Micro-Rate Constants¶

$$k_{10} = \frac{CL}{V_1}$$ (elimination)

$$k_{12} = \frac{Q}{V_1}$$ (central → peripheral)

$$k_{21} = \frac{Q}{V_2}$$ (peripheral → central)

State Variables¶

State	Symbol	Units	Description
Central amount	A1	mg	Drug in central compartment
Peripheral amount	A2	mg	Drug in peripheral compartment

Differential Equations¶

\[\frac{dA_1}{dt} = -k_{10} \cdot A_1 - k_{12} \cdot A_1 + k_{21} \cdot A_2\]

\[\frac{dA_2}{dt} = k_{12} \cdot A_1 - k_{21} \cdot A_2\]

Bi-Exponential Solution¶

\[C(t) = A \cdot e^{-\alpha t} + B \cdot e^{-\beta t}\]

Where: - α = fast (distribution) rate constant - β = slow (elimination) rate constant - A, B = coefficients (macro-constants)

Hybrid Constants¶

\[\alpha + \beta = k_{10} + k_{12} + k_{21}\]

\[\alpha \cdot \beta = k_{10} \cdot k_{21}\]

Observation¶

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

Derived Parameters¶

Distribution and Elimination Rates¶

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

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

Half-Lives¶

$$t_{1/2,\alpha} = \frac{0.693}{\alpha}$$ (distribution half-life)

$$t_{1/2,\beta} = \frac{0.693}{\beta}$$ (terminal half-life)

Coefficients¶

\[A = \frac{D}{V_1} \cdot \frac{\alpha - k_{21}}{\alpha - \beta}\]

\[B = \frac{D}{V_1} \cdot \frac{k_{21} - \beta}{\alpha - \beta}\]

Volume at Steady State¶

\[V_{ss} = V_1 + V_2 = V_1 \cdot \left(1 + \frac{k_{12}}{k_{21}}\right)\]

AUC¶

\[AUC = \frac{A}{\alpha} + \frac{B}{\beta} = \frac{D}{CL}\]

Julia API¶

Type Definitions¶

# Model kind
struct TwoCompIVBolus <: ModelKind end

# Parameters
struct TwoCompIVBolusParams <: AbstractParams
    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
# CL = 10 L/h, V1 = 20 L
# Q = 15 L/h, V2 = 50 L
params = TwoCompIVBolusParams(10.0, 20.0, 15.0, 50.0)

# Single 500 mg IV bolus
doses = [DoseEvent(0.0, 500.0)]

# Create specification
spec = ModelSpec(
    TwoCompIVBolus(),
    "twocomp_example",
    params,
    doses
)

# Fine time grid to capture distribution phase
grid = SimGrid(0.0, 48.0, collect(0.0:0.1:48.0))

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

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

# Observe bi-exponential decline
conc = result.observations[:conc]
println("C at t=0: ", round(conc[1], digits=2), " mg/L")
println("C at t=1h: ", round(conc[11], digits=2), " mg/L (distribution)")
println("C at t=24h: ", round(conc[241], digits=2), " mg/L (terminal)")

Calculate Derived Parameters¶

# Micro-rate constants
k10 = params.CL / params.V1
k12 = params.Q / params.V1
k21 = params.Q / params.V2

# Alpha and beta
sum_k = k10 + k12 + k21
prod_k = k10 * k21
discriminant = sqrt(sum_k^2 - 4*prod_k)

alpha = (sum_k + discriminant) / 2
beta = (sum_k - discriminant) / 2

# Half-lives
t_half_alpha = log(2) / alpha
t_half_beta = log(2) / beta

println("Distribution t½: ", round(t_half_alpha, digits=2), " h")
println("Terminal t½: ", round(t_half_beta, digits=2), " h")

# Vss
Vss = params.V1 + params.V2
println("Vss: ", Vss, " L")

Distribution Phase Analysis¶

# Identify distribution vs elimination phases
conc = result.observations[:conc]
t = result.t

# Log-transform for visual inspection
log_conc = log.(conc)

# Find inflection point (approximate end of distribution)
# Where second derivative changes sign
diff1 = diff(log_conc)
diff2 = diff(diff1)

# Transition occurs where distribution phase ends
# Typically when |dC/dt| stabilizes

Clinical Example: Vancomycin¶

# Vancomycin typical parameters
# CL ≈ 4.5 L/h, V1 ≈ 15 L, Q ≈ 4 L/h, V2 ≈ 40 L
params = TwoCompIVBolusParams(4.5, 15.0, 4.0, 40.0)

# 1g over 1 hour (infusion)
doses = [DoseEvent(0.0, 1000.0, 1.0)]

spec = ModelSpec(TwoCompIVBolus(), "vancomycin", params, doses)
grid = SimGrid(0.0, 24.0, collect(0.0:0.25:24.0))

result = simulate(spec, grid, solver)

# Therapeutic monitoring
# Target peak (2h post-start): 30-40 mg/L
# Target trough: 15-20 mg/L
idx_2h = findfirst(x -> x ≈ 2.0, result.t)
idx_24h = findfirst(x -> x ≈ 24.0, result.t)

println("Peak (2h): ", round(result.observations[:conc][idx_2h], digits=1), " mg/L")
println("Trough (24h): ", round(result.observations[:conc][idx_24h], digits=1), " mg/L")

Population Simulation¶

# Typical parameters
typical_params = TwoCompIVBolusParams(10.0, 20.0, 15.0, 50.0)

# IIV on all parameters
# 30% CV on CL, 25% CV on V1, 35% CV on Q, 30% CV on V2
omega = OmegaMatrix([
    0.09 0.0  0.0  0.0;    # ω²_CL
    0.0  0.0625 0.0 0.0;   # ω²_V1
    0.0  0.0  0.1225 0.0;  # ω²_Q
    0.0  0.0  0.0  0.09    # ω²_V2
])

doses = [DoseEvent(0.0, 500.0)]
base_spec = ModelSpec(TwoCompIVBolus(), "pop", typical_params, doses)

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

grid = SimGrid(0.0, 48.0, collect(0.0:1.0:48.0))
result = simulate_population(pop_spec, grid, solver)

# Summary statistics
summary = result.summaries[:conc]
println("Median at 24h: ", summary.median[25])

Comparison with One-Compartment¶

# Same total volume and clearance
CL = 10.0
Vtotal = 70.0

# One-compartment
params_1comp = OneCompIVBolusParams(CL, Vtotal)

# Two-compartment (same Vss)
params_2comp = TwoCompIVBolusParams(CL, 20.0, 15.0, 50.0)

doses = [DoseEvent(0.0, 500.0)]
grid = SimGrid(0.0, 48.0, collect(0.0:0.1:48.0))

# Simulate both
spec_1comp = ModelSpec(OneCompIVBolus(), "1comp", params_1comp, doses)
spec_2comp = ModelSpec(TwoCompIVBolus(), "2comp", params_2comp, doses)

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

# Two-comp has:
# - Higher initial concentration
# - Faster initial decline
# - Same terminal AUC

Equations Summary¶

Quantity	Formula
k10	$CL/V_1$
k12	$Q/V_1$
k21	$Q/V_2$
α	$\frac{1}{2}[(k_{10}+k_{12}+k_{21}) + \sqrt{\Delta}]$
β	$\frac{1}{2}[(k_{10}+k_{12}+k_{21}) - \sqrt{\Delta}]$
C(t)	$A \cdot e^{-\alpha t} + B \cdot e^{-\beta t}$
Vss	$V_1 + V_2$
Terminal t½	$0.693/\beta$

Quantity	Formula
k10	\(CL/V_1\)
k12	\(Q/V_1\)
k21	\(Q/V_2\)
α	\(\frac{1}{2}[(k_{10}+k_{12}+k_{21}) + \sqrt{\Delta}]\)
β	\(\frac{1}{2}[(k_{10}+k_{12}+k_{21}) - \sqrt{\Delta}]\)
C(t)	\(A \cdot e^{-\alpha t} + B \cdot e^{-\beta t}\)
Vss	\(V_1 + V_2\)
Terminal t½	\(0.693/\beta\)