One-Compartment IV Bolus¶

The simplest pharmacokinetic model representing instantaneous drug administration into a single well-mixed compartment with first-order elimination.

Model Overview¶

graph LR
    D((Dose)) -->|Bolus| C[Central<br/>Compartment<br/>V]
    C -->|CL| E((Elimination))

Clinical Applications¶

Simple IV drugs with rapid distribution
Initial PK characterization studies
Drugs without significant tissue distribution
Loading dose calculations
TDM (Therapeutic Drug Monitoring)

When to Use¶

Use When	Don't Use When
Mono-exponential decline	Bi/tri-exponential decline
Rapid equilibration	Significant distribution phase
Small molecules	Large molecules with tissue binding
Initial screening	Final model development

Mathematical Formulation¶

Parameters¶

Parameter	Symbol	Units	Description	Constraints
Clearance	CL	L/h	Volume of plasma cleared per time	CL > 0
Volume	V	L	Apparent volume of distribution	V > 0

Micro-Rate Constant¶

\[k_{el} = \frac{CL}{V}\]

State Variable¶

State	Symbol	Units	Description
Amount in central	A	mg	Drug amount in compartment

Differential Equation¶

\[\frac{dA}{dt} = -k_{el} \cdot A = -\frac{CL}{V} \cdot A\]

Analytical Solution¶

For a single bolus dose \(D\) at \(t = 0\):

\[A(t) = D \cdot e^{-k_{el} \cdot t}\]

\[C(t) = \frac{D}{V} \cdot e^{-k_{el} \cdot t} = C_0 \cdot e^{-k_{el} \cdot t}\]

Where \(C_0 = D/V\) is the initial concentration.

Observation¶

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

Derived Parameters¶

Half-Life¶

\[t_{1/2} = \frac{\ln(2)}{k_{el}} = \frac{0.693 \cdot V}{CL}\]

AUC (Single Dose)¶

\[AUC_{0-\infty} = \frac{D}{CL}\]

Steady-State (Multiple Dosing)¶

For dose \(D\) given every \(\tau\) hours:

\[C_{ss,max} = \frac{D/V}{1 - e^{-k_{el} \cdot \tau}}\]

\[C_{ss,min} = C_{ss,max} \cdot e^{-k_{el} \cdot \tau}\]

\[C_{ss,avg} = \frac{D}{CL \cdot \tau}\]

Accumulation Factor¶

\[R = \frac{1}{1 - e^{-k_{el} \cdot \tau}}\]

Julia API¶

Type Definitions¶

# Model kind
struct OneCompIVBolus <: ModelKind end

# Parameters
struct OneCompIVBolusParams <: AbstractParams
    CL::Float64    # Clearance (L/h)
    V::Float64     # Volume (L)
end

Basic Simulation¶

using NeoPKPD

# Define parameters
params = OneCompIVBolusParams(5.0, 50.0)  # CL=5 L/h, V=50 L

# Single 100 mg dose at t=0
doses = [DoseEvent(0.0, 100.0)]

# Create model specification
spec = ModelSpec(
    OneCompIVBolus(),
    "onecomp_iv_example",
    params,
    doses
)

# Define time grid (0 to 24 hours, hourly)
grid = SimGrid(0.0, 24.0, collect(0.0:1.0:24.0))

# Configure solver (high precision for validation)
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

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

# Access results
println("Time: ", result.t)
println("Concentration: ", result.observations[:conc])
println("Amount: ", result.states[:A_central])

Expected Output¶

Time: [0.0, 1.0, 2.0, ..., 24.0]
Concentration: [2.0, 1.8097, 1.6375, 1.4816, 1.3406, ...]
Amount: [100.0, 90.48, 81.87, 74.08, 67.03, ...]

Multiple Dosing¶

# 100 mg every 12 hours for 3 days
doses = [
    DoseEvent(0.0, 100.0),
    DoseEvent(12.0, 100.0),
    DoseEvent(24.0, 100.0),
    DoseEvent(36.0, 100.0),
    DoseEvent(48.0, 100.0),
    DoseEvent(60.0, 100.0),
]

spec = ModelSpec(OneCompIVBolus(), "multiple_dose", params, doses)
grid = SimGrid(0.0, 72.0, collect(0.0:0.5:72.0))

result = simulate(spec, grid, solver)

# Find steady-state trough (just before 6th dose)
trough_idx = findfirst(x -> x ≈ 59.5, result.t)
println("Steady-state trough: ", result.observations[:conc][trough_idx])

IV Infusion¶

# 100 mg infused over 1 hour (duration parameter)
doses = [DoseEvent(0.0, 100.0, 1.0)]

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

result = simulate(spec, grid, solver)

# Cmax at end of infusion
idx_1h = findfirst(x -> x ≈ 1.0, result.t)
println("Cmax at end of infusion: ", result.observations[:conc][idx_1h])

Parameter Estimation¶

Fitting from Data¶

# Observed data
data = EstimationData(
    ids = [1, 1, 1, 1, 1],
    times = [0.5, 1.0, 2.0, 4.0, 8.0],
    dv = [1.8, 1.6, 1.3, 0.9, 0.4],
    doses = [DoseEvent(0.0, 100.0)],
    dose_ids = [1, 1, 1, 1, 1]
)

# Initial estimates
init = InitialEstimates(
    theta = [5.0, 50.0],     # CL, V
    omega = [0.09, 0.04],    # IIV on CL, V
    sigma = [0.01]           # Proportional error
)

# Fit with FOCE
result = estimate(data, OneCompIVBolus(), init, FOCEConfig())

println("Estimated CL: ", result.theta[1])
println("Estimated V: ", result.theta[2])

Population Simulation¶

# Typical parameters
typical_params = OneCompIVBolusParams(5.0, 50.0)

# IIV: 30% CV on CL, 20% CV on V
omega = OmegaMatrix([
    0.09 0.0;
    0.0  0.04
])

doses = [DoseEvent(0.0, 100.0)]
base_spec = ModelSpec(OneCompIVBolus(), "pop", typical_params, doses)

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

grid = SimGrid(0.0, 24.0, collect(0.0:1.0:24.0))
result = simulate_population(pop_spec, grid, solver)

# Population summary
summary = result.summaries[:conc]
println("Median Cmax: ", maximum(summary.median))
println("90% PI: ", maximum(summary.quantiles[0.05]), " - ",
        maximum(summary.quantiles[0.95]))

Validation¶

Analytical Verification¶

# Compare simulation to analytical solution
D = 100.0   # Dose
CL = 5.0    # Clearance
V = 50.0    # Volume
k = CL / V  # Elimination rate

t = collect(0.0:0.1:24.0)

# Analytical solution
C_analytical = (D / V) .* exp.(-k .* t)

# Simulation
params = OneCompIVBolusParams(CL, V)
doses = [DoseEvent(0.0, D)]
spec = ModelSpec(OneCompIVBolus(), "validation", params, doses)
grid = SimGrid(0.0, 24.0, t)
result = simulate(spec, grid, solver)

C_simulated = result.observations[:conc]

# Check agreement
max_error = maximum(abs.(C_analytical .- C_simulated))
println("Maximum error: ", max_error)  # Should be < 1e-10

Clinical Examples¶

Example 1: Aminoglycoside Dosing¶

# Gentamicin: CL ≈ 5 L/h, V ≈ 15 L
params = OneCompIVBolusParams(5.0, 15.0)

# Calculate dosing for target Cmax of 8 mg/L
target_cmax = 8.0
dose = target_cmax * params.V  # 120 mg

doses = [DoseEvent(0.0, dose)]
spec = ModelSpec(OneCompIVBolus(), "gentamicin", params, doses)
grid = SimGrid(0.0, 24.0, collect(0.0:0.5:24.0))

result = simulate(spec, grid, solver)
println("Achieved Cmax: ", maximum(result.observations[:conc]))

Example 2: Loading Dose Calculation¶

# Target steady-state average: 10 mg/L
# CL = 5 L/h, dosing interval τ = 8 h
target_css_avg = 10.0
CL = 5.0
tau = 8.0

maintenance_dose = target_css_avg * CL * tau  # 400 mg

# Loading dose for immediate effect
V = 50.0
loading_dose = target_css_avg * V  # 500 mg

println("Loading dose: ", loading_dose, " mg")
println("Maintenance dose: ", maintenance_dose, " mg q", tau, "h")

Equations Summary¶

Quantity	Formula
Rate constant	\(k_{el} = CL/V\)
Concentration	\(C(t) = (D/V) \cdot e^{-k_{el} \cdot t}\)
Half-life	\(t_{1/2} = 0.693/k_{el}\)
AUC	\(AUC = D/CL\)
Steady-state average	\(C_{ss,avg} = D/(CL \cdot \tau)\)
Accumulation	\(R = 1/(1 - e^{-k_{el} \cdot \tau})\)