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Population Modeling

Population pharmacokinetic/pharmacodynamic (PopPK/PD) modeling accounts for variability between and within individuals, enabling more accurate predictions and personalized dosing.

Overview

NeoPKPD supports comprehensive population modeling including:

  • Inter-Individual Variability (IIV) - Parameter differences between subjects
  • Inter-Occasion Variability (IOV) - Parameter changes within a subject over time
  • Covariate Effects - Patient characteristics affecting parameters
  • Residual Error Models - Unexplained variability in observations
graph LR
    A[Typical Parameters] --> B[IIV]
    B --> C[Individual Parameters]
    D[Covariates] --> C
    E[IOV] --> C
    C --> F[Simulation]
    F --> G[Predictions]
    H[Residual Error] --> G

Documentation

  • Inter-Individual Variability

    Log-normal distribution of parameters across subjects

    IIV Reference

  • Inter-Occasion Variability

    Parameter variation between dosing occasions

    IOV Reference

  • Covariate Models

    Weight, age, renal function effects

    Covariates

  • Residual Error

    Additive, proportional, combined error models

    Residual Error

Quick Start

Basic Population Simulation

using NeoPKPD

# Typical parameters
typical_params = OneCompIVBolusParams(5.0, 50.0)  # CL=5, V=50

# Omega matrix (variance of random effects)
# Diagonal: 30% CV on CL, 20% CV on V
omega = OmegaMatrix([
    0.09 0.0;   # ω²_CL
    0.0  0.04   # ω²_V
])

# Base specification
doses = [DoseEvent(0.0, 100.0)]
base_spec = ModelSpec(OneCompIVBolus(), "pop_sim", typical_params, doses)

# Population specification
pop_spec = PopulationSpec(
    base_spec,
    n = 100,           # Number of subjects
    omega = omega,
    seed = 12345       # For reproducibility
)

# Simulate
grid = SimGrid(0.0, 24.0, collect(0.0:1.0:24.0))
solver = SolverSpec(:Tsit5, 1e-10, 1e-12, 10_000_000)

result = simulate_population(pop_spec, grid, solver)

# Access results
println("Number of individuals: ", length(result.individuals))
println("Median Cmax: ", maximum(result.summaries[:conc].median))

Adding Covariates

# Weight-based allometric scaling
covariate_model = CovariateModel([
    CovariateEffect(:CL, :WT, 70.0, :power, 0.75),  # CL ~ WT^0.75
    CovariateEffect(:V, :WT, 70.0, :power, 1.0)     # V ~ WT^1.0
])

# Generate subject weights
using Random
Random.seed!(42)
weights = 50.0 .+ 40.0 .* rand(100)  # 50-90 kg
covariates = [Dict(:WT => w) for w in weights]

# Population with covariates
pop_spec = PopulationSpec(
    base_spec,
    n = 100,
    omega = omega,
    seed = 12345,
    covariate_model = covariate_model,
    covariates = covariates
)

result = simulate_population(pop_spec, grid, solver)

Key Concepts

Random Effects

Individual parameters are derived from typical values and random effects:

\[P_i = \theta \cdot e^{\eta_i}\]

Where: - \(P_i\) = Individual parameter - \(\theta\) = Typical (population) value - \(\eta_i\) = Random effect, \(\eta_i \sim N(0, \omega^2)\)

Omega Matrix

The omega matrix defines the variance-covariance of random effects:

# Diagonal (independent parameters)
omega = OmegaMatrix([
    0.09 0.0;
    0.0  0.04
])

# Full (correlated parameters)
omega = OmegaMatrix([
    0.09 0.03;   # Correlation between CL and V
    0.03 0.04
])

Coefficient of Variation

For log-normal distribution, the relationship between omega and CV is:

\[CV = \sqrt{e^{\omega^2} - 1} \approx \omega \text{ (for small ω)}\]
ω² ω Approximate CV
0.04 0.20 20%
0.09 0.30 30%
0.16 0.40 42%
0.25 0.50 53%

Population Result Structure

struct PopulationResult
    individuals::Vector{SimResult}        # Individual simulation results
    realized_params::Vector{Dict}         # Realized parameter values
    etas::Matrix{Float64}                 # Random effects (n × p)
    summaries::Dict{Symbol, Summary}      # Population summary statistics
    metadata::Dict                        # Additional information
end

struct Summary
    mean::Vector{Float64}
    median::Vector{Float64}
    sd::Vector{Float64}
    quantiles::Dict{Float64, Vector{Float64}}
end

Summary Statistics

# Access population summaries
summary = result.summaries[:conc]

# Central tendency
summary.mean      # Mean concentration at each time
summary.median    # Median concentration

# Variability
summary.sd        # Standard deviation

# Percentiles
summary.quantiles[0.05]   # 5th percentile
summary.quantiles[0.25]   # 25th percentile
summary.quantiles[0.50]   # Median
summary.quantiles[0.75]   # 75th percentile
summary.quantiles[0.95]   # 95th percentile

Next Steps