Individual and Population Simulation

Simulate typical individuals and virtual populations to understand variability and prediction.
Tip

Big picture: A single simulation represents one expected profile. Population simulation shows how individuals may differ.

Learning Objectives

By the end of this lesson, you will be able to:

  • distinguish individual and population simulation
  • explain why variability matters
  • generate virtual populations
  • interpret simulated variability
  • compare typical and population predictions

Key Ideas

  • individual simulation represents one expected profile
  • population simulation introduces variability
  • populations produce distributions rather than single curves
  • uncertainty and variability are different concepts

Setup

library(tidyverse)
library(rxode2)

This lesson uses simulated population PK examples.

No estimation is performed.

Why Simulate Populations?

A single profile can be useful.

But real patients differ.

Questions become:

  • will everyone respond similarly?
  • how much variability should we expect?
  • are extreme responses possible?

Conceptually:

Typical Subject
↓
Many Virtual Subjects
↓
Distribution of Outcomes

Worked Example 1: Simulate a Typical Individual

Define a simple PK model.

pk_model <-
  rxode2({

    ka <- 1
    cl <- 3
    v <- 30

    d/dt(depot) = -ka * depot
    d/dt(center) = ka * depot - cl / v * center

    cp = center / v

  })

Create events.

ev <-
  et(amt = 100, cmt = "depot") %>%
  et(seq(0, 24, by = 0.25))

Simulate.

sim_typical <-
  rxSolve(pk_model, ev) %>%
  as_tibble()

Plot.

ggplot(sim_typical, aes(time, cp)) +
    geom_line() +
    labs(
        title = "Typical Individual",
        x = "Time",
        y = "Concentration"
        )

Interpretation:

This profile represents:

Expected Response
↓
No Between-Subject Variability

Worked Example 2: Add Population Variability

Now introduce variability.

pk_pop <-
  rxode2({

    ka <- exp(log(1) + eta.ka)
    cl <- exp(log(3) + eta.cl)
    v <- exp(log(30) + eta.v)

    d/dt(depot) = -ka * depot
    d/dt(center) = ka * depot - cl / v * center

    cp = center / v

  })

Specify variability.

omega <-
  lotri(
    eta.ka ~ 0.10,
    eta.cl ~ 0.20,
    eta.v ~ 0.10
  )

omega
       eta.ka eta.cl eta.v
eta.ka    0.1    0.0   0.0
eta.cl    0.0    0.2   0.0
eta.v     0.0    0.0   0.1

The object omega defines between-subject variability.

Each diagonal value represents the variance of a random effect.

Conceptually:

eta.ka → variability in absorption rate
eta.cl → variability in clearance
eta.v  → variability in volume

Larger values produce greater differences between virtual subjects.

In population PK models, this matrix is often called the Ω (omega) matrix.

Simulate multiple subjects.

sim_pop <-
  rxSolve(pk_pop, ev, nSub = 100, omega = omega) %>%
  as_tibble()

Unlike the previous lesson, the model is no longer generating a single concentration-time profile.

Instead, rxSolve() repeatedly samples random effects from the specified omega matrix and generates a profile for each virtual subject.

Inspect.

sim_pop %>%
  count(sim.id)
# A tibble: 100 × 2
   sim.id     n
    <int> <int>
 1      1    97
 2      2    97
 3      3    97
 4      4    97
 5      5    97
 6      6    97
 7      7    97
 8      8    97
 9      9    97
10     10    97
# ℹ 90 more rows

Interpretation:

Each sim.id represents one virtual subject.

sim_pop %>%
  distinct(sim.id) %>%
  nrow()
[1] 100

The simulation generated 100 virtual subjects.

Each subject has a unique set of PK parameters sampled from the specified variability distribution.

Worked Example 3: Visualize Population Profiles

Plot all individuals.

ggplot(sim_pop, aes(time, cp, group = sim.id)) +
    geom_line(alpha = 0.15) +
    labs(
        title = "Population Simulation",
        x = "Time",
        y = "Concentration"
        )

Interpretation:

Notice:

  • profiles are no longer identical
  • peaks differ
  • decline rates differ

Question:

Does one line represent truth?

No.

The collection represents possible outcomes.

Worked Example 4: Compare Individual and Population Simulation

Overlay typical and population summaries.

pop_summary <-
  sim_pop %>%
  group_by(time) %>%
  summarise(
    median = median(cp),
    low = quantile(cp, 0.05),
    high = quantile(cp, 0.95),
    .groups = "drop"
  )

ggplot() +
    geom_ribbon(data = pop_summary, aes(time, ymin = low, ymax = high), alpha = 0.2) +
    geom_line(data = pop_summary, aes(time, median)) +
    geom_line(data = sim_typical, aes(time, cp), linetype = 2) +
    labs(
        title = "Typical vs Population",
        x = "Time",
        y = "Concentration"
        )

Interpretation:

Compare:

Single Prediction
↓
Range of Predictions

The dashed line represents:

Typical Subject

The band represents:

Population Variability

Worked Example 5: Variability versus Uncertainty

Students often confuse:

Variability ≠ Uncertainty

Interpretation:

Concept Meaning
variability individuals differ
uncertainty estimates are imperfect

Population simulation primarily introduces:

Variability

not parameter uncertainty.

Connecting to Decisions

Population simulation helps answer:

  • what fraction exceeds exposure targets?
  • how variable are outcomes?
  • how sensitive are predictions?

Simulation supports decisions because real populations are heterogeneous.

Looking Ahead

So far we changed:

Who receives treatment

Next we change:

What treatment is given

We will explore:

  • dose changes
  • dosing schedules
  • exposure-response differences

Strategies

  • inspect distributions
  • compare individuals and populations
  • summarize variability

Common Mistakes

  • simulating too few subjects
  • overinterpreting extremes
  • assuming average equals everyone

Practice Problems

  1. Why simulate populations?

  2. What creates differences between subjects?

  3. What does the dashed line represent?

  4. Why simulate many subjects?

  5. What is variability?

Problem 1

Real patients differ.

Problem 2

Random effects.

Problem 3

Typical prediction.

Problem 4

To evaluate distributions.

Problem 5

Biological differences across individuals.

Summary

  • individual simulation produces one profile
  • population simulation produces many profiles
  • variability creates distributions
  • simulation supports decisions
  • Typical ≠ population
  • Variability ≠ uncertainty
  • Simulate before deciding