Direct Effect Models

Build the first PK/PD models by connecting concentration directly to response.
Tip

Big picture: Direct effect models assume response changes immediately as concentration changes.

Learning Objectives

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

  • explain direct response models
  • distinguish linear and Emax relationships
  • interpret Emax and EC50
  • simulate exposure–response relationships
  • recognize saturation

Key Ideas

  • concentration drives effect
  • response may increase nonlinearly
  • saturation is common
  • PD parameters have biological interpretation

Setup

library(tidyverse)

This lesson uses simulated examples.

What Is a Direct Effect Model?

Direct effect assumes:

Concentration → Response

Response changes immediately as concentration changes.

There is no delay.

Examples:

  • rapid receptor binding
  • immediate biomarker change

Worked Example 1: Linear Effect Model

A simple possibility is a linear relationship.

\[ E = E_0 + S C \]

Interpretation:

Parameter Meaning
\(E_0\) baseline effect
\(S\) sensitivity
\(C\) concentration

Question:

Does each increase in concentration produce the same effect increase?

Simulate.

conc <-
  tibble(
    C = seq(0, 100, by = 1)
  )

lin_effect <-
  conc %>%
  mutate(
    E = 20 + 0.8 * C
  )

ggplot(
  lin_effect,
  aes(C, E)
) +
geom_line() +
labs(
  title = "Linear Effect",
  x = "Concentration",
  y = "Effect"
)

Interpretation:

Effect increases proportionally.

Worked Example 2: Why Linear Models Fail

Suppose concentration continues increasing.

Question:

Should response increase forever?

Usually not.

Many systems saturate.

Conceptually:

More Exposure
↓
Smaller Additional Effect

This motivates Emax.

Worked Example 3: Emax Model

Introduce saturation.

\[ E = E_0 + \frac{E_{max} C}{EC_{50} + C} \]

Interpretation:

Parameter Meaning
\(E_0\) baseline
\(E_{max}\) maximum effect
\(EC_{50}\) concentration producing half-maximal effect

Question:

How much exposure is needed to generate effect?

Simulate.

emax_effect <-
  conc %>%
  mutate(
    E = 20 + (100 * C / (10 + C))
  )

ggplot(
  emax_effect,
  aes(C, E)
) +
geom_line() +
labs(
  title = "Emax Relationship",
  x = "Concentration",
  y = "Effect"
)

Interpretation:

Effect rises rapidly.

Then approaches a maximum.

Extension: Sigmoid Emax (Hill) Models

The Emax model assumes a smooth transition toward maximum effect.

Sometimes response changes more abruptly.

A common extension is the sigmoid Emax (Hill) model.

\[ E = E_0 + \frac{ E_{max} C^{\gamma} }{ EC_{50}^{\gamma} + C^{\gamma} } \]

Interpretation:

Parameter Meaning
\(E_0\) baseline
\(E_{max}\) maximum effect
\(EC_{50}\) half-maximal concentration
\(\gamma\) Hill coefficient

The Hill coefficient controls curve shape.

Higher γ → steeper transition

Lower γ → smoother transition

Question:

Does response switch gradually or sharply?

Examples:

  • receptor cooperativity
  • steep concentration–effect relationships

Simulate different Hill coefficients.

hill_tbl <-
  crossing(
    C = seq(0, 100, by = 0.1),
    gamma = c(0.5, 1, 5)
  ) %>%
  mutate(
    E =
      100 *
      C^gamma /
      (10^gamma + C^gamma)
  )

ggplot(
  hill_tbl,
  aes(
    C,
    E,
    linetype = factor(gamma)
  )
) +
  geom_line(linewidth = 1) +
  labs(
    title = "Effect of the Hill Coefficient",
    x = "Concentration",
    y = "Effect",
    linetype = expression(gamma)
  )

Interpretation:

  • \(\gamma = 1\) produces the standard Emax model
  • larger values of \(\gamma\) produce steeper, more switch-like behavior
  • smaller values of \(\gamma\) produce more gradual transitions

For simplicity, this course primarily uses the standard Emax model.

Worked Example 4: Understanding EC50

EC50 controls horizontal position.

Example:

Lower EC50 → less concentration needed

Higher EC50 → more concentration needed

EC50 is often interpreted as a measure of potency.

Conceptually:

Lower EC50 → higher potency

Higher EC50 → lower potency

A more potent drug achieves the same effect at a lower concentration.

Simulate.

ec_tbl <-
  crossing(
    C = seq(0, 100, by = 1),
    EC50 = c(5, 10, 20)
  ) %>%
  mutate(
    E = 100 * C / (EC50 + C)
  )

ggplot(
  ec_tbl,
  aes(C, E, group = EC50)
) +
geom_line(
  aes(
    linetype =
      factor(EC50)
  )
) +
labs(
  title = "Effect of EC50",
  x = "Concentration",
  y = "Effect",
  linetype = "EC50"
)

Interpretation:

Smaller EC50 shifts the concentration–effect relationship to the left.

At any given effect level, less concentration is needed.

This is often described as higher potency.

Worked Example 5: Direct Effect Thinking

Direct effect models answer:

Exposure

↓

Immediate Response

Questions become:

  • how large?
  • how sensitive?
  • when does saturation occur?

Direct models are useful.

But many responses are delayed.

That motivates the next lesson.

Strategies

  • visualize curves
  • interpret parameters biologically
  • compare shapes

Common Mistakes

  • assuming larger concentration always helps
  • treating EC50 as efficacy
  • ignoring limits

Practice Problems

  1. What assumption defines direct effect?

  2. What does Emax represent?

  3. What does EC50 represent?

  4. Why can linear models fail?

  5. What happens when EC50 decreases?

Problem 1

Concentration → Response

Problem 2

Maximum achievable effect.

Problem 3

Half-maximal concentration.

Problem 4

Response often saturates.

Problem 5

Response shifts left.

Summary

  • direct models connect exposure and response
  • Emax introduces saturation
  • EC50 controls sensitivity
  • parameters explain biology
  • Direct ≠ delayed
  • EC50 ≠ efficacy
  • Saturation matters