Introduction to Pharmacodynamic Models

Author

Vijay Ivaturi

1 Learning Objectives

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

  • Understand the purpose and clinical relevance of pharmacodynamic (PD) modeling
  • Distinguish between pharmacokinetics (PK) and pharmacodynamics (PD)
  • Recognize the four major PD model paradigms
  • Select the appropriate model type for different drug-effect scenarios
  • Navigate the PD modeling tutorial series effectively
Recommended Learning Path

New to PD modeling? Start with this introduction, then proceed to Direct Response (01). Familiar with basics? Jump to the model type relevant to your application. Model selection challenges? Go directly to Tutorial 05.

2 What is Pharmacodynamic Modeling?

Pharmacodynamics (PD) is the study of what the drug does to the body, in contrast to pharmacokinetics (PK), which describes what the body does to the drug.

flowchart LR
    subgraph PK["Pharmacokinetics"]
        Dose --> ADME[Absorption<br/>Distribution<br/>Metabolism<br/>Excretion]
        ADME --> Cp[Concentration<br/>at Site of Action]
    end
    subgraph PD["Pharmacodynamics"]
        Cp --> Receptor[Drug-Receptor<br/>Interaction]
        Receptor --> Effect[Pharmacological<br/>Effect]
    end
Figure 1: The relationship between PK and PD in drug action

2.1 Why Model Pharmacodynamics?

PD modeling serves several critical purposes in drug development and clinical practice:

  1. Dose Optimization: Determine the dose that maximizes efficacy while minimizing toxicity
  2. Efficacy Prediction: Predict therapeutic effects for new dosing regimens
  3. Biomarker Development: Link drug exposure to measurable biomarkers
  4. Clinical Trial Design: Inform dose selection for Phase II/III trials
  5. Personalized Medicine: Account for patient-specific factors affecting drug response
The PK-PD Disconnect

A common misconception is that higher drug concentrations always lead to greater effects. In reality, the concentration-effect relationship can be complex, involving delays, saturation, tolerance, and feedback mechanisms. PD modeling helps characterize these relationships quantitatively.

3 The Concentration-Effect Relationship

At the heart of PD modeling is the relationship between drug concentration and its effect (Holford and Sheiner 1981). The most fundamental model is the Emax model:

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

Where:

Parameter Description
E Observed effect
E_0 Baseline effect (without drug)
E_{max} Maximum drug-induced effect
C Drug concentration
EC_{50} Concentration producing 50% of E_{max} (potency measure)
Figure 2: The Emax model showing the concentration-effect relationship

3.1 Key Concepts

Potency vs. Efficacy
  • Potency (EC_{50}): How much drug is needed to produce an effect. Lower EC_{50} = more potent.
  • Efficacy (E_{max}): The maximum effect the drug can produce. Higher E_{max} = more efficacious.

A drug can be highly potent but have low efficacy, or vice versa.

4 The Four Paradigms of PD Response

PD models can be broadly categorized into four paradigms based on the temporal relationship between drug concentration and effect:

flowchart TD
    Start[PD Data Available] --> Q0{Concentration<br/>data available?}
    Q0 -->|No| KPD[<b>K-PD Model</b><br/>Virtual compartment]
    Q0 -->|Yes| Q1{Immediate<br/>equilibrium<br/>between Cp and Effect?}
    Q1 -->|Yes| DR[<b>Direct Response</b><br/>Effect = f Cp]
    Q1 -->|No| Q2{Hysteresis<br/>observed in<br/>Effect vs Cp?}
    Q2 -->|Yes| EC[<b>Effect Compartment</b><br/>Effect = f Ce]
    Q2 -->|No| IDR[<b>Indirect Response</b><br/>dR/dt = kin - kout·R]

    IDR --> Q4{Drug effect<br/>direction?}
    Q4 -->|Inhibits production| IDR1[IDR Type I]
    Q4 -->|Inhibits removal| IDR2[IDR Type II]
    Q4 -->|Stimulates production| IDR3[IDR Type III]
    Q4 -->|Stimulates removal| IDR4[IDR Type IV]

    style DR fill:#e1f5fe
    style EC fill:#fff3e0
    style IDR fill:#e8f5e9
    style KPD fill:#fce4ec
Figure 3: Decision flowchart for selecting PD model type

4.1 Paradigm 1: Direct Response Models

When to use: The drug effect is in immediate equilibrium with the plasma concentration.

flowchart LR
    subgraph PK
        Dose --> Cp[Plasma<br/>Concentration]
    end
    subgraph PD
        Cp --> Effect[Effect<br/>E = f Cp]
    end
Figure 4: Direct response model structure

Characteristics:

  • Effect rises and falls in parallel with concentration
  • No time delay between Cp and effect
  • Maximum effect occurs at peak concentration

Clinical examples:

  • Anesthetics (rapid onset/offset)
  • Bronchodilators
  • Some antihypertensives

4.2 Paradigm 2: Effect Compartment Models

When to use: There is a temporal disconnect (hysteresis) between plasma concentration and effect (Sheiner et al. 1979).

flowchart LR
    subgraph PK
        Dose --> Cp[Plasma Conc<br/>Cp]
    end
    subgraph Biophase
        Cp -->|ke0| Ce[Effect Conc<br/>Ce]
    end
    subgraph PD
        Ce --> Effect[Effect<br/>E = f Ce]
    end
Figure 5: Effect compartment model structure

Characteristics:

  • Effect lags behind plasma concentration
  • Hysteresis loop observed when plotting Effect vs. Cp
  • The k_{e0} parameter governs the rate of equilibration

Clinical examples:

  • Digoxin (cardiac effects)
  • Warfarin (initial anticoagulant effect)
  • Morphine (CNS effects)

4.3 Paradigm 3: Indirect Response Models

When to use: Drug effects are mediated through modulation of physiological processes with inherent turnover (production and elimination) (Dayneka, Garg, and Jusko 1993).

flowchart TD
    subgraph Turnover
        kin[k_in<br/>Production] --> R[Response<br/>R]
        R --> kout[k_out<br/>Removal]
    end
    Drug -->|Inhibit or<br/>Stimulate| kin
    Drug -->|Inhibit or<br/>Stimulate| kout
Figure 6: Indirect response model structure

The Four IDR Types:

Type Drug Effect Response Direction Example
I Inhibits k_{in} Decreases Corticosteroid suppression of cortisol
II Inhibits k_{out} Increases Proton pump inhibitors
III Stimulates k_{in} Increases Erythropoietin stimulation
IV Stimulates k_{out} Decreases Enzyme induction

Characteristics:

  • Effect persists after drug concentration declines
  • Baseline is maintained by production/removal balance
  • Drug perturbs this equilibrium

4.4 Paradigm 4: K-PD (Kinetics without PK Data) Models

When to use: When pharmacodynamic data is available but plasma concentration measurements are not (Jacqmin et al. 2007).

flowchart LR
    subgraph Input
        Dose[Dose]
    end
    subgraph Virtual["Virtual Compartment"]
        A["Drug Amount A(t)"]
    end
    subgraph PD["PD Model"]
        Effect["Effect E(t)"]
    end
    Dose -->|Input| A
    A -->|KDE| Elimination
    A -->|EDK50| Effect
Figure 7: K-PD model structure with virtual compartment

Characteristics:

  • No plasma concentration data required
  • Uses virtual compartment driven by dose
  • Parameters: KDE (apparent elimination), EDK50 (apparent potency)
  • Cannot predict concentrations or exposure metrics

When applicable:

  • Phase I studies with limited sampling
  • Exploratory dose-finding studies
  • Retrospective analysis of response-only data

5 Comparison of PD Model Types

Feature Direct Response Effect Compartment Indirect Response K-PD
Equilibration Instantaneous Delayed Via turnover Via virtual compartment
Hysteresis None Yes Variable N/A (no Cp measured)
Effect duration Parallels Cp Lags Cp Outlasts Cp Predicted from dose only
Key parameter EC_{50} k_{e0} k_{in}, k_{out} KDE, EDK50
Data required Cp + Effect Cp + Effect Cp + Effect Dose + Effect only

6 Tutorial Series Overview

This PD modeling tutorial series is designed to build your skills progressively:

Tutorial Title Content
00 Introduction (this tutorial) Overview and model selection
01 Direct Response Models Emax, sigmoid Emax, full workflow
02 Effect Compartment Models Hysteresis, k_{e0}, biophase
03 Indirect Response Models All 4 IDR types, turnover concepts
04 K-PD Models Virtual compartment, modeling without PK
05 Model Selection Comparison, diagnostics, case studies

7 Prerequisites

Before diving into the PD tutorials, you should be comfortable with:

  • Basic Pumas model structure (@model, @param, @dynamics, @derived)
  • Fitting models using fit() and FOCE
  • Basic simulation with simobs()
  • Data manipulation with DataFrames
PK Foundation Required

All PD tutorials assume a linked PK model. If you need a refresher on PK modeling in Pumas, we recommend completing the Introduction to PK Modeling tutorials first.

8 Quick Reference: Mathematical Formulations

8.1 Emax Model (Direct Response)

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

8.2 Sigmoid Emax Model

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

Where \gamma is the Hill coefficient controlling steepness.

8.3 Effect Compartment

\frac{dC_e}{dt} = k_{e0} \cdot (C_p - C_e)

8.4 Indirect Response (General Form)

\frac{dR}{dt} = k_{in}(C) - k_{out}(C) \cdot R

At baseline steady-state: R_{ss} = \frac{k_{in}}{k_{out}}

8.5 K-PD Model (Virtual Compartment)

\frac{dA}{dt} = \text{Input}(t) - \text{KDE} \cdot A

Where effect is typically: E = E_0 + \frac{E_{max} \cdot A \cdot \text{KDE}}{\text{EDK50} + A \cdot \text{KDE}}

9 Summary

In this introduction, we covered:

  1. Pharmacodynamics studies what the drug does to the body

  2. The Emax model is the foundation of concentration-effect relationships

  3. Four paradigms exist for PD modeling:

    • Direct response (immediate equilibrium)
    • Effect compartment (delayed equilibrium)
    • Indirect response (turnover-mediated)
    • K-PD (virtual compartment when no PK data)

  4. Model selection depends on the temporal relationship between concentration and effect

  5. This tutorial series provides comprehensive coverage of all paradigms

In the next tutorial, we will explore Direct Response Models in detail, including implementation in Pumas, simulation, fitting, and diagnostics.

References

Dayneka, N. L., V. Garg, and W. J. Jusko. 1993. “Comparison of Four Basic Models of Indirect Pharmacodynamic Responses.” Journal of Pharmacokinetics and Biopharmaceutics 21 (4): 457–78.
Holford, N. H. G., and L. B. Sheiner. 1981. “Understanding the Dose-Effect Relationship: Clinical Application of Pharmacokinetic-Pharmacodynamic Models.” Clinical Pharmacokinetics 6 (6): 429–53.
Jacqmin, P., E. Snoeck, E. A. van Schaick, R. Bentzen, Q. Ooi, D. Faber, C. W. Alvey, and K. Jorga. 2007. “Modelling Response Time Profiles in the Absence of Drug Concentrations: Definition and Performance Evaluation of the K-PD Model.” Journal of Pharmacokinetics and Pharmacodynamics 34 (1): 57–85.
Sheiner, L. B., D. R. Stanski, S. Vozeh, R. D. Miller, and J. Ham. 1979. “Simultaneous Modeling of Pharmacokinetics and Pharmacodynamics: Application to d-Tubocurarine.” Clinical Pharmacology & Therapeutics 25 (3): 358–71.

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