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Review Article

Theoretical Basis, Experimental Design, and Computerized Simulation of Synergism and Antagonism in Drug Combination Studies

Preclinical Pharmacology Core Laboratory, Molecular Pharmacology and Chemistry Program, Memorial Sloan-Kettering Cancer Center, New York, New York

Abstract

I. Introduction

A. Why Drug Combination?

B. Pitfalls in Drug Combination Studies

1. Synergism versus Enhancement or Potentiation.

2. The Most Common Errors.

C. Truth or Fallacy and Its Consequences

D. An Approach for Extinguishing Controversies

II. Theoretical Basis for Dose-Effect Analysis

A. An Approach of Merging the Mass-Action Law with Mathematical Induction and Deduction

1. The Power of Mathematical Induction and Deduction.

2. Nature's Law.

3. Dealing with Diversified Biological and Pharmacological Systems.

B. The Derivation of Equations and Theorems Based on the Mass-Action Law

1. The Median-Effect Equation.

2. The Unified Theory.

C. Extension of Mass-Action Law to Multiple Drug-Effect Systems

1. The Multiple Drug-Effect Equation.

2. The Combination Index Theorem and Plot.

3. The General Equation for Combination of n Drugs.

4. Algorithms for Determining Synergism and Antagonism.

5. Main Features of the General Equation.

6. The Fa-Combination Index Plot and Isobologram Are Two Sides of the Same Coin.

7. How Much Synergism Is Synergy?

D. The Dose-Reduction Index Equation and Plot

E. The Polygonogram

III. Experimental Design for Drug Combinations

A. The Prerequisite and Theoretical Minimum Requirements for Drug Combination Studies

B. Constant Ratio Drug Combinations, Dose Range, Dose Density, and Experimental Scheme

C. The Nonconstant Ratios of Drug Combinations

D. The Optimal Combination Ratio for Maximal Synergy

E. Combination Designs for Three or More Drugs

F. Drug Combination in Vitro, in Vivo, and in Clinics

G. Schedule Dependence

H. Condition-Dependent Synergism or Antagonism and Combination of Drugs with Different Modalities, Different Units, and Mechanisms

IV. Computerized Automation, Graphic Simulation, and Informatics

A. Computer Software

B. The Median-Effect Plot and the Simulation of Dose-Effect Curve

C. Simulation of the Fa-Combination Index Plot

D. Construction of the Classic and Normalized Isobologram

E. Simulation of the Fa-Drug-Reduction Index Plot

F. Step-by-Step Use of CompuSyn Software for Single Drug and for Drug Combination Studies

G. Statistical Considerations

V. Selected Examples of Cited Applications

A. Cited Methods and Evaluation of Single Drug and Drug Discovery

1. Exploration of Potency, Toxicity, Parameters, and Structure-Activity Relations for New Compounds.

2. Low-Dose Risk Assessment for Carcinogens and Radiation:

3. Calculation of Ki from the IC50 Value:

4. Exclusive and Nonexclusive Inhibitors and Topology of Binding Sites:

5. Drug Resistance Evaluation and Other Applications:

6. Cellular Pharmacological Studies:

7. Tissue Pharmacological Studies:

8. Cardiovascular Pharmacological Studies:

9. Pharmacological Studies on Animals:

10. Behavioral Studies:

11. Cancer Prevention Agents:

B. Examples of Cited Applications in Drug Combinations

1. Anticancer Agent Combinations.

2. Antiviral Agent Combinations.

3. Immunosuppressant Combinations for Organ Transplantations.

4. Schedule Dependence of Combinations.

5. Drug Combinations That Highlight Antagonism.

6. Topological Analysis of Multiligand Bindings.

7. Selectivity of Synergism.

8. Gene Therapy or Molecular Biology by Combinations.

9. Combinations of Other Anti-Infectious Disease Agents.

10. Cardiovascular Drug Combinations:

11. Combination for Animal Growth:

12. Anesthetic Combinations:

13. Radiation and Drug Combinations:

14. Antiparasitic Combination:

15. Segmental Reviews for Median-Effect Principle and Combination Index Methods.

VI. Illustrations of Real Data Analysis with Mass-Action Law-Based Computer Software

A. Single-Drug, Two-Drug, and Three-Drug Combination Analysis with Computer Software

1. Single-Drug Analysis and Two-Drug Combinations.

2. Topological Analysis for the Multiple Ligand Sites in the Steady-State System.

3. Two- and Three-Drug Combinations against Cancer Cell Growth and the Construction of Polygonograms.

B. Other Applications of the Median-Effect Principle of the Mass-Action Law

1. Estimating Low-Dose Risk of Carcinogens.

2. Risk Assessment for Radiation.

3. Therapeutic Index and Safety Margin

4. Age-Specific Cancer Incidence Rate Analysis.

5. Epidemiological Applications (Chou, 1978; Chou and Miller, 1980):

6. Calculation of Ki from IC50.

C. Sample Analysis of Drug Combination Data with Computerized Summaries

1. Synergism of Two Insecticides on Houseflies.

2. Antagonism between Methotrexate and Arabinosylcytosine.

3. Seven-Drug Combination against Human Immunodeficiency Virus and Their Polygonograms.

a. Introduction.

b. Summaries of results.

c. Conclusions.

D. Approaches for the Conservation of Laboratory Animals

1. The Median-Effect Principle.

2. Experimental Design.

3. Serial Deletion Analysis.

4. Polygonogram.

Appendix I: Derivation of the Multiple Drug-Effect Equation

A. Summation of the Effects

B. Alternative Equations for Multiple Inhibitors in First-Order Systems

C. Inhibition of Higher-Order Kinetic Systems by a Single Inhibitor

D. Inhibition of the Higher-Order Kinetic Systems by Mutually Exclusive Inhibitors

E. Multiple Inhibitions by Mutually Nonexclusive Inhibitors

1. First Order.

a. Case 1.

b. Case 2.

2. Multiple Inhibitions by Inhibitors with Different Kinetic Orders.

3. Higher-Order Multiple Mutually Nonexclusive Inhibitors.

Glossary

The median-effect equation derived from the mass-action law principle at equilibrium-steady state via mathematical induction and deduction for different reaction sequences and mechanisms and different types of inhibition has been shown to be the unified theory for the Michaelis-Menten equation, Hill equation, Henderson-Hasselbalch equation, and Scatchard equation. It is shown that dose and effect are interchangeable via defined parameters. This general equation for the single drug effect has been extended to the multiple drug effect equation for n drugs. These equations provide the theoretical basis for the combination index (CI)-isobologram equation that allows quantitative determination of drug interactions, where CI < 1, = 1, and > 1 indicate synergism, additive effect, and antagonism, respectively. Based on these algorithms, computer software has been developed to allow automated simulation of synergism and antagonism at all dose or effect levels. It displays the dose-effect curve, median-effect plot, combination index plot, isobologram, dose-reduction index plot, and polygonogram for in vitro or in vivo studies. This theoretical development, experimental design, and computerized data analysis have facilitated dose-effect analysis for single drug evaluation or carcinogen and radiation risk assessment, as well as for drug or other entity combinations in a vast field of disciplines of biomedical sciences. In this review, selected examples of applications are given, and step-by-step examples of experimental designs and real data analysis are also illustrated. The merging of the mass-action law principle with mathematical induction-deduction has been proven to be a unique and effective scientific method for general theory development. The median-effect principle and its mass-action law based computer software are gaining increased applications in biomedical sciences, from how to effectively evaluate a single compound or entity to how to beneficially use multiple drugs or modalities in combination therapies.

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