Current Methods for Quantifying Drug Synergism

Abstract

The effectiveness of drug combinations for treatment of a variety of complex diseases is well established. “Drug cocktail” treatments are often prescribed to improve the overall efficacy, decrease toxicity, alter pharmacodynamics, etc in an overall treatment strategy. Specifically, if when combined, drugs interact in some way that causes the total effect to be greater than that predicted by their individual potencies, then drugs are considered synergistic. While there are established ways to quantify the impact of drug combinations clinically, it is an open challenge to quantitatively summarize a synergistic interaction. In this paper, we discuss an overview of the current statistical and mathematical methods for the study of drug combination effects, especially drug synergy quantification (where the interaction effect is not just detected, but quantified according to its magnitude). We first introduce two popular reference models for testing to null hypothesis of non-interaction for a combination, including the Bliss independence model and the Loewe additivity model. Then we discuss several methods for quantifying drug synergism. The advantages and disadvantages with these methods are also provided, and finally, we discuss important next directions in this area.

INTRODUCTION

For a variety of complex diseases, it is an accepted paradigm that drugs are given in combination [14]. A drug interaction is a situation in which another drug affects the activity of a drug when both are administered together. This action can be synergistic (when the drug’s effect is increased) or antagonistic (when the drug’s effect is decreased) [12]. The evaluation of combination effects between biological or chemical agents plays a significant role in pharmacology and biomedicine. Combination therapies, often referred to as “cocktail” therapies have revolutionized patient outcomes in diseases such as HIV [1], asthma[21], breast cancer [6, 20], hypertension [13], and cancers such as melanoma [16]. The impact of chemical mixtures is also increasingly appreciated in the toxicology space as well, as people are not exposed to chemicals in isolation[7]. A recent review discusses the concept of synergy as used in a variety of fields[23].

The interaction of biologically or chemically active agents is often grouped into three categories: synergy, additivity (no interaction) and antagonism, based on the degree of departure of observed combination effects from the expected response without interaction [3, 12]. Specifically, if drugs when combined interact with each other and cause the total effect greater than that predicted by their individual potencies, then this is synergistic drug combination[8]. Such synergistic interactions can often reduce host toxicity and adverse side effects, given that doses of drug combinations are typically lower than that of single drugs to achieve desired efficacy. Additionally, it can also reduce the development of drug resistance and other complications [15, 26].

While the concept of synergy has been appreciated for a century, recent methods development and computational advances have allowed for new approaches for quantifying this phenomenon [10]. There are an emerging set of modeling approaches for quantifying synergism. These reference models have been developed based on distinctive biological and chemical assumptions. In addition, different methods are also developed to further detect and quantify synergistic effects specifically. It is typical in dose response assays to collect measures of drug response at multiple dose points. Such experiments can be expanded to assay both the individual drug responses from a pair or combination of drugs to include both the individual responses to the drug, and then to the combinations. Such an experimental design provides the baseline information about response from a single drug, to compare to the synergistic combinations.

In this review, we provide an overview of the current statistical and mathematical methods for the study of drug combination effects, especially drug synergy quantification. We first introduce two popular reference models for null hypothesis of non-interaction, which serve as the baseline to define synergy. Any deviation from the reference models will be regarded as synergy or antagonism. Subsequently, we discuss several statistical and mathematical approaches to quantify drug synergism. Finally, the common issues and opportunities with these methods are also provided. Although this paper mainly covers drug synergy, the concepts and methods mentioned in this review can be applied to other disciplines as well, such as toxicology and epidemiology.

TWO REFERENCE MODELS

To properly define synergy, it is of great importance to formulate a reference model for null hypothesis of non-interaction first, which suggests that the effects of drugs simply add up, not affecting each other (Additivity)[14]. Any deviation from the reference models will be regarded as synergy or antagonism, depending on the directions of departure. As shown in Figure 1, if the drug combination X and Y achieves the same response level with less dose than that of additive case (the reference model), the combination is said to by synergistic. Currently, there are two popular reference additivity models, Bliss independence model and the Loewe additivity model, which have different biological and chemical assumptions.

Figure 1.

Comparison of additive, synergy and antagonism at the same response level. If the drug combination X and Y achieves the same response level with less dose than that of additive case (the reference model), the combination is said to by synergistic.

Bliss Independence Model.

One of the oldest methods for quantifying synergy is the Bliss Independence model, dating back to the 1930s [4]. This model assumes that drugs do not interact with each other and elicit their responses independently [14]. According to independence probability theory, the expected response of drug combination can be written in terms of individual drug responses [4]:

$$R_c=R_1+\left(1-R_1\right)\times R_2=R_1+R_2-R_1\times R_2$$

Where drug 1 at dose y₁ produced a response R₁, drug 2 at dose y₂ produced a response R₂, and R_c is expected response of drug combination 1 and 2 at dose y₁ and y₂, respectively.

Loewe Additivity Model.

The more recently developed Loewe additivity model, on the other hand, assumes that drugs have similar modes of action on the same pathway [14]. To formulate a Loewe additivity model, the dose-response relationship of individual drugs needs to be modeled first. Let the dose of drug 1 = y₁ and the dose of drug 2 =y₂. Then the Loewe additivity model can be expressed as the following equation [14]:

$$\frac{y_1}{Y_1}+\frac{y_2}{Y_2}=1$$

Where Y₁ is the dose of drug 1 that achieve the same response level as the drug combination and Y₂ is the dose of drug 2 that achieve the same response level as the drug combination.

The major differences of the two reference models come from their underlying assumptions. The Bliss independence model assumes that drugs do not interact with each other and elicit their responses independently, whereas the Loewe additivity model assumes that drugs have similar modes of action on the same pathway. In fact, none of these models hold true for all cases of drug combinations. As a result, the model selection has become a personal preference[25].

THE METHODS FOR QUANTIFYING DRUG SYNERGISM

Next, we will discuss methods for actually directly quantifying synergy. The methods are briefly introduced here, with references provided for a more detailed description of each approach.

Response Surface

Response surface modeling is an approach to represent effects of drug combinations in three-dimensional plot where the doses of individual drugs are plotted as a horizontal x-y-plane, and the expected effect of drug combination is plotted on the z-axis, as shown in Figure 2 [14]. Both Bliss independence models and Loewe additivity model can be used to calculate the expected effect of drug combination [14]. The experimental effect of drug combination can then be plotted on this surface. Any departure from the 3D surface is classified into synergism or antagonism, depending on the measurement of the z-axis. [22].

Figure 2.

Example of response surface for two drugs, X and Y. The doses of individual drugs are plotted as a horizontal x-y-plane, and the expected effect (response) of drug combination is plotted on the z-axis.

Chou-Talalay method

The Chou-Talalay method is by far the most commonly used approach to quantify drug combination, especially synergistic interactions [10, 11]. This method adopts the median-effect equation, which is derived from the unified theory mass-action law principle[9]. The median-effect equation is written below [11]:

$$\frac{f_a}{f_u}={\left(\frac{D}{D_m}\right)}^m$$

Where f_a is the percent of cells killed, f_u is the percent of cells living, D is dose of drugs given, D_m is the median-effect dose, and m is a parameter in dose -response curve.

One disadvantage of the Chou-Talalay method is that raw data must be preprocessed, including scaling the data and taking the log of a function of the scaled data [10].

In addition to the median-effect equation, Chou-Talalay also developed the combination index (CI) theorem, where additive effect (CI = 1), synergism (CI < 1), and antagonism (CI > 1) in drug combinations [9, 10].

MixLow method

More recently, Boik, Newman, and Boik (2008) developed the Mixlow method as an alternative to the Chou-Talalay method. The term MixLow means Mixed-effects Loewe, which has three components: a nonlinear mixed-effects model, the Loewe index, and a method to calculate confidence intervals for the index. The MixLow method uses nonlinear mixed effects model for estimating sigmoidal curve parameters from concentration-response data, and associated confidence intervals [5]. Compared with the Chou-Talalay method, the MixLow method produces more precise parameter estimation, and has improved coverage of confidence intervals. In addition, the use of a nonlinear fixed-effects model in the MixLow method also eliminates the need for data preprocessing in the Chou-Talalay method [5].

Drug synergy quantification using a Bayesian approach

In 2010, Hennessey et al. proposed a Bayesian approach to dose-response assessment and synergy quantification. Briefly, they use a Bayesian hierarchical nonlinear regression model to explain the “variability between-experiments, variability within experiments, and variability in the observed responses of the controls” [17]. They first use Markov chain Monte Carlo (MCMC) to fit the model to the data. The second step is to carry out posterior inference on quantities of interest. Finally, they assess the presence of synergy while accounting of uncertainty using a modified version Loewe additivity. Simulation results suggest that this method is more reliable in drug synergy estimation than the Chou-Talalay method, which often ignores important sources of variability and uncertainty that is generally the rule, instead of the exception in biology [17].

CURRENT PROBLEMS AND FUTURE DEVELOPMENTS IN DRUG SYNERGY QUANTIFICATION

Drug combinations provide many advantages in the treatment of complex disease. The search for drug combinations has been widely recognized as one of the most important things in finding successful treatment of cancer and other diseases [10]. Although recent methods development has improved, there are still a number of open challenges and issue that need to be addressed.

First, there are still a number of challenges related to even defining synergy, much less quantifying it. In the current literature, the term synergy is not often clearly defined. Research papers usually use different reference models to quantify synergy in diffident cases, which causes confusion[18, 25].. Thus, a standard reference framework should be developed and provide a clear definition of additivity, synergy, and antagonism. Additionally, the standard framework should also be general enough to cover rare and specific cases so that researchers can use a universal method to quantify drug synergy. Our group has recently reviewed some of the challenges and differences in the terminology related to synergy[23].

Additionally, there are outstanding challenges in experimental design that need to be considered and advanced. One of the most important aspects of any study that will study synergy is the selection of dose and dose ratio in drug combination studies. The advantages of combination therapy not only depends on the property of the drugs but also depend on the dose ratio[19, 24]. Considering that two drugs combined at a given ratio are often treated as a new drug with its own dose–effect relationship in cells and tissues, we not only need to study whether a particular combination is synergistic, we also need to consider what dose ratio optimizes the synergistic interaction[19]. This is important in both experimental studies, and in clinical application.

Finally, we need to keep advancing more rigorous statistical methodology to interpret the variation in drug synergy quantification. Current methods quantify synergy, but do not ascribe a statistical confidence level with those estimates. Biological systems always carry experimental error and inherent biological variation. However, the most commonly used combination indexes based on Bliss Independence and Loewe Additivity are often calculated without a suitable error function to measure the degree of uncertainty. The lack of a formal statistical framework in these approaches makes it difficult to interpret drug combination effects, especially for borderline cases.

CONCLUSION

In the current review, we discuss an overview of the current statistical and mathematical methods for the study of drug combination effects, especially drug synergy quantification. We introduce two popular reference models for non-interaction of a combination, including the Bliss independence model and the Loewe additivity model. Then we discuss several methods for quantifying drug synergism. The advantages and disadvantages with these methods are also provided, and finally, we discuss current problems and future developments in drug synergy quantification

Addressing these limitations represent an important methodological research direction. Recently there have been a number of new approaches to quantify dose response curves using machine learning methods, including evolutionary algorithms[2]. Such an approach could be extended to the drug combination effects as well.

Advances in the statistical methods will allow researchers to estimate the variability in biological or clinical experiments with sufficient accuracy and further improve the degree of confidence in drug synergy detection. Moreover, these advances will also benefit high-throughput drug combination screening greatly. The integration of automated screening techniques with robust statistical methods will facilitate the discovery of reliable synergistic drug interactions, ultimately improving the sensitivity and specificity of the screening process. Although we mainly discuss drug synergy here, these advances in statistical methods can be easily applied to other disciplines as well, such as environmental toxicology and epidemiology. For instance, we can detect the combination effects of multiple environmental chemicals for risk assessment purposes with a high degree of confidence.

Table 1.

Summary of advantages and disadvantages of current methods.

	Advantages	Disadvantages	Data type
Response Surface	Characterize the full concentration-response relationship	No formal quantification of the intensity of a synergistic interaction	Raw data
Chou-Talalay method	Linear regression can be applied	Raw data preprocessing and no statistical inference	Preprocessed raw data
MixLow method	More accurate estimation	Not easy to understand and use the method	Raw data
Bayesian approach	More accurate estimation	Potential complexity in Bayesian statistics	Raw data