Jun Ma and Alison MotsingerReif* 
Corresponding Author: Alison MotsingerReif, PhD Department of Statistics, Bioinformatics Research Center, North Carolina State University, USA 
Received: February 05, 2019; Accepted: February 21, 2019; Published: July 22, 2019 ; 
Citation: Ma J & MotsingerReif A. (2019) Current Methods for Quantifying Drug Synergism. Proteomics Bioinformatics, 1(2): 4348. 
Copyrights: ©2019 Ma J & MotsingerReif A. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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 noninteraction 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 [1]. 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) [2]. 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 [3],
asthma [4], breast cancer [5,6], hypertension [7] and cancers such as melanoma
[8]. The impact of chemical mixtures is also increasingly appreciated in the
toxicology space as well, as people are not exposed to chemicals in isolation
[9]. A recent review discusses the concept of synergy as used in a variety of
fields [10].
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 [2,11]. Specifically, if drugs when combined interact with
each other and cause a total effect that is greater than that predicted by
their individual potencies, then this is considered a “synergistic drug”
combination [12]. Such synergistic interactions can often reduce host toxicity
and adverse side effects, given those doses of each drug in the combination are
typically lower than that of single drugs to achieve desired efficacy.
Additionally, such combination therapies can also reduce the development of
drug resistance and other complications [13,14].
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 the null hypothesis of
noninteraction, 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
noninteraction first, which suggests that the effects of drugs simply add up,
not affecting each other (Additivity) [15]. 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.
Bliss independence model
One
of the oldest methods for quantifying synergy is the Bliss Independence model,
dating back to the 1930s [16]. This model assumes that drugs do not interact with
each other and elicit their responses independently [1]. According to
independence probability theory, the expected response of drug combination R_{c}
(0 ≤ R_{c} ≤ 1) can be written in terms of individual drug responses
[16]:
R_{c} = R_{1 }+ (1  R_{1})
× R_{2} = R_{1 }+ R_{2 } R_{1 }× R_{2}
Where
drug 1 at dose y_{1} produced a response R_{1}, drug 2 at dose
y_{2} produced a response R_{2} and R_{c} is the
expected response of drug combination 1 and 2 at dose y_{1} and y_{2},
respectively. As the drug’s effects R_{1}, R_{2} and R_{c}
are measured as the percentage of biological response, 0 ≤ R_{1} ≤ 1, 0
≤ R_{2} ≤ 1 and 0 ≤ R_{c} ≤ 1. Any observed response of drug
combination greater than the expected response R_{c} can be interpreted
as synergy and antagonism otherwise.
This
null model is classically known in toxicology as “simple independent action”
and is based on probabilistic independence. The paradigm is where there are two
disjoint and independent causal pathways on which the two drugs act. The above
equation can alternatively simply be rewritten as additivity in the logarithms
of the two probabilities of nonresponse.
Loewe additivity model
An alternative null model is the additivity model, which assumes that drugs have similar modes of action on the same pathway [1,17]. In classical toxicology, this model is known as “simple similar action.” It specifies that one drug’s dose has the same effect on response as a scaling factor times the other drug’s dose. To formulate this as specified in the Loewe additivity model, the doseresponse relationship of individual drugs needs to be modeled first. Let the dose of drug 1=y_{1} and the dose of drug 2=y_{2}. Then the Loewe additivity model can be expressed as the following equation [1,17]:_{}
(y_{1} / Y_{1}) + (y_{2}
/ Y_{2}) = 1
Where
Y_{1} is the dose of drug 1 that achieves the same response level as
the drug combination, y_{1}+y_{2} and Y_{2} is the dose
of drug 2 that achieve the same response level as the drug combination. The
left side of this equation is the widely used combination index. If a
combination index is less than 1, synergy is declared. Similarly, a combination
index greater than 1 can be interpreted as antagonism.
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. Note that if the response is rare, the two formulations are asymptotically
equivalent. In practice, the selection of one of those two to serve as the null
model for assessing synergy and antagonism becomes largely a matter of personal
preference [18]. To address the concerns raised in the two reference models, Yadav
et al. [19] recently proposed a new reference model called zero interaction
potency (ZIP). It evaluates drug interaction by comparing the change in dose
response relationships between single drugs and their combinations. The results
show that this new scoring method is able to keep the advantages of the two
popular reference models mentioned above while overcome their limitations.
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
threedimensional plots where the doses of individual drugs are plotted as a
horizontal xyplane, and the expected effect of drug combination is plotted on
the zaxis, as shown in Figure 2
[1]. Both the Bliss independence model and the Loewe additivity model can be
used to calculate the expected effect of drug combination [1]. The experimental
effect of a drug combination can then be plotted on this surface. Any departure
from the 3D null surface is classified into synergism or antagonism, depending
on the sign of the discrepancy as measured on the zaxis. [20,21].
ChouTalalay method
The ChouTalalay method is by far the most commonly
used approach to quantify effects of drug combinations, especially synergistic
interactions [15,21]. This method adopts the medianeffect equation, which is
derived from the unified theory massaction law principle [9]. The
medianeffect equation is written below [21]:
f_{a} / f_{u} = (D / D_{m})^{m}
Where
f_{a} is the fraction affected by dose, f_{u} is the fraction
unaffected by dose (f_{a}+f_{u}=1), D is dose of drugs given, D_{m}
is the medianeffect dose (e.g. IC50) and m is a parameter used to describe the
shape of the doseresponse curve.
The
medianeffect equation can be linearized by taking logarithm of both sides of
the equation, as shown below:
Log
(f_{a} / f_{u}) = m × log (D) – m × log (D_{m})^{m}
Then
the values of m and D_{m} can be estimated using linear regression.
With this linear model, we can estimate the expected drug response values given
specific drug doses, which will be used in the calculation of combination index
(CI):
CI = D_{1} / D_{x1} + D_{2}
/ D_{x2}
Where
D_{1} and D_{2} are the doses of two single drugs and D_{x1}
and D_{x2} is the theoretical individual drug doses needed in order to
achieve the same drug effect as the drug combination, which can be calculated
based on the linear model mentioned above. CI<1 suggests synergism, CI=1
suggests additivity and CI>1 suggests antagonism [15,22].
One
disadvantage of the ChouTalalay method is that raw data must be preprocessed,
including scaling the data and taking the log of a function of the scaled data
[15].
MixLow method
More
recently, Boik et al. [23] developed the MixLow method as an alternative to the
ChouTalalay method. The term MixLow means Mixedeffects Loewe, which has three
components: a nonlinear mixedeffects model, the Loewe index and a method to
calculate confidence intervals for the index. The MixLow method uses a
nonlinear mixed effects model for estimating sigmoidal curve parameters from
concentrationresponse data, and associated confidence intervals [23]. Compared
with the ChouTalalay method, the MixLow method produces more precise parameter
estimation, and has improved coverage of confidence intervals. In addition, the
use of a nonlinear fixedeffects model in the MixLow method also eliminates
the need for data preprocessing in the ChouTalalay method [23].
Drug synergy quantification using a Bayesian
approach
In
2010, Hennessey et al. proposed a Bayesian approach to doseresponse assessment
and synergy quantification. Briefly, they use a Bayesian hierarchical nonlinear
regression model to explain the “variability betweenexperiments, variability within
experiments, and variability in the observed responses of the controls” [24].
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 for uncertainty
using a modified version of Loewe additivity. Simulation results suggest that
this method is more reliable in drug synergy estimation than the ChouTalalay
method, which often ignores important sources of variability and uncertainty
that is generally the rule, instead of the exception in biology [24].
Summary of advantages and disadvantages of current methods (Table 1)_{}
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 strategies for finding successful treatments of cancer and other
diseases [15]. Although recent methods development has improved, there are
still a number of open challenges and issues 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 particular cases, which causes lack of comparability and
confusion [18,25]. Thus, a standard reference framework should be developed to
address the concerns raised in the current reference models 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 [10].
Additionally,
there are outstanding challenges in experimental design that need to be
considered and advanced. One of the most important challenges of any study that
will study synergy is the selection of dose and dose ratios in drug combination
studies. The advantages of combination therapy not only depends on the
properties of the drugs but also depend on the dose ratios [26,27]. Considering
that two drugs combined at a given ratio are often treated as a new drug with
its own doseeffect 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 [26]. 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. Data
from biological systems always carry experimental error and there is also
inherent biological variation. However, the most commonly used combination
indexes based on Bliss Independence and Loewe Additivity are often calculated
without a suitable error assessment 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
noninteraction 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 associated with these methods
are also provided, and finally, we discuss current problems and future
developments in drug synergy quantification.
Addressing
these limitations represents 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 [28]. 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 highthroughput 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.
ACKNOWLEDGEMENT
This
research was supported by the Intramural Research Program of the NIH, National
Institute of Environmental Health Sciences.
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