# Mathematical modeling of multi-drugs therapy: a challenge for determining the optimal combinations of antiviral drugs

- Yoshiki Koizumi
^{1}and - Shingo Iwami
^{2}Email author

**11**:41

https://doi.org/10.1186/1742-4682-11-41

© Koizumi and Iwami; licensee BioMed Central Ltd. 2014

**Received: **7 June 2014

**Accepted: **15 September 2014

**Published: **25 September 2014

## Abstract

In the current era of antiviral drug therapy, combining multiple drugs is a primary approach for improving antiviral effects, reducing the doses of individual drugs, relieving the side effects of strong antiviral drugs, and preventing the emergence of drug-resistant viruses. Although a variety of new drugs have been developed for HIV, HCV and influenza virus, the optimal combinations of multiple drugs are incompletely understood. To optimize the benefits of multi-drugs combinations, we must investigate the interactions between the combined drugs and their target viruses. Mathematical models of viral infection dynamics provide an ideal tool for this purpose. Additionally, whether drug combinations computed by these models are synergistic can be assessed by two prominent drug combination theories, Loewe additivity and Bliss independence. By combining the mathematical modeling of virus dynamics with drug combination theories, we could show the principles by which drug combinations yield a synergistic effect. Here, we describe the theoretical aspects of multi-drugs therapy and discuss their application to antiviral research.

## Keywords

## Background

Several landmark mathematical modeling studies of anti-retroviral therapy were reported in 1995[1–4]. Thereafter, mathematical modeling has provided quantitative insights into antiviral drugs targeting HIV[5–13], HCV[14–18], HBV[19, 20] and influenza virus[21, 22]. These studies have elucidated the key steps of viral replication cycles and viral infection dynamics, such as the half-life of viruses and how viruses replicate in cells (reviewed in[23–27]). Although mathematical viral models could also search for drug combinations that improve antiviral effects and prevent the emergence of drug-resistant viruses, this potential of viral modeling has yet to be properly explored. Multi-drug administration is a standard treatment for HIV and HCV infection, and is also required in HBV and influenza. The established combination drug therapy for HIV, known as highly active anti-retroviral therapy (HAART), remains prohibitively expensive in developing countries. Additionally, HIV patients receiving long-term HAART may experience severe side effects such as lipodystrophy, hepatotoxicity, renal dysfunction, peripheral neuropathy and cardiovascular diseases (reviewed in[28]). By optimizing drug combinations, we could reduce the doses of individual drugs, thereby lowering the cost of treatment, reducing side effects, enhancing the antiviral effects and reducing the risk of drug-resistant viruses.

To accelerate the above benefits of drug combinations, we need to know whether or not a drug combination exerts a synergistic effect (reviewed in[29, 30]). A non-synergistic drug combination requires a larger than expected dose to achieve a regular antiviral effect. Although multi-drug therapy is standard practice, the complexity of the combined drugs’ responses is improperly understood, and rendered counter-intuitive by drug absorption, distribution, metabolism, and excretion through drug transporters[28]. To overcome these difficulties, comprehensive trials (or experiments) of various combinations and doses of drugs have been required. The combined effect has been investigated by empirical methods such as Loewe additivity[29, 31] and Bliss independence[29, 32]. Both theories have evolved from pharmacological research, and are used to classify combination effects as synergistic, additive or antagonistic[29, 33, 34]. These empirical frameworks have become widely accepted in pharmacology and have provided quantitative insight into drug combinations. However, because their underlying mechanics are unknown, the frameworks cannot provide reasons for a given combination effect (e.g. why is this drug combination additive, whereas that combination is antagonistic?). Showing the mechanisms of drug combination effects is expected to remarkably advance the development of multicomponent therapeutics. Thus, in addition to judging drug combination effects, knowledge of the mechanism is crucial to fundamental antiviral research.

Here we discuss two primary concepts of drug combination theory: Loewe additivity and Bliss independence. We apply these models to viral replication in a host cell, and discuss their utility for understanding two-drug interactions and viral dynamics. Finally, we discuss how drug combination theory might be applied to cancer chemotherapies and antiviral therapies. Combining mathematical models of viral replication with drug combination theory, we can establish an efficient framework for optimizing drug combinations.

### Basic drug combination theory: Loewe additivity

*C*

_{ A }and

*C*

_{ B }are the respective concentrations of drugs A and B that produce the same effect as the drug combination. That is, the Loewe additivity specifies the concentration ratio of a single drug and its combination with another drug. Note that Loewe additivity assumes that two drugs target the same molecule or pathway. If two drugs do not mutually interact, they can be related through the combination index (

*CI*) based on the mass action law derived by Chou and Talalay[35]:

where the right hand side of Eq. (2) is identical to the left side of Eq. (1). When CI < 1, the relationship is synergetic, when CI = 1, it is additive, and when CI > 1, it is antagonistic. For example, suppose that 4 μM of drug A and 5 μM of drug B exert the same effect as a combination of 1 μM of A and 2 μM of B. Substituting these concentrations into Eq. (2), we obtain *CI* = 0.65 (=1/4 + 2/5 < 1), implying that the drug combination is synergistic. Note that the *CI* does not express the extent of synergy or antagonism; it is merely a criterion that separates the two behaviors without quantifying them.

*CI*, we require dose–response curves of drugs A and B, from which we can determine${c}_{A}^{*}$ and${c}_{B}^{*}$. The effect of a single drug

*E*is modeled by the Hill function as follows:

*E*

_{ max }is the maximum effect,

*c*is the drug concentration,

*h*is the Hill coefficient that determines the steepness of the dose–response curve, and

*IC*

_{50}is the concentration at which

*E*exerts 50% of its maximum effect. Substituting Eq. (3) into Eq. (1) and rearranging in the case of

*E*

_{ max }= 1, we obtain:

Numerically solving Eq. (4) for *E*, we can predict the additive effect at any drug concentrations${c}_{A}^{*}$ and${c}_{B}^{*}$[35, 36]. The following section explains an example of applying Loewe additivity to estimate rational drug combinations and optimal doses.

### Application of Loewe additivity for estimating rational drug combinations and optimal doses

### The alternative drug combination theory: Bliss independence

*U*

_{ AB }as the product of the unaffected probabilities

*U*

_{ A }and

*U*

_{ B }of drugs A and B:

*U*by 1 -

*P*, where

*P*denotes the probability of an intercepted target, Eq. (5) can be expressed as (1 -

*P*

_{ AB }) = (1 -

*P*

_{ A }) × (1 -

*P*

_{ B }), or

If the combined effect of the single drug doses is consistent with Eq. (6), this drug combination is evaluated as additive according to the Bliss independence measure. For instance, suppose that drugs A and B exhibit 50% and 60% inhibition, respectively. The Bliss independence predicts that drugs A and B will exert a combined inhibitory effect of 80%, calculated as *P*_{
AB
} = 0.5 + 0.6 - 0.5 × 0.6 = 0.8.

Unlike the Loewe additivity, the Bliss independence can evaluate the effect of a drug combination without requiring dose–response curves of the single drugs. If we attempt to assess a drug combination by the Loewe additivity, we must measure the effects at several single doses for constructing the dose–response curve. In contrast, the Bliss independence can assess the effects of a drug combination from a single point. Thus, the Bliss independence is useful if experimental data are limited. Additionally, in specific cases, the Bliss independence and Loewe additivity can be rendered equivalent using the Hill function Eq. (3)[41]. The following section explains an example of applying Bliss independence to quantifying the antiviral effects of intrinsic factors on HIV replication.

### Quantifying the antiviral effects of APOBEC3G on HIV replication by Bliss independence

*f*

_{ WT }and

*f*

_{ CI }respectively, in terms of the unaffected fractions of viral infectivity by reverse transcription inhibition and C-to-U mutation,

*f*

_{ RT }and

*f*

_{ Mu }, as follows:

where *x* is the expression level of APOBEC3G, *h*_{
RT
} and *h*_{
Mu
} are Hill coefficients, and$I{C}_{{50}_{\mathit{RT}}}$ and$I{C}_{{50}_{\mathit{Mu}}}$ are the expression levels required to achieve 50% inhibition by reverse transcription inhibition and C-to-U mutation, respectively. We estimated 4 parameters (*h*_{
RT
}, *h*_{
Mu
},$I{C}_{{50}_{\mathit{RT}}}$, and$I{C}_{{50}_{\mathit{Mu}}}$) by fitting to experimental data of WT and mutated APOBEC3G (Figure 2B). Thus, the anti-HIV effect of C-to-U mutation by APOBEC3G can be quantitatively estimated by a Bliss independence-based modeling approach.

### Potential applications of drug combination theory

The Loewe additivity and Bliss independence form the basis of sophisticated protocols for assessing drug combinations, especially in cancer-targeted chemotherapy[33, 34]. Research on biological network systems, as well as the biochemical characteristics of drugs, has also unraveled the mechanisms by which drug combinations produce synergistic effects (reviewed in[41, 43–45]). Such research is important because (1) it facilitates rational drug discovery and (2) it predicts unknown connectivities in biological networks. Regarding the first point, the effects of a developing drug combined with conventional drugs are maximized by investigating biological networks, and rational drug targets are decided. For example, using computational modeling of cancer signaling networks and animal experiments, Kirouac *et al.*[46] identified the optimal inhibitor combination for suppressing the growth of *ERBB2*-amplified breast cancer. They adopted Bliss independence as a synergy criterion. Supplementing this computational approach with experimental data is useful for developing theoretically effective novel drugs. Regarding point (2), the biological networks of signaling pathways constructed by various components can be predicted from the dose–response shapes of drug combinations (reviewed in[47]). Lehar *et al.*[48] used several drug combination criteria, including Loewe additivity and Bliss independence, to connect biological components from the results of cellular responses to chemical combinations. This approach assumes that the dose–response shape of a drug combination depends on how drug targets are connected in a biological system.

*et al.*[49] showed how HCV replication responds to chemical combinations that inhibit some enzymes involved in the sterol biosynthesis pathway. Supplementing computational simulation with

*in vitro*experiments, they identified a rational inhibitor combination as a novel drug candidate. Mathematical models of HCV dynamics during drug therapy have also been investigated at the intracellular level[18], the intercellular level[14–17, 50, 51], and on a scale encompassing both levels[51–54]. These studies have quantitatively elucidated the properties of anti-HCV drugs such as interferon and protease inhibitors. Furthermore, by inputting clinical data to a mathematical model of HCV dynamics, Rong

*et al.*[55] derived a mechanism by which drug-resistant viruses can rapidly emerge during single-drug therapy. Applying drug combination theories to these studies, mathematical models of HCV dynamics could be used to optimize the anti-HCV drug dosage and combination, thereby enhancing the anti-viral effect and reducing the risk of drug-resistant viruses (Figure 3).

## Conclusion

We have reviewed two major drug combination theories, Loewe additivity and Bliss independence, and discussed how combining these theories with mathematical modeling of viral dynamics might assist antiviral drug therapy. In addition, we have proposed an efficient framework for optimizing drug combinations and quantifying the anti-viral effect. Based on previous studies of computational virology, the integration of drug combination theory and dynamic modeling is a new approach with great potential for showing viral responses to drug combinations, and accelerating novel antiviral drug discovery.

## Declarations

### Acknowledgments

This research is partly supported by a Grants-in-Aid for Young Scientists B25800092 (to S.I.) from the Japan Society for the Promotion of Science (JSPS); the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development (to S.I.); the Aihara Innovative Mathematical Modeling Project, JSPS through the “Funding Program for World-Leading Innovative R & D on Science and Technology (FIRST program)”, initiated by Council for Science and Technology Policy (to S.I.).

## Authors’ Affiliations

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