A semiparametric response surface model for assessing drug interaction

Real-Time Cell Migration Monitoring to Analyze Drug Synergism in the Scratch Assay Using the IncuCyte System

Drug-mediated interference with metastasis represents a key approach to improve cancer therapy. In this regard, appropriate in vitro assays are needed to identify drugs, which inhibit cell migration as one feature for metastatic potential of cancer cells. One such migration assay is the wound healing or scratch assay, designed to allow cells for closure of an artificially generated gap (wound/scratch) in the monolayer. To identify possibly effective anti-migratory drugs as monotherapy or as synergistic drug combination, novel screening tools besides viability measurements at the experimental endpoint are needed. In this context, particularly drug combinations allow to increase treatment efficacy paralleled by lowered side effects. Here, a protocol for real-time monitoring cellular motility and its inhibition by anti-migratory drugs and combinations by the IncuCyte system and a 96-well scratch assay is described. A pipetting scheme allowing data collection for synergy calculation using one plate per replicate is provided. Using the IncuCyte System 2, drug combinations built of three biological replicates each using three technical replicates can be tested in parallel within hours to few days to accelerate identification of efficient antimetastatic drugs.

Semiparametric estimation of the attributable fraction when there are interactions under monotonicity constraints

BACKGROUND

The population attributable fraction (PAF) is the fraction of disease cases in a sample that can be attributed to an exposure. Estimating the PAF often involves the estimation of the probability of having the disease given the exposure while adjusting for confounders. In many settings, the exposure can interact with confounders. Additionally, the exposure may have a monotone effect on the probability of having the disease, and this effect is not necessarily linear.

METHODS

We develop a semiparametric approach for estimating the probability of having the disease and, consequently, for estimating the PAF, controlling for the interaction between the exposure and a confounder. We use a tensor product of univariate B-splines to model the interaction under the monotonicity constraint. The model fitting procedure is formulated as a quadratic programming problem, and, thus, can be easily solved using standard optimization packages. We conduct simulations to compare the performance of the developed approach with the conventional B-splines approach without the monotonicity constraint, and with the logistic regression approach. To illustrate our method, we estimate the PAF of hopelessness and depression for suicidal ideation among elderly depressed patients.

RESULTS

The proposed estimator exhibited better performance than the other two approaches in the simulation settings we tried. The estimated PAF attributable to hopelessness is 67.99% with 95% confidence interval: 42.10% to 97.42%, and is 22.36% with 95% confidence interval: 12.77% to 56.49% due to depression.

CONCLUSIONS

The developed approach is easy to implement and supports flexible modeling of possible non-linear relationships between a disease and an exposure of interest.

Drug Combinations: Mathematical Modeling and Networking Methods

Treatments consisting of mixtures of pharmacological agents have been shown to have superior effects to treatments involving single compounds. Given the vast amount of possible combinations involving multiple drugs and the restrictions in time and resources required to test all such combinations in vitro, mathematical methods are essential to model the interactive behavior of the drug mixture and the target, ultimately allowing one to better predict the outcome of the combination. In this review, we investigate various mathematical methods that model combination therapies. This survey includes the methods that focus on predicting the outcome of drug combinations with respect to synergism and antagonism, as well as the methods that explore the dynamics of combination therapy and its role in combating drug resistance. This comprehensive investigation of the mathematical methods includes models that employ pharmacodynamics equations, those that rely on signaling and how the underlying chemical networks are affected by the topological structure of the target proteins, and models that are based on stochastic models for evolutionary dynamics. Additionally, this article reviews computational methods including mathematical algorithms, machine learning, and search algorithms that can identify promising combinations of drug compounds. A description of existing data and software resources is provided that can support investigations in drug combination therapies. Finally, the article concludes with a summary of future directions for investigation by the research community.

Parametric modeling and optimal experimental designs for estimating isobolograms for drug interactions in toxicology

In toxicology and related areas, interaction effects between two substances are commonly expressed through a combination index [Formula: see text] evaluated separately at different effect levels and mixture ratios. Often, these indices are combined into a graphical representation, the isobologram. Instead of estimating the combination indices at the experimental mixture ratios only, we propose a simple parametric model for estimating the underlying interaction function. We integrate this approach into a joint model where both the parameters of the dose-response functions of the singular substances and the interaction parameters can be estimated simultaneously. As an additional benefit, this concept allows to determine optimal statistical designs for combination studies optimizing the estimation of the interaction function as a whole. From an optimal design perspective, finding the interaction parameters generally corresponds to a [Formula: see text]-optimality resp. [Formula: see text]-optimality design problem, while estimation of all underlying dose response parameters corresponds to a [Formula: see text]-optimality design problem. We show how optimal designs can be obtained in either case as well as how combination designs providing reasonable performance in regard to both criteria can be determined by putting a constraint on the efficiency in regard to one of the criteria and optimizing for the other. As all designs require prior information about model parameter values, which may be unreliable in practice, the effect of misspecifications is investigated as well.

Bliss and Loewe interaction analyses of clinically relevant drug combinations in human colon cancer cell lines reveal complex patterns of synergy and antagonism

We analyzed survival effects for 15 different pairs of clinically relevant anti-cancer drugs in three iso-genic pairs of human colorectal cancer carcinoma cell lines, by applying for the first time our novel software (R package) called COMBIA. In our experiments iso-genic pairs of cell lines were used, differing only with respect to a single clinically important KRAS or BRAF mutation. Frequently, concentration dependent but mutation independent joint Bliss and Loewe synergy/antagonism was found statistically significant. Four combinations were found synergistic/antagonistic specifically to the parental (harboring KRAS or BRAF mutation) cell line of the corresponding iso-genic cell lines pair.

COMBIA offers considerable improvements over established software for synergy analysis such as MacSynergy™ II as it includes both Bliss (independence) and Loewe (additivity) analyses, together with a tailored non-parametric statistical analysis employing heteroscedasticity, controlled resampling, and global (omnibus) testing.

In many cases Loewe analyses found significant synergistic as well as antagonistic effects in a cell line at different concentrations of a tested drug combination. By contrast, Bliss analysis found only one type of significant effect per cell line.

In conclusion, the integrated Bliss and Loewe interaction analysis based on non-parametric statistics may provide more robust interaction analyses and reveal complex patterns of synergy and antagonism.

Monotonic single-index models to assess drug interactions

Although single-index models have been extensively studied, the monotonicity of the link function f in the single-index model is rarely studied. In many situations, it is desirable that f is monotonic, which results in a monotonic single-index model that can be very useful in economics and biometrics. In this article, we propose a monotonic single-index model in which the link function is constructed using penalized I-splines along with constraints on coefficients to achieve monotonicity of the link function f. An algorithm to estimate the single-index parameters and the link function is developed, and the sandwich estimate of the variance of the index parameters is provided. We propose to apply this monotonic single-index model to estimate the dose-response surface and assess drug interactions while considering the variability of the observed data. An extensive simulation study was carried out to evaluate the performance of the proposed monotonic single-index model. A case study is provided to illustrate the application of the proposed model to estimate the dose-response surface and assess drug interactions. Both the simulation and case study show that the proposed monotonic single-index model works very well. Copyright © 2016 John Wiley & Sons, Ltd.

Advances in Synergistic Combinations of Chinese Herbal Medicine for the Treatment of Cancer

The complex pathology of cancer development requires correspondingly complex treatments. The traditional application of individual single-target drugs fails to sufficiently treat cancer with durable therapeutic effects and tolerable adverse events. Therefore, synergistic combinations of drugs represent a promising way to enhance efficacy, overcome toxicity and optimize safety. Chinese Herbal Medicines (CHMs) have long been used as such synergistic combinations. Therefore, we summarized the synergistic combinations of CHMs used in the treatment of cancer and their roles in chemotherapy in terms of enhancing efficacy, reducing side effects, immune modulation, as well as abrogating drug resistance. Our conclusions support the development of further science-based holistic modalities for cancer care.

A simple test for synergy for a small number of combinations

A method for detecting deviations from the Loewe additive drug combination reference model for in vitro drug combination experimentation is described. It is often difficult to fit a response surface model to drug combination data, especially in situations where the experimental design contains a sparse set of combinations. The literature does contain good response surface modeling approaches, but they tend to be complex and can be difficult to execute. It is especially difficult to check model quality when fitting to more than two combined agents. A simple method based on sound statistical principles is proposed that examines the mean response deviation of each combination from the predicted response under Loewe additivity. The method can readily handle any number of combined agents, does not require sophisticated modeling, and can even be programmed into Microsoft Excel without the use of macros. Several potential extensions to the method are discussed in detail. Computer-generated simulations demonstrate the statistical capabilities of the approach, and a real-data example is given to illustrate the method.

Assessing interactions for fixed-dose drug combinations in subcutaneous tumor xenograft studies

Drug combinations in preclinical tumor xenograft studies are often assessed using fixed doses. Assessing the joint action of drug combinations with fixed doses has not been well developed in the literature. Here, an interaction index is proposed for fixed-dose drug combinations in a subcutaneous tumor xenograft model. Furthermore, a bootstrap percentile interval of the interaction index is also developed. The joint action of two drugs can be assessed on the basis of confidence limits of the interaction index. Tumor xenograft data from actual two-drug combination studies are analyzed to illustrate the proposed method.

Assessing interactions for fixed-dose drug combinations in tumor xenograft studies

Statistical methods for assessing the joint action of compounds administered in combination have been established for many years. However, there is little literature available on assessing the joint action of fixed-dose drug combinations in tumor xenograft experiments. Here an interaction index for fixed-dose two-drug combinations is proposed. Furthermore, a regression analysis is also discussed. Actual tumor xenograft data were analyzed to illustrate the proposed methods.

A minimal model of tumor growth inhibition in combination regimens under the hypothesis of no interaction between drugs

One important issue in the preclinical development of an anticancer drug is the assessment of the compound under investigation when administered in combination with other drugs. Several experiments are routinely conducted in xenograft mice to evaluate if drugs interact or not. Experimental data are generally qualitatively analyzed on empirical basis. The ability of deriving from single drug experiments a reference response to the joint administrations, assuming no interaction, and comparing it to real responses would be key to recognize synergic and antagonist compounds. Therefore, in this paper, the minimal model of tumor growth inhibition (TGI), previously developed for a single drug, is reformulated to account for the effects of noninteracting drugs and simulate, under this hypothesis, combination regimens. The model is derived from a minimal set of basic assumptions that include and extend those formulated at cellular level for the single drug administration. The tumor growth dynamics is well approximated by the deterministic evolution of its expected value that is obtained through the solution of an ordinary and several partial differential equations. Under suitable assumptions on the cell death process, the model reduces to a lumped parameter model that represents the extension of the very popular Simeoni TGI model to the combined administration of noninteracting drugs.

Combined use of biocontrol agents to manage plant diseases in theory and practice

Effective use of biological control agents (BCAs) is a potentially important component of sustainable agriculture. Recently, there has been an increasing interest among researchers in using combinations of BCAs to exploit potential synergistic effects among them. The methodology for investigating such synergistic effects was reviewed first and published results were then assessed for available evidence for synergy. Correct formulation of hypotheses based on the theoretical definition of independence (Bliss independence or Loewe additivity) and the subsequent and statistical testing for the independence-synergistic-antagonistic interactions have rarely been carried out thus far in studies on biocontrol of plant diseases. Thus, caution must be taken when interpreting reported "synergistic" effects without assessing the original publications. Recent theoretical modeling work suggested that disease suppression from combined use of two BCAs was, in general, very similar to that achieved by the more efficacious one, indicating no synergistic but more likely antagonistic interactions. Only in 2% of the total 465 published treatments was there evidence for synergistic effects among BCAs. In the majority of the cases, antagonistic interactions among BCAs were indicated. Thus, both theoretical and experimental studies suggest that, in combined use of BCAs, antagonistic interactions among BCAs are more likely to occur than synergistic interactions. Several research strategies, including formulation of synergy hypotheses in relation to biocontrol mechanisms, are outlined to exploit microbial mixtures for uses in biocontrol of plant diseases.

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.

Validation of a novel statistical model for assessing the synergy of combined-agent cancer chemoprevention

Lung cancer is the leading cause of cancer death, developing over prolonged periods through genetic and epigenetic changes induced and exacerbated by tobacco exposure. Many epigenetic changes, including DNA methylation and histone methylation and acetylation, are reversible. The use of agents that can modulate these aberrations are a potentially effective approach to cancer chemoprevention. Combined epigenetic-targeting agents have gained interest for their potential to increase efficacy and lower toxicity. The present study applied recently developed statistical methods to validate the combined effects of the demethylating agent 5-aza-2-deoxycytidine (5-AZA-CdR, or AZA, or decitabine) and the histone deacetylase inhibitor suberoylanilide hydroxamic acid (SAHA or vorinostat). This validation compared AZA alone with SAHA alone and with their combinations (at later or earlier time points and in varying doses) for inhibiting the growth of cell lines of an in vitro lung carcinogenesis system. This system comprises isogenic premalignant and malignant cells that are immortalized (earlier premalignant), transformed (later premalignant), and tumorigenic human bronchial epithelial cells [immortalized BEAS-2B and its derivatives 1799 (immortalized), 1198 (transformed), and 1170-I (tumorigenic)]. AZA alone and SAHA alone produced a limited (<50%) inhibition of cell growth, whereas combined AZA and SAHA inhibited cell growth more than either agent alone, reaching 90% inhibition under some conditions. Results of drug interaction analyses in the E(max) model and semiparametric model supported the conclusion that drug combinations exert synergistic effects (i.e., beyond additivity in the Loewe model). The present results show the applicability of our novel statistical methodology for quantitatively assessing drug synergy across a wide range of doses of agents with complex dose-response profiles, a methodology with great potential for advancing the development of chemopreventive combinations.

Emax model and interaction index for assessing drug interaction in combination studies

Applying the Emax model in a Lowe additivity model context, we analyze data from a combination study of trimetrexate (TMQ) and AG2034 (AG) in media of low and high concentrations of folic acid (FA). The Emax model provides a sufficient fit to the data. TMQ is more potent than AG in both low and high FA media. At low TMQ:AG ratios, when a smaller amount of the more potent drug (TMQ) is added to a larger amount of the less potent drug (AG), synergy results. When the TMQ:AG ratio reaches 0.4 or larger in low FA medium, or when the TMQ:AG ratio reaches 1 or larger in high FA medium, synergy is weakened and drug interaction becomes additive. In general, synergistic effect in a dilution series is stronger at higher doses that produce stronger effects (closer to 1-Emax) than at lower dose levels that produce weaker effects (closer to 1). The two drugs are more potent in the low compared to the high FA medium. Drug synergy, however, is stronger in the high FA medium.

Applying Emax model and bivariate thin plate splines to assess drug interactions

We review the semiparametric approach previously proposed by Kong and Lee and extend it to a case in which the dose-effect curves follow the Emax model instead of the median effect equation. When the maximum effects for the investigated drugs are different, we provide a procedure to obtain the additive effect based on the Loewe additivity model. Then, we apply a bivariate thin plate spline approach to estimate the effect beyond additivity along with its 95 per cent point-wise confidence interval as well as its 95 per cent simultaneous confidence interval for any combination dose. Thus, synergy, additivity, and antagonism can be identified. The advantages of the method are that it provides an overall assessment of the combination effect on the entire two-dimensional dose space spanned by the experimental doses, and it enables us to identify complex patterns of drug interaction in combination studies. In addition, this approach is robust to outliers. To illustrate this procedure, we analyzed data from two case studies.

Confidence Intervals of Interaction Index for Assessing Multiple Drug Interaction

Studies of interactions among biologically active agents have become increasingly important in many branches of biomedical research. We consider that the Loewe additivity model is one of the best general reference models for defining drug interactions. Based on the Loewe additivity model, synergy occurs when the interaction index is less than one, and antagonism occurs when interaction index is greater than one. Starting from the Loewe additivity model and the marginal dose-effect curve for each drug involved in a combination, we first present a procedure to estimate the interaction index and its associated confidence interval at a combination dose with observed effects. Following Chou and Talalay's method for assessing drug interaction based on the plot of interaction indices versus effects for combination doses at a fixed ray, we then construct a pointwise (1-alpha)x100% confidence bound for the curve of interaction indices versus effects. We found that these methods work better on the logarithm transformed scale than on the untransformed scale of the interaction index. We provide simulations and case studies to illustrate the performances of these two procedures, and present their pros and cons. We also provide S-Plus/R code to facilitate the implementation of these two procedures.