Statistical determination of synergy based on Bliss definition of drugs independence

Abstract

Although synergy is a pillar of modern pharmacology, toxicology, and medicine, there is no consensus on its definition despite its nearly one hundred-year history. Moreover, methods for statistical determination of synergy that account for variation of response to treatment are underdeveloped and if exist are reduced to the traditional t-test, but do not comply with the normal distribution assumption. We offer statistical models for estimation of synergy using an established definition of Bliss drugs' independence. Although Bliss definition is well-known, it remains a theoretical concept and has never been applied for statistical determination of synergy with various forms of treatment outcome. We rigorously and consistently extend the Bliss definition to detect statistically significant synergy under various designs: (1) in vitro, when the outcome of a cell culture experiment with replicates is the proportion of surviving cells for a single dose or multiple doses, (2) dose-response methodology, (3) in vivo studies in organisms, when the outcome is a longitudinal measurement such as tumor volume, and (4) clinical studies, when the outcome of treatment is measured by survival. For each design, we developed a specific statistical model and demonstrated how to test for independence, synergy, and antagonism, and compute the associated p-value.

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1. Geometrical illustration of drug independence according to Bliss (M = mortality, S = survival, M = 1 − S).

The two circles depict surviving fractions in the population of cells upon single-agent administration of either drug A or drug B, denoted SA and SB, respectively. When administered in combination, drug B only affects cells that survived drug A (and vice versa). Therefore, if drugs act independently, the surviving fraction of cells following treatment with the drug combination is the product: SA × SB.

Fig 2. Daphnia acute tests with single-dose stressors NiCl and CuSO4.

Although surviving fraction when the two stressors are applied simultaneously is smaller than if they were acting independently, the “detected” synergy of 17% is not statistically significant.

Fig 3. Drug interaction for testing synergy of BYL and GSK at single doses in breast cancer cells.

Fig 4. Illustration of the two-drug copula mortality function using EC50 contours.

When drugs have moderate effect on mortality (m = 1) synergy will be underestimated by the Loewe approach and overestimated if drugs have a strong effect (m = 2).

Fig 5. Fit of 1:5 ratio of rotenone to pyrethrins data by the probit-based two-drug copula mortality function.

The bold curve depicts the combination of doses of the two drugs that leads to the expected 50% fly kill (its projection on the xy-plane is depicted as the dashed line). The synergy between drugs is statistically significant with p-value = 0.0086.

Fig 6. The output of program nls for estimation of the probit-based two-drug copula model (11).

Fig 7. Exponential growth in four groups of mice for testing synergy between EHT1864 and fulvestrand for breast tumor treatment estimated by linear mixed model (see details in supporting information).

The data are plotted on the log scale; the tickmarks correspond to the original volume in mm3. There is a statistically significant synergy between the drugs with p-value = 0.006.

Fig 8. Kaplan-Meier survival curves in “Intention-to-treat population” for treatment metastatic melanoma patients with ipilimumab (drug A) and/or nivolumab (drug B); see [49].

The survival curve “Drug independence” is computed by formula (12). This curve basically overlaps with the observed survival curve SAB (blue) which means that the hypothesis of independence cannot be rejected.

Fig 9. The hypothesis that drugs act independently for patients with PD-L1-negative tumors cannot be rejected because the expected (black) and observed (blue) curves are close; see [49].