Rebuttal to the Response of Chou
J. Jack Lee, Maiying Kong · Cancer Research · 2011
We appreciate the opportunity to clarify and refute the serious issues raised by Dr. Chou in the rebuttal to our response to his letter. The main question he posed is how a probabilistic/statistical approach can lead to a quantitative/deterministic equation based on the mass-action law. He also stated, “synergism is a physicochemical mass-action law issue, not a statistical issue, so synergism is determined with combination index (CI) values, not with P values.” As noted in our original response, the reference model for assessing drug interaction can be derived mathematically by understanding the nature of drug interactions. In this step, no statistics are required. However, statistics come into play when drug effects are measured with error (uncertainty). One cannot determine synergism based on the point estimate of the interaction index (or CI) alone. For example, an interaction index of 0.9 with a 95% confidence interval of 0.83–0.97 is considered to show synergy, whereas if the corresponding confidence interval is 0.77–1.05, the drug combination cannot be considered to show synergy, statistically speaking . We do not claim to have derived the reference model by using a probabilistic/statistical approach. Instead, as stated in the abstract of our article, we “… expound the meaning of the interaction index and propose a procedure to calculate the interaction index and its associated confidence interval under the assumption that the dose–effect curve for a single agent follows Chou and Talalay's median effect equation” (1). The P value and confidence interval are two sides of a coin, that is, a P value of less than 0.05 indicates that the 95% confidence interval does not contain 1. When the upper end of a 95% confidence interval is smaller than 1 (and the P value will be <0.05), the data show synergy with statistical rigor. The confidence interval we constructed (1) is based on asymptotic theory and is calculated directly. This process of construction is different from that of the confidence interval for the CI method, serial deletion analysis, which uses an iterative analysis that deletes one data point at a time (2). In addition, the framework for constructing the interaction index does not depend on the assumption that the dose–response curve follows the median effect equation. To set the record straight, the original definition of the CI by Chou and Talalay (ref. 3, p. 35) is different from the interaction index. They defined ${\rm CI} = {\frac{( D)_1}{(D_x})_1}+ {\frac{( D)_2} (D_x)_2}$ for mutually exclusive drugs and ${\rm CI} = {\frac{( D)_1} {(D_x)_1}}+ {\frac{( D )_2}{(D_x )_2}} +{ \frac{( D)_1( D)_2} {(D_x )_1(D_x)_2}}$ for mutually nonexclusive drugs, where the combination dose [(D)1, (D)2] produces the same effect as agent 1 alone at dose level (Dx)1 and agent 2 alone at dose level (Dx)2. The CI is exactly the same as the interaction index when the 2 agents are mutually exclusive, whereas the CI is different from the interaction index when the 2 agents are mutually nonexclusive. In 2006, Dr. Chou proposed a revised definition for the CI (2) as the first expression, which is exactly the same as the interaction index when the individual dose–response curves follow the median effect equation.In his rebuttal, Dr. Chou made 3 related claims, namely, that (i) Eqs. 1 and 2 defined in our initial response letter are inappropriately referred to as the Loewe additivity (1920s) based on the work of Fraser (1870s); (ii) additivity, synergism, and antagonism were mathematically derived and clearly defined for the first time by Chou and Talalay; and (iii) Chou's works included the first explicit derivation of the isobologram equation. A survey of the literature does not support these claims. As noted in our initial response, Eq. 1 was provided in Loewe's article published in 1928 (ref. 4, p. 179); derivation of the isobologram equation can be dated to Fraser in the 1870s (5, 6) and was expanded upon by Loewe and Muischnek in 1926 (7) and by Loewe in 1953 (8). In his 1977 review, Berenbaum made the following statements:The use of Eq. 2 to define synergy, additivity, and antagonism can be traced back at least to the work of Berenbaum in 1977 (ref. 9, p. 6). Assessments of drug interaction based on Eq. 2 are exactly same as the assessments obtained from the isobole approach (9–11). Chou asserts that we have merely replicated the CI theory for drug combination synergy quantification with new terminology but without any actual derivation of the synergy quantification. For these general equations, we have never claimed that we derived them but rather cited the literature, including important work from Chou and Talalay (3) along with other articles (9–14). Our contribution has been to give statistical rigor to the assessment of drug interaction by constructing confidence intervals for the interaction index based on asymptotic properties (1, 15).Equation 3 in the Supplementary Material was said to raise problems for the other equations. This equation depicts the Emax model, which is well known in the literature (16–18). For example, the sigmoid Emax model $E = {\frac {E_{\rm max} C^N} {EC_{50}^N + C^N}} $ was defined by Holford and Sheiner (ref. 17, p. 434; ref. 18, p. 150) after the Hill equation (16). The Hill equation also cited by Chou (ref. 2, p. 630; ref. 19, p. 67) as ${\rm log}[v/(V_{\rm max}} - v)] = n{\rm log}(S) - {\rm log}(K)$, that is, $v = (V_{{\rm max}}S^n )/(K + S^n )$, which is essentially the sigmoid Emax model with 3 parameters. It is easy to verify, by setting Emax = 1 or Vmax = 1, that the resulting 2-parameter models share the same mathematical formula as the median effect equation presented by Chou himself and in work with his collaborator (2, 3, 19). In our article (1), we used y as the response variable for consistency with the other models in the same article. Thus, Chou's assertion that “Equation 3 was copied from the MEE 62 with incorrect denotation by arbitrarily adding the coefficient 63 ‘Emax,’ which is not justified” is incorrect.Chou stated that we added an “irrelevant coefficient ‘β’ … that has no physicochemical bearing to the analogous combination index equation of Chou-Talalay.” We sought to make valid statistical inferences for assessing drug synergy by accounting for biological variation and measurement errors. Thus, we expressed the interaction index at (d1, d2) as the function of the regression coefficients of median effect plots. From this expression, we derive the expression for the asymptotic variance of the interaction index, thus allowing us to construct the confidence interval for the estimated interaction index.Finally, Chou challenges the analysis results and concludes that a contradiction occurs when the fraction survived instead of fraction killed is used as the endpoint. We answer that the interaction index calculated at any combination dose (d1, d2) is exactly the same no matter which endpoint is used. In our initial response to his letter, we did not intend to imply that “y = S = K” but rather the interaction index as calculated on the basis of the fraction survived is the same as the interaction index as calculated on the basis of the fraction killed for any combination dose (d1, d2). When we use the fraction survived as the endpoint, the response variable y = S, whereas when we use the fraction killed as the endpoint, the response variable y = K. Toillustrating this point clearly, if one assumes that a dose–response curve based on the fraction survived is $S = (d/D_m^S )^{m_S } /[ {1 + (d/D_m^S )^{m_S}} ]$, with $D_m^S$ = 1, 2, 4 for agent 1, a combination dose with d1:d2 = 1:9, and agent 2, respectively, and mS = −2 for all dose-response curves. At the combination dose of (0.1, 0.9), we have 80% cell survival. On the basis of the dose–response curves for single agents, agent 1 alone at dose 0.5 or agent 2 alone at dose 2 also yields 80% survival. Thus, the interaction index at the combination dose (0.1, 0.9) based on the fraction survived is 0.1/0.5 + 0.9/2.0 = 0.65. Alternatively, we can derive the dose–response curve based on the fraction killed. It is clear that $K = 1 - S = 1/(1 + (d/D^S_m )^{m_S})= (d/D_m^S )^ {- m_S} /(1 + (d/D_m^S )^{- m_S})}$. By comparing this to $K = (d/D_m^K )^{m_K} /}(1 + (d/D_m^K )^{m_K})$, we have mK = −mS = 2 and $D_m^S = D_m^K$ because the dose for 50% killed is the same as the dose for 50% survived. The doses required to produce 20% killed for agent 1, the combination dose, and agent 2 are 0.5, 1, and 2.0, respectively. Hence, the interaction index at combination dose (0.1, 0.9) based on the fraction killed is 0.1/0.5 + 0.9/2.0 = 0.65, which is and should be exactly the same as that calculated on the basis of the fraction survived. As shown, the interaction index for the combination dose (d1, d2) is the same no matter whether one uses the fraction killed or the fraction survived as the endpoint as long as the endpoint is consistently defined over all drugs.In conclusion, we reject the assertion that our studies merely replicate Dr. Chou's work but with the use of different terminology. We also disagree with Dr. Chou's claim regarding his originality of the additivity reference model, which we and others have termed the Loewe additivity model. Along with the related isobologram, the Loewe additivity model represents a totally unique work in its originality and predates Dr. Chou's work. As biostatisticians, our contribution has been to provide a statistical assessment for determining drug synergy to account for the uncertainty caused by biological variation and measurement errors. In addition, we have developed response surface models for assessing drug interactions when combinations are made between 2 drugs (20, 21) and 3 drugs (22). A myopic perspective of the rich field of drug combination studies obscures the benefits that accrue from scientists working across different disciplines, which are essential for the advancement of science and medicine.See the Response, p. 2796No potential conflicts of interest were disclosed.