7994 Vol. 10, 7994 – 8004, December 1, 2004
`
`Clinical Cancer Research
`
`Evaluation of Combination Chemotherapy: Integration of Nonlinear
`Regression, Curve Shift, Isobologram, and Combination
`Index Analyses
`
`Liang Zhao, M. Guillaume Wientjes, and
`Jessie L-S. Au
`College of Pharmacy and James Cancer Hospital and Solove
`Research Institute, The Ohio State University, Columbus, Ohio
`
`ABSTRACT
`Isobologram and combination index (CI) analyses are
`the two most popular methods for evaluating drug interac-
`tions in combination cancer chemotherapy. As the com-
`monly used CI-based software program uses linear regres-
`sion, our first objective was to evaluate the effects of
`logarithmic data transformation on data analysis and con-
`clusions. Monte-Carlo simulations were conducted with ex-
`perimentally relevant parameter values to generate error-
`containing effect or concentration-effect data of single
`agents and combinations. The simulated data were then
`analyzed with linear and nonlinear regression. The results
`showed that data transformation reduced the accuracy and
`precision of the regression-derived IC50, curve shape param-
`eter and CI values. Furthermore, as neither isobologram nor
`CI analyses provide output of concentration-effect curves
`for investigator evaluation, our second objective was to de-
`velop a method and the associated computer program/algo-
`rithm to (a) normalize drug concentrations in IC50 equiva-
`lents and thereby enable simultaneous presentation of the
`curves for single agents and combinations in a single plot for
`visual inspection of potential curve shifts, (b) analyze con-
`centration-effect data with nonlinear regression, and (c) use
`the curve shift analysis simultaneously with isobologram
`and CI analyses. The applicability of this method was shown
`with experimentally obtained data for single agent doxoru-
`bicin and suramin and their combinations in cultured tumor
`cells. In summary, this method, by incorporating nonlinear
`regression and curve shift analysis, although retaining the
`attractive features of isobologram and CI analyses, reduced
`the potential errors introduced by logarithmic data trans-
`formation, enabled visual inspection of data variability and
`goodness of fit of regression analysis, and simultaneously
`
`Received 6/3/04; revised 8/31/04; accepted 9/14/04.
`Grant support: NIH Grant R01-CA100922 (M. G. Wientjes) and
`R01-CA97067 (J. L. Au.)
`The costs of publication of this article were defrayed in part by the
`payment of page charges. This article must therefore be hereby marked
`advertisement in accordance with 18 U.S.C. Section 1734 solely to
`indicate this fact.
`Requests for reprints: M. Guillaume Wientjes, College of Pharmacy,
`500 West 12th Avenue, Columbus, OH 43210. Phone: (614) 292-4244;
`Fax: (614) 688-3223; E-mail: Wientjes.1@osu.edu.
`©2004 American Association for Cancer Research.
`
`provided information on the extent of drug interaction at
`different combination ratios/concentrations and at different
`effect levels.
`
`INTRODUCTION
`Evaluation of drug-drug interaction is important in all areas
`of medicine and, in particular, in cancer chemotherapy where
`combination therapy is commonly used. The nature and quan-
`titative extent of drug interaction is usually determined in in
`vitro studies. Two recent reviews describe the various evalua-
`tion methods (1, 2). These methods fall in three categories, each
`based on a different model of drug interaction. The Bliss inde-
`pendence model assumes that the combined effect of two agents
`equals the multiplication product of the effects of individual
`agents. This assumption is valid only for linear drug concentra-
`tion-effect relationship (i.e., drug effect increases linearly with
`concentration) and not for nonlinear drug concentration-effect
`relationship such as the commonly observed sigmoidal curve.
`Hence, this model has limited applicability. The additivity en-
`velope model was developed to describe the log-linear cell
`survival relationship observed in radiation studies and, because
`this relationship is not observed for cytotoxic agents, is not
`widely used. The Loewe additivity model is based on the as-
`sumption that a drug cannot interact with itself. The model
`additionally takes into account the sigmoidal shape of the con-
`centration-effect relationship and is, therefore, more appropriate
`for evaluating drugs demonstrating such a relationship.
`Methods based on the Loewe additivity model include the
`isobologram first described in 1872 (3), the interaction index
`calculation (4), the median effect method (5), and several three-
`dimensional surface-response models (6, 7). The isobologram
`method evaluates the interaction at a chosen effect level and is
`therefore useful to inspect the drug interaction at the corre-
`sponding concentration, often the median effect concentration.
`The surface response methods are more complex in their calcu-
`lations and have not gained wide usage. The median effect
`method is the most commonly used; the original publication by
`Chou and Talalay (5) has ⬎900 citations, and the derived
`software program to calculate combination indices (CI) is
`widely used. The following provides an overview of the isobo-
`logram and CI analyses of drug interaction based on concentra-
`tion-effect data.
`The drug-induced effect, E, is described by the Hill Equa-
`tion (equation A; refs. 8, 9):
`
`E ⫽ Emax ⫻
`
`Cn
`n ⫹ Cn
`IC50
`
`,
`
`(A)
`
`where E is the measured effect; C is the drug concentration;
`Emax is the full range of drug effect, usually at or near 100%;
`IC50 is the drug concentration producing the median effect of
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`Clinical Cancer Research
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`50%; and n is the curve shape parameter describing the steep-
`ness of the concentration-effect relationship. The two key pa-
`rameters in the Hill Equation are IC50 and n.
`The isobologram analysis evaluates the nature of interac-
`tion of two drugs, i.e., drug A and drug B, as follows (10). First,
`the concentrations of drugs A and B required to produce a
`defined single-agent effect (e.g., IC50), when used as single
`agents, are placed on the x and y axes in a two-coordinate plot,
`corresponding to (CA, 0) and (0, CB), respectively. The line
`connecting these two points is the line of additivity. Second, the
`concentrations of the two drugs used in combination to provide
`the same effect, denoted as (cA, cB), are placed in the same plot.
`Synergy, additivity, or antagonism are indicated when (cA, cB)
`is located below, on, or above the line, respectively.
`CI analysis, similar to isobologram analysis, provides qual-
`itative information on the nature of drug interaction, and CI, a
`numerical value calculated as described in equation B, also
`provides a quantitative measure of the extent of drug interaction.
`
`CI ⫽
`
`CA,x
`ICx,A
`
`⫹
`
`CB,x
`ICx,B
`
`(B)
`
`CA,x and CB,x are the concentrations of drug A and drug B
`used in combination to achieve x% drug effect. ICx,A and ICx,B
`are the concentrations for single agents to achieve the same
`effect. A CI of less than, equal to, and more than 1 indicates
`synergy, additivity, and antagonism, respectively.
`In the Chou and Talalay method, the concentration-effect
`curve described by equation A is linearized by logarithmic
`transformation as shown by equation C (5):
`
`log共fu⫺1 ⫺ 1兲 ⫽ log共fa⫺1 ⫺ 1兲⫺1 ⫽ nlog共C兲 ⫺ nlog共Cm兲,
`
`(C)
`
`where fu is the fraction of cells left unaffected after drug
`exposure, fa is the fraction of cells affected by the exposure, C
`is the drug concentration used, Cm is the concentration to
`achieve the median effect, and n is the curve shape parameter.
`Cm and n are equivalent to IC50 and n, respectively, in the Hill
`Equation. The values of n (obtained from the slope), nlog(Cm)
`(obtained from the absolute value of the intercept), and, there-
`fore, Cm are obtained by plotting log(fu⫺1 ⫺ 1) versus log(C).
`The effects of logarithmic data transformation on data distribu-
`tion and analysis results are not known. However, because
`errors in low and high drug effect levels (e.g., ⬍10% or ⬎90%)
`are exaggerated because of logarithmic transformation, it is
`conceivable that data transformation affects the precision and
`accuracy of IC50, n, and CI obtained with linear regression
`analysis. In contrast, nonlinear regression analysis does not
`require data transformation and presents a theoretical advantage
`over linear regression. The first goal of the present study was to
`evaluate the effects of logarithmic data transformation on data
`analysis and conclusions.
`Although isobologram and CI analyses provide informa-
`tion on the nature and extent of drug interaction at different
`concentrations of the drugs used in combination and/or at dif-
`ferent effect levels, neither method provides the conventional,
`investigator-friendly plots of drug concentration-effect curves
`commonly used in pharmacological studies. In isobologram
`analysis, a separate plot is presented for each effect level and
`includes only the concentrations of the drugs in combination to
`
`produce the specified effect. The typical plots provided by CI
`analysis as used in the Chou and Talalay method show CI as a
`function of effect levels and do not include the corresponding
`drug concentrations either as single agents or combinations.
`Furthermore, isobologram and CI plots, because they are based
`on values (e.g., CI) calculated with the IC values derived from
`the concentration-effect curves, do not provide information on
`the variability of the actual data. Accordingly, an investigator
`would not be able to decide with confidence that the extent of
`synergy or antagonism indicated by these plots is significant
`compared with the data variability.
`On the other hand, plots of effects as a function of con-
`centrations enable an investigator to visually inspect data vari-
`ability, goodness of fit by regression analysis. Hence, the second
`goal of the present study was to develop a nonlinear regression-
`based method and the associated computer program/algorithm
`that enable curve shift analysis and capture the strengths of
`isobologram and CI analyses. An earlier version of the computer
`program had been published (11).
`
`MATERIALS AND METHODS
`Experimental Drug Concentration-Effect Data. The
`experimental data were obtained with previously described
`methodologies (12). Briefly, rat prostate MAT-LyLu tumor cells
`were cultured and treated with suramin, doxorubicin, or combi-
`nations. Drug effect was measured as inhibition of bromode-
`oxyuridine incorporation. We used the bromodeoxyuridine as-
`say because the results indicate the overall drug effects,
`including inhibition of cell growth and induction of cell death,
`and, in addition, indicate the residual replication ability. The
`latter is not provided by other cell growth assays such as
`microtetrazolium reduction or sulforhodamine assays. Further-
`more, we found similar results with these three assays in doxo-
`rubicin-treated rat prostate MAT-LyLu tumor cells, whereas the
`bromodeoxyuridine results yielded the lowest data variability
`and greatest data reproducibility.
`The rationale for using suramin was to enhance the tumor
`sensitivity to doxorubicin based on our earlier observations
`(12–14). This study used the fixed ratio method, where the
`doxorubicin and suramin concentrations were present in fixed
`ratios of concentrations corresponding to the IC50 equivalents of
`single agents. The stock solutions contained 0, 160, 320, 640,
`and 1280 ␮mol/L suramin combined with 10,000 nmol/L doxo-
`rubicin, representing approximate suramin-to-doxorubicin IC50-
`equivalent ratios of 0, 1:400, 1:200, 1:100, and 1:50, respec-
`tively (referred to as S1D400 and so on). Cells were treated with
`serial dilutions (10- to 100,000-fold diluted) of the stock solu-
`tions. Controls were processed similarly but without drugs. The
`concentrations of single-agent suramin treatment were 0, 10, 50,
`100, 500, and 1000 ␮mol/L. The results were analyzed with
`linear and nonlinear regressions to obtain the corresponding
`IC50 and n (see below).
`General Strategy for Simulations. We examined the
`effects of data transformation on regression-derived IC50 and n
`values, sensitivity of these parameters to data variability at low
`and high effect levels (i.e., ⬍10% and ⬎90%), and the calcu-
`lated CI values. These studies were done with computer simu-
`lations. The parameters used to generate simulated data were
`
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`7996 Computer-assisted Analysis of Drug-Drug Interaction
`
`the analysis results with true values indicated the precision and
`accuracy of the two regression methods. Note that simulation of
`a drug concentration-effect curve requires only IC50 and n
`values.
`Effect of Logarithmic Data Transformation on Accu-
`racy and Precision of IC50 and n Values Obtained from
`Regression Analyses. Fig. 1 outlines the procedures. For this
`study, the concentration-response curves were generated with
`arbitrarily chosen IC50 and n values. Monte-Carlo simulations
`were used to generate variability or error-containing concentra-
`tion-effect curves for single agents according to equations A and
`B, with equation D:
`
`simulated effect ⫽ preselected effect ⫹ ␴
`
`(D)
`
`where ␴ is the normally distributed error with a mean value of
`0. The SD for ␴ ranged from 0.1 to 5%. The simulations used 10
`concentrations, which cover the conventional six to eight con-
`centrations used in concentration-response experiments (typi-
`cally performed in 96-well plates).
`Effect of Logarithmic Data Transformation on Accu-
`racy and Precision of Calculated Combination Indices.
`Fig. 2A outlines the procedures. In contrast to the study on IC50
`and n determination for single agents, which was accomplished
`with arbitrarily chosen values, the determination of CI required
`using experimentally relevant concentration-response data. For
`this purpose, we used parameter values, including IC50, n, and
`CI values, which were based on the experimental results ob-
`
`Fig. 1 Outline of Monte-Carlo simulations to study the effects of
`logarithmic data transformation on the precision and accuracy of regres-
`sion-derived IC50 and n values.
`
`selected based on or derived from experimental data, where
`appropriate. The general simulation strategy was to first select
`appropriate values for the parameters (i.e., IC50, n, effect data
`variability expressed as ␴, and CI). These values, referred to as
`true values, were then used together with simulations to generate
`sets of concentration-effect curves, which were subsequently
`analyzed with linear or nonlinear regression. A comparison of
`
`Fig. 2 Outline of Monte Carlo simulations to evaluate effects of
`logarithmic data transformation on precision and accuracy of the
`calculated CI values. A, simulated data without change. B, simu-
`lated data with artificially adding or subtracting 1 SD at ⬍10% or
`⬎90% effect level.
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`Clinical Cancer Research
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`tained for the doxorubicin and suramin study described above.
`The SD values were varied according to the drug effect levels,
`as observed experimentally.
`Effect of Logarithmic Data Transformation on Sensi-
`tivity of Regression-derived IC50 and n Values to Data Var-
`iability.
`In linear regression analysis, IC50 and n values are
`calculated based on log(fa⫺1 ⫺ 1)⫺1 (equation 3). Because of the
`logarithmic transformation, the errors in fa are especially magnified
`at low or high fa levels or the asymptotic regions of the sigmoidal
`concentration-effect curve (e.g., ⬍10% and ⬎90%). We therefore
`evaluated the effects of data transformation under these conditions.
`For a two-drug combination, there are four potential permutations
`to study the effects of changes in these data points, i.e., data
`variability at low and high effect levels for each of the two drugs.
`We evaluated the effect of variability in the data for drug A at low
`(⬍10%) and high (⬎90%) effect levels. Note that similar studies
`can be done by introducing variability in the data for drug B at low
`and/or high effect levels. These analyses are not presented here
`because of space limitation.
`For this purpose, the IC50 and n values obtained from the
`experimental data for single-agent doxorubicin (70 nmol/L and 0.6,
`respectively) were used to simulate a concentration-effect curve
`and thereby identify the effects at 1 and 10,000 nmol/L doxorubicin
`concentration (equaling 7.2 and 95.5%, respectively). For compar-
`ison and to show the substantial effects of data variation at these
`high and low drug effect levels, we also calculated the concentra-
`tion where the effect is near the median value, i.e., 100 nmol/L
`producing 55.5% effect. These effect values were then altered to
`include an error of up to one experimentally observed SD, which
`was 4% at ⬍10% effect level, 3% at 55% effect level, and 1% at
`⬎90% effect level, and thereby generated effect levels between 3.2
`and 11.2% at 1 nmol/L, between 52.3 and 58.3% at 100 nmol/L,
`
`and between 94.5 and 96.5% at 10,000 nmol/L. These error-
`containing effects levels and the corresponding concentrations were
`substituted into the original data set, and the resulting concentra-
`tion-effect curves for single agents and combinations were ana-
`lyzed with linear and nonlinear regressions to obtain IC50 and n
`values.
`Effect of Logarithmic Data Transformation on Sensi-
`tivity of CI Values to Data Variability. Fig. 2B outlines the
`procedures. Note that the methods are nearly identical to those
`outlined for the study of effects of logarithmic data transforma-
`tion on the accuracy and precision of the calculated CI (Fig. 2A),
`with the exception of adding and subtracting from the effect
`(5%) a value (4%) equal to one experimentally observed SD.
`Also note that subtracting ⬎5% SD value will result in negative
`drug effect, and it will not be possible to obtain the transformed
`effect by log(fa⫺1 ⫺ 1)⫺1.
`Development of Curve Shift Analysis and Its Incorpo-
`ration with Isobologram and CI Analyses to Analyze Drug-
`Drug Interaction. We developed a curve shift method, in
`conjunction with isobologram and CI analyses, to analyze drug
`interaction. A computer program, written in SAS language and
`published elsewhere (6), was implemented to capture the
`strengths of all three analyses. The algorithm is outlined in Fig.
`3. The applicability of this new method was shown with, as an
`example, experimentally obtained results of the doxorubicin/
`suramin combination study. Furthermore,
`the results of the
`studies outlined above indicated that logarithmic data transfor-
`mation compromised data distribution and analysis, thereby
`introducing errors in regression-derived IC50, n, and CI values,
`whereas these problems were avoided by using nonlinear re-
`gression analysis. Hence, we elected nonlinear regression for
`subsequent studies and method development.
`
`Fig. 3 Algorithm for develop-
`ing a nonlinear regression-based
`method to integrate curve shift,
`isobologram, and CI analyses of
`drug interaction data.
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`7998 Computer-assisted Analysis of Drug-Drug Interaction
`
`typically provides the
`A drug interaction experiment
`concentration-effect data for single agents and their combina-
`tions. Because of the differences in the effective concentrations
`for the different treatments (e.g., lower drug concentrations for
`combinations as compared with single agents), multiple plots of
`concentration-effect curves would be required if the x axis is in
`absolute drug concentration terms (e.g., ng/mL). This limitation
`was overcome by normalizing the concentrations of drugs in
`combinations to their respective single-agent IC50; drug concen-
`trations were converted to fractions or multiples of the IC50
`equivalents. Equation E states the IC50-equivalent concentration
`of drug A or drug B, used alone or in combination with each
`other, required to produce x% effect. Note that for a single
`agent, one of the two terms (CA,x or CB,x) on the right side of the
`equation becomes 0.
`
`IC-equivalent concentration ⫽
`
`CA,x
`IC50,A
`
`⫹
`
`CB,x
`IC50,B
`
`(E)
`
`Substituting equation E into equation A yields equation F,
`which describes the effects of combination therapy as a function
`
`of IC50-equivalent concentrations. IC50,combo and ncombo are the
`values for the combination therapy.
`
`Combination therapy effect
`
`冊
`
`n
`
`combo
`
`⫹
`
`CB,x
`IC50,B
`
`⫹ 共IC50,combo兲combo
`
`n
`
`(F)
`
`Emax冉 CA,x
`冊
`
`⫽
`
`冉 CA,x
`
`IC50,A
`
`IC50,A
`n
`CB,x
`IC50,B
`
`⫹
`
`combo
`
`Plotting the effects of single agents and combinations
`against IC50-equivalent drug concentrations enabled the simul-
`taneous presentation of these concentration-effect curves in a
`single plot.
`Computer Software Packages and Procedures. All
`programming codes, graphical representations and calcula-
`tions used SAS language and procedures (SAS, Cary, NC).
`Linear and nonlinear regressions were done with the SAS/
`STAT Proc REG routine and the SAS/STAT Proc NLIN
`routine with the Marquardt iteration method, respectively.
`
`Fig. 4 Concentration-effect curves
`of single-agent doxorubicin and sur-
`amin and their combinations in rat
`prostate tumor cells. Effects of sin-
`gle agent doxorubicin/suramin and
`their combinations were measured.
`The four combinations of suramin
`and doxorubicin, S1D400, S1D200,
`S1D100, and S1D50, correspond to
`the suramin-to-doxorubicin concen-
`tration ratios. The experimental data
`were fitted with equation 1 or 5.
`Solid line, fitting results with non-
`linear regression. Dotted line, fitting
`results with linear regression. Note
`the inverse relationship between
`survival and drug effect, i.e., 10%
`survival is equivalent to 90% effect
`level.
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`The output of the drug-drug interaction results obtained from
`the 96-well microplate reader was used as input to simulta-
`neously perform curve shift, isobologram, and CI analyses, as
`outlined in Fig. 3.
`A pilot study evaluated whether the quality of data fitting
`could be improved by weighing the data with 1/concentration as
`the weight. The results showed no significant improvement.
`Accordingly, simple regression was used.
`
`RESULTS
`Experimental Data. Fig. 4 shows the concentration-
`effect curves for single-agent doxorubicin and suramin and their
`combinations, analyzed with linear and nonlinear regressions. In
`all cases, nonlinear regression provided better-fitted curves, as
`indicated by the higher R2 values (range, 0.9959 to 0.9998;
`mean, 0.9977), compared with linear regression (range, 0.8892
`to 0.9883; mean, 0.9499). Similar results were obtained with
`Akaike Information Criterion analysis (data not shown). Single-
`agent suramin and doxorubicin showed different curve shapes,
`with a steeper curve for suramin.
`The nonlinear regression analysis yielded, for single-agent
`doxorubicin and suramin, IC50 of 70 nmol/L and 450 ␮mol/L
`and n values of 0.5 and 1.6, respectively. The CI values were
`between 0.1 and 0.9.
`Effect of Logarithmic Data Transformation on Accu-
`racy and Precision of Regression Results: IC50 and n Values
`for a Single Agent. Table 1 summarizes the results. The
`nonlinear regression method overestimated the IC50 and n val-
`ues in 16 of 20 cases, whereas the linear regression underesti-
`mated these values in 15 of 20 cases. There are also substantial
`qualitative and quantitative differences between the linear and
`nonlinear regression results as follows.
`With respect to accuracy, the nonlinear regression results
`showed relatively small deviations from true values (i.e., aver-
`age deviation was 3.1% for IC50 and 2.2% for n). In comparison,
`the linear regression results were of poorer quality, with 2.5- to
`12-fold greater deviations (i.e., average deviation of 9.0% for
`
`IC50 and 26.2% for n). The accuracy was also affected, differ-
`ently for the two methods, by the steepness of the concentration-
`effect curves (i.e., n values of 0.5 to 2.0 to reflect shallow and
`steep curves, respectively) and data variability (i.e., SD). Devi-
`ations of the regression results from true values were random
`and not dependent on the curve shape for nonlinear regression
`results (e.g., compare the deviations at the four n values with
`identical SD of either 2 or 5%). In contrast, in linear regression,
`the deviations in the n values increased with the curve steepness
`reaching as high as 55%, whereas the deviations in IC50, al-
`though they also increased with the curve steepness, reached the
`maximum value of ⬃14% at n of 2.0. For both regression
`methods, the deviations for IC50 and n were higher at higher
`SDs. At 2 versus 5% SD, the respective average deviations were
`0.6 versus 5.7% and 0.6 versus 3.9% for the nonlinear regres-
`sion results and ⫺8.0 versus ⫺10.0 and ⫺22.0 versus ⫺30.5%
`for the linear regression results.
`Precision of the regression results is indicated by the co-
`efficient of variation (CV) of the mean values. The average CV
`was comparable for both regression methods, at ⬃8% (Table 2),
`but greater CV values were observed for the nonlinear regres-
`sion results for the steeper curves at n value of ⱖ1.5.
`Effect of Logarithmic Data Transformation on Accu-
`racy and Precision of Regression Results: CI for Combina-
`tion Therapy. Table 2A summarizes the results. With respect
`to accuracy, both regression methods yielded similar deviations
`(⬍10% deviations for both methods), although there was a trend
`of higher deviations at the lower effect levels for the linear
`regression results. With respect to precision, the linear regres-
`sion results showed significantly higher CV values compared
`with the nonlinear regression results (mean SD of 15.8 versus
`8.9%).
`Effect of Logarithmic Data Transformation on Sensi-
`tivity of Regression Results to Data Variability: IC50 and n
`Values for a Single Agent. The results are shown in Table 3.
`Both linear and nonlinear regression results showed higher
`deviations from true values (i.e., 70 nmol/L for IC50 and 0.6 for
`
`Table 1 Effects of logarithmic data transformation on accuracy and precision of IC50 and n values for a single agent
`Regression results
`
`True values
`
`Nonlinear
`
`Linear
`
`Nonlinear
`
`Linear
`
`IC50
`
`N
`
`SD in
`effect, %
`
`Mean ⫾ SD
`(CV, %)
`
`Dev,
`%
`
`Mean ⫾ SD
`(CV, %)
`
`Dev,
`%
`
`Mean ⫾ SD
`(CV, %)
`
`Dev,
`%
`
`Mean ⫾ SD
`(CV, %)
`
`DEV, %
`
`Experiment
`no.
`IC50 N
`Representative values
`0.0 0.51 ⫾ 0.02 (4.0) ⫹2.0
`10.07 ⫾ 0.57 (5.7) ⫹0.7 10.12 ⫾ 0.57 (5.7) ⫹1.2 0.50 ⫾ 0.01 (2.0)
`1
`10
`10.84 ⫾ 1.39 (13.9) ⫹8.4
`9.71 ⫾ 1.14 (11.4) ⫺2.9 0.51 ⫾ 0.03 (6.0) ⫹2.0 0.50 ⫾ 0.04 (4.0)
`0.0
`2
`10
`10.25 ⫾ 0.23 (2.3) ⫹2.5
`9.23 ⫾ 0.83 (8.3) ⫺7.7 1.01 ⫾ 0.02 (2.0) ⫹1.0 0.86 ⫾ 0.07 (7.0) ⫺14.0
`3
`10
`10.74 ⫾ 0.61 (6.1) ⫹7.4
`9.01 ⫾ 0.25 (2.5) ⫺9.9 1.03 ⫾ 0.06 (6.0) ⫹3.0 0.75 ⫾ 0.08 (8.0) ⫺25.0
`4
`10
`10.17 ⫾ 0.17 (1.7) ⫹1.7
`8.87 ⫾ 0.98 (9.8) ⫺11.3 1.52 ⫾ 0.04 (2.7) ⫹1.3 1.03 ⫾ 0.11 (7.3) ⫺31.3
`5
`10
`10.44 ⫾ 1.14 (11.4) ⫹4.4
`8.64 ⫾ 1.41 (14.1) ⫺13.6 1.59 ⫾ 0.50 (33.3) ⫹6.0 0.87 ⫾ 0.11 (7.3) ⫺42.0
`6
`10
`9.76 ⫾ 1.91 (19.1) ⫺2.4
`8.55 ⫾ 1.23 (12.3) ⫺14.5 2.00 ⫾ 0.38 (19.0)
`0.0 1.11 ⫾ 0.13 (6.5) ⫺44.5
`7
`10
`10.24 ⫾ 1.51 (15.1) ⫹2.4
`8.66 ⫾ 1.98 (19.8) ⫺13.4 2.09 ⫾ 0.65 (32.5) ⫹4.5 0.90 ⫾ 0.12 (6.0) ⫺55.0
`8
`10
`⫹3.1
`⫺9.0
`⫹2.2
`⫺26.2
`(9.4)
`(10.5)
`(12.9)
`(6.3)
`Average SD or Dev
`NOTE. Simulated data were analyzed using linear and nonlinear regressions to obtain IC50 and n values, which were compared with true values
`used to generate the simulations.
`Abbreviations: Dev, deviation from true values. ⫹, overestimation; ⫺, underestimation; CV, coefficient of variation.
`
`0.5
`0.5
`1
`1
`1.5
`1.5
`2
`2
`
`2
`5
`2
`5
`2
`5
`2
`5
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`Table 2 Effect of logarithmic data transformation on accuracy, precision, and sensitivity of CI values to data variability
`
`NOTE. Simulated concentration-effect data were generated with IC50 and n values of 70 nmol/L and 0.6 for drug A (doxorubicin), values of 450
`␮mol/L and 1.4 for drug B (suramin), and analyzed as described in Materials and Methods. S1D50, S1D100, S1D200, and S1D400 refer to the
`combinations containing suramin-to-doxorubicin ratios (in IC50 units) of 1:50, 1:100, 1:200, and 1:400, respectively. Results for 20, 30, 70, and 80%
`effect levels are in line with the results at the other effect levels and are not shown because of space limitation. For simulated data, Dev represents
`deviation from true CI values. The effect of data transformation on the accuracy and precision of regression-derived CI values as a function of data
`variability at 5% effect level was studied with simulated data generated with error-free effect data (0%, top panel) or error-containing data (generated
`by increasing or decreasing the effect of drug A by 1 SD or 4%).
`
`Table 3 Effect of data transformation on sensitivity of IC50 and n values of a single agent to data variability
`IC50, nmol/L
`N
`
`Concentration,
`nmol/L
`1
`
`100
`
`10,000
`
`Nonlinear regression
`
`Linear regression
`
`Nonlinear regression
`
`Linear regression
`
`Mean
`effect, %
`(no. of SD)
`Dev, %
`Estimate
`Dev, %
`Estimate
`Dev, %
`Estimate
`Dev, %
`Estimate
`⫹13.42
`⫹4.90
`⫹36.98
`⫹3.69
`0.68
`0.63
`95.88
`72.58
`3.2 (1)
`⫹5.72
`⫹2.78
`⫹15.48
`⫹2.03
`0.63
`0.62
`80.84
`71.42
`5.2 (0.5)
`0.00
`0.60
`0.00
`0.60
`0.00
`70.00
`0.00
`70.00
`7.2 (0)
`⫺6.63
`⫺4.52
`⫺17.22
`⫺3.85
`0.56
`0.57
`57.94
`67.31
`9.2 (0.5)
`⫺10.62
`⫺8.23
`⫺27.10
`⫺7.37
`0.54
`0.55
`51.03
`64.84
`11.2 (1)
`⫺2.87
`⫺0.82
`⫹4.11
`⫹10.84
`0.58
`0.60
`72.88
`77.59
`52.3 (1)
`⫺0.47
`⫹2.03
`⫹5.27
`0.00
`0.60
`0.60
`71.42
`73.69
`53.8 (0.5)
`0.00
`0.60
`0.00
`0.60
`0.00
`70.00
`0.00
`70.00
`55.3 (0)
`⫹0.58
`⫺2.01
`⫺5.00
`0.00
`0.60
`0.60
`68.59
`66.50
`56.8 (0.5)
`⫹1.30
`⫺4.00
`⫺9.74
`0.00
`0.60
`0.61
`67.20
`63.18
`58.3 (1)
`⫺2.87
`⫺0.95
`⫹5.91
`⫹0.64
`0.58
`0.59
`74.14
`70.45
`94.2 (1)
`⫺1.50
`⫺0.47
`⫹3.00
`⫹0.32
`0.59
`0.60
`72.10
`70.22
`94.7 (0.5)
`0.00
`0.60
`0.00
`0.60
`0.00
`70.00
`0.00
`70.00
`95.2 (0)
`⫹1.65
`⫹0.47
`⫺3.11
`⫺0.31
`0.61
`0.60
`67.82
`69.78
`95.7 (0.5)
`⫹3.50
`⫹0.93
`⫺6.36
`⫺0.61
`0.62
`0.61
`65.55
`69.57
`96.2 (1)
`NOTE. A concentration-effect curve for doxorubicin was generated with true values of 70 nmol/L for IC50 and 0.6 for n (see Materials and
`Methods) and showed average effect of 7.2% at 1 nmol/L, 55.5% at 100 nmol/L, and 95.5% at 10,000 nmol/L. With variability of up to 1 SD (1 SD
`was 4% at 1 nmol/L, 3% at 100 nmol/L, and 1% at 10,000 nmol/L; see Fig. 6), the range of effect values was 3.2 to 11.2% for 1 nmol/L, 52.3 to
`58.3% for 100 nmol/L, and 94.2 to 96.2% for 10,000 nmol/L. These values were substituted into the original concentration-effect curves, which were
`then analyzed with linear and nonlinear regression. The regression-derived IC50 and n values (estimates) were compared with true values of 70 nmol/L
`and 0.6, to calculate the deviations (Dev).
`Abbreviations: ⫹, overestimation; ⫺, underestimation.
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`n) at higher SDs (e.g., compare the deviations at 0.5⫻ and 1⫻
`the SD). The average deviations for the nonlinear regression
`results at low and high effect levels (i.e.,7.2 and 95.2%) re-
`mained negligible at an average of 3.4% for IC50 and 4.1% for
`n. In contrast, the deviations for the linear regression results
`were ⬃5.7-fold higher at 19.4% for IC50 and 1.8-fold higher at
`7.3% for n. In comparison, the parameter values derived from
`linear regression were less sensitive to the change at the mid
`effect level of 55% (average deviations of 2.4% for IC50 and
`0.57% for n).
`Effect of Logarithmic Data Transformation on Sensi-
`tivity of Combination Indices to Data Variability. Fig. 5B
`and Table 2B show the results when the 5% effect level was
`reduced by 4% (1 SD), and Fig. 5C and Table 2C shows the
`results when the 5% effect was increased by 4%. A compar-
`ison of these results to the results generated with the perfect
`data (i.e., 0% variation; Fig. 5A and Table 2A) showed two
`
`major differences. First, for perfect data, the two regression
`methods yielded regression-derived CI values that were
`closely aligned with true CI values and with similar accuracy.
`On the other hand, data variation at the 5% effect level in
`either direction yielded larger deviations between the regres-
`sion results and true CI values for either regression methods.
`Second, the linear regression results showed substantially
`lower accuracy and precision compared with the nonlinear
`regression results. These results are additionally described
`below.
`With respect to accuracy, irrespective of true CI values and
`regression methods, a 4% reduction at the 5% effect level led to
`overestimated CI values at 10 to 70% effect levels and slight
`underestimation at 90% effect level. In general, the magnitude
`of overestimation increased with decreasing effect level and was
`⬃3-fold greater for linear regression. On the contrary, a 4%
`increase in the 5% effect level resulted in underestimated CI
`
`Fig. 5 Effect of logarithmic data trans-
`formation on regression-derived CI
`values. The CI values were obtained as
`outlined in Fig. 2. True CI values were
`0.2, 0.5, or 0.8 (indicated by dotted
`lines). The four combinations of sura-
`min and doxorubicin, S1D400 (E),
`S1D200 (), S1D