(CANCER RESEARCH 50. 5318-5327. September 1. 1990]
`
`Application of a New Approach for the Quantitation of Drug Synergism
`to the Combination of m-Diamminedichloroplatinum and
`l -ß-D-ArabinofuranosyIcytosine '
`
`William R. Greco,2 Hyoung Sook Park, and Youcef M. Rust um
`
`Grace Cancer Drug Center [ W. R. G., H. S. P., Y. M. R.J and Department of Biomathematks
`of Health. Buffalo. New York 14263
`
`[W. R. G.J, Roswell Park Memorial
`
`Institute, New York State Department
`
`ABSTRACT
`
`the universal
`This report describes the application of a new approach,
`response surface approach,
`to the quantitative assessment of drug inter
`action, i.e., the determination of synergism, antagonism, additivity, po-
`tentiation,
`inhibition, and coalitive action. The specific drug combination
`and experimental growth system for this introductory application was
`that of I-/3-r>arabinofuranosylcytosine (ara-C) and cisplatin with simul
`taneous drug exposure (1, 3, 6, 12, or 48 h) against I.I21(1 leukemia in
`vitro. To quantitate the type and degree of drug interaction, a model was
`fitted using nonlinear regression to the data from each separate experi
`ment, and parameters were estimated (K. C. Syracuse and W. R. Greco,
`Proc. Biopharm. Sect. Am. Stat. Assoc., 127-132,1986). The parameters
`included the maximum cell density over background in absence of drug,
`the background cell density in presence of infinite drug, the 50% inhibi
`tory concentrations and concentration-effect slopes for each drug, and a
`synergism-antagonism parameter, a. A positive a indicates synergism, a
`negative a, antagonism, and a zero a, additivity. Maximal synergy was
`found with a 3-h exposure of ara-C + cisplatin, with a = 3.08 ±0.96
`(SE) and 2.44 ±0.70 in two separate experiments. Four different graphic
`representations of the raw data and fitted curves provide visual indications
`of goodness of fit of the estimated dose-response surface to the data and
`visual
`indications of the intensity of drug interaction. The universal
`response surface approach is mathematically consistent with the tradi
`tional isobologram approach but is more objective, is more quantitative,
`and is more easily automated. Although specifically developed for in
`vitro cancer chemotherapy applications,
`the universal response surface
`approach should prove to be useful
`in the fields of pharmacology,
`toxicology, epidemiology, and biomedicai science in general.
`
`INTRODUCTION
`
`of a new approach,
`the application
`report describes
`This
`URSA,3 to the quantitative
`assessment of drug interaction,
`i.e,
`the determination
`of synergism, antagonism,
`additivity, poten-
`tiation,
`inhibition, and coalitive action. The specific drug com
`bination-experimental
`system for this introductory application
`was that of ara-C and DDP against L1210 leukemia in vitro.
`This report describes in detail
`the analysis of data from one 48-
`h growth experiment which began with a 3-h incubation of
`
`revised 2/28/90.
`Received 4/14/89;
`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.
`'Supported
`by National Cancer
`Institute Grants CA18420. CA50456. and
`CA46732.
`2To whom requests for reprints should be addressed.
`' The abbreviations used are: URSA,
`universal response surface approach;
`cisplatin or DDP. m-diamminedichloroplatinum;
`ara-C.
`l-f)-D-arabinofuranosyl-
`cytosine; 2-D. two-dimensional; 3-D, three-dimensional: PC. personal computer.
`Mathematical
`symbols used are: E. measured cell density; [DRUG], [ara-C], and
`(DDP], drug, ara-C, and cisplatin concentration respectively: £m„.maximum cell
`density over background at zero drug concentration:
`/(. background cell density
`at infinite drug concentration; Dm, IDJO,or IC50. median effective concentration
`
`of drug or concentration of drug which inhibits growth (£„,„)by 50%; D„/O|0,
`
`and ¡On,concentration of drug which inhibits growth (£„,„)by -V%.by 10%, and
`by 90%, respectively; D„.„_cand O„.DDP.median effective concentration of ara-
`C and cisplatin; DXmJ:. DX.DOF.concentration
`of ara-C. DDP which inhibits
`growth (£„„)by X"i: m. mmf,
`and mDor. slope parameters
`for concentration-
`effect curves: «.synergism-antagonism parameter; y. interaction parameter
`the case in which one drug has no effect as a single agent;^,
`fraction affected.
`
`for
`
`In
`of ara-C and DDP.
`LI210 cells with various combinations
`addition,
`this report briefly describes the results from 11 addi
`tional separate growth experiments, with incubation times rang
`ing from 1 to 48 h and explores the dynamics (time course) of
`the synergistic interaction between DDP and ara-C. Since the
`emphasis of this report
`is on the new data analysis approach,
`discussion of the biomédicalimplications of the results is kept
`to a minimum. The biological and clinical
`implications of this
`study are only briefly described under "Discussion."
`The situation which gave impetus to the development of this
`new approach for assessing drug interactions
`included:
`(a) the
`determination
`of synergism, antagonism, and additivity among
`drugs is widespread and important
`in biomedicine (over 20,000
`articles
`in the biomedicai
`literature
`from 1981 to 1988 used
`"synergism" as a key word, and of these, over 2400 were cancer
`related);
`(¿>)there is widespread disagreement
`over concepts
`and terminology;
`(c) there exist many different approaches
`for
`assessing drug interactions which will result
`in different conclu
`sions for the same data set; (d) most older approaches
`include
`limitations;
`(e) conclusions
`regarding
`drug interactions
`are
`often suboptimally used.
`Put more succinctly, quantitative drug interaction assessment
`is widely done, differently done, often poorly done, and yet
`important.
`URSA was developed by adapting and combining elements
`from many well-established approaches
`for assessing drug in
`teractions. The fundamental concept
`from Loewe (1) of isobol-
`ograms underpins
`the whole approach. Many of the equations
`and symbols were adapted from those of Chou and Talalay (2,
`3). The guidelines for the derivation of drug interaction models
`were adapted from Berenbaum (4) but also share features of
`earlier work from Hewlett and Plackett
`(5, 6) and Finney (7).
`The statistical
`concept of generalized
`linear
`(or nonlinear)
`models by McCullagh and Neider
`(8) provides URSA with its
`universal nature. Finally,
`the use of response surface techniques
`in assessing drug interactions
`has gained wide acceptance be
`cause of the work by Carter's group (9).
`The general approach used in the present study to quantitate
`synergism has been reported previously (10-13),
`and the spe
`cific application to the data presented in this paper has been
`reported previously in an abstract
`(14). However,
`this is the
`first report of the application of URSA to real laboratory data
`published in a peer-reviewed journal. A brief description of the
`advantages of this method over traditional methods is included
`in the "Discussion";
`a more extensive
`comparative
`critical
`review is in preparation. Mathematical
`derivations
`of drug
`interaction models are provided in Appendix 1. More extensive
`mathematical
`descriptions
`and statistical
`characterizations
`of
`the drug interaction models are included in a paper
`in the
`statistical
`literature
`(11). A brief description of the statistical
`approach,
`the mathematical models, and the derivation of the
`models is included in "Materials
`and Methods."
`
`5318
`
`1 of 10
`
`Alkermes, Ex. 1044
`
`

`
`QUANTITATION OF DRUG SYNERGISM
`
`1
`
`[ara-C]
`E - B
`
`\'
`
`[DDP]
`E- B
`—E + B
`Ã(cid:173)-
`'
`",
`
`7
`-max
`
`I/»»DDP
`
`MATERIALS AND METHODS
`Chemicals. Cisplatin was obtained from the Bristol-Myers Company
`(Syracuse, NY). ara-C was supplied by Sigma Chemical Company (St.
`Louis, MO). The purity of the compounds was determined by high
`pressure liquid chromatography
`to be 98%. Drugs were dissolved and
`serial dilutions made in RPMI 1640 plus 20 mM 4-(2-hydroxyethyl)-l-
`piperazineethanesulfonic
`acid (Gibco, Buffalo, NY). Drug solutions
`were sterilized by passage through a 0.2-nm Acrodisc from Gelman
`Sciences, Inc. (Ann Arbor, MI).
`Exposure of Cells. Murine leukemia LI210 cells were grown in
`suspension culture in RPMI 1640 supplemented with 10% heat-inac
`tivated fetal bovine serum and 20 mM 4-(2-hydroxyethyl)-l-piperazine-
`ethanesulfonic
`acid, pH 7.3, at 37°C.To assess drug effects on cell
`growth, L1210 cells in log phase with a doubling time of about 12 h
`were utilized. L1210 cells at a final concentration of 50,000 cells/ml
`in
`4 ml of the above medium supplemented with 10% heat-inactivated
`fetal bovine serum (complete medium) were exposed to six logarith
`mically spaced concentrations
`of ara-C and cisplatin centered at the
`predicted IDM,for each drug in a 62 factorial design, for a total of 36
`different combinations. From 4 to 8 control
`(0 ¿IMara-C plus O n\ì
`DDP)
`tubes and 2 tubes of each of the other 35 drug combinations
`were used. Tubes were stoppered,
`randomly placed in racks, and incu
`bated in an upright position at 37"C during the drug exposure (1-48 h)
`and the subsequent drug-free growth period. Drug exposure was fol
`lowed by two washes with sterile 0.9% NaCl solution and final resus-
`pension in complete medium (free of drug). Growth was determined by
`counting the number of cells in tubes with a Coulter electronic cell
`counter, Model ZBI (Coulter Electronics,
`Inc., Hialeah, FL) 48 h after
`the start of drug exposure.
`Data Analysis. Equation 1 was fitted to the complete data set from
`an experiment
`(74-78 measurements) with unweighted least squares
`nonlinear regression, and parameters were estimated (10-14). Equation
`1 contains
`the respective drug concentrations
`[ara-C] and [DDP] as
`inputs; and the measured cell density, /;. as the output. The 7 estimable
`
`parameters include: £„,„,the maximum cell density, over background,
`at 0 drug concentration; B, the extrapolated background cell density in
`the presence of an infinite drug concentration;
`the respective II )..„sor
`median effective concentrations, £>™,.r.-cand Dm.oar; the respective
`concentration-effect
`slopes, mm.c and mDDP;and the synergism-antag-
`onism parameter, a. When a is positive, synergism is indicated; when
`a is negative, antagonism is indicated; and when a is 0, no interaction
`or additivity is indicated.
`
`for both ara-C and DDP. The form of
`later shown) to be appropriate
`Equation 2 (without B) and with fa (fraction affected) = E/Em,„is
`simply a rearrangement
`of the median-effect equation of Chou and
`Talalay (2, 3, 15) or of the Hill model (16-19). Also,
`the terms and
`symbols Dmand m are from Ref. 2.
`Equation 1 was fitted to data using custom software called SYNFIT,
`which was written in the computer
`language, MicroSoft C (Microsoft
`Corp., Bellevue, WA). SYNFIT uses a version of the Marquardt algo
`rithm (20) for nonlinear
`regression as described by Nash (21). The
`output of the program includes parameter estimates, asymptotic stand
`ard errors, 95% confidence limits for the parameters,
`and residual
`analyses. All comparisons
`for statistical
`significance were performed
`with a type I error rate of 0.05. Since Equation 1 is not in closed form,
`a one-dimensional
`bisection root
`finder
`(e.g., Ref. 22) was used to
`calculate predicted values of E. Initial parameter estimates for the D„
`and m parameters for the nonlinear
`regression were obtained by fitting
`the median-effect equation of Chou and Talalay (2) to the single drug
`data with weighted linear regression. The SAS/PC software package,
`Version 6 (23), was used to generate the 3-D graph of Fig. 1. The
`graphs in Figs. 2-4 were made by simulating data from Equation 1,
`using the estimated parameters
`from the best fit of Equation 1 to the
`observed data, with custom FORTRAN programs,
`and plotting the
`simulated data by hand. All software was run on IBM PC/AT.
`IBM
`PC/XT,
`and Leading Edge Model M microcomputers.
`Inquiries re
`garding distribution of the custom software package, SYNFIT, should
`be addressed to W. R. Greco.
`URSA could be implemented with many commercial statistical soft
`ware packages. We have used URSA with the mainframe version of
`BMDP (24), the PC version of SAS (23), and PCNONLIN (25). To be
`suitable for implementation of URSA, a package must include a non
`linear regression procedure which does not require the coding of ana
`lytical derivatives and, in order to code the root finder, does allow the
`function definition to include IF statements, and either GOTO state
`ments and/or
`loops. An example of a set of model-definition statements
`for fitting Equation 1 to data is listed in Appendix 2. This example
`consists of control
`language for SAS and will be appropriate
`for SAS
`running on microcomputers, minicomputers,
`and mainframe
`com
`puters. By adapting this example,
`interested researchers should be able
`to implement URSA into many other commercial statistical packages
`which include a nonlinear
`regression module.
`We are implementing URSA into custom user-friendly software in
`C language for microcomputers
`running the MicroSoft Disk Operating
`System,
`instead of emphasizing the use of commercial software pack
`ages for several reasons:
`(a) many researchers may be unacquainted
`with the use of general nonlinear
`regression software; (b) researchers
`may be hesitant
`to spend several hundred dollars to purchase general
`statistical software for implementing a new approach to data analysis;
`(c) the most current concepts and approaches can be implemented into
`a custom software package, whereas commercial
`statistical packages
`may present
`limitations and restrictions;
`(</) custom software can be
`custom engineered for a specific use and audience, whereas general
`packages must accommodate wide areas of application.
`General Description of URSA. An unambiguous,
`logical, and func
`tional definition for the term, "synergism,"
`is crucial before progress
`can be made in its assessment and good use can be made of its claim.
`An intuitive definition is that synergism occurs between two agents
`when the observed pharmacological
`effect (growth inhibition in this
`report) of a combination is more than what would be predicted from a
`good knowledge of the individual effects from each agent alone. A
`specific, complete,
`functional definition was provided above in the
`description of Equation 1, but a more general, succinct definition is
`provided here. Given specific, appropriate models for each agent alone,
`such as Equation 2, and given a logically derived model for the combi
`nation of the two agents which includes an interaction term with an
`estimable interaction parameter, such as «in Equation I; when the true
`interaction parameter
`is positive, synergism exists; when the true inter
`action parameter
`is negative; antagonism exists; and then the true
`interaction term is zero, no interaction (additivity) exists. To estimate
`the true interaction parameter, a combination model, such as Equation
`response at infinite drug concentration). Equation 2, was assumed (and
`1, is fitted to the complete data set by an appropriate
`statistical
`5319
`
`q[ara-C][DDP]
`E - B
`.x-E + B
`
`E - B
`m„- E + B
`
`£=
`
`l +
`
`U)
`
`(2)
`
`curves for
`Equation I allows the slopes of the concentration-effect
`the two drugs to be unequal. Equation I was derived (see "Appendix
`1") using the guidelines of Berenbaum (4) for defining the predicted
`additivity surface for a combination when the concentration-effect
`models are known (or assumed) for each individual drug, but with the
`addition of a first order interaction term. A convention used in Equa
`tions 1 and 2, is that as drug concentration(s)
`increases,
`the measured
`response (cell density) decreases;
`the slope parameter, m, is negative.
`Equations
`1 and 2 could be easily adapted so that
`the measured
`pharmacological effect (growth inhibition) would increase with increas
`ing drug concentration.
`The general sigmoid-/;',,,.,, or logistic equation (with a background
`
`2 of 10
`
`Alkermes, Ex. 1044
`
`

`
`QUANTITATION OF DRUG SYNERGISM
`
`surface. The vertical or Z axis is
`estimated concentration-effect
`the unnormalized measured cell density, and the X and Y axes
`are drug concentrations
`on a linear scale. Solid data points lie
`above the fitted surface, and open points
`lie below. Vertical
`lines are drawn from the data points to the fitted surface. Data
`points which would be hidden by the surface have been excluded
`from this figure. The parameter estimates ±SE from the fit of
`Equation 1 to the data in Fig. 1 are: £max= 176,000 ±7,500
`cells/ml; B = 20,000 ±7,500 cells/ml; Dm.,,^ = 14.6 ±1.8
`UM, /Wara-c= "0.916 ±0.010; AH.DDP= 9.81 ±0.87 ^M; mDDP
`= -1.58 ±0.21; and a = 3.08 ±0.96. The 95% confidence
`interval for a is from 1.14 to 5.02. Since this interval does not
`encompass 0, one can conclude that synergism between ara-C
`and DDP was demonstrated
`in this experiment. The 95%
`confidence interval for wara.cis —0.936to —0.896;that for moof
`is —2.00to —1.16.Since these intervals do not overlap, one can
`conclude that the concentration-effect
`curve for DDP is steeper
`than for ara-C;
`the individual concentration-effect
`curves are
`not parallel. The 95% confidence intervals for flm.ara-Cis from
`
`11.0 to 18.2 Ã(cid:141)Ã(cid:141)M;that for Dm,DDPis 8.05 to 11.6 UM. Since the
`intervals overlap, one cannot conclude that DDP is more potent
`than ara-C under the specified experimental
`conditions.
`The estimated background cell density (in the presence of an
`infinite drug concentration),
`20,000 cells/ml,
`is reasonably
`close to but less than the seeded number of cells, 50,000 cells/
`ml. Both the death and the disintegration
`of cells caused by
`high concentrations
`of drugs and artifacts caused by washing
`and clumping may account
`for the fact that B is less than the
`seeded 50,000 cells/ml. This background can be noted in Fig.
`1 as the height of the surface at the highest concentrations
`of
`drugs, 50 AIMara-C plus 20 ^M DDP. The background param
`eter could have been better characterized
`if higher concentra
`tions had been used. The £maxparameter
`represents
`the differ
`ence in cell density resulting from an exposure of cells to 0 UM
`
`200-
`
`is
`and the true parameter
`regression,
`such as nonlinear
`approach,
`estimated, along with a measure of uncertainty in the estimate. The
`word "potentiation"
`is reserved for the case in which one drug has no
`effect by itself, but increases the effect of an individually effective second
`drug; "inhibition"
`is reserved for the case in which one drug has no
`effect by itself but decreases the effect of a second drug, and "coalitive
`action" is reserved for the case in which two drugs have no effects by
`themselves, but the combination does have an effect. It should be noted
`here that
`there are many useful, functional definitions of synergism
`that are less restrictive:
`they do not require a particular mathematical
`model
`to describe the drug interaction; but rather,
`they require only
`that
`the observed effect of the combination be greater
`than that pre
`dicted from a specific, simpler noninteraction
`(additivity) model (e.g.,
`Refs. 2, 4, 26-28). However,
`if an appropriate,
`full interaction model
`can be adequately fitted to data for a particular experimental
`system,
`then quantitative approaches which utilize this model should be supe
`rior to approaches which do not.
`The full general universal response surface approach consists of eight
`steps: (a) the functional
`form of the individual concentration-effect
`models for each drug is characterized (e.g., median effect, median effect
`with a background,
`logistic, exponential, exponential with a shoulder,
`linear, etc.) from past experience,
`theoretical considerations,
`and pre
`liminary data;
`(b) a logical model
`for the joint action of the drug
`combination is derived using an adaptation of the guidelines of Beren-
`baum (4). Briefly, with Berenbaum's
`approach the isobol constraint
`equation
`
`Dm
`
`Dm
`
`is assumed to be correct, specific mathematical models for the individual
`drug concentration-effect
`curves are assumed to be correct, and a
`composite model for joint drug effect for the case of no interaction
`(additivity) is derived. An adaptation of this approach (See Appendices
`1C and ID) consists of deriving composite models for joint drug effect
`for the cases of interaction (synergism, antagonism), which include
`interaction parameters;
`(c) the experiment
`is designed; (d) the experi
`ment
`is conducted;
`(?) this model
`is fit
`to the full data set by an
`appropriate curve-fitting technique (e.g., weighted nonlinear regression,
`maximum likelihood estimation,
`etc.) which takes into account
`the
`statistical nature of the data (e.g., continuous
`responses, binary re
`sponses, counts, etc.) and data variation; (/)
`the goodness of fit of the
`model to the data is assessed by examining the 95% confidence intervals
`around the parameter estimates and by visually assessing the concord
`ance of the fitted surface to the observed data points; (g) if the fit is
`good, the model is accepted, parameter estimates along with measures
`of uncertainty in the estimates are reported, and conclusions are made;
`(In if the fit is not good,
`logical changes are made to the model, and
`steps 5-7 (or possibly steps 3-7), are repeated. If no logical model can
`be found which adequately fits the full data set, a model is derived for
`additivity (no interaction) using the guidelines of Berenbaum (4). this
`model
`is fit to all of the single drug data,
`the combination data are
`superimposed upon the fitted surface, and departures
`from additivity
`are noted by visual inspection.
`the above new approach is mathe
`As shown in the "Discussion,"
`matically consistent with the traditional
`isobologram approach but is
`more objective, is more quantitative, and is more easily automated.
`
`RESULTS
`
`exposure of L1210 with
`The results of a 3-h simultaneous
`ara-C plus DDP are shown in Figs. 1-4. Each of these four
`figures illustrates
`a different view of both the measured data
`and the concentration-effect
`surface
`estimated
`from fitting
`Equation 1 to the data.
`It should be emphasized that none of
`the curves shown in Figs. 1-4 are merely hand-drawn curves
`intended to connect data points;
`rather
`they are all curves
`simulated from the best fit of Equation 1 to the data.
`Fig. 1 is a 3-D representation
`of
`the raw data and the
`5320
`
`surface for a 3-h drug exposure
`concentration-effect
`Fig. 1. Three-dimensional
`to ara-C plus DDP. Fishnet surface, predicted concentration-effect
`surface, esti
`mated from Titting Equation I to the data with nonlinear
`regression as described
`in the text; points, measured cell densities from single tubes. Solid points (•)are
`above the surface; open points (O) fall below the surface.
`
`3 of 10
`
`Alkermes, Ex. 1044
`
`

`
`QUANTITATION OF DRUG SYNERGISM
`
`CD
`
`concen
`Fig. 2. Families of 2-dimensional
`tration-effect curves for a 3-h drug exposure to
`ara-C and DDP. The cunes are predicted con
`centration-effect curves, estimated as described
`in Fig. 1 and in the text. This set of curves is
`a 2-dimensional
`representation of the 3-dimen-
`sional surface in Fig. 1, but with the ordinate
`transformed
`to a percentage of (£„,„+ B\.
`Points, transformed measured cell densities.
`
`2
`\
`[Aro C]
`
`20
`10
`5
`(/¿M,log scale)
`
`90
`
`2
`1
`[Cisplatin]
`
`5
`
`20
`tO
`, log scale)
`
`50
`
`Talalay (2), who use a positive slope for both inhibitory and
`stimulatory drugs. A larger absolute value of the slope param
`eter results in a steeper concentration-effect
`curve. Although
`for the experiment
`shown in Fig. 1, the estimated mDDPis 1.7-
`fold greater than wara.c,slope differences are not clearly seen in
`the 3-D plot of Fig. 1. Since the synergism-antagonism param
`eter, a, is positive, synergism is indicated. The magnitude of a,
`3.08,
`is reasonably large, but
`like the difference in slopes,
`is
`difficult
`to appreciate
`from Fig. 1. Thus, although
`the 3-D
`concentration-effect
`surface in Fig. 1 provides a good overall
`picture, Figs. 2-4 which are three different 2-D representations
`of the results of the same experiment,
`are necessary to provide
`visual indications of goodness of fit of the estimated surface to
`the data and visual indications of the intensity of drug interac
`tion.
`Fig. 2 consists of two sets of families of 2-D concentration-
`effect curves. The same raw data are shown in both the left and
`right panels. The curves are sections of the best
`fit surface
`estimated from the fit of Equation
`1 to the data.
`In fact,
`the
`left set of six curves are transformations
`of slices of the full
`surface depicted in Fig. 1, expressed as a percentage of the
`predicted control
`[Em^ + A], from the face of the cube nearest
`the viewer, and continuing
`through five higher DDP levels.
`Analogously,
`the right panel of Fig. 2 could be constructed as
`transformations
`of slices of the surface in Fig. 1, starting at the
`left face, and continuing toward the right
`face at five higher
`ara-C levels. Note that
`the concentration
`scales in Fig. 2 are
`logarithmic. Note also that
`the predicted effect at 0 fiM ara-C
`plus 0 MMDDP is normalized to 100%. In Fig. 2, the relative
`magnitude of £maxand B can be seen. The estimated Dm values
`are designated by horizontal bars. Note that
`the D„(or ID50)
`does not appear at the 50% level, but rather at the midpoint of
`the £maxrange.
`It
`is clear that DDP has the steeper
`sloping
`concentration-effect
`curves. The goodness-of-fit of the data by
`the fitted surface can be visually assessed in Fig. 2 (29). The
`points are reasonably close to the fitted surface and reasonably
`random about the surface. Note that
`if each of 12 curves in Fig.
`2 were simply drawn by hand to connect
`the points,
`the figure
`would appear quite different. However,
`like Fig. 1, Fig. 2 lacks
`a good visual
`impression of the degree of interaction between
`ara-C and DDP.
`of the 3-D
`representation
`Fig. 3 is a 2-D isobolographic
`surface in Fig. 1. Isoeffect contours are shown at 10, 50, and
`90% pharmacological
`effect, corresponding
`to cell densities of
`178,400, 108,000, and 37,600 cells/ml,
`respectively. Note:
`
`% of pharmacological
`
`effect =
`
`08
`
`06
`0.2 04
`[AraC]/DXiAraC
`Fig. 3. Families of two-dimensional
`isobols for a 3-h drug exposure to ara-C
`and DDP. The curves were estimated as described in Fig. 1 and in the text. The
`set of isobol contours
`is another 2-dimensional
`representation
`of the 3-dimen-
`sional surface in Fig. 1, but with the ara-C and DDP concentrations
`transformed
`by division by the appropriate D, value.
`
`1.0
`
`120
`
`100
`
`80
`
`'S
`»e
`
`40
`
`20-
`
`S
`
`Aro C
`Cupial
`I I ratio
`
`10
`
`20
`
`50
`
`I Drug] ¡/¿M,log scale)
`
`curves for 3-h drug exposure to ara-C
`Fig. 4. Predicted concentration-effect
`alone, to DDP alone, to ara-C plus DDP in a 1:1 ratio, and to ara-C plus DDP
`in a 1:1 ratio which would show no interaction (additivity). The curves were
`simulated as described in Figs. 1-3 and in the text. This set of curves is yet
`another
`informative 2-dimensional
`representation
`of the 3-dimensional
`surface
`in Fig. 1. Points, transformed measured cell densities.
`
`ara-C plus 0 p\t DDP versus infinite ara-C plus infinite DDP.
`Or,
`in other words, Emaxis the range of response that can be
`affected by drugs. The sum of Em*xplus B is the estimated cell
`density at 0 ^M ara-C plus O /¿MDDP. The estimated median
`effective concentrations Z>m.ara.cand Dm,DDPare those concentra
`tions necessary to reduce £max(not
`the sum of EmM + B) by
`50%. Negative slope parameters of concentration-effect
`curves
`would indicate inhibitory drugs, e.g., mara.c and WDDPin this
`study, whereas positive slope parameters
`indicate stimulatory
`drugs. This is not
`the same convention as that of Chou and
`5321
`
`4 of 10
`
`Alkermes, Ex. 1044
`
`

`
`QUANTITATION OF DRUG SYNERGISM
`
`is the
`left to the lower right
`line from the upper
`The diagonal
`line of no interaction (additivity). No observed data points are
`shown because none appeared at exactly 10, 50, or 90% effect.
`The ordinate and abscissa are drug concentrations
`normalized
`by the respective Dx values (D10, A.O, Dw values; estimated
`concentration
`resulting in X% inhibition;
`e.g., 10, 50, 90%
`inhibition). The 3 curves in Fig. 3 are slices of Fig. 1 obtained
`by coming down from the top face of the cube, cutting at the
`10, 50, and 90% levels (of £max,not [£max+ B]), and normalizing
`by the respective Dx values. The degree of bowing of the isobol
`contours
`is a visual
`indication of the degree of synergism.
`It
`should be emphasized that
`the curves in Fig. 3 are sections of
`the fitted concentration-effect
`surface and not handdrawn iso-
`bols derived from handdrawn concentration-effect
`curves (27).
`Note that all of the isobols in Fig. 3 are smooth and symmetrical
`(any roughness
`in the curves is due to the difficulty of drawing
`the curves from simulated data), even though mara-c^ WDDP.
`There have been many suggested geometric indices of the degree
`of bowing of an isobol (6). The synergism-antagonism param
`if
`eter, a, is algebraically related to these indices. For example,
`the distance between the origin (0, 0) and the crossing of the
`diagonals
`(0.5, 0.5)
`is designated
`as ON, and the distance
`between the origin and the point where the rising (left to right)
`diagonal meets the 50% isobol
`is designated as OM,
`then the
`ratio S (S = ON/OM), will be an index of synergism. A large
`ratio will indicate a lot of bowing and a large synergism. This
`ratio, 5,
`is related to «by Equation
`3. The derivation
`of
`Equation 3 is included in Appendix 1C.
`
`«= 4(S2 - S)
`
`(3)
`
`study, S = 1.51.
`for a = 3.08, as in the present
`Note that
`This could be verified by the interested reader by using a ruler
`to measure the required distances
`in Fig. 3 and then making
`the required calculations. A form for the general
`isobol equation
`can be derived from Equation 1 by setting
`
`and
`
`= [1 - 0.01*]£„„+ B
`
`A»
`
`=
`
`X
`'\100-A-
`
`in Equa
`(from Equation 2) and substituting these expressions
`tion 1. After some algebra. Equation 4, a general equation for
`an isobologram,
`results.
`
`l -
`
`1 +
`
`Dx
`
`(4)
`
`It is the equation which describes
`Equation 4 is an hyperbola.
`the curves in Fig. 3. Note that at the ID50, where E = 0.5Emax
`+ B, the term
`
`100 - X
`
`raised to the power
`
`is more bowed than
`centage effects. In Fig. 3 the 90% isobol
`the 50% isobol, which is more bowed than the 10% isobol. Also
`note that a in Equation 4 also effects the degree of bowing, i.e.,
`as a positive a increases,
`the degree of bowing will increase.
`Although the isobol representation
`does give a visual indica
`tion of the degree of interaction,
`it lacks two main desirable
`features:
`(a)
`raw data cannot be superimposed
`on the fitted
`curve to provide a visual measure of goodness of fit; and (b) a
`good indication of the vertical distance between the synergism
`surface and the predicted additivity surface is not provided.
`While Fig. 2 includes the first feature, Fig. 4 includes both of
`these features. Fig. 4 is another 2-D representation
`of Fig. 1.
`The layout of the axes of Fig. 4 is the same as that of Fig. 2.
`Four concentrations-effect
`curves are included in Fig. 4: Curve
`1, ara-C alone; Curve 2, DDP alone; Curve 3, ara-C plus DDP
`in a 1:1 ratio; and Curve 4, the predicted additivity curve for
`ara-C plus DDP in a 1:1 ratio. The first 3 curves are appropriate
`slices through the full concentration-effect
`surface of Fig. 1.
`Corresponding
`raw data are superimposed on the 3 curves. The
`fourth curve was simulated with Equation 1 after setting a = 0.
`It is clearly evident from Fig. 4 that DDP has a steeper concen
`tration-effect
`curve than does ara-C. The additivity curve for a
`1:1 concentration
`ratio of ara-C plus DDP lies between the
`respective curves for ara-C and DDP. The fitted curve for the
`1:1 ratio lies below and to the left of the other 3 curves. The
`estimated Dm values for the four curves are: Curve 7, ara-C
`alone, 14.6 ¿¿M;Curve 2, DDP alone; 9.81 pM; Curve 3, ara-C
`plus DDP in a 1:1 ratio, 7.86 pM (or 3.93 J/M concentrations
`of each drug); and Curve 4, additivity curve of ara-C plus DDP
`ina 1:1 ratio, 10.6 p\t (or 5.28 fiM concentrations of each drug).
`Two logical measures of synergistic effect would include the
`horizontal distance between Curves 3 and 4 at the median effect,
`2.74 ¿¿M,and the ratio of Dm values for Curves 3 and 4, 1.35.
`Note that
`this ratio, 1.35, is not
`the same as the ratio 5, 1.51
`calculated from the isobol representation of Fig. 3. Other logical
`measures of synergistic
`effect
`include: horizontal
`differences
`and ratios at other effect
`levels; the maximum horizontal dif
`ference and maximum ratio; vertical differences and ratios at
`various drug levels; and the maximum vertical difference and
`maximum ratio.
`A total of 12 individual experiments were performed, 3 with
`a 1-h drug exposure time, 2 with 3 h, 2 with 6 h, 2 with 12 h,
`and 3 with 48 h exposure, each with a 74-78-tube