(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-Arabinofuranosy
Icytosine '
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 [W. R. G.J, Roswell Park Memorial Institute, New York State Department
of Health. Buffalo. New York 14263
ABSTRACT
This report describes the application of a new approach, the universal
response surface approach, to the quantitative assessment of drug inter
action, i.e., the determination of synergism, antagonism, additivity, potentiation, 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
This report describes the application of a new approach,
URSA,3 to the quantitative assessment of drug interaction, i.e,
the determination of synergism, antagonism, additivity, potentiation, 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 48h growth experiment which began with a 3-h incubation of
Received 4/14/89; revised 2/28/90.
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-arabinofuranosylcytosine; 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„.„_c
and O„.DDP.
median effective concentration of araC and cisplatin; DXmJ:. DX.DOF.concentration of ara-C. DDP which inhibits
growth (£„„)
by X"i: m. mmf, and mDor. slope parameters for concentrationeffect curves: «.synergism-antagonism parameter; y. interaction parameter for
the case in which one drug has no effect as a single agent;^, fraction affected.
LI210 cells with various combinations of ara-C and DDP. In
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 isobolograms 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."
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QUANTITATION
OF DRUG SYNERGISM
later shown) to be appropriate for both ara-C and DDP. The form of
Equation 2 (without B) and with fa (fraction affected) = E/Em,„is
Chemicals. Cisplatin was obtained from the Bristol-Myers Company
simply a rearrangement of the median-effect equation of Chou and
(Syracuse, NY). ara-C was supplied by Sigma Chemical Company (St.
Talalay (2, 3, 15) or of the Hill model (16-19). Also, the terms and
Louis, MO). The purity of the compounds was determined by high
symbols Dmand m are from Ref. 2.
pressure liquid chromatography to be 98%. Drugs were dissolved and
Equation 1 was fitted to data using custom software called SYNFIT,
serial dilutions made in RPMI 1640 plus 20 mM 4-(2-hydroxyethyl)-lwhich was written in the computer language, MicroSoft C (Microsoft
piperazineethanesulfonic acid (Gibco, Buffalo, NY). Drug solutions
Corp., Bellevue, WA). SYNFIT uses a version of the Marquardt algo
were sterilized by passage through a 0.2-nm Acrodisc from Gelman
rithm (20) for nonlinear regression as described by Nash (21). The
Sciences, Inc. (Ann Arbor, MI).
output of the program includes parameter estimates, asymptotic stand
Exposure of Cells. Murine leukemia LI210 cells were grown in ard errors, 95% confidence limits for the parameters, and residual
suspension culture in RPMI 1640 supplemented with 10% heat-inac
analyses. All comparisons for statistical significance were performed
tivated fetal bovine serum and 20 mM 4-(2-hydroxyethyl)-l-piperazineethanesulfonic acid, pH 7.3, at 37°C.To assess drug effects on cell 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
growth, L1210 cells in log phase with a doubling time of about 12 h calculate predicted values of E. Initial parameter estimates for the D„
were utilized. L1210 cells at a final concentration of 50,000 cells/ml in and m parameters for the nonlinear regression were obtained by fitting
4 ml of the above medium supplemented with 10% heat-inactivated
the median-effect equation of Chou and Talalay (2) to the single drug
fetal bovine serum (complete medium) were exposed to six logarith
data with weighted linear regression. The SAS/PC software package,
mically spaced concentrations of ara-C and cisplatin centered at the
Version 6 (23), was used to generate the 3-D graph of Fig. 1. The
predicted IDM,for each drug in a 62 factorial design, for a total of 36
graphs in Figs. 2-4 were made by simulating data from Equation 1,
different combinations. From 4 to 8 control (0 ¿IM
ara-C plus O n\ì
using the estimated parameters from the best fit of Equation 1 to the
DDP) tubes and 2 tubes of each of the other 35 drug combinations
observed data, with custom FORTRAN programs, and plotting the
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) simulated data by hand. All software was run on IBM PC/AT. IBM
PC/XT, and Leading Edge Model M microcomputers. Inquiries re
and the subsequent drug-free growth period. Drug exposure was fol
garding distribution of the custom software package, SYNFIT, should
lowed by two washes with sterile 0.9% NaCl solution and final resusbe addressed to W. R. Greco.
pension in complete medium (free of drug). Growth was determined by
URSA could be implemented with many commercial statistical soft
counting the number of cells in tubes with a Coulter electronic cell ware packages. We have used URSA with the mainframe version of
counter, Model ZBI (Coulter Electronics, Inc., Hialeah, FL) 48 h after
BMDP (24), the PC version of SAS (23), and PCNONLIN (25). To be
the start of drug exposure.
suitable for implementation of URSA, a package must include a non
Data Analysis. Equation 1 was fitted to the complete data set from
linear regression procedure which does not require the coding of ana
an experiment (74-78 measurements) with unweighted least squares
lytical derivatives and, in order to code the root finder, does allow the
nonlinear regression, and parameters were estimated (10-14). Equation
1 contains the respective drug concentrations [ara-C] and [DDP] as function definition to include IF statements, and either GOTO state
ments and/or loops. An example of a set of model-definition statements
inputs; and the measured cell density, /;. as the output. The 7 estimable
for fitting Equation 1 to data is listed in Appendix 2. This example
parameters include: £„,„,
the maximum cell density, over background,
consists of control language for SAS and will be appropriate for SAS
at 0 drug concentration; B, the extrapolated background cell density in
running on microcomputers, minicomputers, and mainframe com
the presence of an infinite drug concentration; the respective II )..„s
or
puters. By adapting this example, interested researchers should be able
median effective concentrations, £>™,.r.-c
and Dm.oar; the respective
to implement URSA into many other commercial statistical packages
concentration-effect slopes, mm.c and mDDP;and the synergism-antagwhich include a nonlinear regression module.
onism parameter, a. When a is positive, synergism is indicated; when
We are implementing URSA into custom user-friendly software in
a is negative, antagonism is indicated; and when a is 0, no interaction
C language for microcomputers running the MicroSoft Disk Operating
or additivity is indicated.
System, instead of emphasizing the use of commercial software pack
ages
for several reasons: (a) many researchers may be unacquainted
[ara-C]
[DDP]
1
E - B
\'
I/»»DDP
with the use of general nonlinear regression software; (b) researchers
E- B
may be hesitant to spend several hundred dollars to purchase general
7-max —E
B
Ã-- +' ",
statistical software for implementing a new approach to data analysis;
U)
(c) the most current concepts and approaches can be implemented into
q[ara-C][DDP]
a custom software package, whereas commercial statistical packages
E - B
E - B
may present limitations and restrictions; (</) custom software can be
.x-E + B
m„
- E+ B
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
£=
(2) can be made in its assessment and good use can be made of its claim.
MATERIALS AND METHODS
l +
An intuitive definition is that synergism occurs between two agents
when the observed pharmacological effect (growth inhibition in this
Equation I allows the slopes of the concentration-effect curves for report) of a combination is more than what would be predicted from a
the two drugs to be unequal. Equation I was derived (see "Appendix
good knowledge of the individual effects from each agent alone. A
1") using the guidelines of Berenbaum (4) for defining the predicted
specific, complete, functional definition was provided above in the
additivity surface for a combination when the concentration-effect
description of Equation 1, but a more general, succinct definition is
provided here. Given specific, appropriate models for each agent alone,
models are known (or assumed) for each individual drug, but with the
such as Equation 2, and given a logically derived model for the combi
addition of a first order interaction term. A convention used in Equa
nation of the two agents which includes an interaction term with an
tions 1 and 2, is that as drug concentration(s) increases, the measured
estimable interaction parameter, such as «in Equation I; when the true
response (cell density) decreases; the slope parameter, m, is negative.
interaction parameter is positive, synergism exists; when the true inter
Equations 1 and 2 could be easily adapted so that the measured
action parameter is negative; antagonism exists; and then the true
pharmacological effect (growth inhibition) would increase with increas
interaction term is zero, no interaction (additivity) exists. To estimate
ing drug concentration.
The general sigmoid-/;',,,.,, or logistic equation (with a background
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
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QUANTITATION
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approach, such as nonlinear regression, and the true parameter is
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 Berenbaum (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.
As shown in the "Discussion," the above new approach is mathe
estimated concentration-effect surface. The vertical or Z axis is
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 ÕÕM;
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 £max
parameter represents the differ
ence in cell density resulting from an exposure of cells to 0 UM
200-
matically consistent with the traditional isobologram approach but is
more objective, is more quantitative, and is more easily automated.
RESULTS
The results of a 3-h simultaneous exposure of L1210 with
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
Fig. 1. Three-dimensional concentration-effect surface for a 3-h drug exposure
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.
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QUANTITATION
OF DRUG SYNERGISM
CD
Fig. 2. Families of 2-dimensional concen
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-dimensional surface in Fig. 1, but with the ordinate
transformed to a percentage of (£„,„
+ B\.
Points, transformed measured cell densities.
\
2
5
10
20
90
[Aro C] (/¿M,log scale)
0.2 04 06 08
[AraC]/DXiAraC
1.0
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-dimensional surface in Fig. 1, but with the ara-C and DDP concentrations transformed
by division by the appropriate D, value.
120
100
'S
»e
80
40
S
Aro C
Cupial
I I ratio
20-
10
20
50
I Drug] ¡/¿M,log scale)
Fig. 4. Predicted concentration-effect curves for 3-h drug exposure to ara-C
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 /¿M
DDP. The estimated median
effective concentrations Z>m.ara.c
and 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
1
2
[Cisplatin]
5
tO
20
50
, log scale)
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 1.7fold 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 concentrationeffect 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 £max
and 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 £max
range. 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.
Fig. 3 is a 2-D isobolographic representation of the 3-D
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 =
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QUANTITATION
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The diagonal line from the upper left to the lower right is the
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 isobols 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
eter, a, is algebraically related to these indices. For example, if
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.
centage effects. In Fig. 3 the 90% isobol is more bowed than
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
«= 4(S2 - S)
(3)
ina 1:1 ratio, 10.6 p\t (or 5.28 fiM concentrations of each drug).
Note that for a = 3.08, as in the present study, S = 1.51. Two logical measures of synergistic effect would include the
horizontal distance between Curves 3 and 4 at the median effect,
This could be verified by the interested reader by using a ruler
2.74 ¿¿M,
and the ratio of Dm values for Curves 3 and 4, 1.35.
to measure the required distances in Fig. 3 and then making
Note that this ratio, 1.35, is not the same as the ratio 5, 1.51
the required calculations. A form for the general isobol equation
calculated from the isobol representation of Fig. 3. Other logical
can be derived from Equation 1 by setting
measures of synergistic effect include: horizontal differences
= [1 - 0.01*]£„„
+B
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
and
maximum ratio.
X
A total of 12 individual experiments were performed, 3 with
A» =
'\100-Aa 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 design.
(from Equation 2) and substituting these expressions in Equa
Fig. 5 illustrates the effect of drug exposure time on the a
tion 1. After some algebra. Equation 4, a general equation for
parameter estimated by fitting Equation 1 to the data sets from
an isobologram, results.
the 12 separate experiments. The plotted points in Fig. 5 are
displaced slightly from the true exposure times to allow a visual
l comparison of the 95% confidence intervals from repeat exper
(4)
iments. There is a suggestion of a pattern to the time course of
synergism in Fig. 1. Mild synergism occurs at a 1-h drug
1+
Dx
exposure, synergism seems to peak at 3-6 h, and then synergism
becomes mild again from 6 to 48 h drug exposure. However,
Equation 4 is an hyperbola. It is the equation which describes
the curves in Fig. 3. Note that at the ID50, where E = 0.5Emax because the 95% confidence intervals for a all overlap and
because the number of experiments conducted at the suggested
+ B, the term
peak interval of synergism for drug exposure (3-6 h) was small,
100 - X
and because one of the experiments conducted with a 6-h drug
exposure showed enhanced synergism while one did not, more
experiments will have to be conducted in the range of 1-12 h
raised to the power
drug exposure time to conclusively characterize the time course
of synergism.
1
1
2m\
2ni
DISCUSSION
becomes 1 and thus disappears from Equation 4. Also, it is this
term which increases the degree of bowing for increasing per-
The present study is the first report of a specific application
of a new method, the universal response surface approach or
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QUANTITATION
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concentration
5B
41".0W^
»1.<16•1•tZ
2E.fir-_•ii41o3•t
_ä
"
Drug Exposure Time
4"-_-II
(hrs)
Fig. 5. Effect of drug exposure time on the synergism-antagonism interaction
parameter, a. Each point represents the a estimate from a complete separate
experiment obtained by fitting Equation 1 to the data with nonlinear regression
as described in the text. Bars, 95% confidence intervals for «.Even though the
drug exposure times were at exactly 1, 3, 6, 12, or 48 h, the plotted n estimates
are displaced slightly to the right or left to avoid confusion in interpretation of
the 95% confidence bars.
URSA (10, 11), to quantitatively assess drug interactions with
real, previously unpublished data.
The advantages of the new method over older methods in
clude: (a) objective, quantitative definitions are provided for
synergism, antagonism, additivity, potentiation, inhibition, and
coalitive action; (b) a logical framework is provided for many
useful quantitative measures of the intensity of synergism and
antagonism; (c) precision estimates are provided for parameter
estimates; (d) a unitless, estimable synergism-antagonism pa
rameter, a, is provided for objectively quantitating the intensity
of interaction; (e) differences in precision among the data are
taken into account in the analysis; (/) because of points c-e,
objective decisions can be made regarding the existence of drug
interactions and the effects of other variables, such as drug
exposure time, on the degree of interaction; (g) a set of useful
graphic representations is provided which aid in the interpre
tation of the results; (/;) because a logical equation(s) is used to
model drug interaction, many objective experimental statistical
design criteria, such as D-optimality (30), can be used to gen
erate useful experimental designs for laboratory experiments;
(i) the method can be extended to more complex situations by
expanding the set of models.
One important situation which commonly arises in cancer
studies is one in which only one drug of a two-drug combination
is cytotoxic individually, and the other drug modulates its
effects; e.g., the case of 5-fluorouracil plus folinic acid (31).
This kind of situation is both easier to evaluate (often by visual
inspection of the results) and possibly easier to explain biolog
ically. A logical mathematical model for it is Equation 5, where
it is assumed that ara-C is not cytotoxic, and that the individual
concentration-effect curve for DDP is in the logistic form of
Equation 2.
[DDP]
1=
E- B
D„.DD
I /"•
DDP
.»-E+B.
(5)
7[ara-CHDDP|
DmDDP\
=-E + B)
\£-„,„
In Equation 6, 7, is a modulation
\£m„
- E—
+
parameter with units of
'. When y is positive, there is potentiation, when
7 is negative, there is inhibition. Equation 5 was derived from
Equation 1, a model for drug interaction where both drugs are
individually cytotoxic, by replacing <x/Dm,„,.c
with 7, and by
letting Dm.ara.<be infinity. The slope parameter for ara-C, m„,.
c, is retained to adjust the effective slope of the concentrationeffect curve for the drug combination for possible differences
from mone. Equation 5 is a fundamentally different model than
is Equation 1. Since both ara-C and DDP are cytotoxic in our
system when used as individual agents, Equation 5 would not
be an appropriate model for the present study. Note that a
logical equation for the situation where neither drug is cytotoxic
individually, but where the two-drug combination is cytotoxic,
would be Equation 5 without the first term on the righthand
side of the equation, and with the AH.DDPdeleted from the
second term. This type of interaction is commonly called coa
litive action (5).
A detailed description of URSA from a statistical viewpoint
has been published (11). Briefly, each complete model for an
experiment can be dissected into two components: a random
component, which describes the variation in the responses; and
a nonlinear structural component, which describes the relation
ship between a set of drug concentrations and the expectation
(true mean) of the random component (8). Equations 1, 2, and
5 are examples of nonlinear structural components. In the
present study, a normal distribution with equal variance for the
responses at all points of the concentration-effect surface was
assumed for the random component. Therefore, unweighted
nonlinear regression was used to fit the structural models to
the data.
The assumption of a normal distribution with equal variance
for the data in this study was justified with the following
arguments: (a) the cell counts ranged from about 20,000 cells/
ml to about 200,000 cells/ml. Therefore, because the numbers
of counts were large, the normal approximation to the Poisson
distribution (32) for count data was considered appropriate; (b)
because there was only a 10-fold range in the magnitude of the
observations, the need for unequal weighting of the data to
balance any heterogeneity of variance was considered unneces
sary; (c) if one examines the distribution of the raw data about
the fitted curves in Fig. 2, it is visually apparent that these
assumptions were reasonable.
For other types of data, random models for data distributions
other than the normal distribution may be more appropriate.
For example, for data sets consisting of small numbers of counts
(e.g., from colony formation assays), the Poisson distribution
might be used. Or, for data sets consisting of proportions of
successes, failures or deaths (e.g., in acute toxicology experi
ments), the binomial distribution might be used. For data with
distributions other than the normal, general maximum likeli
hood procedures (8) would be used in place of nonlinear regres
sion procedures for fitting models to data.
In principle, one could have fit a comprehensive model to the
data from the complete set of 12 experiments (920 total data
points, ~48 parameters), which incorporated extra parameters
to model the asymmetrical humped shape of the a time course
seen in Fig. 5. Or, in a somewhat simpler approach, one could
have fit all of the data from replicate experiments at each
exposure time with a model which included a common «but
separate Dm.ara.c,Dm.DDP,m„,-c,/WDDP,Em,K,and B parameters
(e.g., a total of 19 parameters for the set of 3 experiments at a
1-h drug exposure time). However, the sophistication in the
data analysis approach as presented in this paper was judged to
be already well beyond the sophistication usually seen in the
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QUANTITATION
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analysis of drug combination studies; more complex approaches
did not seem warranted. The seemingly large confidence inter
vals surrounding each «estimate in Fig. 5 serve as a reminder
of the difficulty in quantitatively assessing joint drug action;
other approaches which do not include a quantitative measure
of uncertainty merely mask the difficulty.
Since in the LI210 system, DDP and ara-C are both potent
growth inhibitors, any synergistic interaction could be the result
of ara-C directly enhancing the effects of DDP, DDP directly
enhancing the effects of ara-C, ara-C indirectly enhancing the
effects of DDP, DDP indirectly enhancing the effects of ara-C,
independence of action between ara-C and DDP (effect multi
plication), DDP plus ara-C causing an effect together which
could not be caused by either drug alone, or any combination
of the above. A comprehensive search for a biochemical or
molecular mechanism to explain the synergistic interaction
would include all of the proposed mechanisms of action for
both drugs used individually plus other more fanciful possibil
ities. A comprehensive mechanistic mathematical model to
relate mechanism(s) to cytotoxicity would be necessary for a
complete understanding of the phenomena. The classic report
which presented simulations of mathematical models of antineoplastic drug interactions is that of Werkheiser et al. (33).
Although only a handful of publications in cancer research have
appeared in the last 17 years which have used the approach
pioneered in Ref. 33, it is anticipated that the recent surge in
the wide availability of computer hardware and simulation
software will spur an acceleration of the use of this approach
to solve complex pharmacological problems.
At least 12 in vitro (34-45) and four in vivo (46-49) studies
have previously reported a synergistic interaction or the lack of
a synergistic interaction between ara-C and DDP in a wide
variety of experimental systems. These studies have used a wide
variety of definitions of synergism, experimental designs, and
methods to analyze data. A comprehensive critical review of
these studies is in preparation.
The most comprehensive search reported to date for the
mechanism(s) for synergism between ara-C and DDP was con
ducted in LoVo colon carcinoma cells by Fram et al. (34). It
was found that ara-C had no effect on the binding of DDP to
DNA, the induction of DDP-induced interstand DNA cross
links, the excision of DDP-induced interstand DNA cross-links,
and the excision of total platinum from DNA. Also, ara-C had
a less than additive effect with DDP on the inhibition of DNA
synthesis. However, ara-C markedly delayed the recovery of
DNA synthesis inhibited by DDP. DDP did not inhibit the
incorporation of ara-C into normally replicating DNA. DDP
enhanced incorporation of ara-C into DNA undergoing repair
synthesis. The direct effect of DDP on either the synthesis or
retention of ara-CTP pools was not studied. Future work in our
laboratory will involve a systematic look at these possible
mechanisms in LI210, but with a careful interpretation of
results which will reflect the temporal and concentration-effect
patterns observed for cytotoxicity.
In summary, the above set of terms, definitions, models
(equations) and procedures for fitting models to data comprise
a new empirical approach, URSA, for assessing drug interac
tions. The terms synergism, antagonism, additivity, potentiation, inhibition, and coalitive action include all important po
tential types of empirical interactions and are each associated
with an explicit mathematical model (equation). The models
form a hierarchical set. The general approach to data analysis
is always the same. Adaptations can be easily introduced to
reflect additional complications, such as background parame
ters for only one drug. The general approach is consistent with
the traditional isobologram approach (1, 27), but also incor
porates modern statistical procedures for both parameter esti
mation and parameter uncertainty estimation.
The main disadvantage of the new method is its requirement
for extensive calculations. Practical use of the new method
would be impossible without user-friendly computer software
and accessible computer hardware. Work is in progress to
complete the development of software which implements the
method for the IBM PC family of microcomputers.
ACKNOWLEDGMENTS
The authors thank Robin Susice for extraordinary computer pro
gramming and data analysis work. We also thank Drs. Morris Berenbaum. Peter Gessner, Ting-Chao Chou, Lawrence Emrich, and Harry
Slocum for helpful discussions concerning the assessment of synergism.
APPENDIX
1: DERIVATION
OF EQUATION
1
Equation 1, that for interaction between two inhibitory drugs, each
having a background response which remains even in the presence of
infinite drug concentrations, is derived here. Symbol definitions are
presented both in the "Introduction" and in Footnote 3. Although araC and DDP are specific drugs used here, the derivation is a general one
for two inhibitory drugs. The derivation consists of five parts, labeled
A-E. The five parts are: (A) derivation of an equation, using the
guidelines of Berenbaum (4), for additivity (no interaction) for two
drugs, the individual concentration-effect curves of which follow Equa
tion 2, and whose concentration-effect slope parameters are equal; (B)
derivation of an equation, using the guidelines of Berenbaum (4), for
additivity for two drugs, the individual concentration-effect curves of
which follow Equation 2, and the concentration-effect slope parameters
of which are unequal; i.e., /n„..c^ mDDP;(C) justification of an equation
for drug interaction where the slope parameters of the concentrationeffect curves are equal; (D) proof of the validity of a constraint equation
to be used to derive a logical equation (a generalized linear model) for
drug interaction for two drugs, the individual concentration-effect
curves of which follow Equation 2, and the concentration-effect slope
parameters of which are equal; (E) derivation of an equation, Equation
1, using the constraint derived in D, for drug interaction for two drugs,
the individual concentration-effect curves of which follow Equation 2,
and the concentration-effect slope parameters of which are unequal;
(A) Derivation of an Equation,
(4), for Additivity (No Interaction)
centration-Effect Curves of Which
tration-Effect Slope Parameters of
Equation 2 adequately describes
behavior of both drugs.
£m,x
E =
\
Using the Guidelines of Berenbaum
for Two Drugs, the Individual Con
Follow Equation 2, and the Concen
Which Are Equal. It is assumed that
the individual concentration-effect
Dm
!
+B
(2)
The guidelines of Berenbaum (4) define a set of mathematical pro
cedures for the derivation of an additivity (no interaction) concentra
tion-effect model for a drug combination. Hewlett (6) earlier summa
rized a similar set of procedures, including work by Finney (7), which
can also be used to derive equations similar to Equation 1. These sets
of procedures assume the validity of the classical isobologram approach
of Loewe (1) for assessing drug interactions. The basis of the classical
isobologram approach is that a logical definition for additivity between
two drugs is where one drug acts as if it were merely a dilution of the
other. In other words, the sham combination of one drug with itself
will result in a conclusion of additivity. Berenbaum's guidelines start
with the listing of an isobol constraint equation; e.g.. Equation Al,
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QUANTITÄTEN
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where Z„..c,ZDDPare the concentrations of agents that individually
would produce the same effect as the combination, and [ara-C] and
[DDP| are the concentrations of drugs in the combination.
[ara-C]
1=
[DPP]
(Al)
Equation B3 results, which is analogous to Equation A6. However,
Equation B3, that for additivity, m,„.c^ "IDDP,cannot be further
simplified and must be left in unclosed form.
(ara-C]
1=-
[PPP]
Dm
Then each individual concentration-effect equation is written with
Z.T..C,ZDDPsubstituted for the expression, [DRUG], in Equation 2.
An.,
E = Em
+B
"•DDI
(C) Justification of an Equation for Drug Interaction Where the Slope
Parameters of the Concentration-Effect Curves Are Equal. It is assumed
that Equation Cl is a logical, useful equation for drug interaction, for
cases where concentration-effect slopes are equal.
(A2)
[ara-C]
[DPP]
An.ara-C
An.ODP
ara-C]
A,.DD
E = £m
An.ara-
(A3)
Zppp
I +
E-B
(A4)
£m>,- £+ B
E-B
(AS)
Then, Equations A4, AS are substituted into Equation Al, yielding
Equation A6.
[ara-C]
E-B
[PPP]
E-B
£m«
- E + B
£„,„-£
+ B
TA»
(A6)
Equation A6 is then rearranged by simple algebra to yield Equation
A7.
+ B
(Al)
Equation A7 is very similar to the mutually exclusive model of Chou
and Talalay (2). Equation A7 is a logical equation for additivity (no
interaction) for two inhibitory drugs, the individual concentration-effect
slopes of which are equal, and which share a common background
parameter.
(B) Derivation of an Equation, Using the Guidelines of Berenbaum
(4), for Additivity for 2 Drugs, the Individual Concentration-Effect Curves
of Which Follow Equation 2, and the Concentration-Effect Slope Param
eters of Which Are Unequal; i.e., mM„,,
, m,,,,,.. Equations Bl, B2 are
substituted in place of Equations A2, A3 to model the case where the
concentration-effect slopes are not necessarily equal; i.e., m„,.c
& mDop.
An.ara
cAn.DDP
«[ara-C][PPP]1m
[PPP]
Dm.ara-cDfn.DDP
An.DDP
" ''
J
«[ara-C]|DPP]
An.ara-cAn.DDP
in both the upper and lower bracketed expressions. The justifications
of this maneuver include:
1. Equation A7 and Cl can be formally shown to be among the class
of generalized linear models (8), a class of empirical statistical models
with useful properties.
2. The addition of a first order interaction term into a generalized
linear model such as the linear model, the linear logistic model, or the
proportional hazards model (8) is a common, widely accepted, exten
sively validated practice, used to empirically model phenomena with
covariate interaction.
3. The addition of a first order interaction term is simpler than the
addition of a higher order interaction term; thus Occam's razor is being
adhered to.
4. Equation Cl, or its analogue for the case where concentrationeffect slope parameters are unequal, Equation 1, have been successfully
fit to hundreds of data sets (this paper, Refs. 10-14; unpublished
results).
5. Equation Cl incorporates the appealing intuitive idea that with
drug synergism (or antagonism) one effectively has more (or less) drug
in a system than one intentionally added.
6. Equation Cl includes a unitless, interaction parameter, a, with a
useful interpretation; i.e., when a > 0, one has synergism, when a < 0,
one has antagonism, and when «= 0, one has additivity (no interaction).
When a = 0, the interaction term drops out of Equation Cl, and
Equation A7 results, an equation similar to one derived by Chou and
Talalay (3) from basic biochemical principles for mutually exclusive,
noninteracting enzyme inhibitors. Because a is unitless, this parameter
can be logically compared across different agents and different experi
mental systems.
7. A form for the general isobol equation. Equation C2, where
»We= "»DDP,
can be derived from Equation Cl by setting
£= [1 - 0.01 A-]£m.x
+ B
and
£m„
+ B
l +
a[ara-C][PPP]
Equation Cl was derived from Equation A7 by merely adding a first
order interaction term
AM.DDI
The set of 3 equations is now solved simultaneously for £with the
elimination of Z„a.c,
ZDDP-First, by simple algebra. Equations A2, A3
are rearranged to Equations A4, A5, respectively.
1
(B3)
E-B
t,IT — DJ?
(Bl)
Zara-c
öm>ara-<
(from Equation 2), and substituting these expressions in Equation Cl.
""\A„.DD,
£=
+ B
1+
(B2)
[PPP]
_ [ara-C]
~
Av.ar.-C
(ZDDP
Om.DDI
AV.DDP
q|ara-C] | 100 - X
Av.„.-C
L
(C2)
X
Then, Equations Bl, B2, along with the isobol constraint. Equation
Al, are solved simultaneously in a manner analogous to Perivation A.
Equation C2 is an hyperbola and has the regular shape of a drug
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QUANTITATION
OF DRUG SYNERGISM
combination isobol found in many studies. (Note that Equation 4 in
"Results," is a more general form of Equation C2. where marac 5¿
mDDP.)
Also, as explained in "Results" and shown below, the synergism-
The result is Equation P2.
j
[ara-C]
antagonism parameter, a, is related to geometric indices of the degree
of bowing of isobols (6).
At the 50% effect isobol. X = 50, and the term in Equation C2,
[DPP]
E- B
£•„.,
- E + B
Dâ„¢"-C(E^-/+ B.
(D2)
«[ara-C][DDP]
(100-JO.
Vt-
or the term in Equation 4,
(100 - A-)
becomes 1, and thus is eliminated. Also, Dm.DDPand An.arac can be
substituted for DI-.DDPand Dx.,,,.c, respectively. Note that at points
along the diagonal line going from (0,0) to (1,1) in Fig. 3,
[DPP] _ [ara-C)
Dm.DDP
An.araC
E-B
E-B
Dm.nt-cDm.DDP
Equation D2 can now be simplified with simple algebra to Equation
Cl. Thus, the correctness of Equation Dl as a proper constraint for
drug interaction, when each individual concentration-effect curve fol
lows Equation 2, has been shown.
(E) Derivation of an Equation, Equation 1, Using the Constraint
Derived in D, for Drug Interaction for 2 Drugs, the Individual Concen
tration-Effect Curves of Which Follow Equation 2, and ConcentrationEffect Slope Parameters of Which Are Unequal; i.e., m.„.c5e inDDP.
Equation Bl, B2 are substituted into Equation Dl, and Equation I
results. Equation I cannot be simplified further and must be used in
unclosed form.
Thus, the expression, [ara-C]/Dm.„„
c can be substituted for
[DPP)Dm.DDPin Equation C2, which yields, after solving a quadratic
equation and simplifying, Equation C3.
[ara-C]
1=
[DPP]
1/«DOF
[ara-C] _ -1 + Vl + a
(C3)
g[ara-C][DDP]
(~"¿"
E-B
Now let the point ON be the point in Fig. 3 where the two straight
diagonal lines meet at the center, and let the point OM be the point
where the 50% isobol cuts the diagonal line going from (0,0) to (1,1).
The distance between the point (0,0) and ON is the hypotenuse of a 45degree right triangle. Since the^wo equal sides are each 0.5 unit long,
the hypotenuse is equal to 0.5V2. The distance between the point (0,0)
and OM is also the hypotenuse of a 45-degree right triangle, where the
two equal sides are each equal to the expression on the righthand side
of Equation C3. Thus,
(E^-
OM =
Appendix
S = ON/OM =
(C5)
v/2
Equation C5 can be simplified to Equation 3, which shows the
relationship between a and the geometric measure of bowing of the
isobologram, S.
„= 4(52 - S)
(3)
(D) Proof of the Validity of a Constraint Equation to be Used to Derive
a Logical Equation (a Generalized Linear Model) for Drug Interaction
for 2 Drugs, the Individual Concentration-Effect Curves of Which Follow
Equation 2, and the Concentration-Effect Slope Parameters of Which
Are Equal. It will first be assumed that Equation PI is the proper
constraint to replace the isobol constraint. Equation Al, for deriving
Equation Cl from the equations for individual concentration-effect
curves. Equation 2, for the case in which m = ma,.c = /WDDP.
[ara-C| | [DPP] {
typed between the slash-asterisk
symbols are comments
The following statements in the SAS language comprise the model
definition for Equation 1. from the paper, Greco et al.. An
application of ..., in Cancer Research, 1990. This set of statements
was designed to run under PROC NLIH in the PC version of SAS Ver. 6,
but should run with minor modifications under any version of SAS on
any computer.
(C4)
0.5V2
-1 + Vl + a
2
Statements
-1 +
The ratio, S= ON/OM, a geometric measure of the degree of bowing
at the 50% isobol, is equal to the expression in Equation C5.
(i)
* The title is defined, and the MS DOS filename of the datafile with
* the full pathname, if necessary,is assigned to a SAS system file.
Title ' Estimation of Synergism - Antagonism
Filename synfile 'D:\WP50\SYNDAT2.DAT1;
Effect of Drugs
';
/,..«»....«.,»..,...«.....«.................,,..»,.»...,.,,..,.,...».,
* The the SAS system file is brought into the working environment, and
* assigned the name of dataset A, and three columns of the datafile
* are assigned variable names, E for measured cell density, Dl for
* the cone, of drug 1, and D2 for the cone, of drug 2. This dataset
* is the one which is extensively analyzed in the paper.
*/
Data A;
infile synfile;
input E Dl D2;
* The nonlinear regression procedure is called, and the 7 parameters
*
* are assigned initial guesses.
•
,,...,....,...........,...«««,*,»,,,,,.».,,,,,,**.,.,,»,.,,..,,,«......
Proc NLIN data-A method=dud;
parms Emax=4810.662109
Dml-15.046
ml=-0.7B6
Dm2=9.498
m2=-1.204
Alpha-3.027
B=211.953003;
g[ara-C)[DDP]
(PI)
Equations A2, A3 are rearranged to Equations A4, A5, respectively.
The expressions for Zara_c,ZDDPare then substituted into Equation Dl.
5326
The actual model definition starts here. Note that it involves a
one dimensional bisection rootfinder, which uses an iterative
procedure. Consult (Thisted, R.A., Elements of Statistical Computing
pp. 170, algorithm 4.2.5, Chapman and Hall, New York, 1988) or some
other text on numerical analysis for an explanation of the bisection
root finder.
This first part screens the datafile for the cases where both Dl and
D2 are zero ; i.e., the controls, and prevents the algorithm from
bombing because of the division of 0 by 0. The value of the slope
parameters of both drugs are assumed to be negative. This would be
appropriate for a growth inhibition experiment, where survival
decreases monotonically with increasing drug concentrations. For
monotonically increasing dose-response curves, alterations must be
made in the code.
if Dl=0 and D2=0
then do;
Model.E=Emax+B;
end;
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1990 American Association for Cancer Research.
QUANTITATION
OF
DRUG
SYNERG1SM
21. Nash. J. C. Compact Numerical Methods for Computers: Linear Algebra
and Function Minimisation, pp. 136-137. New York: John Wiley & Sons.
Inc.. 1979.
22. Thisted. R. A. Elements of Statistical Computing, p. 170. Algorithm 4.2.5.
else
do;L=B;U=Emax+B;do
New York: Chapman and Hall. Ltd.. 1988.
23. SAS Institute. Inc. SAS/STAT Guide for Personal Computers, Version 6
25;H
1=1 to
Ed. Cary, NC: SAS Institute. Inc.. 1987.
(L-»-U)/2;Parti
=
24. Dixon. W. J.. Brown. M. B.. Engelman. L.. Hill. M. A., and Jennrich. R. I.
(I/ml);)
**
(Part2 - Dml *
(eds.) BMDP Statistical Software Manual. Berkeley. CA: University of
(Part3 - Dm2 *
(l/m2);)
**
California Press. 1988.
(Part4 - Diri *
(0.5/ml);)
**
= Dm2 * ((M-B)/(Emax-M*B)(M-B)/(Emax-m-B)(M-B)/(Emax-M+B)(M-B)/(Emax-M*B))
••
(0.5/m2);
25. Statistical Consultants Inc. PCNONLIN and NONLIN84 software for the
G = Dl/Partl -»D2/Part2 + Alpha*Dl*D2/(Part3*Part4)
-1;
statistical analysis of nonlinear models. Am. Statistician, 40: 52, 1986.
if G<0
26. Webb, J. L. Effect of more than one inhibitor. In: Enzyme and Metabolic
then L-M;
Inhibitors. Vol. 1, pp. 66-79, 488-512. New York: Academic Press, 1963.
else U-M;
end;
27. Elion, G. B., Singer. S.. and Hitchings, G. H. Antagonists of nucleic acid
Model.E-M;
and derivatives. Vili. Synergism in combinations of biochemically related
antimetabolites. J. Bio). Chem., 20«:477-488. 1954.
28. Berenbaum. M. C. Criteria for analysing interactions between biologically
* Different weighting options could be inserted here. Uniform
*
active agents. Adv. Cancer Res.. 35: 269-335, 1981.
* weighting, weight = 1, was used in the work reported in Greco et
*
29. Bates. D. M.. and Watts, D. G. Nonlinear Regression Analysis and Its
* al, 1990.
•
.......«,..»..,,.».«,..,«.,......,....................«............../
Applications, p. 91. New York: John Wiley & Sons, Inc., 1988.
_Height_ =•
1 ;
30. Landaw, E. M. Optimal design for individual parameter estimation in pharmacokinetics. In: M. Rowland, L. B. Sheiner, and J.-L. Steimer (eds.).
/..................*»...„...».......,...•.....««•»«»»*•.«•••»*.*.•*«...
Variability in Drug Therapy: Description, Estimation and Control, pp. 187* SAS has many options for numerical and graphical output. Here are *
* commands for a minimal set of useful numerical output options.
*
200. New York: Raven Press. 1985.
•»*«..i*«*«***»«*««*..«..*.**»*.****«**.****...»..»..»..»*«.*»*«.««.*/
31. Laskin. J. D., Evans, R. M., Slocum, H. K.. Burke, D.. and Hakala. M. T.
output out=outNLIN
predicted=E_predic
Residual=Res;
Basis for natural variation in sensitivity to 5-fluorouracil in mouse and human
Data Final;
cells in culture. Cancer Res., 39: 383-390. 1979.
set A;
set outNLIN;
32. Larson, H. J. Introduction to Probability Theory and Statistical Inference,
Proc Print;
Ed. 3. New York: John Wiley & Sons, Inc., 1982.
33. Werkheiser, W. C.. Grindey, G. B.. and Moran, R. G. Mathematical simu
.,,...,,.,,,,,..,.,».,..,,.........,,,....,.....,.........,,...,.,....
* Here is the actual run command which starts the analysis.
*
lation of the interaction of drugs that inhibit deoxyribonucleic acid biosyn
.....................................E..,..,.,,.,,.,.,,......,.»..,../
thesis. Mol. Pharmacol.. 9: 320-329, 1973.
Run;
34. Fram, R. J.. Robichaud, N., Bishov. S. D., and Wilson. J. M. Interactions of
f/î-diamminedichloroplatinum(II) with 1-rf-D-arabinofuranosylcytosine in
LoVo colon carcinoma cells. Cancer Res., 47: 3360-3365, 1987.
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* Here is the main iterative routine of the bisection root finder
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* parts, so as to make the code easier to read.
5327
Downloaded from cancerres.aacrjournals.org on June 17, 2017. © 1990 American Association for Cancer Research.
Application of a New Approach for the Quantitation of Drug
Synergism to the Combination of cis
-Diamminedichloroplatinum and 1-β-d-Arabinofuranosylcytosine
William R. Greco, Hyoung Sook Park and Youcef M. Rustum
Cancer Res 1990;50:5318-5327.
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