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A Two-Stage Response Surface Approach to Modeling
Drug Interaction
a
a
a
Wei Zhao , Lanju Zhang , Lingmin Zeng & Harry Yang
a
a
MedImmune, LLC, Gaithersburg, MD, 20878
Version of record first published: 01 Oct 2012.
To cite this article: Wei Zhao, Lanju Zhang, Lingmin Zeng & Harry Yang (2012): A Two-Stage Response Surface Approach to
Modeling Drug Interaction, Statistics in Biopharmaceutical Research, 4:4, 375-383
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A Two-Stage Response Surface Approach
to Modeling Drug Interaction
Wei ZHAO, Lanju ZHANG, Lingmin ZENG, and Harry YANG
Studies of drug combinations have become increasingly important, especially in treating malignant cancers. Researchers are interested in identifying compounds
that act synergistically when combined. Such synergy
is usually measured through an interaction index. The
existing statistical methods, in general, estimate the
interaction index using pooled data from compounds administered individually and in combination. In this article,
we propose a two-stage response surface approach. Parameters of monotherapy dose–response curves are estimated and then incorporated in estimating the interaction
index through a quadratic response surface model. Using
multiple simulation studies, we demonstrate that the new
method gives less biased estimates for both monotherapy
dose–response curves and interaction index. Also developed is a bootstrapping method that allows constructing
a confidence interval for interaction index at any combination dose levels. An example is provided to illustrate
the method.
Key Words: Bootstrap; Drug combination; Loewe additivity;
Nonlinear least squares; Simplex algorithm.
1. Introduction
Advanced tumors are often resistant to single agents.
There is an increasing trend to combine drugs to
achieve better treatment effect and reduce safety issues
(Ramaswamy 2007; Sun et al. 2011). It is desirable that
the combination drugs are synergistic; that is, optimal
treatment effect is achieved at lower dose levels when
drugs are combined. The interaction effect of combination drugs can be synergistic, additive, or antagonistic.
Commonly used models are Bliss independence (Bliss
1939) and Loewe additivity model (Loewe and Muischnek 1926). Greco, Bravo, and Parsons (1995) discussed
these two reference models in detail. Generally, the Bliss
independence is less desirable because it may incorrectly
claim synergy or antagonism when two identical drugs
are combined (sham experiment). Therefore, the reference model that we use in this article is Loewe additivity,
which has been preferred by many authors (Chou and
Talalay 1984; Greco, Bravo, and Parsons 1995; Kong and
Lee 2006; Fang et al. 2008). Based on Loewe additivity,
Berenbaum (1985) proposed to use interaction index to
quantify the interaction between two drugs. The interaction index is defined as
d2
d1
+
,
τ=
D y,1
D y,2
where τ is the interaction index at combination dose level
(d 1 , d 2 ) for drugs 1 and 2, and D y,i is the monotherapy
dose level for drug i (i = 1, 2) that achieves treatment
effect y. The ratio di /D y,i can be thought intuitively to
represent a standardized dose of drug i, and then τ can be
interpreted as the standardized combination dose. τ < 1
means that the same treatment effect can be achieved at
a lower combination dose level; τ > 1 means that more
drugs have to be given to achieve the same treatment
effect; and τ = 1 means that the treatment effects are
C American Statistical Association
Statistics in Biopharmaceutical Research
November 2012, Vol. 4, No. 4
DOI: 10.1080/19466315.2012.707087
375
Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4
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additive and there is no advantage or disadvantage to
combine them. More concisely, the three scenarios are
summarized as
⎧
⎨ < 1 synergy ,
τ = 1 additivity,
⎩
> 1 antagonism.
Many models have been proposed to estimate the
interaction index. Some models use just one parameter
to describe drug interaction across all combination dose
levels. For example, methods proposed by Greco, Park,
and Rustum (1990), Machado and Robinson (1994), and
Plummer and Short (1990) are all in this category. On
the other hand, saturated models (Lee and Kong 2009;
Harbron 2010) calculate the interaction index separately
for every combination. In this case, the number of parameters is as many as the number of combination doses.
Other modeling approaches use fewer parameters than
saturated models. For example, the response surface
model of Kong and Lee (2006) includes six parameters to
describe the interaction index. Harbron (2010) provided a
unified framework that accommodates a variety of linear,
nonlinear, and response surface models. Some of these
models can be arranged in a hierarchical order so that a
statistical model selection procedure can be performed.
This is advantageous because, in practice, simple models
tend to underfit the data and saturated models may use
too many parameters and overfit the data.
In existing models, monotherapy and drug combination data are pooled together for model fitting. Parameters
describing monotherapy dose–response curves and those
modeling drug interaction indexes are estimated by fitting
a single model. We refer to these as one-stage models.
All existing models are one-stage models. In many situations, pooling data is a good statistical practice because
it increases precision of parameter estimation. However,
in the situation of drug combination, pooling data may
compromise the accuracy of estimating drug interaction
as will be demonstrated in this article. More specifically,
we will show through simulation studies that the parameters estimated using one-stage models may significantly
deviate from their true values, and thus potentially lead
to false claims of synergy, additivity, and antagonism.
In practice, effects of monotherapy treatments are
usually evaluated before designing a drug combination
study; the knowledge gained is used to optimize drug
combination study designs. Fang et al. (2008) developed
a method of interaction analysis befitting this two-stage
paradigm. In their article, the monotherapy data are used
to determine the drug–response curve for each drug.
Then an experiment for combination study is designed
in some optimal sense, using the information from
individual dose–response curves. Finally, the interaction
376
index is modeled using combination data only. However,
the variability of individual dose–response curves (or
monotherapy data) is ignored in constructing a confidence interval for the interaction index. In this article, we
propose a two-stage method to estimate the interaction
index using a quadratic response surface model. The
parameters in the interaction index model are estimated
conditional on the estimates of monotherapy dose–
response parameters. The variances of model parameters
are calculated using the bootstrap technique (Davison
and Hinkley 2006) and their confidence intervals can be
constructed at any combination dose levels.
We also tackle a computational issue frequently encountered in drug combination analysis. Most models are
nonlinear: monotherapy dose–response models are nonlinear and interaction index models are nonlinear as well.
Nonlinear least-square (NLS) method is commonly used
to fit such models (Bates and Watts 1988; Lee et al. 2007;
Harbron 2010). NLS is available in many existing statistical packages such as R and SAS and is easy to use
once the models are correctly specified. However, NLS
becomes unreliable for complicated nonlinear problems.
We have experienced difficulties in parameter estimation using NLS, due to the singularity of Hessian matrices. Our investigation suggests that this issue is caused
by the program’s failing to calculate numerical derivatives. Since NLS uses a Newton–Raphson type of procedure to perform parameter estimation, it relies heavily
on the accuracy of numerical derivatives when the explicit form is not available. However, depending on the
complexity of nonlinear equations, numerical derivatives
may be difficult to get and NLS may stop working at
any iteration. To improve the robustness of the model
fitting, we develop a simplex method to search for parameters that maximize the likelihood function. Simplex
method is a widely used direct search method to minimize or maximize an objective function (Zhao et al.
2004).
The simplex algorithm, originally proposed by Nelder
and Mead (1965), provides an efficient way to estimate
parameters, especially when the parameter space is large.
It is a direct search method for nonlinear unconstrained
optimization. It attempts to minimize a scalar-valued nonlinear function using only function values, without any
derivative information (explicit or implicit). The simplex
algorithm uses a linear adjustment of the parameters until
some convergence criterion is met. The term “simplex”
arises because the feasible solutions for the parameters
may be represented by a polytope figure called a simplex.
The simplex is a line in one dimension, a triangle in two
dimensions, and a tetrahedron in three dimensions. Since
no division is required in the calculation, the “divided by
zero” runtime error is also avoided.
Two-Stage Response Surface Approach
The rest of the article is organized as follows. We describe the two-stage method in Section 2. In Section 3,
we provide extensive simulation studies to demonstrate
the performance characteristic of our method and show
the consequence of pooling monotherapy and combination data. An example is provided in Section 4 and a
discussion is given in Section 5.
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2. Statistical Method
In this section, parameter estimation and calculation of bootstrap variances are described in detail for
the two-stage model. In Section 2.3, we use Greco’s
model to illustrate how data are fitted using the one-stage
method.
2.1
= f ( | β̂0,i , β̂1,i , dA , dB ),
First Stage
In the first stage, the monotherapy dose effect is modeled using a median-effect (Greco, Bravo, and Parsons
1995) equation for each drug i.
i m i
E i,max Ddm,i
yi =
i m i ,
1 + Ddm,i
where Dm,i is median-effect dose of drug i, mi is a slope
parameter, yi is the monotherapy drug effect at dose di ,
and E i,max is the maximal effect of drug i. We assume
E i,max = 1 in this article and the logistic regression model
for the monotherapy dose effect data can be conveniently
written in a linear regression form
log
yi
= β0,i + β1,i log di + εi ,
1 − yi
Second Stage
In the second stage, interaction index τ is modeled
using a quadratic response surface model
di
β0,i y β 1
1,i
i=A,B exp − β
1−y
1,i
= exp γ0 + γ1 δ1 + γ2 δ2 + γ12 δ1 δ2 + γ3 δ12 + γ4 δ22
= exp (X ) ,
(2)
τ =
dA dB = 0.
The model for combination dose response can be written as
y
log
= f ( | β̂0,i , β̂1,i , dA , dB ) + e.
(3)
1−y
Assuming e follows iid normal distribution N (0, σ 2 ),
the log-likelihood function for the combination data is
obtained as
N log (2π )
− N log σ
l=−
2
2
y
N
log 1−yj j − f ( | β̂0,i , β̂1,i , d1, j , d2, j )
−
,
2σ 2
j=1
(4)
(1)
where β0,i = −m log Dm,i and β1,i = m i . In R, parameters can be easily estimated using simple linear regression function lm(). The distribution of (β̂0,i , β̂1,i )
can be approximated by a bivariate normal distribution,
ˆ i ), where ˆ i is the estimated covariance
N (( β̂β̂0,i ), 1,i
matrix for β̂0,i and β̂1,i .
2.2
where y is the response at the combination doses (d A , d B ),
= {γ0 , γ1 , γ2 , γ12 , γ3 , γ4 }, δi = log di , and X is the design matrix. The response surface model we use in this
article has six parameters so that it requires a dataset
with at least six combination dose levels; otherwise, the
model is not identifiable. If in fact there are fewer than six
combination dose levels, one can remove some quadratic
terms to make the model identifiable. If there is adequate number of combinations, higher-order terms can
also be added. Conditional on β̂0,i and β̂1,i estimated in
the first stage, Equation (2) defines an implicit function,
f , between the expected logit response, y, and the combination doses, (d A , d B ). Since f is implicit, we solve y
numerically using a bisection method. This relationship
can be symbolically written as
y E log
β̂0,i , β̂1,i
1−y where N is the number of total combination data points.
It is worth noting that this likelihood function is a conditional likelihood function, depending on β̂0,i and β̂1,i estimated in the first stage. The unknown interaction parameters, , can be estimated by maximizing Equation (4).
We propose to use the simplex method to estimate .
Conditional on β̂0,i and β̂1,i , ˆ approximately follows
ˆ ˆ | β̂0,i , β̂1,i ).
a multivariate normal distribution, N (,
Let j be the index of combination dose levels and
F = {F j,k } be the Jacobian matrix,
F j,k =
ˆ d1 j , d2, j |β̂0,i , β̂1,i )
∂ f (,
.
∂ γ̂k
Since f is implicit, its partial derivatives F j,k are also
implicit and can only be calculated numerically. Then the
estimated conditional asymptotic covariance matrix of ˆ
can be written as
ˆ = var(ˆ | β̂0,i , β̂1,i ) = σ 2 (F F)−1 .
(5)
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Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4
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By the theorem of conditional variance (Mood, Graybill, and Boes 1974), the covariance matrix of ˆ can be
written as
var ˆ = E var ˆ | β̂0,i , β̂1,i
(6)
+ var E ˆ | β̂0,i , β̂1,i .
ˆ does not exAlthough an explicit form of var()
ist, it can still be calculated numerically. We propose a
bootstrap method (Davison and Hinkley 2006) to calcuˆ First, a sample of (β̂ ∗ , β̂ ∗ ) is drawn from
late var().
0,i
1,i
β̂0,i
ˆ i ) distribution. Conditional on (β̂ ∗ , β̂ ∗ ), inN (( β̂ ), 0,i
1,i
1,i
teraction index parameters ˆ are calculated by maximizing Equation (4) using the simplex method and then
∗
∗
, β̂1,i
) is calculated using
conditional variance var(ˆ | β̂0,i
Equation (5). Repeat this exercise for a number of times
ˆ can be approximated by replacing expectation
and var()
and variance using sample mean and sample variance in
ˆ becomes stabiEquation (6). In our experience, var()
lized after 50 repeats. The 100(1 − α)% confidence interval for interaction index τ at any combination dose (d A ,
d B ) can then be calculated using the following formula:
ˆ
CI = τ̂ exp −z 1−α/2 x var()x ,
ˆ
× τ̂ exp z 1−α/2 x var()x ,
where x = (1, δ1 , δ2 , δ1 δ2 , δ12 , δ22 ) and z α/2 is the
100(α/2)th percentile of the standard normal distribution.
The framework to calculate the interaction index
is very general and other models such as Harbron’s
unified approach can be plugged in Equation (2) as
well.
2.3
One-Stage Model
There are many one-stage models in the literature; in
this article, we use the one by Greco, Park, and Rustum
(1990) as an example. The model is written as
1=
Dm,1
d1
y m1 +
d2
y m21
Dm,2 1−y
ηd1 d2
+
y 2m1
y 2m1 ,
2
Dm,1 1−y 1 Dm,2 1−y
1−y
1
where Dm,i and mi are monotherapy parameters for drug
i (i = 1, 2), the same as in Section 2.1; η is one single parameter to describe drug interaction at different
dose levels. All the parameters are estimated using the
pooled data. It is easily understood that the estimation of
monotherapy parameters depends not only on how well
the model fits the monotherapy data but also on the combination therapy data. On the other hand, the estimation of η
378
depends on monotherapy data as well, which is not desirable since monotherapy and combination therapy effects
are coupled and there is no way to quantify the interaction
effect solely due to drug combination.
3. Simulation Study
The simulation studies are designed to show that a
one-stage model is not adequate when drug interactions
are constant or varying across combination dose levels
and also to demonstrate how the second-stage response
surface model behaves when the first-stage model variances are increased. Six simulation studies are included.
Logistic regression dose–response curve is assumed for
monotherapy drugs A and B. Two sets of parameters
are taken for the monotherapy dose responses and three
sets of parameters are taken for the combination dose
response. The monotherapy models for drugs A and B
are
yA
= −2 + log dA + εA ,
log
1 − yA
yB
log
= −2.5 + 2 log dB + εB .
1 − yB
To make the simulation simple, var(εA ) and var(εB )
are assumed to be equal when generating the data, but
they are estimated separately in the actual calculation.
Two variances are used in the simulation, var(εA ) =
var(εB ) = 0.2 and 0.4. Theoretical monotherapy dose–
response curves for drugs A and B are shown in Figure 1.
Three response surface model setups for interaction indices are
Setup 1: τ = exp(−0.7),
Setup 2: τ = exp(−0.5 − 0.2δ1 − 0.2δ2 − 0.1δ1 δ2
+ 0.1δ12 + 0.1δ22 ),
Setup 3: τ = exp(0.1 − 0.2δ1 − 0.5δ2 + 0.1δ12 − 0.2δ22 ).
Six dose levels, 0.1, 0.5, 1, 2, 3, and 10, are assumed
for each of the drugs. The total data points for each simulated data are 48, including 36 data points for combination doses and another 12 for monotherapy doses. We
assume σ 2 = 0.3 as the error variance in Equation (3)
in all six simulations. Schematic plots for the interaction
index model setups 2 and 3 are shown in Figure 2.
Tables 1 and 2 show the estimations of the secondstage response surface model and their corresponding
square root mean square errors (RMSEs). The first row in
each setup contains the true parameters used in that simulation. We ran 1000 repetitions for each simulation and
recorded the estimated interaction index parameters. The
initial values of the response surface parameters were
derived from the least-square fit of the interaction index model using the combination data. The parameters
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Two-Stage Response Surface Approach
Figure 1.
Theoretical monotherapy dose–response curves for drugs A and B in simulation studies.
Figure 2. Theoretical interaction index contours for simulation setups 2 and 3. White is for strong synergy and black is for antagonism.
379
Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4
Table 1. Estimated response surface parameters when var(εA ) =
var(εB ) = 0.2. The first row in each setup is the true parameters used in
the simulation; the second row is the mean of parameter estimations over
1000 repeats; and the third row is their corresponding square RMSEs
γ1
γ2
−0.7
−0.703
0.094
0
−0.002
0.043
Setup 1
0
−0.001
0.043
γ12
γ3
γ4
0
0.002
0.02
0
0.001
0.018
0
−0.001
0.019
−0.1
−0.099
0.02
0.1
0.099
0.018
0.1
0.1
0.02
0
0.003
0.021
0.1
0.101
0.018
(d A , d B )
(0.1, 0.1)
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
−0.2
−0.201
0.025
converged to the true values in all simulations. When the
variance in the monotherapy model increased from 0.2 to
0.4, the RMSE estimated parameters also increased. For
example, RMSE of ν0 in setup 1 increased from 0.094 in
Table 1 to 0.118 in Table 2.
Tables 3 and 4 demonstrate how much the estimated
interaction indexes deviate from the true values. The first
row at the top of each table lists six combination dose
levels with equal component and the first row (τ ) in
each setup contains the corresponding true interaction
index; τ̂RS is the interaction index estimated using the
two-stage response surface model and RMSERS is the
corresponding RMSE. Also, τ̂GR and RMSEGR are for
Greco’s one-stage model and are given in rows 3 and 4 in
each setup. For setup 1 with constant interaction index, the
two-stage model is unbiased but Greco’s one-stage model
gives completely wrong calculation. For setups 2 and 3
True
Mean
RMSE
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γ0
Table 3. Estimated interaction index using two-stage response surface
model and Greco’s one-stage model at equal doses, with var(εA ) =
var(εB ) = 0.2; τ̂ is the average of the estimated interaction index over
1000 repeats and RMSE is their corresponding rooted mean square
error. Subscript RS is used to denote response surface model and GR is
used to denote Greco’s model
True
Mean
RMSE
−0.5
−0.497
0.092
−0.2
−0.204
0.045
Setup 2
−0.2
−0.2
0.042
True
Mean
RMSE
0.1
0.099
0.085
−0.2
−0.203
0.044
Setup 3
−0.5
−0.502
0.051
Table 2. Estimated response surface parameters when var(ε A ) =
var(ε B ) = 0.4. The first row in each setup is the true parameters used
in the simulation; the second row is the mean of parameter estimations
over 1000 repeats; and the third row is the corresponding square RMSEs
γ0
γ1
γ2
True
Mean
RMSE
−0.7
−0.704
0.118
0
−0.003
0.054
Setup 1
0
−0.001
0.055
True
Mean
RMSE
−0.5
−0.503
0.112
−0.2
−0.205
0.055
−0.2
−0.206
0.057
True
Mean
RMSE
380
0.1
0.098
0.102
γ12
γ3
γ4
0
0.004
0.022
0
0
0.019
0
−0.001
0.02
Setup 2
−0.2
−0.203
0.054
−0.1
−0.096
0.022
0.1
0.101
0.019
0.1
0.1
0.024
Setup 3
−0.5
−0.504
0.067
0
0.004
0.024
0.1
0.101
0.019
−0.2
−0.201
0.03
(0.5, 0.5)
(1, 1)
(2, 2)
(3, 3)
(10, 10)
0.5
0.51
0.1
0.77
0.29
Setup 1
0.5
0.5
0.5
0.5
0.04
0.05
0.72
0.72
0.23
0.23
0.5
0.5
0.05
0.73
0.25
0.5
0.5
0.05
0.74
0.26
0.5
0.51
0.09
0.79
0.31
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
2.59
2.73
0.65
2.34
0.43
Setup 2
0.84
0.61
0.85
0.61
0.07
0.06
0.63
0.43
0.22
0.19
0.48
0.48
0.05
0.32
0.16
0.44
0.44
0.05
0.29
0.16
0.41
0.42
0.08
0.25
0.17
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
3.26
3.45
0.86
1.87
1.46
Setup 3
1.71
1.11
1.72
1.11
0.15
0.09
1.29
1.05
0.45
0.11
0.65
0.65
0.06
0.86
0.22
0.45
0.46
0.05
0.77
0.32
0.13
0.13
0.03
0.54
0.42
with more complicated interaction index contours, it is
easier to observe that the one-stage model is inadequate.
When the variance in the first stage increases, the mean
estimated interaction index remains unchanged for both
Table 4. Estimated interaction index using two-stage response surface
model and Greco’s one-stage model at equal doses with var(εA ) =
var(εB ) = 0.4; τ̂ is the average of the estimated interaction index over
1000 repeats and RMSE is their corresponding rooted mean square
error. Subscript RS is used to denote response surface model and GR is
used to denote Greco’s model
(d A , d B )
(0.1, 0.1)
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
(0.5, 0.5)
(1, 1)
(2, 2)
(3, 3)
(10, 10)
0.5
0.52
0.11
0.78
0.31
Setup 1
0.5
0.5
0.5
0.5
0.05
0.06
0.72
0.72
0.23
0.23
0.5
0.5
0.07
0.72
0.24
0.5
0.5
0.07
0.73
0.26
0.5
0.51
0.11
0.78
0.31
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
2.59
2.8
0.84
2.36
0.46
Setup 2
0.84
0.61
0.85
0.61
0.08
0.07
0.63
0.43
0.22
0.19
0.48
0.48
0.07
0.32
0.16
0.44
0.44
0.07
0.29
0.16
0.41
0.42
0.1
0.25
0.17
τ
τ̂RS
RMSERS
τ̂GR
RMSEGR
3.26
3.55
1.13
1.85
1.47
Setup 3
1.71
1.11
1.73
1.11
0.18
0.11
1.28
1.05
0.45
0.12
0.65
0.65
0.08
0.86
0.22
0.45
0.46
0.07
0.77
0.32
0.13
0.14
0.04
0.53
0.41
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Two-Stage Response Surface Approach
Figure 3. Monotherapy dose–response curves for drugs A and B fitted by the two-stage response surface model and Greco’s one-stage model
using Harbron’s (2010) data.
models, but RMSEs increase for the two-stage model and
remain roughly unchanged for the one-stage model.
4. Example
We use an example studied by Harbron (2010) to
demonstrate how our method is performed in practice.
Briefly, two drugs A and B were each studied under
monotherapy dosing for nine dose levels with threefold
spacing. They were studied in combinations in a factorial
design for all of the lowest six doses, 36 combination
doses in total (data are not shown). Table 5 gives
parameter estimates and their corresponding estimated
standard errors using the two-stage response surface
Table 5.
data
Estimated response surface parameters for Harbron’s (2010)
γ0
γ1
γ2
γ12
γ3
γ4
Estimate
−0.102 −0.291 −0.046 −0.003 −0.044 −0.001
Standard error 0.145 0.066 0.048 0.016 0.024 0.017
model. If needed, information in the table can be used to
do model selection.
We compare the fitted monotherapy curves using the
two-stage model with that using the one-stage model. Figure 3 shows the comparisons for both drugs A and B. The
x-axis is dose level in log scale. Depending on drugs and
dose levels, the monotherapy dose–response fitting using one-stage models deviates from the two-stage model.
Thus, the monotherapy doses may be overestimated (underestimated) and the interaction index will be underestimated (overestimated,) leading to false positive (false
negative) claims. For drug A, the monotherapy doses using Greco’s model may be underestimated by as much
as 40% at high response levels; deviation from the twostage model can be 40% as well for drug B at low response
levels.
We present a contour plot of the upper 95% confidence interval for the interaction index in Figure 4. A
region of combination doses can be claimed synergistic if
all values in the region are less than one. It is easy to observe that the region in light gray consists of statistically
significant synergistic drug combinations. The level of
drug interaction depends primarily on drug A. Synergistic
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Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4
Figure 4. Contour plot of upper 95% confidence interval of interaction index for Harbron’s (2010) data.
interaction is observed at high doses of drug A, while the
upper bound of the confidence interval for the interaction
index remains relatively constant at all doses of drug B.
5. Discussion
Pooling data for analysis is a good statistical practice
in many situations. However, blindly following this practice in other situations such as drug combination analysis
may in fact lead to less accurate results and false conclusions. We propose a new method that estimates the
drug combination interaction index through a two-stage
approach using a response surface model. The method
first fits monotherapy models and then carries the estimated parameters and their variances to the second stage
of estimating the drug combination interaction index. For
the two-stage model, the variance in the first stage is able
to propagate to the second stage for estimating the interaction index, which may not be true for the one-stage
model. This is because in general there are far more data
points in the combination study than in the monotherapy
study so that the variances from the monotherapy part
are suffocated by the variances from the combination
part. The first-stage monotherapy data serve as a foundation to evaluate drug interactions. Failure to properly
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incorporate the variance from the first stage to the second stage easily leads to false claims of drug synergy or
antagonism.
Using simulations and an example, we compare the
performance of the proposed method with that of a onestage model. It is shown that the method produces more
accurate results than the one-stage model. The monotherapy doses estimated using one-stage models deviate significantly from true values (data not listed), leading to
biased estimation of interaction index and potential false
claims. We also propose a simplex method to estimate
the interaction index. It overcomes some of the pitfalls
of the NLS procedures used in R and other statistical
software packages. All calculations in the simulations
and example converged to proper values and no error
message was observed. Convergence means that the calculation converges both in parameters and in likelihood.
Calculation is said to reach convergence when the difference between two iterations is less than a prespecified convergence limit. However, different software may
have different ways to specify convergence limits. For
example, fminsearch() function in MATLAB allows specifying convergence limits for both parameters and likelihood, but optim() function in R allows this only for
likelihood. Nevertheless, convergence in likelihood is often equivalent to convergence in parameters. The default
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Two-Stage Response Surface Approach
tolerance limit in optim() is 1e-8 and we use 1e-20 in our
calculation.
To demonstrate, we use median response model as
the monotherapy model to calculate monotherapy doses.
This model can be easily replaced by other appropriate
models. The response surface model we use to describe
drug interaction is sufficient in most situations. However,
this model can be replaced by other models if necessary. All models described by Harbron (2010) should
apply. Using the parameter variances estimated from
Equation (6), one can further do model selection to simplify the response surface model.
Ideally, drug combination studies should be conducted in a stagewise fashion with insights gained with
monotherapy experiments being fully used to optimize the
design and analysis of drug combination experiments. In
such a context, the two-stage analysis method proposed
in this article becomes a natural method of choice to provide proper guidance and estimation of the effect of drug
combination.
[Received July 2011. Revised March 2012.]
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About the Author
Wei Zhao, Lanju Zhang, Lingmin Zeng, and Harry Yang,
MedImmune, LLC, Gaithersburg, MD 20878. (E-mail for correspondence: [email protected]).
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