Download Experimental Designs for Drug Combination Studies

Experimental Designs for Drug
Combination Studies
B. Almohaimeed & A. N. Donev
First version: 1 October 2011
Research Report No. 8, 2011, Probability and Statistics Group
School of Mathematics, The University of Manchester
Experimental Designs for Drug Combination
Studies
B. Almohaimeed and A. N. Donev
Abstract
The interest in drug combinations is growing rapidly due to the
opportunities they create to increase the therapeutic effect and to
reduce the frequency or magnitude of undesirable side effects when
single drugs fail to deliver satisfactory results. Considerable effort in
studying the mechanisms leading to the benefits of the joint action
of drugs has been matched by the development of relevant statistical methods and tools for statistical analysis of the data obtained in
such studies that allow important statistical assumptions to be taken
into account, i.e. the appropriate statistical model and the distribution of the response of interest (e.g. Gaussian, Binomial, Poisson).
However, much less attention has been given to the choice of suitable
experimental designs for such studies, while only high quality data
can ensure that the objectives of the studies will be fulfilled. We propose methods for construction of such experimental designs which are
economical and make most efficient use of the available resources. We
also propose simple but flexible experimental designs, which we call
ray-contour designs, which are particularly useful when the use of low
or high doses is undesirable and hence a standard statistical analysis
of the data is not possible.
Keywords: additivity; antagonism; ray design; ray-contour design; synergy.
1
Introduction
Combinations of drugs have been known to be highly successful in treating
different diseases for many years [1]. They provide additional opportunities
over single drug therapies to achieve sufficient therapeutic effect at a lower
1
and potentially safer dose. Extensive discussions of the scientific background
of such studies, include [1, 2, 3, 4, 5]. There are many examples of successful
combination studies in the literature, e.g. [6, 7, 8, 9, 10, 11]. Recently there
has been substantial interest in the appropriate statistical methods for analysis of data obtained in combination studies. These publications, including
[9, 12, 13, 14, 15, 16, 17, 18, 19], discuss typical methods for statistical analysis of combination studies. Some of these methods have been implemented
in statistical packages. Software developed for design and analysis of data
from bioassay is also useful. Packages recently developed using the free statistical software R [20] include drc, grofit, dosefinding [21, 22, 23, 24, 25, 26].
They are regularly improved and updated. Common feature of these packages is that they allow for a variety of statistical models to be estimated and
different assumptions about the distribution of the response of interest (e.g.
Gaussian, Binomial, Poisson) to be made. In Section 2 we summarize the
common statistical models used in the statistical analysis of data collected
in a bioassay and present ways of assessing useful statistical properties of the
results that depend on the choice of the experimental design.
Combination studies are a great deal more complex than studying the
effect of individual drugs separately. They typically take up longer time and
substantial resources compared to studies of single drugs. Therefore, it is
important that the experimental designs for such studies make use of all relevant available information in order to ensure statistically viable results and
most efficient use of the available resources. Frequently used experimental
designs require to collect data for the response of interest at different combinations (rays) of doses of the studied drugs. These ray designs [27], are
easy to implement and the results obtained using them are easy to interpret.
We discuss how economical but efficient ray designs can be constructed using information obtained in previous studies of the individual components in
Section 3. We propose an approach for design of combination studies that
takes into account the statistical analysis that will be carried out, i.e. the
type of the model that will be estimated as well as the distribution of the
experimental errors: we consider the cases when these distribution can be
Gaussian, inverse Gaussian, Binomial, Poisson and Gamma. Designs constructed this way can be useful for many experimental situations and are
particularly useful for preclinical studies. Finally, in Section 4 we provide
an extension of the ideas discussed previously and propose the use of raycontour designs that permit a variety of combinations studies to be designed
in such a way that low or high doses are possible to avoid and the statistical
2
analysis can be simplified. For example, ray-contour designs are particulary
useful for combination studies where small doses should be avoided due to
ethical considerations, e.g. in animal studies and in early studies in men
when the use of higher doses may not be desirable because of a concern for
their safety. We illustrate some benefits of using ray-contour in an example
based on the results of a pilot studied carried out at the Paterson Institute
for Cancer Research, Manchester, UK.
All discussions in the paper are limited to cases where the combination
of 2 drugs is required. However, the presented methodology can be easily
extended to more general situations.
Computer programs implementing all methods proposed in this paper
have been written using the computer language R and are available at http://
www.maths.manchester.ac.uk/ adonev/combinations.
2
Statistical analysis of combination studies
When a ray design has been used, the statistical analysis of the data involves
the estimation of the model relating the response of interest Yij and the doses
xij ,
δ−γ
Yij = η(xij , θ) = γ + ¡
(1)
¢λi + ²ij ,
1 + 10(xij −αi ,)βi
where i, i = 1, . . . , r, denotes the drug combination (ray), j, j = 1, . . . , c,
denotes the number of dose level at which the response Yij is measured, xij
is the dose, while θ = (αi βi γ δ λi ), i = 1, . . . , r, is a vector of all model
parameters. The dose xij is obtained by combining the doses dij1 and dij2
d
of the studied drugs. We denote Ri = dij1
the ratios of the drugs in the
ij2
d
ij2
combinations and pi = dij1 +d
the corresponding proportions of drug 2.
ij2
They both are the same for all rays.
The model parameters have a useful and easy interpretation: αi represents the logarithm to base 10 of the dose leading to a response which is
the average of the minimum and maximum possible response γ and δ, respectively. This dose is also known as the logIC50, or logEC50. In (1) βi
is related to the rate at which the response changes with the dose and it is
called Hill’s slope. If λi = 1, the relationship between the response and the
dose is symmetric around the dose αi . However, including λi in the model,
allows it to capture departure from such symmetry. For simplicity, in this
3
paper we do not consider the case when one of the drugs or any drug combinations are not able to achieve 100% inhibition, but the extension of our
results in such cases is straightforward.
In (1) ²ij is the experimental error which we assume to be additive. Assumptions about the distribution of ²ij are usually made based on the knowledge of the nature of the response variable. When the response is continuous
a common assumption is that ² is normally distributed with zero mean and
a variance that may depend on the response. When the experimental errors
are expected to be asymmetric around zero, a more common assumption is
that their distribution is inverse Gaussian or Gamma. When the response
of interest is counts, more suitable assumption for ²ij is that it has a Poison
distribution, while if it is binary ²ij has a Binomial distribution.
Often existing knowledge about the studied response allows for simpler
models to be used. When λ is assumed to be one, Hill model [28] is obtained.
In many cases it is possible to make further assumptions that γ = 0, or that
γ = 0 and δ = 1, resulting in simpler models. Thus, models obtained by
simplifying (1) that have 2, 3, or 4 parameters may be used.
Depending on the distributions of the experimental errors model (1) is
nonlinear or generalized nonlinear model and the estimation of its parameters
θ can be done using available software. More complicated analysis required
when more factors affect the results is also possible but is not discussed in
this paper. A case when it may impractical or not possible to collect data to
fit model (1) or a 2, 3 or 4 parameter simplification, is considered in Section
4.
We use Loewe’s [29] definition for additivity. Two compounds are considered to have additive joint action when the combination index
CI =
d1
d2
+
Dy0 ,1 Dy0 ,2
(2)
is equal to 1, where Dy0 ,1 and Dy0 ,2 are doses of compounds 1 and 2, respectively, required to achieve a defined level of response y0 , or equivalently
an inhibition level I, 0 ≤ I ≤ 100. In (2) d1 and d2 denote doses of the
two compounds which when used in combination lead to the same level of
the response. For example, y0 can be chosen to correspond to the IC50 (or
EC50) dose, when I = 50, i.e. the dose required to obtain 50% of the maximum possible effect. In a study looking for an improvement of efficacy, it
is desirable for the two compounds to have synergistic effect. Then CI < 1.
4
Alternatively, when the response is related to the drugs’ safety (e.g. existence, number or severity of undesirable side effects) it is hoped that the joint
action of the drugs will be antagonistic, hence CI > 1. Desirable effect may
exist only for some drug combinations. The estimation of the combination
index and the evaluation of its statistical and biological interpretation plays
an important role in combination studies.
The values of the parameters αj in (1) are used to calculate the combination indices (2) for all studied drug combinations. [30, 31] show that when
² ∼ N (0, σ 2 ), the distribution of the logarithm of the combination index has
approximately a normal distribution. This makes the interpretations of the
results easy: a test for the statistical hypothesis that the combination ratio
for a particular drug combination is zero is equivalent to testing whether or
not the joint action of the drugs in that combination is additive.
The accuracy of the estimate of the combination index depends on the
experimental design used to collect the data and on the variability in the
study. The Fisher’s information matrix for the model parameters holds useful
information about the statistical properties of the estimated model. Firstorder approximation of this matrix is given by
M (θ) = X T W X,
(3)
where X is the n × p design matrix, n is the total number of observations and
p is the number of the model parameters. The columns of M are given by the
partial derivatives of the model with resect to the model parameters evaluated
at the observations. Appendix B lists the formulas for these calculations when
model (1) is used.
The information matrix (3) depends on the matrix W . When the observations are independent, W is diagonal with diagonal elements the variances
of the observations, i.e.
¡
¢
W = diag w1 w2 · · · wn .
(4)
The ith diagonal element of the matrix W depends on the distribution
of the experimental errors and is given by
µ ¶
dµ
−1
wi = V (µ)
,
(5)
dη
where µ is the mean of Y evaluated for the model parameters values. Table
5 in Appendix A provides the formulas for these entries when the distribution of ² is assumed to be Gaussian, inverse Gaussian, Binomial, Poisson or
5
Gamma. When simpler models with 2, 3 and 4 parameters are needed, the
corresponding elements of the matrix W are obtained by setting γ = 0, δ = 1
and λ = 1 as necessary.
The approximate standardized prediction variance at a dose xij is
d(xij , θ) = nwx−1
f T (xij , θ)M −1 (θ) f (xij , θ),
ij
(6)
where f (xij , θ) for model (1) is defined by (14) given in Appendix B. An
experimental design that minimizes the maximum of (6) over the design
region is called G-optimum.
The volume of the confidence region for the estimates of the model parameters is proportional to the inverse of the determinant of the Fisher’s
information matrix (3). The criterion requiring minimization of this volume,
and hence the maximization of the determinant of (3), is called D-optimality.
However, as the elements of (3) depend on the true values of θ, the determinant can only be calculated for specific values of θ. The same extends also
to d(xij , θ). This creates difficulties for comparing and constructing experimental designs with respect to these useful criteria. We discuss how they
can be overcome in the next section. For a general review of the theory of
design optimality, see [32].
3
Experimental designs
We consider a number of typical experimental scenarios and present statistical methodology and tools for the construction of suitable designs for combination studies. However, the choice of experimental designs for combination
studies is affected by many considerations. The final choice of a design may
be influenced by various additional scientific and practical considerations.
We divide the experimental designs in three groups, each suitable for
specific experimental situations:
• serial dilutions ray designs,
• small efficient ray designs,
• ray-contour designs.
Ray-contour designs are the subject of the next section. A serial dilution
design is specified by the maximum dose (MD) that is used, the number
6
of doses (ND= c) and the dilution factor (DF). In the case of combination
studies, the dose is the total of the doses of the ingredients of the drug combination. These designs ensure good coverage of the different doses of the
studied combinations and excellent statistical properties when the parameters of the design are suitably chosen. [33] provide advice about how MD,
ND and DF can be chosen so that the required resources are minimized.
They also provide computer code for their construction. However, the use of
such experimental designs for combination studies is expensive. Also, these
designs lack the sophistication to fully take into account the relevant model
assumptions (e.g. model and distribution of experimental errors) and therefore may use badly the experimental resources.
We devote the rest of this section to the construction of small and efficient ray designs which can be obtained using a specific design criterion
for their construction. Similar designs have been used in monotherapeutic
studies. For example, [34] propose the use of an optimality criterion for the
construction of such designs and the observations are normally distributed
with homogenous variance. As it was shown in Section 2 the elements of the
information matrix depend on the unknown parameter values θ. One possibility to construct a D-optimum design for a specific combination study is to
use plausible values for the model parameters in order to calculate the design
criterion. The resulting designs are called locally D-optimum experimental
designs. Typically locally D-optimum designs for estimating nonlinear models are saturated, i.e. require as many distinct observations as the number
of the estimated parameters. In the case of a combination study with r rays
and c = p doses used for each ray, c × r drug combinations need to be tested,
including a suitable number of positive and negative controls, i.e. measurements of the response when no drug is used and when a maximum effect can
be seen. Clearly such designs are very economical and at the same time they
make most efficient use of the experimental resources. We extend this idea
to the cases when generalized nonlinear models are needed and consider the
specific challenges that the combinations studies pose.
A combination study will typically follow a careful study of the individual
affects of the components of the studied drug combinations. As [35] point
out, the estimates of the parameter values α1 and α2 obtained at that stage
can be used to calculate the values αi , i = 3, . . . , r, for all studied drug
7
combinations. If the joint action of the drugs is additive,
α̂i =
α1 α2 (1 + pr )
,
α1 + pr α2
i = 3, . . . , r.
(7)
In order to estimate the remaining corresponding parameters in the case
of additivity a functional relationship has to be assumed about how they
change with the ratio of the components. In our experience, the values for βi
and αi , i = 3, . . . , r can be successfully estimated using a linear interpolation
between the corresponding values for the individual drugs, while γ, δ and
λi can be chosen in the same way as in the study of the individual drugs.
However, it is clear that the parameter values used to obtain locally Doptimum designs may not be accurate and therefore, the optimum will only
be approximately optimum.
An alternative approach to construct small efficient ray designs is to
choose the doses in such a way that the designs are approximately D-optimum
for ranges of parameter values, or for their prior distribution. Such designs
are called pseudo-Bayesian D-optimum experimental designs. In general they
may require a larger number of doses than the local D-optimum designs
but they are more robust against possible discrepancy between the assumed
model parameter values and those observed in the experiment. PseudoBayesian D-optimum experimental designs are usually also approximately
D-optimum. They are generally considered more robust against misspecification of the model parameters.
Whatever choice is made, i.e. to use locally or pseudo-Bayesian Doptimum designs, the solution of the resulting optimization problem is helped
by the result of the General Equivalent Theorem of Kiefer and Wolfowitz [36]
as it provides a way to verify whether the constructed design is optimum with
respect to the chosen criterion of optimality for each studied drug combination. The theorem states that when the design is D-optimum, the maximum
of the standardized prediction variance (6) is equal to the number of the
model parameters, p and it is attained at the tested doses. In this case the
experimental design is both D- and G-optimum.
3.1
Example 1. Locally D- and G-optimum designs
We illustrate the construction of locally and pseudo-Bayesian D-optimum
experimental designs for combination studies for a variety of statistical models and possible error distributions with examples. Table 2 lists the locally
8
D- and G-optimum designs for model 1 and the values of the model parameters as given in Table 1 under an assumption for additive joint action of
two drugs when Gaussian, Poisson, Binomial, Gamma, and inverse Gaussian
distribution for the experimental errors are assumed. The ratios of the drug
combinations for seven rays are chosen to be (0, 0.2, 0.3, 0.5, 0.7, 0.8, 1),
respectively. The dose range was chosen to include doses up to 10 units. The
designs can be easily recalculated for other choices. In all cases the maximum
available dose was included in the design. Therefore, 0 max0 is used in the
tables of results to indicate that the optimum design point is located at the
maximum available dose.
Figure 1 shows the standardized prediction variance of the response 6
over the design region. In order to save space, this is done only for one of the
rays (Ray 4). The plots for the remaining rays are similar. The standardized
prediction variance do not exceed the number of model parameters, and
it achieves its maximum at the design points, indicated by the dots (•).
Therefore, based on the result of the General Equivalence Theorem it is
clear that the designs are both locally D- and G-optimum.
3.2
Example 2. Pseudo Bayesian D- and G-optimum
designs
Similarly to Example 1, Table 3 lists the pseudo Bayesian D- and G-optimum
designs for model (1). The values of the model parameters used to construct
the designs are given in Table 1, while Figure 2, shows the corresponding
standardized prediction variances of the response for Ray 4. For the chosen
design regions, the pseudo-Bayesian D-optimum designs are saturated, but
this may not be the case or other design regions and model parameters values.
4
Ray-contour designs
The aim of these designs is to
• allow for local synergy or antagonism to be detected;
• avoid the use of too small or too high doses if necessary;
• simplify the statistical analysis of the data.
9
Table 1: Parameters values for Example 1 and Example 2.
4
6
8
10
5
4
3
2
1
0
4
6
6
8
10
4
3
4
6
Dose
5
4
3
2
1
4
6
8
10
Dose
Figure 1: Standardized variance for local D- and G-optimum design for combination ray 4.
10
10
2
2
0
2
8
1
0
Inverse Gaussian errors
0
10
0
4
3
4
8
5
Gamma errors
Dose
Standardized prediction variance
2
Binomial errors
2
2
0
Dose
1
0
Poisson errors
Dose
Standardized prediction variance
2
5
0
Standardized prediction variance
0
1
2
3
4
5
Gaussian errors
0
Standardized prediction variance
Standardized prediction variance
Parameters
α1
α2
β1
β2
γ
δ
λ σ2 κ υ
n
Example 1
0.20
0.18
0.94
0.89
0.1 1.1 -1.5 1 0.5 1.5 2 × p
Example 2 [1.31,3.01] [1.12,2.81] [0.50,1.15] [0.30,1.11] 0.1 1.1 -1.5 1 0.5 1.5 2 × p
Table 2: Locally D- and G-optimum designs for
c = 7).
Doses
Distribution
Ray 1
2
3
1
0 0.29 2.25
2
0 0.30 2.26
3
0 0.30 2.27
Gaussian
4
0 0.32 2.31
5
0 0.33 2.35
6
0 0.34 2.37
7
0 0.35 2.40
1
0 0.09 0.66
2
0 0.10 0.67
3
0 0.10 0.68
Poisson
4
0 0.11 0.70
5
0 0.11 0.71
6
0 0.12 0.72
7
0 0.13 0.74
1
0 0.12 0.79
2
0 0.12 0.80
3
0 0.12 0.81
Binomial
4
0 0.13 0.83
5
0 0.14 0.85
6
0 0.14 0.86
7
0 0.15 0.87
1
0 0.20 1.40
2
0 0.20 1.41
3
0 0.21 1.43
Gamma
4
0 0.22 1.44
5
0 0.23 1.48
6
0 0.23 1.50
7
0 0.24 1.51
1
0 0.17 1.25
2
0 0.18 1.26
3
0 0.18 1.27
Inverse Gaussian
4
0 0.19 1.30
5
0 0.20 1.33
6
0 0.21 1.34
7
0 110.22 1.36
combination studies (r =
4
6.56
6.57
6.58
6.61
6.65
6.66
6.69
3.14
3.15
3.16
3.19
3.21
3.22
3.25
3.35
3.37
3.38
3.40
3.42
3.43
3.46
5.14
5.15
5.16
5.18
5.20
5.23
5.26
4.83
4.84
4.85
4.88
4.91
4.92
4.95
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
5
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
Table 3: Bayesian D- and G-optimum designs for
7).
Doses
Distribution
Ray 1
2
3
1
0 0.61 3.07
2
0 0.61 3.08
3
0 0.62 3.09
Gaussian
4
0 0.62 3.10
5
0 0.62 3.11
6
0 0.63 3.11
7
0 0.63 3.12
1
0 0.26 1.27
2
0 0.26 1.29
3
0 0.27 1.30
Poisson
4
0 0.27 1.31
5
0 0.27 1.33
6
0 0.28 1.34
7
0 0.28 1.36
1
0 0.31 1.47
2
0 0.32 1.49
3
0 0.32 1.50
Binomial
4
0 0.32 1.52
5
0 0.33 1.54
6
0 0.33 1.57
7
0 0.34 1.57
1
0 0.46 2.21
2
0 0.47 2.22
3
0 0.47 2.23
Gamma
4
0 0.48 2.24
5
0 0.48 2.26
6
0 0.48 2.27
7
0 0.48 2.29
1
0 1.04 4.30
2
0 1.04 4.30
3
0 1.04 4.30
Inverse Gaussian
4
0 1.03 4.29
5
0 1.03 4.29
6
0 1.03 4.29
7
0 121.03 4.29
combination studies (r =
4
7.16
7.17
7.17
7.18
7.19
7.19
7.20
4.29
4.32
4.34
4.37
4.41
4.43
4.47
4.54
4.58
4.60
4.64
4.68
4.70
4.75
6.06
6.08
6.09
6.12
6.14
6.15
6.17
7.97
7.97
7.97
7.97
7.96
7.96
7.96
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
5
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
(max)
4
6
8
10
5
4
3
2
1
0
2
4
6
6
8
10
4
3
4
6
Dose
5
4
3
2
1
4
6
8
10
Dose
Figure 2: Standardized variance for Bayesian D- and G-optimum design for
combination 4.
13
10
2
2
0
2
8
1
0
Inverse Gaussian errors
0
10
0
4
3
4
8
5
Gamma errors
Standardized prediction variance
Binomial errors
Dose
Standardized prediction variance
0
Dose
2
2
Poisson errors
Dose
1
0
Standardized prediction variance
5
4
3
2
1
0
2
5
0
0
Standardized prediction variance
Standardized prediction variance
Gaussian errors
This is achieved by including in the experimental design doses in such a way
that if the joint action of the drugs is additive, the expected response to the
j th dose of each ray is the same and equal to a predetermined inhibition
level Ii %, i = 1, . . . , r. A similar suggestion and a discussion of the possible
benefits of using such an experimental design is given in [37]. However, a
simple situation is only considered.
The doses of a ray-contour design can be calculated by first calculating the
expected values of α̂i , i = 1, ... . . . , r, using equation (7). Then, the combined
dose for each drug combination can be calculated as
IC(Ij ) = (Ij /(100 − Ij ))−βj 10αj ,
(8)
where βi , i = 1, . . . , r, are estimated by interpolation between β1 and β2
obtained in previous studies. The individual doses are obtained using the
chosen proportions pi of the drugs in the combinations. Therefore, the rays
of a ray-contour design are specified by the proportions pi , i = 1, . . . , r, of the
studied drug combinations, while the contours are specified by the selected
inhibition levels.
The choice of the inhibition levels Ii %, i = 1, . . . , c, as well as their number, can be made by taking into account practical considerations and research
objectives. For example, in an in vitro study, a ray contour design would allow the experimenter to see whether synergy or antagonism is exhibited only
at specific dose ranges. Also, these designs will be useful in early animal
combination studies where in the absence of reliable information about the
suitable dose range, the experimenter may want to avoid using too high
doses. This approach also provides an easy way to design dose escalating
clinical trials for combination studies. In other clinical trials, for example
when the drug-combination is an anaesthetic, the experimenter may prefer
to avoid unethically low doses. In this case the contours can be chosen to correspond to relatively high inhibition levels. No existing experimental designs
are suitable in these cases.
The statistical analysis of data obtained using a ray-contour design is simple and does not require fitting model (1). As a result of the construction of
the experimental design, when the joint action is additive the expectations of
the observations of the response taken at each contour (i.e. inhibition level)
should be the same for all studied drug combinations. Therefore, the analysis of variance of the data (ANOVA) can be used to the test for synergy or
antagonism at each inhibition level. ANOVA is a method well described in
14
many textbooks; see for example [38]. It is also implemented by all statistical
packages.
Example 3. Cancer Study. The aim of the study was to investigate the
combined efficacy of a conventional cytotoxic drug (drug 1) and a novel
mechanism based agent (drug 2) in colorectal cancer cell lines, with the ultimate intension of combining the two agents in the clinic. The ratios of the
doses of Drug 2 to Drug 1 used in the combinations were: 0.5, 1.0, 2.0, 4.1
and 8.2 (these rays were numbered in reverse order), while the percentages
inhibition to be studied were chosen to be 20, 35, 50, 60, 70, and 80. Data
were also collected for the individual drugs (Ray 1 and Ray 2). The response
was measured also at positive and negative control. The ray-contour experimental design that was used is given in Table 4 and was obtained as described
above using the results of previous studies where the IC50 and Hill slopes
were estimated as 4.62 µM and 9.47 µM , and -1.12 and -3.82, for Drug 1
and Drug 2, respectively. The design was replicated on 3 plates.
A plot of the means of the observations is shown in Figure 3. The benefits
of using a ray-contour design are obvious - one does not need to perform a
complicated statistical analysis in order to see that antagonism is observed
at Ray 3 where Drug 2 is used 8.2 time more than Drug 1 in the combination.
The synergy reduces and then diminishes as the ratio reduces.
Many research laboratories that use ray designs with just a single drug
combination been defined by the ratio of the IC50 values of the two drugs
would have missed this useful result. Clearly, the choice of the drug combinations is very important to the success of the study. Usually it is not known in
advance what drug ratio may possess useful properties. Therefore studying
as large as possible number of ratios appears to be desirable. However, the
experimental resources are usually limited. A compromise could be reached
by keeping the number of rays large, while reducing the number of contours
(i.e. the doses on each ray). Note that the contour corresponding to 50%
inhibition ensures largest sensitivity to detect non-additive joint drug action.
We trust the ray-contour designs will find many useful traditional and new
applications that will make them popular with experimenters. For example,
they will be well suited to new research devoted to identifying interaction
thresholds [39].
15
100
80
60
0
20
40
Inhibition
1
2
3
4
5
6
7
Ray
Figure 3: Means with error bars at one standard deviation for the observations at different inhibition levels.
16
Table 4: Ray-contour design for Example 3.
Ray
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
Drug ratio
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
inf
inf
inf
inf
inf
inf
inf
inf
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.2
4.1
4.1
4.1
4.1
4.1
4.1
4.1
4.1
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Inhibition (%)
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
0
20
30
40
50
65
80
100
17
Dose 1
100.0
15.9
9.8
6.6
4.6
2.7
1.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
10.9
1.5
1.3
1.1
0.9
0.7
0.5
0.0
19.6
2.7
2.2
1.8
1.5
1.1
0.7
0.0
32.8
4.7
3.6
2.9
2.3
1.6
0.9
0.0
49.3
7.2
5.3
4.0
3.1
2.0
1.1
0.0
66.1
9.9
6.9
5.0
3.7
2.3
1.2
0.0
Dose 2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
100.0
13.6
11.8
10.5
9.5
8.1
6.6
0.0
89.1
12.3
10.3
8.8
7.6
5.9
4.1
0.0
80.4
11.2
9.1
7.6
6.3
4.6
3.0
0.0
67.2
9.6
7.4
5.9
4.7
3.3
1.9
0.0
50.7
7.4
5.4
4.1
3.2
2.0
1.1
0.0
33.9
5.1
3.5
2.6
1.9
1.2
0.6
0.0
Acknowledgements. The authors are grateful to Dr. Christopher Morrow,
Paterson Institute for Cancer Research, Manchester, UK, for providing the
data of Example 3.
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21
Appendix A.
Table 5: The ith ,i = 1, . . . , n, diagonal element of the W matrix for model
(1).
Distribution
Normal with mean µ and variance σ 2 ,
Expression
i.e.
Yi ∼ N (µ, σ 2 )
wi = σ 2
Poisson with mean µ, Yi ∼ Pois (µ)
wi = eη(xi ,θ)
Binomial with n trials, and p probability of success,
i.e. Yi ∼ B (n, p)
and the variance is Var (Yi ) = np (1 − p)
wi = neη(xi ,θ)
Gamma with scale ϑ and shape κ parameters
i.e. Yi ∼ Γ (κ, ϑ)
and variance Var (Yi ) = κϑ2
wi = κη(xi , θ)−2
Inverse-Gaussian with mean µ and shape parameter υ
i.e. Yi ∼ IG (µ, υ)
and variance Var (Yi ) =
22
wi =
µ3
υ
υ
4
(η(xi , θ))3/2
Appendix B. The derivatives of model (1) with respect to the parameters
(αi , βi , γ, δ and λi , respectively) are
∂ηk (xij , θ)
(δ − γ) λi 10(xij −αi )βi βi ln (10)
=¡
¢λi ¡
¢,
∂αi
1 + 10(xij −αi )βi
1 + 10(xij −αi )βi
(9)
∂ηk (xij , θ)
(δ − γ) λi 10(xij −αi )βi (xij − αi ) ln (10)
=− ¡
¢λi ¡
¢,
∂βi
1 + 10(xij −αi )βi
1 + 10(xij −αi )βi
(10)
¢λi ´−1
∂ηk (xij , θ) ³¡
= 1 + 10(xij −αi )βi
,
∂δ
(11)
³¡
¢λi ´−1
∂ηk (xij , θ)
= 1 − 1 + 10(xij −αi )βi
∂γ
(12)
¡
¢
(δ − γ) ln 1 + 10(xij −αi )βi
∂ηk (xij , θ)
.
=−
¢λi
¡
∂λi
1 + 10(xij −αi )βi
(13)
and
Therefore
f T (xij , θ) =
³
∂ηk (xij ,θ)
∂αi
∂ηk (xij ,θ)
∂βi
∂ηk (xij ,θ)
∂δ
23
∂ηk (xij ,θ)
∂γ
∂ηk (xij ,θ)
∂λi
´
.
(14)