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Estimating pA2 values for different designs
Estimating pA2 Values for Different
L. OTT, 5.
Designs
WEINER, F-i. CHENG, AND J. WOODWARD
The method for analyzing data leading to the estimation of a pA2 depends on the experimental design. Three different experimental designs are discussed. For each design, we provide a brief description of experimental situations that give rise to the design, a means for verifying relevant assumptions, and sample data with summa~ calculations.
Key Words:
Estimate; Experimental design; Agonist; Antagonist; Prediction lim-
its; pA2; DR2
INTRODUCTION Since its introduction, pA, (Schild, 1947) has become a popular way to express the potency of a competitive antagonist and a useful tool to study drug receptor interaction. The term pA, is defined as the negative logarithm of the molar concentration of an antagonist that produced an x-fold shift in the agonist dose-response curve. Based on the mass action law, Schild (1957) derived the following equation: log(dose ratio -
I) = -log
KB + log [Bl
where KB is the dissociation constant of the antagonist for the receptor and [Bl is the concentration of the antagonist. A plot of log (dose ratio - 1) against -log [B], called a Schild plot, gives a straight line and allows one to calculate pAz when the dose ratio is 2; in general, a plot of log[dose ratio -(x - 1)J versus -log [B] can be used to calculate PA, when the dose ratio is x. Although pA, values have been used as measures of competitive antagonism in vitro and in vivo, some scientists (Brittain and Levy, 1976) use DR, values (e.g., dose of antagonist in mg/kg, to produce an x-fold shift in dose response curves) to express potencies of competitive antagonists. When used in vivo, DR, and PA, can be interchangeable by converting the dose of antagonist in mglkg into mole/kg (or vice versa). In the past, the computed pAz value from an experiment was often taken as an absolute rather than an estimated value. Recent publications have signaled a change in this thinking by presenting methods that provide an interval estimate of the PA,, From the Departments
of Biostatistics,
Preclinical
Pharmacology,
Merrell
Research
Center,
Cincinnati,
Ohio. Address
reprint
requests
to Dr.
5, 1980;
revised
Lyman
Ott,
Department
of Biostatistics,
2110 E. Calbraith
Rd.,
Cincin-
nati, OH 45215. Received
April
and accepted July 28, 1980.
75 iournal of
Pharmacoio~ical
G 1981 Elseviei
North
Methods
Holland,
5, 75-92 11981i
Inc., 52 Vanderbilt
Avenue. New York. NY 10017
76
1. Ott, et al. or an estimate
of the pA, with a standard
Stone and Angus, with the situation
for each agonist-antagonist This
paper expands
three
commonly we will
verifying tions
methods
combination
discuss
the relevant
for each design.
order
to estimate
(Tallarida
et al., 1979; MacKay, 1978;
combination.
upon these
used experimental
one agonist-antagonist design,
error
1978; Waud and Parker, 1971). However these reports deal only where different experimental units (dogs, tissues, etc.) are used
estimation
pA2 values
In two of these
Sample
modifications
any pA, (or DR,)
by providing
designs,
is applied to each experimental of the pA, and statistical
assumptions. Simple
for estimating
designs.
tests
than
For each
appropriate
data are used to illustrate can be made to these
more
unit.
for
the calculaprocedures
in
value.
METHODS
Notation B, : jth level of antagonist (j = 1, 2, . . . . b) in moles/kg; A,: ith level of agonist (i = 1, 2, . . . . a); unit (animal or tissue) R,, : response when experimental
4,:
agonist and jth level of antagonist; number of experimental units treated with
is treated with
ith level of agonist
ith level of
and jth level of
antagonist.
General Assumptions Assumption dose-response pressed
using
for a pA, Analysis
7. For a fixed level of B,, particularly curve, a linear
R, is linearly regression
related
within
to log A,. This
the middle
equation.
R,, = ci, + B, log A; + E;,
FIGURE 1.
portion
relationship
Parallel trend lines for different levels of Bi.
of the
can be ex-
Estimating pAa Values Bl
I
I
log dl
log d2
(Control)
log da
108 agonist FIGURE 2.
B2
B4
B3
log 4, or
log
Ai
Relationship between di and response of magnitude R on antagonist j.
where pi is the slope of the linear regression equation and cyiis the intercept. The symbol eij represents a random error that accounts for the fact that not all observations fall on the linear trend line, aj + pi log A;. Assumption 2. The trend lines for different levels of Bj are parallel (ie, Pi = f3) (see Figure 1). Assumption 3. The common slope p for the separate dose-response lines is different from zero (1 p ) # 0). Assumption 4. If dj denotes the predicted dose of agonist corresponding to a response of magnitude Ron antagonist j (j = 1,2, . . . . b) and d, denotes the predicted dose of agonist for the control group (no antagonist), the variable log(dj/d, - 1) is linearly
related to log(l/Bj) (see Figures 2 and 3). 5. The slope p’ in the trend line (Y’ + p’ log(l/B$
Assumption
log
dj/dl-1 (
>
I FIGURE 3.
is equal to -1.
linear relationship
- log[antagonistI or log(llB,)
between variable log (did,
-
1) and log (l/Bi).
77
78
1. Ott, et al.
Estimation of the pA2 The pA2 value of an antagonist for a given agonist lOg(l/Bj) corresponding to a two-fold shift to the right in curve of Figure 2. Or, equivalently, the pAz value is line a’ + j3’ lOg(l/Bj) intersects the log(l/B$ axis (see The original equation log(dose ratio -
is defined to be the value of the log agonist dose-response the point at which the trend Figure 3).
1) = pA1 + log B
derived by Schild (1957) and expanded upon by Arunlakshana and Schild (1959) is based upon the mass law for drug-receptor interaction. This equation is equivalent to the equation log(dj/dl
-
1) = (Y’ + p’ lOg(l/Bj)
where pAZ = -_(y’/p with p’ = -1. In practice the constants “jr p, (Y’ and p’ for the models R, = 01, + /3 log A; + E,j and log(dj/d,
-
1) = (Y’ + p’ log(l/Bj)
+ E’
are unknown and must be estimated using experimental data. Consequently, the pAz value computed from sample data is also an estimate of the actual unknown value and should be accompanied by a measure of variability in the form of a prediction interval or standard error. In this paper we will consider estimation of the pA, for a given agonist/antagonist relationship under the following three experimental settings: Case I: Different experimental units (eg, animals or tissues) for each combination of levels of agonist and antagonist. Case II: Different experimental units for each level of antagonist. Case III: Each experimental unit receives all combinations of agonist and antagonist. For each of the three experimental situations we will present the tools necessary for analyzing sample data leading to an estimate of pAz. Since the experimental designs are different, the analyses (which reflect the experimental designs) are also different. We will find that not all of the general Assumptions (I-5) can be verified rigorously for each of the designs. For some situations, a biological judgment must be made; in others, the assumptions can be put under more careful analytical scrutiny. The presentation for each case will contain the following: 1. A brief description of experimental situations for which the design is appropriate. 2. A checklist for verification of assumptions l-5. 3. Sample data with summary calculations for obtaining an estimate of the pA, with a measure of its variability. 4. Discussion
Estimating pA2 Values RESULTS Case I: Different Experimental Units feg, Animals or Tissue) for Each Combination of levels of Agonist and Antagonist Case I in which each individual animal receives only a single dose of agonist and a single dose of antagonist is commonly used for pharmacological evaluation involving quanta1 estimation of drug antagonism. Examples of this experimental design include grouped amphetamine toxicity and its antagonism by drugs, apomorphine- or amphetamine-induced stereotypy and its antagonism by drugs, antagonism of analgesic effects of morphine by naloxone, and antagonism of physostigmineinduced death by anticholinergies. The analyses of data from a study where different experimental units are used for each agonist/antagonist combination is straightforward and provides tests for all assumptions underlying the competitive antagonism model. We have included a flow chart for the analysis in Figure 4 to illustrate the steps of the analysis which lead to estimation of a pAZ. The analysis incorporates an analysis of variance for a parallel line assay (Finney, 1964) and inverse prediction from a linear regression equation (Ott, 1977). Example (Case I) In Case I, each individual animal receives only one dose of agonist and one dose of antagonist. The data shown for this example were obtained from a graphical representation by McGilliard and Takemori (1978) who measured the analgesic effect of morphine in mice and the antagonism by naloxone. The analgesic effect of a subcutaneous injection of morphine was assessed by the tail-flick method as an increase in the reaction time of the animal (tail flick) to avoid a painful stimulus. A reaction time of 1.61 2 0.02 seconds (mean + SE) was recorded in the vehicletreated mice. For this experiment, an increase in reaction time greater than three SD was considered a significant analgesic effect (Table 1). Assumptions l-3 were verified for these data using tests appropriate for a parallel
TABLE 1
Percentage of Mice with a Significant Increase in Reaction Time
AGONISI (mgikg)
ANTAGONISTDOSE (moles/kg) --.-_--
~.___. CONTROL
O.11x1O-b
0.22x10-"
_.
0.44~10-~
0.88~10~"
__” -3.52~10-~
2.5
10 WIO)"
5
40 (4110)
18 (2/11)
IO
60 (6110)
40 (4110)
25 121'8)
70 (7110)
50 (5110)
12 (118)
89 (8/9)
40 (4/10)
25 (218)
80 (8110)
50 (S/IO)
12 (l/8)
70 (7/10)
40 (4110)
20 40 80 160
60 E/IO)
320 640 a Numbers
in parentheses
effect of the total
number
indicate of animals
the number tested.
of animals
showing
a significant
analgesic
79
00
1. Ott, et al.
TEST ON ONIATICN .=Rm LIhEIRIlY (AI) IWE RlEs=msE CUWES NWIN*R?
-
FIGURE 4. Flow chart for the estimation of pA2: s = statistically significant at the 0.05 level; ns = not statistically significant (continued on following page). line assay (Finney slope
1964). Table 2 gives the estimated
intercepts
k$‘s) and common
(P) for the model R, = OL, + p log A; + E,,
In addition, the predicted log agonist antagonist based on a fixed response umn 5 of Table 2. Column
dose (log d,) was computed for each level of R = 50%. These results are displayed in Col-
6 gives values log(djldl
of the transformed -
1)
data
Estimating pA, Values
x No
STW
FIGURE 4.
Flow chart for estimation of pA, (continued from previous page).
TABLE 2 Case I Data: Estimated Intercepts and Common Slope for Log Agonist Dose-Response Curves ANTAGONIST
6,
(moles/kg)
l/B, -
lNTERCEpT
ir,
COMMON
SLOPE
f3
log
d, at R = 50%
log(d,/dI- I)
-26.6059
90.5225
0.8463
0.11 x 10
h
9090909.09
-47.8559
90.5225
1.0810
0.22 x 10
b
4545454.55
--63.1059
90.5225
1.2495
0.1848
0.44 x 10
b
2272727.27
-101.0225
90.5225
1.6683
0.7511
0.88 x lomb
1136363.64
-123.9392
90.5225
1.9215
1.0370
3.52 x IO-"
284090.91
-189.4392
90.5225
2.6451
1.7918
Control
-0.1446
81
82
L. Ott, et al.
which are needed to fit the line
- 1) = CX’+ f3’ lOg(l/Bj) + E’
log(d,/d,
The test on deviation from linearity for this model was nonsignificant and hence we assumed Assumption 4 was verified. The estimated intercept (&) and slope (6’) for this model were 89106 and -1.3002 with corresponding standard errors equal to 0.4081 and 0.0646, respectively. Although the slope of the line is significantly different from -1, the difference was considered sufficiently small to be of no consequence; thus we substituted into the formula for the estimate of pAZ to obtain PA:! =
- 8.9106
= 6.85 or 0.14 x 10m6 in moles/kg
- 1.3002
The corresponding 95% prediction limits for the pAZ are 6.74 to 6.99 (or 0.10 x lO-‘j to 0.18 x low6 moles/kg). The estimated value of the pA2 and corresponding prediction limits can be reexpressed as an estimate of the DR, with appropriate prediction limits. For these data, with a molecular weight of 363.83 for naloxone hydrochloride we have Dl& = antilog( -PA,)
x 363.83 x IO3
= 0.051 mg/kg Transforming each of the limits in the same way we find the 95% prediction limits fin mgikg) on the DRz to be (0.037, 0.066). To summarize our calculations for Case I, we used the sample data (Table 1) to check Assumptions l-3. Having encountered no difficulties with these assumptions, we fit the model R,, = a, + I3 Iog A, I- E;j to obtain estimates of the individual intercepts and common slope. These results, summarized in Table 2, were used to calculate the log d, based on a response of R = 50%. Data from Table 2 were then used to fit the model log(djd,
-
I) = cxt + 8’ IogWB,)
+ E’
Because there was no significant deviation from linearity and the estimated slope 6’ was near - 1, we computed the estimate of pA, as p& = - &‘/fi’ and re-expressed it as a D& in mg/kg of the antagonist dose. Details of the calculations for obtaining 95% prediction limits were not presented because the formulas can be quite tedious (Snedecor and Cochran 1967; Ott, 1977). Fortunately, some linear regression software packages offer an option for computing these limits (eg, SAS,1979 and MODI, 1978). One point may be of particular concern as you examine the sample data for Case I. That is, how did we verify Assumption 57 Of all the assumptions required for examining competitive antagonism, verification of Assumption 5, that the slope l3’ is equal to -1, is probably the most difficult. There are several reasons for this.
Estimating pA2 Values First, we frequently are working with experiments for which there are a limited number of dose levels of antagonist. Because of this, the slope of the line log(di/d,
-
1) = 01’ + B’ log(l/B,)
+ E
is estimated with only a few data points (values of dj/d,) and as a consequence, a statistical test of the hypothesis B’ = -1 is quite unreliable. Second, if the number of agonist doses is increased, there is more likelihood that we will detect nonparallelism of the lines RI, = aj + B log A; + E (see Schild, 1957). Because of this dilemma, we seldom conduct a formal test to verify Assumption 5. Rather, as long as B’ is near (in a very general sense) -1, we recommend estimating the pA2 using the formula
Case II: Different
Experimental
Units for Each level of Antagonist
Case II, in which each animal receives multiple doses of agonist but only one dose of antagonist is commonly used when the effect of an antagonist on the graded response to an agonist, ie, parametric measurements, is desired. Examples currently used in our laboratory include antagonism of histamine- or serotonin-induced skin wheals in various species. The analysis of data from an experimental design of this type can be diagrammed in the same way as the analysis for Case I data. The differences, which are not apparent from examining a flow chart (Figure 4), arise from the need to use different statistical tests for the various assumptions to account for the fact that more than one measurement is obtained on each animal (tissue). The underlying model for competitive antagonism and the corresponding assumptions remain the same. The tests of the assumption do change and are derived from the analysis of repeated measures data. As with Case 1, a worked example is provided with appropriate references. Example (Case II) In this model, each animal received multiple doses of agonist but only one dose of antagonist. The data shown in Table 3 were obtained from an experiment to examine the antihistamine effect of chlorpheniramine against histamine-induced skin wheals in guinea pigs. Eight guinea pigs per group were dosed orally with vehicle (water) or chlorpheniramine (1.023 x 10-6, 2.046 x 10e6, and 4.094 x 10e6 moles/kg) one hour before intradermal injections of histamine (five different concentrations, 0.1 ml/wheal) to the skin of the shaved back of guinea pigs. Immediately following the histamine injection, the animals were injected intracardially with 1 ml of 1% Evans blue dye to make the wheal more visible. Twenty minutes later, the guinea pigs were killed and wheal diameters were measured in mm on the reflected
83
84
1. Ott, et al. TABLE 3
Case fl Data: Skin Wheal Area (mm*)
ACWIST (moles/kg) (I%) 1 _ .-._._______~~____._~. 0.125 78.5 0.25 113.1 Control 0.5 176.7 1.0 254.5 2.0 254.5
CUINtA
ANTAGONIST
2
3
63.6 113.1 153.9 201.1 201.1
78.5 95.0 176.7 201.7 254.5
5
6
7
8
63.6 113.1 .176.7 201.1 201.1
95.b 153.9 176.7 227.0 283.5
19.6 95.0 276.7 176.7 227.0
78.5 153.9 227.0 254.5 314.2
95.0 153.9 227.0 254.5 283.5
IO
11
12
13
14
15
16
711 12.6 19.6 38.5 113.1
1916 63.6 78.5 132.7 153.9
95.0 132.7 153.9 227.0 227.0
63.6 113.1 176.7 254.5 254.5
78.5 153.9 201.1 227.0 346.4
50.3 95.0 95.0 201.1 176.7 22..
63.6 95.0 113.1 176.7 201.1 23
59.1 2 11.5 99.8 2 16.0 124.0 -e 20.9 182.3 -t 24.3 212.5 2 24.9
18
19
20
21
95.0 132.7 153.9 201.1 227.0 24 --
17
19.6 38.5 95.0 132.7 176.7
63.6 132.7 153.9 227.0 346.4
78.5 95.0 153.9 201.1 254.5
19.6 78.5 95.0 132.7 176.7
38.5 28.3 43.6 78.5 78.5 113.1 176.7 153.9 201 .'l 201.1
95.0 95.0 201.1 201.1 283.5
43.8 Ir 11.2 73.6 f 13.5 112.9 T 20.3 155.6 + 22.8 208.5 + 33.1
25
26
27
28
29
30
31
32
7.1 78.5 113.1 176.7
7.1 7.1 28.3 132.7
19.6 28.3 132.7 176.7
7.1 38.5 153.9 176.7
19.6 19.6 113.1 201.1
7.1 7.1 50.3 132.7
38.5 95.0 227.0 201.1
3.1 7.1 95.0 113.1
9
1.023 x TO-'
2.046 x IO-' __--~____
0.25 .5 1.0 2.0 4.0
0.5 2.0 2.0 4.0 8.0
1.0 4.094 x IO -lj 2.0 8.0 16.0
7.1 7.2 22.6 19.6 28.3
4
PI<;
MFAY 2 SEM 71.6 i 8.8 123.9 % 9.2 186.4 i 9.3 221.3 !I 10.8 252.4 -c 14.4 _
13.7 2 4.2 35.2 5 12.1 174.2 -t 21.8 163.9 t 11.9
skin surface. The area of each wheal was then calculated according to the equation: Area = (Diameter/2~2 T. Assumptions 1 to 3 were verified using an analysis of variance for a two-factor experiment with repeated measures on one factor, the agonist (see Winer, 1971). Table 4 gives the estimated intercepts (&J and common slope (8) for the log agonist dose response curves. Table 4 can also be used to fit the model lOg(djld7 -
1) = 0~’ + 6’ log(l/B,)
+ E’
Since there are only three data points (values of d,/d,), we cannot test for deviation from linearity to verify Assumption 4. Assuming Assumption 4 holds, the estimated intercept and slope for this model are 11.5900 and -1.8668 with corresponding standard errors of 0.5135 and 0.0902, respectively. Although the estimated value of 8’ is significantly different from - 1, we feel this significance was due primarily to the sensitivity of the assay and not to the difference
Estimating pA2 Values TABLE 4 Case II Data: Estimated Intercepts and Common Slope for Log Agonist Dose-Response Curves ANTAGONIST6, (moles/kg)
COMMON SLOPE
Control 1.023 x IO-' 2.046 x 10 b 4.094 x lomb
977517.11 488758.55 244259.89
log d, at R = 50 mm2'
212.100
136.149
-1.1906
log(d,/d,- I) _ -
135.535 77.898 -0.2638
136.149 136.149 136.149
-0.6282 -0.2049 0.3692
0.4234 0.9383 1.5477
INTERCEPT U,
l/B,
b
a R isthe variable,skin wheal area (mm')
between the estimated value of p’ and the theoretical 1.8668, the estimate of the pAZ is -11.5900
- 1.8668 The 95% prediction
value of -1.
Using 6’ =
= 6.21 or 0.617 x 10e6 moles/kg
limits are 5.98 to 7.07 (or 0.085 x lo-6
kg). Using a molecular weight of 390.86 for chlorpheniramine the pAz estimate to an estimate of the DRz in mg/kg. Dl?, = antilog( - p&)
x
to 1.047 x 10e6 moles/ maleate, we can convert
390.86 x IO3
= 0.241 mg/kg When each of the limits for the pAz is transformed using the same formula, we obtain the 95% prediction limits for the DR, (0.033, 0.409). Several points are of interest in this example. First, although the analysis seems identical to that for Case I, the actual statistical tests that are used for validating Assumptions l-3 are quite different. In Case I, because only one measurement was obtained from each experimental unit, we made use of the analysis for a parallel line assay. For Case II, more than one measurement was obtained for each experimental unit. Hence we employed statistical tests that are appropriate for this type of repeated measures data. Second, methods for verifying Assumptions 4 and 5 are identical for Cases I and II. Third, these sample data illustrate a situation where we must extrapolate beyond the data points in the Schild plot in order to estimate the value of pA2. Ideally, doses of the antagonist should be chosen so that values of dj/dl bracket 2. While this is not always practical, care should be taken to employ enough antagonist doses to approximate d/d, = 2. Otherwise, as the degree of extrapolation increases, the limits on the estimated pAz become wider. Fourth, as with the data used to illustrate Case I, we require very little verification of Assumption 5 in order to estimate the pAz. Recall the estimated slope 6’ was -1.8674 with standard error 0.0902. While these data seem to suggest the actual slope p’ is different from -1, the estimated slope is based on so few data points (three) as to be somewhat unreliable. Also, from a practical standpoint we feel that
85
86
1. Ott, et al. TABLE 5
Alternate Analysis for Case II Data
1. Check the control
data by eye for linearity.
2. Average the control 3. Fit a s;raight
responses
4. For each experimental obtain d,. (Note: 5. Form
at each level of agonist.
line to the’control
the ratios
There d,/d,.
data to obtain
unit
in 6, (j > I)
will
be n, values
d,:
fit a straight
line to
of d,).
For each level of antagonist,
there
will
be n, ratios. 6. Proceed
to fit the model log(d,/d,
7. Estimate
-
I)
= a’ + B’ log(l/B,)
the pAz (with appropriate
limits)
+ E.
as was done in the
Case II analysis.
less than a two-fold difference between the estimated value of p’ and the theoretical value of - 1 is not important to detect. For these reasons we were not uncomfortable proceeding with estimating the pA2. Finally, there is another way to analyze data from Case II studies that utilizes more biological judgement in examining whether the underlying assumptions (especially Assumptions I-3) hold. The steps in the analysis are recorded in Table 5. Note that for this approach, no statistical checks are made for Assumptions l-3. Rather one must depend on a visual, intuitive inspection of the data. One can check Assumptions 4 and 5 as was done for the original analysis of Case II data.
TABLE 6 Data
Sequence of Steps in the Analysis of Case III
1. For each experimental 2. Obtain
a trend
3. Make a biological 4.
unit
plot the agonist/antagonist
data.
line for each level of antagonist. judgement
If the data “satisfy”
regarding
assumptions
l-3,
assumptions obtain
l-3.
the predicted
d,
value. 5. Form
the ratios
d,id, and fit the model
log(d,/d, 6. Estimate
-
I)
= a’ + B’ log(l/B,)
+ t.
the pA> as
I PA* = +. 7. Combine
the separate
mate and standard
estimates
of pA, to form
the final esti-
error.
Final estimate: ZPAL n where
n is the number
value was computed.
2 SEM
of experimental
units
for which a p&
Estimating pA2 Values Case III: Each Experimental Antagonist
Unit Receives All Combinations
of Agonist and
Case III, in which each animal or tissue receives multiple doses of both agonist and antagonist, is also used when the effect of an antagonist on the graded response to an agonist is desired. Examples currently used in our laboratory include antiadrenergic, antihistaminic, or anticholinergic experiments in anesthetized dogs or in isolated tissues. The analysis of data resulting from Case III is unlike the analyses proposed for Cases I and II. In particular, data from each experimental unit (eg, tissue or animal) is examined separately to obtain an estimate of the pAz. Because none of the underlying assumptions can be rigorously checked, we must depend on one’s biological judgement as to whether data from an individual tissue can be used to obtain a pAZ value. The final estimate of pAz is the average of the individual estimates. A measure of variability is given by the standard error of the mean estimate. The steps in the analysis of Case III data are outlined in Table 6. Example
(Case III)
In Case III, the antihistaminic effect of chlorpheniramine (antagonist) was studied in vitro against histamine-induced contractile responses of the isolated guinea pig ileum (Table 7). A piece of the guinea pig ileum (2-3 cm) was set up in a 37” C organ bath filled with physiological salt solution (Tyrodes). The solution was constantly aerated with 95% O,-5% COZ. The dose-response experiment was performed by adding the agonist (histamine) cumulatively to the organ bath in the absence (control) and the presence of different concentrations of chlorpheniramine (Van Rossum, 1963). Responses were expressed as percentage of the maximal contractile effects produced by histamine. In this example, five dose-response curves were generated from each strip of guinea pig ileum. Since each experimental unit (eg, tissue) receives all agonist/antagonist combinations, Case III data requires that we treat data from each tissue as a separate experiment to be analyzed. Hence for each tissue we must obtain a trend line for each level of antagonist and after visually “checking” Assumptions 1-3, obtain the d, for R = 50%. The results of these separate calculations are displayed by tissue in Table 8. The data for each tissue
in Table 8 are then
log(dj/dl
-
used to fit the line
I) = IX’ + B’ log(l/B,)
+ E’
and then to estimate the pA2 value. These analyses are summarized in Table 9. Note the consistency of the data in Table 9, particularly the magnitudes of the estimated slopes which are all near -1. The estimate of the pA2 value in this competitive antagonism experiment combines the separate estimates to obtain the mean pA2. The measure of variability for this estimate is the standard error of the mean (SEMI. Unlike the estimates of variability generated for Case I and Case II data, which are based on how well the mean data fit the log(d,/d, - 1) model, this SEM reflects animal to animal variability and not variability associated with the model
87
TABLE 7
Case III Data: Percent of Maximal Contractile Effect TISSUENO.
AGONIST ANTACONI~T B, (M)
Control
0.3162 x 10 a
1x10
?'
IogWA,)
1
2
'
5
6
MEAN
k SEM
0
5
0
0
0
2
1.2 2 0.8 -
7.5
0
15
0
2
3
4
4.0 t- 2.3 11.2 t- 2.4
7.0
10
22
6
13
6
10
6.5
25
39
16
26
16
27
24.8 i- 3.5
6.0
45
56
35
48
34
52
45.0 i- 3.7
5.5
75
78
61
68
61
85
71.3 -+ 4.0
5.0
100
93
81
89
78
100
90.2 ? 3.8
1
2
3
4
5
6
7.0
0
7
0
2
2
8
6.5
0
I5
3
7
3
a
6.0 k 2.2
6.0
15
27
10
15
6
19
15.3 -t 3.0
5.5
30
41
23
37
28
31
31.7 k 2.6
5.0
60
59
42
59
58
67
57.5 k 3.4
4.5
80
78
68
81
78
81
77.7 t 2.0
4.0
70
78
84
89
88
88
82.8 + 3.1
3.5
80
90
94
89
91
96
90.0 + 2.3
1
2
3
4
5
6
3.2 t 1.4
6.0
10
15
4
9
4
8
8.3 rf-1.7
5.5
20
24
14
19
11
23
18.5 -+ 2.1
5.0
40
41
26
37
28
46
36.3 ? 3.2
4.5
60
61
46
59
54
69
58.2 i 3.1
4.0
70
68
71
74
75
85
73.8 ? 2.5
3.5
75
80
81
85
88
92
83.5 It 2.5
3.0
75
98
100
96
91
96
92.7 f 3.7
4
5
3
6
_
5.5
12
14
6
11
6
12
10.2 t 1.4
5.0
20
32
13
22
13
27
21.2 ? 3.1
4.5
35
51
23
34
31
54
38.0 t 4.9
4.0
55
85
39
56
63
92
15.0 + a.1
3.5
70
-
71
74
81
100
79.2 2 5.5
3.0
70
-
87
93
94
100
88.8 k 5.1
2.5
75
-
94
93
88
100
90.0 +- 4.2
4
5
6
5.0
20
22
6
16
6
8
l3.0 2 3.0
4.5
28
37
13
22
16
15
21.8 2 3.8
4.0
50
48
26
41
50
31
41.0 + 4.2
3.5
75
61
74
52
75
73
68.3 + 3.9
3.0
80
a5
87
74
81
92
83.2 + 2.6
2.5
80
95
94
81
81
103
89.0 2 3.9
12
1 x 10-7
4
8.0
12
0.3162 x 10
3
3
_
_ 0.78976 1.313M
-5.16667
-4.68750
-4.11589
-3.97790
.3162 x 10'
1 x ld
.3162 x lOa
1 x 10'
2.03990
1.9KM2
log(d,/d,- 11
'og d,
-6.02174
l/B,
Control
1
1.48258 1.54470 2.21045
-4 71329 -4.65302 -3.99682 -3.75oOa
-3.87179
-4.46386
~4.88083
086822
-5.28655
log d, -5.73846
log(d,/d, 11
-6.20994
log d,
2
198398
1.86073
1.25089
0.79274
logtd,/d,
3
Case III Data: Separate Calculations for Each Tissue
ANTAGONISTDOS~
TABLE 8
1, log d,
-3.66766
-4.16875
-4.69412
-5.20455
-5.96907
2.29923
1.79319
1.25126
0.67045
_
logid,id,
4 II
loi+d,
~3.87500
-4.22000
~4.57838
-5.09756
-5.70899
1.82758
1.47467
1.09721
0.48957
_
logtd,/d,
5 II
log d,
3.mO37
-4 64523
~4.90500
-5 23874
-6 10084
2.29829
1.44013
I16725
0.79792
_
logid,fd,- 11
6
90
1. Ott, et al. TABLE 9
Case III Data: Separate Estimate of pA2 TISWE
ESTIMATE
1
, ix
fitting
2
8.23370
3
7.86417
4
7.95672
5
9.91782
6
8.02914
~-
8.82564
PA(
~0.86746
-0.81777
-0.83673
-1.08571
-0.87831
-0.95481
P&
9.49
9.62
9.51
9.13
9.14
9.24
procedure.
For these
data
Average pA2 = 9.36 or 4.365 and SEM
x 1O--‘o M
= 0.09.
DISCUSSION In this
paper we have presented
one might be interested pA, values. petitive
First
we discussed
antagonism
data from
We found
of the underlying to verify
methodology
Assumptions
sumptions
1-3
l-3.
settings
and illustrated different
analysis
using
that form the basis of the comanalyses
the analysis
statistical that offers
for
by way of
of Case I and Case II data both allowed
even though
where
antagonists
(1957). Next we considered
settings
An alternative
was provided
experimental
of competitive
assumptions
of Schild
that analyses
assumptions,
different
the potencies
certain
each of the experimental
an example.
three
in estimating
checks
tests were employed fewer
checks
on As-
for Case II data.
The analyses of data for Case III provides no formal verification for Assumptions l-3. Rather the experimenter must visually check the data and proceed accordingly. An analysis
is performed
on data from each experimental
of pA2. The final estimate form of a mean (*SEMI. There are other possible paper. For example, animal (tissue)
combines
information
experimental
one could conceive
receives
a single
from
designs
unit to obtain an estimate all experimental
than the three
of an experimental
dose of agonist
but multiple
units
in the
discussed
situation
in this
where
antagonist
each
doses.
The
authors have examined this design and would not recommend its use owing to difficulties in verifying the assumptions and in obtaining a valid measure of variability
associated
with
pA2. Another
for each level of antagonist
design,
where
there is a different
has been used by some.
control
We are certain other
group designs
are possible. In this paper the analyses for three of the more common designs were outlined with the hope that experimenters will recognize that one cannot necessarily apply the same pA2 analysis to data generated from different designs. The reader should note that we have not spent much time discussing alternate procedures mula
for estimating
pA, values.
Thus
we have recommended
use of the for-
Estimating pA2 Values
where &’ and pf are obtained
from a fit of the model
log(dj/dl There
are, of course,
discusses prediction
1) = CX’ + B’ log(l/B,)
-
other formulas
one could employ.
+ E For example,
1. Assuming
B’ =
estimated
-1
intercept
and using
the modified
formula
pA, = (Y’ where
B’ =
-1
and fitting
the model
log(dj/d,
-
1) = (Y’ -
log(l/B,)
2. Assuming
3. Obtaining
an estimate -
no positive
(or negative)
authors
1) + log(l/B,).
provided
= &‘. the formula
a plot of pA, versus
the estimates
feel that the estimate
procedures
Tallarida
can be combined
pA,
lOg(l/Bj)
has
as an average
using
method
p&
=
-&l/B’
is the most
of data for Cases I-111 would
of estimation
et al. (1979) have provided
and underlying checks
the formula
to the analysis
is chosen.
These
different
not es-
only alter the final stages of the analysis. assumptions
posed by these authors formal
Then
slope,
but the approaches
change even if an alternative
ground
+ E’ to obtain p&
5 SEM.
appropriate, timation
+ E’.
of pA2 for each value of d,/d, through
= log(dj/d, PA,
&’ is the
for the model
lOg(dj/dl -1) = a’ + B’ lOg(llBj)
The
MacKay (1978)
several different methods for estimating pA, values with corresponding limits for data that we have classified as Case I. These methods include:
addresses
on Assumptions
an excellent
review
for pAZ analyses. the estimation
1-3.
The
of the theoretical statistical
analysis
backpro-
of the pA, for Case I data without
In particular,
the authors
recommend
fitting
the model -
log(d,& where
the slope
1) = (Y’ -
B’ is assumed to be -1, -1. The estimate
lOg(l/Bj)
unless some pharmacological of the pA2 and corresponding
can be given for B’ #
limits are then derived from a fit of this model. The importance of our work is recognition of the fact that different settings
give rise to different
timation
of the pA2, the estimate
of variability. a pA, (or DR,). pA, (or DR,),
The
analyses
procedures
In order one must
should
and no matter what formula in this
for these methods
to be applicable for estimating
+ ej
equation (X -
I)]
measure
paper have been used to estimate
replace the equation
log[d,/d, -
experimental is used for es-
be accompanied by an appropriate
presented
log(d,/d, - 1) = (Y' + p' lOg(l/Bj) with the more general
justification prediction
= (Y’ + B’ log(l/Bj)
+ E,
a general
91
92
1. Ott, et al. Then fit
for estimating
a pAlo (or DRlo)
log(djd, Everything
-
else in the analysis
one substitutes
x = 10 into this
9) = ty’ -t @’ IogfliB,)
remains
equation
to
+ E,
the same as was done for estimating
a pA,
or DR,.
REFERENCES Arunlakshana tative
0,
uses
Schild
of drug
HO
(1959)
Some
antagonists.
quanti-
Rr / Pharmacol
14148858. Brittain
Levy GP (1976) A review of the animal
pharmacology
of labetalol,
macol 3(4) (Suppl D]
(1964) D (1978)
affinity
OL- and
Br ) C/in
Phar-
3):681-694. Hafner How
constants
tive antagonists
a combined drug.
Statistical
Assay. New York: MacKay
McGilliard
in Biological
Press.
should
values
of pA, and
for pharmacological be estimated?
KL, Takemori
naloxone
of
depression
competi-
1 Pharm Pharmaco~
AE (1978) Antagonism
narcotic-induced
and analgesia.
by
respiratory
/ Pharmacol
Exp Ther
207:494-503. WODI User’s
Guide
(1978) Statistical
Lexington,
Ott L (1977) Introduct/on
Data Analysis.
North
Computing
Kentucky.
HO
(1957) Drug
Snedecor
Br
/
Pharmacof
antagonism
and PA,.
G, Cochran W (1967) Stafjsfical
6th ed. Ames,
Scituate,
Stone
Iowa:
Methods
Mass.:
and
Duxbury
M, Angus
puter-based
The
Guide (1979) SAS fnstitute
inc.,
Raleigh,
NC. HO 11947) pA, a new scale for the measure-
JA (1978)
estimation
ciated analysis. Tallarida
Phar-
Iowa State
~efhods, University
receptor Van Rossum curves. uation
antagonism.
IMW (1979)
A statistical
Techniques curves
of drug
pAL and
analysis
of
Life Sci 25:637-654.
JM (1963) Cumulative
II.
response
of comand asso-
A, Adler
differentiation:
competitive
Developments of pAL values
f Pharmacof Exp Ther 207:705-718.
R, Cowan
dose
response
for the making
in isolated parameters.
organs
of dose
and the eval-
Arch inf Pharmaco-
dyn Ther 143 : 299-330. DR,
mation Statistical
Parker
Winer
RB (1971) Pharmacological
of drug-receptor evaluation
/. Pharmacol
SAS User’s
antagonism.
macol Rev 9:242-246.
Waud
to Statistical
Press.
Schild
drug
Press.
Method
30:312-313.
Service,
of
Schild
RT,
~-adrenoceptor-blocking Finny
ment
2:189-206.
dissociation
II, Competitive
esti-
constants. antagonists.
Fxp Ther 177:13-24.
BJ 11971) Statisticaf Principals in Experimental Design. 2nd ed. New York: McGraw-Hill, pp. 518-520.
L. OTT, 5.
Designs
WEINER, F-i. CHENG, AND J. WOODWARD
The method for analyzing data leading to the estimation of a pA2 depends on the experimental design. Three different experimental designs are discussed. For each design, we provide a brief description of experimental situations that give rise to the design, a means for verifying relevant assumptions, and sample data with summa~ calculations.
Key Words:
Estimate; Experimental design; Agonist; Antagonist; Prediction lim-
its; pA2; DR2
INTRODUCTION Since its introduction, pA, (Schild, 1947) has become a popular way to express the potency of a competitive antagonist and a useful tool to study drug receptor interaction. The term pA, is defined as the negative logarithm of the molar concentration of an antagonist that produced an x-fold shift in the agonist dose-response curve. Based on the mass action law, Schild (1957) derived the following equation: log(dose ratio -
I) = -log
KB + log [Bl
where KB is the dissociation constant of the antagonist for the receptor and [Bl is the concentration of the antagonist. A plot of log (dose ratio - 1) against -log [B], called a Schild plot, gives a straight line and allows one to calculate pAz when the dose ratio is 2; in general, a plot of log[dose ratio -(x - 1)J versus -log [B] can be used to calculate PA, when the dose ratio is x. Although pA, values have been used as measures of competitive antagonism in vitro and in vivo, some scientists (Brittain and Levy, 1976) use DR, values (e.g., dose of antagonist in mg/kg, to produce an x-fold shift in dose response curves) to express potencies of competitive antagonists. When used in vivo, DR, and PA, can be interchangeable by converting the dose of antagonist in mglkg into mole/kg (or vice versa). In the past, the computed pAz value from an experiment was often taken as an absolute rather than an estimated value. Recent publications have signaled a change in this thinking by presenting methods that provide an interval estimate of the PA,, From the Departments
of Biostatistics,
Preclinical
Pharmacology,
Merrell
Research
Center,
Cincinnati,
Ohio. Address
reprint
requests
to Dr.
5, 1980;
revised
Lyman
Ott,
Department
of Biostatistics,
2110 E. Calbraith
Rd.,
Cincin-
nati, OH 45215. Received
April
and accepted July 28, 1980.
75 iournal of
Pharmacoio~ical
G 1981 Elseviei
North
Methods
Holland,
5, 75-92 11981i
Inc., 52 Vanderbilt
Avenue. New York. NY 10017
76
1. Ott, et al. or an estimate
of the pA, with a standard
Stone and Angus, with the situation
for each agonist-antagonist This
paper expands
three
commonly we will
verifying tions
methods
combination
discuss
the relevant
for each design.
order
to estimate
(Tallarida
et al., 1979; MacKay, 1978;
combination.
upon these
used experimental
one agonist-antagonist design,
error
1978; Waud and Parker, 1971). However these reports deal only where different experimental units (dogs, tissues, etc.) are used
estimation
pA2 values
In two of these
Sample
modifications
any pA, (or DR,)
by providing
designs,
is applied to each experimental of the pA, and statistical
assumptions. Simple
for estimating
designs.
tests
than
For each
appropriate
data are used to illustrate can be made to these
more
unit.
for
the calculaprocedures
in
value.
METHODS
Notation B, : jth level of antagonist (j = 1, 2, . . . . b) in moles/kg; A,: ith level of agonist (i = 1, 2, . . . . a); unit (animal or tissue) R,, : response when experimental
4,:
agonist and jth level of antagonist; number of experimental units treated with
is treated with
ith level of agonist
ith level of
and jth level of
antagonist.
General Assumptions Assumption dose-response pressed
using
for a pA, Analysis
7. For a fixed level of B,, particularly curve, a linear
R, is linearly regression
related
within
to log A,. This
the middle
equation.
R,, = ci, + B, log A; + E;,
FIGURE 1.
portion
relationship
Parallel trend lines for different levels of Bi.
of the
can be ex-
Estimating pAa Values Bl
I
I
log dl
log d2
(Control)
log da
108 agonist FIGURE 2.
B2
B4
B3
log 4, or
log
Ai
Relationship between di and response of magnitude R on antagonist j.
where pi is the slope of the linear regression equation and cyiis the intercept. The symbol eij represents a random error that accounts for the fact that not all observations fall on the linear trend line, aj + pi log A;. Assumption 2. The trend lines for different levels of Bj are parallel (ie, Pi = f3) (see Figure 1). Assumption 3. The common slope p for the separate dose-response lines is different from zero (1 p ) # 0). Assumption 4. If dj denotes the predicted dose of agonist corresponding to a response of magnitude Ron antagonist j (j = 1,2, . . . . b) and d, denotes the predicted dose of agonist for the control group (no antagonist), the variable log(dj/d, - 1) is linearly
related to log(l/Bj) (see Figures 2 and 3). 5. The slope p’ in the trend line (Y’ + p’ log(l/B$
Assumption
log
dj/dl-1 (
>
I FIGURE 3.
is equal to -1.
linear relationship
- log[antagonistI or log(llB,)
between variable log (did,
-
1) and log (l/Bi).
77
78
1. Ott, et al.
Estimation of the pA2 The pA2 value of an antagonist for a given agonist lOg(l/Bj) corresponding to a two-fold shift to the right in curve of Figure 2. Or, equivalently, the pAz value is line a’ + j3’ lOg(l/Bj) intersects the log(l/B$ axis (see The original equation log(dose ratio -
is defined to be the value of the log agonist dose-response the point at which the trend Figure 3).
1) = pA1 + log B
derived by Schild (1957) and expanded upon by Arunlakshana and Schild (1959) is based upon the mass law for drug-receptor interaction. This equation is equivalent to the equation log(dj/dl
-
1) = (Y’ + p’ lOg(l/Bj)
where pAZ = -_(y’/p with p’ = -1. In practice the constants “jr p, (Y’ and p’ for the models R, = 01, + /3 log A; + E,j and log(dj/d,
-
1) = (Y’ + p’ log(l/Bj)
+ E’
are unknown and must be estimated using experimental data. Consequently, the pAz value computed from sample data is also an estimate of the actual unknown value and should be accompanied by a measure of variability in the form of a prediction interval or standard error. In this paper we will consider estimation of the pA, for a given agonist/antagonist relationship under the following three experimental settings: Case I: Different experimental units (eg, animals or tissues) for each combination of levels of agonist and antagonist. Case II: Different experimental units for each level of antagonist. Case III: Each experimental unit receives all combinations of agonist and antagonist. For each of the three experimental situations we will present the tools necessary for analyzing sample data leading to an estimate of pAz. Since the experimental designs are different, the analyses (which reflect the experimental designs) are also different. We will find that not all of the general Assumptions (I-5) can be verified rigorously for each of the designs. For some situations, a biological judgment must be made; in others, the assumptions can be put under more careful analytical scrutiny. The presentation for each case will contain the following: 1. A brief description of experimental situations for which the design is appropriate. 2. A checklist for verification of assumptions l-5. 3. Sample data with summary calculations for obtaining an estimate of the pA, with a measure of its variability. 4. Discussion
Estimating pA2 Values RESULTS Case I: Different Experimental Units feg, Animals or Tissue) for Each Combination of levels of Agonist and Antagonist Case I in which each individual animal receives only a single dose of agonist and a single dose of antagonist is commonly used for pharmacological evaluation involving quanta1 estimation of drug antagonism. Examples of this experimental design include grouped amphetamine toxicity and its antagonism by drugs, apomorphine- or amphetamine-induced stereotypy and its antagonism by drugs, antagonism of analgesic effects of morphine by naloxone, and antagonism of physostigmineinduced death by anticholinergies. The analyses of data from a study where different experimental units are used for each agonist/antagonist combination is straightforward and provides tests for all assumptions underlying the competitive antagonism model. We have included a flow chart for the analysis in Figure 4 to illustrate the steps of the analysis which lead to estimation of a pAZ. The analysis incorporates an analysis of variance for a parallel line assay (Finney, 1964) and inverse prediction from a linear regression equation (Ott, 1977). Example (Case I) In Case I, each individual animal receives only one dose of agonist and one dose of antagonist. The data shown for this example were obtained from a graphical representation by McGilliard and Takemori (1978) who measured the analgesic effect of morphine in mice and the antagonism by naloxone. The analgesic effect of a subcutaneous injection of morphine was assessed by the tail-flick method as an increase in the reaction time of the animal (tail flick) to avoid a painful stimulus. A reaction time of 1.61 2 0.02 seconds (mean + SE) was recorded in the vehicletreated mice. For this experiment, an increase in reaction time greater than three SD was considered a significant analgesic effect (Table 1). Assumptions l-3 were verified for these data using tests appropriate for a parallel
TABLE 1
Percentage of Mice with a Significant Increase in Reaction Time
AGONISI (mgikg)
ANTAGONISTDOSE (moles/kg) --.-_--
~.___. CONTROL
O.11x1O-b
0.22x10-"
_.
0.44~10-~
0.88~10~"
__” -3.52~10-~
2.5
10 WIO)"
5
40 (4110)
18 (2/11)
IO
60 (6110)
40 (4110)
25 121'8)
70 (7110)
50 (5110)
12 (118)
89 (8/9)
40 (4/10)
25 (218)
80 (8110)
50 (S/IO)
12 (l/8)
70 (7/10)
40 (4110)
20 40 80 160
60 E/IO)
320 640 a Numbers
in parentheses
effect of the total
number
indicate of animals
the number tested.
of animals
showing
a significant
analgesic
79
00
1. Ott, et al.
TEST ON ONIATICN .=Rm LIhEIRIlY (AI) IWE RlEs=msE CUWES NWIN*R?
-
FIGURE 4. Flow chart for the estimation of pA2: s = statistically significant at the 0.05 level; ns = not statistically significant (continued on following page). line assay (Finney slope
1964). Table 2 gives the estimated
intercepts
k$‘s) and common
(P) for the model R, = OL, + p log A; + E,,
In addition, the predicted log agonist antagonist based on a fixed response umn 5 of Table 2. Column
dose (log d,) was computed for each level of R = 50%. These results are displayed in Col-
6 gives values log(djldl
of the transformed -
1)
data
Estimating pA, Values
x No
STW
FIGURE 4.
Flow chart for estimation of pA, (continued from previous page).
TABLE 2 Case I Data: Estimated Intercepts and Common Slope for Log Agonist Dose-Response Curves ANTAGONIST
6,
(moles/kg)
l/B, -
lNTERCEpT
ir,
COMMON
SLOPE
f3
log
d, at R = 50%
log(d,/dI- I)
-26.6059
90.5225
0.8463
0.11 x 10
h
9090909.09
-47.8559
90.5225
1.0810
0.22 x 10
b
4545454.55
--63.1059
90.5225
1.2495
0.1848
0.44 x 10
b
2272727.27
-101.0225
90.5225
1.6683
0.7511
0.88 x lomb
1136363.64
-123.9392
90.5225
1.9215
1.0370
3.52 x IO-"
284090.91
-189.4392
90.5225
2.6451
1.7918
Control
-0.1446
81
82
L. Ott, et al.
which are needed to fit the line
- 1) = CX’+ f3’ lOg(l/Bj) + E’
log(d,/d,
The test on deviation from linearity for this model was nonsignificant and hence we assumed Assumption 4 was verified. The estimated intercept (&) and slope (6’) for this model were 89106 and -1.3002 with corresponding standard errors equal to 0.4081 and 0.0646, respectively. Although the slope of the line is significantly different from -1, the difference was considered sufficiently small to be of no consequence; thus we substituted into the formula for the estimate of pAZ to obtain PA:! =
- 8.9106
= 6.85 or 0.14 x 10m6 in moles/kg
- 1.3002
The corresponding 95% prediction limits for the pAZ are 6.74 to 6.99 (or 0.10 x lO-‘j to 0.18 x low6 moles/kg). The estimated value of the pA2 and corresponding prediction limits can be reexpressed as an estimate of the DR, with appropriate prediction limits. For these data, with a molecular weight of 363.83 for naloxone hydrochloride we have Dl& = antilog( -PA,)
x 363.83 x IO3
= 0.051 mg/kg Transforming each of the limits in the same way we find the 95% prediction limits fin mgikg) on the DRz to be (0.037, 0.066). To summarize our calculations for Case I, we used the sample data (Table 1) to check Assumptions l-3. Having encountered no difficulties with these assumptions, we fit the model R,, = a, + I3 Iog A, I- E;j to obtain estimates of the individual intercepts and common slope. These results, summarized in Table 2, were used to calculate the log d, based on a response of R = 50%. Data from Table 2 were then used to fit the model log(djd,
-
I) = cxt + 8’ IogWB,)
+ E’
Because there was no significant deviation from linearity and the estimated slope 6’ was near - 1, we computed the estimate of pA, as p& = - &‘/fi’ and re-expressed it as a D& in mg/kg of the antagonist dose. Details of the calculations for obtaining 95% prediction limits were not presented because the formulas can be quite tedious (Snedecor and Cochran 1967; Ott, 1977). Fortunately, some linear regression software packages offer an option for computing these limits (eg, SAS,1979 and MODI, 1978). One point may be of particular concern as you examine the sample data for Case I. That is, how did we verify Assumption 57 Of all the assumptions required for examining competitive antagonism, verification of Assumption 5, that the slope l3’ is equal to -1, is probably the most difficult. There are several reasons for this.
Estimating pA2 Values First, we frequently are working with experiments for which there are a limited number of dose levels of antagonist. Because of this, the slope of the line log(di/d,
-
1) = 01’ + B’ log(l/B,)
+ E
is estimated with only a few data points (values of dj/d,) and as a consequence, a statistical test of the hypothesis B’ = -1 is quite unreliable. Second, if the number of agonist doses is increased, there is more likelihood that we will detect nonparallelism of the lines RI, = aj + B log A; + E (see Schild, 1957). Because of this dilemma, we seldom conduct a formal test to verify Assumption 5. Rather, as long as B’ is near (in a very general sense) -1, we recommend estimating the pA2 using the formula
Case II: Different
Experimental
Units for Each level of Antagonist
Case II, in which each animal receives multiple doses of agonist but only one dose of antagonist is commonly used when the effect of an antagonist on the graded response to an agonist, ie, parametric measurements, is desired. Examples currently used in our laboratory include antagonism of histamine- or serotonin-induced skin wheals in various species. The analysis of data from an experimental design of this type can be diagrammed in the same way as the analysis for Case I data. The differences, which are not apparent from examining a flow chart (Figure 4), arise from the need to use different statistical tests for the various assumptions to account for the fact that more than one measurement is obtained on each animal (tissue). The underlying model for competitive antagonism and the corresponding assumptions remain the same. The tests of the assumption do change and are derived from the analysis of repeated measures data. As with Case 1, a worked example is provided with appropriate references. Example (Case II) In this model, each animal received multiple doses of agonist but only one dose of antagonist. The data shown in Table 3 were obtained from an experiment to examine the antihistamine effect of chlorpheniramine against histamine-induced skin wheals in guinea pigs. Eight guinea pigs per group were dosed orally with vehicle (water) or chlorpheniramine (1.023 x 10-6, 2.046 x 10e6, and 4.094 x 10e6 moles/kg) one hour before intradermal injections of histamine (five different concentrations, 0.1 ml/wheal) to the skin of the shaved back of guinea pigs. Immediately following the histamine injection, the animals were injected intracardially with 1 ml of 1% Evans blue dye to make the wheal more visible. Twenty minutes later, the guinea pigs were killed and wheal diameters were measured in mm on the reflected
83
84
1. Ott, et al. TABLE 3
Case fl Data: Skin Wheal Area (mm*)
ACWIST (moles/kg) (I%) 1 _ .-._._______~~____._~. 0.125 78.5 0.25 113.1 Control 0.5 176.7 1.0 254.5 2.0 254.5
CUINtA
ANTAGONIST
2
3
63.6 113.1 153.9 201.1 201.1
78.5 95.0 176.7 201.7 254.5
5
6
7
8
63.6 113.1 .176.7 201.1 201.1
95.b 153.9 176.7 227.0 283.5
19.6 95.0 276.7 176.7 227.0
78.5 153.9 227.0 254.5 314.2
95.0 153.9 227.0 254.5 283.5
IO
11
12
13
14
15
16
711 12.6 19.6 38.5 113.1
1916 63.6 78.5 132.7 153.9
95.0 132.7 153.9 227.0 227.0
63.6 113.1 176.7 254.5 254.5
78.5 153.9 201.1 227.0 346.4
50.3 95.0 95.0 201.1 176.7 22..
63.6 95.0 113.1 176.7 201.1 23
59.1 2 11.5 99.8 2 16.0 124.0 -e 20.9 182.3 -t 24.3 212.5 2 24.9
18
19
20
21
95.0 132.7 153.9 201.1 227.0 24 --
17
19.6 38.5 95.0 132.7 176.7
63.6 132.7 153.9 227.0 346.4
78.5 95.0 153.9 201.1 254.5
19.6 78.5 95.0 132.7 176.7
38.5 28.3 43.6 78.5 78.5 113.1 176.7 153.9 201 .'l 201.1
95.0 95.0 201.1 201.1 283.5
43.8 Ir 11.2 73.6 f 13.5 112.9 T 20.3 155.6 + 22.8 208.5 + 33.1
25
26
27
28
29
30
31
32
7.1 78.5 113.1 176.7
7.1 7.1 28.3 132.7
19.6 28.3 132.7 176.7
7.1 38.5 153.9 176.7
19.6 19.6 113.1 201.1
7.1 7.1 50.3 132.7
38.5 95.0 227.0 201.1
3.1 7.1 95.0 113.1
9
1.023 x TO-'
2.046 x IO-' __--~____
0.25 .5 1.0 2.0 4.0
0.5 2.0 2.0 4.0 8.0
1.0 4.094 x IO -lj 2.0 8.0 16.0
7.1 7.2 22.6 19.6 28.3
4
PI<;
MFAY 2 SEM 71.6 i 8.8 123.9 % 9.2 186.4 i 9.3 221.3 !I 10.8 252.4 -c 14.4 _
13.7 2 4.2 35.2 5 12.1 174.2 -t 21.8 163.9 t 11.9
skin surface. The area of each wheal was then calculated according to the equation: Area = (Diameter/2~2 T. Assumptions 1 to 3 were verified using an analysis of variance for a two-factor experiment with repeated measures on one factor, the agonist (see Winer, 1971). Table 4 gives the estimated intercepts (&J and common slope (8) for the log agonist dose response curves. Table 4 can also be used to fit the model lOg(djld7 -
1) = 0~’ + 6’ log(l/B,)
+ E’
Since there are only three data points (values of d,/d,), we cannot test for deviation from linearity to verify Assumption 4. Assuming Assumption 4 holds, the estimated intercept and slope for this model are 11.5900 and -1.8668 with corresponding standard errors of 0.5135 and 0.0902, respectively. Although the estimated value of 8’ is significantly different from - 1, we feel this significance was due primarily to the sensitivity of the assay and not to the difference
Estimating pA2 Values TABLE 4 Case II Data: Estimated Intercepts and Common Slope for Log Agonist Dose-Response Curves ANTAGONIST6, (moles/kg)
COMMON SLOPE
Control 1.023 x IO-' 2.046 x 10 b 4.094 x lomb
977517.11 488758.55 244259.89
log d, at R = 50 mm2'
212.100
136.149
-1.1906
log(d,/d,- I) _ -
135.535 77.898 -0.2638
136.149 136.149 136.149
-0.6282 -0.2049 0.3692
0.4234 0.9383 1.5477
INTERCEPT U,
l/B,
b
a R isthe variable,skin wheal area (mm')
between the estimated value of p’ and the theoretical 1.8668, the estimate of the pAZ is -11.5900
- 1.8668 The 95% prediction
value of -1.
Using 6’ =
= 6.21 or 0.617 x 10e6 moles/kg
limits are 5.98 to 7.07 (or 0.085 x lo-6
kg). Using a molecular weight of 390.86 for chlorpheniramine the pAz estimate to an estimate of the DRz in mg/kg. Dl?, = antilog( - p&)
x
to 1.047 x 10e6 moles/ maleate, we can convert
390.86 x IO3
= 0.241 mg/kg When each of the limits for the pAz is transformed using the same formula, we obtain the 95% prediction limits for the DR, (0.033, 0.409). Several points are of interest in this example. First, although the analysis seems identical to that for Case I, the actual statistical tests that are used for validating Assumptions l-3 are quite different. In Case I, because only one measurement was obtained from each experimental unit, we made use of the analysis for a parallel line assay. For Case II, more than one measurement was obtained for each experimental unit. Hence we employed statistical tests that are appropriate for this type of repeated measures data. Second, methods for verifying Assumptions 4 and 5 are identical for Cases I and II. Third, these sample data illustrate a situation where we must extrapolate beyond the data points in the Schild plot in order to estimate the value of pA2. Ideally, doses of the antagonist should be chosen so that values of dj/dl bracket 2. While this is not always practical, care should be taken to employ enough antagonist doses to approximate d/d, = 2. Otherwise, as the degree of extrapolation increases, the limits on the estimated pAz become wider. Fourth, as with the data used to illustrate Case I, we require very little verification of Assumption 5 in order to estimate the pAz. Recall the estimated slope 6’ was -1.8674 with standard error 0.0902. While these data seem to suggest the actual slope p’ is different from -1, the estimated slope is based on so few data points (three) as to be somewhat unreliable. Also, from a practical standpoint we feel that
85
86
1. Ott, et al. TABLE 5
Alternate Analysis for Case II Data
1. Check the control
data by eye for linearity.
2. Average the control 3. Fit a s;raight
responses
4. For each experimental obtain d,. (Note: 5. Form
at each level of agonist.
line to the’control
the ratios
There d,/d,.
data to obtain
unit
in 6, (j > I)
will
be n, values
d,:
fit a straight
line to
of d,).
For each level of antagonist,
there
will
be n, ratios. 6. Proceed
to fit the model log(d,/d,
7. Estimate
-
I)
= a’ + B’ log(l/B,)
the pAz (with appropriate
limits)
+ E.
as was done in the
Case II analysis.
less than a two-fold difference between the estimated value of p’ and the theoretical value of - 1 is not important to detect. For these reasons we were not uncomfortable proceeding with estimating the pA2. Finally, there is another way to analyze data from Case II studies that utilizes more biological judgement in examining whether the underlying assumptions (especially Assumptions I-3) hold. The steps in the analysis are recorded in Table 5. Note that for this approach, no statistical checks are made for Assumptions l-3. Rather one must depend on a visual, intuitive inspection of the data. One can check Assumptions 4 and 5 as was done for the original analysis of Case II data.
TABLE 6 Data
Sequence of Steps in the Analysis of Case III
1. For each experimental 2. Obtain
a trend
3. Make a biological 4.
unit
plot the agonist/antagonist
data.
line for each level of antagonist. judgement
If the data “satisfy”
regarding
assumptions
l-3,
assumptions obtain
l-3.
the predicted
d,
value. 5. Form
the ratios
d,id, and fit the model
log(d,/d, 6. Estimate
-
I)
= a’ + B’ log(l/B,)
+ t.
the pA> as
I PA* = +. 7. Combine
the separate
mate and standard
estimates
of pA, to form
the final esti-
error.
Final estimate: ZPAL n where
n is the number
value was computed.
2 SEM
of experimental
units
for which a p&
Estimating pA2 Values Case III: Each Experimental Antagonist
Unit Receives All Combinations
of Agonist and
Case III, in which each animal or tissue receives multiple doses of both agonist and antagonist, is also used when the effect of an antagonist on the graded response to an agonist is desired. Examples currently used in our laboratory include antiadrenergic, antihistaminic, or anticholinergic experiments in anesthetized dogs or in isolated tissues. The analysis of data resulting from Case III is unlike the analyses proposed for Cases I and II. In particular, data from each experimental unit (eg, tissue or animal) is examined separately to obtain an estimate of the pAz. Because none of the underlying assumptions can be rigorously checked, we must depend on one’s biological judgement as to whether data from an individual tissue can be used to obtain a pAZ value. The final estimate of pAz is the average of the individual estimates. A measure of variability is given by the standard error of the mean estimate. The steps in the analysis of Case III data are outlined in Table 6. Example
(Case III)
In Case III, the antihistaminic effect of chlorpheniramine (antagonist) was studied in vitro against histamine-induced contractile responses of the isolated guinea pig ileum (Table 7). A piece of the guinea pig ileum (2-3 cm) was set up in a 37” C organ bath filled with physiological salt solution (Tyrodes). The solution was constantly aerated with 95% O,-5% COZ. The dose-response experiment was performed by adding the agonist (histamine) cumulatively to the organ bath in the absence (control) and the presence of different concentrations of chlorpheniramine (Van Rossum, 1963). Responses were expressed as percentage of the maximal contractile effects produced by histamine. In this example, five dose-response curves were generated from each strip of guinea pig ileum. Since each experimental unit (eg, tissue) receives all agonist/antagonist combinations, Case III data requires that we treat data from each tissue as a separate experiment to be analyzed. Hence for each tissue we must obtain a trend line for each level of antagonist and after visually “checking” Assumptions 1-3, obtain the d, for R = 50%. The results of these separate calculations are displayed by tissue in Table 8. The data for each tissue
in Table 8 are then
log(dj/dl
-
used to fit the line
I) = IX’ + B’ log(l/B,)
+ E’
and then to estimate the pA2 value. These analyses are summarized in Table 9. Note the consistency of the data in Table 9, particularly the magnitudes of the estimated slopes which are all near -1. The estimate of the pA2 value in this competitive antagonism experiment combines the separate estimates to obtain the mean pA2. The measure of variability for this estimate is the standard error of the mean (SEMI. Unlike the estimates of variability generated for Case I and Case II data, which are based on how well the mean data fit the log(d,/d, - 1) model, this SEM reflects animal to animal variability and not variability associated with the model
87
TABLE 7
Case III Data: Percent of Maximal Contractile Effect TISSUENO.
AGONIST ANTACONI~T B, (M)
Control
0.3162 x 10 a
1x10
?'
IogWA,)
1
2
'
5
6
MEAN
k SEM
0
5
0
0
0
2
1.2 2 0.8 -
7.5
0
15
0
2
3
4
4.0 t- 2.3 11.2 t- 2.4
7.0
10
22
6
13
6
10
6.5
25
39
16
26
16
27
24.8 i- 3.5
6.0
45
56
35
48
34
52
45.0 i- 3.7
5.5
75
78
61
68
61
85
71.3 -+ 4.0
5.0
100
93
81
89
78
100
90.2 ? 3.8
1
2
3
4
5
6
7.0
0
7
0
2
2
8
6.5
0
I5
3
7
3
a
6.0 k 2.2
6.0
15
27
10
15
6
19
15.3 -t 3.0
5.5
30
41
23
37
28
31
31.7 k 2.6
5.0
60
59
42
59
58
67
57.5 k 3.4
4.5
80
78
68
81
78
81
77.7 t 2.0
4.0
70
78
84
89
88
88
82.8 + 3.1
3.5
80
90
94
89
91
96
90.0 + 2.3
1
2
3
4
5
6
3.2 t 1.4
6.0
10
15
4
9
4
8
8.3 rf-1.7
5.5
20
24
14
19
11
23
18.5 -+ 2.1
5.0
40
41
26
37
28
46
36.3 ? 3.2
4.5
60
61
46
59
54
69
58.2 i 3.1
4.0
70
68
71
74
75
85
73.8 ? 2.5
3.5
75
80
81
85
88
92
83.5 It 2.5
3.0
75
98
100
96
91
96
92.7 f 3.7
4
5
3
6
_
5.5
12
14
6
11
6
12
10.2 t 1.4
5.0
20
32
13
22
13
27
21.2 ? 3.1
4.5
35
51
23
34
31
54
38.0 t 4.9
4.0
55
85
39
56
63
92
15.0 + a.1
3.5
70
-
71
74
81
100
79.2 2 5.5
3.0
70
-
87
93
94
100
88.8 k 5.1
2.5
75
-
94
93
88
100
90.0 +- 4.2
4
5
6
5.0
20
22
6
16
6
8
l3.0 2 3.0
4.5
28
37
13
22
16
15
21.8 2 3.8
4.0
50
48
26
41
50
31
41.0 + 4.2
3.5
75
61
74
52
75
73
68.3 + 3.9
3.0
80
a5
87
74
81
92
83.2 + 2.6
2.5
80
95
94
81
81
103
89.0 2 3.9
12
1 x 10-7
4
8.0
12
0.3162 x 10
3
3
_
_ 0.78976 1.313M
-5.16667
-4.68750
-4.11589
-3.97790
.3162 x 10'
1 x ld
.3162 x lOa
1 x 10'
2.03990
1.9KM2
log(d,/d,- 11
'og d,
-6.02174
l/B,
Control
1
1.48258 1.54470 2.21045
-4 71329 -4.65302 -3.99682 -3.75oOa
-3.87179
-4.46386
~4.88083
086822
-5.28655
log d, -5.73846
log(d,/d, 11
-6.20994
log d,
2
198398
1.86073
1.25089
0.79274
logtd,/d,
3
Case III Data: Separate Calculations for Each Tissue
ANTAGONISTDOS~
TABLE 8
1, log d,
-3.66766
-4.16875
-4.69412
-5.20455
-5.96907
2.29923
1.79319
1.25126
0.67045
_
logid,id,
4 II
loi+d,
~3.87500
-4.22000
~4.57838
-5.09756
-5.70899
1.82758
1.47467
1.09721
0.48957
_
logtd,/d,
5 II
log d,
3.mO37
-4 64523
~4.90500
-5 23874
-6 10084
2.29829
1.44013
I16725
0.79792
_
logid,fd,- 11
6
90
1. Ott, et al. TABLE 9
Case III Data: Separate Estimate of pA2 TISWE
ESTIMATE
1
, ix
fitting
2
8.23370
3
7.86417
4
7.95672
5
9.91782
6
8.02914
~-
8.82564
PA(
~0.86746
-0.81777
-0.83673
-1.08571
-0.87831
-0.95481
P&
9.49
9.62
9.51
9.13
9.14
9.24
procedure.
For these
data
Average pA2 = 9.36 or 4.365 and SEM
x 1O--‘o M
= 0.09.
DISCUSSION In this
paper we have presented
one might be interested pA, values. petitive
First
we discussed
antagonism
data from
We found
of the underlying to verify
methodology
Assumptions
sumptions
1-3
l-3.
settings
and illustrated different
analysis
using
that form the basis of the comanalyses
the analysis
statistical that offers
for
by way of
of Case I and Case II data both allowed
even though
where
antagonists
(1957). Next we considered
settings
An alternative
was provided
experimental
of competitive
assumptions
of Schild
that analyses
assumptions,
different
the potencies
certain
each of the experimental
an example.
three
in estimating
checks
tests were employed fewer
checks
on As-
for Case II data.
The analyses of data for Case III provides no formal verification for Assumptions l-3. Rather the experimenter must visually check the data and proceed accordingly. An analysis
is performed
on data from each experimental
of pA2. The final estimate form of a mean (*SEMI. There are other possible paper. For example, animal (tissue)
combines
information
experimental
one could conceive
receives
a single
from
designs
unit to obtain an estimate all experimental
than the three
of an experimental
dose of agonist
but multiple
units
in the
discussed
situation
in this
where
antagonist
each
doses.
The
authors have examined this design and would not recommend its use owing to difficulties in verifying the assumptions and in obtaining a valid measure of variability
associated
with
pA2. Another
for each level of antagonist
design,
where
there is a different
has been used by some.
control
We are certain other
group designs
are possible. In this paper the analyses for three of the more common designs were outlined with the hope that experimenters will recognize that one cannot necessarily apply the same pA2 analysis to data generated from different designs. The reader should note that we have not spent much time discussing alternate procedures mula
for estimating
pA, values.
Thus
we have recommended
use of the for-
Estimating pA2 Values
where &’ and pf are obtained
from a fit of the model
log(dj/dl There
are, of course,
discusses prediction
1) = CX’ + B’ log(l/B,)
-
other formulas
one could employ.
+ E For example,
1. Assuming
B’ =
estimated
-1
intercept
and using
the modified
formula
pA, = (Y’ where
B’ =
-1
and fitting
the model
log(dj/d,
-
1) = (Y’ -
log(l/B,)
2. Assuming
3. Obtaining
an estimate -
no positive
(or negative)
authors
1) + log(l/B,).
provided
= &‘. the formula
a plot of pA, versus
the estimates
feel that the estimate
procedures
Tallarida
can be combined
pA,
lOg(l/Bj)
has
as an average
using
method
p&
=
-&l/B’
is the most
of data for Cases I-111 would
of estimation
et al. (1979) have provided
and underlying checks
the formula
to the analysis
is chosen.
These
different
not es-
only alter the final stages of the analysis. assumptions
posed by these authors formal
Then
slope,
but the approaches
change even if an alternative
ground
+ E’ to obtain p&
5 SEM.
appropriate, timation
+ E’.
of pA2 for each value of d,/d, through
= log(dj/d, PA,
&’ is the
for the model
lOg(dj/dl -1) = a’ + B’ lOg(llBj)
The
MacKay (1978)
several different methods for estimating pA, values with corresponding limits for data that we have classified as Case I. These methods include:
addresses
on Assumptions
an excellent
review
for pAZ analyses. the estimation
1-3.
The
of the theoretical statistical
analysis
backpro-
of the pA, for Case I data without
In particular,
the authors
recommend
fitting
the model -
log(d,& where
the slope
1) = (Y’ -
B’ is assumed to be -1, -1. The estimate
lOg(l/Bj)
unless some pharmacological of the pA2 and corresponding
can be given for B’ #
limits are then derived from a fit of this model. The importance of our work is recognition of the fact that different settings
give rise to different
timation
of the pA2, the estimate
of variability. a pA, (or DR,). pA, (or DR,),
The
analyses
procedures
In order one must
should
and no matter what formula in this
for these methods
to be applicable for estimating
+ ej
equation (X -
I)]
measure
paper have been used to estimate
replace the equation
log[d,/d, -
experimental is used for es-
be accompanied by an appropriate
presented
log(d,/d, - 1) = (Y' + p' lOg(l/Bj) with the more general
justification prediction
= (Y’ + B’ log(l/Bj)
+ E,
a general
91
92
1. Ott, et al. Then fit
for estimating
a pAlo (or DRlo)
log(djd, Everything
-
else in the analysis
one substitutes
x = 10 into this
9) = ty’ -t @’ IogfliB,)
remains
equation
to
+ E,
the same as was done for estimating
a pA,
or DR,.
REFERENCES Arunlakshana tative
0,
uses
Schild
of drug
HO
(1959)
Some
antagonists.
quanti-
Rr / Pharmacol
14148858. Brittain
Levy GP (1976) A review of the animal
pharmacology
of labetalol,
macol 3(4) (Suppl D]
(1964) D (1978)
affinity
OL- and
Br ) C/in
Phar-
3):681-694. Hafner How
constants
tive antagonists
a combined drug.
Statistical
Assay. New York: MacKay
McGilliard
in Biological
Press.
should
values
of pA, and
for pharmacological be estimated?
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