Heuristic and practical graphing of dose-response relationships

26

TIPS - January 1985 Reddy, V. V., Flores, F., Petro, Z., Kuhn, M., White, R. J., Takaoka, Y. and Wolin, I. (1975) Recent Prog. Horm. Res. 31, 291315

13 Gorski, R. A. (1979) in The Neurosciences: 4th Study Program (Schmitt, F. O. and Worden, F., eds), pp. 969-982, MIT Press, Cambridge

Heuristic and practical graphing of dose-response relationships When two drugs are acting together, the analogue of a dose-response relationship becomes a three dimensional entity, a dose-dose-response curve. In turn, just as an overall dose-response relationship can be broken down into a series of connected graphs representing underlying steps, a system involving two drugs can be represented by an equivalent three dimensional model. An "interaction plot" can be helpful in practical summarization of results when two drugs are used. D. R. W a u d and B. E. W a u d have tried to outline an approach to analysis of drug interactions both from the point of view of the somewhat complex underlying conceptual framework and from the more practical aspect of dealing with real data in a focused efficient and human-engineered manner. In particular, they present a format for clear graphical summarization and compact statistical analysis of experiments comparing effects of mixtures of drugs with those of their individual components.

D. R. Waud is a Professor in the Department of Pharmacology, University of Massachusetts Medical School, and B. E. Waud is a Professor in the Departments of Anesthesiology and Pharmacology, University of Massachusetts Medical School, 55 Lake Avenue North Worcester, Ma 01605, USA. 1985, Elsevier Science Publishers B.V., A m s t e r d a m

the usual expression 1 we shall formulate this whole discussion in classical terms; the generalization to other models is reasonably straightforward 2. YA ----[A]/([A] + KA)

D. R. Waud and B. E. Waud

We have recently become interested in w h e t h e r mixtures of competitive n e u r o m u s c u l a r blocking agents like tubocurarine and p a n curonium s h o w simple additive kinetics, or, instead, s h o w p o t e n tiation. (We shall use the term additivity to refer to p r o d u c t i o n b y a mixture of an effect consistent w i t h w h a t w o u l d be expected from two drugs reacting w i t h a c o m m o n receptor a n d the term p o t e n t i a t i o n to refer to production of a greater than a d d i t i v e effect.) The traditional w a y to distinguish potentiation from a d d i t i v i t y is to d e t e r m i n e d o s e - r e s p o n s e curves to drugs A and B i n d i v i d ually, identify equivalent concentrations of each, and then to see w h e t h e r a mixture of one-half equivalent of A w i t h one-half of B will p r o d u c e an effect matching a full equivalent of either. The basic analysis, while easy to u n d e r s t a n d is inefficient experimentally since a two-step process is involved and there exists the p o s s i b i l i t y that the error in the p r e l i m i n a r y assay might be magnified in the next steps. Furthermore, the approach does not readily generalize. While

14 Toran-Allerand, C. D. (1981) in Bioregulation of Reproduction (Jagie~ll-o, G. and Vogel, H. J., eds), pp. 43-57, Academic Press, New York

one could easily look at other than 50:50 mixtures (however, any other c o m b i n a t i o n will p r o b a b l y reduce the detection level b y app r o a c h i n g more closely conditions w h e n a single drug is given) it is not easy to examine a range of resp o n s e levels. We have generally preferred an experimental design which makes more complete and integral use of the experimental results both because of the usual advantage of a good statistical analysis that lets us get more precise error limits for a given expenditure of experimental effort, a n d because subjective biasses are more readily avoided if a c o m p u t e r is used objectively to analyse the results in toto. Furthermore, the fewer constraints on choice of experimental values the w i d e r will be the applicability of the results - a notion inherent in the p h i l o s o p h y of the factorial design. All these considerations led us to reconsider the approach to analysis of a p h a r m a c o d y n a m i c drug interaction. The dose--dose--response curve The first p r o b l e m is to generalize the d o s e - r e s p o n s e curve to include a second drug. This is best done in two steps. First, we can consider getting the receptors occupied and then proceed to relating this to effect. For a single drug, the fractional receptor occupany will be

0165 - 6147185/$02.00

(1)

with [A] as the (molar) concentration of the drug A, and KA its d r u g receptor dissociation constant. We n o w seek the analogous expression for a mixture. Although the two drugs will generally have different affinities for the c o m m o n receptor, the solution is straightforward. The molar concentrations of each drug can be converted to natural units of concentration b y d i v i d i n g through b y the corresp o n d i n g d r u g - r e c e p t o r dissociation constant. Thus we can replace the molar concentration of A b y the mathematically and chemically natural concentration CA with CA = [A]/KA

(2)

Similarly C8 = [B]/KB. (3) Graphically, these transformations can be represented as in Fig. 1. If equation (1) is rewritten in terms of (2) we get the more compact form YA = CA/CA + 1)

(4)

The p o i n t of all this is that the total fractional receptor occupancy, YA+B, b y a mixture of drugs A and B is given 1 b y YA+B --(CA + CB)/(CA + CB + 1)

(5)

which can be written Yc = YA+B = Cc/(Cc + 1) (6) to indicate that, w h e n one thinks in natural units of concentration, the combination behaves like a single drug (i.e. equation 6 is of the same form as equation 4) given at a concentration Cc, equal to the sum of the i n d i v i d u a l concentrations CA and CB. H o w do we graph this? W e can break it d o w n into two stages. First, let us represent the sum CA + CB. This requires a three d i m e n sional d i a g r a m (we have three variables - the sum, CA and CB) and the relationship itself is simply a plane t h r o u g h the origin. Fig. 2 illustrates this.

27

TIPS - J a n u a r y 1985

(a)

~) 1

cA

CB

slope

F m ~ o n a l

/ • =s].ope " 1/KB

[A]

twitch

[dTc]

[B]

Fig. 1. Graphical representation of equations (2) and (3) for the conversion of[A] and[B] to CA and CB. Abscissae: molar concentrations; ordinates: natural units of concentration.

(~)

YdT~

0

l

Fig. 4. Representation of (a) an exponential fall of concentration with time and (b) the margin of safety of neuromuscular transmission. In the latter graph the twitch response to indirect stimulation is plotted against the fractional receptor occlusion, Y=c, by the antagonist.

~

YdTc [,~-

I I I I I J I I~1

0 t

Yd'rc

f

~

(~)

h (iii)

Y~(ii)

CB

Cc

[dTc] Fractionaltwitch height

c

C^+Cd Plane 'rJ~e

CA

Fig. 2. Plot of Cc = CA + CB. A dose a of A and b ofB define pointp in the CA--Osplane. Passing vertically you hit the CA+B plane at q and then proceed horizontally to reach the Cc axis at c.

01

5 c

Fig. 3. Graph of Y = C/(C + I).

r'

1

I l l l i J l l l l

CB

CA+B

[

0

Fig. 6. Variant of Fig. 5 taking advantage of logarithmic transformation to improve the representation. The path t,c, y, h traces an example of how one gets to response h that would be produced at time t given a dose that would put us on the second highest curve in panel (i).

Fig. 5. Graphical summary of the behavior of tubocurarine. Pane/(i) (= Fig. 4a) gives antagonist concentration as a function of time. Pane/ (ii) (= Fig. 3) converts that concentration into fractional receptor occlusion. Pane/ (ill) (= Fig. 4b) converts that occlusion into the end response, the effect on twitch height.

'

[A]

10 J

Fractionaltwitchheight Fig. 7. Overall graph to summarize the behavior of a combination of two drugs. See text.

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T I P S - J a n u a r y 1985

To represent the step from Cc to Yc we simply graph e q u a t i o n 6 as i n Fig. 3.

J tape

Graphical catenation

j,i~ge b

C^+ 1 ('a0

i

t;

at

ri b'

Fig. 8. Do-it-yourself model of Fig. 7. To assemble connect (i) and (ii) via slots a and a'. The clear plastic sheet representing the CA + CB surface is cut so that the apex 0 falls at the origin 0 on sheet (ii). Next slide sheet (iii) into (i) via slots b and b ". the paths from a and b through a; b ', p, q, c" y" to h' in Fig. 7can be represented by lengths of string glued over the corresponding paths of panels (ii) and (iii), with a length running from point p on (ii) up through hole q on (i) and over to point r on (iii).

1

)

• ) 0

0

1

1

i

i

0

I

,

0

Concentration

C.oncenWa~on

Fig. 9. "Interaction plots" of model data. Ordinates: fractional tw#ch height. Abscissae: concentrations scaled so that EDsa will be 1 for the reference curve. Circles: Drug A (KA = 2), Squares: Drug B ( K B = 10), Triangles: Drug A + Drug B, interaction parameter C = I, 0.9, 0.8 and 0.6 in panels (a) through (d) respectively. Lefthand curve is drawn ~arallel to reference curve and shifted to the left by a factor equal to the estimated value of C.

I

Years ago, in an i n - h o u s e dep a r t m e n t discussion at the Shattuck Street School, Oleg Jardetsky wanted to emphasize the idea that the overall dose-response relationship was, in general, the expression of a sequence of steps each of which could be expressed b y a graph relating the o u t p u t of each step to the input. He therefore pictured a chain of graphs in which the scale of ordinates on the first, became the scale of abscissae on the second graph and so on. In the general form this m a n o e u v r e did not catch on. However, we have found over the years that the format can provide a very convenient way to summarize behavior of specific systems. Furthermore, it can be expanded to include a pharmacokinetic step(s) as well. To illustrate, consider a drug like tubocurarine. We can summarize the pharmacokinetics with a diagram like Fig. 4a. (We have d r a w n a single exponential fall-off to keep the a r g u m e n t simple. The reader can easily generalize to a more realistic multiexponential case or to other variants such as inclusion of a zero order c o m p o n e n t w h e n appropriate for some other drug.) The next step, reaction with the receptor, would be s u m m a r i z e d by a curve of the form of that already given i n Fig. 3. The last step, interference with the twitch response to indirect stimulation, would be represented b y a relationship like that d i a g r a m m e d in Fig. 4b. This plot is a schematic s u m m a r y of empirical m e a s u r e m e n t 3 of the relation between twitch height and receptor occlusion. Here, we have simply d r a w n a curve such that the twitch is normal until about 75--80% of the receptors are occluded a n d then falls to zero a r o u n d 90-95% receptor occlusion. A g a i n the reader can substitute whatever variant may apply i n some other system. We can n o w chain these three graphs together as in Fig. 5 to obtain a very compact s u m m a r y of what is going on w h e n you give a drug like tubocurarine. There is a variant of this diagram which can be even more useful. It follows from the fact that an exponential curve is more easily visualized i n a semilogarithmic plot

TIPS - January

29

1985

and from the c o n v e n t i o n of plotting dose-response curves with a logarithmic concentration scale to spread out the low end. Fortunately the logarithmic conversion comes in the right place to be comm o n to the first two sections of Fig. 5 which can therefore be redrawn as in Fig. 6. Fig. 6 also shows an additional advantage of the logarithmic format. In the first section one can represent different doses b y a family of curves which, in the simple case we're considering, first order kinetics, will all be parallel and differ only in height along the (log)concentration axis. Fig. 6 also shows how the graph can be used. Suppose the dose corresponding to the second highest curve has b e e n given and we are n o w at time t. Following the arrows we have concentration c, fractional receptor occlusion y and finally a twitch response given by h.

A conceptual interaction graph We can return n o w to the original problem of s u m m a r i z i n g an interaction b e t w e e n two drugs. To avoid m a k i n g what the reader will agree is already a very complicated diagram still worse, we shall leave off the initial (pharmacokinetic) step. It can easily be added later b y those who so desire. Fig. 7 is the end result. Since a three d i m e n sional diagram is not easily represented on a page we shall go through Fig. 7 stepwise. First pick out Fig. 2 in the top righthand part of Fig. 7. Then note that Figs l a and l b have been attached to the CA and CB axes respectively. Next identify Fig. 3 attached to the CA+B axis a n d finally, pick out Fig. 4b at the end of the chain (the final scale of ordinates is a broken line because it lies b e h i n d and u n d e r the plane of the CA V. [A] section). It will help n o w to walk through the specific example represented b y the smaller arrows. We start by giving concentrations a and b of drugs A and B. Each is converted to natural units a' and b'. These in turn define a point p on the CA -- CB plane. We n o w pass vertically to the point q on the plane representing the sum of CA and CB and from there reach c' on the CA+B axis. Now we proceed as in the last two steps of Fig. 6 to get y' and then h' the final resultant twitch response. Fig. 7 is obviously not easy to

foUow. Part of the reason is simply that the u n d e r l y i n g concepts are not trivial, b u t a lot of the reason is simply the confusion that results w h e n a three d i m e n s i o n a l structure is flattened out. The reader may wish to construct the original structure (it makes an entertaining mobile/conversation piece to hang in your office). To this end construct and assemble the three cardboard sheets depicted in Fig. 8.

The 'interaction plot' Although Fig. 8 or some variant thereof can be a useful heuristic model, it is a long way from a useful format for s u m m a r i z i n g actual experimental results. We turn n o w to this issue. We have approached the problem as follows. We may illustrate with a simple system, the twitch response of an isolated nerve-muscle preparation stimulated indirectly. The experimental results will be sets of twitch responses in the presence of various concentrations of antagonist A, antagonist B or a mixture of both. First we have recognized that, if the simple classical competitive model holds (our null hypothesis), we can replace KA and KB in the preceding analysis by the EDs0 values of A and of B respectively. We have then represented the relationship between twitch height, y, and concentration by the function (chosen empirically and for computational convenience) y = 1 -

xS/(x s +

1)

(7)

and have taken x to be [A]/ED50A, [B]/EDs0B or their sum as appropriate. S is a parameter which determines the steepness of the curve. We have n o w simply told the computer to look at all the points and find the values of the parameters EDs0A, EDsoB and S most consistent with the experimental observations, and to determ i n e the error variance associated with the resultant fitted curve. Next we have repeated the process but, for responses w h e n two drugs are present, we have allowed the computer to use one more multiplicative parameter C to adjust the sum [A]/EDs0A + [B]/EDs0B to get a closer fit than in the first run. This second run then gives two interesting n u m b e r s , C with its standard error and the error mean square. If C is significantly different from u n i t y we can conclude the simple additive model does not fit

the data adequately. Alternatively, we can approach the same question b y comparing the reduction in the error sum squares between runs I and 2 with the error for r u n 2. The resultant variance ratio will again test for deviation from the model represented by the null hypothesis. A significant variance ratio will indicate experimental behavior inconsistent with simple additive kinetics. Now we can summarize the results compactly in a graph. First we have the computer draw as a reference curve function 7 with the value of S determined from the least-squares regression (run 2). Next we can have the computer plot the points at positions on the concentration axis d e t e r m i n e d by d i v i d i n g the original molar concentration by the EDs0 of A or B determined in the least-squares fit. However, we do not allow the computer to use the parameter C to bring the c o m b i n a t i o n points into line. Therefore, we expect that the points representing administration of A or B alone should cluster on the reference curve and the scatter will give a c o n v e n i e n t subjective visual impression of background noise. If there is simple additivity, the points for the combinations curve will also be buried in this cluster whereas if the simple model is inadequate, they will lie to the left of the cloud formed by the other two sets of points. Thus, if one can see daylight between the two groups, one can conclude there is potentiation b e y o n d additivity. Finally, we have had the computer draw a second curve shifted from the reference curve by the factor c. This will simply show the line of best fit to the c o m b i n a t i o n points. Fig. 9 shows the sort of picture one gets. For this illustration we had the computer generate data with EDs0A = 2, ED50B = 10 and S = 5 (a reasonable value for a nerve-muscle twitch response). We added an error of 10% of maxi m u m twitch response (we chose 10% because the results including the error limits on C then resembled actual experimental values reasonably; for similar reasons, we limited fractional twitch responses to the range 0-1) and looked at values of C of 1, 0.9, 0.8 and 0.6 i.e. 10%, 20% and 40% off unity. In the 'control' panel (a) C is unity, there is true additivity and no more than

30

TIPS - January 1985

a glance at the figure is n e e d e d to confirm this. Similarly, w h e n C is m a r k e d l y different from one, i.e. there is marked potentiation, this fact is clearly evident from the figure (panels c and d). Finally, panel b leaves the correct impression that it's hard to tell. In such a case you go back to the error limits on C or the analysis of variance. In this particular case, C was estimated as 0.877 + 0.023 (SE) and the variance ratio was 20.15 (1, 43 degrees of freedom). For simplicity of presentation we have first described only the m i n i m a l analysis. In a real experimental setting one can a d d appropriate bells and whistles. Thus, for example, k n o w i n g that nerve degeneration p r o b a b l y begins the m o m e n t one isolates a n e r v e muscle preparation, w e set u p the actual analysis in a factorial design involving six animals with the drugs given in the orders ABC CBA

BCA ACB

CAB BAC

In other w o r d s d r u g A comes first second a n d third twice. Drug B a n d the c o m b i n a t i o n C do likewise. Furthermore, in this example one can look at the effect of order of administration by comparing three columns ('permutation') or two rows ('direction'). O n e gets the effect of ' t i m e ' b y c o m p a r i n g all the first a d m i n i s t r a t i o n s with all those second a n d all those third. The details, which involve a split plot factorial design, n e e d not concern us here (although w e m a y note in p a s s i n g that ' p e r m u t a t i o n ' and 'direction' were w i t h o u t effect while ' t i m e ' was, as anticipated, quite significant); the p o i n t is first that the basic approach is easily generalized (we have also h a d occasion to a d a p t it to experiments involving parallel shifts of d o s e response curves from a control curve) a n d second, that the final plot will always come d o w n to the

form of Fig. 9 since the nature of the plot is such that the c o m p u t e r s i m p l y includes all extraneous factors (such as ' t i m e ' above, or parameters reflecting ' a n i m a l variation') in the calculation of w h e r e the points should finally lie. We have illustrated our approach w i t h p o t e n t i a t i o n as the alternative to simple addition. Note, however, that the format of the 'interaction plot' of Fig. 9 w o u l d also allow one to look at cases w h e r e less than a d d i t i v i t y m i g h t occur i.e. the triangles fall to the right of the reference cluster. While the present p a p e r was in preparation, another article on synergism a p p e a r e d in this journal 4. A few comments to relate the two approaches m a y be helpful. Chou and Talalay linearized sigm o i d d o s e - r e s p o n s e curves with a Hill plot (their Fig. 2), fitted straight lines, and c o m p a r e d xintercepts. These intercepts are measures of the ED50 so they could compare the EDs0 for a combination w i t h that expected from simple additive effects. Thus w e ' r e all going through the same general procedure (as of course one might expect). In particular, we both normalize concentrations by dividing molar concentrations b y a reference concentration ~,2 and we both fit the data to a logistic function. We prefer, however, to take advantage of m o d e r n computing p o w e r to eliminate the need for linearization, to facilitate analysis of the experiment as an unit and to p r o v i d e a final presentation that gives the a n s w e r directly complete with error limits and a convenient pictorial representation. Thus we believe our interaction plot (cf. Fig. 9) gives a more focused s u m m a r y d i a g r a m than their set of parallel linearized d o s e - r e s p o n s e curves (their Fig. 2). Similarly, our C parameter gives a direct index of the extent of n o n - a d d i t i v i t y , whereas the approach of Chou and Talalay

give us n u m b e r s (0.890 and 0.989 in their specific example) whose error limits/level of significance is not i m m e d i a t e l y clear. They imply some advantage in that 'conclusions could be d r a w n w i t h o u t k n o w l e d g e of the conventional kinetic constants (Kin, Vmax or Ki) and involved no a s s u m p t i o n s w h e t h e r the mechanism was competitive, non-competitive or uncompetitive'. In the example they use, their reference concentration D m is effectively an EDs0 analogous to a Km and to calculate their fractional effects they have to assume some m a x i m u m analogous to a Vmax so the first advantage seems more a p p a r e n t than real. The ind e p e n d e n c e from m e c h a n i s m follows from the normalization of concentration and so is inherent to b o t h our approaches. Finally we m a y note that, if for some reason one w a n t e d to present the d o s e response curves in the linear form of the Hill plot, this w o u l d simply represent just one particular choice of functional relationship that could be p l u g g e d in the computer p r o g r a m carrying out our analysis. Thus the interaction plots w o u l d become straight lines but more importantly, error limits and significance levels of the key parameters w o u l d then be directly available.

Acknowledgements The authors appreciate support in the form of Grant NS12255 from NINCDS of the National Institutes of Health.

References 1 Waud, D. R. (1968)Pharmacol. Rev. 20, 4988 2 Colquhoun, D. (1973) in Drug Receptors (Rang, H.P., ed), pp. 149-182, University Park Press, London 3 Paton, W. D. M. and Waud, D. R. (1967) J. Physiol. (London) 191, 59-90 4 Chou, T. C. and Talalay, P. (1983) Trends Pharmacol. Sci. 4, 450-454