potency measures for developmental toxicity

Environmetrics; 1993; 4 (4): 507-51 8

Potency Measures for Developmental Toxicity

LOUISE RYAN

ABSTRACT

Potency measures provide a useful means of ranking chemicals, drugs and other substances with respect to the danger they pose to humans. While several authors have proposed estimators of carcinogenic potency, the task of ranking developmental toxicants has received little attention. The purpose of this paper is to develop a quantitative measure of developmental potency, based on a correlated multinomind model that allows for dose effects on foetal death, variation and malformation rates. The proposed estimator is illustrated with data from several studies conducted through the National Toxicology Program.

KEY WORDS: Generalized estimating equations; clustered data; teratology.

1. INTRODUCTION

Governmental agencies are becoming increasingly aware of the need for improved and more sophisticated methods of risk assessment for developmental and reproductive toxicology. Unlike cancer where ideas of dose- response modelling are well established, quantitative risk assessment for developmental and reproductive toxicity is a relatively new field of study. While the topic of dose-response modelling for developmental toxicants has received some attention in recent years, few authors have addresed the related question of how to measure teratogenic potency. One measure of tere- togenic potency was proposed by Fabro et al. (1982). These authors suggested fitting a dose-response curve to the data using a probit model, then measuring teratogenic potency with the tDo5, that is the dose required to

Department of Biostatistics, Dana Farber Cancer Institute, and Harvard School of Public Health, Boston, MA 02115, USA.

1180-4009/93/040507-12$11.00 @ 1993 by John Wiley & Sons, Ltd.

Received 10 March 1993 Revised 30 Jdy 1993

508 L. RYAN

induce an additional 5 per cent malformation rate among live pups above background. The term tD stands for teratogenic dose, as opposed to TD which traditionally stands for toxic dose. The authors chose 5 per cent above background because they felt that this corresponded to a minimal level of teratogenic activity that could be reliably detected in most experimental situations. In many respects the tDos is similar to the benchmark dose which has gained in popularity for characterizing the dose effect for developmental toxicants (Crump, 1984). Wang and Schwetz (1987) suggested an alternative potency measure that weights a variety of factors including type and nature of malformations, incidence of variations (atypical features that are not serious enough to count as malformations), maternal toxicity and so on. While the idea of accounting for multiplicity and different endpoints has appeal, the Wang and Schwetz method does not use quantitative dose- response modelling in its calculations. The purpose of the present paper is to propose a quantitative measure of teratogenic potency that takes into account some of the multiple outcomes considered by Wang and Schwetz.

The problem of multiple outcomes is one of the most challenging and interesting aspects of developmental toxicity. Unlike cancer, where the primary outcome is presence or absence of a tumour, exposure to environmental toxicants can affect many different aspects of reproduction and development, including reproductive functioning (e.g. sperm count and ovulation), fertilization and implantation. Once implantation has occurred, exposures can result in early pregnancy loss, malformation, lowered foetal weight or subsequent developmental problems. Ryan et al. (1991) discuss the usefulness of building a dose-response model to characterize the effect of exposure on the entire developmental process. However, doing so raises several challenging statistical issues. For example, some outcomes such as foetal resorption, death and malformation are hierarchically related, in the sense that a malformation can only be observed if resorption or death have not occurred. Furthermore, the analysis of developmental toxicity data is complicated by the presence of the litter effect, or the tendency for litter mates to respond similarly (Haseman and Kupper 1979). A number of authors have discussed the problem of accounting for litter effects in the context of single binary outcomes. For example, Williams (1975) d' iscusses the use of beta-binomial models which assume that litter-specific response rates are distributed according to a beta distribution. Kupper et al. (1986) discuss the importance of allowing the beta parameters characterizing the litter effect to vary with dose level. In a reader reaction to the Kupper et al. paper, Williams (1988) agrees that litter efects are likely to depend on dose, but argues that the problem can be addressed more easily by using quasi-likelihood-based methods. Ryan (1992a) also argues for the use of quasi-likelihood and related methods for the analysis of the correlated binary outcomes that arise in developmental toxicity. In many ways, the paper of Fabro et al. (1982) was ahead of its time because a simple quasi-likelihood method (Finney 1947, Section 42, p. 160) was used there to adjust for the

POTENCY MEASURES 509

litter effect. Several authors have recently discussed the problem of analysing multiple outcomes (Ryan 199213; Krewski and Zhu 1993; Chen et al. 1991). This paper uses similar ideas to develop an estimate of teratogenic potency that takes into account several stages of foetal development, including foetal deaths, variations and malformation. The estimate will be applied to several National Toxicology Program (NTP) data sets, and the results compared with other approaches. Finally, some issues for further statistical research will be briefly discussed.

2. MODELLING HIERARCHICALLY RELATED OUTCOMES

Chen e t al. (1991), Ryan (1992b) and Krewski and Zhu (1993) discussed the use of multinomial modelling techniques to characterize the effects of exposure on hierarchically related outcomes such as foetal death and malformation. In calculating teratogenic potency, it is also desirable to take into account the incidence of variations. Consider an experiment with K dose groups (dl = O,&, . . .,&) and Ii dams per dose group. One can characterize the observable data for the jth dam in the ith dose group by the vector Yij = ( Yijl, Yij2, Yij3, Yij4), whose elements denote the number of dead foetuses, malformed foetuses, non-malformed foetuses with variations, and normal foetuses, respectively, in the litter. The sum of the four elements for each dam equals Naj, the total number of implantation sites for that litter. Possible outcomes at a single implantation site can be schematically represented by Figure 1.

Ignoring for the moment the question of litter effects, it is natural to model YQ as a multinomial random variable since every foetus falls into one and only one of the four designated categories. Strictly speaking, it will be possible for some foetuses to have both variations and malformations. However, we assume that the category “malformation” overrides the category “variation”, and a foetus with both would fall into the former. The natural hierarchy of the observable data suggests a continuation- ratio formulation (McCullagh and Nelder 1989, Section 5.2.5; Agresti 1990, Section 9.3.3). That is, the probability vector P(4) associated with the outcome Yij will have components

where 81 ( d ) denotes the probability that an implanted embryo at dose level d does not survive the gestational period; &(d) denotes the conditional prob ability of observing a malformation, given that a foetus at dose level d survives to be examined; and & ( d ) denotes the conditional probability of

510 L. RYAN

Implantation

Normal Offspring

Figure 1. Foetal development.

observing a variation given that a foetus at dose level a! was examined and found to have no malformations.

In the context of building models for only the first two stages (death and malformation), Chen et al. (1991), Ryan (1992b) and Krewski and Zhu (1993) each consider slightly different formulations of the dose-response relationship on the risk functions 4 ( d ) and &(d). Ryan considers a general class of dose-response models of the form

W d ) = Sk + (1 - Sk)Ft(% + PklOS(&I, (2)

where F is a cumulative distribution function. Cheri e t at. consider a special case of the family (2), the Weibull model, obtained by setting F(z) = 1 - e q [ - e q ( z ) ] . Krewski and Zhu generalized this approach by adding number of implants n as a covariate and a threshold parameter ( d k o ) at each stage,

e k ( d I n>= 1 - e x p [ - ( a k + p k n + ( ~ k + ~ k n ) ( d - d k o ) 9 . (3)

Ryan (1992b) considered a different class of doseresponse models,

&Ad) = F(ak + PkdYk) , (4)

where F is a cumulative distribution function, as in (2). Putting F ( z ) =

1 - exp(-x), for instance, leads to the familiar Weibull or extreme value model. Putting F ( z ) = @J(z), where @(z) denotes the cumulative density function of the normal distribution, yields a modified probit model. Ryan


e t al. (1991) and Catalan0 and Ryan (1992) apply modified probit models, arguing that malformation and death can be thought of as being associated with some underlying continuous phenomena. In all the models discussed here, the subscripts i allow the baseline rates, dose effects, and power parameters to vary for each of the different types of outcomes.

As discussed by Ryan et al. (1993), the methods of Chen et al. (1991), Ryan (1992b) and Krewski and Zhu (1993) also differ with respect to the way they adjust the models to allow for the litter efect. Just as for binary outcomes in developmental toxicity, clustering within litters will tend to induce more variability in the data than can be explained by a multinomial model with response probabilities depending solely on dose (Chen and Kodell 1989). Chen et al. (1991) and Krewski and Zhu (1993) both describe parametric approaches based on Dirichlet-multinomial models, which are natural extensions of the beta-binomial distribution (Williams 1975) to account for the litter effect. Ryan (1992b) and Krewski and Zhu (1993) also consider quasi-likelihood methods. Ryan (1992b) applies the method suggested by McCullagh and Nelder (1989) for multinomial data, modeling the probability vector P(4) as a function of dose, but inflating the standard multinomial variance by a scale factor. Krewski and Zhu (1993) apply a different quasi-likelihood approach using marginal moments derived from a Dirichlet-multinomial model. This paper uses generalized estimating equations (GEEs) (Liang and Zeger 1986; Zeger and Liang 1986). GEEs provide a mathematically tractable and appealing solution to the problem of adjusting for litter effects. Like the quasi-likelihood approaches, GEE methods require only specification of the mean and covariance structure of the observed data. However, GEE methods have an additional benefit of incorporating an empirical variance “fix-up” that ensures consistent estimation of the parameters characterizing the mean, even if the covariance structure is misspecified. Ryan (1992b) describes the use of GEEs to model binary outcomes in developmental toxicity, and shows how the approach can work well for a wide class of dose-response models. Ryan et al. (1993) describe the application of GEE models to the analysis of foetal death and malformation. In brief, the approach involves the solution to a set of estimating questions, constructed as follows. First, associate with the multinomial vector Yij = ( Yijl, Yij2, Yij3, Yij4) three different vectors whose elements are the binary indicators of death, malformation and variation status corresponding to the totals YG,, Yij2 and Yij3. That is, for Ic = i ,2 or 3, Zi j , = (Zij,,,Zij,,, . . . ,Zijbuk), where for k = 1, each of the Zijll are binary indicators of death status for the rijl = Nij implants in the zjth litter; for k=2, the Zij21 are binary indicators of malformation status for the rij2 = Nij - Yijl live foetuses, and the Zij31 are binary indicators of variation status for the rij3 = Nij - Yijl - Yijz non-malformed live foetuses, in the ajth litter. Conditional estimating equations can be constructed by considering the marginal moments of Zij,, the conditional moments of Z,, given Zip These are, for k = 1,2 and 3:

512 L. RYAN

i-I j - l

where 1, is a p-vector of ones, D i j k represents the r4ik x p design matrix a e k ( & ) / d P k , where P k represents the p-vector of unknown parameters characterizing &(&), and v i j k is the r i jk x r i jk assumed covariance matrix of &jk. Following Liang and Zeger (1986), V i j k can be written as

where A i j k is the diagonal matrix with elements 6+(di)(l - O,(dJ), and R i j k is a suitable correlation matrix. It is straightforward to show that a constant correlation matrix is appropriate if the multinomial vectors for each dam follow a Dirichlet-multinomial model. The set, of equations defined at (5) can be solved using the public-domain software GEE available from Johns Hopkins University, and implementable in either SAS or S. Several approaches can work well. For instance, one can assume that independent sets of parameters characterize 6 k and fit three separate GEE models. Alternatively, by creating a suitable design matrix, one can fit one large GEE model and solve the equations simultaneously. This latter approach also has the advantage of providing a framework where one can test hypotheses such as common dose effects at each stage of development.

3. POTENCY

Several authors (Gold et al. 1984; Pet0 et al. 1984; Portier and Hoel 1987; Bailer and Portier 1993) have discussed the problem of estimating carcinogenic potency. Further discussion may be found in Gart et al. (1986, pp. 120, 121). Generally, potency has been defined in terms of the dose corresponding to a specified risk over the controls, based on a one-hit dose- response model (Pr(event) = 1 - exp(-a - pd)). One popular choice has been the T&, or the dose level at which the excess risk of developing a tumour is 50 per cent higher than the background rate (Sawyer 1984). This quantity has been discussed, evaluated and modified by a number of authors (Gold et al. 1984; Pet0 et al. 1984; Portier and Hoel 1987). Bailer and Portier (1993) extend the TDsO idea to models such as (4) that incorporate power parameters on dose. Following up on the same idea, Meier et al. (1993) discuss problems associated with using the 50th percentile of the response distribution, and recommend the use of a smaller value such as 1 per cent or 10 per cent. Their basic argument is that estimators based on smaller quantiles are likely to be less correlated with the experimental maximum tolerated dose (MTD). They also argue that with more general doseresponse families such as (4), choice of quantile can have a strong


effect on the rankings of chemicals with respect to potency. Thus, it seems advisable to base potency determinations on risk levels closer to levels likely to be encountered by humans.

We propose here that a suitable definition of potency associated with developmental toxicity is the dose at which the probability of having a abnormal offspring is increased by q per cent from control levels. We call this measure the aD, with aD denoting abnormal dose (as opposed to teratogenic dose suggested by Fabro et al.). As recommended by Fabro et al. and Meier et al., small values of q will be considered, specifically q = 0.05.

From Figure 1, it follows that the probability of an abnormal foetus will be

Potency can then be measured by the solution to r(d) = q , where r(d) denotes a suitable measure of the increase in the chance of having an abnormal offspring at dose level d , compared with controls. As discussed by Gart et al. (1986), the risk function r(d) can be defined on the additive scale

or on a multiplicative scale,

The use of the multiplicative scale has a special advantage for the family of doseresponse models (2), since the parameters Si factor out of (9). For some choices of F,rmdt(d) simplifies even further under model (2). For example, when F(z) = 1 - exp[-exp(z)], and PI = p2 - p3 = p, the same doseresponse parameter for each stage of development, then

where T = [exp(aI) + exp(a2) + exp(a3)] and y = exp(p). It immediately follows that the uD, is

aD, = (-lnq/T)'/Y.

Since the true r(d) is unknown, it can be estimated by 4, the solution to P(d) = q. A lower confidence limit for the uD, could be calculated using the delta method, and exploiting the asymptotic normality of the estimated parameters from the model (see Gart et al. 1986). A related method that tends to have greater numeric stability was described by Kimmel and Gaylor (1988). These authors suggested finding a lower confidence limit on aDq by determining the dose corresponding to an upper confidence limit on the

514 L. RYAN

excess risk function rather than a lower confidence limit on dose producing a given excess risk. That is, the idea is to solve the following expression for d:

P(d ) + 1.645SE[C(d)] = q.

Using the delta method, an approximate variance of the excess risk function is given by

where p represents the vector of all the unknown parameters, including those characterizing & ( d ) , & ( d ) and 03(d). While this expression becomes algebraically messy, it is quite straightforward to calculate using the estimated parameters and their variances obtained from the GEE output.

4. APPLICATIONS

Data from several developmental toxicity studies in mice will be used as illustration. These studies were conducted by the Research Triangle Institute under contract to the National Toxicology Program (NTP) , and investigated the chemicals ethylene glycol (EG) (Price e t al. 1987), triethylene glycol dimethyl ether (TGDM) (George et al. 1987) and diethylene glycol dimethyl ether (DYME) (Price e t al. 1985). Summary statistics from these studies appear in Table 1. The table shows the number of deaths at each dose group, and their percentage as a function of the number of implants, malformations among live pups, and variations among non-malformed live pups. While not used directly for risk assessment, incidence of variations is thought to be an important and sensitive indicator that tends to show up at lower exposure levels than malformations.

The final column of Table 2 shows the estimated sass based on the multinomial model that has been discussed in this pa,per. The number in parentheses are lower confidence limits on the true potencies. For compar- ison, the table also shows potency estimators obtained by considering only deaths, malformations or variations as the endpoint of interest. For malformations, this is essentially the procedure proposed by Fabro et al. Figure 2 is a graphical representation of the results in Table 2. The figure plots the estimated potency values, with the axis represented on the left side denoting mg/kg ranging from 0 to 3000. Table 2 and Figure 2 both show how if potency is based on a single outcome, then the relative rankings of the chemicals will change, depending on which outcome is chosen. The ranking based on the multinomial model provides a way to synthesize the information from all the endpoints. Of the three chemicals considered, DYME is the most


Study/ Dams dose Imp Dths (%) Live Malf (%) No Malf Var (%)

EG 0 750 1500 3000

TGDM 0 250 500 1000 DYME 0.0 62.5 125.0 250.0 500.0

25 24 23 23

27 26 26 28

21 20 24 23 23

334 37 (11) 297 1 310 34 (11) 276 26 266 37 (14) 229 89 283 57 (20) 226 129

342 23 (07) 319 1 296 21 (07) 275 0 296 34 (11) 262 2 327 41 (11) 286 33

297 15 (05) 282 1 242 17 (07) 225 0 312 22 (07) 290 7 299 38 (03) 261 59 285 144 (51) 141 124

296 250 140 97

318 275 260 253

281 225 283 202

9

16 (05) 12 (04) 14 (05) 31 (12)

Table 1. Summary statistics.

Binomial models Multinomial model Study Death Malformation Variation

EG 2086 537 764 460

TGDM 69 1 798 764 350

DYME 262 156 81 66

(809) (222) (544) (210)

(91) (734) (544) (179)

(156) (143) (22) (43)

Table 2. uDo5 potency estimates in mg/kg (lower 95 per cent confidence limit).

potent for all three separate outcomes, and this is reflected by the very low value of the uD05. EG is very potent if one considers malformations alone, but not very potent at all with regard to deaths or variations. TGDM, on the other hand, has a moderately high potency with respect to all three endpoints. It makes intuitive sem that the overall measure of potency ranks these two chemicals quite closely.

516

E 3

TG

DY

3000.

2000

1000

0

3M

ME

ETJ

M TG

[E

Figure 2. Relative potencies.

5.

I M

DYVE

L. RYAN

Malformation Variation.

TC €

D’I

CONCLUSIONS

T t DE

&DO5

EM ME

This paper has proposed a simple measure of teratogenic potency ,ased on the dose required to increase the probability of an a,bnormal offspring by 5 per cent compared with unexposed individuals. The proposed measure applies concepts similar to those used to define carcinogenic potency, but takes multiple outcomes into account. The properties of the estimator have been illustrated empirically for several data sets.

There are several areas where further research is needed. First, the models described address only the effect of exposure on foetal death, malformation and variation rates. Potency should ideally take into account the effects of exposure on all aspects of development. Catalan0 et al. (1993) discuss dose-response models that incorporate fetal weight in addition to death and malformation. Potency estimators based on such multivariate model provide much applied potential.

REFERENCES

Agresti, A.A. (1990) Analysis of Ordered Categorical Data. Wiley, New York. Bailer, A.J. and C.J. Portier (1993). “An index of tumorigenic potency”. Biometrics 49,

Catalano, P.J. and L.M. Ryan (1992). “Bivariate latent variable models for clustered discrete and continuous outcomes”. Journal of the America1 Statistical Association 87, 651-658.

Catalano, P.J., D. Scharfstein, L.M. Ryan, C. Kimmel and G. Kimmel (1993). “Statistical model for fetal death, fetal weight and malformation in developmental toxicity studies”. Temtology 47, 281-290.

357-366.


Chen, J.J. and R.L. Kodell (1989). “Quantitative risk assessment for teratologic effects”. Journal of the American Statistical Association 84, 966-971.

Chen, J.J., R.L. Kodell, R.B. Howe and D.W. Gaylor (1991). “Analysis of trinomial responses from reproductive and developmental toxicity experiments”. Biometrika 47, 1049-1058.

Crump, K.S. (1984) “A new method for determining allowable daily intakes”. Fundamentals of Applied Toxicology 4, 854-871.

Fabro, S., G. Schull and N.A. Brown (1982). “The relative teratogenic index and teratogenic potency: proposed components of developmental toxicity risk assesment” . Terntogenesis, Carcinogenesis and Mutagenesis 2, 16-76.

Finney, D.J. (1947). Probit Analysis. Cambridge University Press, Cambridge. George, J.D., J.C. Price, C.A. Kimmel and M.C. Marr. “The developmental toxicity of

tirethylene glycol dimethyl ether in mice”. Fundamentals of Applied Toxicology 9, 173- 181.

Gart, J.J., D. Krewski, P.N. Lee, R.E. Tarone and J. Wahrendorf (1986). Statistical Methods in Cancer Research. Volume III: The Design and Analysis of Long-Term Animal Experiments. International Agency for Research on Cancer, Lyon.

Gold, L.S., C.B. Sawyer, R. McGraw, G.M. Buckman, M. devecidna, R. Levinson, N.K. Hooper, W.R. Havender, L. Bernstein, R. Peto, M.C. Pike and B.N. Ames (1984). “A carcinogenic potency database of the standardized results of animal biosassays” . Environmental Health Perspectives 58, S319.

Haseman, J.K. and L.L. Kupper (1979). “Analysis of dichotomous response data from certain toxicological experiments”. Biometrics 35, 281-293.

Kimmel, C.A. and D.W. Gaylor (1988). “Issues in qualitative and quantitative risk analysis for developmental toxicology”. Risk Analysis 8, 15-20.

Krewski, D. and Y. Zhu (1993). “Applications of multinomial dose response models in developmental toxicity risk assessment”. Risk Analysis to appear.

Kupper, L.L., C. Portier, M.D. Hogan and E. Yamamoto (1986). “The impact of litter effects on dose-response modeling in teratology” . Biometrics 42, 85-98.

Liang, K.-Y. and S.L. Zeger (1986). “Longitudinal data analysis using generalized linear models. Biometrika 73, 13-22.

McCullagh, P. and J.A. Nelder (1989). Generalized Linear Models. Chapman and Hall, London.

Meier, K.L., A.J. Bailer and C.J. Portier (1993). “A measure of tumorigenic potency incorporating dose reponse shape”. Biornetrics in press.

Peto, R., M.C. Pike, L. Bernstein, L.S. Gold and B.N. Ames (1984). “The TD50: A proposed general convention for the numerical description of the carcinogenic pot.ency of chemicals in chronic exposure animal experiments”. Environmental Health Perspectives 58, 1-8.

Price, C.J., C.A. Kimmel, R.W. Tyl and M.C. Marr (1985). “The developmental toxicity of ethylene glycol in rats and mice”. Toxicology and Applied Pharmacology 81, 113-127.

Price, C.J., C.A. Kimmel, J.D. George and M.C. Marr (1987). “The developmental toxicity of diethylene glycol dimethyl ether in mice”. Fundamentals of Applied Toxicological 8, - .

1 15- 126. Portier, C.J. and D. Hoe1 (1987). “Issues concerning the estimation of the TD50”. Risk . . -

Analysis 7, 437-447. Ryan, L.M., P.J. Catalano, C. Kimmel and G. Kimmel (1991). “Relationship between fetal

weight and malformation in developmental toxicity studies”. Terntology 44, 215-223. Ryan, L.M. (1992a) “The use of generalized estimating equations for risk assessment in

developmental toxicity”. Risk Analysis 12, 439-447. Ryan, L.M. (1992b) “Quantitative risk assessment for developmental toxicity”. Biometrics

Ryan, L.M., D. Scharfstein and P.J. Catalano (1993) “Modeling fetal death and malformation in developmental toxicity”. Risk Analysis to appear.

Sawyer, C., R. Peto, L. Bernstein and M.C. Pike (1984). “Calculation of carcinogenic potency from long term animal carcinogenesis experiments”. Biometrics 40, 27-40.

Wang, G.M. and B.A. Schwetz (1987). “An evaluation system for ranking chemicals with teratogenic potential”. Terntogenesis, Carcinogenesis and Mutagenesis 7, 133-139.

48, 163-174.

518 L. RYAN

Williams, D.A. (1975). “The analysis of binary response from toxicological experiments

Williams, D.A. (1988). “Reader reaction: estimation bias using the beta-binomial

Zeger, S.L. and K.-Y. Liang (1986). “Longitudinal data analysis for discrete and continuous

involving reproduction and teratogenicity” . Biometrics 31, 94-952.

distribution in teratology” . Biometrics 44, 305-307.

outcomes”. Biometrics 42, 121-130.

potency measures for developmental toxicity

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