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childhood and adult cancers. Progress is therefore genuine with the possibility of comparing very closely connected different ethnic groups and lor chemical products. What about low doses? First, one comment must be made. Extrapolation has almost always been from models known as tolerance models which presupposes the absence of effect below a certain dose. As continuous functions do not prevent extrapolation below this dose, mathematicians have noticed that according to the models, at origin the slope goes from 0 to infinity but if what is known about carcinogenesis and the kinetics and metabolism of the chemical product are taken into account, the latter argument leads to linear methods of extrapolation towards low doses, whatever the model. It therefore seems essential to require biomathematicians to adopt a less contradictory attitude towards the significance, omission and evaluation of a threshold: Extrapolation from what? If it concerns cancers which are very rare in the general population, their appearance defines an absolute risk and makes it possible to establish a dose/effect relationship from an accumulation even limited to exceptional cancers (absolute risk). If the number of cancers is greater than three this suffices in principle to define the risk in a human group and to research the part played by genetics and acquisition. The essential problem is the bringing together of cases, achieved through a toxicovigilance program examining scattered cases (speregic phenomena). If it concerns common cancers the added risk from the chemical product is only relative. Multiplication of a relative risk by an appreciable factor is only possible with large size cohorts and comparable populations: 1000 people are needed to guarantee confirmation of a risk x 3.5, about 5000 for a risk x 2 and about 10,000 for a risk x 1.5. 2501171205 10

Now such investigations often undertaken in professional pathology require guarantees of good epidemiological practice whose details are still under discussion, which means that many already published studies risk suffering from bias or procedural error and should be considered with caution. Ekpext consensus There are two kinds of expert consensus: a) The most frequently encountered kind brings together experts provided with secondhand documents or already drafted summaries. The conclusions of such meetings are simple and result in a genuine consensus. In other words everyone agrees to reduce the reference indicated by a factor of 1000 (10 for species, 100 for the highest rate without cancers, NOEL). We are in the habit of accepting a regulatory attitude from such information because the number of experimental cancers observed in the current anti- vivisectionist conditions (40 to 50 animals per group) corresponds to a high proportion, several cancers per hundred human beings. Such a prediction, which is very disturbing, justifies the two stages: recognition of an NEL rate (the observable term limiting confirmation by observation of an unlimited population) and moving to a rate said to be acceptable (10-1 x 10-2) whilst knowing that this rate ought never be observed in the present environment of the general population. b) the other expert attitude is described as the Delphi method based on the anonymity of contributors and the progressive interaction of a group of experts. The question defining the objective is posed in successive "rounds" until the appearance of a convergence, a little like convergent sequences in mathematics. Of course, the sequence may not converge, or may aim at two 2501171206 11

different and incompatible points, but it is a process used more or less consciously with regard to modern regulation. In practice, regulatory bodies are content with an extremely crude dose- effect relationship, most commonly limited to comparison of the effects of two doses. It no longer concerns models. The reduction coefficients usually applied by groups of experts in chronic toxicology (1/100 NOEL if there is neither mutagenesis nor experimental carcinogenesis, 11200 to 11500 if there is only mutagenesis, 1/1 000 if there is carcinogenesis) well represent the average result of current considerations regarding cancer prevention. When part of the conclusion is disliked, they start again. This is a quasi- Delphi. Perhaps it would be useful to add to each product a real elemental model adapted to toxicokinetics and the experimental criteria of a complete carcinogen, an initiator, a promoter and its fate in the organism? c) Other contributors will probably wish to discuss the beneficial and adverse effects of low doses if reputedly toxic products are involved. . This point, the traditional basis of homeopathy, has been evoked in the face of leucose graphs as a function of the radiation dose suggested by a slope which is slightly negative at origin. The positive, negative or complex quality of the coefficients of representative functions permits the suggestion that models of this type and the Belle group are forced to give a scientific basis to this type of reasoning. True cellular protection within narrow limits can be envisaged if the genes preventing cellular access or repair are more sensitive to the product than pro-oncogenes. Would a first reference be greater affinity, a larger number of identifiable adducts? The formation of antibodies is another possible effect of low doses. N LA ~ ~ ~ -%j 0 -w.!

d) What has to be weighed is the risk of presence and the risk due to banning. We should at least admit that linear extrapolation toward the origin is a theoretical artefact, that numerous arguments are opposed to a simplification which eliminates the obvious idea of a tolerance-threshold, which animals demonstrate with not small doses administered throughout their lives without apparent adverse effect. What also has to be admitted is that the rates deemed acceptable with a risk in the order of 10-6 are guarantees which it is especially advisable to weigh against the risk associated with a ban on the product. In outline, three illustrative cases may become apparent: the adverse risk (appearance of over-representation of cancers) exceeds the adverse risk associated with a ban, it is less, it is comparable. This point is always tackled belatedly when the regulatory bodies try to reverse a manifestly erroneous decision. In general conclusion: We have the means to bring together medical observation of human cancers and assessment of a cumulative exposure (concentration x years of exposure). We have the means to bring together the most detailed observation of animal cancers and a fairly precise assessment of an exposure (concentration or dose x months of exposure) of a very small animal population. The concentrations used for animals are usually clearly greater than those corresponding to human exposure. We have experimental tests with a semi-quantitative predictive value regarding the initiation, promotion and formation of cancers. These tests 2501171208 13

refer to a range of concentrations usually much higher than the two previous concentrations. The experts do not agree on the simplest definitions: For example, the European term, Guide-Line, means an expression of a principle to be followed categorically. In Japan it is interpreted as the minimum demand required and in the USA as a reference open to discussion case by case. And each body is primarily organized around its own doctrine which it refuses to modify on the grounds that the system has worked until now. Most of the regulations only accept the notion of a threshold if there is no argument in favour of genotoxicity. Now the most obvious test, relating to a very large population, that of B, Ames demonstrates from the evidence, that for most molecules tested nothing is observed below a concentration which has to be called the threshold concentration. Under these conditions, the "worldwide" extrapolation to low doses appears to be a purely intellectual exercise which does not rely on any biological argument but which has the merit of reminding us that the essential mathematical operation in the life sciences is the rule of three. Other approaches Perhaps it would be more effective to move closer to the analyses of the engineers in charge of the complex systems which define the reliability- probability of a system which does not break down within a given period or in the course of accomplishing a defined task - and operating safety, a complementary aspect of the risk of breakdown: P (safety + risk) = 1 If we assume while simplifying considerably that the sole animal risk (spontaneous and variable) in laboratory rodents is cancer, and that in man this risk predominates with regard to the epidemiology of mortality and the 14 2501171209

evaluation of an "extra cost" of chemical origin, it will be possible to individualize and evaluate a) cellular systems evolving in parallel (the global risk is the product of the risks on each element), b) systems evolving in series (the global risk is the sum of the elemental risks) for high risks reaching several % of the population, the only ones accessible to mathematical epidemiology. c) the mean time between failures (MTBF). Evaluation of small risks will remain difficult, precisely because the "chemical cause" for very low doses will never be the principal cause - if not, it is this dose which has to be considered as the primary reference - but only as one element amongst scattered and fragmented causes. Paris April 1993 References ArmiCage P., DoI1 R . The age distribution of cancer and a multistage theory of th e carcinogenesis Br . J. Cancer 1954; 8; 1-12.; Birnbaum LS. Age-related changes in drug disposition In: N Ln Zenser T.V.& Coe R.M. ed +~ . ~ Cancer and aging Springer Verlag 1989 pp25-138 -r Q ~ N 0 0 15

Crump K.S. Hoel D. G.,,.Langley C.H. Peto R. Fundamental carcinogenic processes and their applications for low dose risk assessment Cancer Res ].976; 36 ; 2973-2979 . Hartley H.O., Sielken J.R., Estimation of safe doses in carcinogenic experiments Biometrics 1977 ;33 ;1-30 LP.C.S. Principles for evaluating chemical effects on the aged population Fanv Health Crit. 144 1993 W.H. O. Geneva Moolgavkar S.H., Venzon D.J. Two-event models for carcinogenesis: Incidence curves for childhood and adult tumors Math Biosciences 1979 47, 55-77 Rai K. Van Ryzin J.A. . A generalized multi-hit dose response model for low dose extrapolation B3oinetrics 1981 ; 37 1; 341-352 Sankaranarayanan K. determination and evaluation of genetic risks to humans from exposure to chernicals. Prog Mut Res. 1982; 3.; 289-321 Valieron Aj, Bignon J., Hughes J.M., Hesterberg T.W. & al Low dose exposure to natural and man-made fibres and the risk of cancer ; towards a collaborative European epidemiology : Br . J; Ind. Med . 1992 ;49 ; 606-614 . Valleron Aj'. Thomas G. Methodology of carcinogenic risk assessment at low doses 1993 (to be published) Vljg J., Papaconsiantiriou J. Aging and longevity genes strategies for Identifying DNA sequences controlling life span J. Geront . 1990 , 45 (5), B179-B182

Table 1. Tolerance dose response models Tolerance distribution Model Probability of response at dose d x2 normal Lo Probit _ J i131ogd 1 1, 2 dx (~3> 0) g 0 2n Lo lo istic it Lo > 0 g g g i a ) Gamma Gamma multi-hit oQ l A i+e'("Lt _ ~ e-xXk-1 { k>0 0 ' fl F(k) ) ,a> Extreme value Weibuil 1 - exp(-Od') (0 > 0, m > 0)

2.2.1. Empirical models 2.2.1.1. Log-llriear models Any probability distribution with support the positive real line may be postulated for T. lt is thus obviously convenient to consider log-linear models, of the form : T = exp(a + (i logd + vW) where W is a real random variable. Table 2 presents several possibilities. Table 2. Log-linear models Distribution of W Distribution of T Normal Lognormal Extreme value Weibuil Logistic Log-logistic More generally, one can consider the mdefs : T = exp(OtZ + aW) where 0 is a p-vector of parameters and Z a p-vector of functions of dose alone.