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is strictly spaaking unrealistic (Krewoki et al 1984). vhicil is not to say that to propose and tolerate au added acceptable r.is}s (between 10-5 and 10'8) makes more sea9e: 10-5 x 70 ycaars: about 6 ttours in terms of lifo Qxpect.ancy. The biologist acCustomed to margina of error othervise large has a poor gracp of the practical value of attitudee, vhich may be compared to a proposition of Pure Behavioral Act: Art. 1"the designated population ehould livp without sin". Should the risk of 51.nning be aqrezd to be 10'S or l0'8? Should attempts at QvalUatinq a carcinogcnic effect ha similarly rejected vholesala? The answer is certainly no, proviCed.:tLe limita of models are knovn. * N_B. I sxou1Q especially like to thank ProfessorG A.J. Valleroa and G. Thomas, who describrd to me the methods uFerl and thc limits of uce. a) The model with threshold (tolerance model) assumes that a subject exposed tv a dose (cumulative) of a carcinogen vill develop a cancerous tumour if the dose exceeds a thrweholQ called a tolerance_ Various approaches are suggested (see Appendix). These models are only valid for binary situations exe-TuCirig all interference from otner factors, eliednating the time tac(.or to the advantage of thee single cumulative Cose. Elemental toxicology, throughout life, p®rmits thesp calculations_ 1a X

These models are however little used, for it is rare that human observations concern more than three situations: A lot, a little, or no chemical product. Experiments on animals rarely involve more than three to four doses: one close to the maxiTUum dose more or less well tolerated (in the general sense) by the animal; another fairly low dose -;,& selected in the reasonable expectation that nothing will be observed; and one or two intermediate doses which are the only ones genuinely compatible with a sub-normal life expectancy_ In these conditions it is difficult to draw up a graph with a single point - or two - and the regulations most frequently allow for the lowest dose which shoWed no effect (NOEL). Nodels allovi.ng for the effect of time Time is a fundamental variable of carcinogenesis but its introduction necessitates a biological unity such as average life span or the extreme life expectancy of the species or ethnic group, or that of the appearance of perceptible phenomena of which cancers form a part. There is no consensus about the mechanism of the increase in the prevalence of cancer according to age (accumulation of errors, progressive chromosomal abnormality, perigenic abnormality of the histones, epigenetic abnormalities of cellular regulators (hormones, adenylcyclases, calcic mediators etc.), but an experimental gain is confirmed by 1/~

monitoring animals throughout their lives_ Little by little the notion is taking hold that in certain mammals the prevalence of cancerous mortality becomes preponderant in excess of 70%. In man the situation is evolving in the same direction although the part played by degenerative causes with cellular death remains high. If the average life span reaches 80 years cancerous morbidity should become considerable. N.B. This discussion is different from that about the sensitivity of elderly subjects to exposure to carcinogens. Models using time refer to empirical models called log- linear, of the type T (probability - distribution according to observation) + exp' (Bl Z + 6W) p. vector p. vector actual random variable of parameters of functions of a single dose or of regression (Lox) y (t:d) = yo (t) exp (o'Z(t))) and to models based on biological hypotheses: multi-hits, multi-stage. These already old models like those of Fisher and Bolloman (1951) have had the merit of taking parallel events into account (more than six cells transformed together - abandoned- ) or more DNA disorders (6-7 0 ~?

successive mutations on the same cell). The latter argument was essential to explain why the-incidence of many human cancers would grow with age to the power of 5 or 6. We acknowledge that currently the appearance of a cancer supposes at least two, and probably fewer than seven predisposing factors affecting one cell. The model derived f roai the work of Moolgavkar, Venzon and Knudson (1981) results in an- outline consisting of normal and intermediate cells and those proliferating out of control, capable of reproducing themselves as"they are, of leading to the later stages, or dying. Recent models associated vith validated experimental or epidemiological data, studies of absorption or metabolisms, encompass usable results for childhood and adult cancers_ Progress is therefore genuine with the possibility of comparing very closely connected different ethnic groups and /or chemical products. What about low doses? First, one cornment must be made. Extrapolation has almost always been from models kinown 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 4 to infinity but if N what is knovn about carcinogenesis and the kinetics and 0 ~ ~ ~

metabolism of the chemical product are taken into account, the latter argument leads to linear methods of extrapolation tovards loW doses,.vhatever 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 tbree 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 phencxaena). 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. Now such investigations often undertaken in professional ~1~

pathology require guarantees of good epidemiological practice whose details are still under discussion, vbich means that many already published studies risk sufferinq from bias or procedural error and should be considered with caution. F.xpert 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'Z) whilst knoving 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 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 vith 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, 1/200 to 1/500 if there is only mutagenesis, 1/100 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-afld to each product a real elemental model adapted to toxicokinetics and the experimental criteria of a corsolete carcinogen, an initiator, a promoter and its fate in the organism2 c) Other contributors will probably wish to discuss the ;~E

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. 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, /1 ~-

acceptable vith a risk in the order of 10-6 are guarantees uhich it is especially advisable to veigh against the ri.sk associated with a ban on the product. In outline, three iliustrative cases may become apparent: the adverse risk (appearance of over-representation of cancers~ exceeds the adverse risk associated uith a ban, it is less, it is comparable. This point is alvays tackled belatedly vhen 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 brit~g together the most detai3ed 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 anf3ma~,'s are usually clearly greater than those correspor~ding to ht~tan exposure _ We have experimental tests urith a semi-quantitative predi.ctive value regarding the #.nitiati.on, promotion and formation ot cancers. These tests refer to a range of concenttations usually much higher than the tvo previous concentrations. N O ~ V O ~ ~.

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 opery 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. Nov the most obvious test, relating to a: very large population, that of B, Ames demonstrates from the evidence, that for most molecules tested rpthing,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 N analyses of the engineers in charge of the complex systems ~ which define the reliability-probability of a system which X W d'oes not break down within a given period or in the course ~ UT %I 06