[CANCER RESEARCH 36, 2973-2979, September 1976] Fundamental Carcinogenic Processes and Their Implications for Low Dose Risk Assessment K. S. Crump,1 D. G. Hod,2 C. H. Langley, and R. Peto National Institute of Environmental Health Sciences, National Institutes of Health, Research Triangle Park, North Carolina 27709 (K. S. C., 0. G. H.. C. H. L.J, and University of Oxford, Oxford, England (A. P.) Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 Summary 1. Cancers are believed to he single cell in origin (6, 7). Of a large number of cells at risk in the individual organism, 1 Various possible models of carcinogenesis are analyzed undergoes certain changes that allow it to divide and grow with respect to low dose kinetics. The importance of back into a tumor. Thus we can view the carcinogenic process as ground carcinogenesis upon the shape of the dose-re mechanistically single cell in origin even though, by the sponse curve at low dose is emphasized. It is shown that, if time a cancer is pathologically recognizable, very extensive carcinogenesis by an external agent acts additively with any changes may have developed. already ongoing process, then under almost any model the 2. It will be shown that it is important to know whether the response will be linear at low dose. Measures of the degree causal processes associated with the particular carcinogen of linearity are obtained for multistage models of carcino of interest are common to those involved in carcinogenesis genesis, where it is shown that throughout the dose range due to other causes, either “spontaneous― or from other where the extra risk is less than the spontaneous risk linear carcinogens. In other words, we need to know whether or extrapolation must be quite accurate. not carcinogenesis due to a particular carcinogen is inde pendent of other modes of carcinogenesis. In the 1st section of this paper, the consequences of the Introduction manner of combining “spontaneous― and “ induced―carci nogenesis will be explored. We will show that, if the addi The presence of carcinogenic agents in the environment tion of the test carcinogen merely increases the rates of is an accepted fact. Although most agents can be avoided processes that were occurring anyway, then dose-response once they are identified as carcinogenic, some may be relationships will be linear at low dose levels. In the 2nd avoided only at great expense or alternative risk, in which section several models will be considered and their low case “risk versus benefit―must be evaluated. One impor dose properties will be identified. We will find that every tant aspect of the determination of risk is the estimation reasonable model of carcinogenesis is linear or sublinear at from animal experiments conducted at high doses on small low dose. Finally, in the 3rd section we will look more to moderate numbers of animals of the risks to such animals closely at this linearity and determine the accuracy of linear of cancer associated with very low levels of exposure. This approximations in ‘ ‘ multistage―models of cancer. is likely to be a principal element in risk estimation for the It should be recognized that there may be agents that myriad of chemicals that must be evaluated. Relevant hu indirectly affect the carcinogenic process. An example man data are usually not available. might be some dietary alteration that led to a modification The estimation of attributable risk at a dose very much of gut flora that may change the carcinogenic process in a lower (say 1/1000th) than the smallest practical experimen qualitative way. Although our analysis and conclusions tal dose involves the interpolation between 2 dose levels: might be appropriate for some of these indirect carcino the control and the experimental dose levels. Interpolation genic processes as well, we are chiefly discussing direct necessitates an assumption about the behavior of the risk carcinogenic processes in which the compound or its me with increasing dose. The assumption can be specified tabolite acts at the cellular level to produce an irreversible arbitrarily or it can be deduced from reasonable models of and heritable (genetic or epigenetic) change. the carcinogenic process. An “estimate― of risk is as arbi trary as the interpolation scheme that produced it. We will Significance of the Relation of a Carcinogen to Occur attempt in what follows to relate the properties of various rence of Cancer due to Other Causes risk estimation procedures to several observations and as sumptions about carcinogenesis. Throughout this paper we shall concentrate upon the Two properties of carcinogenesis are critical to low dose case of a population chronically exposed to carcinogens at risk estimation. constant dose rates. We are interested in the individual response when the population is exposed to a particular I Present address: Louisiana Tech university, Ruston, La. 71270. carcinogen at an approximately constant dose rate d per 2 To whom requests for reprints should be addressed, at National Institute unit time. This response can be described by the age-spe of Environmental Health Sciences, P. 0. Box 12233, Research Triangle Park. N. C.. 27709. cific cancer incidence rate l(t,d) which is the expected rate Received September 25. 1975; accepted June 1. 1976. per unit time at which cancer will be discovered in individ SEPTEMBER1976 2973 K. S. Crump et al. uals of age t who were previously cancer free. In consider by using the formula of Abbott (1) for correcting for re ing this response, it is important to keep in mind that mdi sponse due to extraneous causes. In terms of the above viduals at risk will ordinarily be exposed to a large number formulation, this is equivalent to supposing that the 2nd of carcinogens, and we are interested in the effect upon the group of carcinogens contains only the primary one, or, in response of a single one of these which we shall for conven other words, the carcinogen of interest acts in some man ience call the primary carcinogen. As we shall see, the ner completely independent of the mechanism by which all independence or equivalence of the mechanism(s) of action other cancers are formed . If this does, in fact, turn out to be of the primary carcinogen and the other carcinogens can be the true biological situation, then the response function 1(d) important in determining the response due to a low dose is still represented by Equation C except now 0, = 0. Thus rate of the primary carcinogen. we see that even in this case 1(d) can still be linear at low We can reasonably divide the totality of carcinogens into dose provided the slope of the function H is positive at zero. Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 2 groups: Group 1, containing all of those carcinogens that This turns out to be true for some models of carcinogenesis, cause a response of cancer in a way that is completely e.g., 1-hit models (2), but not for others, e.g. , some multihit independent of the mechanisms by which the primary car models (15) and the Mantel-Bryan model (11). However, cinogen causes a response; and Group 2, consisting of when other carcinogens act in conjunction with the primary those carcinogens (including spontaneous biochemical ac one (D,, > 0), the linearity of the response merely depends cidents) that somehow act in conjunction with the primary upon H having a positive slope at the point D,,. This seems carcinogen in causing cancer. Let l@be the incidence rate of intuitively likely and, in fact, is the case in all models of new cancers at a fixed time t due to a carcinogen in Group 1 carcinogenesis with which we are familiar. or via an inherent spontaneous phenomenon that is There are, of course, a number of questions about the mechanistically related to the effects of the carcinogens of biological validity of the assumptions. Group 1. Let l@be the incidence rate of new cancers at time t 1. Do all carcinogens act independently or do certain due either to a carcinogen in Group 2 or to an inherent subgroups act in conjunction with each other? It should not spontaneous phenomenon that is mechanistically related to be difficult to answer this question experimentally with re the effects of the carcinogens of Group 2. Then, because of gard to the effects of specific carcinogens on specific can the assumed independence, we can write cers. 2. Given that at least some carcinogens act in conjunction 1(d) = I + 19 (A) with the primary carcinogen, is it reasonable to assume that their individual effects are additive in the sense of Equation where, as shall be done throughout the paper when conven B? This question is probably much more difficult than the ient, the argument t has been omitted. 1st one. However, the idea of complete additivity of effects Now suppose that Group 2 consists of m carcinogens at is not essential to our arguments, and a variety of other dose levels d1@ d―in addition to the primary carcino assumptions would lead to effectively the same conclusion. gen at a dose level d. The simplest assumption with regard 3. Is the assumption that H'(D,,) is positive valid? For to the interactive effect of these carcinogens would be to example, if there were some type of threshold effect operat suppose that the effect is additive, i.e. , the rate I, at which ing so that H(D) = 0 for D less than some threshold value cancer occurs due to a carcinogen in Group 2 is a function DIh, then if D,, were less than D,, the argument would break of an effective dose rate down and 1(d) would, in fact, not be linear at the lowest dose rates. On the other hand, if cancer is single cell in D = D,, + @3d (B) origin, then the threshold D1@, is a property of a single cell rather than of the whole organism. Viewed in this light, it is @ whereD,, issomefunction ofd1 Nowwe can write entirely plausible that, even if a threshold effect does exist for each cell, nevertheless in the entire organism the proba /9 = H(D) bility of response may be linear at low dose rates. These cellular thresholds will not all be constant but will be distrib and we will assume that H is a nondecreasing analytical uted over some range of doses. If this range includes D,,, the function. (This merely implies that any increase in the dose organismic response will, generally speaking, be linear at rate does not decrease the age-specific cancer incidence D,, + 0. (The same conclusion would follow from postulating — l@ +H(D0) +f3H'(Djd +0(d) rate.) Now we can write that each person in a large population has a particular threshold but that individual thresholds have a random dis l(d)=11+H(D,,+f3d) (C) tribution.) 4. Even if one is willing to accept the fact that the re sponse curve is linear for low dose rates, this in itself may as the dose rate d approaches zero where o(d) denotes a be of little value unless there is some knowledge about function with the property o(d)/d approaches zero as d “how― linear and “how― low the dose rates must be. To approaches zero. Thus we see that 1(d) will be a linear answer such quantitative questions as these, one must function of the dose rate d at low dose rates provided H'(D,,) make more specific assumptions than are incorporated into > 0. the very general discussion presented here. We shall return Other authors (11) have allowed for cancers in the models to this question in the light of some particular models for due to causes other than the primary carcinogen of interest carci nogenesis. 2974 CANCER RESEARCH VOL. 36 Carcinogenic Processes and Low Dose Risk Assessment Particular Models for Carcinogenesis where Experimental evidence (6, 7) indicates that cancers origi @ Sk(t) = nk J f(t —U)Uk du (H) nate from a single cell. The models we shall look at will be based upon this premise. First, consider the time to re sponse of a single cell, where the response might possibly At low dose rates this response is linear in the dose rate. To be detection of or death due to a cancer originating with see this, we note that it is possible to write this cell. This time to response can be written as the sum of the period of genetic and/or epigenetic alteration of the cell l(t,d) = SktQkd) = S@t @A+ Bd + o(d)} to a malignant phenotype, plus the growth period from the where Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 time at which the cell is completely altered to the time of the observed response. The time to cell alteration is presumably /.. dependent upon the dose rate d. This may also be true for A = ir a@and B = @.f3@ @r @ the growth time, but we shall assume that the latter effect is I = I i = .1 I negligible. If these 2 times are independent of each other, we can in general write The incidence rate will be linear in dose rate at low dose rates whenever the constant B is positive. In order that there be both background carcinogenesis and also some effect of lr(t,d) = ji@@(t _ u,d)f(u)du (D) the dose rate d, it is necessary that all of the (51Sand at least 1 of the [3's be positive. However, under these conditions the constant B is seen to be positive, and thus the incidence where I,, (t,d) is the incidence rate of the alteration of a rate is linear at low dose. single cell, f(t) is the density of cancer growth time, and If k = 1 (1-hit model) then I@(t,d)is the observed incidence rate for cancer response. The approximation in Equation D is valid because I,,t,d) applies to a single cell and will be very small. Now the 1(d) = (a + f3d)S(t) (I) observed incidence rate is for an entire tissue and, as pointed out by Armitage and Doll (4), insufficient attention and the incidence rate is exactly linear in dose for all dose has been given in some earlier models to the distinction rates. between cell response and tissue response. If a tissue is Alternatively, one could consider the time to cellular alter composed of n cells, then the time to response of a tissue is ation to be the result of a multistage process. This process the minimum of the associated n cell response times. If we has been applied to carcinogenesis by Armitage and Doll suppose that these n cells all respond in the same manner (5). As in the multihit process, k events must occur in a cell but independently, then we arrive at the formula to initiate cancer, these events occurring with fixed rate constants. The only difference between the multistage and 1t,d) = flI,.(t,d) (E) the multihit processes is that the k-initiating events in a multistage process must occur in some particular time Se 1. Multihit and Multistage Models. Suppose k 1 differ quence. It can be shown (3) that Expression G still holds ent events (hits) must occur in a cell before it is sufficiently provided the right side of this equation is divided by k. altered and suppose the ith event occurs at a constant rate Consequently, the comments on the linearity of the re xi, I = 1 ,2 k. Suppose further that cells that have suf sponse for the multihit model hold true for the multistage fered some, but not all, of the relevant events have no model also. Our comments would, of course, also hold for selective advantage or disadvantage relative to normal intermediate models in which some stages must occur in a cells. Nordling (15) used the multihit model for the total fixed sequence while others may occur in various orders. response time, time to alteration, plus growth time, but here 2. Generalized Multievent Model. More generally, a we generalize his approach by using the multihit process to large number of events related to the initiation of cancer model only the time to cellular alteration. The effect of dose could occur in a cell. However, rather than its being neces rate is introduced in the manner of Neyman and Scott (14) sary for all of the events to occur to initiate cancer, there by taking x1 = a + f3d (subject to a 0 and /3 0). The could be a (possibly quite large) number of subcollections incidence rate of the alteration of a single cell is [see Armi A ,..., A,, of these events so that cancer is initiated as soon tage and Doll (4)] as all of the events in any 1 of these subcollections occur. Thus, in this model there are many paths through which cancer can be initiated, a single path corresponding to a @ l,1t,d) kth - I { (@ @1d)}= ktk - ‘Qk(d) (F) particular subcollection A of events. A path containing k events can be called a k-hit path. The subcollections need not be disjoint so that 1 particular event could be included where Qkd is a kth degree polynomial in d with constant in a number of different paths. The general consequences coefficients. For the observed incidence rate of cancer we of this model are readily understood without going through obtain, using Equations D, E, and F, all of the details. It makes no difference for our purposes whether or not some of the events must occur in a specified l(t,d) = Qk(d)Sk(t) (G) order since adjustments necessary to go from one case to SEPTEMBER1976 2975 K. S. Crump et a!. the other are quite like the adjustment necessary for going 1+p from a multihit to a multistage model. It is again supposed 1 + log 1 + P) that the ith event occurs at a constant rate (5, + /3@d where the a65 may be functions of dose rates of other carcinogens as k —@@. Thus, we have the bounds but are notfunctions ofd. In the terminology of the previous section, all of the paths in which the rates are not functions @ 1 r (1 + p)/{1 + log (1 + p)} (M) of d (that is in which the f3@'sare all zero) represent the “1 st group' ‘mechanisms, which are independent of the mecha nisms by which the primary carcinogen causes cancer. The This upper bound forr holds for any k, and for any multipath incidence rate associated with the union of these paths model with different values of k for different paths. The represents I of Equation A, and the incidence rate associ upper bound for the ratio is increasing in p . For a “doubling Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 @ ated with the union of those paths in which at least 1 of the dose― p = 1 we have r 1.18 regardless of the value of k, @ rates is a function of d represents 19.It can be shown that the and if k = 2 we have r 1.09. For a 10% increase in @ response will be linear in d for small dose rate d unless all incidence over background, we setp = 0.1 and find thatr paths depending upon d contain at least 2 events that occur 1.004. These results are quite interesting and useful for 2 only in the presence of the specific primary carcinogen. reasons: (a ) they indicate the closeness to linearity for a very general class of models; (b ) the results depend only on How Linear Is “Linear―? the proportion over background. If a finer bound is desired, the number of stages of the carcinogenic process or an In this section we attempt to describe quantitatively the upper bound to the number of stages is needed. range of dose rates for which the linear approximations are Our next approach to the question “ how linear?―could valid. To do this one must obviously be somewhat model be of interest in the following situation. Suppose estimates specific, and we will assume the hit models. Two different for Rd) and 1(o) are available from experiments where d is a approaches to the question of ‘ ‘ how linear?' ‘ will be consid known experimental dose and information is desired about ered. the incidence rate curve at dose rates much lower than d. For a k-hit model with an arbitrary distribution for cancer There are 2 possible problems to consider. First of all, one induction time, we found (Equation G) that !t,d) can be might ask what dose rate d,, would yield a prescribed mci expressed as the product of a function of age t only, and the dence rate I,, which may represent ‘ ‘ a given acceptable― increase in the incidence rate over the background mci polynomial @r(a1 + f31d)in the dose rate d. It is of interest dence rate 1(o). It should be expected that, if d is so small as to compare this exact expression with its linear approxima to be on the linear portion of the dose-response curve, then tion d( can be approximated by fitting a straight line through the points [o,!(o)] and [d,!(d)] and using the dose rate corre = !o) + d!'o) (J) sponding to I,, on the line. The dose computed using this linearization process is where !d is given exactly by Equation G. To do this we shall consider the ratio r of !d to the linear approximation I,, —1o) 11d when 1(d) is a certain prescribed proportional excess p d,.= d (N) 1(d) —1o) of 1(o), the age-specific incidence rate at a zero dose rate. Symbolically, we have Since!,, — 1(o)will usually be very small, d,. = d,,!'(o)d/41d — r—@@— 1o)1+p) /(o)), and thus the ratio of the true dose d,, and the approxi @ — l,d@) !o) + I' (K) mation d1.is — 1+p — @! -@- 1d 1o) R 1+ (0) 1(0) d,. !‘o)d where d,, satisfies 1(d,,)= (1 + p)/(o). On the other hand, one may be interested in estimating the It is easily seen that r 1 is independent of t. Moreover, it incidence rate l@which corresponds to a very small environ can be shown using the method of Lagrange multipliers that mental dose d@.If l@is approximated by l@. , the incidence r assumes its largest value when a@/f31 a9//39 . . . rate corresponding to d@on the line joining [o,l(o)] and f3@..When this condition holds, we find that [d,!(d)], then the ratio of the approximation I,. —1(o) of the increase over background incidence to the true value 1E 1+p 1(o), is the same R, since @ r= k {(1+ p)l/k 1} (L) @ This is an appealing result in that r depends only on the l@.—lo) ______ IE ho) d!'o) (P) number of stages k and the proportion of background p, parameters that are easily interpretable. This expression is increasing in k and approaches Let us now consider the k-hit model and express R in 2976 CANCER RESEARCH VOL. 36 Carcinogenic Processes and Low Dose Risk Assessment terms of p, the excess response over background, deter sumed to arise from events associated with or occurring mined by 1 + p = l(d)/!(o). It can be shown that inside single cells. We note again here that this analysis is appropriate only IQ' for those agents that affect cancer incidence through the 1-R< — — k{(1 + p)lIk 1 } ‘ I alteration of single cells in an irreversible and hereditable manner (e.g. , chronic exposure to low-level ionizing radia the upper bound being attained when all f3/a1 are equal. iton). Those agents that increase cancer by anatomical and/ This upper bound increases steadily from 1 for a 1-hit model or physiological alteration of whole tissues and organs (e.g. , dietary modification of gut flora) may or may not be @ (k = 1) to — as k —@ 7L,We have tabulated the upper log(1 +p) described by these models. Since we do not know the Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 bound for R for different values ofp and k in Table 1. We see relative proportion of these 2 types of carcinogens and that the linear approximations are reasonable over a wide often do not know into which category a particular agent range of values of k and p . The only circumstances in which falls, we must stress the importance of understanding basic linear approximation might be inappropriate are seen to be carcinogenic mechanisms. those where the background rate is vanishingly small and Relationship between “Spontaneous― and Induced Car thusp is very large. An example of this might be the induc cinogenesis. As we have shown, the independence or de tion of angiosarcomas by vinyl chloride; however, if recent pendence of “spontaneous― and “induced― carcinogenesis evidence (13) that common tumors are also caused by vinyl is critical to the shape of the low dose-response curve. Two chloride is confirmed, p will not be extreme, and linear types of evidence indicate that these 2 processes share approximation will be adequate even in this case. The ex many common mechanistic steps if they are not identical. cess p of experimental incidence rate over background Cancers thought to be induced are generally indistin incidence 1(o) thus plays a key role in the accuracy of the guishable from “spontaneous― cancers. This obviously linear approximations. In particular, the linear approxima does not demonstrate that the cancers arise by a common tions improve asp decreases towards zero. mechanism, but it is consistent with a common pathway to “ induced― and “spontaneous― carcinogenesis. DISCUSSION The view of carcinogenesis as a fundamentally mutational phenomenon, as recently reviewed by Knudson (8, 9), sup ports the assumption that induced and spontaneous steps We have shown under some reasonable assumptions are mechanistically identical. That is, experimental induc about carcinogenic mechanisms and processes that dose tion of cancer is the speeding up or the increasing of the responses will be approximately linear at low doses. Let us probability of the various steps. examine the evidence in favor of these assumptions and The most important observation relevant to the relation review the generality of models considered. shipbetween“induced― and “spontaneous―isthathumans Single-Cell Origin. If individual cancers arise from an demonstrate a high background incidence of cancer. original, single, “transformed―cell, then the statistical na Whether these are due totally to “ induction―by environ ture of the carcinogenic dose response will be governed by mental agents or also to some truly spontaneous process is the extreme tail of the “transformation― response distribu immaterial when considering the effects of a small amount tion. The effect of this is to make virtually any process of of increased human exposure to a particular carcinogen. discrete events approximately linear at low dose. Approximately 1 of 5 Americans develops a cancer, and Two primary observations indicate the single-cell origin for any particular environmental carcinogen we are inter of cancers. In women who are heterozygous for electropho ested in a very small associated increase in risk. This 20% retic variants of X-Iinked glucose-6-phosphate dehydrogen background must surely provide some significant ‘ ‘sponta ase, cancers are unitormly of one phenotype or the other neous―processes that are shared with carcinogenesis by (6), whereas a comparable amount of normal tissue is com the carcinogen in question; from a public health standpoint posed of a mixture of cells of the 2 phenotypic classes. the assumption that “induced―and “spontaneous― are not Further evidence for the single-cell origin of cancers comes independent is conservative, as well as being biologically from experimental efforts in which “transformed― cells are plausible. Small extra doses of a carcinogen will therefore transplanted into whole animals. Although there is much elicit linear increases in risk for virtually any response controversy associated with various aspects of this line of model. research, it seems that the ability of a single cell to give rise One practical implication of the fact that different carcin to a cancer is well demonstrated (7). Thus, 2 lines of evi ogens share many mechanistic steps is that enhancement dence indicate that cancer can be most reasonably as of certain carcinogenic processes may have a more readily detectable effect on cancer incidence in animals with high Table 1 background levels of all other carcinogenic processes. Values of the upper bound of the ratio A from Equation Q Therefore carcinogenicity tests of various substances should possibly include tests on high-spontaneous-mci 0.1 0.5 4 10 100 dence strains or experiments to see whether the test sub 1 1 stance enhances the carcinogenic effect of a standard car 1.02 1.11 1.21 1.62 2.16 5.52 cinogen. 1.04 1.18 1.35 2.11 3.25 13.18 Induction Time and Dose. In our discussion of stochastic 1.05 1.23 1.44 2.49 4.17 21.67 models, we assumed that induction time is variable but SEPTEMBER 1976 2977 K. S. Crump et a!. independent of dose. This assumption is unfortunately might not be excluded from the class of “linear at low weak in that high doses could well affect induction time. dose.― This, then, is an area of research that is in need of further We have also attempted to answer the question of how effort. If we consider the low doses at which individual linear is “linear at low dose' ‘ for particular models. For the environmental carcinogens are experienced, however, it multihit and multistage models, linearity is dependent on seems reasonable to us to assume relatively little effect on background incidence. If the background is within a typm induction times. cally observable range, then the linear model provides a Now let us examine the generality of the result: linear reasonable estimate of the true state of nature, while ap dose response at low dose. Given the uncertainties and proaching this estimate from the conservative side. complexities of carcinogenesis, it is conceivable that sev All these considerations clearly demonstrate the impor eral distinct mechanistic phenomena will eventually be dis tance of explicit and realistic modelling in the development Downloaded from http://aacrjournals.org/cancerres/article-pdf/36/9_Part_1/2973/2397820/cr0369p12973.pdf by guest on 12 June 2026 covered to contribute to the appearance of cancer. Thus, of low-dose extrapolation schemes. Many may feel that we we must have an open mind about our modelling and at have not considered certain biological observations or hy tempt to present the least model-dependent result that we potheses in the models presented above. We have tried to can. This we have done. embrace as much relevant information about carcinogene Virtually all models of carcinogenesis that depict the ex sis as possible and to obtain results that were the least posure as affecting an already ongoing process will lead to model dependent. linearity at low dose. We have discussed the validity of this The weight of these results for human risk assessment is assumption above. This result then implies that, no matter difficult to judge. It is likely that the error in the acceptable what the biological mechanism we might imagine, if the dose associated with simple linear extrapolation will be carcinogen increases some part of the already ongoing much less than that associated with the species-to-species process, then we should expect the response to be approxi extrapolation to man from the laboratory animal data. The mately linear at low dose. BEIR report (16) recommended linear extrapolation on As pointed out above, this assumption of dependence or pragmatic grounds. The theoretical conclusions of the common mechanism is not trivial. It can make orders of present paper are that linear extrapolation to low dose magnitude differences in the estimated risk associated with levels is generally valid as a realistic yet slightly conserva low dose exposure. tiveprocedure. If we conceive of the cell alteration process as a series of Practical Implications. Our results may be crudely sum discrete single-cellular events that can occur in sequence or marized by the observation that, in environments already randomly in any given cell and that a dose-independent containing appreciable amounts of carcinogenic proc induction period follows, then we should expect dose re esses, the effects of any slight addition to these processes sponse over background to be linear. We have required will be proportional to the amount added. Both control neither that all steps be affected by the carcinogen (only laboratory animals and wild humans already suffer a con some) nor that these steps be all mechanistically similar in siderable incidence of cancer; thus the extra incidence quantity or quality. This general class incorporates most of caused by a small amount of a new carcinogen will be the reasonable models that have been proposed. The keys proportional to the dose rate of that carcinogen. This to this result are the assumptions of the single-cell origin thought is not particularly remarkable, but its implications and the lackof any appreciable dose dependence in the are that much previous investigation of the form of the induction period. dose-response relationship at infinitesimal doses is irrele A further extension of this group of models allows the vant to the interpretation of animal studies for the formula incorporation of threshold models into the class of ‘ ‘linear tion of social policy. at low dose.―We have indicated that, if we conceive of Unfortunately, the implications of linear extrapolation are single cells as the biological unit at risk and that the initia bleak. Mantel et a!. (10-12) have proposed that safe doses tion response is a threshold phenomenon, then by assum be defined on the basis of “probit―extrapo'ation from upper ing that the threshold is randomly distributed in dose we confidence limits defined by the experimental results, argu find that the low dose response of the whole tissue over ing that such a procedure would reward good experimental background will be approximately linear. If, rather implausi investigations (by allowing industry bigger permitted doses) bly, we do suppose that some sort of cellular thresholds while enabling regulators to guarantee to the public that exists, then clearly all cells do not have the same threshold permitted doses were so small that they would cause cancer since all cells do not all become cancers simultaneously. in less than 1 person in 108. Probit extrapolation may be Here again, we have assumed that the carcinogen acts in scientifically valid for a few very special ‘ ‘ indirect―carcino conjunction with the “spontaneous― or background effects. genic processes, but our arguments suggest that in general The Mantel-Bryan procedure (11) may be interpreted as a it is not correct. The social implications of our results are random threshold model (albeit without our assumption best understood by considering, as an extreme case, the that substances equivalent to the suspect carcinogen are use of linear extrapolation to define a “safe' ‘ level after already present in the environment), although this interpre doing a large experiment in which no carcinogenic effects tation was not made by Mantel and Bryan. This requires that were observed. The most definite such negative experiment the whole organism or tissue be interpreted as the biologi that is practical might compare animals fed with the order of cal unit with a threshold. However, if the single cell is the 10% of the test substance in their diet with a control group unit at risk, it must be tentatively accepted that even the and might conclude that the extra risk of cancer was less threshold concept of carcinogenesis (if it were appropriate) than something like 1%. Linear extrapolation, even from 2978 CANCER RESEARCH VOL. 36 Carcinogenic Processes and Low Dose Risk Assessment such ideal results as these, implies that a dose level below publicly agreed for such substances, as it was for radiation 100 ppb is needed for the risk to be less than 10 ‘@, and in 20 years ago. real experiments dose levels below 10 or even 1 ppb are likely to be indicated by linear extrapolation in order to REFERENCES guarantee a risk below 10 ‘@. Our arguments that linear extrapolation is generally appropriate at least suffice to 1. Abbott, W. S. A Method of Computing the Effectiveness of an Insecti demonstrate that linear extrapolation may be appropriate; cide. J. Econ. 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