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### CHAPTER 14 Shape of the Dose-Response Relationship, and Low-Dose Cancer-Yields Based on the Best-Fit Curve

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This chapter is arranged in six parts:

1. Supra-Linear Shape of the Dose-Response Relationship, p.1
2. Basis for Ruling-Out a Concave-Upward Shape, p.2
3. Purely Low-LET Radiation versus Mixed (Gamma + Neutron), p.5
4. Basis for Generalizing from the A-Bomb Study, p.6
5. Low-Dose Cancer-Yields Derived from the Best-Fit Curve, p.6
6. The Bottom Line from Regression Analysis, p.8
Then tables.
Then figures.

In the previous chapter, the Cancer Difference Method gives us Minimum Fatal Cancer-Yields in both the T65DR and DS86 dosimetries, and the results indicate that the cancer-risk is more severe per centi-sievert (rem) at low doses than at high doses. In other words, the findings in the previous chapter strongly suggest that shape of dose versus cancer-response is presently supra-linear. (Other terms for supra-linear, including "concave-downward," "upward convex," and even "sub-linear," are discussed in Chapter 23, Part 4.)

In this chapter, we will use the technique of curvilinear regression analysis for three purposes:   (A) to depict the shape of dose-response in the A-bomb survivors, (B) to determine whether or not the supra-linear shape meets the test of statistical significance, and (C) to calculate Cancer-Yields based on the best-fit equation.

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1.   Supra-Linear Shape of the Dose-Response

In Chapter 29, we have used the data from Table 13-A, Rows 1 through 6, to demonstrate the technique of curvilinear regression. The steps of input, output, writing the best-fit equation, plotting graphs, and statistical testing are all presented in detail in that chapter. Readers who consult Chapter 29 will see for themselves exactly how we obtain the findings which are discussed in this chapter and elsewhere.

Findings from Chapter 29 are brought forward into this chapter. For instance, the equation which best fits the observations, in the T65DR dosimetry, is brought forward from Table 29-B and is presented in the Upper Notes of Table 14-A of this chapter.

Using the equation, we have calculated the predicted cancer-rates in Table 14-A, Column C, for dose-intervals of 10 cSv -- and for even smaller intervals at very low doses. In addition, Column C includes best-fit cancer-rates calculated for the specific organ-doses where we have the observed cancer-rates, so that the observed rate (in Column D) and the rate predicted by the curve (in Column C) can be compared. (Readers can ignore Columns E, F, and G until Part 5 of this chapter.)

Since the best-fit equation can provide predicted cancer-rates at any dose-level, of course Table 14-A includes estimates for 2, 5, and 10 cSv -- doses which lie between the mean doses received by Dose-Group 2 and Dose-Group 3. These estimated rates are interpolations between two actual observations;   they are not extrapolations in a direction beyond any observed datapoint.

Figures 14-E and 14-F :

The information in Columns A, C, and D is plotted in Figure 14-E, which shows the cumulative cancer death-rate per 10,000 initial persons versus T65DR dose. The boxes are the actual observations, while the smooth curve says:   This is what one would most probably see if one had more observations and less sampling variation.

Table 14-B and Figure 14-F provide the comparable information for the DS86 dosimetry.

Figures 14-E and 14-F look very much alike. Indeed, in both dosimetries, the equations which best fit the observations turn out to have the same dose-exponent:   Dose^0.75. From Figures 14-E and F, it is self-evident that the dose-response curves are presently concave-downward (supra-linear) in both dosimetries.

(Our analysis has been made in terms of cancer-deaths per 10,000 initial persons. Some readers may be curious about the shape of dose-response if response is measured in cancers per 10,000 person-years. The analysis is provided for them in Chapter 30.)

RERF's Treatment of Dose-Group 8 :

The actual dose-response must be somewhat more supra-linear than we can know. The basis for our statement is a fact found in RERF's report TR-9-87 (p.7). In Dose-Group 8 (the highest Dose-Group), " . . . the T65D total kerma is set equal to 6 Gy for all survivors whose T65D total kerma estimate is greater than 6 Gy." With this sentence, RERF refers to both TR-1-86 and TR-12-80, so apparently RERF has been throwing out some part of the dose not only in the 1950-1982 follow-up, but also in the previous 1950-1978 follow-up.

It follows that, in our own analysis, the combined Dose-Group 6+7+8 must really have a somewhat higher mean organ-dose than we can know. But, of course, the observed cancer death-rate would not change. Therefore, in Figure 14-E, the uppermost datapoint really needs some sliding to the right (toward higher dose), a move which would operate in the direction of greater supra-linear curvature. We are confident that the effect would be small. However, we do not see how RERF's handling of Dose-Group 8 can improve anyone's analysis of dose-response.

In Part 3 of this chapter, we identify another factor which also will operate in the direction of underestimating the supra-linearity of low-LET dose-response.

Males and Females Tested Separately :

By definition, the general public includes both sexes. It is impossible to have "population exposure" without irradiating both men and women. Therefore, if analysts are evaluating the dose-response from exposure of a general population, what matters is the net dose-response. When they treat males and females as a unit in their analyses, the shape they obtain for dose versus cancer-response necessarily incorporates and reflects whatever difference may exist in male versus female response.

For other purposes, however, we may want to know if males and females are alike in the shape of dose-response. Of course, the moment analysts start subdividing the database, they increase the small-numbers problem, and findings are necessarily less reliable.

Using exactly the steps demonstrated in Chapter 29, we did regression analyses for males and females separately. The input data for cancer-rates and mean organ-doses were obtained from Table 11-G. The results are summarized below, in the equations of best fit. All the equations have supra-linear dose-exponents (below 1.0).

o -- MALES :

T65DR:   Ca-deaths per 10,000 initial persons = (5.986)(Dose^0.75) + 796.389
DS86:      Ca-deaths per 10,000 initial persons = (7.248)(Dose^0.70) + 792.248

o -- FEMALES :

T65DR:   Ca-deaths per 10,000 initial persons = (10.086)(Dose^0.70) + 540.838
DS86:      Ca-deaths per 10,000 initial persons = (9.463)(Dose^0.70) + 538.102

In other words, examined separately, males and females each show a supra-linear dose-response. The values of R-Squared for males are lower than for females, which means the finding is statistically weaker for males.

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2.   Basis for Ruling-Out a Concave-Upward Dose-Response

Chapter 29 demonstrates the technique of achieving curvilinear regression by raising a single dose-term, serially, to a variety of dose-exponents. We vary the exponent from Dose^2 (the quadratic dose-response), to Dose^1.4 and Dose^1.16 (linear-quadratic shapes), to Dose^1.0 (the linear dose-response), to Dose^0.85 and lower (supra-linear curves). Let us be explicit about the cancer-risks associated with these terms.

Supra-Linear Dose-Response :

This model of dose-response predicts that, with increase in total dose, the increase in cancer death-rate per cSv of dose will decrease. Each additional cSv of exposure will be less hazardous than the previous cSv. The plot of cancer-rate versus dose is concave-downward (illustrated by Figure 14-A), and the dose-exponent is less than 1.0.

Linear Dose-Response :

Here a plot of cancer death-rate versus dose yields a straight line -- hence the name "linear." The increase in cancer-rate per additional unit of dose is the same over the entire dose-range (illustrated by Figure 14-B), and the dose-exponent is 1.0.

When the quadratic dose-term (Q) has a positive coefficient, this model predicts that the increase in cancer-rate, per unit increase in dose, will increase as total dose increases. Each additional cSv of exposure will be more hazardous than the previous cSv. The plot of cancer-rate versus dose is concave-upward (illustrated by Figure 14-C). When a single dose-exponent is used, the exponent must be greater than 1.0, but less than 2.0.

However, as emphasized elsewhere (Go89b;   also Chapters 23 and 29 of this book), when the quadratic term has a negative coefficient, the net result is a concave-downward, supra-linear dose-response (see Figure 23-H).

This model, whose plot is also concave-upward, bends away even more than the linear-quadratic dose-response from a straight line (illustrated by Figure 14-D), and the dose-exponent is 2.0.

Figures 14-A, B, C, D, for males and females combined, come from the input provided in Table 14-D. The four figures depict how the actual observations in the T65DR dosimetry relate to the values calculated by best-fit equations having the four shapes described above. Comparable figures are not included for the DS86 dosimetry simply because Figure 14-F already reveals that they would look like the T65DR figures.

Curve Fitting -- Supra-Linear Fit Is Significantly Better

In a good fit, not only should the weightiest observations lie close to the calculated curve, but their scatter (if any) should fall to both sides of it. In addition, it is a sign of poor fit if the observations on both ends lie on the same side of the curve while the observations in the middle all lie on the opposite side.

Inspection of Figures 14-A, B, C, and D shows the greatly inferior fit of both the linear-quadratic (Dose^1.4) and the pure quadratic (Dose^2) models. Indeed, such inspection predicts the results of the formal statistical testing in Tables 29-D and E.

The results in Tables 29-D and 29-E show that the supra-linear dose-response in the A-Bomb Study (1950-1982), in both the T65DR and the DS86 dosimetries, is significantly better than the linear relationship (p = 0.01).

As for a concave-upward dose-response, statistical testing in Tables 29-D and 29-E simply rules out such a relationship as the plausible choice. Even in the absence of any formal statistical testing, this conclusion is evident from inspection of Figures 14-C and 14-D, compared with Figures 14-A and 14-B.

As an independent check on the statistical significance of the supra-linear fit, we also used the power polynomial method of curve-fitting. It shows that there is both a statistically significant linear dose-term (Dose^1.0) and a statistically significant quadratic dose-term (Dose ^2.0), and that the coefficient of the quadratic term is negative. The equation of best fit from the power polynomial method produces a plot of cumulative cancer-rate versus dose which is virtually identical with the plot produced by the best-fit equation containing the Dose^0.75 term (Figures 14-E and 14-F).

Comparison with Statements from RERF :

Readers are in a position to evaluate our analysis of the shape of dose-response, step-by-step, from start to finish. They will not be able to compare it directly with RERF reports, however. RERF analysts are determining dose-response from input which is different from ours. For instance, in TR-5-88, Shimizu and co-workers discard the evidence between 1950-1955, and use only the observations from 1956 onwards. They are using only 75,991 of the initial 91,231 persons. For their 75,991 persons, they have additional observations out to 1985. They are using newly constructed cohorts, not a constant-cohort analysis. In effect, they are using a different database.

Nonetheless, there is a key similarity between our analysis and the analysis by Shimizu and co-workers:   The RERF analysts do not find a concave-upward dose-response either. They find the following:

1.   When they examine all cancers combined except leukemia as we do, and when they include all the Dose-Groups as we do, they find that their data fit linearity and supra-linearity equally well (Shi87, pp.28-30, and Shi88, pp.50-51).

2.   When they examine males and females separately and include all the evidence, as we do, again they indicate that they find a linear or supra-linear dose-response (Shi88, p.53, Table 19).

However, Shimizu and co-workers never use the term, supra-linear. Their important points about the supra-linear shape might even be missed by any readers who assume that LQ (linear-quadratic) and LQ-L models are always concave-upward. The assumption would be mistaken. If the Q-coefficient for dose is negative in an LQ model, the net LQ curvature is concave-downward (Figure 23-H, again). Therefore readers of RERF reports need to pay close attention to RERF statements and footnotes such as:

"For those sites other than leukemia and colon, the fitted curve associated with the LQ model is invariably concave downwards, not upwards . . . " (Shi87, top p.29).

" . . . since the curvature is invariably downwards when a curvilinear model gives an acceptable fit, this would imply a higher risk at low doses than that which obtains under a linear model" (Shi87, p.30).

"Coefficient for the Q-term is negative" for the LQ model;   this is the footnote which applies to analysis of the full dose-range in Table 19 of TR-5-88 (Shi88, p.53).

A Possible Route to Error :

Having found the dose-response to be linear or supra-linear (concave-downward), Shimizu and co-workers propose an alternative way to determine the dose-response. We quote:

"For all cancers except leukemia, although the L model fits well for both the total dose range and the dose range excluding high doses, the LQ model can not be shown to be inappropriate statistically. It should be noted that Q term in the LQ model is negative when the entire dose range is used, reflecting the level off of the dose-response curve at the higher dose range. In order to obtain useful risk estimates in the low-dose range with the LQ model, we have estimated the risk limiting doses to under 2 Gy, so as to obtain a positive Q term" (Shi88, p.50-51).

Although the paper (Shi88) is unclear on whether 2 Gy is kerma dose or internal organ-dose unadjusted for RBE, the statement quoted above means that they threw away the high Dose-Groups (probably 6,7,8) because, in someone's opinion, supra-linearity (the negative Q-coefficient) is not "useful." Not useful to whom? And for what?

Where does such an approach to evidence end?

It may easily end in error. For instance, it is self-evident from Figures 13-A and 13-B in the previous chapter, that if one discarded Dose-Group 5 as well as Dose-Group 6-8, one would end up with the opposite result:   The dose-response would be based on the four residual datapoints, and it would be more supra-linear, not less. (This statement is supported by regression analysis which excludes Dose-Groups 5-8).

Moreover, if we look objectively at the entries in Table 13-A, Column E, we see that the absolute number of cancer-deaths observed in Dose-Group 4 is about the same as in Dose-Group 5, and also in the combined Dose-Group 6+7+8. This means that the statistical reliability of each of these three observations is about the same. If analysts are willing to discard one, then on an objective basis, why should they not discard all three?

Suppose the first discarding of data (Dose-Groups 6-8) would result in decreasing the study's supra-linear curvature, but suppose the next, equally justifiable discarding of data (Dose-Group 5) would result in exaggerating its supra-linear curvature. What is the appropriate choice?

In my opinion, the curvature which is most likely to be right is the curvature which comes from using all the available evidence. It would certainly not be science at all, if I were to keep the evidence which leads to answers I may like, while throwing out the evidence which produces answers I may not like.

In my judgment, analysts will be most likely to obtain the right answer about dose-response when they use all of the observations. The reason is this. The jaggedness observed in Figures 13-A and 13-B has virtually no chance of being biologically meaningful. Such jaggedness is almost certainly the result of sampling variation, which means that it would not be there (the dose-response would be smooth) if the study had included a billion persons instead of only 91,231. One of the great scientific virtues of the A-Bomb Study is its inclusion of such a vast range of doses. If we use all the data in regression analysis, the additional observations are likely to help "correct" the jaggedness of sampling variation. But if we start throwing away any of the valuable evidence without a very good reason indeed, we will almost certainly increase the chance of mistaken results.

Conclusion about Shape in the A-Bomb Study :

Our analysis of dose-response is based on all the evidence. No Dose-Groups (and no follow-up years) have been thrown out. Our findings fit concavity-downward (supra-linearity) provably better than linearity, and fit concavity-downward enormously better than concavity-upward.

We are forced to conclude, not by preference but by the evidence currently available to us, that concavity-upward is not credible as the shape of dose-response in the A-Bomb Study. The credible choice is presently supra-linearity.

Will supra-linearity persist to the end of the study, decades from now? No one can know. (As we said in Chapter 12, no one can rule out even remote possibilities -- like there being no excess cancer anymore at the end of the study, which would mean a flat dose-response. Of course, in that unlikely case, the interim excess of cancer-deaths would represent a major misery for those who died from the disease 10, 20, 30, 40 years earlier than otherwise.)

Meanwhile, analysts must report whatever is ruled in and out by the currently available evidence. This chapter and Chapter 29 rule out the concave-upward shape as a good fit for the 1950-1982 observations. The data say that the dose-response curvature is concave-downward.

It should be emphasized that the findings about dose-response -- like the findings for minimum Fatal Cancer-Yields -- involve no forward projections and no hypotheses about radiation carcinogenesis. Our findings simply amount to an objective description of what the present evidence is on the shape of the dose-response.

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3.   Purely Low-Let Radiation versus Mixed (Gamma + Neutron)

The dose-response curve which fits the observations best is presently concave-downward or supra-linear (Figure 14-E for the T65DR dosimetry;   Figure 14-F for the DS86 dosimetry). In each dosimetry, the equation which generates the best-fit has a dose-exponent of 0.75.

The dose-input for the regression-analysis (Chapter 29) was composed of two types of radiation:   Gamma and neutron. (Tables 9-C and 10-E, Row 14, show the small fraction of the internal organ-dose, in rems, which was contributed by neutrons.) Therefore, the curves depict dose-response for a mixture of the two radiations.

Nonetheless, one must conclude that the concave-downward curvature is caused by the low-LET (gamma) component of the exposure, not by the high-LET (neutron) component.

The basis for this conclusion is clear if we start by imagining that the A-bomb survivors received only neutron-exposure, but no gamma exposure. For neutron-exposure, the experimental observation is that dose-response is linear, at least up to 10 rads of total neutron dose (Chapter 8, Part 5). And if we adjust for the greater carcinogenic potency of neutrons, by multiplying neutron doses (below 10 rads) by a constant RBE of 20 to obtain rems, a plot of cancer-rate versus pure neutron doses in rems would still be linear. We have shown (page 8-8) that the highest mean neutron organ-dose was about 4.369 rads in the DS86 dosimetry, where such doses are supposed to be correct;   4.369 is a dose well below 10. Therefore, if the A-bomb survivors had received only neutron-exposure, our plots of cancer-rates versus dose in rems would be linear.

Now, we return to the real situation. The A-bomb survivors also received a gamma dose. And when cancer-rate is plotted versus dose in rems, for the combined neutron and gamma doses, the best fit for the observations becomes supra-linear, even though it would have been linear if only the neutrons had been present. It follows that the curvature is caused by the gamma exposure, not by the neutrons.

Underestimation of the Low-Let Curvature :

Our analyses must somewhat underestimate the true degree of supra-linearity for low-LET (gamma) dose-response. Table 10-E, Row 14, shows that the fraction of total dose, in cSv, contributed by neutrons rises with rising total dose. The rising share from neutrons (from 5.4 % in Dose-Group 2, up to 18 % in Dose-Group 8) prevents the supra-linear curvature for gamma-exposure from being fully seen.

The gamma's supra-linear dose-response means that the percent increase in spontaneous cancer-rate, per average rem of gamma dose, falls as gamma-dose rises. By contrast, the neutron's linear dose-response means that the carcinogenicity of neutrons is constant in all eight dose-groups. As the combined dose from gammas and neutrons is rising, the average carcinogenicity of the gamma rems is falling whereas the carcinogenicity of the neutron rems is not falling.

Therefore, when neutrons contribute an "extra" share of the combined dose as the combined dose is rising (Table 10-E, Rows 11 and 14), it means that the observed cancer-rates at the higher doses are somewhat elevated above the rates which would have occurred if the fraction contributed by neutrons had not risen. The result is that the "extra" share from neutrons progressively "lifts" the right-hand half of the curve of Cancer-Rate versus Combined Dose, in the direction of linearity. In other words, the supra-linear curvature would be more pronounced if the fraction of combined dose coming from neutrons had not risen. Thus the supra-linearity of low-LET dose-response is somewhat underestimated in our DS86 analyses.

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4.   Basis for Generalizing from the A-Bomb Study

This chapter and Chapter 29 have confirmed what Table 13-B so strongly suggested:   The dose-response is presently supra-linear throughout the dose-range. The result does not depend on high-dose data. If analysts threw out Dose-Groups 5-8, the supra-linearity would be even more pronounced. We would emphatically not approve of throwing out data, however.

The finding, that dose-response for low-LET exposure is concave-downward (supra-linear), is based on observation of all fatal cancers combined, with only leukemia excluded. In other words, the finding does not rest on a small study involving just leukemia or a few cancer-sites, or on a study resting on incidence instead of mortality. And the finding is based on two dosimetries. And the finding is not based on a single sex- or age-group. It is broadly based on both sexes and all ages.

In other words, the finding of supra-linearity at low doses is based on excellent human epidemiological evidence -- in our judgment, the best which is available at this time anywhere.

Therefore, it is scientifically reasonable to generalize from the A-Bomb Study, 1950-1982:   In humans, the dose-response for induction of fatal cancer by low-LET ionizing radiation is most probably supra-linear in shape, even at low doses. The risk per rem rises as total dose falls.

This finding is directly at variance with the widely applied presumption -- not based on human epidemiology -- that the human cancer-hazard per centi-sievert of low-LET exposure would go down with decreasing total doses. Readers are referred to Un77, p.414, para.318;   Un86, p.191, para.153;   Beir80, p.190;   Ncrp80, pp.5-9;   Nih85, p.iv;   Nrc85, p.II-101-103;   Doe87, p.7.3, 7.4;   and others. Some of these sources use the presumption, while also acknowledging that the available human epidemiological data do not support it (see Chapter 22).

The Past and Future of Supra-Linearity :

The supra-linear shape of dose-response has been showing up in the A-Bomb Study for at least three consecutive follow-ups:   1950-1974, 1950-1978, and 1950-1982 (Go81;   Go89a;   Ncrp80 -- details in our Chapter 22, Part 2). In other words, supra-linearity is not a characteristic which appeared only with the addition of the 1978-1982 observations. And, according to RERF analysts (Shi87;   Shi88), it is still showing up in the revised database when they add some observations through 1985 (see this chapter, Part 2).

Although no one can be sure that supra-linearity will continue its persistence through all future follow-ups, the only reasonable forward projection is the one which rests on the best available evidence. And the best available evidence, from at least three consecutive follow-ups, suggests that supra-linearity will persist.

On the other hand, if the A-Bomb Study itself does not persist with a continuous "constant-cohort, dual-dosimetry" database, it will be hard for anyone to sort out which future findings on dose-response result from extension of the time-interval since the exposure, and which future findings result from perpetual revision of the DS86 doses and cohorts.

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5.   Low-Dose Cancer-Yields Based on the Best-Fit Curve

When analysts seek to estimate the cancer-hazard from exposing populations of mixed ages to ionizing radiation, the doses received by Dose-Group 3 in the A-Bomb Study are considerably higher than the relevant levels suggested by nuclear accidents like Chernobyl, for example. We should be asking, what are the likely Cancer-Yields if people receive total doses like 5 cSv (or less)?

Tables 14-A and 14-B provide the probable values for the minimum Fatal Cancer-Yields, in Column G. We have starred the entries calculated from 5 cSv of total exposure, because we think those are the appropriate ones to use for low-dose exposures up to 5 cSv and for slow exposures. (We closely examine the issue of slow dose-rates in Chapter 23, Parts 6 and 7.)

The notes of Tables 14-A and 14-B explain exactly how the values were obtained. Readers will see that this is still the Cancer Difference Method:   A difference in cancer-rate is divided by the corresponding difference in dose. However, in this version, the cancer-rates are not the direct observations;   instead, they are the rates predicted after the actual observations have produced an equation of best fit. In Tables 14-A and 14-B, the division-step for the starred entries at 5 cSv amounts to the approximation that every rem (cSv) between 0 and 5 rems is equally potent.

Table 14-C assembles the low-dose Cancer-Yields from Table 13-B as well as from Tables 14-A and 14-B, so that they can be easily compared with each other. The net effect of regression-analysis is to reduce the T65DR risk-estimate below its value in Table 13-B, and to render it almost identical with the DS86 estimate.

The Basing of Values on a Total Dose of 5 Centi-Sieverts :

Readers may wonder why we suggest using Cancer-Yields calculated from a total dose of 5 cSv, even for use with population-exposure which might be lower (say, one centi-sievert or less). At first glance, it may look as if we are deliberately underestimating the likely Minimum Fatal Cancer-Yields from very low-dose exposure, since we are using linearity instead of supra-linearity between 0 dose and 5 rems.

Our reasoning is as follows. The technique of curvilinear regression provides the values of low-dose Cancer-Yield which are most likely to be true, given the evidence at hand. And the equation which has the highest R-Squared value in regression analysis is the equation which is most likely to make the best predictions. Therefore the equation which we should use, and which we do use, is the one in which the dose-exponent is 0.75. Objectivity requires use of results from available evidence, rather than use of preconceptions about how the curvature "ought" to behave at low doses (see Chapter 23). Unlike the BEIR-3 Committee (see Chapter 22), we do not constrain any regression in order to make it support a pre-judgment.

On the other hand, as we pointed out in Chapter 29, while we know that 0.75 is significantly better than the dose-exponent 1.0, we do not know that 0.75 is significantly better than 0.80, 0.85, 0.70, or 0.65. Yet the shape of the curve is such that small changes in the dose-exponent have a big effect, at one or two cSv, on the values for Cancer-Yield in Columns F and G of Tables 14-A and 14-B. In view of this sensitivity, we want to avoid using any values for Cancer-Yield derived directly from the curve at one or two cSv.

We regard our decision as a scientifically reasonable judgment which simultaneously (A) avoids the irresponsibility of throwing away the low-dose results of regression analysis down to 5 cSv, and (B) avoids the introduction of any unstable element into an analysis which has been securely based in reality.

A Comment by RERF about BEIR Choices :

The shape of dose-response is central to obtaining risk-estimates at low (and slow) doses. If analysts choose unrealistic versions of the dose-response relationship, they will provide unrealistic estimates of cancer-hazard. RERF analysts, in trying to figure out why their own current risk-estimates are so much higher than those of the BEIR-3 Committee, comment (TR-5-88, p.51):

[Some of the disparity] " . . . may be ascribed to the fact that in BEIR III, the curvature in dose response for leukemia was used for all cancers except leukemia instead of the actual curvature which probably is much closer to linearity, and this may cause much smaller estimates to be produced than if the actual dose-response curve were to be applied."

Venturing below 10 Rems :

Now that we have examined the logic and results of regression analysis, as a tool for obtaining a smooth and probable dose-response at all doses, we can discuss a matter which puzzles us and may puzzle readers too.

In its 1980 report, the BEIR-3 Committee declined to make risk-estimates for acute exposures lower than 10 rems (Beir80, p.144). RERF analysts appear to be split on this issue. In TR-9-87 (Pr87b, p.35), Preston and Pierce present their estimates of Lifetime Fatal Cancer-Yield as cancer deaths per 10 milli-sieverts (per rem). By contrast, in TR-5-88 (Shi88, Table 19, p.53), Shimizu and co-workers explicitly constrain their estimates of Lifetime Fatal Cancer-Yields to acute exposures of 0.1 Sievert (10 rems).

It is puzzling to us that Shimizu and co-workers make a big effort to determine what the dose-response relationship is, starting at zero dose, and then they seem unwilling to use it in the low dose-range. As we pointed out in Part 1 of this chapter, estimates below 10 rems are not extrapolations in a direction beyond any actual observations. Such estimates are interpolations between actual observations in Dose-Group 3 and Dose-Group 2. Indeed, Dose-Groups 1-3 provide the most reliable observations in the whole study, in terms of cancer-cases (not necessarily in dosimetry). If analysts will not use the section of the dose-response below Dose-Group 3, it would seem they should have no reason to use it above Dose-Group 3 either, where the datapoints are based on far fewer cancer-deaths.

By contrast, to us it seems highly reasonable -- almost obligatory -- for analysts to presume that the dose-response which derives from the dose-range as a whole also characterizes the little segment between zero dose and 10 rems.

On the other hand, refusal to make estimates below 10 rems could be a way of suggesting that maybe the risk of radiation-induced cancer just disappears somewhere between 10 rems and zero dose.

The human evidence against any harmless dose of ionizing radiation, with respect to carcinogenesis, is examined in detail in the Threshold section of this book (Section 5). Here we shall limit our comments to the A-Bomb Study (see also Chapter 35, Part 9).

The A-Bomb Study, properly handled, certainly offers no basis for belief in a threshold, or a lesser hazard per rem either, anywhere below 10 rems. On the contrary. Its present supra-linear curvature indicates the risk per rem is growing steadily higher as dose approaches zero. Even if its present dose-response were linear (instead of supra-linear), this would be no basis for belief either in a safe threshold somewhere below 10 rems, or a lesser effect per rem.

In short, even if there were no additional evidence in Section 5 against a threshold, and even if the dose-response in the A-Bomb Study were linear instead of supra-linear, we would consider the basis for making risk-estimates below 10 rems to be scientifically compelling.

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6.   The Bottom Line from Best-Fit Curves

1.   This chapter and Chapter 29 show that the relationship between dose and cancer-response per 10,000 initial persons is presently supra-linear (concave-downward). Statistical testing demonstrates that the evidence fits a concave-downward curvature significantly better than the evidence fits a linear dose-response, and very much better than it fits a concave-upward shape. See Figures 14-A, B, C, and D. In short, the present evidence from the A-bomb survivors is that cancer-risk is greater per rem (centi-sievert) at low doses than at high doses, in both dosimetries. (Chapter 30 shows the same finding in cancer-response per 10,000 person-years.)

2.   The finding of supra-linearity is solidly based in the existing evidence, and does not rely on any forward projections, hypotheses, or models. We have simply presented an objective description of what the available evidence is showing in a database which covers all cancers (leukemia excluded), all doses, all ages, and both sexes. This direct and comprehensive human epidemiological evidence carries great scientific weight compared with observations from other species, of course, or from laboratory experiments.

3.   The evidence is at variance with the assumption, almost universally used by the radiation community, that the cancer-risk should be less severe per rem at low acute doses than at high acute doses. With regard to low doses delivered slowly, we show in Chapter 23, Part 7, that there is no reason to reduce the low-dose Cancer-Yields in Table 14-C when exposure is slow instead of acute.

4.   Although no one can be certain that the supra-linear curvature will persist through all future follow-ups, the only reasonable forward projection is the one which rests on the best available evidence. And the best available evidence, from at least three consecutive follow-ups, is that supra-linearity is persistent. However, if the A-Bomb Study itself does not persist with a continuous "constant-cohort, dual-dosimetry" database, it will be hard for anyone to sort out which future findings on dose-response result from extension of the time-interval since the bombings, and which future findings result from perpetual revision of the DS86 doses and cohorts.

5.   Regression analysis provides the best-fit equation for dose-response, and the equation can predict cancer-rates at any dose-level, including doses like 2, 5, and 10 rems which lie between the mean dose received by Dose-Group 2 and Dose-Group 3. The estimated cancer-rates at these doses are interpolations between two actual observations -- they are not extrapolations in a direction beyond any observed data-point.

6.   Unlike some current analysis at RERF, our analysis of dose-response uses all of the observations, high-dose and low-dose, and all of the follow-up years, in order to obtain the most reliable results. We do not approve of throwing away evidence without a very good reason indeed. It should be noted that the supra-linear curvature of dose versus cancer-response occurs throughout the dose-range. In fact, if the high-dose evidence from Dose-Groups 5-8 were discarded, the low-dose evidence from Dose-Groups 1-4 would produce greater supra-linearity -- not less.

7.   The best-fit equation from our regression analysis is used to obtain another set of Minimum and Lifetime Fatal Cancer-Yields by the Cancer Difference Method, for low-dose exposure. Table 14-C compares the new set with the first set, in both T65DR and DS86 dosimetries. The net effect of regression analysis is to reduce the estimate in the T65DR dosimetry. In the new set of estimates, the Lifetime Cancer-Yields remain probable underestimates, as they were in Table 13-B. The Lifetime Fatal Cancer-Yield from the best-fit curve is 12.90 in the T65DR dosimetry, and 12.03 in the current version of the DS86 dosimetry. By contrast, the lifetime values commonly used by the radiation community for statements about low-dose exposure are between 1.0 and 2.0 (see Chapter 24, Part 7, and Chapter 34, Wolfe).

```
```

Table 14-A
T65DR Dosimetry (RBE = 2):   Comparison of Calculated Cancer-Rates
with Observed Cancer-Rates.
T65DR Dosimetry (RBE = 2):   Minimum Fatal Cancer-Yields
per Centi-Sievert Among 10,000 Persons.
```|=============================================================================================|
|  Col.A      Col.B        Col.C        Col.D  |                 Col.E       Col.F    Col.G   |
|                                              |                                              |
|   Dose  Dose^0.75  Cancer-Rate  Cancer-Rate  |    Calculated Ca-Rate   Avg.Incr.  MINIMUM   |
|    cSv              per 10,000   per 10,000  |  MINUS the Calculated  in Ca-Rate    FATAL   |
|  T65DR              Calculated     Observed  |   Spontaneous Ca-Rate     per cSv CA-YIELD   |
|==========================================================================================   |
|      0    0.00000     649.5440       649.31  |                                              |
|      1    1.00000     656.5968               |                 7.053       7.053    8.675   |
|  1.511    1.36260     659.1542       651.89  |                 9.610       6.362    7.825   |
|      2    1.68179     661.4053               |                11.861       5.931    7.295   |
|      5    3.34370     673.1265               |                23.582       4.716    5.801 * |
|     10    5.62341     689.2048               |                39.661       3.966    4.878   |
| 10.994    6.03762     692.1261       712.02  |                42.582       3.873    4.764   |
|     20    9.45742     716.2453               |                66.701       3.335    4.102   |
|     30   12.81861     739.9511               |                90.407       3.014    3.707   |
| 35.361   14.50098     751.8165       723.72  |               102.273       2.892    3.557   |
|     40   15.90541     761.7217               |               112.178       2.804    3.449   |
|     50   18.80302     782.1579               |               132.614       2.652    3.262   |
|     60   21.55825     801.5900               |               152.046       2.534    3.117   |
|     70   24.20045     820.2250               |               170.681       2.438    2.999   |
| 71.308   24.53891     822.6120       836.27  |               173.068       2.427    2.985   |
|     80   26.74961     838.2037               |               188.660       2.358    2.901   |
|     90   29.22011     855.6276               |               206.084       2.290    2.816   |
|    100   31.62278     872.5731               |               223.029       2.230    2.743   |
|    110   33.96601     889.0995               |               239.555       2.178    2.679   |
|    120   36.25650     905.2539               |               255.710       2.131    2.621   |
|    130   38.49971     921.0748               |               271.531       2.089    2.569   |
|    140   40.70015     936.5940               |               287.050       2.050    2.522   |
|    150   42.86161     951.8383               |               302.294       2.015    2.479   |
|    160   44.98731     966.8305               |               317.286       1.983    2.439   |
|    170   47.08003     981.5900               |               332.046       1.953    2.402   |
|176.662   48.45714     991.3025       988.45  |               341.759       1.935    2.379   |
|    180   49.14218     996.1339               |               346.590       1.925    2.368   |
|    190   51.17587     1010.477               |               360.933       1.900    2.337   |
|    200   53.18296     1024.633               |               375.089       1.875    2.307   |
|=============================================================================================|
```
UPPER NOTES:   -----
Entries in Col.A come from Table 13-A, with many doses added between observations.
Entries in Col.C for the predicted rates are calculated, both for observed doses and interpolated doses, with the equation derived from Table 29-D:
Cancer-Rate = (7.0528)(Dose^0.75) + 649.544.
Values for the term (Dose^0.75) are obtained from Col.B above. Entries in Col.D come from Table 13-A, and lie near the calculated values. Columns A, C, and D are plotted in Figure 14-E.

--------------------   Right-Hand Side of Table   --------------------

FATAL CANCER-YIELD = NUMBER OF RADIATION-INDUCED CANCER-DEATHS AMONG 10,000
INITIAL PERSONS OF MIXED AGES, PER CENTI-SIEVERT OF WHOLE-BODY INTERNAL ORGAN-DOSE.

Entries in Col.E are Col.C minus 649.544 (which is the calculated spontaneous rate / 10,000).
Entries in Col.F are Col.E / Col.A. The entries correspond to the Min. Fatal Cancer Yield calculated by the Cancer Difference Method, before the 1.23-fold correction used by RERF for underascertainment of cancer-deaths (see Chapter 11). The progressive decline of Col.F entries with rising dose reflects the supra-linearity of dose-response.
Entries in Col.G are Col.F entries times 1.23, the underascertainment correction. The starred value is the one which we use for low-dose exposure. In subsequent chapters also, we use values per cSv based on best-fit at 5 cSv.

```
```

Table 14-B
DS86 Dosimetry:   Comparison of Calculated Cancer-Rates With ObservedD Cancer-Rates.
DS86 Dosimetry:   Minimum Fatal Cancer-Yields per CentiI-Sievert Among 10,000 Persons.
```|==============================================================================================|
|  Col.A      Col.B        Col.C        Col.D  |                 Col.E       Col.F     Col.G   |
|                                              |                                               |
|   Dose  Dose^0.75  Cancer-Rate  Cancer-Rate  |    Calculated Ca-Rate   Avg.Incr.   MINIMUM   |
|    cSv              per 10,000   per 10,000  |  MINUS the Calculated  in Ca-Rate     FATAL   |
|   DS86              Calculated     Observed  |   Spontaneous Ca-Rate     per cSv  CA-YIELD   |
|===========================================================================================   |
|      0     0.0000      647.693               |                                               |
|  0.089     0.1634      648.768       649.31  |                                               |
|      1     1.0000      654.272               |                 6.579       6.579     8.093   |
|  1.890     1.6121      658.299       651.89  |                10.606       5.611     6.902   |
|      2     1.6818      658.758               |                11.065       5.533     6.805   |
|      5     3.3437      669.692               |                21.999       4.400     5.412 * |
|     10     5.6234      684.691               |                36.998       3.700     4.551   |
| 14.564     7.4553      696.744       712.02  |                49.051       3.368     4.143   |
|     20     9.4574      709.916               |                62.223       3.111     3.827   |
|     30    12.8186      732.030               |                84.337       2.811     3.458   |
|     40    15.9054      752.339               |               104.646       2.616     3.218   |
| 40.625    16.0915      753.564       723.72  |               105.871       2.606     3.205   |
|     50    18.8030      771.404               |               123.711       2.474     3.043   |
|     60    21.5582      789.531               |               141.838       2.364     2.908   |
|     70    24.2005      806.915               |               159.222       2.275     2.798   |
| 74.238    25.2911      814.091       836.27  |               166.398       2.241     2.757   |
|     80    26.7496      823.687               |               175.994       2.200     2.706   |
|     90    29.2201      839.941               |               192.248       2.136     2.627   |
|    100    31.6228      855.749               |               208.056       2.081     2.559   |
|    110    33.9660      871.166               |               223.473       2.032     2.499   |
|    120    36.2565      886.235               |               238.542       1.988     2.445   |
|    130    38.4997      900.994               |               253.301       1.948     2.397   |
|    140    40.7002      915.472               |               267.779       1.913     2.353   |
|    150    42.8616      929.692               |               281.999       1.880     2.312   |
|    160    44.9873      943.678               |               295.985       1.850     2.275   |
|    170    47.0800      957.447               |               309.754       1.822     2.241   |
|    180    49.1422      971.014               |               323.321       1.796     2.209   |
|197.054    52.5943      993.727       988.45  |               346.034       1.756     2.160   |
|    200    53.1830      997.600               |               349.907       1.750     2.152   |
|==============================================================================================|
```
UPPER NOTES:   -----
Entries in Col.A come from Table 13-A, with many doses added between observations.
Entries in Col.C for the predicted rates are calculated, both for observed doses and interpolated doses, with the equation derived from Table 29-C:
Cancer-Rate = (6.5793)(Dose^0.75) + 647.693.
Values for the term (Dose^0.75) are obtained from Col.B above. Entries in Col.D come from Table 13-A, and lie near the calculated values. Columns A, C, and D are plotted in Figure 14-F.

--------------------   Right-Hand Side of Table   --------------------

FATAL CANCER YIELD = NUMBER OF RADIATION-INDUCED CANCER-DEATHS AMONG
10,000 PERSONS OF MIXED AGES, PER CENTI-SIEVERT OF WHOLE-BODY INTERNAL ORGAN-DOSE.

Entries in Col.E are Col.C minus 647.693 (which is the calculated spontaneous rate / 10,000).
Entries in Col.F are Col.E / Col.A. The entries correspond to the Min. Fatal Cancer Yield calculated by the Cancer Difference Method, before the 1.23-fold correction used by RERF for underascertainment of cancer-deaths (see Chapter 11). The progressive decline of Col.F entries with rising dose reflects the supra-linearity of dose-response.
Entries in Col.G are Col.F entries times 1.23, the underascertainment correction. The starred value is the one which we use for low-dose exposure. In subsequent chapters also, we use values per cSv based on best-fit at 5 cSv.

```
```

Table 14-C
Cancer-Yields at the Low Doses, by the Cancer Difference Method,
with and without Curvilinear Regression.

Basis:  A-bomb survivors, all ages combined, 1950-1982. Both T65DR and DS86 dosimetries.
Cancer-hazard from X-rays may be underestimated by the A-Bomb Study. See Chapter 13, Part 4.

Cancer-Yields are radiation-induced cancer-deaths among 10,000 persons of mixed ages, per cSv of whole-body internal organ-dose.
All Cancer-Yields below are corrected for underascertainment of cancer-deaths.

```|==================================================================================|
|                          ||                          ||                          |
|                          ||  --Fatal Cancer-Yield--  ||  --Fatal Cancer-Yield--  |
|                          ||     T65DR Dosimetry      ||      DS86 Dosimetry      |
|                          ||    Neutron RBE = 2.0     ||     Neutron RBE = 20     |
|                          ||                          ||                          |
|==================================================================================|
|            Col.A         ||    Col.B         Col.C   ||    Col.D         Col.E   |
|                          ||                          ||                          |
|      Source of Estimate  ||    FATAL         FATAL   ||    FATAL         FATAL   |
|      for Minimum Fatal   ||  CANCER-       CANCER-   ||  CANCER-       CANCER-   |
|      Cancer-Yields       ||    YIELD         YIELD   ||    YIELD         YIELD   |
|==================================================================================|
| Row                      ||                          ||                          |
|      Dose-Group 3        ||                          ||                          |
|      versus Ref. Group   ||                          ||                          |
| 1    Table 13-B.         ||     7.29         16.20   ||     5.50         12.23   |
|                          ||                          ||                          |
|                          ||                          ||                          |
|      Best-Fit Curve      ||                          ||                          |
|      Tables 14-A and     ||                          ||                          |
| 2    14-B.               ||     5.80         12.90   ||     5.41         12.03   |
|                          ||                          ||                          |
|==================================================================================|
```
NOTES -----
1. The minimum values above in Row 1 come from Table 13-B, Columns B and H. If there had been no dose-groups higher than Dose-Group 3 in the A-bomb experience, these are the only values which would exist in Table 13-B. The entries in Row 1 above are the values before regression analysis provides a smooth best-fit curve.
2. The minimum values in Row 2 above come from the best-fit curves provided by regression analysis, using all dose-groups.
The minimum value in Row 2, Column B, comes from Table 14-A, Column G (the starred value).
The minimum value in Row 2, Column D, comes from Table 14-B, Column G (the starred value).
3. lifetime values ( Columns C and E ) are always the minimum value times 2.223, in the Cancer Difference Method. These lifetime entries are probably underestimates (see text and subsequent chapters). The factor 2.223 comes from Table 28-D, Row 14.

```
```

Table 14-D
Input Values for Figures 14-A, 14-B, 14-C, and 14-D.
T65DR Dosimetry with Neutron RBE = 2.
```|=====================================================================================|
|       Input for Figure 14-A               |       Input for Figure 14-B             |
|       ========================            |       ========================          |
|  Equation for Dose^0.75 from Table 29-B:  |  Equation for Dose^1 from Table 29-B:   |
|  Ca-Rate = (7.0528)(Dose^0.75) + 649.544  |  Ca-Rate = (1.947)(Dose^1) + 661.153    |
|                                           |                                         |
|      Dose      Dose   Ca-Rate   Ca-Rate   |     Dose      Dose   Ca-Rate   Ca-Rate  |
|       cSv  cSv^0.75     Calc.  Observed   |      cSv     cSv^1     Calc.  Observed  |
|                                           |                                         |
|     0.000    0.0000   649.544    649.31   |    0.000     0.000   661.153    649.31  |
|     1.511    1.3626   659.154    651.89   |    1.511     1.511   664.094    651.89  |
|    10.994    6.0376   692.126    712.02   |   10.994    10.994   682.558    712.02  |
|    35.361   14.5010   751.817    723.72   |   35.361    35.361   730.002    723.72  |
|    71.308   24.5388   822.611    836.27   |   71.308    71.308   799.990    836.27  |
|   130.000   38.4997   921.075             |  130.000   130.000   914.263            |
|   176.662   48.4571   991.303    988.45   |  176.662   176.662  1005.114    988.45  |
|=====================================================================================|
|                                           |                                         |
|       Input for Figure 14-C               |       Input for Figure 14-D             |
|       ========================            |       ========================          |
|  Equation for Dose^1.4 from Table 29-B:   |  Equation for Dose^2 from Table 29-B:   |
|  Ca-Rate = (0.242)(Dose^1.4) + 671.922    |  Ca-Rate = (0.01047)(Dose^2) + 680.048  |
|                                           |                                         |
|      Dose      Dose   Ca-Rate   Ca-Rate   |     Dose      Dose   Ca-Rate   Ca-Rate  |
|       cSv   cSv^1.4     Calc.  Observed   |      cSv     cSv^2     Calc.  Observed  |
|                                           |                                         |
|     0.000     0.000   671.922    649.31   |    0.000      0.00   680.048    649.31  |
|     1.511     1.782   672.353    651.89   |    1.511      2.28   680.072    651.89  |
|    10.994    28.682   678.863    712.02   |   10.994    120.87   681.313    712.02  |
|    35.361   147.212   707.547    723.72   |   35.361   1250.43   693.140    723.72  |
|    71.308   393.001   767.028    836.27   |   71.308   5084.83   733.286    836.27  |
|   130.000   911.005   892.385             |  130.000   6900.00   856.991            |
|   176.662  1399.591  1010.623    988.45   |  176.662  31209.54  1006.812    988.45  |
|=====================================================================================|
```
The construction of Table 14-D is described, step by step, in Chapter 29.

When the above values are plotted for dose (cSv), calculated cancer-rate, and observed cancer-rate, they demonstrate graphically how closely or how distantly the observed points lie to the corresponding curve calculated by regression analysis.

In a good fit, not only should the weightiest observations lie close to the calculated curve, but their scatter (if any) should fall to both sides of it. In addition, it is a sign of poor fit if the observations on both ends lie on the same side of the curve while the observations in the middle all lie on the opposite side.

Because the dose-response is so similar in T65DR and DS86 (compare Figure 14-E with Figure 14-F), we have not shown graphs comparable to 14-B, 14-C, and 14-D for the DS86 analysis.

```
```

Figures 14-A,B,C,D
Distribution of Datapoints Relative to Four Dose-Response Curves. T65DR Dosimetry:   Cumulative Cancer-Deaths versus Dose.

Input for the four figures below is provided by Table 14-D. Each figure depicts the same observations (indicated by the boxy symbol):   Cancer-mortality versus dose in the A-Bomb Study, 1950-1982. What differs is the dose-response curve in each figure.

o -- In Figure 14-A, the "fit" between the observations and the supra-linear curve is good, with datapoints either lying on the curve or falling to both sides of it.

o -- In Figure 14-B, the fit between the observations and the linear "curve" is inferior to the fit in Figure 14-A.

o -- In Figure 14-C, the fit between the observations and the linear-quadratic (Q-positive) curve is very poor, with the observations at both ends lying on the same side of the curve, and the observations in the middle all lying on the opposite side.

o -- In Figure 14-D, the fit between observations and the quadratic dose-response curve is even worse than in Figure 14-C.

o -- Statistical testing (Chapter 29) establishes that the supra-linear dose-response fits the evidence significantly better than the linear dose-response.

Figure 14-A, below:   Fit Relative to Supra-Linear (Dose^0.75)

Figure 14-B, below:   Fit Relative to Linear (Dose^1.0)

Figure 14-C, below:   Fit Relative to Linear-Quadratic (Dose^1.4)

Figure 14-D, below:   Fit Relative to Quadratic (Dose^2.0).

```
```

Figure 14-E
T65DR Dosimetry:  Best-Fit Curve for Cumulative Cancer-Deaths versus Dose.

For the atomic-bomb survivors, this plot shows cumulative cancer-deaths (1950-1982) per 10,000 initial persons, versus mean whole-body internal organ-dose in the T65DR dosimetry (RBE = 2). Input for this figure comes from Table 14-A, Columns A, C, and D.

o -- The boxy symbols, which show the observed cancer death-rate per 10,000 initial persons versus dose, come from Columns A and D of Table 14-A.

o -- Points along the best-fit curve come from Column C of Table 14-A, and show calculated cancer death-rates per 10,000 initial persons versus dose, based on the equation of best fit, shown below. This curve is the same as the curve in Figure 14-A, of course.

Figure 14-F will show the best-fit curve for the same cohorts of survivors in the supplemental DS86 dosimetry.

```
```

Figure 14-F
DS86 Dosimetry:  Best-Fit Curve for Cumulative Cancer-Deaths versus Dose.

For the atomic-bomb survivors, this plot shows cumulative cancer-deaths (1950-1982) per 10,000 initial persons, versus mean whole-body internal organ-dose in the DS86 dosimetry (RBE = 20). Input for this figure comes from Table 14-B, Columns A, C, and D.

o -- The boxy symbols, which show the observed cancer death-rate per 10,000 initial persons versus dose, come from Columns A and D of Table 14-B.

o -- Points along the best-fit curve come from Column C of Table 14-B, and show calculated cancer death-rates per 10,000 initial persons versus dose, based on the equation of best fit.

With our "constant-cohort, dual dosimetry" approach to the DS86 dosimetry, the T65DR cohorts of survivors remain undisturbed, and merely receive a second dose-estimate. Super-imposition of Figures 14-F and 14-E would show that the DS86 dosimetry shifts the boxy symbols somewhat to the right (higher dose) -- as predictable from comparing Column A in Tables 14-A and 14-B. Consequently, the equations of best fit for the T65DR and DS86 dosimetries are somewhat different.

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