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CHAPTER 23
Proper Risk-Estimates for "Low and Slow" Exposures :
No Conflict Between Human Epidemiology
and the Linear-Quadratic Hypothesis from Radiobiology
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This chapter is arranged in seven parts:
1. Introduction, p.1
2. The Unmodified LQ Model, As Used by Radiation Committees, p.2
3. The Modified LQ Model, As Described by Radiation Committees, p.4
4. The Modified LQ Model, Fitting Supra-Linear, Linear, Sigmoid,
and Other Shapes, p.5
5. Proper Risk-Estimates for Low Acute Exposures, p.9
6. Risk-Estimates for Slow Delivery of High Total Doses, p.10
7. Risk-Estimates for Slow Delivery of Low Total Doses, p.13
Then tables.
Then figures.
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1. Introduction
================
If a cherished hypothesis comes into apparent conflict with
valid human epidemiological evidence -- and note the term, "valid"
-- then we clearly do NOT throw away valid evidence and insist on
the hypothesis. The proper approach is to find out how and why
such a hypothesis is inadequate to explain the valid
epidemiological evidence.
Risk-Estimates for Low Acute Doses :
------------------------------------
As shown in the previous chapter, those who support the
existence of risk reduction-factors for acute low-dose exposures
-- in the presence of human evidence against such factors --
commonly invoke "radiobiological considerations" as the
justification.
Radiobiological considerations include the hypothesis that
both intra-track and inter-track events can result in fully
competent carcinogenic lesions, and that in a linear-quadratic
(LQ) equation, the linear term represents intra-track
carcinogenesis and the quadratic term represents inter-track
carcinogenesis (see Chapter 19, Part 4). Thus, the sign should be
positive for both the linear (L) and quadratic (Q) terms, and
dose-response should be concave-UPWARD.
Even if one accepts this hypothesis, the hypothesis can be
readily reconciled with the real-world observation that, for
carcinogenesis in the A-Bomb Study, human dose-response is
supra-linear and concave-DOWNWARD throughout the dose-range.
The reconciliation is achieved by modifying the LQ equation
with an exponential modifier which represents a cell-INACTIVATION
function operating throughout the entire dose-range. By
inactivation, we do NOT necessarily mean cell-killing, as we will
explain in Part 4.
Recognition that the concave-upward shape, of an unmodified LQ
equation, can be changed by an exponential modifier is nothing
new. It has been recognized for decades. In fact, the format which
we shall use in this chapter was displayed by NCRP in 1980
(Ncrp80, p.19) and by UNSCEAR-86 (p.188).
NCRP and all of the radiation committees recognized that an
unmodified LQ equation was inherently false for human
dose-response, when they all acknowledged that the curve flattens
out "at high doses." Ncrp80 (p.160) even conceded that the
supra-linear curvature was seen "at relatively low doses in the
Hiroshima data." Various reports and analysts have explored use of
"cell-killing" terms to modify their LQ equations -- but only to
change their curvature at very HIGH doses.
Since we all agree that the unmodified LQ equation does not
match human dose-response, the task which remains is to modify an
LQ equation so that it has what the human evidence requires: A
supra-linear or a linear dose-response THROUGHOUT the dose-range.
If this can be done -- with both the L and Q terms positive -- of
course it means that "radiobiological reasons" do NOT always
predict a concave-upward dose-response at all.
It can be done, as this chapter demonstrates to any doubters.
Risk-Estimates for Slow Exposures :
-----------------------------------
Then, in Parts 6 and 7, we shall turn to the possible
implications for dose-RATE.
It is possible (though not certain) that a HIGH acute dose is
more carcinogenic than the same dose delivered slowly. We shall
explore a range of possible values.
However, there is no reason at LOW total doses to expect more
than trivial protection (reduction in risk) from slow delivery.
ICRP itself acknowledged this, as noted in our Chapter 22, Part 3.
In Part 7 of the present chapter, we have evaluated the maximum
amount of protection which might occur if 5 rads, for instance,
were given slowly instead of acutely. The amount would be very
small indeed.
Therefore, the low-dose Cancer-Yields in Section 4 of this
book are definitely valid for SLOW delivery as well as for acute
delivery.
2. The Unmodified LQ Model,
As Used by Radiation Committees
===================================
For the unmodified LQ model, depicted in Figures 23-B and
23-C, we use the following notation:
A cancers = intra-track cancers.
B cancers = inter-track cancers.
(A+B) cancers = total excess cancers.
D^1 = dose in cSv to the first power.
D^2 = dose in cSv squared.
(a cancers per cSv) = coefficient of the L term.
(b cancers per cSv^2) = coefficient of the Q term.
The units cancel out when we write:
-----------------------------------
o -- EQUATION (1) :
A cancers = (a cancers cSv^-1) x (D^1 cSv^1)
A = aD^1
o -- EQUATION (2) :
B cancers = (b cancers cSv^-2) x (D^2 cSv^2)
B = bD^2
o -- EQUATION (3) : A+B = aD^1 + bD^2
In the radiation literature, one sees suggestions that the
linear and quadratic terms may make equal contributions to total
excess cancer (radiation-induced cancer) at doses between 50 and
150 rads (cGy). These are purely speculative limits. We shall
explore limits of 50 and 400 cSv.
A = B at 50 cSv :
-----------------
If A = B at 50 cSv, we would write from Equations (1) and (2):
(a cancers cSv^-1) x (50 cSv) = (b cancers cSv^-2) x (2500 cSv^2)
50a = 2500b
a = 50b.
This is a ratio (a/b = 50), and we can set the value of b
equal to 1.0 for the sake of clarity in producing Table 23-B.
Table 23-B shows the values of A, B, and (A+B) when b = 1.0,
if A = B at 50 cSv. Because we have shown (above) that a = 50b in
these circumstances, we can substitute (50b) for (a) in the
equation A = aD^1. Thus the entries for Column A in Table 23-B are
calculated from the equation A = (50b)(D^1).
In the row for 50 cSv, readers will find equal
cancer-contributions from the intra-track and inter-track terms:
2,500 from each.
From Table 23-B, Columns A, B, and (A+B) are depicted by
Figure 23-B, BB, BBB. This is a single figure in which 23-B
examines the plots out to 400 cSv, 23-BB examines them out to 200
cSv, and 23-BBB examines them out to 100 cSv. In Figure 23-BBB, it
is clear that A = B at 50 cSv. This is visible also in Figure
23-BB.
A = B at 400 cSv :
------------------
If A = B at 400 cSv, we would write a = 400b. Table 23-C shows
the values of A, B, and (A+B) when b = 1.0, if A = B at 400 cSv.
In the row for 400 cSv, readers will find equal
cancer-contributions from A and from B: 160,000 from each.
From Table 23-C, Columns A, B, and (A+B) are depicted by
Figure 23-C, CC, and CCC. Figure 23-C examines the plots out to
400 cSv, 23-CC examines them out to 200 cSv, and 23-CCC examines
them out to 100 cSv. In Figure 23-C, it is clear that A = B at 400
cSv.
The Mistaken Risk-Reduction Factors for Acute Exposures :
---------------------------------------------------------
It is self-evident from Figures 23-B and 23-C that the
unmodified LQ model produces a concave-UPWARD curvature for total
excess cancers -- Plot (A+B). This matches the presumption
discussed in Chapter 22. Indeed, Figure 23-BB closely resembles
the model displayed by Ncrp80 (p.16, Figure 3.4). Comparison of
Figures 23-B and 23-C also shows that the concave-upward curvature
is much greater if A = B at 50 cSv than if A = B at 400 cSv.
We shall calculate risk reduction-factors for both limits, for
acute exposures, as if dose-response were actually concave-upward.
Let us assume that we have observation of risk-rates only at 160
cSv of acute exposure and at zero dose (in other words, two
datapoints), and we decide to estimate the risk from acute
exposure at very low doses, by making a linear interpolation
between 160 cSv and zero dose.
On Figures 23-BB and 23-CC, we have drawn a dashed straight
line from Plot A+B at 160 cSv (the high-dose observation) to the
origin. This dashed line lies above the A+B curve at low doses,
and so linear interpolation would OVERestimate risk at low doses
-- if dose-response were truly concave-upward.
When the dose is very low, readers can see for themselves that
Plot B (the quadratic term) generates about zero excess cancer. At
low doses, essentially all the excess cancers are coming from
intra-track carcinogenesis (Plot A). This is not in dispute.
Indeed, the Figures show that Plot A+B and Plot A are just about
on top of each other, and their slopes are about the same in the
low dose-range.
DREFS -- The Ratio of Slopes :
------------------------------
Under these circumstances of interpolation, it would be
appropriate for everyone to estimate risk reduction-factors by
dividing one slope by another: The slope of Plot A+B shown as the
dashed line between zero and 160 cSv, by the slope of Plot A --
because the slope of Plot A approximates the actual slope of Plot
A+B at low doses. As stated in Chapter 22, Part 1, a DREF is the
ratio of the steeper slope over the lower slope (Ncrp80, p.176).
Since slope in these circumstances means excess cancers per cSv,
the centi-sieverts cancel out in such a division, and the ratio of
excess cancers remains.
If A = B at 50 cSv, the slope of the dashed line is 33600 /
160, or 210. The value 33600 need not be read off Figure 23-B; it
comes from Table 23-B, the A+B column at 160 cSv. The slope of
Plot A is 8000 / 160, or 50. The ratio of the slopes is 210 / 50,
or 4.2. Therefore, if the plots shown in Figure 23-B were actually
observed plots matching real-world data, linear interpolation from
160 cSv would overestimate risk at low doses by 4.2-fold. Under
these circumstances, a risk reduction-factor of 4.2 would be
appropriate.
If A = B at 400 cSv, the slope of the dashed line in Figure
23-CC is 89600 (from Table 23-C) divided by 160, or 560. The slope
of Plot A is 64000 / 160, or 400. The risk reduction-factor under
such circumstances would be 560 / 400, or 1.4-fold, to make
interpolations from 160 cSv.
Visual comparison of Figure 23-BB with 23-CC shows why the
presumption of A = B at 50 cSv "requires" a much larger risk
reduction-factor than the presumption that A = B at 400 cSv, if
linear interpolation is used. In Figure 23-BB, the linear term
(Plot A) accounts for only a small share of Plot A+B. In Figure
23-CC, the linear term (Plot A) is not very distant from Plot A+B,
and so linear interpolation from Plot A+B at 160 cSv "requires"
less of a correction.
Nature of the Mistake :
-----------------------
We put "requires" in quotes because no risk reduction-factors
are required for acute exposures at all, for two reasons: (1)
dose-response in the human is NOT concave-upward, and (2) direct
human observations at low doses leave no need to use high-dose
data, in order to estimate low-dose risks.
The LQ Model -- Positive vs. Negative Q-Coefficient :
-----------------------------------------------------
The underlying assumption of this model is that combined
action of intra-track carcinogenesis and inter-track
carcinogenesis gives rise to the total radiation-induced cancers
(A+B). Therefore, both the linear and quadratic terms are presumed
to be positive. It follows that at every dose, the points along
Plot A+B are necessarily the sum of the corresponding points along
Plot A and Plot B. If inter-track carcinogenesis is occurring at
all, Plot A+B must always lie above Plot A. Moreover, the shape of
Plot B guarantees that the shape of A+B is concave-upward.
However, when analysts try to fit the LQ model to a
supra-linear set of datapoints, regression analysis produces a
best-fit equation in which the quadratic term is NEGATIVE (see
Chapters 14 and 22). This means that the points along the combined
linear and quadratic plot are the A values MINUS the corresponding
B values. The result is necessarily a supra-linear,
concave-DOWNWARD plot, as illustrated by Figure 23-H.
The findings that (1) the Q-term is negative and (2) the
dose-response is concave-downward, do not require any presumption,
however, that intra-track lesions are carcinogenic and inter-track
lesions are protective. Such a hypothesis would be utterly
implausible, as shown in Chapter 19, Part 4.
The concave-downward dose-response, which is observed in the
evidence, is perfectly consistent with POSITIVE signs for both the
linear and quadratic terms -- as we shall show in Part 4 of this
chapter.
3. Modified LQ Model,
As Described by Radiation Committees
========================================
No one disputes that the concave-upward shape of an LQ model
can be altered by adding an exponential modifier. Indeed, we will
use the type of modifier suggested by Ncrp80 (p.19, Figure 3.5):
o -- EQUATION (4) :
I = [aD^1 + bD^2] x [exp-(m'D^1 + m"D^2)]
where I = incidence of effect being studied (total excess
cancers).
Equation (4) is clearly the unmodified Equation (3), except
for the righthand term between brackets. The quantity between the
righthand pair of brackets is the "exponential modifier" by which
the quantity between the lefthand pair of brackets is multiplied.
Some readers will want to review the meaning of "exp." See Chapter
19, Part 2, Poisson Equation, and the note below Table 23-A in
this chapter. (We shall defer examination of Table 23-A, for its
other purposes, until later.)
In Equation (4), the cSv-unit cancels out in the modifier,
because the units of the m' and m" coefficients are cSv^-1 and
cSv^-2, respectively.
Contribution of the NCRP :
--------------------------
Discussions of radiation carcinogenesis commonly refer to the
necessity of modifying the LQ model, by some additional term, in
order to fit the "leveling off" and "flattening" and "falling" of
the incidence curve with rising dose. Almost invariably, these
phenomena are described as occurring at HIGH doses. For instance,
Ncrp80 (p.17) states:
"The phenomenon of the dose-response curve leveling off and
then falling at high doses ... is seen frequently in radiobiology
and specifically in curves for mutagenesis and carcinogenesis. Its
cause, although incompletely understood, is frequently ascribed to
cell killing. Since it is still seen in cell transformation
experiments in tissue culture in which the results are normalized
to surviving cells ... it could be due at least in part to
intracellular processes that prevent the presumed `induction'
phenomenon from becoming manifest."
The NCRP statement above is very useful, and it is clear that
NCRP does not consider cell-killing as necessarily the only
explanation for the "leveling off" effect. We would agree on that
point.
But we fault the NCRP statement for its general suggestion
that the "leveling off" effect is necessarily a HIGH-dose effect.
The understanding of why radiobiology and epidemiology are NOT in
conflict depends upon not pre-judging how intense the "leveling
off" effect is and at what dose it becomes appreciable.
The NCRP formulation using an exponential term, in order to
take account of the "leveling off" effect, is quite reasonable.
But after introducing the expression [exp-(m'Dose^1 + m"Dose^2)],
it is totally unreasonable to pre-judge what the appropriate
values for m' and m" are going to be. (It might be noted that
Ncrp80 uses the gamma and delta symbols instead of m' and m".)
The values of the m' and m" coefficients in the exponential
term must be determined by curve-fitting with real data, not by
some pre-judgment which can totally distort the reality of
epidemiological evidence.
Pre-Judgment in NIH Report :
----------------------------
Other discussions in the literature, of the linear and
quadratic terms, also tend to suggest that the "leveling off"
effect can occur ONLY at very high doses -- a suggestion which I
am sure has added wholly unnecessary confusion to the scene.
For example, in the 1985 NIH Report, the authors refer to how
various "official bodies" (their term) have handled risk-reduction
factors, and in this discussion, they say (Nih85, p.26):
"The BEIR III Committee did not incorporate the competing
effect of cell inactivation, mainly at high dose levels, into its
risk calculations, although it did consider the problem
theoretically ..." The NIH Report also states, in discussing the
Japanese A-bomb survivors (p.26-27):
"Doses high enough to reduce the carcinogenic response
appreciably through the competing effect of cell inactivation
might well be in the lethal range for man when delivered to his
whole body."
The NIH Report clearly takes an extremely prejudicial position
on how very high the doses might have to be, before cell
inactivation could influence the dose-response for cancer. The NIH
committee has decided, contrary to the evidence available in 1985
from the A-Bomb Study, that whole-body doses might have to be high
enough to be lethal to man.
Implications of Breast-Cancer Data :
------------------------------------
Moreover, it would appear that the NIH committee did not
consider the implications of its own acceptance of the conclusion
that the dose-response for human breast-cancer is linear over the
entire dose-range (Chapter 22, Part 2).
The acceptance of linearity at all doses implies either (1)
inter-track carcinogenesis is negligible over the entire
dose-range, or (2) the "leveling off" effect from cell
inactivation is appreciable enough at low doses to off-set the
concave-UPWARD curve of Plot B. Indeed, where whole-body exposure
occurred (the A-bomb survivors), the "leveling off" effect was
appreciable enough to make the dose-response for breast-cancer
concave-DOWNWARD rather than linear (Ncrp80, p.144; Go81, Chaps.
10,11).
"Radiobiologic Findings" :
--------------------------
It is my opinion that pre-judgments or "blind-spots" about the
interaction of the linear term, quadratic term, and the
cell-INACTIVATION term, have led to much nonsense about
radiobiology being in conflict with human epidemiological
evidence, and to suggestions that we must accept dose-response
curves "based on radiobiology" even though the proposed curves are
totally at variance with real human epidemiological data of good
quality.
For instance, "radiobiologic findings" were named as the basis
when BEIR-3 endorsed UPWARD curvature (Beir80, p.261), and
substituted a leukemia curve for the all-cancer curve; see our
Chapter 22, Part 3. This substitution was noted by RERF analysts
in explaining why their own risk-estimates are higher than
BEIR-3's:
[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" (Shi88, p.51).
4. A Modified LQ Model,
Fitting Supra-Linear, Linear,
Sigmoid, and Other Dose-Responses
=====================================
Unlike the radiation committees cited in Part 3, we shall NOT
assume that the cell-inactivation term -- which is the exponential
modifier in Equation (4) -- applies only at some arbitrarily high
radiation dose. Instead, we shall investigate how this term can be
used to fit actual human epidemiological data having supra-linear
and linear dose-responses.
Moreover, we shall not pre-judge how prominent the linear term
(A, or intra-track carcinogenesis) is in comparison with the
quadratic term (B, or inter-track carcinogenesis). We shall
explore two limits: A = B at 50 cSv, and A = B at 400 cSv.
When A = B at 50 cSv, the unmodified LQ dose-response has a
markedly concave-upward curvature (Figure 23-B), but even such
curvature becomes concave-DOWNWARD with the appropriate
cell-inactivation function -- as we shall show.
Cell-Inactivation, and Table 23-A :
-----------------------------------
To explain the observed "leveling off" of radiation
carcinogenesis with rising dose, some analysts refer to the need
for a "cell-killing" term. We do not regard cell-killing as the
only possible explanation. Some other analysts make the
presumption that "cell sterilization" occurs with rising dose, and
that inability of cells to reproduce accounts for the observation.
Also, it is possible that what accounts for the observation is
redundancy of carcinogenic lesions, with rising dose, or
dose-dependent changes in the biochemical milieu.
In the absence of evidence on the cause or causes of
supra-linear and linear dose-responses, we prefer to use the term
"cell-inactivation."
"Cell-inactivation" is a term compatible with all
possibilities. It implies that, as dose rises, (1) a decreasing
fraction of irradiated cells remains capable of providing the
precursors of a clinical cancer, and (2) an increasing fraction of
irradiated cells becomes "inactivated" with respect to developing
into clinical cancers, even though radiation may have produced
carcinogenic alteration in such cells.
It must be emphasized that there are no rules and no
radiobiological principles which prevent variation of the
cell-inactivation term, from one species to another. We shall
return to this issue early in Part 5.
Table 23-A evaluates the cell inactivation term for various
pairs of m' and m" values in Equation (4) -- which is the modified
LQ model. For convenience, it is provided again below.
o -- EQUATION (4) : Incidence = [aD^1 + bD^2] x [exp-(m'D^1 +
m"D^2)]
In Table 23-A, one finds that the "active" fraction is 100 %
at zero dose, and falls with rising dose. Figure 23-A shows that
the fractions fall in a non-linear manner.
A = B at 400 cSv.
Modified LQ Model Yields Supra-Linearity :
------------------------------------------
In Part 2, we showed how Table 23-C yields Figure 23-C -- a
concave-upward dose-response with no "leveling off" or flattening
at high doses. We plotted only the A, B, and A+B columns.
Now we return to Table 23-C, where A = B at 400 cSv, and we
call attention to Column C. Column C shows dose-response (A+B) as
modified by the cell-inactivation term when m' = 0.005 and m" =
-0.000004. In other words, Column C is the evaluation of Equation
(4) when those are the values of m' and m".
For instance, if we use 100 cSv as an example, the entry in
Col. C is 50,000 excess cancers (from Column A+B, unmodified, at
100 cSv) times 0.63128 (the value of the cell inactivation term,
from Table 23-A, Column D, at 100 cSv), or 31,564 excess cancers.
The cell-inactivation term operates upon the intra-track and
inter-track terms alike. One arrives at the same entry by
modifying the A and B entries at 100 cSv separately: (40,000 x
0.63128) + (10,000 x 0.63128) = 31,564.
In Figure 23-D, we have plotted Column C as Plot M (for
Modified). Otherwise, Figure 23-D is exactly the same as Figure
23-C.
It is self-evident, in Figure 23-D, DD, and DDD that Plot M
has a supra-linear, concave-downward shape throughout the
dose-range.
Figure 23-D and the A-Bomb Study :
----------------------------------
Plot M of Figure 23-D, DD, DDD is in harmony with the human
dose-response observed in the A-bomb survivors.
Examination of Figure 23-DDD shows that Plot M starts
diverging from the unmodified LQ dose-response (which is Plot A+B)
at very low doses. In other words, the cell-inactivation term is
already operating below 30 cSv to produce a supra-linear bend
which is appreciable and detectable. This is in accord with the
curvature noted in Chapter 14 for the A-bomb survivors.
On the other hand, no one should expect PERFECT matching
between Figure 23-D, DD, DDD and the dose-response depicted in
Chapter 14. Figure 23-D is constrained only by its equation,
whereas Figures 14-E and 14-F are constrained not only by
different equations, but also by real-world datapoints. The plots
in Chapter 14 are empirical best-fits to actual data, and thus do
not pre-judge the interplay of factors such as possible population
heterogeneity, possible lethargy of repair-systems at very low
doses, possible effects of biochemical milieu, possible redundancy
of injury, or other factors which may affect the steepness of
slope in the very low dose-region.
A Spurious Conflict :
---------------------
If we return to Figure 23-D, DD, DDD, the key point is that a
modified LQ model resolves any alleged conflict between the
real-world observation -- that supra-linearity (starting at very
low doses) occurs throughout the entire dose-range of the A-Bomb
Study -- and "radiobiology."
We cannot explain why the assumption is made, by so many, that
the modifying term in an LQ model cannot operate below very high
doses. There seems to be no logical or scientific basis for such
an assumption. When we permit the modifying term in Equation (4)
to have effect at quite low doses, we are not assaulting the LQ
model. More importantly, we can attain a curve which is compatible
with human epidemiology, instead of substituting a curve which
assaults such evidence.
The Graphic Meaning of "Supra-Linear" and "Sub-Linear" :
--------------------------------------------------------
Examination of Figure 23-DD shows clearly that Plot M has a
concave-downward bend. If a straight line were to connect any two
points along Plot M, the curve of Plot M would lie ABOVE the
straight line. That is why it is correct to characterize a shape
like Plot M as SUPRA-linear.
If confusion is to be avoided, the standard for describing the
shape of a curve must be the curve itself, and not some other
curve which may happen to be present in the same figure. Thus, the
fact that Plot M lies beneath the linear Plot A does NOT make it
appropriate to call it "sub-linear" -- a term encountered in the
NRC's 1985 report (Gilb85, p.II-102). If Plot A were not in our
figure at all, it would be impossible to guess the meaning of
"sub-linear." By contrast, the term supra-linear is unambiguous
with respect to shape.
As noted repeatedly in this book, another term commonly used
for the supra-linear shape is "concave-downward." Far less common
is the term "upward convex" -- a term encountered in Sho86
(p.693).
A = B at 400 cSv.
Modified LQ Model Yields Linearity :
------------------------------------
Next we will show that, with certain other values of m' and m"
in the cell-inactivation term, the LQ model is also fully
consistent with observing a LINEAR dose-response in human
epidemiological studies.
We turn attention to Table 23-C again, where the linear and
quadratic terms contribute equally to radiation-induced cancer at
400 cSv.
Column D modifies the LQ equation by using the following
values in the cell-inactivation term: m' = 0.0025, and m" =
-0.0000025. The details are in Note 2 of that table.
Column D is shown as Plot M (for Modified) in Figure 23-E.
Except for Plot M, Figure 23-E is exactly the same as Figure 23-D.
It is self-evident, in Figure 23-E, that Plot M (the boxy
symbol) looks linear from the origin out to about 250 cSv.
Comparison of the entries in Table 23-C, Column A versus Column D,
shows just how close the match is between the linear component (A)
and the modified sum of A+B (Column D), out to 200 cSv. Indeed,
Plot M is so very close to Plot A that one cannot distinguish the
two plots from each other in Figure 23-EE, or EEE. In Figure 23-E,
Plots A and M separate enough that, by about 250 cSv, we can
discern that there are actually TWO plots, superimposed on each
other, over most of the dose-range.
Thus, with a shift in the values of m' and m" in the
cell-inactivation term, we can convert the concave-downward curve
in Figure 23-D to a virtually perfect linear dose-response in
Figure 23-E -- while the linear and quadratic terms are preserved
intact.
Indeed, Figure 23-E shows that the effect of the exponential
modifier (the cell-inactivation term), acting upon BOTH the linear
and quadratic terms, can produce a "pseudo-linear" result -- as if
no quadratic term existed at all.
Linearity and Supra-Linearity --
Different Faces of the Same Coin ?
----------------------------------
We pointed out at the end of Part 2 that, if there is a
positive linear term and a positive quadratic term in radiation
carcinogenesis, then, without modification, the sum of these two
types of terms must necessarily give a concave-upward
dose-response.
It follows that observation of a linear dose-response means
that a cell-inactivation term (or some equivalent) must be
operating to convert the concave-upward dose-response to
linearity. In other words, observed linearity is a step on the way
from concave-upward curvature to supra-linearity with just the
intensity of the modification separating linearity from
supra-linearity. Ncrp80 (p.18) also points out that linearity can
be derived in this way, and that several workers have made this
suggestion in the past.
Nonetheless, the insight has been ignored for almost a decade,
while "radiobiology" was mistakenly invoked to support a
concave-upward dose-response and the popular risk
reduction-factors -- in the face of contrary human evidence.
Figures 23-D and 23-E reconcile radiobiology with reality.
They are based on assuming a positive linear term, a positive
quadratic term, and an exponential modifying term.
Next, we will show that the reconciliation is feasible also at
the other limit -- when A = B at 50 cSv.
A = B at 50 cSv.
Modified LQ Model Yields Supra-Linearity
----------------------------------------
We now return to Table 23-B, where the linear term (A) and the
quadratic term (B) are equally prominent at 50 cSv. The unmodified
concave-upward shape of A+B was depicted in Figure 23-B.
We shall modify Column A+B of Table 23-B twice, as we did in
Table 23-C. However, this time we shall illustrate conversion of
Col. A+B into a sigmoid dose-response and then into a supra-linear
dose-response.
Conversion to Sigmoid Shape :
-----------------------------
Column C from Table 23-B has been depicted as Plot M in Figure
23-F, FF, FFF. Otherwise, Figure 23-F and Figure 23-B are the
same.
Figure 23-FF and FFF show that Plot M is concave-upward below
100 cSv. Since dose-response in the A-Bomb Study is NOT
concave-upward in this dose-range, it is clear that the values of
m' and m" tested by Column C are at variance with reality.
Nonetheless, Figure 23-F is provided here in order to help
demonstrate that the method itself is not ruling out the sigmoid
shape. We rule out the sigmoid shape on the basis of the human
epidemiological evidence. The method itself is capable of fitting
just about any dose-response one has ever seen. We shall return to
this point.
Conversion to Supra-Linearity :
-------------------------------
In order to convert Figure 23-F to supra-linearity, we need
only to change the values of m' and m". This has been done in
Table 23-B, Column D.
Column D from Table 23-B has been depicted as Plot M in Figure
23-G, GG, GGG. Except for Plot M, Figure 23-G and Figure 23-F are
the same.
In Figure 23-GGG, the particular pair of values for m' and m"
in Column D happens to make Plot M very nearly linear between 0
and 50 cSv. Beyond 50 cSv, Plot M becomes supra-linear.
It is evident, from Figure 23-GG, that Plot M (which is A+B
modified) diverges from the unmodified Plot A+B at very low doses.
Indeed, Figure 23-GGG shows that, by the time dose rises to 50
cSv, the cell-inactivation term has cut the excess cancer depicted
by Plot M in about half, compared with the unmodified Plot A+B.
A = B at 100 cSv.
Additional Conversions of the LQ Model :
----------------------------------------
Readers need not depend on our assertion that the modified LQ
model is capable of fitting a vast variety of observed
dose-responses.
For instance, Figure 23-I (Eye) shows how merely changing the
values of m' and m" converts the unmodified concave-upward model
(where m' = 0, and m" = 0 also) into the other five shapes shown
on that page. Using Tables 23-A and 23-B as examples, readers
could generate the input and output for all six of those curves by
using A = B at 100 cSv and by using the values of m' and m" shown
in each figure.
A very important point is that shapes in Figure 23-Eye have
actually been reported in the literature for certain
dose-responses.
The shape shown by Figure 23-Eye-5, for instance, closely
resembles the shape of dose-response shown in Ncrp80 Figure 4.12
for specific-locus mutation frequencies versus X-ray dose in mouse
spermatogonia. It also resembles, in shape (not in scale), the
dose-response shown in Ncrp80 Figure 9.1 for the incidence of
myeloid leukemia in male RF mice versus dose, over a wide range of
X-ray or gamma-ray doses.
Figure 23-Eye-6 -- with its initial rise followed by a
flattening which is then followed by a secondary rise -- has the
same pattern as a number of cell-transformation studies, such as
those done by Hall and Miller with C3H10T1/2 mouse cells (Ha81). A
similar shape was reported by Preston and Brewen in studies of
translocations in mouse spermatogonial cells (Pres73).
A Warning about Other Species and about Cell-Studies :
------------------------------------------------------
In the previous chapter, we showed that, (through mid-1989)
the radiation committees have over-ruled direct human
epidemiological evidence on the shape of dose-response for solid
cancers, in favor of generalizations from NON-human evidence and
from cell studies. (Breast-cancer is the significant exception to
this practice.)
The errors which may be inadvertently introduced, by
extrapolating from one species to another, are well known. What
may be less fully appreciated by some readers are the serious
confounding variables even within a single species. So we will
provide an illustration, described by Little (Li81), in which an
experimental dose-response was converted from concave-upward to
linear, by changing the chemical milieu of the irradiated cells.
Little described experiments (Terz76; Kenn78) using mouse
10T1/2 cells to study cell transformation (from normal to
cancerous). When the mouse cells were irradiated with various
doses of X-rays, a clearly concave-upward dose-response was
observed.
However, when the irradiated cells were exposed to the
promoting agent commonly known as TPA
(12-O-tetradecanoyl-phorbol-13-acetate) during the
post-irradiation expression-period, two changes were observed.
First, an enormous increase in transformation-yield per surviving
cell occurred at all radiation doses. And second, the shape of
dose-response changed from concave-upward to perfectly LINEAR.
Such cell-transformation data confirm that the biochemical
milieu in which cell cultures are grown can profoundly influence
the shape of the dose-response which investigators will observe.
This is well known now, and probably explains much of the apparent
inconsistency of results reported from experimental work, even
within a single species.
In short, biochemical milieu makes a huge difference in
dose-response, and if the biochemical milieu of human cells in a
laboratory is unnatural -- and it is -- there is no guarantee that
cells in intact human beings will have the same dose-response
which they have in someone's laboratory.
Without denigrating the valuable experimental work done in
laboratories, we simply remind readers that the only reliable data
on human dose-response for radiation carcinogenesis are
necessarily the human epidemiological data themselves.
5. Proper Risk-Estimates for Low Acute Exposures
=================================================
We have worked our way back, now, to the title of this
chapter.
We have demonstrated that there is no conflict between human
epidemiology (which shows a supra-linear or linear dose-response,
but not a concave-upward one) and the hypothesis from radiobiology
that intra-track carcinogenesis can be expressed by a positive
linear term, and inter-track carcinogenesis by a positive
quadratic term.
The absence of conflict follows from the fact that the
linear-quadratic hypothesis does NOT necessarily predict a
concave-upward dose-response. It can predict ANY of the shapes
(and more) which are depicted in Figure 23-Eye. It can certainly
predict a supra-linear or linear dose-response, as we have shown
-- provided that analysts exclude artificial constraints, such as
the presumption that the cell-inactivation term can operate only
at very high doses.
Curve-Fitting to the Real Evidence :
------------------------------------
There is a great deal of experimental evidence (from other
species and cell studies), as well as the direct human evidence
itself, which confirms that the LQ model needs an exponential
modifier -- as shown in Equation (4) -- in order to fit actual
observations.
It is crucial that the values and signs (positive, negative)
for m' and m" in the exponential modifier be chosen in order to
fit the observations. It is the antithesis of objectivity for
anyone to pre-judge these values, and then to discard
reality-based observations if they do not fit the resulting curve.
Moreover, Figures 23-D through 23-Eye show that the LQ
hypothesis would not be violated if it should turn out that
dose-response for radiation carcinogenesis is concave-downward for
some species (e.g., the human) and concave-upward or some other
shape for other species. Such variation would be perfectly
consistent with species-specific variation in values of m' and m".
It is difficult to understand why this point has not been
emphasized by the radiation committees for the past ten years.
Instead, they have repeatedly suggested that analysts must choose
between "radiobiology" and human epidemiological evidence, as if a
conflict existed. We have shown that radiobiology and epidemiology
can be in complete harmony with each other regarding the LQ
hypothesis.
Risk Increase-Factors Needed :
------------------------------
We and the radiation committees (see for instance Un88, p.415,
para.62) are in agreement that the best human epidemiological
evidence on the shape of dose-response comes from the A-Bomb
Study, for the reasons described in our Chapter 4. And within the
A-Bomb Study, analysis for all cancers combined is, of course, far
more reliable than single-site analysis.
For three consecutive follow-ups (1950-74, 1950-78, 1950-82),
the A-Bomb Study has shown that the dose-response for acute
exposure is not concave-upward. Within the data, dose-response for
all cancers combined is supra-linear throughout the dose-range.
(Beyond 1982, the data are not yet available for anyone to do a
"constant-cohort, dual-dosimetry" analysis.)
The absence of a concave-upward dose-response means that risk
reduction-factors are completely inappropriate for making
risk-estimates at low acute exposures. Supra-linearity means that
risk INCREASE-factors would be needed, if one insisted on
estimating low-dose risks from high-dose data.
Illustrative Risk Increase-Factors :
------------------------------------
Like risk reduction-factors, risk increase-factors derive from
the ratio of the steeper linear slope over the lower linear slope
(see Part 2).
Supra-Linearity, with A = B at 400 cSv :
----------------------------------------
Suppose Figure 23-D described reality, but one had datapoints
only at 160 cSv and at zero dose. A linear interpolation, between
160 cSv on Plot M and the origin, would have the slope of 44,600.9
(from Table 23-C, Column C at 160 cSv) over 160 cSv, or 278.8
cancers per cSv. At low doses, the actual slope of Plot M,
however, is almost identical with the unmodified slope of Plot A.
This can be verified by comparing low-dose entries in Column A
with the corresponding entries in Column C of Table 23-C. The
congruence at very low doses is expected, since the unmodified
Column B makes only a small contribution to cancers, and the value
of the cell-inactivation term is close to 1.0 at low doses (see
Table 23-A).
Thus one can use the slope of the unmodified Plot A to
approximate the slope of Plot M at low doses. The slope is 64,000
/ 160, or 400. The appropriate risk INCREASE-factor, under these
particular circumstances and values of m' and m", would be 400 /
278.8, or 1.43.
Supra-Linearity, with A = B at 50 cSv :
---------------------------------------
Under these circumstances and with the values of m' and m"
used in Table 23-B, the linear slope of Plot M between 160 cSv and
the origin would be 4361.97 (from Table 23-B, Column D) over 160,
or 27.26. The slope of the unmodified Plot A would be 8000 / 160,
or 50. The risk INCREASE-factor would be 50 / 27.26, or 1.83.
No Need to Use "Factors", Up or Down :
--------------------------------------
Section 4 of this book clearly shows that direct observations
exist along the dose-response curve right down to 10-15 cSv (rems)
of acute internal organ-dose. There is simply no need for anyone
to use "factors" (up, or down) to estimate low-dose cancer-risk
from high-dose data.
If "factors" are invoked anyway, we have shown that risk
INCREASE-factors are needed for low acute exposures -- not risk
REDUCTION-factors.
6. Risk-Estimates for Slow Delivery of High Doses
==================================================
The observation of a supra-linear dose-response from acute
exposure rules out risk reduction-factors for low acute exposures,
but supra-linearity does NOT automatically rule out the
possibility that a high dose is less carcinogenic if it is
delivered slowly, than if it is delivered all at once. The various
possibilities receive some quantitative examination in Part 6,
here, for slow delivery of a high dose. Slow delivery of LOW doses
is examined separately, in Part 7.
Lack of Conclusive Evidence for Moderate and High Doses :
---------------------------------------------------------
I am not convinced that existing human epidemiological data
are capable of reliably quantifying a dose-rate effect -- if one
exists at all. It will not be an easy question to settle. If
different human studies involve exposure to different
distributions of gamma or X-ray energies, and if there is no
reliable way to evaluate a valid RBE between such radiations, then
a uncertainty factor of about 2 could be introduced on this basis
alone. Moreover, dosimetry would have to be excellent.
The breast-cancer fluoroscopy studies cited in Chapter 21 are
very reliable for testing whether or not all cancer-response was
eliminated by flawless repair, but that is a totally different
matter from asking those studies to tell us the exact magnitude of
cancer-risk per rem -- well enough to discern a possible effect
from dose-rate.
As for experimental data on a dose-rate effect, they are far
from conclusive. Some results clearly challenge the conventional
assumption that dose-fractionation or very slow delivery reduces
the carcinogenic risk, compared with the risk from acute delivery
of the same dose. For illustrative purposes, we will mention one
such report on cell transformation following irradiation, by Hall
and Miller (Ha81). They describe the complicated responses which
have been observed with fractionation of X-ray dose (emphasis in
the original):
"Experiments with fractionated doses of X-rays indicate that
dividing the dose into two equal fractions separated by 5 hr
results in a decrease of transformation incidence compared with a
single exposure of the same total dose for doses above 1.5 Gy, but
that at lower dose levels splitting the dose ENHANCES
transformation incidence. In a further series of experiments, it
has been shown that the transformation incidence resulting from a
dose of 1 Gy delivered in two, three, or four equal fractions
spread over 5 hr increases progressively with the number of
fractions compared with the same total dose delivered in a single
exposure. The same is true for continuous low-dose-rate
irradiation, where 1 Gy delivered over 6 hr resulted in more
transformations than an acute exposure of the same dose. Thus it
has been clearly demonstrated with this IN VITRO system that the
temporal distribution of dose, in particular its protraction over
a period of time, significantly enhances transformation incidence
at relatively low doses" (Ha81, p.208).
In my opinion, the combination of experimental work, human
epidemiology, and hypothetical considerations, does not yet tell
us whether dose-rate has any effect on human cancer-risk, when the
total dose delivered is moderate or high.
As we shall see in Part 7, however, when the total dose is
low, no basis exists for postulating a reduced cancer-risk from
slow delivery.
Some General Considerations :
-----------------------------
The presumption of the LQ model is that the Q term represents
inter-track carcinogenesis. When we compare Column B (the Q term)
with Column A+B in Tables 23-B and 23-C, we see that -- at low
doses -- inter-track carcinogenesis contributes very little to
total excess cancer. The linear term (single tracks, acting
independently) accounts for virtually all of the radiation-induced
cancer from low acute doses.
Therefore, if a high total dose like 100 cSv is delivered as a
series of low doses -- say, one cSv per exposure -- it is widely
presumed that the cancer-risk per cSv is reduced, because the
inter-track contribution is virtually eliminated (see, for
instance, Ncrp80, p.15).
As we illustrate how this presumption operates, it will become
evident that a great deal depends upon the relative prominence of
the linear and the quadratic terms in the LQ equation. We will
evaluate slow versus fast delivery of 100 cSv, for the case where
A = B at 400 cSv and, separately, where A = B at 50 cSv.
We will illustrate a range of risk reduction-factors, first by
using the common -- but mistaken -- presumption that human
dose-response is concave-upward, and then by using the real-world
observation that it is NOT.
With a Concave-UPWARD
Dose-Response and 100 cSv Total Dose :
--------------------------------------
If A = B at 400 cSv and b = 1.0, then Table 23-C, Column A+B,
shows that excess cancers = 50,000 at 100 cSv of acute dose. The
inter-track term (Column B) is contributing substantially --
10,000 out of 50,000. But the same table shows that cancer-risk
from one cSv of dose is 401, to which the linear term contributes
400 cancers and the inter-track term contributes only 1.0.
We can neglect the inter-track term and can say that the risk,
from 100 doses of one cSv each, would be (100 x 400 cancers), or
40,000 cancers -- instead of 50,000. Therefore, the risk from slow
delivery would be (40,000 / 50,000), or 80 % of the risk from the
same dose acutely delivered.
If A = B at 50 cSv and b = 1.0, then Table 23-B, Column A+B,
shows that excess cancers = 15,000 at 100 cSv of acute dose. The
inter-track term (Column B) is contributing very substantially --
10,000 out of 15,000. But the same table shows that cancer-risk
from one cSv of dose is 51, to which the linear term contributes
50 cancers and the inter- track term contributes only 1.0.
Again we can neglect the inter-track term and can say that the
risk, from 100 doses of one cSv each, would be (100 x 50 cancers),
or 5,000 cancers -- instead of 15,000. Therefore, the risk from
slow delivery would be (5,000 / 15,000), or 33.3 % of the risk
from the same dose acutely delivered.
With a Concave-DOWNWARD
Dose-Response and 100 cSv Total Dose :
--------------------------------------
If one accepts the linear-quadratic hypothesis, it follows
that when dose-response is concave-downward, one is dealing with
Equation (4) -- the modified LQ equation with the
cell-inactivation term. At low doses, the quadratic term virtually
disappears and only the linear term operates -- and so the cell
inactivation term operates essentially on the linear term alone.
But we must ask, "Which cell-inactivation term?" It seems
clear that the appropriate choice of cell-inactivation term is
dependent upon the rate of delivery of the radiation. Therefore,
if 100 cSv is delivered acutely, then the cell-inactivation term,
for any particular choice of m' and m", will be found on the line
for 100 cSv in Table 23-A. However, if 100 cSv is delivered in the
form of 100 doses, each of one cSv, then the cell-inactivation
term, for those same values of m' and m", will be found on the
line for 1 cSv in Table 23-A.
The consequences of these presumptions will be demonstrated
with specific examples.
If A = B at 400 cSv :
---------------------
Excess cancers will be calculated for acute delivery of 100
cSv, and for 100 cSv delivered in 100 separate doses, each of 1
cSv. The entries from Table 23-C, Column C, will be used. The cell
inactivation values will be obtained from Table 23-A, Column D.
Those values for the active fraction of cells remaining are:
For dose = 1 cSv: Active fraction = 0.99502
For dose = 100 cSv: Active fraction = 0.63128
For acute delivery of 100 cSv:
A + B = 50,000 (from Table 23-C)
Excess cancers = (50,000)(0.63128)
= 31564
This value is in Column C of Table 23-C.
The calculation for "slow" delivery -- one hundred separate
doses of 1 cSv -- requires more detail, now provided.
The first dose of 1 cSv has an associated active fraction of
0.99502 . Each subsequent dose also has an associated active
fraction of 0.99502 BUT these subsequent doses are operating on
cells which already have had their active fraction reduced by the
operation of prior 1 cSv doses. Thus, the total active fraction
for the FIRST 1 cSv dose-increment is 0.99502. The total active
fraction for the SECOND 1 cSv dose is (0.99502) x (0.99502), or
0.990064. The total active fraction for the THIRD 1 cSv dose is
(0.990064) x (0.99502), or 0.985134 .
This procedure is repeated 100 total times, and provides the
active fraction left for carcinogenesis from each of the 100
separate 1 cSv doses. The 100th active fraction has the value
(0.99502)^100, or 0.606989. The AVERAGE active fraction, for these
100 doses of 1 cSv each, is 0.785248.
Since each 1 cSv dose provides 401 excess cancers, and since
the average active fraction is 0.785248, it follows that the total
excess cancers from 100 doses will be (100) x (401 cancers) x
(0.785248), or 31,488 excess cancers.
The result is extremely close to the 31,564 excess cancers
produced by a single dose of 100 cSv. The single dose is
associated with a lower modifier (0.63128), operating on a larger
number of cancers (50,000). The serial doses are associated with a
higher average modifier (0.785248), operating on a lower number of
cancers (100 x 401, or 40,100).
No Reduced Cancer-Risk :
------------------------
The conclusion is that there is NO protection (no reduced
cancer-risk) from slow delivery of 100 cSv IN THIS CASE. We stress
"IN THIS CASE" because this result is obtained for the case where
A = B at 400 cSv, and for a particular set of m' and m" values,
chosen to illustrate a concave-downward dose-response.
Intuitively, we are not surprised at the result. When the
linear term is so much more important than the quadratic term --
that is, where B does not reach a contribution equal to that of A
until a dose of 400 cSv -- the lessening of the quadratic response
with dose-delivery in small increments is far smaller than it
would be in the case where A = B at some much lower value, e.g. 50
cSv. Indeed, we shall now explore that case.
If A = B at 50 cSv :
--------------------
Excess cancers will be calculated for acute delivery of 100
cSv, and for 100 cSv delivered in 100 separate doses, each of 1
cSv. The entries from Table 23-B, Column D, will be used. The cell
inactivation values will be obtained from Table 23-A, Column H.
Those values for the active fraction of cells remaining are:
For dose = 1 cSv: Active fraction = 0.98513
For dose = 100 cSv: Active fraction = 0.25666
For acute delivery of 100 cSv:
A + B = 15,000 (from Table 23-B)
Excess cancers = (15,000)(0.25666)
= 3849.9
This value is in Column D of Table 23-B.
For the "slow" delivery -- 100 separate doses of 1 cSv each --
we must go through the same type of iterative procedure as was
done for the case of A = B at 400 cSv.
The active fraction for the FIRST 1 cSv dose-increment is
0.98513. The total active fraction for the SECOND 1 cSv dose is
(0.98513) x (0.98513), or 0.970481 . The total active fraction for
the THIRD 1 cSv dose is (0.970481) x (0.98513), or 0.956049 . This
procedure is repeated 100 total times, and provides the active
fraction left for carcinogenesis from each of the 100 separate 1
cSv doses. The 100th active fraction has the value (0.98513)^100,
or 0.223539 . The AVERAGE active fraction, for these 100 doses of
1 cSv each, is 0.514401 .
Since each 1 cSv dose provides 51 excess cancers, and since
the average active fraction is 0.514401, it follows that the total
excess cancers from 100 doses will be (100) x (51 cancers) x
(0.514401), or 2623.445 excess cancers.
Yes, Reduced Cancer-Risk :
--------------------------
By contrast, the expectation from acute delivery of 100 cSv,
for this case, is 3849.9 excess cancers (above).
Therefore, under these circumstances and assumptions, we would
invoke a risk-reduction factor of (2623.445 / 3849.9), or 0.68,
for slow delivery of 100 cSv.
Summary on Slow Delivery of 100 cSv :
-------------------------------------
We have shown in Part 6 that estimates of risk
reduction-factors, for slow delivery of 100 cSv compared with
acute delivery, are affected not only by the shape of the ACUTE
dose-response, but also by the dose at which the linear and
quadratic terms (A,B) are presumed to be equal.
It may not be appropriate to invoke any risk reduction-factors
at all, for slow delivery. Above, under the concave-downward
dose-response, we provided one illustration where the presumption
of risk reduction-factors would not be warranted under the
linear-quadratic model, and one illustration where it would be
warranted.
Pending more evidence on this issue, we regard it as premature
for anyone to count on a much lower cancer-risk from 100 cSv
slowly delivered than from 100 cSv acutely delivered. As we said
at the outset of Part 6, we do not believe the issue can yet be
settled, for moderate and high total doses, on an objective
scientific basis.
7. Risk-Estimates for Slow Delivery of Low Doses
=================================================
We have already shown, in Section 5 of this book, that there
is no dose or dose-rate which is SAFE, with respect to induction
of fatal human cancer.
However, Section 5 did not examine the possibility that low
doses received slowly might be LESS carcinogenic than the same low
doses received acutely. Now, in Part 7, we shall show why risk
reduction-factors are NOT appropriate for slow delivery of low
total doses.
Doses in the Range of Millirems :
---------------------------------
It is unnecessary to look at this issue below 100 millirems.
Readers who refer back to Chapter 20 (especially Part 3 and Table
20-M) will see that, at the level of the cell-nucleus, a few
hundred millirems can be regarded as the slowest conceivable
dose-rate.
Since in the very low dose-range, we are dealing with
essentially a single track through a nucleus, there is no
difference in dose-rate between a few hundred millirems delivered
all at once, and the same total dose spread out over years. In
both cases, the dose-rate is virtually instantaneous delivery of
the entire dose to those cell-nuclei which receive any dose at
all.
It follows that in the entire dose-range between zero dose and
a few hundred millirems, the issue is already settled about
possible risk-reduction from slow delivery: There is NO reduction
of risk to be considered because there is no reduction in
dose-rate.
Next, we shall consider a somewhat higher dose.
Slow Delivery of 5 cSv :
------------------------
Using the approach demonstrated in Part 6, we will examine the
possible risk-reduction if a total dose of 5 cSv is delivered in
five fractions of one cSv each.
If A = B at 400 cSv :
---------------------
Excess cancers will be calculated for acute delivery of 5 cSv,
and for 5 cSv delivered in 5 separate doses, each of 1 cSv. The
entries from Table 23-C, Column C, will be used. These are entries
for a dose-response which is concave-DOWNWARD between zero and 5
cSv. The cell-inactivation values will be obtained from Table
23-A, Column D. Those values for the active fraction of cells
remaining are:
For dose = 1 cSv: Active fraction = 0.99502
For dose = 5 cSv: Active fraction = 0.97541
For acute delivery of 5 cSv:
A + B = 2,025 (from Table 23-C)
Excess cancers = (2,025)(0.97541)
= 1975.20
This value is in Column C of Table 23-C.
For the "slow" delivery -- 5 separate doses of 1 cSv each --
we must do what we did in Part 6.
The first dose of 1 cSv has an associated active fraction of
0.99502 . Each subsequent dose also has an associated active
fraction of 0.99502 BUT these subsequent doses are operating on
cells which already have had their active fraction reduced by the
operation of prior 1 cSv doses. Thus, the total active fraction
for the FIRST 1 cSv dose-increment is 0.99502. The total active
fraction for the SECOND 1 cSv dose is (0.99502) x (0.99502), or
0.990064.
Since there are only 5 doses of 1 cSv involved, we can show
the full set of active fractions, and the excess cancers
calculated for each 1 cSv dose. Each modifier operates on an A+B
value of 401 cancers per cSv. Therefore, in each line below,
Excess Cancers = 401 x 1 x Active Fraction:
Dose- Active Excess
Increment Fraction Cancers
------------------------------------------------
First 0.995020 399.0030
Second 0.990064 397.0157
Third 0.985134 395.0387
Fourth 0.980228 393.0714
Fifth 0.975346 391.1137
Total Excess Cancers 1975.243
We could have made the same calculation here as we did for the
case of 100 separate 1 cSv doses:
The average active fraction = 0.985158
Excess cancers = 5 x 401 x 0.985158
= 1975.242
For A = B at 400 cSv, the fast delivery and the slow delivery
of 5 cSv yield identical results.
We shall now examine this same comparison, of slow versus fast
delivery of 5 cSv, for the case where the quadratic component is
more prominent.
If A = B at 50 cSv :
--------------------
Excess cancers will be calculated for acute delivery of 5 cSv,
and for 5 cSv delivered in 5 separate doses, each of 1 cSv. The
entries from Table 23-B, Column D, will be used. These are entries
for a dose-response which is slightly concave-UPWARD between zero
and 5 cSv. The cell inactivation values will be obtained from
Table 23-A, Column H. Those values for active fraction of cells
remaining are:
For dose = 1 cSv: Active fraction = 0.98513
For dose = 5 cSv: Active fraction = 0.92807
For acute delivery of 5 cSv:
A + B = 275 (from Table 23-B)
Excess cancers = (275)(0.92807)
= 255.219
This value is in Column C of Table 23-C.
For the "slow" delivery -- 5 separate doses of 1 cSv each --
we can show the full set of active fractions, and the excess
cancers calculated for each 1 cSv dose. Each modifier operates on
an A+B value of 51 cancers per cSv. Therefore, in each line below,
Excess Cancers = 51 x 1 x Active Fraction:
Dose- Active Excess
Increment Fraction Cancers
--------------------------------------------------------
First 0.985130 50.24163
Second 0.970481 49.49454
Third 0.956050 48.75855
Fourth 0.941833 48.03351
Fifth 0.927828 47.31926
Total Excess Cancers 243.8475
We could have made the same calculation here as we did for the
case of 100 separate 1 cSv doses:
The average active fraction = 0.956264
Excess cancers = 5 x 51 x 0.956264
= 243.847
By contrast, the expectation from acute delivery of 5 cSv, for
this case, is 255.219 excess cancers (above).
Therefore, under these circumstances and assumptions, we would
expect cancer-risk from slow delivery to be (243.8474 / 255.219),
or 95.5 % of the risk from acute delivery of the same dose -- a
negligible difference by most standards.
Summary on 5 cSv :
------------------
So at one extreme, where A = B at 400 cSv, the LQ model
suggests no risk-reduction factor at all, and at the other
extreme, where A = B at 50 cSv, the LQ model suggests a risk
reduction-factor of approximately 0.95 .
Slow Delivery of 1.0 cSv :
--------------------------
Lastly, we will temporarily ignore our own remarks, "Doses in
the Range of Millirems," in order to push this other approach
below 1.0 cSv.
We will compare 1 cSv of acute delivery with ten doses of 0.1
cSv each, for the case where A = B at 50 cSv. Since Table 23-A,
Column H, and Table 23-B do not include values for 0.1 cSv, we
provide them below:
Active fraction = 0.998501
(A+B) = (5 + 0.01) = 5.01 cancers
In each line below, excess cancers = 5.01 x Active Fraction.
Dose Active Excess
Increment Fraction Cancers
-----------------------------------------------
First 0.99850 5.00249
Second 0.99700 4.99499
Third 0.99551 4.98750
Fourth 0.99402 4.98003
Fifth 0.99253 4.97256
Sixth 0.99104 4.96511
Seventh 0.98955 4.95767
Eighth 0.98807 4.95023
Ninth 0.98659 4.94281
Tenth 0.98511 4.93540
Total Excess Cancers 49.68880
The average active fraction = 0.991792
So we can check the calculation as follows:
Excess cancers = 10 x 5.01 x 0.991792
= 49.68880
For comparison, the expectation from acute delivery of 1 cSv,
for this case, is 50.2414 excess cancers (from Table 23-B, Column
D).
Therefore, under these circumstances and assumptions, we would
expect cancer-risk from slow delivery of 1 cSv to be (49.6888 /
50.2414), or 98.9 % of the risk from acute delivery of the same
total dose.
In other words, this final fractionation hardly alters the
expected risk at all. Thus this analysis is in good accord with
the conclusion from track-analysis concerning the meaning of
dose-rate at very low tissue-doses (Chapter 20, Part 3).
Slow Exposure -- Validity of Our Cancer-Yields :
------------------------------------------------
Earlier in this book, we have stated that our low-dose
Cancer-Yields are applicable to both acute-low and slow-low
exposure. Those Cancer-Yields are based on the best-fit curve for
acute exposure at 5 cSv (rems) of internal organ-dose. The
analyses above indicate that this conclusion -- applicability to
both acute-low and slow-low exposure -- is well supported by
considerations related to the linear-quadratic model of
dose-response.
Indeed, the expectation that there is no meaningful difference
between acute and "slow" delivery of radiation dose, in the
dose-region of zero to 5 cSv, is consistent with the near
convergence in Table 23-B of the unmodified linear term alone
(Column A) with the modified linear-quadratic term (Column D), and
with the same near convergence in Table 23-C of Column A with
Column C, in this dose-region.
It is not possible to state, within the evidence available
currently, whether the human data are more consistent with A = B
at 400 cSv, or with A = B at 50 cSv. In either case, the analyses
above, in Part 7, indicate that no meaningful error at all will be
introduced by use of our low-dose Cancer-Yields both for acute and
for "slow" delivery of radiation dose.
A Warning about Risk UNDERestimates :
-------------------------------------
By contrast, meaningful underestimates of aggregate
cancer-risk could develop if low doses, slowly received, were
simply ignored -- as currently discussed under "de minimis" and
"below regulatory concern" notions (see Chapter 24, Parts 9 and
10).
As shown by Sections 5 and 6 of this book, we cannot find any
scientific justification within the evidence for excluding such
exposure from risk-estimates or from associated protective
measures. We have shown that the per-rad risk from acute-low and
slow-low exposure is just as great or even greater than the
per-rad risk from acute-moderate or acute-high exposure.
--------------------------------------
--------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Table 23-A
Evaluation of Cell-Inactivation Function in Equation (4),
for Various Pairs of m' and m" Values.
Equation (4): Incidence = [aDose^1 + bDose^2] x [exp-(m'Dose^1 + m"Dose^2)].
See text, Part 3.
----------------------------------------------------------------------------
m'--> 0.005 0.005 0.005 0.005 0.005 0.00001 0.0025 0.015
Dose^1 Dose^2 m"--> 0.000008 0.000004 0 -0.000004 -0.000006 0.000005 -0.0000025 -0.000014
cSv cSv^2 Col. A Col. B Col. C Col. D Col. E Col. F Col. G Col. H
================================================================================================================
0 0 1 1 1 1 1 1 1 1
1 1 0.99500 0.99501 0.99501 0.99502 0.99502 0.99999 0.99751 0.98513
5 25 0.97511 0.97521 0.97531 0.97541 0.97546 0.99983 0.98764 0.92807
10 100 0.95047 0.95085 0.95123 0.95161 0.95180 0.99940 0.97555 0.86191
20 400 0.90195 0.90339 0.90484 0.90629 0.90701 0.99780 0.95218 0.74498
30 900 0.85453 0.85761 0.86071 0.86381 0.86537 0.99521 0.92983 0.64571
40 1600 0.80832 0.81351 0.81873 0.82399 0.82663 0.99164 0.90846 0.56124
50 2500 0.76338 0.77105 0.77880 0.78663 0.79057 0.98708 0.88803 0.48919
60 3600 0.71979 0.73023 0.74082 0.75156 0.75699 0.98157 0.86849 0.42759
70 4900 0.67760 0.69101 0.70469 0.71864 0.72571 0.97511 0.84980 0.37479
80 6400 0.63686 0.65338 0.67032 0.68770 0.69656 0.96773 0.83194 0.32943
90 8100 0.59762 0.61730 0.63763 0.65863 0.66938 0.95945 0.81485 0.29037
100 10000 0.55990 0.58275 0.60653 0.63128 0.64404 0.95028 0.79852 0.25666
110 12100 0.52372 0.54969 0.57695 0.60556 0.62039 0.94026 0.78290 0.22750
120 14400 0.48909 0.51809 0.54881 0.58135 0.59834 0.92941 0.76797 0.20222
130 16900 0.45603 0.48792 0.52205 0.55856 0.57776 0.91778 0.75371 0.18025
140 19600 0.42452 0.45914 0.49659 0.53708 0.55856 0.90538 0.74008 0.16112
150 22500 0.39455 0.43171 0.47237 0.51685 0.54064 0.89226 0.72706 0.14442
160 25600 0.36612 0.40560 0.44933 0.49778 0.52393 0.87845 0.71462 0.12982
170 28900 0.33919 0.38075 0.42741 0.47979 0.50834 0.86398 0.70275 0.11702
180 32400 0.31374 0.35715 0.40657 0.46283 0.49381 0.84891 0.69143 0.10578
190 36100 0.28973 0.33474 0.38674 0.44682 0.48027 0.83327 0.68062 0.09589
200 40000 0.26714 0.31349 0.36788 0.43171 0.46767 0.81709 0.67032 0.08716
210 44100 0.24591 0.29335 0.34994 0.41745 0.45594 0.80043 0.66051 0.07945
220 48400 0.22600 0.27428 0.33287 0.40398 0.44504 0.78333 0.65116 0.07263
230 52900 0.20738 0.25625 0.31664 0.39125 0.43492 0.76583 0.64227 0.06658
240 57600 0.18999 0.23921 0.30119 0.37923 0.42554 0.74796 0.63381 0.06120
250 62500 0.17377 0.22313 0.28650 0.36788 0.41686 0.72979 0.62578 0.05642
260 67600 0.15869 0.20796 0.27253 0.35715 0.40885 0.71134 0.61816 0.05215
270 72900 0.14468 0.19367 0.25924 0.34701 0.40148 0.69267 0.61094 0.04834
280 78400 0.13170 0.18022 0.24660 0.33743 0.39471 0.67381 0.60411 0.04494
290 84100 0.11970 0.16756 0.23457 0.32837 0.38852 0.65482 0.59765 0.04189
300 90000 0.10861 0.15567 0.22313 0.31982 0.38289 0.63572 0.59156 0.03916
310 96100 0.09839 0.14451 0.21225 0.31174 0.37780 0.61656 0.58582 0.03671
320 102400 0.08899 0.13404 0.20190 0.30410 0.37322 0.59738 0.58042 0.03451
330 108900 0.08036 0.12423 0.19205 0.29689 0.36913 0.57822 0.57537 0.03254
340 115600 0.07245 0.11505 0.18268 0.29008 0.36553 0.55912 0.57064 0.03076
350 122500 0.06522 0.10646 0.17377 0.28365 0.36240 0.54010 0.56623 0.02916
360 129600 0.05861 0.09843 0.16530 0.27759 0.35973 0.52121 0.56214 0.02772
370 136900 0.05259 0.09094 0.15724 0.27188 0.35751 0.50248 0.55836 0.02643
380 144400 0.04711 0.08394 0.14957 0.26649 0.35572 0.48394 0.55488 0.02526
390 152100 0.04214 0.07743 0.14227 0.26143 0.35437 0.46561 0.55170 0.02422
400 160000 0.03763 0.07136 0.13534 0.25666 0.35345 0.44754 0.54881 0.02328
| | | |
See Table See Table See Table See Table
23-B, 23-C, 23-C, 23-B,
Column C Column C Column D Column D
================================================================================================================
Each entry evaluates e (e = 2.71828) raised to the power: -(m'D^1 + m"D^2).
For Lotus 123 spreadsheets, each entry has the form:
@EXP(-((m'D^1)+(m"D^2))).
-------------------------------------------------------------------------
Entries are the active fraction remaining (see text, Part 4). When dose = 0,
the active fraction is 100%. It falls with rising dose, in a non-linear
fashion. Figure 23-A depicts Columns A through F.
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Table 23-B
Contributions from Intra-Track Cancers (A)
and Inter-Track Cancers (B) to the
Total Excess Cancers (A+B), When A = B at 50 cSv (rems).
This table provides the input depicted in Figures 23-B, 23-F, and 23-G.
----------------------------------------------------------------------------
Dose^1 Dose^2 Intra-Track Inter-Track Combined m'--> 0.005 0.015
in in Cancers = Cancers = Excess m"--> 0.000004 -0.000014
cSv cSv^2 (50b)(D^1) (b)(D^2) Cancers
[ ( A+B ) x Exponential Term ]
----------------------------------
A B A+B Col. C Col. D
================================================= Concave-Up ======== Sigmoid == Supra-Linear
Fig.23-B Fig.23-F Fig.23-G
0 0 0 0 0 0.00 0.000
1 1 50 1 51 50.75 50.241
5 25 250 25 275 268.18 255.219
10 100 500 100 600 570.51 517.148
20 400 1000 400 1400 1264.75 1042.970
30 900 1500 900 2400 2058.28 1549.711
40 1600 2000 1600 3600 2928.63 2020.477
50 2500 2500 2500 5000 3855.26 2445.961
60 3600 3000 3600 6600 4819.50 2822.067
70 4900 3500 4900 8400 5804.49 3148.203
80 6400 4000 6400 10400 6795.13 3426.043
90 8100 4500 8100 12600 7777.98 3658.659
100 10000 5000 10000 15000 8741.22 3849.912
110 12100 5500 12100 17600 9674.55 4004.020
120 14400 6000 14400 20400 10569.10 4125.284
130 16900 6500 16900 23400 11417.37 4217.894
140 19600 7000 19600 26600 12213.13 4285.817
150 22500 7500 22500 30000 12951.32 4332.728
160 25600 8000 25600 33600 13627.99 4361.980
170 28900 8500 28900 37400 14240.22 4376.594
180 32400 9000 32400 41400 14786.00 4379.265
190 36100 9500 36100 45600 15264.16 4372.381
200 40000 10000 40000 50000 15674.31 4358.043
210 44100 10500 44100 54600 16016.73 4338.089
220 48400 11000 48400 59400 16292.33 4314.124
230 52900 11500 52900 64400 16502.53 4287.542
240 57600 12000 57600 69600 16649.24 4259.554
250 62500 12500 62500 75000 16734.76 4231.210
260 67600 13000 67600 80600 16761.74 4203.421
270 72900 13500 72900 86400 16733.08 4176.977
280 78400 14000 78400 92400 16651.94 4152.568
290 84100 14500 84100 98600 16521.64 4130.801
300 90000 15000 90000 105000 16345.63 4112.209
310 96100 15500 96100 111600 16127.42 4097.271
320 102400 16000 102400 118400 15870.61 4086.418
330 108900 16500 108900 125400 15578.76 4080.047
340 115600 17000 115600 132600 15255.45 4078.532
350 122500 17500 122500 140000 14904.19 4082.226
360 129600 18000 129600 147600 14528.41 4091.477
370 136900 18500 136900 155400 14131.44 4106.629
380 144400 19000 144400 163400 13716.52 4128.034
390 152100 19500 152100 171600 13286.74 4156.055
400 160000 20000 160000 180000 12845.03 4191.073
================================================================================================
1. When A (intra-track cancers) = B (inter-track cancers) at 50 cSv, then
a = 50b (see text, Part 2). 50b is substituted for a, in the equation
of Col.A. In Col.B, the absolute value of the quadratic
dose-coefficient (b) has been set equal to 1.0, to make the
relationship between A and B very clear.
-----------------------------------------------------------------------
2. Column C is depicted as Plot M in Figure 23-F, FF, FFF. The exponential
modifier for Col.C was evaluated in Table 23-A, Col.B. Thus, when we
use 210 cSv as an example, the entry in Col.C above is 54,600 (from the
A+B column at 210 cSv) times 0.29335 from Table 23-A, Col.B., at 210
cSv.
-----------------------------------------------------------------------
3. Column D is depicted as Plot M in Figure 23-G, GG, GGG. The exponential
modifier for Col.D was evaluated in Table 23-A, Col.H. Thus, when we
use 80 cSv as an example, the entry in Col.D above is 10,400 (from the
A+B column at 80 cSv) times 0.32943 from Table 23-A, Col.H, at 80 cSv.
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Table 23-C
Contributions from Intra-Track Cancers (A)
and Inter-Track Cancers (B) to the
Total Excess Cancers (A+B), When A = B at 400 cSv (rems)."
This table provides the input depicted in Figures 23-C, 23-D, and 23-E."
----------------------------------------------------------------------------
Dose^1 Dose^2 Intra-Track Inter-Track Combined m'--> 0.005 0.025
in in Cancers = Cancers = Excess m"--> -0.000004 -0.000025
cSv cSv^2 (400b)(D^1) (b)(D^2) Cancers
[ ( A+B ) x Exponential Term ]
----------------------------------
A B A+B Col. C Col. D
================================================= Concave-Up ============Supra-Linear====Linear==
Fig.23-C Fig.23-D Fig.23-E
0 0 0 0 0 0.00 0.000
1 1 400 1 401 399.00 400.000
5 25 2000 25 2025 1975.20 1999.970
10 100 4000 100 4100 3901.60 3999.770
20 400 8000 400 8400 7612.81 7998.321
30 900 12000 900 12900 11143.18 11994.849
40 1600 16000 1600 17600 14502.18 15988.967
50 2500 20000 2500 22500 17699.13 19980.670
60 3600 24000 3600 27600 20743.14 23970.305
70 4900 28000 4900 32900 23643.13 27958.539
80 6400 32000 6400 38400 26407.75 31946.335
90 8100 36000 8100 44100 29045.39 35934.929
100 10000 40000 10000 50000 31564.18 39925.811
110 12100 44000 12100 56100 33971.97 43920.703
120 14400 48000 14400 62400 36276.32 47921.549
130 16900 52000 16900 68900 38484.52 51930.494
140 19600 56000 19600 75600 40603.58 55949.880
150 22500 60000 22500 82500 42640.24 59982.227
160 25600 64000 25600 89600 44600.96 64030.230
170 28900 68000 28900 96900 46491.97 68096.752
180 32400 72000 32400 104400 48319.23 72184.813
190 36100 76000 36100 112100 50088.49 76297.588
200 40000 80000 40000 120000 51805.26 80438.406
210 44100 84000 44100 128100 53474.84 84610.740
220 48400 88000 48400 136400 55102.32 88818.214
230 52900 92000 52900 144900 56692.61 93064.598
240 57600 96000 57600 153600 58250.45 97353.805
250 62500 100000 62500 162500 59780.41 101689.902
260 67600 104000 67600 171600 61286.90 106077.100
270 72900 108000 72900 180900 62774.21 110519.767
280 78400 112000 78400 190400 64246.49 115022.426
290 84100 116000 84100 200100 65707.77 119589.763
300 90000 120000 90000 210000 67161.99 124226.627
310 96100 124000 96100 220100 68613.00 128938.041
320 102400 128000 102400 230400 70064.55 133729.209
330 108900 132000 108900 240900 71520.34 138605.520
340 115600 136000 115600 251600 72984.02 143572.556
350 122500 140000 122500 262500 74459.18 148636.107
360 129600 144000 129600 273600 75949.39 153802.173
370 136900 148000 136900 284900 77458.18 159076.981
380 144400 152000 144400 296400 78989.10 164466.992
390 152100 156000 152100 308100 80545.67 169978.916
400 160000 160000 160000 320000 82131.45 175619.724
==================================================================================================
1. When A (intra-track cancers) = B (inter-track cancers) at 400 cSv, then
a = 400b (see text, Part 2). 400b is substituted for a, in the equation
of Col.A. In Col.B, the absolute value of the quadratic
dose-coefficient (b) has been set equal to 1.0, to make the
relationship between A and B very clear.
-----------------------------------------------------------------------
2. Column C is depicted as Plot M in Figure 23-D, DD, DDD. The exponential
modifier for Col.C was evaluated in Table 23-A, Col.D. Thus, when we
use 50 cSv as an example, the entry in Col.C above is 22,500 (from the
A+B column at 50 cSv) times 0.78663 from Table 23-A, Col.D, at 50 cSv.
-----------------------------------------------------------------------
3. Column D is depicted as Plot M in Figure 23-E, EE, EEE. The exponential
modifier for Col.D was evaluated in Table 23-A, Col.G. Thus, when we
use 160 cSv as an example, the entry in Col.D above is 89,600 (from the
A+B column at 160 cSv) times 0.71462 from Table 23-A, Col.G, at 160
cSv.
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-A
Cell-Inactivation Functions. Depiction of Columns A through F of Table 23-A.
----------------------------------------------------------------------------
The cell-inactivation function is the term between the righthand pair of
brackets in the modified linear-quadratic Equation (4): Excess Cancers =
[aD^1 + bD^2] x [exp-(m'D^1 + m"D^2)]. The functions depicted here were
evaluated in Table 23-A. Thus "Col.A", "Col.B", etc. in this figure refer to
Column A and Column B in that table.
In this figure, all six curves have a value of 1.0 or 100 % at zero
dose, and the fraction falls below 1.0 in a non-linear fashion as dose
increases. The fractions at 400 cSv of dose in the figure correspond with
the bottom entries in Columns A through F of Table 23-A.
[Figure 23-A]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-B
Unmodified LQ Model of Dose-Response. Contributions from Intra-Track Cancers
(A) and from Inter-Track Cancers (B) to the Total Excess Cancers (A+B), When
A = B at 50 cSv.
Total Excess = aD^1 + bD^2 (see text, Part 2), and a = 50b.
----------------------------------------------------------------------------
Input for these figures comes from Table 23-B :
Column A = Plot A, Column B = Plot B, and Column A+B = Plot A+B.
Depicted: Dose-range out to 400 cSv, out to 200 cSv, and out to 100
cSv.
The dashed slope in Figure 23-BB represents a linear extrapolation from
160 cSv to the origin.
[Figures 23-B, 23-BB, & 23-BBB]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-C
Unmodified LQ Model of Dose-Response. Contributions from Intra-Track Cancers
(A) and from Inter-Track Cancers (B) to the Total Excess Cancers (A+B), When
A = B at 400 cSv.
Total Excess = aD^1 + bD^2 (see text, Part 2), and a = 400b.
----------------------------------------------------------------------------
Input for these figures comes from Table 23-C : Column A = Plot A, Column B
= Plot B, and Column A+B = Plot A+B. Depicted: Dose-range out to 400 cSv,
out to 200 cSv, and out to 100 cSv. The dashed slope in Figure 23-CC
represents a linear extrapolation from 160 cSv to the origin.
[Figure 23-C, 23-CC, 23-CCC]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-D
Example of a Linear-Quadratic Model with a
Supra-Linear Dose Response at Low Doses. A = B at 400 cSv.
----------------------------------------------------------------------------
* Input comes from Table 23-C. Col.A = Plot A. Col.B = Plot B. Col.A+B =
Plot A+B. Col.C = Plot M.
* Col.C (Plot M) is the original equation (Plot A+B), after modification
by a cell inactivation term.
* Comparison of Plot A+B, with Plot M, shows that a cell inactivation
term can convert the shape of a linear-quadratic equation from
concave-UPWARD to concave-DOWNWARD (supra-linear). Figure 23-DDD shows
that Plot A+B and Plot M diverge at very low doses.
* Except for the addition of Plot M, Figure 23-D is the same as Figure
23-C.
[Figure 23-D, 23-DD, 23-DDD]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-E
Example of a Linear-Quadratic Model with a Linear Dose-Response.
A = B at 400 cSv.
----------------------------------------------------------------------------
* Input comes from Table 23-C. Col.A = Plot A. Col.B = Plot B. Col.A+B =
Plot A+B. Col.D = Plot M.
* Col.D (Plot M) is the original equation (Plot A+B), after modification
by a cell inactivation term.
* Comparison of Plot A+B, with Plot M, shows that a cell inactivation
term can convert the shape of a linear-quadratic equation from
concave-UPWARD to LINEAR. Between zero and 200 cSv of dose, Plot A (the
linear term) and Plot M are virtually on top of each other. Compare
entries in Table 23-C.
* Except for the altered nature of Plot M, Figure 23-E is the same as
Figure 23-D.
[Figure 23-E, 23-EE, 23-EEE]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-F
Example of a Linear-Quadratic Model with a Sigmoid Dose-Response.
A = B at 50 cSv.
----------------------------------------------------------------------------
* Input comes from Table 23-B. Col.A = Plot A. Col.B s Plot B. Col.A+B =
Plot A+B. Col.C = Plot M.
* Col.C (Plot M) is the original equation (Plot A+B), after modification
by a cell inactivation term.
* Comparison of Plot A+B, with Plot H, shows that a cell inactivation
term can convert the shape of a linear-quadratic equation from
concave-UPWARD to SIGMOID. Between zero and 130 cSv of dose, Plot M is
concave-upward, and beyond 130 cSv, Plot M is concave-downward.
* Except for the addition of Plot M (and some vertical scaling), Figure
23-F is the same as Figure 23-B.
[Figure 23-F, 23-FF, 23-FFF]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-G
Example Of a Linear-Quadratic Model with a
Supra-Linear Dose-Response at Low Doses. A - B at 50 cSv.
----------------------------------------------------------------------------
* Input comes from Table 23-B. Col.A = Plot A. Col.B = Plot B. Col.A+B =
Plot A+B. Col.D = Plot M.
* Col.D (Plot M) is the original equation (Plot A+B), after modification
by a cell inactivation term.
* Comparison of Plot A+B, with Plot M, shows that a cell inactivation
term can convert the shape of a linear-quadratic equation from
concave-UPWARD to concave-DOWNWARD. Between zero and 20 cSv of dose,
Plot M is very nearly linear, and beyond 20 cSv, Plot M is
concave-downward.
* Except for the altered nature of Plot M, Figure 23-G is the same as
Figure 23-F.
[Figure 23-G, 23-GG, 23-GGG]
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
----------------------------------------------------------------------------
Figure 23-H
Example of a Negative Quadratic Term, in the LQ Model,
Producing a Supra-Linear Curve. A = B at 200 cSv.
See text, Part 2, "LQ Model: Positive vs. Negative Q-Coefficient."
----------------------------------------------------------------------------
* Input for Figure 23-H is like the input for Figure 23-B and 23-C,
except that here, A = B at 200 cSv.
* When both the linear and quadratic terms are POSITIVE, the plot of an
unmodified linear-quadratic equation is necessarily concave-upward,
because it is the sum of the points along Plot A (always a straight
line) and along Plot B (always a concave-upward curve).
* When the quadratic term has a NEGATIVE sign, the plot of an unmodified
LQ equation is necessarily concave-downward (supra-linear), because it
is the points along Plot A (always a straight line) MINUS the points
along Plot B. In the example below, A = B at 200 cSv. Thus at 200 cSv,
Plot (A minus B) would be (200 minus 200), or zero excess cancers -- if
the Q-term (Plot B) were negative.
* A curve is supra-linear if it lies ABOVE a straight line, drawn between
any two points along itself.
* Ordinarily, the sign (positive, negative) of the Q-term is determined
by the evidence, not by the preference of a set of analysts. The sign
(positive, or negative) "falls out" of a regression analysis, in which
analysts permit actual datapoints to "say" which shape fits them best
and which sign is appropriate. [EXCEPTION: The BEIR-3 Committee -- in
its analysis of dose-response for cancer in the A-Bomb Study --
constrained the equation so that the Q-term could NOT turn out
negative: The Q-term was "constrained to be nonnegative" (Beir80,
p.186). See Chapter 22, Part 2.]
[Figure 23-H, 23-HH, 23-HHH]
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Figure 23-Eye
Variety of Dose-Responses Fit by a Single Linear-Quadratic Equation,
with Changes in Its Modifier.
See text, Part 4, "Additional Conversions of the LQ Model."
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* The equation for all six figures here is Equation (4) from Part 3 of
the text: Excess Cancer = [aD^1 + bD^2] x [exp-(m'D^1 + m"D^2)] .
* For all six figures on this page, A = B at 100 cSv. Thus in all the
figures alike, a = 100b, as shown in the text, Part 2. The X-axis is
Dose in cSv, and the Y-axis is Excess Cancers (in thousands) when b =
1.0.
* The ONLY input which changes from one figure to the next is the value
of m' and m". Those values are indicated within each figure.
* Figure 23-Eye-1 represents the unmodified LQ equation, because m' and
m" both are zero.
[Figure 23-Eye {1-6}]
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