Development of

Probability Distribution of DDREF 

for Solid Cancers

David C. Kocher, A. Iulian Apostoaei,

F. Owen Hoffman, and John R. Trabalka

Oak Ridge Center for Risk Analysis

Oak Ridge, Tennessee

April 11, 2018
Purpose of Study

[1] Develop probability distribution of dose and 

dose-rate effectiveness factor (DDREF) for solid 

cancers for use in future revision of Interactive 

RadioEpidemiological Program (IREP)

Probability distribution of possibly true values to 

represent uncertainty in DDREF

[2] DDREF distribution intended to be suitable for 

use in any risk assessments for solid cancers that 

account for uncertainty

2
What Is a DDREF?

[1] Adjustment factor used in estimating cancer risks 

from exposure to low-LET radiation (photons and 

electrons) at low acute doses or low dose rates

[2] Represents assumption that risk per unit dose (Gy) 

at low acute doses or low dose rates (RL) may differ 

from risk per Gy at higher acute doses (RH)

DDREF defined as RH/RL –

RL = RH/DDREF

RH estimated from studies of Japanese atomic-bomb 

survivors [Life Span Study (LSS) cohort)]

3
When Is a DDREF Used?

DDREF is used in estimating cancer risks at low 

acute doses or low dose rates of low-LET radiation 

whenever linear dose-responses are assumed –

Risk = αD

Use of DDREF means that risk per Gy (α) may 

depend on dose or dose rate

Risks of most specific solid cancers and all solid 

cancers combined in LSS cohort can be described by 

linear models at doses up to about 2–3 Gy

4
Original Bases for Use of DDREF

[1] “True” dose-response from acute exposure to 

low-LET radiation is linear-quadratic (LQ)

Risk = αD + βD2

, β > 0

Dose-response may level off at high doses due to 

effect of cell sterilization [exp(−γD) term]

[2] Risk from chronic exposure at low dose rates is 

determined by linear term in LQ dose-response 

from acute exposure

Firm basis for assumption of LQ model is lacking

5
Illustration of Concept of DDREF for 

Low-LET Radiation

6
Implications of LQ Dose-Responses

[1] When LQ dose-response is assumed, risk per Gy

[(α + βD), β > 0] increases with increasing dose 

(DDREF > 1)

[2] LQ models incorporate dose-dependent DDREF –

DDREF = RH/RL = (α + βD)/α

= [1 + (β/α)D]

β/α referred to as “curvature parameter”

Since LQ model lacks firm basis, other dose-responses 

(e.g., supra-linear, DDREF < 1) not ruled out

7
Solid Cancers vs Leukemias

DDREFs for solid cancers and leukemias should be 

considered separately –

[1] Linear dose-responses for most solid cancers in 

LSS cohort adjusted by possible DDREF at low 

acute doses or low dose rates

[2] LQ dose-responses for all leukemias in LSS 

cohort (incorporate dose-dependent DDREF)

Dose-responses for all leukemias do not inform 

DDREF for solid cancers

8
Why Use a DDREF for Solid Cancers?

[1] Linear fits to dose-responses for solid cancers in

LSS cohort may not describe responses at doses

below limits of detection (< 0.1–0.2 Gy)

Linear fits may conceal small nonlinearities in

dose-responses at low acute doses

[2] Linear fits to dose-responses for solid cancers in

LSS cohort may not describe dose-responses

(risks per Gy) at low dose rates

9
Components of DDREF (1)

DDREF represents two distinct concepts –

[1] Low-dose effectiveness factor (LDEF)

Effect of dose from acute exposure on risk per Gy

Estimated by analyzing possible nonlinearities in

dose-responses from acute exposure

Usually estimated from fits to dose-responses for

all solid cancers in LSS cohort using LQ models

10
Components of DDREF (2)

[2] Dose-rate effectiveness factor (DREF)

Effect of dose rate on risk per Gy

Usually estimated by comparing risks of all solid

cancers or all cancers except leukemias from

chronic exposure of workers or public with risks in

LSS cohort assuming linear dose-responses11
Importance of Uncertainty in DDREF

[1] IREP and other state-of-the-art cancer risk

assessments account for all sources of uncertainty

Objective is to obtain subjective confidence

intervals (CIs) to represent uncertainty in estimated

risks (“state of knowledge”)

[2] Uncertainty in DDREF can be important source of

uncertainty in estimating risks of solid cancers at

low acute doses or low dose rates

12
Probability Distributions of DDREF

in IREP

13
Basic Issues in Developing DDREF 

Distribution for Solid Cancers

[1] Which studies are relevant in developing DDREF 

distribution for solid cancers? 

Human epidemiologic studies only, or inclusion of 

radiobiologic (cell and animal) studies?

Can radiobiologic data be used to quantify (inform) 

uncertainty in DDREF for solid cancers in humans?

Selection of relevant epidemiologic studies

[2] How should selected DDREFs be combined?

What weights should be given to relevant studies?

14
Present Study to Develop DDREF for 

Solid Cancers

Developed state-of-knowledge probability 

distribution of DDREF for solid cancers using data 

from human epidemiologic studies only

Analysis of LDEFs and DREFs for incidence and 

mortality from all solid cancers combined (four 

data sets in all; unique)

DDREFs for specific solid cancers estimated but not 

used in analysis

Distribution intended to apply to specific solid 

cancers when linear dose-responses are assumed

15
Estimation of LDEF from Analyses of 

Dose-Responses in LSS Cohort

Two ways of estimating LDEF from fits to acute 

dose-responses in LSS cohort using LQ models –

[1] LDEF = RH/RL = [1 + (β/α)D] at D = 1 Gy

Based on ratio of coefficients of quadratic and 

linear terms; β/α is curvature parameter

[2] LDEF = αL/αLQ (independent of dose)

αL = coefficient of linear fit to dose-response

αLQ = coefficient of linear term in LQ fit

16
Studies of LSS Cohort Used to 

Estimate LDEFs

[1] Solid cancer incidence (DS02 dosimetry)

BEIR VII Report (2006), two estimates

Preston et al. (2007) – RERF

[2] Solid cancer mortality (DS02 dosimetry)

Little et al. (2008) – Analysis for UNSCEAR, 

two estimates (ERR and EAR models)

Ozasa et al. (2012) – RERF, four estimates

17
Solid Cancer Incidence in LSS Cohort 

(BEIR VII, 2006)

18
Recent Studies of LSS Cohort Not 

Included in Analysis

Studies at RERF using recently revised dosimetry 

system (DS02R1), including sex-specific LDEFs

[1] Solid cancer incidence – Grant et al. (2017)

Includes longer follow-up of cohort

[2] Solid cancer mortality – Cullings et al. (2017)

Analysis of data used by Ozasa et al. (2012); 

longer follow-up of cohort not included

19
Studies Used to Estimate DREFs from 

Comparisons with Risks in LSS 

Cohort (1)

[1] Solid cancer incidence

U.K. workers, Muirhead et al. (2009)

Risk in LSS cohort from Jacob et al. (2009)

Techa River cohort, Davis et al. (2015)

Risk in LSS cohort assuming BEIR VII model

Risks in LSS cohort matched by age at exposure, 

attained age, and male fraction

20
Studies Used to Estimate DREFs from 

Comparisons with Risks in LSS 

Cohort (2)

[2] Solid cancer mortality

U.K. workers, Muirhead et al. (2009)

Risk in LSS cohort from Jacob et al. (2009)

Workers in France, U.K., and U.S. (INWORKS), 

Richardson et al. (2015)

Risk in LSS cohort assuming BEIR VII model

Techa River cohort, Schonfeld et al. (2013)

Risk in LSS cohort assuming BEIR VII model

21
Challenges in Estimating DREFs

[1] Selection of age at exposure and attained age in 

LSS cohort to match age distributions in workers 

or public

[2] Accounting for exposures of workers or public to 

neutrons and alpha particles and nonuniform 

exposures from internal beta emitters

Exposures to lower-energy x rays and 3H beta 

particles (increased biological effectiveness)

[3] Risk transfer between populations (should be 

unimportant for all solid cancers combined)

22
Studies Not Used to Estimate DREFs

[1] Workers in 15 countries (IARC, Cardis et al. 2007)

Positive dose-response due entirely to estimated 

risk in Canadian cohort now believed to be invalid

Non-significant dose-responses in workers in other 

14 countries, some with undefined lower bounds

[2] Mayak workers, Chernobyl emergency workers 

(concerns about reliability of dosimetry)

[3] Any studies with non-significant dose-responses 

(uninformative about uncertainty in DDREF)

23
Interpretations of Data from 

Individual Studies

[1] Reported central values of risks, risk coefficients 

(α, β) and curvature parameters (β/α) assumed to 

be maximum likelihood estimates (MLEs)

[2] Reported MLEs and CIs of risk quantities were 

represented by Weibull distributions with modes 

at MLEs (flexible; allow values < 0)

Normal distributions assumed in some analyses

[3] Our estimates of LDEFs, DREFs, and DDREFs 

are 50th percentiles (medians) and 90% CIs

24
25
Approach to Combining Estimates of 

LDEF and DREF (1)

[1] Approach based on concept of multi-model 

inference (model averaging) (UNSCEAR 2012)

Each estimate of LDEF or DREF considered to 

be distinct model to represent DDREF (not

repeated measurement of same quantity)

[2] Uncertain estimates of LDEF and DREF are 

combined by assigning subjective weights to 

represent their relevance to estimating DDREF

Weights account for quality of underlying studies

26
Approach to Combining Estimates of 

LDEF and DREF (2)

Important property of model averaging –

CI obtained by combining multiple probability 

distributions of LDEFs or DREFs is always wider

than narrowest CI of individual distributions 

Differs from CI using inverse-variance weighting of 

LDEFs or DREFs [statistical (random) uncertainties 

in model fits to dose-responses only]; deflation of 

uncertainty compared with individual CIs

Approach used in meta-analyses of epidemiologic studies

27
Assumptions in Combining Estimates 

of LDEF (1)

[1] Combined LDEF for solid cancer incidence

50% weight to LDEF from Preston et al. (2007)

25% weight to each LDEF from BEIR VII report 

(2006)

Equal weight to results from two expert groups

28
Assumptions in Combining Estimates 

of LDEF (2)

[2] Combined LDEF for solid cancer mortality

25% weight to each LDEF from Little et al. 

(2008)

15% weight to each LDEF from Ozasa et al. 

(2012) at 0–2 Gy colon dose

10% weight to each LDEF from Ozasa et al. 

(2012) at 0–4 Gy shielded kerma

Equal weight to results from two expert groups

29
Assumptions in Combining Estimates 

of DREF (1)

[1] Combined DREF for solid cancer incidence

80% weight to DREF based on risk in U.K. 

workers (Muirhead et al. 2009)

20% weight to DREF based on risk in Techa

River cohort (Davis et al. 2015)

Only public cohort in which risks of all solid 

cancers are estimated; lower weight reflects 

concerns about uncertainties in estimated doses

30
Assumptions in Combining Estimates 

of DREF (2)

[2] Combined DREF for solid cancer mortality

40% weight to DREF based on risk in U.K. 

workers (Muirhead et al. 2009)

40% weight to DREF based on risk from 

INWORKS (Richardson et al. 2015) [data in U.K. 

workers from Muirhead et al. (2009)]

20% weight to DREF based on risk in Techa

River cohort (Schonfeld et al. 2013)

31
Assumptions in Combining Estimates 

of LDEF and DREF

DDREF distribution obtained from distributions of 

combined LDEFs and DREFs for solid cancer 

incidence and mortality assuming –

[1] Relative weights of 2:1 to incidence- vs 

mortality-based LDEFs and DREFs

[2] Equal weights to resulting LDEF and DREF for 

incidence and mortality combined

[3] Distribution truncated to range of 0.2–20 (about 

1.3% of values removed, mostly < 0.2)

32
Final DDREF Distribution 

[1] Properties of probability distribution of DDREF 

Median and CIs –

1.3 (0.47, 3.6), 90% CI

(0.39, 5.6), 95% CI

Probability about 27% to DDREF < 1 and 17% 

to DDREF > 2

[2] Distribution intended to represent state of 

knowledge of DDREF

33
Comparisons with Other Estimates

[1] BEIR VII (2006) – 1.5 (0.91, 2.46), 90% CI

Accounts for data in LSS cohort and animals

Essentially LDEF for solid cancer incidence

[2] Jacob et al. (2009) – Comparisons of risks in 

workers/public and LSS cohort (meta-analyses)

Solid cancer mortality (7 studies; main result) –

DREF = 0.83 (0.53, 2.0), 90% CI

Solid cancer incidence (3 studies) –

DREF = 1.0 (0.65, 2.4), 90% CI

34
35
Dependence of LQ Fit on Dose Range 

(Ozasa et al. 2012; LSS mortality)

36
Harmonic Mean of DDREF 

Distribution

[1] DDREF appears in denominator of equation to 

estimate risks at low acute doses or low dose rates 

Arithmetic mean (average) of uncertain risk at 

low acute doses or low dose rates is proportional 

to reciprocal of harmonic mean (HM) of DDREF

HM = n/Σ(1/ai), HM < arithmetic mean

[2] HM of DDREF distribution = 1.1 (mean = 1.6)

Reduction in average risks at low acute doses or 

low dose rates only about 10% (not ≈ 40%)

37
Estimates of LDEFs from Recent 

Analyses of Data in LSS Cohort (1)

LDEFs for solid cancer incidence (MLEs and CIs) –

[1] Preston et al. (2007), 0–2 Gy to colon –

1.3 (1.01, 1.90), 90% CI

[2] Grant et al. (2017), 0–4 Gy kerma, DS02R1 

dosimetry, longer follow-up –

1.22 (1.01, 1.60), 90% CI

38
Estimates of LDEFs from Recent 

Analyses of Data in LSS Cohort (2)

LDEFs for solid cancer mortality (MLEs and CIs) –

[1] Ozasa et al. (2012), preferred estimate –

1.8 (1.1, 9.6), 95% CI, 0–2 Gy to colon, [1+(β/α)]

[2] Cullings et al. (2017), DS02R1 dosimetry –

2.0 (1.2, 18), 95% CI, 0–2 Gy to colon, [1+(β/α)]

No significant curvature at 0–4 Gy kerma (similar 

to previous analysis using DS02 dosimetry)

39
Sex-Dependence of LDEFs (MLEs)

[1] Grant et al. (2017), solid cancer incidence –

2.3 (M) 1.1 (F) (0–4 Gy kerma)

[2] Cullings et al. (2017), solid cancer mortality –

1.1 (M) 1.1 (F) (0–4 Gy kerma)

2.1 (M) 2.0 (F) (0–2 Gy to colon)

[3] Ozasa et al. (2012), solid cancer mortality –

1.1 (M) 1.1 (F) (0–4 Gy kerma)

4.2 (M) 1.5 (F) (0–2 Gy to colon)

40
Issues with Recent Estimates of LDEF 

Based on Data in LSS Cohort

[1] Dependence of LDEF for solid cancer mortality 

on dose range in analyzing dose-response

Effect of dose range on LDEF for solid cancer 

incidence not reported (but probably small)

Should same dose range be used to estimate 

LDEF and risk per Gy (RH)?

[2] Are LDEFs in males and females significantly 

different; is sex-dependence different for solid 

cancer incidence vs mortality?

41
Recent Estimates of DREFs by ICRP 

Task Group (Shore et al. 2017) (1)

Estimates of DREFs from meta-analyses of studies of 

solid cancer mortality (19) and incidence (3) in 

workers or public; inverse-variance weighting

DREFs (medians and 95% CIs) –

2.8 (1.8, 7.1), mortality only

3.0 (1.9, 7.7), mortality and incidence

Estimates dominated by DREF for solid cancer 

mortality in Mayak workers (weights of 91 and 80%)

42
Recent Estimates of DREFs by ICRP 

Task Group (Shore et al. 2017) (2)

DREFs most comparable to our DDREF distribution 

exclude Mayak workers (medians and 95% CIs) –

0.89 (0.54, 2.5), mortality only

1.9 (1.0, 11), mortality and incidence

1.3 (0.39, 5.6), our DDREF (includes LDEFs)

DREFs based on studies with mean dose < 100 mGy –

0.94 (0.55, 3.3), mortality only (excl. Mayak)

1.7 (0.94, 10), mortality and incidence

43
Critique of Inverse-Variance 

Weighting (Meta-Analyses)

Assumptions in using inverse-variance weighting to 

combine estimated DREFs or LDEFs (meta-analyses)

[1] Each estimate represents direct (repeated) 

measurement of same quantity

[2] Each estimate is free from bias (systematic error), 

or biases in different estimates cancel out

Assumptions generally not met in combining 

estimates from different epidemiologic studies

CIs are too narrow to represent state of knowledge

44
Concluding Remarks (1)

Estimation of DDREF and its uncertainty always

involves important subjective judgments –

Epidemiologic data to be included; approach to 

combining estimates of LDEF and DREF

There is no purely objective approach (“right 

way”) to selecting data and combining estimates

Our approach of subjective weighting based on 

relevance and use of model averaging differs from 

inverse-variance weighting used in meta-analyses

Our analysis and all others are works in progress!

45
Concluding Remarks (2)

[1] Increasing tendency over time toward central 

estimate of DDREF ≈ 1

[2] Unresolved issues with LSS data

Significant dependence of LDEF on sex (?)

Differences in LDEFs for solid cancer incidence 

and mortality (?)

Dependence of LDEF on dose range analyzed

[3] Important challenges in estimating DREFs

[4] Uncertainties in LDEFs and DREFs matter!

46
Concluding Remarks (3)

[1] Our analysis is unique in including LDEFs and 

DREFs for solid cancer incidence and mortality 

[2] Weighting of LDEFs and DREFs based on 

relevance and quality of studies and use of model 

averaging also unique

[3] We believe that both approaches are necessary to 

obtain probability distribution of DDREF that 

represents state of knowledge

Basing DDREF distribution only on statistical 

uncertainties in underlying estimates of risk does 

not fully represent state of knowledge

47
Documentation of Study

Report providing documentation of entire 

study available at –

https://www.cdc.gov/niosh/ocas/pdfs/

dps/orcra-lowletrad-r0.pdf

48