[ RadSafe ] ERR query: the exact form of LNT

garyi at trinityphysics.com garyi at trinityphysics.com
Mon Jan 14 14:45:38 CST 2008

So you are saying that the kD approximation is only good for a given cancer type if the 
background probability is low - is that it?

-Gary Isenhower

On 14 Jan 2008 at 12:11, Strom, Daniel J wrote:

Date sent:      	Mon, 14 Jan 2008 12:11:44 -0800
From:           	"Strom, Daniel J" <strom at pnl.gov>
To:             	<radsafe at radlab.nl>
Copies to:      	"Strom, Daniel J" <strom at pnl.gov>, Rainer.Facius at dlr.de,
	blc at pitt.edu
Subject:        	[ RadSafe ] ERR query: the exact form of LNT

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Regarding Bernard Cohen's <blc at pitt.edu> query on excess relative
risk, <Rainer.Facius at dlr.de> wrote:


RELATIVE RISK: Relative risk is a ratio (the risk of cancer for a
given radiation dose, divided by the background risk)

EXCESS RELATIVE RISK: Another way of expressing risk is excess
relative risk. This is figured by subtracting 1 (that is, the
background risk) from the relative risk.


R(D) = R(0) [ 1 + k D ]

RR(D) (relative risk) =: R(D)/R(0)

ERR(D) (excess relative risk) =: RR(D) - 1

ERR(D)= RR(D) -1 = R(D)/R(0) - 1 = k*D;  q.e.d.


What our German colleague writes is correct, but is not the whole
story. Please remember that the [kD] term is, and always has been,
just the first order approximation of the general form of the model as
D approaches 0. 

To understand this, take k to be the probability per unit dose of a
single occurrence, e.g., a single tumor, in an individual. Bear in
mind that there are possibilities of multiple occurrences (e.g., some
people get dozens of skin cancers). The universe of possible outcomes
in an individual is zero, one, two, or more tumors.

The probability of n = zero tumors is 
P(n=0 | k, D) = EXP(-kD). 

The probability of 1 or more tumors (that is, the rest of the universe
of possible outcomes) is P(n>0 | k, D) = 1 - P(n=0 | k, D) = [1 -
EXP(-kD)]. Under the linear, non-threshold (LNT) dose-response model,
[1 - EXP(-kD)] is the exact form of ERR, not simply kD.

The Taylor series expansion for the exponential function is
EXP(-kD) = 1 + (-kD) + ((-kD)^2)/2! + ((-kD)^3)/3! + ...
              ~ 1 - kD if kD << 1.

So, as kD approaches zero, we have the probability of n = 1 or more
tumors P(n>0 | k, D) = [1 -1 -(-kD)] = kD

For common cancers such as skin cancer in Caucasians or other
neoplastic growths, such as thyroid abnormalities, it is necessary to
use the exact form.

- Dan Strom

The opinions expressed above, if any, are mine alone and have not been
reviewed or approved by Battelle, the Pacific Northwest National
Laboratory, or the U.S. Department of Energy.

Daniel J. Strom, Ph.D., CHP
Energy and Environment Directorate, Pacific Northwest National
Mail Stop K3-56, PO BOX 999, Richland, Washington 99352-0999 USA
Overnight: Battelle for the U.S. DOE, 790 6th St., Richland WA 99354
ATTN: Dan Strom K3-56 Telephone (509) 375-2626 FAX (509) 375-2019
mailto:strom at pnl.gov Radiological Sciences and Engineering:
http://radiologicalsciences.pnl.gov/ Brief Resume:
http://www.pnl.gov/bayesian/strom/strombio.htm Online Publications:
http://www.pnl.gov/bayesian/strom/strompub.htm Pagemaster for 
http://www.pnl.gov/bayesian   http://qecc.pnl.gov http://bidug.pnl.gov

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Gary Isenhower, M.S.

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