# [ RadSafe ] ERR query: the exact form of LNT

Strom, Daniel J strom at pnl.gov
Mon Jan 14 14:11:44 CST 2008

```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.
(http://www.doh.wa.gov/Hanford/publications/overview/risk.html)

Thereby:

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.

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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
Laboratory
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
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