Biology Contributions
The assessment of RBE effects using the concept of biologically effective dose

https://doi.org/10.1016/S0360-3016(98)00364-2Get rights and content

Abstract

Purpose: To modify existing linear-quadratic (LQ) equations in order to take account of relative biological effectiveness (RBE) using the concept of biologically effective dose (BED).

Methods and Materials: Clinically useful forms of the LQ model have been modified to incorporate RBE effects in such a way as to allow comparison between high- and low-LET (linear energy transfer) radiations in terms of similar biological dose units. The new parameter in the formulation is RBEM, the intrinsic (or maximum) RBE at zero dose. The principal assumption (following Kellerer and Rossi; ref. 1) is that high-LET radiation modifies the α-coefficient of damage while leaving the β-coefficient unaltered.

Results: The equations allow a quantitative estimation of how the apparent RBE will change with changes in dose/fraction or dose-rate and of how the magnitude and rate of change is governed by the low-LET α/β ratio of the irradiated tissue. The modifications are applicable to all types of radiotherapy (fractionated, continuous low dose-rate, therapy with decaying sources, etc.). In cases where the normal tissue RBEM is greater than that for the tumor, the revised formulation helps explain why there will be situations where therapeutic index will be adversely affected by use of high-LET radiation. Such clinical advantages as have been observed are more likely to result from favorable geometrical sparing of critical normal tissues and/or the fact that slowly growing tumors may have α/β values more typical of late-responding normal tissues.

Conclusions: The incorporation of RBE into existing LQ methodology allows quantitative assessment of clinical applications of high-LET radiations via an examination of the associated BEDs. On the basis of such assessments high-LET radiations are shown to confer few advantages.

Introduction

The theory of dual radiation action (1) predicts that high linear energy transfer (high-LET) radiation increases the linear (α) component of radiation damage, while the quadratic (β) component remains unchanged. As a consequence it is generally assumed that the “intrinsic” relative biological effectiveness (RBE) of high-LET radiation compared to a reference (low-LET) quality is the ratio of the initial slopes (at zero dose) of the associated cell-survival curves 2, 3. The measured RBE, as determined from the ratio of doses required to achieve a specific endpoint, is dependent on the chosen biological endpoint, and is therefore variable.

In terms of the linear-quadratic (LQ) model, the slope at zero dose is determined entirely by the α-coefficient of cell kill. Using the nomenclature of Carlsson (4) the apparent value of the linear constant (αH) at high LET is related to the low-LET radiosensitivity (αL) by: αHLRBEM where RBEM is the RBE in the highest limit of surviving fraction, i.e., at zero dose.

This article explores how RBEM can be accommodated within the LQ model, using the principal assumption that the β-coefficient of damage is unaltered by changes in LET, i.e., βH = βL.

Section snippets

RBE and fractionated irradiation

Adopting the general LQ symbolism whereby the negative of the logarithm of the surviving fraction is written as E, we have: For single doses (dL) of the reference (low-LET) radiation: ELLdLLdL2 where αL and βL are the respective linear and quadratic coefficients at low LET for the particular cell line.

If a high-LET radiation is used instead: EHHdHLdH2 where αH reflects the higher radiosensitivity to the higher LET radiation and dH is the lower dose consequently required. Then, from Eq. 1

Discussion

The idea that RBE influences predominantly only the linear-dose coefficient in the LQ model is not new. In a study on the interaction between X-rays and 3 MeV neutrons in the skin of mouse foot, Joiner et al. (2) found that increasing the proportion of photon contamination from 11% to 100% reduced both the α- and β-coefficients, by factors of 4.8 and 1.44 respectively (19). Stenerlöw et al. (19) have also shown experimentally that the β-values for some cell lines appear LET-dependent. No

Acknowledgements

We are greatly indebted to Professors Pierre Scalliet, Andre Wambersie, Jack Fowler (Leuven), Dr. Pat Price (London), and Drs. Lowell Anderson and Clifton Ling (New York) for helpful comments and discussions. Dr. Roland Hawkins (New Orleans) is thanked for providing much useful background material in relation to his own work.

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