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LQ

To the Editor,

The linear quadratic (LQ) formalism is now widely used, for analyzing cell survival in vitro,^1-4 for comparing radiotherapy regimens,^5-11 and for studying other radiobiological endpoints.^12-14 Consequently, its foundations are of some importance. In a recent letter,¹⁵ Professor M. Zaider argued that the LQ formalism applied to cell survival has no mechanistic basis, being just empirical. He considers that the formalism fits the relevant cell survival data well, and also passes two mechanistic tests which he imposes, but concludes that agreement between predictions of the formalism and data should be regarded as serendipitous, not mechanistically grounded. His critique has considerable force for high-LET radiations and for very high doses of low-LET radiations, but we argue here that the LQ formalism, when applied to the doses and radiations of primary interest in radiotherapy, does have a significant mechanistic basis.

The formalism describes dose-response relations produced by acute or protracted dose delivery regimens. It considers lethal lesions produced by one or two tracks, i.e. one or two independent radiation events. For example, a lethal point mutation in a vital gene is typically one-track, while a chromosome aberration is often a two-track lethal lesion. The LQ equations express surviving fraction S in terms of coefficients a and b, respectively characterizing the average yield Y of lethal lesions made by one- or two-track action:

S = exp (- Y ), (1a)

where

Y = aD + GbD². (1b)

Here D is the total dose, and G, the generalized Lea-Catcheside dose-protraction factor, is given for any dose rate R(t) (including any sequence of fractions) by¹⁶

(1c)

G systematically accounts for the effects of protracting dose delivery in any way, with G=1 for a single acute dose and G<1 for any other dose-delivery pattern. Essentially, Eq. (1c) describes the situation when one primary lesion (often, but not necessarily, identified with a DNA double-strand break) is made at time t', is subject to first-order repair with rate constant l, and, if not repaired, can interact with another primary lesion made at time t to produce a two-track lethal lesion.

Professor Zaider gives several related reasons for supposing this LQ formalism has no mechanistic basis. First, he considers that the formalism cannot be derived from a particular biophysical model, the theory of Dual Radiation Action¹⁶; we will argue that Eqs. (1abc) can be derived from other common biophysical models. Second, he points out the mechanistic assumption behind Eq. (1a), that the yield of lethal lesions is Poisson-distributed from cell to cell, and maintains that microdosimetric theory shows the distribution cannot be Poisson; we will argue that for low-LET radiations at the doses relevant to radiotherapy the Poisson assumption is a reasonable approximation. Finally, he suggests that at sufficiently large doses the LQ formalism is inconsistent; for low-LET radiations we will calculate the doses involved, and argue that they are much too large to be relevant.

Regarding attempts to derive the LQ formalism from the theory of Dual Radiation Action, the latter¹⁶ represents only one of several different approaches to describing radiobiological damage mechanistically. An approach originally devised by Lea and his coworkers¹⁷ deals instead with the kinetics of damage development. Typical kinetic models use non-linear ordinary differential equations to track the temporal evolution of lesions as a cell gradually repairs or misrepairs initial damage. This highly mechanistic approach has been widely pursued, as summarized in various reviews^18-22. Many different molecular mechanisms have been studied kinetically, such as pairwise misrepair of DNA double strand breaks, direct one-hit induction of lethal lesions, saturable repair pathways in which the repair enzyme system can be overloaded, etc.

It is now known that a surprisingly broad spectrum of such kinetic models lead to the LQ formalism, giving Eqs. (1abc) via first-order time-dependent perturbation theory when the dose or dose rate are not too high, with most clinically and experimentally relevant doses and dose rates allowed.^5,22-24 The various kinetic models supply mechanistic interpretations of the LQ parameters, with a and b usually determined by different damage processing pathways (a conceptual approach somewhat different from that of Dual Radiation Action, in which the ratio a/b is determined solely by physical energy deposition properties and site size^16,25). Thus the LQ formalism (Eqs. 1abc) is mechanistically based on lesion kinetics; can utilize molecular interpretations of a and b; and expresses robust aspects of cellular survival, common to different pathways for production, repair, and misrepair of radiobiological damage.

Professor Zaider also argues, correctly in our opinion, that assuming lethal lesions to be Poisson distributed is a key step in a mechanistic derivation of the LQ equations. Indeed, this assumption, embodied in Eq. (1a), is used in all current models which calculate the fraction of surviving cells from the average yield of lethal lesions. Professor Zaider says that on microdosimetric grounds the distribution cannot be Poisson. For high-LET radiations this point is well taken.²⁶ But for low-LET radiations in the dose range of radiotherapeutic interest, the Poisson assumption is a reasonable approximation. Because kinetic models also use the Poisson assumption, its rationale and the degree to which it holds were examined at length in a recent review.²² For low-LET radiations at doses per fraction less than about 5 Gy it was concluded that deviations from a Poisson distribution for lethal lesions are minor. Indeed Professor Zaider allows that "…differences between the distribution of lesions in the cell and a Poisson distribution ... are undetectably small".¹⁵

Finally, with regard to the inconsistency Professor Zaider considers as arising for the LQ formalism, he shows that the doses involved are greater than [(1/z_Fb) - (a/b)], where z_F is the average specific energy deposited per event per cell nucleus. For cell killing by low-LET radiation, b~0.1 Gy ^-2 or less, and a/b~10 Gy.²² For gamma irradiation and a typical nucleus diameter of ~7 mm, z_F~10^-3 Gy,²⁷ so the lowest dose per fraction at which an inconsistency putatively arises is ~10,000 Gy. Even if a much smaller site diameter of, say, 1 mm were assumed, the inconsistency would still arise only at doses per fraction above ~200 Gy - far beyond the LQ model’s accepted domain of validity²², and far beyond the range of radiotherapeutic relevance.

Thus we would suggest that Professor Zaider’s arguments are quite important for some situations but not for the doses and types of radiations of main interest in radiotherapy. In the latter domain, the LQ formalism is mechanistically grounded, being based on molecular repair and misrepair kinetics. The formalism incorporates dominant, virtually ubiquitous low-LET radiobiological effects, such as curvilinear dose-response relations and dose-rate sparing, into an overall picture, giving radiotherapists and radiobiologists a common ground for quantitative analyses. As befits equations with sufficiently few parameters to be of practical use in radiotherapy, Eqs. (1abc) do involve various intentional oversimplifications, assumptions (including the Poisson assumption), and approximations.²² In contexts where it is appropriate to study more deeply any particular aspect even at the price of introducing extra parameters (which may decrease the formalism’s general utility), the LQ approach can be extended^28-30, in some cases using Monte-Carlo techniques which do not require the Poisson assumption^31,32.

To sum up, the LQ formalism has a mechanistic, molecular kinetics basis. It does have very significant limitations, but in many situations it is the best tool currently available for drawing practical conclusions from fundamental information on radiobiological damage.

a) electronic mail: sachs@math.berkeley.edu

1. M.J. Marchese, M. Zaider and E.J. Hall, "Dose-rate effects in normal and malignant cells of human origin," Br. J. Radiol. 60, 573-576 (1987).
2. P.J. Deschavanne, B. Fertil, N. Chavaudra, and E.P. Malaise, "The relationship between radiosensitivity and repair of potentially lethal damage in human tumor cell lines with implications for radioresponsiveness," Radiat. Res. 122, 29-37 (1990).
3. J.H. Peacock, M.R. De Almodovar, T.J. Mcmillan, and G.G. Steel, "The nature of the initial slope of radiation cell survival curves," Brit. J. Radiol. 24, S57-S60 (1992).
4. G.W. Barendsen, "Parameters of linear-quadratic radiation dose-effect relationships, dependence on LET and mechanisms of reproductive cell death," Int. J. Radiat. Biol. 71, 649-655 (1997).
5. H.D. Thames and J.H. Hendry, Fractionation in radiotherapy, (Taylor and Francis, London, 1987).
6. J.F. Fowler, "The linear-quadratic formula and progress in fractionated radiotherapy," Brit. J. Radiol. 62, 679-694 (1989).
7. B. Jones, L.T. Tan and R.G. Dale, "Derivation of the optimum dose per fraction from the linear quadratic model," Br. J. Radiol. 68, 894-902 (1995).
8. J.H. Hendry, S.M. Bentzen, R.G. Dale, J.F. Fowler, T.E. Wheldon, B. Jones, A.J. Munro, N.J. Slevin NJ and A.G. Robertson, "A modelled comparison of the effects of using different ways to compensate for missed treatment days in radiotherapy," Clin. Oncol. (Royal College of Radiologists) 8, 297-307 (1996).
9. C. Robertson, A.G. Robertson, J.H. Hendry, S.A. Roberts, N.J. Slevin and W.B. Duncan, "Similar decreases in local tumor control are calculated for treatment protraction and for interruptions in the radiotherapy of carcinoma of the larynx in four centers," Int. J. Radiat. Oncol. Biol. Phys. 40, 319-329 (1998).
10. D. J. Brenner and E.J. Hall, "Conditions for the equivalence of continuous to pulsed low dose rate brachytherapy," Int. J. Radiat. Oncol. Biol. Phys. 20, 181-190 (1991).
11. D. Dodds, R.P. Symonds, C. Deehan, T. Habeshaw, N.S. Reed and D.W. Lamont, "A linear quadratic analysis of gynaecological brachytherapy," Clin. Oncol. (R. Coll. Radiol.) 8, 90-96 (1996).
12. M. Zaider, "Evidence of a neutron RBE of 70 (+/- 50) for solid-tumor induction at Hiroshima and Nagasaki and its implications for assessing the effective neutron quality factor," Hlth. Phys. 61, 631-636 (1991).
13. D.J. Brenner, P. Hahnfeldt, S.A. Amundson and R. K. Sachs, "Interpretation of inverse dose-rate effects for mutagenesis by sparsely ionizing radiation," Int. J. Radiat. Biol. 70, 447-458 (1996).
14. E. Schmid, H. Roos, G. Rimpl and M. Bauchinger, "Chromosome aberration frequencies in human lymphocytes irradiated in a multi-layer array by protons with different LET," Int. J. Radiat. Biol. 72, 661-5 (1997).
15. M. Zaider, "There is no mechanistic basis for the use of the linear-quadratic expression in cellular survival analysis," Med. Phys. 25, 791-2 (1998).
16. A.M. Kellerer and H.H. Rossi, "The theory of dual radiation action," Curr. Top. Radiat. Res. Q. 8, 85-158 (1972).
17. D.E. Lea, Actions of Radiations on Living Cells, (Cambridge University Press, London, 1946).
18. C.A. Tobias, "The repair-misrepair model in radiobiology: comparison to other models," Radiat. Res. Suppl. 8, S77-S95 (1985)
19. J. Kiefer, Quantitative Mathematical Models in Radiation Biology, (Springer, New York, 1988).
20. W. Sontag, "Comparison of six different models describing survival of mammalian cells after irradiation," Radiat. Environ. Biophysics 29, 185-201 (1990).
21. B. Fertil, I. Reydellet and P.J. Deschavanne, "A benchmark of cell survival models using survival curves for human cells after completion of repair of potentially lethal damage," Radiat. Res. 138, 61-69 (1994).
22. R.K. Sachs, P. Hahnfeldt, and D.J. Brenner, "Review: The link between low-LET dose-response relations and the underlying kinetics of damage production/repair/misrepair," Int. J. Radiat. Biol. 72, 351-374 (1997).
23. S.B. Curtis, "Lethal and potentially lethal lesions induced by radiation--a unified repair model," Radiat. Res. 106, 252-270 (1986).
24. D. J. Brenner, L.R. Hlatky, P.J. Hahnfeldt, Y. Huang, and R.K.Sachs, "The linear-quadratic and most other common radiobiological models predict similar time-dose relationships," Radiat. Res. 150, 83-88 (1998).
25. H. H. Rossi and M. Zaider, Microdosimetry and its Applications, (Springer-Verlag, Berlin, 1996).
26. N. Albright, "A Markov formulation of the repair-misrepair model of cell survival," Radiation Research 118, 1-20 (1989).
27. P. Kliauga and R. Dvorak, "Microdosimetric measurements of ionization by monoenergetic photons," Radiat. Res. 73, 1-20 (1978).
28. S.L. Tucker and E.L. Travis, "Comments on a time-dependent version of the linear-quadratic model," Radiotherapy and Oncology 18, 155-63 (1990).
29. M. Zaider, "The combined effects of sublethal damage repair, cellular repopulation and redistribution in the mitotic cycle. II. The dependency of radiosensitivity parameters a, b and t₀ on biological age for Chinese hamster V79 cells," Radiat. Res. 145, 467-473 (1996).
30. D. J. Brenner, L.R. Hlatky, P.J. Hahnfeldt, E.J. Hall and R.K. Sachs, "A convenient extension of the linear-quadratic model to include redistribution and reoxygenation," Int. J. Radiat. Oncol. Biol. Phys. 32, 379-390 (1995).
31. D. J. Brenner, "Track structure, lesion development and cell survival," Radiat. Res. 124, S29-37 (1990).
32. A.A. Edwards, V.V. Moiseenko and H. Nikjoo, "On the mechanism of the formation of chromosomal aberrations by ionising radiation," Radiat. Environ. Biophys. 35, 25-30 (1996).