Page 1

POINT/COUNTERPOINT

Suggestions for topics suitable for these Point/Counterpoint debates should be addressed to Colin G. Orton, Professor

Emeritus, Wayne State University, Detroit: ortonc@comcast.net. Persons participating in Point/Counterpoint discussions are

selected for their knowledge and communicative skill. Their positions for or against a proposition may or may not

reflect their personal opinions or the positions of their employers.

The linear-quadratic model is inappropriate to model high dose

per fraction effects in radiosurgery

John P. Kirkpatrick, M.D., Ph.D.

Department of Radiation Oncology, Duke University Medical Center, Durham, North Carolina 27710

(Tel: 919-668-7342, E-mail: kirkp001@mc.duke.edu)

David J. Brenner, Ph.D., D.Sc.

Center for Radiological Research, Columbia University, New York, New York 10032

(Tel: 212-305-9930, E-mail: djb3@columbia.edu)

Colin G. Orton, Ph.D., Moderator

?Received 27 May 2009; accepted for publication 28 May 2009; published 1 July 2009?

?DOI: 10.1118/1.3157095?

OVERVIEW

The linear-quadratic ?LQ? model is frequently used for mod-

eling the effects of radiotherapy at low and medium doses

per fraction for which it appears to fit clinical data reason-

ably well. It has also been used at the very high doses per

fraction encountered in stereotactic radiosurgery, but some

have questioned such use because there are little clinical data

to demonstrate that the model is accurate at such high doses.

This is the proposition debated in this month’s Point/

Counterpoint.

Arguing for the Proposition is

JohnP.Kirkpatrick,

Ph.D. Dr. Kirkpatrick is an As-

sociate Professor at the De-

partment of Radiation Oncol-

ogy, Duke University Medical

Center. He has a Ph.D. in

ChemicalEngineering

Rice University, Houston, and

an M.D. degree from the Uni-

versity of Texas Health Sci-

ence Center, San Antonio, TX.

His major research interests in-

M.D.,

from

clude treatment of tumors of the central nervous system, base

of skull and spine, stereotactic brain and body radiosurgery,

IMRT and other highly conformal techniques employing spa-

tiotemporal optimization, tumor hypoxia, and quantitative

modeling of the response of malignant and normal tissue to

ionizing radiation.

Arguing against the Proposi-

tionis David

Ph.D., D.Sc. Dr. Brenner is a

Professor of Radiation Oncol-

ogy and Public Health at the

Columbia University Medical

Center. He focuses on devel-

oping models for the carcino-

genic effects of ionizing radia-

tion on living systems at the

chromosomal, cellular, tissue,

and organism levels. He di-

vides his research time roughly

J.Brenner,

equally between the effects of high doses of ionizing radia-

tion ?related to radiation therapy? and the effects of low

doses of radiation ?related to radiological, environmental,

and occupational exposures?. When not involved in radiation

matters, he supports the Liverpool Football Club.

FOR THE PROPOSITION: John P. Kirkpatrick,

M.D., Ph.D.

Opening Statement

The LQ equation is widely used to describe the effects of

ionizing radiation on normal and neoplastic tissue.1In radio-

therapy, we seek death of malignant cells and, more impor-

tantly, control/cure of disease while avoiding damage to the

surrounding normal tissue. In conventionally fractionated ra-

33813381Med. Phys. 36 „8…, August 20090094-2405/2009/36„8…/3381/4/$25.00© 2009 Am. Assoc. Phys. Med.

Page 2

diotherapy, the LQ model is a useful tool to help predict

isoeffects as a function of the total dose, dose/fraction, and

treatment time.

In stereotactic radiosurgery, damage to the tumor is maxi-

mized and injury to normal tissues is minimized by admin-

istering high dose radiation—typically ?12 Gy—to the tu-

mor in a single fraction while limiting irradiation of adjacent

tissue. At high doses per fraction, it is inappropriate to utilize

the LQ equation because the model does not accurately ex-

plain clinical outcomes, is derived largely from in vitro ob-

servations, and does not consider the impact of radioresistant

clonogen subpopulations.

Clinical outcomes from radiosurgery suggest that a single,

high radiation dose is more efficacious than the “biologically

equivalent” total dose calculated from the LQ model for con-

ventionally fractionated radiotherapy.2–4For example, about

10% of patients with arteriovenous malformations ?AVMs?

treated to 42 Gy in 12 fractions ?biologically equivalent to

one 15 Gy fraction based on the LQ model with ?/?

=3 Gy? exhibited obliteration, and the rate of bleeding is not

different than that in untreated patients.4In contrast, single

fraction radiosurgery at 15 Gy yields an obliteration rate of

about 50%.5

The discrepancy between clinical outcomes and predic-

tions based primarily on in vitro cell survival curves may be

related to radiation-induced changes in supporting tissue.

Much of the data used to generate survival curves and esti-

mate LQ model coefficients comes from in vitro cell culture

experiments, typically at doses/fraction well below those

used in radiosurgery. Preclinically, vascular endothelial dam-

age appears to be triggered in vivo above 10 Gy/fraction.6

Pathological studies of malignant and benign human brain

lesions treated with radiosurgery show profound changes in

the vasculature.7,8For example, for the treatment of AVMs,

obliteration of abnormal vasculature and normal tissue dam-

age are rare below 12 Gy but climb steeply above this dose

threshold. Histopathological studies of AVMs show that the

dominant damage following radiosurgery is loss of vascular

endothelial cells, followed by obliteration of lumens.7

While the LQ model assumes an essentially homogeneous

cell population, the tissue microenvironment is, in fact, quite

heterogeneous. Local hypoxia is present in many tumors,

significantly reducing radioresponsiveness of the overall

tumor.9Moreover, tumors contain a subpopulation of cancer

“stem cells” exhibiting enhanced repair of radiation damage,

which may severely limit curability.10Neither heterogeneity

of mechanism nor target population is reflected in the LQ

model.

It is certainly possible to modify the LQ equation such

that the model fits the dose-response curve and then rational-

ize that the addition of a new parameter reveals some funda-

mental mechanism.1However, one should not extend an em-

pirical model outside the data set from which it has been

derived. By truly understanding the underlying mechanism,

we can create a robust model that both informs us clinically

and aids us in formulating new therapeutic strategies.

AGAINST THE PROPOSITION: David J. Brenner,

Ph.D., D.Sc.

Opening Statement

First, the standard LQ model is an approximation to more

exact ?but more complex? models. LQ generally works fine

at doses per fraction below about 15–20 Gy. At higher doses

per fraction, more exact versions of the LQ are available and

can be used. Second, in order to use the standard LQ model

to predict isoeffect tumor-control doses between high dose

single fractions and multiple-fraction regimens, it is impor-

tant to consider that reoxygenation will generally be different

between the two cases. This can be taken into account with

simple extensions to the LQ model.

1. High doses

It has long been known that the linear-quadratic model is

an approximation to a wide range of damage-kinetic models,

which describe the kinetics of DNA double-strand breaks

?DSBs? and other basic lesions.11In such models, DSBs are

resolved either through restitution or binary misrepair. At

typical radiotherapeutic doses, most DSBs are removed by

restitution, which results in the classic linear-quadratic dose

dependence. At very high doses per fraction, binary misre-

pair can dominate, which results in a linear relation between

effect and dose.11Overall, these mechanisms produce a

linear-quadratic-linear dose-response relationship, as has

been pointed out by many authors.11–15

In fact there have been detailed analyses, both experimen-

tal and theoretical, as to the doses below which the standard

LQ approximation is reasonable to use. Experimentally, in

vivo studies have suggested that the LQ works well up to

about 20–24 Gy for a variety of murine end points,1and

Garcia et al.16recently showed that in vitro cell survival

followed the standard LQ up to about 15 Gy. Theoretically,

Sachs et al.11estimated that the LQ approximation would be

reasonable at doses below about 17 Gy and suggested prac-

tical corrections to the LQ model at somewhat higher doses.

In practice, doses per fraction much above ?20 Gy are rela-

tively unusual in radiosurgery, and so corrections to the LQ

model in the relevant dose range are not major and are not

hard to do.11

Of course one cannot rule out the possibility of other

mechanisms, such as vascular endothelial damage contribut-

ing to radiation-induced tumor control. It is not yet clear how

significant such mechanisms are in the clinic, but it is now

clear that such effects are present at both low and high doses

per fraction17and are not uniquely high-dose phenomena.

2. Reoxygenation

Almost all tumors have a hypoxic component, and one of

the main motivations for fractionated radiotherapy is to per-

mit reoxygenation between fractions. Clearly, this cannot

happen with a single fraction, so if the goal is to produce

isoeffect doses for tumor control between a single and a frac-

tionated dose, one needs to model for reoxygenation. A

simple modification to the LQ model that takes reoxygen-

3382 Kirkpatrick and Brenner: Point/Counterpoint3382

Medical Physics, Vol. 36, No. 8, August 2009

Page 3

ation into account is available for such calculations,18al-

though the rationale for treating malignancies with a single

fraction, and thus losing the benefits of reoxygenation, re-

mains unclear.

Rebuttal: John P. Kirkpatrick, M.D., Ph.D.

The most important goal of modeling dose/response data

is to predict clinical outcome. In conventionally fractionated

radiotherapy, there is often a wealth of clinical data at the

dose/fraction of interest and the clinician is justified in using

the linear-quadratic model—or a modified form of this

model—to interpolate response over a limited range. The

prudent clinician, however, will exercise caution when radi-

cally altering a fraction scheme19no matter how compelling

the radiobiological rationale.20In radiosurgery, clinical data

are much more limited. Thus, “radiosurgeons” are faced with

the task of extrapolating their clinical experience at low

doses per fraction to the high-dose/fraction region utilizing a

model with parameters largely derived from in vitro cell sur-

vival curves and small animal experiments.

Dr. Brenner argues that the modified linear-quadratic

model provides a reasonable fit of isoeffect data up to about

20 Gy/fraction but, in most intracranial radiosurgeries, the

maximum tumor dose is above 20 Gy. I will not argue with

the complex mathematical formalisms and biophysics under-

lying these models, though one would be surprised if the

modified models could not fit these data given the large num-

ber of adjustable parameters. However, as these data are typi-

cally based on cell-suspension experiments, they do not re-

flect changes at the tissue level which become more

important as the dose/fraction increases.21Dr. Brenner al-

ludes to “a simple modification to the LQ model” to account

for reoxygenation but spatial/temporal variations in pO2in

the tumor microenvironment are far more complex. And

whataboutthe effects

radiosensitivity/repair, repopulation, and vascular endothelial

damage ?which is qualitatively different from the low dose

response? at radiosurgical doses?21–23

Our present understanding of these mechanisms and their

impact on tumor control and normal tissue complications at

high doses/fraction is inadequate to model clinical isoeffects.

Fortunately, our knowledge on these mechanisms is growing

and it is incumbent on radiobiologists to incorporate this

knowledge into models that not only predict clinical out-

comes at elevated dose/fraction but also lead physicists and

physicians to enhance treatment planning and biochemo-

therapies.

ofheterogeneousinherent

Rebuttal: David J. Brenner, Ph.D., D.Sc.

The heart of this debate can be summed up in Dr. Kirk-

patrick’s suggestion that LQ is merely an empirical, descrip-

tive model. If this were so, one would indeed be very hesi-

tant about using LQ as a guide for designing new

protocols—the calamitous failure of the empirical NSD

model comes to mind here.24But it was not so. In fact almost

all mechanistically based radiobiological reaction-rate mod-

els reduce to the linear-quadratic model if the dose is not too

high.11The LQ approximation to these radiobiological mod-

els is not merely some empirical power series expansion in

dose; rather, it includes11the generalized Lea–Catcheside

factor for protraction-based sparing,

G = ?2/D2??

−?

−?

?

R?t?dt?

t

e−??t−t??R?t??dt?,

?1?

where R?t? is the temporal dose distribution of the radio-

therapy. Equation ?1? provides a mechanistic description of

the interaction of a DSB ?or other primary lesion? made at

time t?, subject to first-order repair with rate constant ?, with

another DSB made at a later time t—hardly the nonmecha-

nistic empirical model that Dr. Kirkpatrick characterizes LQ

to be.

As described above, LQ is indeed a lower dose

??15–20 Gy? approximation of more detailed mechanistic

models, and these more detailed models can certainly be

used if one is interested in effects at, say, 25–30

Gy/fraction.11But this procedure is certainly not the “exten-

sion of an empirical model” that Dr. Kirkpatrick suggests.

Dr. Kirkpatrick spends some time discussing AVM data.

In fact a recent comprehensive analysis25of essentially all

reported dose-response data forAVM obliteration, with doses

per fraction ranging from 4 to 28 Gy, indicated that the data

over the entire dose range were consistent with a standard

sigmoidal LQ-based dose response, with an ?/? value of

about 2 Gy. No evidence here of different mechanisms at

high versus low doses. Likewise the preponderance of evi-

dence suggests that radiation-induced vascular endothelial

damage to malignancies, while its clinical significance re-

mains unclear, also occurs both at low and high doses.17

1D. J. Brenner, “The linear-quadratic model is an appropriate methodology

for determining isoeffective doses at large doses per fraction,” Semin.

Radiat. Oncol. 18, 234–239 ?2008?.

2J. M. Brown and A. C. Koong, “High-dose single-fraction radiotherapy:

Exploiting a new biology?,” Int. J. Radiat. Oncol., Biol., Phys. 71, 324–

325 ?2008?.

3M. Kocher et al., “Computer simulation of cytotoxic and vascular effects

of radiosurgery in solid and necrotic brain metastases,” Radiother. Oncol.

54, 149–156 ?2000?.

4B. Karlsson et al., “Long-term results after fractionated radiation therapy

for large brain arteriovenous malformations,” Neurosurgery 57, 42–49

?2005?.

5J. C. Flickinger et al., “An analysis of the dose response for arteriovenous

malformation radiosurgery and other factors affecting obliteration,” Ra-

diother. Oncol. 63, 347–354 ?2002?.

6Z. Fuks and R. Kolesnick, “Engaging the vascular component of the

tumor response,” Cancer Cell 8, 89–91 ?2005?.

7B. F. Schneider et al., “Histopathology of arteriovenous malformations

after gamma knife radiosurgery,” J. Neurosurg. 87, 352–357 ?1997?.

8G. T. Szeifert et al., “Cerebral metastases pathology after radiosurgery: A

multicenter study,” Cancer 106, 2672–2681 ?2006?.

9B. J. Moeller, R. A. Richardson, and M. W. Dewhirst, “Hypoxia and

radiotherapy: Opportunities for improved outcomes in cancer treatment,”

Cancer Metastasis Rev. 26, 241–248 ?2007?.

10P. Dalerba, R. W. Cho, and M. F. Clarke, “Cancer stem cells: Models and

concepts,” Annu. Rev. Med. 58, 267–284 ?2007?.

11R. K. Sachs, P. Hahnfeld, and D. J. Brenner, “The link between low-LET

dose-responserelations andthe

production/repair/misrepair,” Int. J. Radiat. Biol. 72, 351–374 ?1997?.

12H. H. Rossi and M. Zaider, in Quantitative Mathematical Models in Ra-

diation Biology, edited by J. Kiefer ?Springer, New York, 1988?, pp. 111–

118.

underlyingkinetics of damage

3383Kirkpatrick and Brenner: Point/Counterpoint 3383

Medical Physics, Vol. 36, No. 8, August 2009

Page 4

13D. J. Brenner, “Track structure, lesion development, and cell survival,”

Radiat. Res. 124, S29–S37 ?1990?.

14T. Radivoyevitch, D. G. Hoel, A. M. Chen, and R. K. Sachs, “Misrejoin-

ing of double-strand breaks after X irradiation: Relating moderate to very

high doses by a Markov model,” Radiat. Res. 149, 59–67 ?1998?.

15M. Carlone, D. Wilkins, and P. Raaphorst, “The modified linear-quadratic

model of Guerrero and Li can be derived from a mechanistic basis and

exhibits linear-quadratic-linear behaviour,” Phys. Med. Biol. 50, L9–L13

?2005?.

16L. M. Garcia, J. Leblanc, D. Wilkins, and G. P. Raaphorst, “Fitting the

linear-quadratic model to detailed data sets for different dose ranges,”

Phys. Med. Biol. 51, 2813–2823 ?2006?.

17B. J. Moeller, M. R. Dreher, Z. N. Rabbani, T. Schroeder, Y. Cao, C. Y.

Li, and M. W. Dewhirst, “Pleiotropic effects of HIF-1 blockade on tumor

radiosensitivity,” Cancer Cell 8, 99–110 ?2005?.

18D. 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 redistri-

bution and reoxygenation,” Int. J. Radiat. Oncol., Biol., Phys. 32, 379–

390 ?1995?.

19W. R. Lee, “The ethics of hypofractionation for prostate cancer,” Int. J.

Radiat. Oncol., Biol., Phys. 73, 969–970 ?2009?.

20J. F. Fowler, “Comment on ‘Magical Protons?’ Editorial by Goitein,” Int.

J. Radiat., Oncol., Biol. Phys. 72, 1270–1271 ?2008?.

21J. P. Kirkpatrick, J. J. Meyer, and L. B. Marks, “The L-Q model is inap-

propriate to model high-dose per fraction effects,” Semin. Radiat. Oncol.

18, 240–243 ?2008?.

22S. F. C. O’Rourke, H. McAneney, and T. Hillen, “Linear quadratic and

tumour control probability modelling in external beam radiotherapy,” J.

Math. Biol. 58, 799–817 ?2009?.

23J. J. Kim and I. F. Tannock, “Repopulation of cancer cells during therapy:

An important cause of treatment failure,” Nat. Rev. Cancer 5, 516–525

?2005?.

24T. D. Bates and L. J. Peters, “Dangers of the clinical use of the NSD

formula for small fraction numbers,” Br. J. Radiol. 48, 773 ?1975?.

25X. S. Qi, C. J. Schultz, and X. A. Li, “Possible fractionated regimens for

image-guided intensity-modulated radiation therapy of large arterio-

venous malformations,” Phys. Med. Biol. 52, 5667–5682 ?2007?.

3384 Kirkpatrick and Brenner: Point/Counterpoint3384

Medical Physics, Vol. 36, No. 8, August 2009