Suggestions for topics suitable for these Point/Counterpoint debates should be addressed to Colin G. Orton, Professor
Emeritus, Wayne State University, Detroit: email@example.com. 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: firstname.lastname@example.org)
David J. Brenner, Ph.D., D.Sc.
Center for Radiological Research, Columbia University, New York, New York 10032
(Tel: 212-305-9930, E-mail: email@example.com)
Colin G. Orton, Ph.D., Moderator
?Received 27 May 2009; accepted for publication 28 May 2009; published 1 July 2009?
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/
Arguing for the Proposition is
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
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-
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
Arguing against the Proposi-
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
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,
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.
diotherapy, the LQ model is a useful tool to help predict
isoeffects as a function of the total dose, dose/fraction, and
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
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
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
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,
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.
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
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-
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
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-
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??
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
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
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