Some implications of linear-quadratic-linear radiation dose-responsewith regard to hypofractionation

Melvin Astrahana�

Department of Radiation Oncology, University of Southern California Keck School of Medicine,1441 Eastlake Avenue, Los Angeles, California 90033

�Received 1 March 2008; revised 18 July 2008; accepted for publication 18 July 2008;published 20 August 2008�

Recent technological advances enable radiation therapy to be delivered in a highly conformalmanner to targets located almost anywhere in the body. This capability has renewed the clinicalinterest in hypofractionation wherein the tumor is delivered a few fractions of very large dose perfraction. Extrapolating clinical experience from conventional regimens to fractions of high dose isimportant to designing hypofractionated treatments. The concept of biologically effective dose�BED� based on the linear-quadratic �LQ� formulation e−��D+�D2� is a useful tool for intercomparingconventional fractionations but may be hampered if the value of � /� is dose range dependentand/or when extrapolating to fractions of high dose because the LQ curve bends continuously onthe log-linear plot. This does not coincide with what is observed experimentally in many clonogeniccell survival studies at high dose wherein radiation dose-response relationships more closely ap-proximate a straight line. Intercomparison of conventional fractionations with hypofractionatedregimens may benefit from BED calculations which instead use a dose range independent linear-quadratic-linear �LQ-L� formulation which better fits the experimental data across a wider range ofdose. The dosimetric implications of LQ-L are explored using a simple model which requires onlythe specification of a dose DT at which the LQ curve transitions to final linearity and the loge cellkill per Gy in the final linear portion of the survival curve at high dose. It is shown that the linetangent to the LQ curve at transition dose DT can often be used to approximate the final slope of thedose response curve. When DT=2� /� Gy, the line tangent to the LQ curve at DT intersects the e−�D

and e−�D2curves at dose � /� Gy and also closely fits the linear response in the high dose region of

some classic in vitro cell survival curves for which the value of � /� is low. It is hypothesized thatDT will increase as the magnitude of � /� increases. Examples are presented illustrating how torecognize LQ-L behavior in multifraction isoeffect studies of late responses such as spinal cord andlung. When planning hypofractionated regimens involving reactions with low � /�, recognizingLQ-L behavior could be important because the dose-response is likely to transition to final linearitywithin the contemplated range of hypofractional doses. © 2008 American Association of Physicistsin Medicine. �DOI: 10.1118/1.2969065�

Key words: biologically effective dose, BED, linear-quadratic, hypofractionation

I. INTRODUCTION

Recent advances such as stereotactic body radiotherapy�SBRT� enable radiation therapy to be delivered in a highlyconformal manner to targets located almost anywhere in thebody. This has renewed clinical interest in hypofractionationwherein the target is delivered a few fractions of very largedose per fraction. Hypofractionation is of perpetual interestto the practice of radiation therapy because of its potential toreduce both the logistical and economic costs of therapycompared to conventional treatment regimens. Generally,however, increasing fraction size has been discouraged1 be-cause when larger fractions are used, late sequelae tend tobecome more severe compared to conventional regimens,even when acute morbidity and tumor response may havebeen matched by an appropriate adjustment in total dose. Theability to extrapolate conventional clinical experience for tu-mor and normal tissues to fractions of high dose will be of

great value when contemplating hypofractionated treatments.

4161 Med. Phys. 35 „9…, September 2008 0094-2405/2008/35

Much of our interpretation of tumor and normal tissueresponse to radiation therapy derives from in vitro andin vivo survival experiments using clonogenic cells of mam-malian and human origin which are exposed to a broad rangeof radiation dose. At present, the most popular method offitting a curve to these experimental results is the linear-quadratic �LQ� formula. The LQ model is based upon theassumption that the biological response to radiation can bedescribed by an equation with two principle components, onethat is proportional to dose and another that is proportional tothe square of the dose. The concept derives from studies inwhich mitotic death of mammalian cells was found to bestrongly associated with asymmetrical exchange-type chro-mosome aberrations �e.g., dicentric or ring� which requiretwo chromosome breaks.2

When the LQ formula is applied to single fraction cellsurvival curves, the surviving fraction �SF� is commonly ex-

pressed as

4161„9…/4161/12/$23.00 © 2008 Am. Assoc. Phys. Med.

4162 Melvin Astrahan: LQ-L implications 4162

SF = e−��D+�D2�, �1�

where D is the fraction dose, the constant � is the loge cellkill per Gy of the linear component, and � is the loge cell killper Gy2 of the quadratic component of the survival curve.

This is illustrated in Fig. 1 where Eq. �1� has been appliedto survival data for Chinese hamster cells in culture exposedto X-rays taken from an early article by Elkind and Sutton.3

The survival curve for these cells exhibits a moderately wideinitial shoulder. Curves for the individual LQ componentse−�D and e−�D2

intersect at the dose where the �D and �D2

components of cell killing are equal. This intersection hap-pens to occur at a dose equal to � /� and is often referred toin the literature simply as � /� with units of Gy. In this workreferences to the unitless ratio of the magnitudes of the con-stants � and � will include the identifier “ratio” �e.g., the� /� ratio�. When � /� appears alone or accompanied by aunit of dose it will implicitly imply the dose at which thecomponent curves intersect.

The LQ formula is not always compatible with cell sur-vival experiments. For example, in Fig. 2, LQ curves �colordashed lines� have been superimposed on a compilation ofsurvival curves for asynchronous cultures of a number of celllines of human and rodent origin. These data �black symbolsand lines� were digitized from Fig. 3.9A in Hall’s radiobiol-ogy textbook.2 Solutions for � and the ratio � /� can befound which allow the LQ equation to closely fit most of theexamples for doses up to 6 Gy, but the LQ model is incom-patible the two most radioresistant examples, EMT-6 mouse

tumor and a glioblastoma of human origin, which are char-

Medical Physics, Vol. 35, No. 9, September 2008

acterized by very broad shoulders. For the glioblastoma data�black squares�, � was estimated by setting the ratio of � to� to a very large number �e.g., 100� and then finding thevalue of � for which the LQ curve passes through the firstdata point at about 2 Gy with the result ��0.05 Gy−1. Thered dashed lines are curves for �=0.05 Gy−1 and variousvalues of the ratio � /�. There exists no solution for the ratio� /� that allows the LQ formula �1� to adequately fit thesedata. Although imperfect, the LQ model does closely fit ex-perimental cell survival and multifraction isoeffect results farmore often than not and therefore has become a popularmethod of modeling radiation dose-response.

The concept of biologically effective dose �BED� basedon the LQ model2,4 has provided a fairly reliable tool forintercomparing fractionation regimens in vivo1–8 that employnominally conventional fraction sizes up to about6 Gy/fraction. For the calculation of isoeffective relation-ships the biological effect �E� per fraction �n� of fractionaldose �D� can be expressed as En= ��D+�D2� and the bio-logically effective dose per fraction is BEDn=En /�. It isgenerally assumed that the fractions are well separated intime so that full repair occurs between fractions and that thebiological effect of each fraction is the same. The formula fortotal BED for n well separated fractions of D per fraction canbe written

BED = nD + �nD2/��/��� , �2�

where � /� implies the dose at which the LQ components−�D −�D2

FIG. 1. On the log-linear plot, the LQ curve closely fitsthese experimental results for Chinese hamster cells inculture up to a dose of 6 Gy, but then continues tobend. The experimental results are observed to becomelinear at high dose.

e and e intersect. It is conventional to express D and

4163 Melvin Astrahan: LQ-L implications 4163

� /� in units of Gy and BED in units of Gy�/� to indicate thereactions to which the result apply.

In addition to being incompatible with the examples ofcell survival characterized by broad shoulders in Fig. 2, thefundamental LQ formulation also exhibits additionalweaknesses,9–12 especially when applied to the high dose re-gion of clonogenic cell survival curves. For instance, Garciaet al.9 recently pointed out that when � /� ratios were deter-mined by fitting the LQ formula to the experimental resultsof four different cell lines in vitro, the resulting solutionsvaried significantly as a function of the dose range overwhich the curve was fit. If the ratio � /� is dose range de-pendent, then BED calculations will also be dose range de-pendent because BED is a function of the dose � /� Gy. Thiscomplicates use of the BED concept for intercomparing dosefractionation regimens. To keep BED calculations simple, adose-range independent approximation for the dose � /� Gyis desirable.

Another characteristic of the formulation of Eq. �1� is thatthe resulting dose-response curve bends continuously on thelog-linear plot �LLP�. This does not coincide with what isobserved experimentally in many clonogenic cell survivalstudies where the dose-response relationships exhibit an ex-ponential decrease of survival at high dose which moreclosely approximate a straight line on the LLP.2,3,13–15 There-fore, in the example illustrated in Fig. 1, the LQ formulationoverestimates the magnitude of cell kill, or, equivalently, un-derestimates the surviving fraction for doses greater than6 Gy. Should in vivo dose-response relationships behavesimilarly, then the LQ formulation will likewise overestimatein vivo response and thus underestimate the dose required toachieve a desired response for fractions of high dose.

These weaknesses of the LQ model could affect the accu-racy of BED calculations if the reactions of tumor, normaltissues, or anatomic organs mimic the dose-response charac-teristics observed for cell survival studies.

To better address high dose per fraction regimens Guer-rero and Li10 proposed a modification to the LQ model whichprovided a better fit at high doses. Their model was noted byCarlone et al.14 to exhibit linear-quadratic-linear �LQ-L� be-havior. Wang et al.11 recently presented a “generalized LQ�gLQ�” formula which also exhibits linear-quadratic-linearbehavior. The formulas employed in these works attempt tomodify the LQ model from a partially mechanistic perspec-tive and are fairly complex. Park et al.12 recently proposed abipartite “universal survival curve” hybrid of the LQ and thehistoric multi-target �MT� model13 in order to better fit cellsurvival experiments. In their implementation the LQ modelswitches to the MT model at a specific transition dose whichthey coined as DT. An inconvenience of this approach is thatthe parameters used by the MT model �n, Dq, and D0� aredetermined by extrapolating the final linear portion of thesurvival curve back to the y axis intercept while the rangedependent LQ terms � and � are determined by fitting theLQ formula to some specified range of the experimentaldata. In order to smoothly fit their experimental data Park

12

et al. solved for � in terms of the other variables and then

Medical Physics, Vol. 35, No. 9, September 2008

solved DT as a function of �, Dq, and D0. They demonstratedthe ability of their model to fit only a single example ofclonogenic cell survival in vitro for a cell line with DT ofabout 6 Gy. One can achieve the same type of LQ-L fit toexperimental data more efficiently by eliminating the MTformalism and simply specifying the loge cell kill per Gy inthe final linear portion of the survival curve and a dose DT atwhich it begins. This generalization eliminates having tosolve n or Dq or � and makes it easier to compute isoBEDcurves and identify LQ-L dose-response behavior in situa-tions where cell survival curves are not applicable such asthe late reactions of spinal cord for which only the ratio � /�can be derived from isoeffect data.

Many extensions to the basic LQ formulation have beendeveloped in an attempt to account for temporal phenomenasuch as repair of sublethal damage and repopulation whichcan occur during lengthy fractionated regimens.2,5,6,10,11,14

For hypofractionated SBRT repopulation will likely not bean issue. Nevertheless, the same methods developed to ad-dress these time related issues could be applied in an identi-cal manner to the model developed here.

II. METHODS AND MATERIALS

In this work a more efficient form of the bipartite methodproposed by Park et al.12 is employed to model LQ-L dose-

FIG. 2. The LQ formula does not adequately model all cell survival experi-ments. Here the LQ model has been applied to a compilation of survivalcurves for asynchronous cultures of a number of cell lines of human androdent origin digitized from Fig. 3.9A in Hall’s2 textbook. The LQ model isnot compatible with the radioresistant examples that exhibit very broadshoulders. See text for further explanation.

response. If ever used clinically, the herein proposed LQ-L

4164 Melvin Astrahan: LQ-L implications 4164

notation should include subscripts for the dose � /� and thedose DT at which the curve becomes linear, perhaps in theform LQ�/�-LDT

. For example, LQ3-L6 would describe amodel which transitions from LQ with � /�=3 Gy to a lineartail for fraction doses greater than 6 Gy.

To transition to an exponential decrease of survival athigh dose the LQ model for single fraction survival can beextended to a bipartite form:

SF = e−��D+�D2� for D � DT �3a�

and

SF = e−��DT+�DT2+��D−DT�� for D � DT. �3b�

For plotting on the LLP, this can equivalently be ex-pressed as

loge�SF� = − ��D + �D2� for D � DT �4a�

and

loge�SF� = − ��DT + �DT2 + ��D − DT�� for D � DT

�4b�

where � is the loge cell kill per Gy in the final linear portionof the survival curve at high dose. The term � is reminiscentof D0, the dose that reduces survival by 1 /e in the final linearportion of the MT survival model, but is more in keepingwith the notation and conventions of the LQ model.

For the bipartite model, the formulation for BED per frac-tion becomes:

BEDn = D + �D2/��/��� for D � DT �5�

and

BEDn = DT + �DT2/��/��� + ���/���D − DT�� for D � DT,

�6�

For intercomparing fractionation regimens, the bipartite formof the BED formula above has the minor disadvantage in thatit requires the ratio � /� in addition to the dose � /�. Imple-mentation of BED calculations would be simpler if an ap-proximation of the ratio � /� could be derived from the dose� /�.

Let us hypothesize that for those dose-response studieswhich the LQ formula fits closely and which smoothly tran-sition to linearity there will exist some dose DT at which �can be approximated as the slope of the line tangent to theLQ curve at dose DT. The concept is illustrated by the solidorange line in Fig. 2 which is the tangent to the LQ solution�dashed orange line� for the human ovary cell line �blacktriangle symbols� assuming DT=5 Gy. The slope of the tan-gent can be calculated from the derivative as

� = �� + 2�DT� = ��1 + �2DT/��/���� . �7�

From the tangent at DT, � /� is

�/� = 1 + �2DT/��/��� . �8�

Assuming that a dose-range independent approximation for

� /� is available, substituting Eq. �8� into Eq. �6� allows

Medical Physics, Vol. 35, No. 9, September 2008

BED to be estimated by specifying just one additional term,the transition dose DT. This is simpler than the method pro-posed by Park et al.,12 which required several terms.

Software was developed which allowed interactive ma-nipulation of the �, � /�, and DT parameters of the LQ-Lmodel for rapid curve fitting by inspection to digitallyscanned images of published dose-response and multifrac-tion isoeffect plots.

III. RESULTS

In Fig. 3, the LQ-L model has been fit by interactive in-spection to examples of published cell survival obtained invitro using colony forming assays and in vivo by histologicalexamination. Figure 3�a� plots surviving fraction for thesame classic Chinese hamster data as in Fig. 1. When �=0.163 Gy−1 and � /�=2.6 Gy, the LQ curve in Fig. 1 fitsthe experimental results closely up to about 6 Gy. In Fig.3�a� the tangent �blue line� to the LQ curve at DT=2� /� Gy�5.2 Gy� intersects the e−�D and e−�D2

curves �green lines� atdose � /� Gy and also closely fits the final linear portion ofthe experimental results. The resulting LQ2.6-L5.2 curve isplotted in red.

In Fig. 3�b� the LQ-L model �red dashed line� has beensuperimposed onto a classic survival curve �black� for hu-man HeLa cells in culture that was digitized from a figure inHall’s textbook,2 which was reprinted from Ref. 15. The sur-vival curve for these cells has a small initial shoulder. For� /�=0.9 Gy and again with DT=2� /� Gy �1.8 Gy� theLQ-L model superimposes almost perfectly over the blackline drawn by Puck and Markus15 across the entire experi-mental dose range. The component curves for e−�D and e−�D2

are plotted in green. The tangent to the LQ-L curve at DT isplotted in blue. The three curves intersect at dose � /� Gy.For historical interest, values are also shown for the MTmodel extrapolation number �n�, quasithreshold dose �Dq�,and final slope �D0� that resulted from the LQ-L curve fit.

In Fig. 3�c� the standard LQ formula �1�, with �=0.208 Gy−1 and � /�=12.0 Gy, has been superimposedonto in vivo survival data for mouse testes stem cells thatwere digitized from another figure in Hall’s textbook2 whichwas, in turn, reprinted from Ref. 16. In the dose range stud-ied it is not clear that these results ever achieve final linearitybecause the LQ curve �dashed red line� appears to fit the datain the high dose region just as well as the black line origi-nally drawn by Thames and Withers.16 If survival curves forcells with high � /� follow a pattern similar to the examplesfor low � /� in Figs. 3�a� and 3�b�, then one can hypothesizethat the LLP linear region for these cells will begin at a doseof about 24 Gy �i.e., �2� /��, which is well beyond therange of these experimental data.

In Fig. 3�d� an LQ-L model with �=0.045 Gy−1, � /�=1.6 Gy, DT=11.0 Gy, and � the tangent at DT has beensuperimposed onto survival data for C3H /10T1/2 cells in cul-ture that were digitized from Chart 2 in Ref. 17. C3H /10T1/2

is a mouse embryo-derived cell line which is fibroblastic innature and which is often used to study oncogenic transfor-

mation. The survival curve is characterized by a very broad

4165 Melvin Astrahan: LQ-L implications 4165

shoulder followed by a sharp bend at about 7 Gy. In fact, theresults appear more bilinear on the LLP rather than linear-quadratic.

The LQ-L model can also fit survival data such as theseby explicitly specifying solutions for DT and �. For instance,in Fig. 3�e� the LQ-L model with �=0.11 Gy−1, � /�=9.6 Gy, DT=7.0 Gy, and �=0.654 Gy−1 better fits the samesurvival data as Fig. 3�d�. An alternative approach that elimi-nates the need to explicitly solve for � is to multiply theslope of the tangent at DT by a constant k in which case Eq.�8� becomes

�/� = k�1 + �2DT/��/���� . �9�

In Fig. 3�e� the same LQ-L curve results when one uses thetangent at DT and k=2.42. The advantage of Eq. �9� is that itmakes BED calculations simpler to implement.

In Fig. 3�f� the LQ-L model is shown to be able tobetter adapt to the radioresistant examples of Fig. 2 thanwas the LQ model. The LQ-L solution used to calculate

−1

FIG. 3. �a� The line tangent to the curve at DT=2� /� Gy fits the final linearin Fig. 1. �b� An LQ-L curve �red dashed line� with DT=2� /� Gy superimp15� for HeLa cells in culture. �c� An LQ curve �red dashed line� superimposedon the curve refer to the number of fractions used to reconstruct that portionrange studied one can not determine if the response has yet become lineartangent at DT superimposed over survival data for mouse C3H/10T1/2 cellsThe LQ-L model �red dashed line� can better fit these data when � and DT acan also adequately fit the radioresistant examples from Fig. 2.

the red curve is �=0.0005 Gy , � /�=0.02 Gy, DT

Medical Physics, Vol. 35, No. 9, September 2008

=6.9 Gy, �=0.667 Gy−1 �or k=1.93� and for the bluecurve �=0.0001 Gy−1, � /�=0.0058 Gy, DT=6.8 Gy, �=0.565 Gy−1 �or k=2.41�.

In Fig. 4, LQ-L solutions are superimposed over in vitrosurvival data for DU145 and CP3 cells that were recentlypublished as Fig. 1 in Garcia et al.9 Both cell lines are ofhuman prostate origin. DU145 was isolated from a brain me-tastasis by Stone et al.18 Leith et al.19 studied this cell line invitro up to doses of 6 Gy and reported radiation survivalparameter values of �=0.155 Gy−1 and � /�=2.97 Gy forthe linear-quadratic model. Garcia et al.9 demonstrated arange dependence to the LQ fit for these cell lines. To maketheir results more meaningful, they elected to divide the doserange in their study into regions of low-dose �LR�, linear-quadratic �LQR�, and high dose �HR�. In Fig. 4�a� the bluedashed line is the best LQ-L fit to the DU145 data that wasachievable by inspection with the constraint that the value of� /� has somehow been determined from a previous study

n of the experimental results for the Chinese hamster cells in culture shownalmost perfectly over the curve originally drawn by Puck and Markus �Ref.r in vivo survival data for mouse testes stem cells from Ref. 16. The numbershe curve from large doses delivered as multiple small fractions. In the doseh dose. �d� An LQ-L curve �red dashed line� with � constrained to be thelture from Han Ref. 17. These cells exhibit an almost bilinear response. �e�constrained. �f� The unconstrained LQ-L model �red and blue dashed lines�

portiooses

oveof t

at higin cure un

�e.g., Ref. 19� and must be exactly equal to 3 Gy, and then

4166 Melvin Astrahan: LQ-L implications 4166

assuming the LQ-L approximations of DT=2� /� Gy and �the tangent at DT. The red dashed line only constrains � to bethe tangent at DT; the values of � /� and DT are allowed toadapt to better fit the data. In Fig. 4�b�, the blue dashed lineis an LQ-L solution for the CP3 data with the single con-straint that DT=2� /� Gy; the values of � /� and � wereunconstrained. The red dashed line is a solution with thesingle constraint that � be the tangent at DT. The two curvesare almost indistinguishable across the experimental datarange.

The point here is that the LQ-L model is flexible enoughto closely fit the entire range of experimental data, even ifvarious simplifying assumptions, approximations, con-straints, or combinations thereof are enforced. The solution

FIG. 4. Comparison of LQ-L models superimposed over recently publishedThe blue dashed line is the best LQ-L fit achievable by inspection with the tdashed line only constraints � to be the tangent at DT. �b� The blue dashedof � /� and � are unconstrained. The red dashed line is an LQ-L solution wsimilar, but not identical results when used to intercompare BED and fracti

TABLE I. Fractionation regimens calculated to have BED equivalent to a stan� /� in the range 2–11 Gy and assuming DT=2� /� Gy and � the tangent a

� /��Gy� BED�/�

LQ

20Fractions

of�cGy�

10Fractions

of�cGy�

5Fractions

of�cGy�

2 120.0 Gy2 261 400 6003 100.0 Gy3 265 418 6394 90.0 Gy4 269 432 6729 73.3 Gy9 280 479 78410 72.0 Gy10 281 485 80011 70.9 Gy11 282 490 815

Medical Physics, Vol. 35, No. 9, September 2008

for � /� will be dose range independent so that BED for aparticular reaction can be intercompared across the entiredose range. The different simplifications, however, generateslightly different solutions for � and hence the solution for� /� will vary as well. This could affect BED calculationsbecause the BED formula requires the � /� term. The resultsof Fig. 4 suggest that variations in � /� of �1 Gy might beexpected depending upon which, if any, simplifying assump-tions were used to fit the data.

Calculations were performed to explore how approximat-ing the value of � /� would affect the intercomparison offractionation regimens. Table I compares fractionations cal-culated to be the BED equivalent of 30 fractions of200 cGy/fraction for � /� in the range 2 to 11 Gy using the

9� survival curves for two human prostate carcinoma cell lines in vitro. �a�constraints that � /�=3 Gy, DT=2� /� Gy and � the tangent at DT. The reds an LQ-L solution with the single constraint that DT=2� /� Gy, the valuese single simplification that � be the tangent at DT. The two solutions yieldn regimens.

course of 30�200 cGy / fraction according to the LQ and LQ-L models for.

LQ-L

3ractions

of�cGy�

2Fractions

of�cGy�

5Fractions

of�cGy�

3Fractions

of�cGy�

2Fractions

of�cGy�

800 1000 640 960 1360861 1084 640 907 1240914 1156 672 920 1220

1100 1421 784 1100 14211128 1462 800 1128 14621154 1500 815 1154 1500

�Ref.ripleline iith thonatio

dardt DT

F

4167 Melvin Astrahan: LQ-L implications 4167

LQ model and the LQ-L model with the approximationsDT=2� /� Gy and � the tangent at DT. Note that the resultscalculated using the LQ-L model for � /� in the range9 to 11 Gy are identical to the LQ results because under theassumption DT=2� /� Gy, the two models do not differ untilfraction doses exceed 18 to 22 Gy. In Tables II and III theresults of Table I have been normalized to the dose per frac-tion calculated for � /�=10 Gy and 3 Gy respectively.

In Table II we see that for high � /� in the range10�1 Gy, all results differ by less than �3% from the doseper fraction for � /�=10 Gy. These results suggest that ap-proximating � /� to within about �1 Gy may be sufficientfor clinical calculations involving high � /�. In Table III, forlower � /� in the range 3�1 Gy, the differences are a littlegreater than were seen in Table II. For nominally conven-tional regimens employing more than 5 fractions, approxi-mating � /� to within about �1 Gy may be sufficient forclinical calculations. The main point here is that extrapolat-ing clinical experience accumulated with conventional regi-mens to hypofractionation will probably have its greatest un-certainty when the magnitude of � /� is low.

In Fig. 5 isoBED curves calculated using the LQ �coloredsolid lines� and LQ-L �colored dashed line� models are su-perimposed onto isoeffect data and curves �black symbolsand lines� for early and late effects in various organs of labo-ratory animals digitized from Fig. 1 in Ref. 1. In the originalfigure late effects were plotted with solid black lines, acuteeffects with dashed black lines. These data are also repro-duced as Fig. 22.7 in Hall’s textbook.2 The biological end-points studied were not specified. The figure was originallyintended to illustrate that isodoses for late effects increasemore rapidly with decrease in dose per fraction than for acuteeffects. The solutions for � /� which resulted in the closestisoBED fit �by inspection� to the experimental results aredisplayed in matching color to the left of each LQ curve.

TABLE II. Data for � /� in the range 9 to 11 Gy from Table I normalized to

� /��Gy� BED�/�

LQ

20Fractions

10Fractions

5Fractions

9 73.3 Gy9 0.996 0.99 0.9810 72.0 Gy10 1 1 111 70.9 Gy11 1.003 1.01 1.02

TABLE III. Data for � /� in the range 2 to 4 Gy from Table I normalized to

� /��Gy� BED�/�

LQ

20Fractions

10Fractions

5Fractions

2 120.0 Gy2 0.98 0.96 0.943 100.0 Gy3 1 1 14 90.0 Gy4 1.01 1.03 1.05

Medical Physics, Vol. 35, No. 9, September 2008

Observe that all of the isoBED curves calculated using theLQ model smoothly and continuously curve downward withincrease in dose per fraction.

Among the tissues and organs with low � /�, late effectsfor spinal cord and kidney do not exhibit LQ-L behavior inthis experimental dose range because the isoeffect data con-tinues to bend downward as dose increases, even above2� /� Gy. However, for lung, the experimental data does notcontinuously bend downwards with increasing dose and canbe fit by an LQ-L model �red dashed line� with � /�=2.4 Gy, DT=4.2 Gy ��2� /��, and � estimated as the tan-gent at DT. The dose range of the data for the other lateeffects is not wide enough enough to determine if LQ-L be-havior applies.

LQ-L behavior will be recognizable on log-log plots ofisoeffect studies by an inflection point in the isoeffect curveas illustrated for the lung data. LQ-L behavior for the acutereactions in this figure would only be noticeable if the tran-sition to linearity began within the plotted dose range. Foracute effects with higher values of � /� �e.g., �10 Gy� thedata are all adequately fit by the LQ model for fraction dosesup to 10 Gy; thus, if any of these acute reactions are LQ-L,the transition to linearity must occur at doses�10 Gy/fraction.

In Figs. 6�a� and 6�b� the reciprocal of the total doserequired to produce a given biological effect is plotted as afunction of the dose per fraction which was delivered asmultiple equal fractions. Douglas and Fowler20 introducedthis method to estimate the ratio � /� from the slope andordinate intercept of the best fit line to the isoeffect data.

The data in Fig. 6�a� for an acute skin reaction in micewere digitized from Fig. 9 in Ref. 20. The hollow and soliddiamond symbols were used to denote experiments per-formed with and without anesthesia. A subset of these data

results for � /�=10 Gy.

LQ-L

3ractions

2Fractions

5Fractions

3Fractions

2Fractions

0.98 0.97 0.98 0.98 0.971 1 1 1 1

1.02 1.03 1.02 1.02 1.03

esults for � /�=3 Gy.

LQ-L

3ractions

2Fractions

5Fractions

3Fractions

2Fractions

0.93 0.92 1.00 1.06 1.101 1 1 1 1

1.06 1.07 1.05 1.01 0.98

the

F

the r

F

4168 Melvin Astrahan: LQ-L implications 4168

are also reproduced as Fig. 18.26 in Hall’s textbook.2 AnisoBED curve calculated using the LQ formula �solid redline� for � /�=10.2 Gy appears as a straight line on this plot.An isoBED curve calculated using the LQ-L model �dashedred line� for � /�=10.2 Gy and assuming DT=2� /� Gy and� the tangent at DT bends at doses greater than DT. For dosesabove 15 Gy/fraction the experimental results are inadequateto confirm LQ behavior or to distinguish between the LQ andLQ-L models. For instance, in the limited range20 to 25 Gy/fraction, the ordinate values of the 7 data pointsare inexplicably linearly decreasing with increasing dose perfraction.

In Fig. 5, the LQ formula was observed to closely fit theisoeffect data of a late reaction of mouse spinal cord forfractions up to about 10 Gy. In Fig. 6�b�, the reciprocal ofthe isoeffect dose for three types of spinal cord damage inrats, digitized from Fig. 6.4 in Ref. 21, are plotted out to20 Gy. The black lines are from the original figure and rep-resent linear fits by regression analysis. The linearity of thecervical white matter necrosis �solid black triangles� data onthis plot demonstrates LQ behavior for that syndrome acrossthe entire dose range. The data for cervical vascular damage�hollow triangles�, however, exhibit the type of bending thatis indicative of LQ-L behavior. An isoBED curve calculatedusing the LQ-L model �red dashed line� with � /�=1.5 Gy,DT=10.5 Gy, and � the tangent at DT appears to fit the cer-vical vascular damage data better than a straight line.

Now consider the hypothetical situation in which a tumorwith � /�=10 Gy and its surrounding tissues or organs with� /�=3 Gy are homogeneously irradiated. When consideringa hypofractionated regimen one could opt to maintainequivalent tumor control, or to maintain equivalent late se-quelae. In Fig. 7 BED is plotted for late reactions as a func-tion of constant acute effect to the tumor of 72 Gy10 and for

Medical Physics, Vol. 35, No. 9, September 2008

acute reactions to the tumor as a function of constant latereactions of 100 Gy3. Increasing BED presumably means in-creasingly severe late reactions and increasing probability oftumor control.

The LQ model �dashed lines� predicts that for constanttumor control the severity of late reactions progressively in-creases as the number of fractions decreases and the dose perfraction increases. Conversely, for constant late reactions, theprobability of tumor control increases as the number of frac-tions increases. If late effects are LQ and only the tumorreactions are LQ-L with the approximations DT�2� /� Gy�i.e., �20 Gy� and � the tangent at DT, the results are virtu-ally identical to the LQ curves �dashed lines�. This occursbecause all of the calculated fraction doses fall below 20 Gyin this study except for a regimen consisting of a single frac-tion, and even in that case the calculated dose is close to20 Gy.

However, under the specific assumptions that both the tu-mor response and late effects happen to exhibit LQ-L behav-ior �solid line� with DT=2� /� Gy and � the tangent at DT,the severity of late sequelae for constant tumor controlpasses through a maximum at 5 fractions and then decreasesas the number of fractions decreases. The calculated dose/fraction that yield constant acute BED=72 Gy10 in this peakregion are also shown. There is an equivalent converse be-havior for constant late reactions. These results must be con-sidered only a demonstration of principle and applicable onlyto the underlying assumptions. They should not be applied toany clinical situation.

IV. DISCUSSION

One limitation of the conventional LQ formulation is thatit may not properly model the exponential decrease of sur-

FIG. 5. IsoBED curves calculated us-ing the LQ model �solid lines, variouscolors� are superimposed onto isoef-fect data �black color� digitized fromFig. 1 in Ref. 1 for early and late ef-fects in various organs and tissues oflaboratory animals. The values for� /� used in the present modeling aredisplayed in matching color to the leftof each curve. The LQ-L model�dashed red line� with DT=4.2 Gy��2� /�� and � the tangent at DT ap-pears to fit the isoeffect data for lungbetter than does the LQ model �solidred line�.

4169 Melvin Astrahan: LQ-L implications 4169

vival with dose that is often observed at high dose in cellsurvival studies. When cell kill is the endpoint, the standardLQ model can overestimate the severity of response at highdose because the curve continues to bend. As a result, cellsurvival will be greater than anticipated and calculations willunderestimate the dose required to match the biological ef-fect of conventional fractionation. LQ-L curve fitting modelsare a simple remedy to this problem.

Garcia et al.9 demonstrated that � /� calculated for vari-ous cell lines in vitro varied widely depending on the doserange to which the LQ model was fit. LQ-L models eliminatethe dose range ambiguity of � /� by requiring the model tofit the entire experimental range of dose. This enforces adose range independent approximation for � /� which in turnallows for dose range independent intercomparison of frac-tionation regimens using the BED concept. One does need tobe consistent, however, if simplifications to the LQ-L modelare assumed �e.g., � being the tangent at DT�.

The LQ-L model described in this work may also be use-ful for fitting and interpreting the survival curves of someatypical and radioresistant cell lines. In Fig. 3�e� responseappears to become linear on the LLP above 7 Gy. Below7 Gy, the LQ-L fit yields an � to � ratio of about 10. Lethal-

FIG. 6. In these figures the reciprocal of the total dose required to produce aas multiple equal fractions. The value of � /� can be estimated from the sloreaction in mice �gray symbols� were digitized from Fig. 9 in Ref. 20. A=10.2 Gy appears as a straight line on this type of plot. An isoBED curv=2� /� Gy, and � the tangent at DT bends at doses greater than DT. The dat�b� These data for three types of spinal cord damage in rats were digitized frlinear fits by regression analysis. The data for cervical white matter necrosisLQ behavior for that syndrome. The data for cervical vascular damage �hollored dashed line is an isoBED curve calculated using the LQ-L model with

ity across the entire dose range therefore appears to be domi-

Medical Physics, Vol. 35, No. 9, September 2008

nated by events that are exponentially proportional to doserather than dose squared. In Fig. 3�f�, on the other hand, thecommon features of fitting LQ-L to the two examples ofradioresistant cells were that the resulting solutions for the �terms yielded very small numbers, � /� ratios 1, and re-sponse again appears to possibly become linear on the LLPbeyond 7 Gy. When the ratio of � to � is much less than 1the quadratic term of the LQ formula dominates, which sug-gests that the mechanism of cell killing for these cells atdoses below 7 Gy is associated with a pair of events. Theoriginal curves drawn to fit these data assumed smooth cur-vature, but it is at least a possibility that the response moreabruptly transitions to linearity at about 7 Gy. An abrupttransition to steeper linearity might indicate that some repairfunctionality of these cell lines becomes overwhelmed atabout 7 Gy.

When DT=2� /� Gy, the line tangent to the curve at DT

intersects the curves of the individual components e−�D ande−�D2

at � /� Gy, the dose at which the linear and quadraticcomponents of cell kill are equal, and also approximates thefinal linear response of some cell lines. It is intriguing tospeculate that that this might be related to some underlying

effect is plotted as a function of the dose per fraction which was deliveredd ordinate intercept of the curve �Ref. 20�. �a� These data for an acute skinoBED curve calculated using the LQ formula �solid red line� with � /�lculated using the LQ-L model �dashed red line� with � /�=10.2 Gy, DT

ve 15 Gy are inadequate to differentiate between the LQ and LQ-L models.ig. 6.4 in Ref. 21. The black lines are from the original figure and representblack triangles� are very linear across the entire dose range, which indicatesngles� exhibit the type of curvature that is indicative of LQ-L behavior. The1.5 Gy, DT=10.5 Gy, and � the tangent at DT.

givenpe an

n ise ca

a aboom F�solidw tria� /�=

biological mechanism. It remains to be determined, however,

4170 Melvin Astrahan: LQ-L implications 4170

how widely the simplifying approximations involving � be-ing the tangent at DT and DT�2� /� Gy that were used toexplore some of the implications of LQ-L in this work can beapplied. For instance, the LQ-L model was able to ad-equately fit the atypical survival data in Figs. 3�d�–3�f� withwhich the LQ model is incompatible, but only when all ofthe simplifying approximations were removed, i.e., both �and DT had to be explicitly provided, they could not be es-timated from other characteristics of the data. It may be thatthese simplifications are only applicable to those reactionsfor which the transition from linear-quadratic to linear issmooth and gradual as seen in Figs. 3�a�, 3�b�, and 4. Addi-tional research will be required to determine to what extentthese simplifications can be applied to the clinic.

As the dose � /� increases, one can hypothesize, however,that DT will increase proportionally as well. For instance, if� /�=10 Gy and DT�2� /� Gy, then doses may have toexceed about 20 Gy before the linear portion of the survivalcurve �on the LLP� begins and the standard LQ model beginsto fail. This was illustrated in Figs. 3�c�, 5, and 6. Few doseresponse studies in the literature extend measurements be-yond 20 Gy. To support the current resurgence of interest inhypofractionation, it will be of value to extend experimentalmeasurements to doses well beyond 2� /� Gy for the reac-tion of interest in order to ascertain if dose response everbecomes linear at hypofraction doses. An interesting impli-cation of DT�2� /� Gy, if applicable, is that Eq. �8� couldbe simplified by the approximation � /��5.

In regards to late sequelae of irradiation, organ function ispresumably related more to the proportion of functional cells

remaining in an irradiated organ than to the proportion of

Medical Physics, Vol. 35, No. 9, September 2008

clonogenic cells.2 Therefore, the ratio � /� for normal tissuesand organs is often inferred from the type of multifractionisoeffect experiments1,2,20,21 depicted in Figs. 5 and 6 ratherthan from single fraction cell survival studies. In Fig. 5 itwas demonstrated that at least one of the late effects �lung�appears to exhibit LQ-L behavior with DT�2� /� Gy and �the tangent at DT. In Fig. 6 one of the three spinal cordsyndromes �cervical vascular damage� appears to exhibitLQ-L behavior with � the tangent at DT. It is reasonable toexpect that other examples of LQ-L behavior for late reac-tions will be found. In Fig. 7, under the assumption that thelate reaction exhibits LQ-L behavior, the model predicts thatwhen cells with very different � /� are homogeneously irra-diated with the goal of maintaining constant biological effectto the higher � /� reaction �e.g., tumor control� that the BEDfor the lower � /� reaction �e.g., late sequelae� initially in-creases as the number of fractions decreases, but eventuallydecreases again when very few fractions are used.

This reason for this can be explained, at least in principle,from the curves plotted in Fig. 8 which are loosely based onFig. 22.6 in the chapter “Time, Dose and Fractionation” inHall’s radiobiology textbook.2 Hall points out that for tumorand other early effects that the � /� ratio is generally large�blue line�, that � dominates at low doses, and that the dose-response curve on the LLP has a marked initial slope anddoes not bend until higher doses. For late effects the � /�ratio is assumed to be small �black line� so the dose-responsecurve appears “more curvy.” In this example the low � /�response is depicted as being relatively less severe per unit ofdose at low dose. The high and low � /� LQ curves eventu-

FIG. 7. A hypothetical study of low � /� reactions �e.g.,late effects� as a function of constant high � /� reac-tions �e.g., acute or tumor response� and high � /� re-actions as a function of constant low � /� effects, as-suming that both low and high � /� reactions exhibitLQ-L behavior with DT=2� /� Gy and � the tangent atDT.

ally intersect at some moderate dose beyond which the low

4171 Melvin Astrahan: LQ-L implications 4171

� /� LQ reactions become progressively more severe withdose compared to the high � /� LQ reactions because thelow � /� LQ curve is bending downward more rapidly withincreasing dose than is the high � /� curve. If the low � /�curve represents late effects, then the late effects will bemore severe when large fractions are used, even if the high� /� reactions �e.g., tumor� are adjusted for constant effect.This therapeutically favors fractionation regimens that em-ploy a large number of small fractions and is one of the basicrationales for conventionally fractionated therapy regimens.

If, on the other hand, the low � /� reactions were to ex-hibit LQ-L behavior �red dashed line� and become linear onthe LLP at some intermediate dose, then the low � /� LQ-Lcurve will straighten in this intermediate dose range ratherthan continuing to bend, while the curve for the high � /�reaction can be expected to continue bending until a muchhigher dose. In this case the high � /� curve and the low � /�LQ-L curve intersect at some moderate dose, and then inter-sect again at some very high dose, beyond which the low� /� LQ-L reactions once more become relatively less severethan the high � /� reactions. It is unknown at this time if thistheory has any clinical applicability, and this upper dose

FIG. 8. Demonstration of principle illustrating how � /� differences mightbe exploited if the low � /� response exhibits LQ-L behavior and DT

�2� /� Gy. The LQ dose-response reactions for low � /� �black line, e.g.,late effects� are initially less severe but more curved than the high � /� �blueline, e.g., tumor� reactions. This therapeutically favors a large umber ofsmall fractions. If the low � /� reactions instead exhibit LQ-L behavior �reddashed line� and the response curve becomes linear at some intermediatedose while the curve for high � /� reactions continues to bend, there couldbe some therapeutic advantage to using a few fractions of high dose. If, onthe other hand, the tumor exhibits LQ-L behavior while the late effectsremain LQ, then dose fractions greater than the tumor DT could be thera-peutically disadvantageous even if both the tumor and late effects are low� /�.

range of relatively less severe reactions for low � /�, if it

Medical Physics, Vol. 35, No. 9, September 2008

exists, could well occur at such a high dose that functionalorgan tolerance is exceeded even though there may be somesurviving cells.

One implication of Figs. 7 and 8, however, is that it mightbe possible to exploit � /� differences by using a few frac-tions of very high dose when treating a tumor with a high� /� that is intermixed with tissues or organs of low � /� if,and only if, the reactions of those low � /� tissues or organsexhibit LQ-L behavior. One advantage of using a few largefractions compared to conventional fractionation, assuminglate sequelae being equal, might be the avoidance of tumorrepopulation. This possibility should be explored in the labo-ratory. A hypothetical experiment could be a tumor with ahigh � /� in the lungs of mice. This idea is based on theobservation in Fig. 5 that a late lung reaction of mice appearsto exhibit LQ-L behavior.

On the other hand, for a tumor such as prostate cancerwhose low � /� of about 3 Gy is similar to the � /� of thesurrounding organs and tissues, speculation exists that theremay be little advantage in terms of late sequelae to any par-ticular fractionation so long as the fraction dose is adjustedto maintain the desired probability of tumor control.22 Thismay not be the situation, however, if the tumor’s dose-response is LQ-L, a possibility suggested by Fig. 4, whileone or more of the related late effects remain LQ as in Fig. 5.In that case, even though both have low � /�, the LQ lateeffects will become relatively more severe compared to theLQ-L reactions of the tumor at doses greater than DT �for thetumor� as also illustrated in Fig. 8. The same basic rationalalso applies to tumors with high � /� that exhibit LQ-L be-havior when late effects are simply LQ. This will be of great-est concern when both the tumor and surrounding tissues areirradiated to the same dose. The steep dose gradients that arenow achievable using intensity modulated planning and de-livery technologies work to reduce the severity of late effectsby reducing the dose to the tissues surrounding the tumor.Still, it may be prudent to consider DT for the tumor reac-tions as the upper limit of fraction doses that should be con-sidered.

When the clinical dose-response reactions of tumor and/orother tissues or organs exhibit LQ-L behavior, intercompari-son of conventional and hypofractionation regimens usingLQ-L models should provide more accurate results than theLQ model because LQ-L better fits a wider range of dose forwhatever experimental data is available. The LQ-L formula-tion proposed here and that of Park et al.12 are attractive forthe clinic because the transition dose DT clearly identifies theupper limit to which the LQ formulation should be applied.Only for fractions of size greater than DT need the transitionto final linearity be applied.

V. CONCLUSIONS

Limitations of the LQ formulation are that it is not com-patible with some examples of clonogenic cell survival andthat it improperly models the experimentally observed expo-nential decrease of cell survival with dose at high dose. Fit-

ting a linear-quadratic-linear curve to the experimental dose-

4172 Melvin Astrahan: LQ-L implications 4172

response data is a simple remedy to this problem. LQ-Lmodels will probably find their greatest clinical utility whenintercomparing conventional and hypofractionated regimensfor those reactions which exhibit LQ-L dose-response andare of low � /� because the transition to final linearity couldwell fall somewhere within the range of doses that might beconsidered for hypofractionation. The LQ-L formulation pro-posed here can fit a wide variety of dose-response data overa very wide range of dose yielding a dose-range-independentapproximation for � /�, which, in turn, enables BED to beintercompared over a wide range of dose. The LQ-L transi-tion dose DT and �, the loge cell kill per Gy in the final linearportion of the survival curve, can be determined by fitting theLQ-L model to experimental dose-response and/or multiplefraction isoeffect studies. The simplification of � being thetangent at DT appears to have wide, but not universal, appli-cability. The term DT carries intuitive clinical significancebecause it describes the dose at which the linear-quadraticmodel ends and the final linear tail of the dose-responsecurve, on the log-linear plot, begins. When late reactions areknown to be LQ, it may be prudent to consider DT for thetumor reactions as the upper limit of fraction doses thatshould be considered.

a�Electronic mail: astrahan@usc.edu; Telephone: 323-865-3091; Fax: 323-865-0000

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