PII S0360-3016(99)00331-4

PHYSICS CONTRIBUTION

DERIVATION OF ISOEFFECT DOSE RATE FOR LOW-DOSE-RATEBRACHYTHERAPY AND EXTERNAL BEAM IRRADIATION

HUAN B. GIAP, M.D., PH.D., AND VINCENT MASSULLO, M.D.

Radiation Oncology, Scripps Clinic and Research Foundation, La Jolla, CA

Purpose: The combination of external beam irradiation and brachytherapy has been used effectively in themanagement of many malignancies. Brachytherapy dose is typically prescribed to an isodose rate line, fromwhich the implant duration is derived. In this study, the linear-quadratic model is used to derive the brachy-therapy dose rate at which biological effectiveness is equivalent to that of external beam irradiation.Methods and Materials: Relative effectiveness per unit dose (RE) for brachytherapy was based on Dale’sformalism. Isoeffect dose rate, defined as the brachytherapy dose rate at which the biological effectiveness isequivalent to that of external beam irradiation, was derived.Results: The functional dependencies of brachytherapy RE on dose rate,a/b ratio, and implant duration wereinvestigated. The isoeffect dose rate depends only on the dose per fraction, sublethal damage repair (SLDR)constant, and the implant duration. The isoeffect dose rate does not depend ona/b ratio. For sufficiently longimplant duration > 10–15 hours, the value for isoeffect dose rate approaches a constant value around 40 to 50cGy/hr.Conclusion: The isoeffect dose rate may be useful in treatment planning and optimization for low-dose-rate(LDR) brachytherapy, especially when brachytherapy is used in combination with external beam irradiation.© 1999 Elsevier Science Inc.

Linear-quadratic model, Biological equivalent dose, Dose rate effect.

INTRODUCTION

The combination of external beam irradiation (EBI) andbrachytherapy (BTX) has been used effectively in the man-agement of head and neck, genitourinary, and gynecologicalmalignancies. Traditionally, doses have been prescribedseparately for each component using various guidelines.Radiobiologically, the dose from EBI is delivered as high-dose-rate (about 100 cGy/min or 6000 cGy/hr) and fraction-ated at 180 to 200 cGy per day over a period of severalweeks to months, whereas the dose from low-dose-rate(LDR) brachytherapy is delivered at a lower dose rate(40–100 cGy/hr) continuously over a period of hours to afew days. The dose prescription for EBI is straightforwardsince the dose distribution to a treatment volume is rela-tively uniform. In other words, the entire tumor volumereceives a similar dose rate. Dose prescription for brachy-therapy is more complex due to the high dose gradientaround the radioactive source. There is significant variety indose rates within tumor and normal tissue volumes. Thebrachytherapy dose is typically prescribed to an isodose rateline, from which the implant duration is derived. The basisfor selection of a prescription dose rate is empirical, and thepractice may vary from institution to institution and even

among physicians within the same institution. Differencesin dose rates may result in varying biological effects totumor and normal tissues. Dose reporting by combining theabsolute doses from the two modalities (EBI and BTX)algebraically without considering the difference in biologiceffects could be misleading. In this study, the linear-qua-dratic model is used to derive the brachytherapy dose rate atwhich the biological effectiveness is equivalent to that ofEBI.

METHODS AND MATERIALS

The effect due to the difference in dose rate of externalbeam irradiation and brachytherapy is modeled using alinear-quadratic formalism described by Dale (1). The basiclinear-quadratic model starts with the following equation:

S5 exp[2(a z d 1 b z d2)] (1)

where S is the fraction of cell survival following anirradiation with dosed (Gy). The constantsa and b aremeasures of two separate cell kill processes: Type A andType B. In Type A damage, two critical target sites are

Reprint requests to: Huan B. Giap, M.D., Radiation Oncology,MSB 1, Scripps Clinic and Research Foundation, 10666 NorthTorrey Pines Rd., La Jolla, CA 92037.Acknowledgment—The authors are grateful to Amy Moghanaki,

Denise Princinsky, and John Spencer for proofreading the manu-script. We are also grateful to Drs. Roger Dale, Jack Fowler, andEric Hall, whose works have inspired this study.

Accepted for publication 29 July 1999.

Int. J. Radiation Oncology Biol. Phys., Vol. 45, No. 5, pp. 1355–1358, 1999Copyright © 1999 Elsevier Science Inc.Printed in the USA. All rights reserved

0360-3016/99/$–see front matter

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hit simultaneously by a single radiation event which leadsto cell death without repair. The Type A damage isrelated toad. In Type B damage, each of the two criticalsites is hit sequentially by different radiation events.When one of the two critical sites is damaged, the cell isconsidered sublethally damaged (SLD). The Type B dam-age is related tobd2. The ratio of the two parameters(a/b) has more clinical significance. The concept ofaverage relative effectiveness per unit dose (RE) pro-posed by Dale can be expressed as:

RE5 1 1Total Type B damage

Total Type A damage(2)

The concept of relative effectiveness per unit dose wasdeveloped for external beam irradiation (REEBI) and brachy-therapy (REBTX), which are defined as follows:

REEBI 5 1 1d

a/b(3)

REBTX 5 1 12R

m

1

a/b H1 21

mT@1 2 exp~2mT!#J

(4)

whered is the dose per fraction (Gy) for external beamirradiation, a/b for particular biological endpoint,R isthe brachytherapy dose rate,m is the time constant (hr21)for sublethal damage repair (SLDR), andT is the implantduration (hour).

There are several hypotheses used in this analysis. First,that the linear-quadratic model is valid. Second, the effect ofrepopulation is not accounted for. This effect is not signif-icant for late tissue reactions, where repopulation is notsignificant, but it may be important for more rapidly divid-ing tissues (tumor). Third, that the sublethal damages repairexponentially.

In this study, the isoeffect dose rate (Ro) is defined as thedose rate at which the biological effectiveness of brachy-therapy is equivalent to that of EBI. This isoeffect dose ratecan be derived by equating the right-hand sides of eqs. 3 and4 or settingREEBI equal toREBTX.

1 1d

a/b5 1 1

2Ro

m

1

a/b H1 21

mT@1 2 exp~2mT!#J

(5)

This equation can be rearranged to solve forRo:

Ro 5d z m

2 H1 21

mT@1 2 exp~2mT!#J21

5d z m

2~time factor! (6)

It is interesting to note that the isoeffect dose rate (Ro)depends only on the dose per fraction, SLDR time constant,

and the implant duration and is independent ofa/b ratio. Asimplant durationT increases, the time factor, which isdefined by the terms enclosed in { }, approaches unity,hence,Ro approaches a constant value of:

Ro,` 5d z m

2(7)

This isoeffect dose rate for “infinite implant duration” onlydepends on dose per fraction and SLDR time constant, butnot a/b or implant duration. As shown in Fig. 1, this“infinite implant duration” is almost reached after 50–60hours.

The value of the SLDR time constantm depends on thehalf-life of SLD repair; for example, if the SLD repairhalf-life is 1.5 hours (2), thenm equals to 0.46 hr21 ascalculated by the following equation:

Fig. 1. Plot of isoeffect dose rate versus implant duration for EBIfraction size of 1.8 and 2.0 Gy per day.

1356 I. J. Radiation Oncology● Biology ● Physics Volume 45, Number 5, 1999

m 50.693

half-life(8)

A value ofm 5 0.46 hr21 is used in this study. Values forSLDR constant have traditionally been derived fromsplit-dose experiments. These experiments are subject toartifacts arising from the effects of radiation-induced cellsynchrony. Several investigators had pointed out that theapparent rate of recovery exhibited by a split-dose cellsurvival curve is slower than the true recovery rate of thedose equivalent of sublethal damage since the dose andcell survival are not linearly related (3, 4). There is apossibility that different tissues may have different ca-pability of SLD repairs, hence differentm values; i.e., theSLD repair half-life may be shorter for early-reactingtissues compared to late-reacting tissue (5). Another pos-sibility is that the value of SLDR constant itself dependson the level of damage, i.e.,m may decrease with higherdose rate or dose (6). Another method to derive SLDRconstant is from intercomparison of isoeffect doses de-termined by various acute and protracted treatment reg-imens as demonstrated in Dr. Dale’s article (1). This typeof analysis is not easy, but it has the advantage ofdecreasing the problem of cell synchrony.

RESULTS AND DISCUSSION

Figure 2 shows a plot of the ratio ofREBTX overREEBI

versus dose rate for two differenta/b ratios assuming avalue ofm 5 0.46 hr21, dose per fraction5 1.8 Gy, andimplant duration5 72 hours. These curves are plottedusing eqs. 3 and 4. This ratio correlates the relativebiological significance (per unit dose) of brachytherapyover external beam irradiation. For a given dose perfraction anda/b ratio, REEBI is a constant; for example,REEBI would be equal to 1.18 for dose per fraction of 1.8Gy and a/b ratio of 10. There are several importantobservations from Fig. 2. First, the relationship of REratio versus dose rate is linear; i.e., doubling the dose ratedoubles relative effectiveness per unit dose when otherparameters are kept constant. Second, the slope of therelationship is greater for smallera/b ratio. Tissues withsmallera/b ratios have greater capability of SLD repair(larger shoulder on the survival curve), hence the effectof dose rate is more remarkable. Third, theREBTX ap-proaches unity as dose rate approaches zero.

The two lines from Fig. 2 intersect at the location wherethe ratio ofREBTX over REEBI is equal to unity. The doserate at this intersection is the isoeffect dose rate (Ro) bydefinition. At dose rates lower thanRo, the biological effec-tiveness of brachytherapy is less than that of EBI, and fordose rate greater thanRo, the biological effectiveness ofbrachytherapy is more than that of EBI.

Figure 1 plots values of isoeffect dose rateRo versusimplant duration (T) for two different dose per fractions of1.8 Gy and 2.0 Gy assuming SLDR constant of 0.46 hr21.

For short implant duration (, 10–15 hours), the time factoris significant, hence the isoeffect dose rate varies signifi-cantly with implant duration. For longer implant duration,the time factor is almost unity, hence, the isoeffect dose rateapproaches a constant valueRo,` described by eq. 7. TheRo,` is equal to 42 and 46 cGy/hr for a dose fraction of 1.8Gy and 2.0 Gy, respectively.

There are several important clinical implications ofRo.First, the brachytherapy dose delivered at the dose rateRo

can be added directly to EBI dose irrespective of biolog-ical endpoints; for example, if a local control for a certaintumor histology and size is 80% using EBI to total doseof 70 Gy and 2 Gy per fraction. If the same tumor istreated by BTX and the implant is done such that thesame target volume is covered by the 46 cGy/hr isodoseline, then similar tumor control is expected for an implantduration of 152 hours that delivers 70 Gy (70 Gy5 0.46Gy/hr * 152 hr). If an implant duration of 152 hours is toolong, the same tumor can be treated with a combinationof EBI to deliver 40 Gy in 20 fractions and BTX to

Fig. 2. The ratio of relative effectiveness of brachytherapy (REBTX)over external beam irradiationREEBI versus the dose rate (cGy/hr)for two differenta/b ratios usingm 5 0.46 hr21, implant durationof 74 hours.

1357Equivalent isodose line for LDR brachytherapy and EBI● H. B. GIAP et al.

deliver 30 Gy in 65 hours (30 Gy5 0.46 Gy/hr * 65 hr).The second clinical implication of the isoeffect dose

rate is that any dose delivered by brachytherapy at a doserate higher thanRo has higher relative effectiveness perunit dose than that delivered by EBI. Vice versa, any dosedelivered by brachytherapy at a dose rate lower thanRo

has lower relative effectiveness per unit dose than thatdelivered by EBI. This concept is demonstrated by anexample in Fig. 3, which shows a tandem and ovoidimplant for a patient with carcinoma of the uterine cervix.The EBI portal is shown. The outlined tumor volume iscovered within the isoeffect dose rate, 42– 46 cGy/hr.Since the tumor volume is treated at the higher dose rate

than the isoeffect dose rate, it receives higher biologicalequivalence dose than EBI for a similar given physicaldose.

CONCLUSION

A concept of isoeffect dose rate is derived in this studybased on the linear-quadratic model, and its clinical impli-cations are discussed. For treatment planning purposes, itmay serve as a reference isodose line for comparing thebiological effectiveness of the LDR brachytherapy to that ofexternal beam irradiation.

REFERENCES

1. Dale R. The application of the linear-quadratic dose–effectequation to fractionated and protracted radiotherapy.Br JRadiol 1987;58:321–332.

2. Liverage WE. A general formula for equating protracted andacute regimes of radiation.Br J Radiol1969;42:432–440.

3. Elkind MM, Sutton H. Radiation response of mammaliancells grown in culture. I. Repair of X-ray damage in sur-viving Chinese hamster cells.Radiat Res1960;13:556 –593.

4. Hall EJ. Radiation dose rate: A factor of importance in radio-biology and radiotherapy.Br J Radiol1972;45:81–97.

5. Thames HD, Withers HR, LJ Peters. Tissue repair capacityand repair kinetics deduced from multifractionated or contin-uous irradiation regimes with incomplete repair.Br J Cancer1984;49(Suppl. 6):263–269.

6. Iliakis G, Pohlit W. Quantitative aspects of repair of poten-tially lethal damage in mammalian cells.Int J Radiat Biol1979;36:649–658.

Fig. 3. Demonstration of isoeffect dose rate for a tandem and ovoid implant. The outlined tumor volume is covered withinthe isoeffect dose rate, 42–46 cGy/hr. Since the tumor volume is treated at a higher dose rate than the isoeffect dose rate,it receives higher biological equivalence dose than EBI for a similar given physical dose.

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