'Harper-Grace Hospitals, Gershenson Radiation Oncology Center, and Wayne State University, ... 'Michael Reese Hospital and Medical Center, Chicago, IL.
0360-3016/88 $3.00 + .Xl Copyright 0 1988 Pergamon Journals Ltd.
Inr. J. Radialion Oncology Biol Phys., Vol. 14, pp. 549-556
Printed in the U.S.A. All rights reserved.
??Oncology Intelligence
A UNIFIED APPROACH TO DOSE-EFFECT RELATIONSHIPS IN RADIOTHERAPY. I: MODIFIED TDF AND LINEAR QUADRATIC EQUATIONS COLIN G. ORTON,
PHD.’ AND LIONEL COHEN, M.D.2
‘Harper-Grace Hospitals, Gershenson Radiation Oncology Center, and Wayne State University, 3990 John R. Detroit, MI 4820 1 ‘Michael Reese Hospital and Medical Center, Chicago, IL A linear quadratic factor analogue (LQF) to the variable-exponent TDF model is introduced. In both of these models, account is taken of the volume of tissue irradiated. Scaling factors are used such that an LQF or a TDF of 100 represents tolerance for each volume or partial volume of each tissue or organ irradiated. These models are sufficient for tissues which are irradiated fairly homogeneously. Examples illustrate the use of these models. TDF, Linear-quadratic model, Time-dose model, Fractionation, LQF, Linear quadratic factor.
INTRODUCTION
for certain end points, either acute or late effects, rarely both.6,7,8,‘3,‘5.‘6.26,33,34 For example, it has been reported that, provided the overall treatment time remains fairly constant, the conventional exponents in the TDF equation are appropriate for early responses in many cases whereas, for late effects, new exponents are needed to make the TDF model approximate reality.14 Furthermore, the advantage that a TDF of 100 corresponds to “tolerance” rarely holds for tissues and organs other than connective tissue stroma and skin. Also, even when the TDF exponents are valid for a certain type of tissue, the equation is only useful over a restricted range of values of time, dose, and fractionation, since the TDF model is predicated upon the assumption that Strandqvist lines are linear.‘2.‘6.37 However recent evidence suggests that Strandqvist lines are concave downwards instead of linear, especially for early reacting tissues, although a linear approximation can, of course, be defined over a restricted range of fraction numbers.14 These observed downward concavities enable the linear-quadratic (L-Q) model to be fitted over a large range of fraction numbers and this is one of the strong points of the L-Q model.
The need for reliable dose-effect relationships in fractionated radiotherapy has increased significantly in recent years with the advent of a variety of innovative fractionation techniques such as hyperfractionation, hypofractionation, accelerated fractionation, high dose-rate fractionated remote afterloading, single fraction intraoperative radiotherapy, and many forms of dynamic fractionation. Unfortunately, there are many problems associated with the use of the two most commonly cited dose-effect relationships, the time-dose factor, TDF (or NSD/CRE), and the linear-quadratic, L-Q, models. Some of these problems are reviewed in the following.
The TDF Model The principal advantage of the TDF model is that it is simple to use. TDFs calculated for different parts of a course of radiotherapy are linearly additive (unlike NSDs and CREs) and account for the effects of all three parameters time, dose, and fractionation.25 There is provision for the calculation of TDFs for brachytherapy and for combined brachytherapy/fractionated therapy modalities.** Finally, the scale of TDFs is convenient, a TDF of 100 being roughly equivalent to normal connective tissue and skin tolerance.23 There are, however, several deficiencies in the TDF model. The exponents used in the TDF equation appear to be appropriate only for certain tissues and organs and
The L-Q model The L-Q model is based upon the apparent linear-quadratic shape of cell-survival curves for which there is some radiobiological and physical supp~rt.‘~*~“~‘~~‘~~‘~~~~ The analog of TDF in the L-Q model is the extrapolated response dose (ERD).’ This is sometimes referred to as
Reprint requests to: Colin G. Orton, Ph.D., Gershenson Radiation Oncology Center, Harper-Grace Hospitals, 3990 John R, Detroit, MI 48201. Acknowledgements-The authors wish to acknowledge the help of Richard L. Maughan, Ph.D. for his constructive reading
of the manuscript at various phases of its development, William E. Powers, M.D. for his continued encouragement and support, and Ms. Hilda Grayson for her work in preparation and typing of the manuscript. Accepted for publication 29 September 1987. 549
550
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extrapolated tolerance dose (ETD), but this latter term really only applies to treatment regimens which take tissues to “tolerance.“’ In this paper it is intended that all levels of tissue response should be addressed, so the term ERD will be used. This model is relatively simple to use, although this is not obvious in many of the publications written about it. Most authors have failed to point out that, like their counterparts TDFs, ERDs are linearly proportional to the number of fractions and hence are also linearly additive. Many of the deficiencies in the L-Q model are similar to those for TDFs. As with TDFs, ERD values for most tissues and organs are not readily related to tolerance. (Visser et aZ,32have recently introduced a new derivative of the ERD model which is the first attempt at redressing this deficiency. This is known as “fractional tolerance,” which is analogous to the “partial tolerance” concept of the NSD model.37) Furthermore, the important parameter (r/p is known only approximately and for only a few human tissues and organs,’ although substantial headway is being made in the determination of (Y/Pvalues for a variety of animal tissues. I4 The appropriate values of (Y/P depend upon the end-point observed (early or late effects) and calculations depend markedly on the (Y/P values used in calculations, especially for the low LY//~ values apparently characteristic of the vitally important late-reacting tissues.14 Finally, to account for the effects of overall treatment time upon proliferation, an additional “time” factor has to be incorporated into the LQ equation. This adds one more “unknown and untested” parameter to the model. The purpose of this paper is to present modified versions of the TDF and L-Q equations which address these deficiencies. METHODS AND MATERIALS ModiJied TDF and L-Q models What is needed are TDF and L-Q models with tissuespecific parameters (exponents in the TDF equation, (Y/ p values for the L-Q method) which, when applied, result in “biologically effective dose” numbers which can be related readily to tissue tolerance. Since tissue tolerance is related to the volume of tissue irradiated, each of these models must include a parameter to account for volume effects. In addition, a “time factor” is needed in the L-Q equation. It is expedient to first review the derivation of a variable-exponent TDF equation which effectively eliminates several of the problems cited earlier and incorporates some of the advantages in a uniform manner. More detailed discussions of this variable-exponent model have been presented elsewhere, so only a brief review will be given here.24 Variable-exponent TDF equation Many investigators have proposed the use of different exponents for the TDF (or NSD/CRE) equation, but
March 1988, Volume 14, Number 3
none have done so in a unified way.6~7~8~‘3~17~‘8~26~33~34 Cona wide variety of TDF values have resulted, apparently unrelated in any systematic way to each other or to tissue tolerance. Many inherent advantages of the TDF concept are lost. The following “unified” variableexponent TDF equation maintains these advantages by the introduction of tissue-specific scaling factors and “volume” correction factors:24 sequently,
TDF
=
JQ,Td6(T/N)-Tu@.
................
(1)
where N = number of fractions; d = dose per fraction; T/N = average time in days between fractions; 6, T and 4 are tissue-specific exponents; u is the partial volume of tissue irradiated and is a fraction of some reference volume (U = 1); and K, is a scaling factor which makes TDF = 100 for the reference volume (v = 1) of tissue irradiated to tolerance. The introduction of the tissue-specific scaling factor, K, , ensures that a TDF of 100 corresponds to tolerance (e.g., 5% probability of complication) for a reference volume of any tissue irradiated. Then the additional incorporation of the tissue-specific volume factor, u6,30makes a TDF of 100 correspond to tolerance for all volumes of all tissues. It is important to note that the so-called “volume effect” in radiotherapy is highly dependent upon what tissues and organs are included within the irradiated volume. No simple volume effect factor can account for all these tissues and organs at once, so it is necessary to determine the effect of the irradiation on each tissue and organ separately. This is why the TDF for each tissue within the irradiated volume has to be calculated separately, using parameters specific for that particular tissue. This power-law volume effect equation has been shown to be a good approximation in clinical practice provided that P(D,, 1) G 1, where P(D,, 1) is the probability of complication if the entire reference volume (v = 1) were to receive the dose D, which is given to partial volume u.~’This limitation is rarely a problem in practice since complication probabilities are normally kept to a minimum. However, appreciable errors may become apparent if there are large “hot-spots” of dose within the irradiated volume. This situation is precluded in the conventional TDF model, since only one value of dose (d) can be inserted into the TDF equation. Thus it is assumed that either the dose distribution is fairly uniform throughout the tissue at risk, or that it is legitimate to use some representative maximum value of d for the estimation of the “tolerance” TDF. value. For the situation where the dose distribution is signihcantly inhomogeneous throughout the tissue or organ at risk, or where the use of a single “hot-spot” dose is not acceptable, some type of “integral” dose-effect model is required. This might be similar to several integral-response models recently introduced for the calculation of probabilities of complication, although these models do not account for
Modified TDF and LQ equations ?? C. G. ORTON AND L. COHEN
fractionation effects.‘0.29”5,36Integral response models which do account for fractionation are the subject of a separate communication. (Orton, C. G., unpublished data, March, 1987). Note that not only are the values of the parameters 6, r,& and K, in Eq. (1) dependent upon the specific tissue irradiated, they are also dependent upon the end-point observed, for example, early or late effects. Examples of the use of this model will be presented later and compared to solutions obtained by the new L-Q model to follow. The linear-quadratic factor model As was the case with TDFs, it is necessary to introduce a volume correction factor and a “time factor” into the L-Q equation. Furthermore, it is expedient to also incorporate a scaling factor to avoid generating a wide variety of extrapolated response doses whose magnitude depends upon the a/p values used and bears no simple relationship to tissue tolerance. The conventional ERD equation is:’ ERD=Nd
l+$ (
>
where CY//~ is a tissue-specific parameter which may be different for early and late reactions. To avoid confusion, a new term, the linear-quadratic factor (LQF), will be used for the modified, scaled, and volume-dependent ERD equation, where: u4.f(T/N).
........
(2)
where kl is a scaling factor which makes LQF = 100 for the reference volume (V = 1) of tissue irradiated to tolerance (e.g. 5% complication probability) andf(T/N) is a “time factor,” which is a function of the average time (T/ N), between fractions. A variety of forms of the “time factor” have been proposed. Some of these are: 1. A power-law factor with tissue-specific exponents similar to that used for the variable-exponent TDF model, that is ( T/N)-7;24 2. A “dose-modification” factor, which specifies an incremental increase in dose for each day that the treatment course exceeds that required for daily fractionation;3*2’,31 3. A function such as (1) or (2) above but with a “latent period” subtracted from the overall treatment time.20 The “latent period” has been introduced because it has been reported that “repopulation” for acutely responding tissues, represented by the time factor, occurs very slowly, if at all, during the first few weeks of a course of therapy.i4 As before, examples of the use of this equation will be
551
given later, but first it is necessary to present values for the various tissue specific parameters of the two models (K, , 6, T, kl, (Y/P, and 4) and to show how these values have been determined.
RESULTS Tissue-specific parameters The cell population kinetic (CPK) model4 includes computer programs that have been used to analyze large clinical data sets to determine the best fitting exponents of N and T for the NSD equation and (Y/Pfor the ERD mode1.5,7,8,9These parameters are readily converted to their TDF and LQF equivalents. Also, the values of K, and kl, which make tolerance for reference volumes of tissue correspond to TDFs and LQFs of 100, can be derived from these clinical data sets, as well as best fitting values of 6. To make the LQF scaling factor k, represent the same level of tolerance as the TDF factor Ki, the same time factor has been applied to the LQF equation as is used for TDFs, namely (T/N)-‘.24 Since the data sets presently available are somewhat sparse, only initial estimates of these parameters for a relatively few tissues can be determined at this time. These are shown in Table 1. A more complete review of the actual data from which these parameters have been derived has been published elsewhere.7,8*9The derivation of the values of $I from the “field size” exponent of the CPK model is given in Appendix I, where a second method of determining 6 (from dose-response data) is also given. The final column of Table 1 gives the average values of 4 derived by these two methods. Following are some examples which illustrate the use of the new TDF and LQF models. Illustrative examples Before showing how these two models may be used to solve practical problems it is important to demonstrate that, due to the incorporation of a power-law volume effect parameter into these equations, TDF and LQF values are linearly additive only if the volume of tissue irradiated does not change for different parts of the course of therapy. This is shown in Appendix II. Of course, this is no different to ignoring volume effects entirely, as is the case with the conventional TDF and ERD equations. What is normally done, and what will be done in illustrative examples which follow, is to consider the irradiation of some representative volume only (often the region of highest dose), and to assume that it is only the tolerance of the tissues in this representative volume that is of importance. It is shown elsewhere that this is frequently an unsatisfactory approximation and that use of some form of integration of TDFs or LQFs over the entire irradiated volume is needed to provide appropriate solutions (Orton, C. G., unpublished data, March, 1987).
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March 1988, Volume 14, Number 3
Table 1. Initial estimates of TDF and LQF parameters for normal tissue tolerance (5% risk of complication). The values of K, and k, are for reference volumes defined by 10 X 10 cm fields. For the derivation of kr values, the same “time-factor” has been used for the LQF model as is used for TDFs (=( T/N)-‘)
k, Tissue
(dose%
Skin Stroma Brain Spinal Cord Lung Gut Kidney
Gy)
0.90 1.19 0.64 1.19 3.42 1.23 5.64
4
7
(doses in Gy)
4
6
al@
(CPK :odel)
(dose response method)
(mean)
1.79 1.54 2.44 1.92 1.41 1.59 1.79
0.16 0.17 0.07 0.13 0.06 0.13 0.34
1.23 1.34 I .09 1.25 2.92 1.47 6.12
7.6 6.9 3.3 2.5 3.6 7.9 3.4
0.15 0.14 0.12 0.14 0.12 0.11 0.08
0.10 0.10 0.05 0.07 0.14 0.10 0.07
0.13 0.12 0.09 0.11 0.13 0.11 0.08
Example 1: Boost to previously irradiated lung A patient’s lung has been irradiated by opposing IO cm X 10 cm fields to a homogeneous dose of 20.00 Gy in 10 daily fractions. What boost dose through opposing 5 cm X 5 cm fields will be just tolerable ifdelivered in 3 extra fractions? (a) TDF solution To maintain linear additivity of TDFs, the TDF solution here will not take into account the field size reduction or the resultant inhomogeneous dose distribution. Then the total TDF is simply the sum of the TDFs for 10 fractions of 2 Gy and 3 fractions of the unknown dose, d Gy. Hence, if the field is assumed to be 5 cm X 5 cm throughout: TDFToTAL = 100 = TDFl + TDF2 = K,( T/N)-‘[( 10 x 26) + (3 x d6)]v+ Using the values of K, (for a 10 cm X 10 cm field), 7, and 6 given for lung in Table 1, with the corresponding value of u (=0.25) and an average value for (T/N) of 1.33 days/ fraction for daily fractionation, the value of d is determined to be: d = 2.19 Gylfraction
Then, solving the above quadratic equation for d using values of k,, a//3, 4, and T for lung from Table 1 and II = 0.25, gives: d = 2.20 Gyjfraction Example 2: Dtflerent fractionation regimen to new volume A patient has just completed a course of therapy in which a 10 cm length of spinal cord received a dose of 42 Gy in 20 daily fractions. A second previously unirradiated 10 cm length of cord is now to be irradiated using a hyperfractionation technique delivering 10 fractions/ week in 48 treatments. What dose/fraction will be tolerable? (a) TDF solution Using values of the TDF parameters for the irradiation of 10 cm of spinal cord given in Table 1, it can be shown
that the first treatment regimen did not exceed cord tolerance (TDF = 95). Since this was close to tolerance, however, it may be reasonable when calculating the tolerance dose to the second 10 cm of cord to assume that the entire 20 cm of cord is taken to tolerance. Then, if the tolerance dose-per-fraction is d: TDF = 100 = 1.19 x 48 x d’.92 x (0.67)-“.‘3 x 2’.”
(b) LQF solution For purposes of comparison of the two models, the same “time factor” (=( T/N)-‘) is used for the LQF solution to this and successive problems as used for the TDF method. As in (a) above, volume effects and dose inhomogeneities are ignored in the simple LQF solution and hence:
Solving for d gives: d = 1.25 Gy/fraction (b) LQF solution As in the above solution, it will be assumed that 20 cm of cord is the appropriate “volume” to consider at risk. Then, if the tolerance dose-per-fraction is d:
LQFTOTAL= 100 = LQFr + LQF, =+0x2(1
LQF = 100 = 1.25 x 48d 1 + &
+$)
(
X 2O.l’ x (0.67)-‘.I3
.>
which yields:
v”.(T/N)-’ d = 1.04 Gy/fraction
553
Modified TDF and LQ equations 0 C. G. ORTON AND L. COHEN
Example 3: High dose-rate remote afterloading A Cervix cancerpatient is receiving whole pelvis treatments delivering 28 fractions of 1.8 Gy to the bowel through 10 cm x 10 cmhelds given 4 fractions/week. On the 5th day of each week the patient receives intracavitary high dose-rate treatments for a total of 6 fractions. Isodose curves show that about one-tenth of the bowel that is included in the external beamjelds receivesa high dose from the intracavitary sources. What dose/fraction fiiom the high dose-rate treatments is tolerable in this small high dose region? (a) TDF solution Using the TDF parameters for gut shown in Table 1 and considering only the small volume of bowel (v = 0.1) to be at risk, TDF = 100 = 1.23 x (1 .33)-“.‘3[(28 x 1.8’.59)+ 6d’.59]x (O.l)‘.” Solving for d gives:
d = 3.17 Gy/fraction (b) LQF solution As above, assuming that it is legitimate to consider only the small volume of bowel (V = 0.1) irradiated to high dose, and using the LQF parameters given in Table
LQF=
lOO= 1.47 x (O.l)O~”x (1.33))0.‘3
Solving this quadratic equation for d gives:
d = 3.39 Gy/fraction
DISCUSSION
AND CONCLUSIONS
By incorporating tissue-specific parameters in a “unified” way it has been demonstrated that the TDF and LQF models are simple to use and to compare in clinical practice. A value of 100 in either model corresponds to tolerance for the irradiation of any volume of any tissue. Here, another parameter (K, or k,) needs to be introduced into the TDF and LQF equations. We believe that the confusion caused by the introduction of this additional parameter will be more than offset by the reduction in confusion resulting from having 100 always representative of “tolerance.” Another major advantage of these new equations is the possible use of them as building blocks for “integral response” models when dose distributions are significantly inhomogeneous (Orton, C. G., unpublished data, March, 1987). Note that, for two of the three illustrative examples used in this paper (examples 1 and 3) the TDF and LQF models give similar answers, the maximum difference being about 7%, which is probably not clinically significant. However, for example 2 the difference is 20%, which probably is clinically significant. Clearly, the lowest estimate ( 1.04 Gy/fraction) should be used for such a patient in the absence of any clinical experience to the contrary. One final question then. Which is the more accurate model: TDF or L-Q? This is a problem which has been debated extensively elsewhere and no unequivocal answer can be given. With neither of the two models are the relevant parameters (6, T, 4, and LY/@) known with any degree of confidence for most tissues. Hence it is important to realize that neither model is accurate and solutions obtained to clinical problems should be considered as rough estimates only. In practice, if there is no clinical experience available on which to base a decision and resort has to be made to a mathematical model, it is probably expedient to solve the problem by both the TDF and L-Q methods, accounting for dose inhomogeneities if significant, (Orton, C. G., unpublished data, March, 1987) and to use the solution which is the most conservative.
REFERENCES 1. Barendsen, G.W.: Dose fractionation, dose rate and iso-
6. Cohen, L. and Awschalom M.: Fast neutron radiation ther-
effect relationships for normal tissue responses. Int. J. Rad-
apy. Ann. Rev. Biophys. Bioeng. 11: 359-390, 1982. 7. Cohen, L. and Creditor, M.: An iso-effect table for radiation tolerance of the human spinal cord. ht. J. Radiat. Oncol. Biol. Phys. 7: 96 l-966, 198 1. 8. Cohen, L. and Creditor, M.: Iso-effect tables for tolerance of irradiated normal human tissues. Int. J. Radial. Oncol.
iat. Oncol. Biol. Phys. 8: 1981-1997,
2. 3. 4.
5.
1982. Chadwick, K.H. and Leenhouts, H.P.: A molecular theory of cell survival. Physics in Medicine and Biology 18: 7887, 1973. Cohen, L.: Theoretical “iso-survival” formulae for fractionated radiation therapy. Brit. J. Radiol. 41: 522-528, 1968. Cohen, L.: A cell population kinetic model for fractionated radiation therapy. I. Normal tissues. Radiology 101: 4 19427,1971. Cohen, L.: Derivation of cell population kinetic parameters from clinical statistical data (program RAD 3). Znt. J. Radiat. Oncol. Biol. Phys. 4: 835-840,
1978.
Biol. Phys. 9: 233-241,
1983.
9. Cohen, L. and Creditor, M.: Iso-effect tables and therapeutic ratios for epidermoid cancer and normal tissue stroma. Int. J. Radiat. Oncol. Biol. Phys. 9: 1065-1071, 1983. 10 Dritschilo, A., Chaffey, J.T., Bloomer, W.A. and Marck, A.: The complication probability factor: A method for selection of radiation treatment plans. Brit. J. Radiol. 51: 370-374,
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J. Radiation Oncology 0 Biology ??Physics
11. Douglas, B.G. and Fowler, J.F.: The effect of multiple small doses of X-rays on skin reactions in the mouse and, a basic interpretation. Radiation Research 66: 401-426, 1976. 12. Ellis, F.: Fractionation in radiotherapy. In Modern Trends in Radiotherapy, Vol. 1, Deeley and Wood, (Eds.). London, Buttenvorth. London, 1967, pp. 34-5 1. 13. Field, S.B., Homsey, S. and Kutsutani, Y.: Effects of fractionated irradiation on mouse lung and phenomenon of slow repair. Brit. J. Radiol. 49: 700-707, 1976. 14. Fowler, J.F.: What next in fractionated radiotherapy? Brit. J. Cancer 49, Suppl. VI, 285-300, 1984. 15. Fowler, J.F. and Stem, B.E.: Dose-rate effects: some theoretical and practical considerations. British Journal of Radiology 31:389-395, 1960. 16. Fowler, J.F. and Stern, B.E.: Dose-time relationships in radiotherapy and the validity of cell survival models. British Journal ofRadiology36: 163-173, 1963. 17. Herrmann, Th., Voigtmann, L., Knorr, A., Lorenz, J., and Johanssen, V.: The time-dose relationship for radiationinduced lung damage in pigs. Radiotherapy and Oncol. 5: 127-135, 1986. 18. Hornsey, S. and White, A.: Isoeffect curve for radiation myelopathy. Brit. J. Radiol. 53: 168-169, 1980. 19. Kellerer, A.M. and Rossi, H.H.: The theory of dual radiation action. Current Topics in Radiation Research 8: 85158, 1972. 20. Kozubek, S.: A simple radiobiological model for fractionated radiation therapy. Int. J. Radiat. Oncol. Biol. Phys. 8: 1975-1980, 1982. 21. Liversage, W.E.: A general formula for equating protracted and acute regimes of radiation. Brit. J. Radiol. 42: 432440,1969. 22. Orton, C.G.: Time-dose factors (TDFs) in brachytherapy. British Journal of Radiology, 47: 603-607, 1974. 23. Orton, C.G.: Bioeffect dosimetry in radiation therapy. In Radiation Dosimetry: Physical and Biological Aspects, Orton, C.G., (Ed.). New York, Plenum Press. 1986, pp. l- 17. 24. Orton, C.G. and Cohen, L.: A variable exponent TDF model. In Optimization of Cancer Radiotherapy, Paliwal, B.R., Herbert, D.E., and Orton, C.G., (Eds.). New York, American Inst. of Physics, 1985, pp. 347-359. 25. Orton, C.G. and Ellis, F.: A simplification in the use ofthe NSD concept in practical radiotherapy. British Journal of Radiology46,529-537, 1975. 26. Pezner, R.D. and Archambeau. J.O.: Brain tolerance unit:
APPENDIX DETERMINATION Using variable volume data
In the CPK model,4 the volume effect is represented by the parameter Z ‘, where Z is the length of the side of the equivalent square ofthe field size (in decimeters), and Y is the “field-size” exponent. The use of “field size” instead of “volume” is necessitated by the data available for analysis, which specifies the tissue irradiated in two (field size) rather than in three dimensions, although the spinal cord is an exception (see later). Almost all available clinical data exhibits this insufficiency. The following analysis describes how the “linear” volume effect parameter, Zy, can be transformed into the three-dimensional TDF parameter, u@.
March 1988, Volume 14, Number 3
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
A method to estimate risk of radiation brain injury for various dose schedules. Int. J. Radiat. Oncol. Biol. Phys. 7: 397-402, 1981. Rubin, P. and Poulter, C.: Principles of radiation oncology and cancer radiotherapy. In Clinical Oncology, a Multidisciplinary Approach, 4th edition, Rubin, P. (Ed.). Am Cancer Sot. 1974, pp. 66-89. Rubin, P. and Poulter, C.: Principles of radiation oncology and cancer radiotherapy. In Clinical Oncology, a Multidisciplinary Approach, 5th Edition. Rubin P. (Ed.). Am. Cancer Sot. 1978, pp. 29-4 1. Schultheiss, T.E. and Orton, C.G.: Models in radiotherapy: Definition of decision criteria. Med. Phys. 12: 183187, 1985. Schultheiss, T.E., Orton, C.G., and Peck, R.A.: Models in radiotherapy: volume effects. Medical Phys. 10:4 1O-4 15, 1983. Spring, E.: A simple formula for calculation of the total dose in fractionated radiotherapy. Cancer Res. 32: 19951997, 1972. Visser, P.A., Moonen, L.M.F., Ven der Kogel, A.J., and Keijser, A.H.: Application of the linear-quadratic concept for prediction of late complications after combined irradiation of the uterine cervix. Radiother. and Oncol. 4: 133141, 1985. Wara, W.M., Phillips, T.L., Margolis, L.W. and Smith, V.: Radiation pneumonitis-A new approach to the derivation of time-dose factors. Cancer 32: 547-552, 1973. Wara, W.M., Phillips, T.L., Sheline, G.E. and Schwade, J.G.: Radiation tolerance of the spinal cord. Cancer 35: 1558-1562,1975. Wolbarst, A.B.: Optimization of radiation therapy II: The critical voxel model. Znt.J. Radiat. Oncol. Biol. Phys. 10: 741-745,1984. Wolbarst, A.B., Stemick, E.S., and Dritschilo, A.: Optimized radiotherapy treatment planning using the complication probability factor (CPF). Int. J. Radiat. Oncol. Biol. Phys. 6: 723-728, 1980. Winston, B.M., Ellis, F., and Hall, E.J.: The Oxford NSD calculator for clinical use. Clinical Radiology 20: 8- 11, 1969. Zaider, M. and Rossi, H.H.: Microdosimetry and its application to biological processes, in Radiation Dosimetry: Physical and Biological Aspects, Orton, C.G. (Ed.). New York, Plenum Press, 1986, pp. 17 l-242.
I OF
4
In the CPK model, the total doses D, and D1 which produce the same biological effect in volumes defined by field sizes with equivalent square sides Z decimeters and 1 decimeter, respectively, are related by the equation: D, = D,ZY Since the data used to determine Yin the CPK model is two dimensional, the third dimension (“thickness” t) has been assumed constant throughout the data set. Hence, the fractional volume irradiated (V in the TDF model), normalized to Z = 1 decimeters, is given by: 2, = (Z2.t)/(1%)
555
Modified TDF and LQ equations 0 C. G. ORTON AND L. COHEN
Combining this with equation (A 1) with d constant gives the same equation as (A4) above:
or v = Z2 Thus the CPK volume effect equation becomes:? D, = D,uy12
q) = -Y/2
..........
(Al)
But the TDF relationship (Eq. 1) for constant fractionation rate gives: K,N,d:(T/N)-‘.v*
= K,N,d;(T/N)-‘.
Using dose-response data The relationship between the total doses D, and D1 which produce the same biological effect to partial volume v and the reference volume (v = 1) respectively, can be represented by the equation:30
l@ D, = D,v?‘~
or Nd6v”= 0 ”
Nd’ I I
********.
642)
Although not explicitly stated in the CPK model papers in which values of Y were determined, most of the clinical data used was characterized by having the dose/fraction approximately constant at about 2 Gy (Cohen L., oral communication, July, 1986). Iso-effect doses for different volumes of tissue irradiated were obtained by varying N. Then combining equations (A 1) and (A2) and setting d constant gives: N, = N,v y/2 and
N,v@ = N,
k is a function of the slope of the logistic dose response curve for the tissue or organ.30 Then, if the dose per fraction, d, is kept constant, and the biological effects are kept equivalent by varying N:
where
N,d = N,dv-‘lk
........
(A5)
where
N, and Nr are the number of fractions required for iso-effect to partial volumes v and 1 respectively at constant dose per fraction d. Combining Eqs. (A3) and (A5) gives:
(from (Al)) (from (A2))
......
$=
(A3)
Hence 4 can be determined by calculating k from the Do5and DSovalues quoted by Rubin and Poulter27,28 using the equation:30
Hence: N,vY12ve = N I or 4 = -Y/2
l/k
,.
k=
(A4)
Analogously, for the LQF model:
ln( 19) ln(&/&s)
(Note: in the clinical data cited by Rubin and Poulter27,28 which has been used to determine @,the dose per fraction and the fraction rate remain constant, so d and (T/N) remain constant).
k,N,d,
APPENDIX
II
Non-additivityof TDFs and LQFs The following demonstrates that, if the volume of tissue treated varies for different parts of a fractionated course of therapy, TDFs and LQFs are no longer linearly additive. Assume that a volume of tissue is divided into two regions of partial volumes vu1and v2. Each of the two parts is treated at the same session (no time gap) to the same
dose d, for the same number of fractions N and for the same overall treatment time T. Clearly the total TDF for the two parts must be equal to the TDF calculated for the whole volume (vi + v2) irradiated with the same treatment parameters. Then, if it is assumed that TDFs are linearly additive:
t The spinal cord is an exception. In this case the field-size (Z) is the irradiated cord length rather than the equivalent square, and the volume is directly proportional to Z, not Z*.
Consequently, for spinal cord the exponent than Y/2 and equation Al reads D, = D~v ‘.
TDFrorAL = TDF,, + TDF,,
Y is used rather
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or, using eq. ( 1): K&d6( T/N)_‘(V, + v# = K, Nd”( T/N)-‘vf + K, Nd6(T/N)-‘v$ Hence (v, + v# = vt + vs This is only true if $J = 1 or either one of VI or ~2 = 0.
March 1988,Volume14,Number3 Since 4 is probably never equal to 1, and o1 or 02 = 0 means that only one volume of tissue is irradiated it has been shown that, if the volume of tissue treated varies for different parts of a course of therapy, the original assumption that TDFs can be added linearly must have been wrong. Similarly, it can be demonstrated that LQFs are also not linearly additive under these circumstances. For inhomogeneously irradiated volumes it is necessary to use the “integral” form of the TDF and LQF models (Orton, C. G., unpublished data, March, 1987).
