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Biologically effective dose for permanent prostate brachytherapy taking into account postimplant edema
Int. J. Radiation Oncology Biol. Phys., Vol. 53, No. 2, pp. 422– 433, 2002 Copyright © 2002 Elsevier Science Inc. Printed in the USA. All rights reserved 0360-3016/02/$–see front matter
PII S0360-3016(02)02755-4
BIOLOGY CONTRIBUTION
BIOLOGICALLY EFFECTIVE DOSE FOR PERMANENT PROSTATE BRACHYTHERAPY TAKING INTO ACCOUNT POSTIMPLANT EDEMA MARION P.R. VAN GELLEKOM, M.SC., MARINUS A. MOERLAND, PH.D., HENK B. KAL, PH.D, JAN J. BATTERMANN, M.D., PH.D.
AND
Department of Radiotherapy, University Medical Center Utrecht, Utrecht, The Netherlands Purpose: To study the influence of radiobiologic and physical parameters and parameters related to edema on the biologically effective dose (BED) for permanent prostate implants and to determine the optimal timing of seed reconstruction for BED calculation. Methods and Materials: On the basis of the linear-quadratic model, an expression for the BED was derived, including the edema parameters. A set of parameter values was defined, and these parameter values were varied one at a time to examine the effect on the BED and the theoretically effective treatment time (teff). A ratio ⑀ was defined to investigate the optimal timing of seed reconstruction. Results: The maximal BED decreases when the extent of lethal damage is smaller, the potential tumor doubling time is smaller, the half-life time of the seeds is shorter, and the magnitude of prostate volume increase is larger. For 125I, the optimal timing of seed reconstruction is 25 days after implantation. Seed reconstruction 1 day after the implantation results in an underestimation of the BED of at most 43%, depending on the magnitude and half-life of edema. An overestimation of the BED of at most 22% is calculated when seed reconstruction took place at the effective treatment time. Conclusion: The maximal BED depends strongly on the value of ␣, the potential tumor doubling time, and the choice of isotope. If prostate volume increase due to edema is not taken into account, the BED will be underestimated shortly after the implantation and overestimated if the calculations are based on images taken several months after implantation. The optimal timing of BED evaluation for 125I seed implants and typical prostate edema values is 25 days after implantation. © 2002 Elsevier Science Inc. BED, Prostate, Brachytherapy, Edema.
INTRODUCTION
ual shrinkage (8). We derived prostate volume changes from radioactive seed distances on radiographs, originally acquired at 3 days, 1 month, and 3 months after implant to check for any loss of seeds, and found that prostate swelling resolves exponentially in time (data not shown). Waterman et al. (7) acquired series of CT scans and also observed that prostate swelling resolved exponentially in time. Willins and Wallner (9) acquired pre- and postimplant CT scans from 11 patients and found that in all patients, the prostate returned to the preimplant size within 2 months of the procedure and was stable at between 2 and 6 months, indicating that edema is probably the most significant contributor to prostate volume changes. From these observations, which were not specific for resolving edema or radiation-induced shrinkage, we conclude that an exponential model adequately describes the prostate volume changes in the first half-year after implant when the dose is actually administered. Postimplant dosimetry may be based on ultrasonography,
Small tumors of the prostate (Stage T1 and T2) can be treated with brachytherapy. 125I, 103Pd, or 198Au seeds are most commonly used for permanent implantation. The prescription doses for permanent prostate implants with 125I, 103 Pd, or 198Au seeds are, in general, 140, 120, or 60 Gy, respectively (1–3). The smaller prescribed physical doses for 103Pd and 198Au implants compared with 125I implants occur from matching biologic end points for nonedematous prostates when the materials implanted have shorter halflives and therefore higher initial dose rates than 125I. The commonly used technique to introduce the radioactive seeds into the prostate is the transrectal ultrasoundguided transperineal technique (4). The dose actually administered to the prostate is influenced by the postimplant volume changes. A short-term effect is prostate swelling from edema caused by piercing the prostate with needles (5–7); a long-term effect may be radiation-induced continReprint requests to: Marion P.R. Van Gellekom, M.Sc., Department of Radiotherapy, University Medical Center Utrecht, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands. Tel: ⫹31-302509863; Fax: ⫹31-30-2581226; E-mail: m.p.r.vangellekom@ radth.med.uu.nl
Supported by the J. A. Cohen Institute for Radiopathology and Radiation Protection, Leiden, The Netherlands. Received Sep 13, 2001, and in revised form Jan 3, 2002. Accepted for publication Jan 24, 2002. 422
Biologically effective dose for permanent prostate implants
MRI, or CT. On ultrasonography, the seeds give rise to artifacts that hamper evaluation of the prostate volume and the seed distribution. A CT-based evaluation method for permanent prostate implants has been developed by Roy et al. (10) and has become common practice thanks to the widespread availability of CT scanners (1). However, several investigators have remarked that CT images frequently overestimate the size of the prostate in the anterior and lateral areas, because the periprostatic fat pad and venous plexus are not well distinguished from the prostate gland itself (11–13). At our institute, we developed a method for postimplant dosimetry that uses the excellent soft-tissue imaging capabilities of MRI to delineate the prostate and the high resolution of isocentric radiographs to identify the seeds (6). Irrespective of the choice of imaging modality, an optimal time of acquiring the images for postimplant dosimetry exists. Moerland (5) derived an equation for the prostate dose as a function of the magnitude and half-life of the prostate volume increase induced by edema and found an optimal time of seed reconstruction for dose calculations, termed implant evaluation. Before this time, an underestimation of the dose will be calculated and, after the optimal time, the dose will be overestimated. Thus, without taking edema into account, the calculated physical total dose is not a good representation of the real dose in the prostate (14, 15). When the radiobiologic parameters (e.g., potential tumor doubling time and repair of sublethal damage) are taken into account, the biologically effective dose (BED), based on the linear-quadratic model (16), can be calculated. Dale (17, 18) derived an equation for the BED taking into account decaying radioactive sources and a tumor repopulation factor. This model has been extended to include tumor volume changes during treatment due to edema and/or shrinkage (8, 19, 20). In this report, we incorporated a physical model of the edema process in implants and studied its influence on BED calculations. The calculations of the BED are based on images taken a few days or weeks after the implantation. As for the physical dose, there is an optimal time for acquiring those images for relevant BED determinations. In addition to the effects of edema, radiobiologic parameter values may influence the optimal timing of the BED evaluation. The purposes of this study were to examine the influence of the type of isotope, radiobiologic parameter values of prostate tumors, and the effect of edema-induced prostate swelling on the BED; and to determine the optimal time of image acquisition for seed reconstruction, so that the error in the calculated BED will be minimal for the chosen set of radiobiologic parameter values.
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dose rate d(t) at a point in the prostate can be expressed as a function of time t: d共t兲 ⫽ d0exp共⫺t兲
Physical dose When decaying radioactive sources (e.g., 125I, 103Pd, 198 Au) are used for the treatment of prostate tumors, the
(1)
where is the decay constant of the radioactive nuclide and d0 is the initial dose rate. A side effect of introducing the needles is edema, which results in swelling of the prostate. If we assume isotropic swelling of the prostate, the distance r of a point in the prostate tissue to any surrounding radioactive seed will change with time according to r共t兲 ⫽ rpre ⫹ 共r0 ⫺ rpre兲exp共⫺edemat兲
(2)
where rpre is the distance preplanned before implantation, r0 is the maximal distance shortly after implantation, and edema is the decay constant of prostate edema. The ratio R of the post- and preimplant distance is R共t兲 ⫽
r共t兲 ⫽ 1 ⫹ 共R0 ⫺ 1兲exp共⫺edemat兲 rpre
(3)
where R0 is the ratio of prostate dimensions shortly after implantation and preimplant dimensions. Because R(t) describes a relative enlargement of the prostate dimensions, the dose rate d(t) in the prostate with edema can be compared with the dose rate when no edema is present (5): d共t兲 ⫽
d0exp共⫺t兲 R2.7共t兲
(4)
where is the decay constant of the radioactive sources and d0 is the initial dose rate. The power 2.7 is explained by the consideration that the radioactive seeds are point sources (d ⬃1/r2) and scatter and absorption g(r) are modeled by g(r) ⬃1/r0.7, which is a valid approximation for 125I, 103Pd, and 198Au seeds. Biologically effective dose The linear-quadratic model (LQ model) of cell inactivation describes the relation between the cell survival fraction S and the dose D (16): S ⫽ exp关⫺共␣D ⫹ D2兲兴
(5)
where ␣ and  are constants. The BED is related to cell survival by (17): S ⫽ exp共⫺␣BED)
METHODS AND MATERIALS
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BED ⫽ D ⫹
(6)
 2 D ␣
For decaying sources, the BED at time t is expressed by (3):
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BED共t兲 ⫽ D兵1 ⫹ 2共d0兲 共/␣兲/共 ⫺ 兲其 ⫺ 0.693t/共␣Tp兲 (7)
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Table 1. Reference set of parameters used for calculation of biologically effective dose Parameter
with
⫽ 关1/共1 ⫺ 兲兴 兵共1 ⫺ 2兲/2 ⫺关1 ⫺ exp共⫺t兲兴/共 ⫹ 兲其 (8) where D is the total dose delivered in time t, d0 is the initial dose rate, is the decay constant of the radioactive source, is the repair rate constant of sublethal damage, Tp is the potential tumor doubling time, and ⫽ exp(⫺t). Equation 7 does not take into account prostate volume changes from edema caused by insertion of the implant needles. For permanent implants, the dose rate in prostate tumors with edema is defined in Eq. 4. Using this equation and the LQ model (Eq. 5), an expression for BED can be derived (Appendix A). The cumulative BED given with permanent implants in the effective treatment time teff and taking into account prostate changes due to edema is expressed by
BEDteff ⫽
冕
exp共⫺t兲 d0 2.7 dt ⫹ R 共t兲
teff
0
冕 冉冕
 2 d02 ␣
teff
0
t2
0
冊
exp共⫺t兲 exp共⫺t2兲 exp共t ⫺ t2兲dt dt2⫺ 2.7 R 共t兲 R2.7共t2兲 0.693teff ␣Tp
(9)
where ␣ is the linear and  the quadratic coefficient of the LQ model, is the repair rate constant of sublethal damage, Tp is the potential tumor doubling time, t2 is the time lapse after induction of sublethal damage, and teff is the effective treatment time. The BED is plotted as a function of the time after the implantation. The BED is a monotonously increasing function until it reaches its maximal value (BEDmax). The point at which the BED is maximum is called the effective treatment time (teff). The terms in Eq. 9 represent two opposite effects contributing to the BED: the first two terms are the rate of tumor cell kill due to irradiation and the third term is the rate of tumor cell repopulation. Those two rates are balanced at a time teff. At times in excess of teff, the effect of repopulation is larger than the effect of cell kill by radiation. Reference parameter set As shown in Eq. 9 many parameters influence the value of teff and the BEDmax achieved at teff. To investigate how the parameters influence BEDmax and teff, the changes in BEDmax and teff are calculated by changing the parameter values one at a time. In Table 1, a reference set of parameters is defined. This set of parameters was used to calculate the BED until the teff.
␣  R0 I125 Pd103 Au198 edema Tp d0(125I)
Value
Reference
0.09 Gy⫺1 0.03 Gy⫺2 0.693 h⫺1 1.09 0.0004798 h⫺1 0.001699 h⫺1 0.010697 h⫺1 0.0032 h⫺1 600 h (25 d) 0.068 Gyh⫺1
Dicker et al. (25) Brenner and Hall (24), Ling (3) Ling (3) Moerland (5), Waterman et al. (7) Nath et al. (21) Nath et al. (21) Ling (3) Moerland (5), Waterman et al. (7) Dicker et al. (25), Ling (3) Moerland (5)
In the case of permanent 125I implants with a half-life of 60 days (21), and the prescribed dose to cover the prostate of 140 Gy in 1 year, the initial dose rate d0 ⫽ 0.068 Gyh⫺1 (5). Other isotopes used for permanent prostate implants are 103 Pd and 198Au with a half-life of 17 and 2.7 days, respectively (3, 21). The d0 values are calculated so that the value of BEDmax for 103Pd and 198Au is equal to the BEDmax for 125 I, in the case of no edema (BEDmax ⫽ 74.9 Gy). Thus, the d0 value for 103Pd and 198Au is 0.16 Gyh⫺1 and 0.634 Gyh⫺1, respectively. These d0 values are somewhat different from values quoted in the literature (3), because we have made the assumption of equal values of BEDmax at teff when no edema is present and with the chosen set of parameters. Part of the radiation-induced damage to the cells will be repaired within 1–3 hours (3, 19, 22, 23). We used for the repair rate constant , the value of 0.693 h⫺1 (T1/2 ⫽ 1 h). In the LQ model, the ␣ and  constants can be related to the induction of lethal and sublethal damage, respectively. In general, for prostate tumors, an ␣/ value of ⱖ10 Gy is applied. For instance, Ling (3) performed BED calculations for an ␣/ value of 10 Gy. However, the results of recent studies have indicated that prostate tumors react like lateresponding tissues (19, 22, 24, 25). An ␣/ value of 3 Gy seems appropriate. To determine the influence of the ␣/ value, ␣/ was varied from 1 to 20 Gy by varying ␣ from 0.03 Gy⫺1 to 0.6 Gy⫺1, while keeping  at 0.03 Gy⫺2 (3, 24). Another radiobiologic parameter is the potential tumor doubling time, Tp. For prostate tumors, Tp may vary from 5 to 67 days (3, 25, 26). In our calculations, the Tp reference value was set to 25 days and the Tp is varied between 10 and 40 days to investigate its influence on the BEDmax and teff. The parameters related to edema, the decay constant of prostate edema edema, and the ratio of post- and preimplant prostate dimensions R0 were also varied to investigate the influence on the BED. Moerland (5) and Waterman et al. (7) found edema-induced prostate volume increases between 0% and 52%, corresponding to R0 variations between 0 and 1.15. The reference value was set to R0 ⫽ 1.09, equivalent to a 30% prostate volume increase, when isotropic enlargement of the prostate is assumed. Postimplant prostate volumes were found to return to preimplant prostate volumes with a mean half-life of about 9 days and variations between 3 and 23 days. This corresponded with a mean edema of
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Fig. 1. BED of the prostate with permanent 125I implants as a function of time for different values of R0 using Eq. 9, and the parameters defined in Table 1. solid line, R0 ⫽ 1.0; large dashes, R0 ⫽ 1.05; short dashes, R0 ⫽ 1.09; and smallest dashes, R0 ⫽ 1.15.
0.0032 h⫺1 and variations between 0.01 and 0.00125 h⫺1, respectively. Optimal timing of implant evaluation In clinical practice, images of the prostate with the implanted radioactive seeds are taken several days or weeks after implantation. Dosimetry calculations are done on the basis of the geometry of the seeds on these images. Because of edema, the volume of the prostate and hence, the distance between the seeds, varies with time. Thus, the moment of taking those images, tm, is important for a correct determination of BED. The optimal timing of implant evaluation is defined as the point after the implantation at which the images are taken and the error in the calculated BED is minimal for the chosen set of radiobiologic parameters values. Moerland (5) has defined, for the physical dose, a parameter ⑀ as the ratio of the estimated cumulative dose and the real cumulative dose as a function of the timing of implant evaluation tm. In analogy, ⑀ can be defined as the ratio of the estimated BED, with a constant volume of the prostate between implantation and teff and the “real” BED, with a variable prostate volume between implantation and teff (for the description of ⑀, see Appendix B). When ⑀ is equal to 1, the optimal time of implant evaluation is achieved for the chosen set of parameters. RESULTS Influence of parameters on BED The BED is calculated as a function of the time after implantation with the parameters in Table 1 (Eq. 9). To investigate the influence of edema on the BED, the extent of edema was varied. The BED was plotted for varying values of R0 in the case of 125I implants in Fig. 1. Without edema (R0 ⫽ 1.0), the BEDmax was equal to 74.9 Gy and teff to 146 days (3504 h). With a prostate volume increase of 30% (R0 ⫽ 1.09), BEDmax decreased to 70.3 Gy, but teff was still 146 days (3504 h). BEDmax decreased further to 67.6 Gy for
a prostate volume increase of 52% (R0 ⫽ 1.15), again the teff was 146 days. The course of prostate swelling is characterized by edema. Varying edema between 0.00125 and 0.01 h⫺1 (equivalent to a half-life of edema between 3 and 23 days), BEDmax varied between 66 and 73.2 Gy and teff remained 146 days (data not shown). In addition to edema, the BED depends on radiobiologic parameter values. The influence of ␣, the parameter of the linear component of the LQ model is shown in Fig. 2. The ␣ values ranged from 0.6 to 0.03 Gy⫺1, with  ⫽ 0.03 Gy⫺2, the ␣/ values ranged from 1 to 20 Gy. The BEDmax ranged from 20 to 120 Gy and teff from 58 days (1392 h) to 310 days (7440 h). Thus, when 125I seeds are implanted, ␣ (or ␣/) influences BEDmax and teff significantly. The  value, correlated with the induction of sublethal damage, on the other hand, hardly changed the BEDmax and teff (data not shown). In Fig. 3, the BED was plotted as a function of time after implantation of 125I seeds for varying values of Tp. A variation in Tp between 10 and 40 days (240 and 960 h) resulted in a variation of BEDmax between 24 and 89 Gy and a variation in teff between 68 and 187 days (1632 and 4488 h). Thus, when the repopulation of prostate tumors is slow, as indicated by Haustermans et al. (26), the BEDmax is large and the t eff is long. The BED also depends on the physical parameters, depending on the choice of the isotope. For instance, 125I has a half-life of 60 days. To cover the prostate with a physical dose of 140 Gy in 1 year, the initial dose rate is equal to 0.068 Gyh⫺1. As shown in Fig. 1, the BEDmax is 74.9 Gy when edema is not taken into account. To relate the BEDmax of 103Pd and 198Au implants to the BEDmax of 125I implants, the initial dose rates are calculated for the implants with the two other isotopes. In the situation without edema, d0 is determined for the BEDmax value of 74.9 Gy. This results in d0 ⫽ 0.16 Gyh⫺1 for 103Pd implants (half-life 17 days) and d0 ⫽ 0.634 Gyh⫺1 for 198Au implants (half-life 2.7 days). With the parameters in Table 1 thus with edema, the BED is calculated as a function of time after implantation for
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Fig. 2. BED of the prostate with permanent 125I implants as a function of time for different values of ␣ using Eq. 9 and the parameters defined in Table 1. 103
Pd and 198Au permanent implants. Figure 4 shows that BEDmax and teff are not the same for those isotopes. For 103 Pd and 198Au, these values are 66.1 Gy and 62 days and 55.6 Gy and 15 days, respectively; without edema, these values are 74.9 Gy and 62 days and 74.9 and 15 days, respectively. This illustrates that edema has more effect on the BED for the short-living isotopes. Optimal timing of implant evaluation Due to edema, the timing of image acquisition is important for seed reconstruction and BED calculations. When implant evaluation takes place at the right time, a realistic BED is calculated. In other words, ⑀ must be equal to 1 Appendix A (Eq. 23). Figures 5 through 7 show ⑀ as a function of the timing of implant evaluation and R0 for 125I, 103 Pd, and 198Au seeds, respectively. The parameters used in Eq. 23 are those defined in Table 1. Implant evaluation right after the implantation always
Fig. 3. BED of the prostate with permanent 125I implants as a function of time for different values of the potential tumor doubling time using Eq. 9 and the parameters defined in Table 1.
results in an ⑀ ⬍1 (for the three isotopes and for all R0 values). For 125I implants, the underestimation of the BED is larger than for 103Pd or 198Au implants. The bigger the R0 value, the larger the underestimation of the dose. Implant evaluation at the effective treatment time results always in an ⑀ ⬎1. The largest overestimation of the BED is calculated when R0 ⫽ 1.15 for all isotopes. Implant evaluation at the effective treatment time results in a larger overestimation of the dose for 198Au implants than for 125I or 103Pd implants. Therefore, there is a point after implantation at which ⑀ ⫽ 1. This optimal timing of implant evaluation depends on the R0 value and the isotope. For small values of R0 and thus small prostate volume increases, ⑀ is close to 1 for the three isotopes at any time after implantation. For a typical prostate volume increase of 30% (R0 ⫽ 1.09) with a half-life of 9 days (edema ⫽ 0.0032 h⫺1), the optimal timing of implant evaluation is 25, 13, and 3 days
Fig. 4. BED of the prostate with permanent implants as a function of time for different isotopes using Eq. 9 and the parameters defined in Table 1.
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Fig. 5. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 125I seeds.
after implantation for 125I, 103Pd, and 198Au implants, respectively. For 125I implants, ⑀ was plotted as a function of R0 and edema for 3 set times after implant: 1 day (Fig. 8A), 25 days (Fig. 8B), and 146 days (Fig. 8C). Implant evaluation 1 day after the seed implantation always results in an underestimation of the BED, because it ignores prostate shrinkage to preimplant dimensions and thus underestimates the dose distribution in the time after implantation. Underestimation depends on the extent of prostate edema and half-life of edema and may amount to 43% for R0 ⫽ 1.15 and edema ⫽
0.0032 h⫺1 (half-life 9 days). When implant evaluation is performed 25 days after seed implantation, a slight overestimation of ⬍1% is found for prostate edema with a half-life of 9 days. At 25 days, the overestimation is at most 3% for prostate edema with a relatively small half-life of 3 days (edema ⫽ 0.01 h⫺1). A relatively large half-life of 23 days (edema ⫽ 0.00125 h⫺1) results in an underestimation of at most 10%. Implant evaluation at teff (146 days after implantation of 125I seeds and the parameter values as defined in Table 1) always results in an ⑀ ⬎1 because of overestimation of the dose contribution delivered in the period when
Fig. 6. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 103Pd seeds.
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Fig. 7. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 198Au seeds.
the prostate was swollen. Depending on R0 and edema, the overestimation can be at most 22%. As mentioned before, ␣ and Tp have a strong influence on BEDmax and teff. A smaller ␣ value (0.045 instead of 0.09 Gy⫺1 or ␣/ value of 1.5 instead of 3 Gy) results in a smaller teff (for other values of ␣, see Fig. 2), and the optimal timing of implant evaluation decreases from 25 to 21 days after implantation (data not shown). An increase of the potential tumor doubling time from 25 to 40 days shifts the optimal timing of implant evaluation by only 1 day from 25 to 26 days (data not shown). Figure 9 shows the influence of R0 and edema on the optimal timing of implant evaluation for different isotopes. Optimal timing does not depend on R0, but edema has some influence, especially in case of implants with isotopes with a half-life that is large compared with the half-life of prostate edema, such as 125I implants. For the range of prostate edema half-lives of 3 to 23 days, the optimal timing for implant evaluation ranges from 12 to 37 days, 8 to 16 days, and 3 to 4 days for 125I, 103Pd, and 198Au, respectively. DISCUSSION BED calculations for prostate brachytherapy have been the subject of studies by several authors (3, 25). Dicker et al. (25) calculated BED values for permanent prostate implants and concluded that the BEDmax is equal to 76 Gy (␣ ⫽ 1.0 Gy⫺1, Tp ⫽ 23 days) and 102 Gy (␣ ⫽ 1.0 Gy⫺1, Tp ⫽ 23 days) for 125I and 103Pd implants, respectively. In these studies, postimplant edema was neglected. We derived a theoretical model to calculate the BED in case of a homogeneous physical dose delivery to the prostate taking into account prostate dimension changes due to postimplant edema. The extent of prostate volume increase due to edema
and the edema half-life may vary from patient to patient. For typical edema values (R0 ⫽ 1.09 and edema ⫽ 0.0032 h⫺1) and radiobiologic parameters (␣ ⫽ 0.09 Gy⫺1, Tp ⫽ 25 days), we found BEDmax values of 70.3 Gy and 66.1 Gy for 125 I and 103Pd implants. Variations of R0 and edema showed a further decrease of BED for larger R0 and smaller edema values. The extent of induced lethal damage, represented by ␣ in the LQ equation, and the potential tumor doubling time, Tp, are the only radiobiologic parameters that strongly influence BEDmax. Dicker et al. (25) and Ling (3) found that an increase of Tp results in an increase of BEDmax. They did not take edema effects into account. This is consistent with the results of our calculations: the longer the potential tumor doubling time, the larger the BEDmax and the longer the teff. We varied the Tpot between 10 and 40 days. Recent literature suggests that the Tpot of prostate tumors is around 40 days (19). The teff also depends on ␣. A small ␣ value means a minor extent of lethal damage, and therefore the BEDmax is relatively small and the teff relatively short. Permanent implants deliver 75% of the physical dose in 2 isotope half-lives, which is 5.4, 26, and 120 days for 198Au, 103 Pd, and 125I implants, respectively. Thus, effective treatment times are shorter for implants with short-living isotopes and the effect of postimplant prostate volume changes are larger for these implants. As pointed out by Antipas et al. (19), the absolute teff and BEDmax values depend on the relative biological effectiveness (RBE) of the applied isotope. However, clinical data on RBE values for 125I, 103Pd, and 198Au as a function of the dose rate in prostate brachytherapy are scarce. Therefore, as a reference dose, we applied the physical dose of 140 Gy as applied in clinical prostate 125I brachytherapy and calculated initial dose rates
Biologically effective dose for permanent prostate implants
Fig. 8. ⑀ as a function of R0 and edema: (a) 1 day after implantation of 125I seeds in the prostate, (b) 25 days after implantation, and (c) 146 days after implantation. Solid line represents edema ⫽ 0.01 h⫺1; dashed line, edema ⫽ 0.0032 h⫺1; and dotted line, edema ⫽ 0.00125 h⫺1.
for 103Pd and 198Au implants to get equivalent BED values in nonedematous prostates, taking into account the radiobiologic parameters and assuming equal RBE. Incorporating different (dose-rate– dependent) RBE values would change the initial dose rates required to get equivalent BEDmax values, but would not change the general conclusions regarding the change of BEDmax as a function of the radiobiologic parameters and edema. In clinical practice, several days or weeks after the implantation, images are taken for
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implant evaluation. Because of induced edema and the decaying dose rate of the seeds, there is an optimal time to take those images. We found an optimal time of 25 days, 13 days, and 3 days after implantation for 125I, 103Pd, and 198 Au implants, respectively. The optimal timing of implant evaluation for the BED is earlier after implantation than the optimal timing for the evaluation of the physical dose. Moerland (5) determined an optimal timing for the physical dose of 28 days after implantation of 125I seeds. However, his calculation of the optimal timing for the physical dose did not take into account repopulation of tumor cells. As shown, the faster the repopulation, the shorter the effective treatment time and so the optimal timing for implant evaluation is earlier after the implantation. Because of logistic reasons, MRI is sometimes done 1 day after seed implantation. Then, the prostate is swollen and therefore an underestimation of the BED of at most 43% is calculated, depending on the extent and the half-life of prostate edema. Yue et al. (14) examined, for a range of half-lives and magnitudes of prostate volume increase, the effect on the optimal time of evaluation for the physical dose. They concluded that the optimal time was highly dependent on the half-life of edema, but the magnitude of prostate volume increase did not change the optimal time significantly for both 125I and 103Pd implants. Our calculations of the BED led to the same conclusion: the optimal timing of image acquisition for the calculation of the BED of 125I, 103Pd, or 198Au implants strongly depends on the half-life of the edema. The larger the half-life of the isotope, the bigger the influence of edema on the optimal timing of the implant evaluation. In addition, the optimal timing is not influenced by the magnitude of the prostate volume increase. For 125I implants and typical prostate edema values (R0 ⫽ 1.09 and edema ⫽ 0.0032 h⫺1), the optimal timing of implant evaluation is 25 days after the implantation. However, in practice, edema will vary from patient to patient. The variations are probably between 0.01 h⫺1 (t1/2 ⫽ 3 days) and 0.00125 h⫺1 (t1/2 ⫽ 23 days), with a mean value of 0.0032 h⫺1 (t1/2 ⫽ 9 days) (7). As a consequence, when the images are taken 25 days after the implantation and edema is either 0.01 h⫺1 or 0.00125 h⫺1, an overestimation of the BED by 3% or an underestimation of the BED by 10% can be calculated, respectively.
CONCLUSION Calculations of the BED after implantation of radioactive seeds in the prostate, based on images taken shortly after implantation, results in an underestimation of the BED of at most 43% for 125I. The BED is overestimated when calculations are based on images taken at teff of at most 9%. Thus, there is an optimal timing of implant evaluation for calculations of the BED, which depends on radiobiologic parameter values and the magnitude and half-life of prostate
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Fig. 9. (a) Optimal timing of implant evaluation as a function of half-life of edema for 125I, 103Pd, and 198Au seeds. (b) Optimal timing of implant evaluation as a function of the extent of edema, R0, for 125I, 103Pd, and 198Au seeds.
volume increase. For the most likely parameter values, we found an optimal time of image acquisition for seed recon-
struction for calculations of the BED of 25 days after seeds implantation.
125
I
REFERENCES 1. Nag S, Beyer D, Friedland J, et al. American Brachytherapy Society (ABS) recommendations for transperineal permanent brachytherapy of prostate cancer. Int J Radiat Oncol Biol Phys 1999;44:789 –799. 2. Nag S, Baird M, Blasko J, et al. American Brachytherapy Society survey of current clinical practice for permanent brachytherapy of prostate cancer. J Brachytherapy Int 1997; 13:243–251. 3. Ling CC. Permanent implants using Au-198, Pd-103 and I-125: Radiobiological considerations based on the linear quadratic model. Int J Radiat Oncol Biol Phys 1992;23:81– 87. 4. Holm HH, Juul N, Pedersen JF, et al. Transperineal iodine125 seed implantation in prostatic cancer guided by transrectal ultrasonography. J Urol 1983;130:283–286. 5. Moerland MA. The effect of edema on postimplant dosimetry of permanent iodine-125 prostate implants: A simulation study. J Brachytherapy Int 1998;14:225–231.
6. Moerland MA, Wijrdeman HK, Beersma R, et al. Evaluation of permanent I-125 prostate implants using radiography and magnetic resonance imaging. Int J Radiat Oncol Biol Phys 1997;37:927–933. 7. Waterman FM, Yue N, Corn BW, et al. Edema associated with I-125 or Pd-103 prostate brachytherapy and its impact on post-implant dosimetry: An analysis based on serial CT acquisition. Int J Radiat Oncol Biol Phys 1998;41:1069 –1077. 8. Dale RG, Jones B, Coles IP. Effect of tumour shrinkage on the biological effectiveness of permanent brachytherapy implants. Br J Radiol 1994;67:639 – 645. 9. Willins J, Wallner K. CT-based dosimetry for transperineal I-125 prostate brachytherapy. Int J Radiat Oncol Biol Phys 1997;39:347–353. 10. Roy JN, Wallner KE, Harrington PJ, et al. A CT-based evaluation method for permanent implants: Application to prostate. Int J Radiat Oncol Biol Phys 1993;26:163–169.
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11. Grimm PD, Blasko JC, Ragde H. Ultrasound-guided transperineal implantation of iodine-125 and palladium-103 for the treatment of early-stage prostate cancer: Technical concepts in planning, operative technique, and evaluation. Atlas Urol Clin North Am 1994;2:113–125. 12. Narayana V, Roberson PL, Pu AT, et al. Impact of differences in ultrasound and computed tomography volumes on treatment planning of permanent prostate implants. Int J Radiat Oncol Biol Phys 1997;37:1181–1185. 13. Narayana V, Roberson PL, Winfield RJ, et al. Impact of ultrasound and computed tomography prostate volume registration on evaluation of permanent prostate implants. Int J Radiat Oncol Biol Phys 1997;39:341–346. 14. Yue N, Chen Z, Peschel R, et al. Optimum timing for imagebased dose evaluation of I-125 and Pd-103 prostate seed implants. Int J Radiat Oncol Biol Phys 1999;45:1063–1072. 15. Chen Z, Yue N, Wang X, et al. Dosimetric effects of edema in permanent prostate seed implants: A rigorous solution. Int J Radiat Oncol Biol Phys 2000;47:1405–1419. 16. Barendsen GW. Dose fractionation, dose rate and iso-effect relationships for normal tissue responses. Int J Radiat Oncol Biol Phys 1982;8:1981–1997. 17. Dale RG. The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy. Br J Radiol 1985;58:515–528. 18. Dale RG. Radiobiological assessment of permanent implants using tumour repopulation factors in the linear-quadratic model. Br J Radiol 1989;62:241–244.
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19. Antipas V, Dale RG, Coles IP. A theoretical investigation into the role of tumour radiosensitivity, clonogen repopulation, tumour shrinkage and radionuclide RBE in permanent brachytherapy implants of I-125 and Pd-103. Phys Med Biol 2001; 46:2557–2569. 20. Dale RG, Jones B. The assessment of RBE effects using the concept of biologically effective dose. Int J Radiat Oncol Biol Phys 1999;43:639 – 645. 21. Nath R, Anderson LL, Luxton G, et al. (GRP) Dosimetry of interstitial brachytherapy sources: Recommendations of the AAPM Radiation Therapy Committee Task Group No. 43. Med Phys 1995;22:209 –234. 22. Fowler J, Chappell R, Ritter M. Is ␣/ for prostate tumors really low? Int J Radiat Oncol Biol Phys 2001;50:1021–1031. 23. King CR. What is the Tpot for prostate cancer? Radiobiological implications of the equivalent outcome with I-125 or Pd-103. Int J Radiat Oncol Biol Phys 2000;47:1165–1167. 24. Brenner DJ, Hall EJ. Fractionation and protraction for radiotherapy of prostate carcinoma. Int J Radiat Oncol Biol Phys 1999;43:1095–1101. 25. Dicker AP, Lin C, Leeper DB, et al. Isotope selection for permanent prostate implants? An evaluation of Pd-103 versus I-125 based on radiobiological effectiviness and dosimetry. Semin Urol Oncol 2000;18:152–159. 26. Haustermans KMG, Hofland I, Van Poppel H, et al. Cell kinetic measurements in prostate cancer. Int J Radiat Oncol Biol Phys 1997;37:1067–1070.
APPENDIX A Dale (17, 18) derived an equation for BED for low-doserate irradiation taking into account repopulation factors. We derived, analogous to Dale’s derivation, an equation for BED, including changing the distance between seeds and the effects of edema. Radiation treatment of tumor with a decaying source The dose rate at any point in the prostate with edema can be compared with the dose rate when no edema is present (see “Physical dose”): d共t兲 ⫽
d0exp共⫺t兲 R2.7共t兲
(10)
with the decay constant of the radioactive sources, d0 the initial dose rate, and R(t) as expressed in Eq. 3. The power 2.7 is explained by the consideration that the radioactive seeds are point sources (d ⬃1/r2), and scatter and absorption g(r) are approximated by g(r) ⬃1/r0.7 (5, 21). Lethal damage (type A damage [17]) induced in the effective treatment time teff after irradiation is expressed as
冕
teff
0
␣d共t兲dt ⫽
冕
teff
0
␣d0exp共⫺t兲 dt R2.7共t兲
(11)
with ␣ the linear constant of the linear-quadratic model. Dale (17) defined lethal damage of type B as the damage resulting from a lethal combination of the damage in the
target pairs induced at two separate sequential irradiations. The probability of sublethal damage in time dt is pd共t兲dt
(12)
with p the probability per unit dose that a sublethal event occurs (i.e., only one target of a pair is hit). Sublethal damage repair can be expressed by e⫺t, with , the sublethal damage repair rate constant. The probability that either target of a pair of targets is hit in one radiation event is equal to 2pd(t)dt. Substituting d(t) (Eq. 10) and taking into account that the damage still exists at time t1, the probability is 2p d0 exp共⫺t兲exp共⫺t1兲 dt R2.7共t兲
(13)
Let t2 ⫽ t ⫹ t1, then t1 ⫽ t2 ⫺ t and then 2p d0 exp共⫺t兲exp关⫺共t2 ⫺ t兲兴 dt R2.7共t兲
(14)
The total probability of sublethal damage in time t2 is presented by
2p d0
冕
t2
0
exp共⫺t ⫺ t2 ⫹ t兲 dt R2.7共t兲
(15)
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The probability that a second undamaged target receives a hit in time dt is pd(t)dt. Therefore, the probability that lethal damage occurs in time dt2 after previous irradiation of duration t2:
冉冕
t2
2 p2 d02
0
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RE ⫽ 1
冕 冉冕 teff
2 d02
0
⫹
冊
exp共⫺t ⫹ t ⫺ t2兲 exp共⫺t2兲 dt dt2 2.7 R 共t兲 R2.7共t2兲
t2
0
冊
exp共⫺t2兲 exp共⫺t ⫹ t ⫺ t2兲 dt dt2 2.7 R 共t兲 R2.7共t2兲
冕
(16)
teff
0
␣ d0 exp共⫺t兲 dt R2.7共t兲 (19)
with the probability that two sublethally damaged targets interact to produce lethal damage. The total probability of lethal damage per target pair in time teff (type B damage) is
This results for the BED dose in
冉冕 冊 teff
BED ⫽ RE
d共t兲dt ⫺
0
冕 冉冕 teff
2d02
0
t2
0
冊
exp共⫺t ⫹ t ⫺ t2兲 exp共⫺t2兲 dt dt2 (17) 2.7 R 共t兲 R2.7共t2兲
with  ⫽ np and n the number of target pairs. According to Dale (17), the average relative effectiveness RE per unit dose (from the linear-quadratic equation BED ⫽ Dtot * RE) is
RE ⫽ 1 ⫹
lethal damage of type B lethal damage of type A
(20)
where the last part of the equation is the result of tumor cell repopulation; Tp is the potential tumour doubling time. Therefore,
冕 冕 冉冕 teff
2
0.693teff ␣Tp
BED ⫽
d0
0
 ⫹ 2 d02 ␣
teff
0
exp共⫺t兲 dt R2.7共t兲 t2
0
冊
exp共⫺t兲 exp共⫺t2兲 exp共t ⫺ t2兲dt dt2 R2.7共t兲 R2.7共t2兲 ⫺
(18)
0.693teff ␣Tp
(21)
APPENDIX B Optimal time of implant evaluation After implantation of the radioactive seeds in the prostate, the prostate is swollen from the induced edema. Because the extent of edema changes with time, the findings of the implant evaluation depend on the timing after implantation. The ratio ⑀ of the calculated BED, with a constant prostate
⑀共tm兲 ⫽
冕
冕 冕 冉冕
exp共⫺t兲 dt R2.7共tm兲
teff
d0
0
teff
0
 d共t兲dt ⫹ 2 ␣
teff
0
t2
0
teff
0
⑀⫽
冕 冉冕
 ␣
teff
0
t2
0
BED共volume constant兲 BED共volume variable兲
(22)
With Eqs. 3, 4 and 9, ⑀ can be expressed by
⫹
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
2
冕
冊
volume, and the “real” BED, with a variable prostate volume, can be expressed by
冊 冊
exp共⫺t兲 exp共⫺t2兲 d0 2.7 exp共t ⫺ t2兲dt d0 2.7 dt2 R 共tm兲 R 共tm兲
冕 冉冕
 d共t兲dt ⫹ 2 ␣
teff
0
t2
0
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
冕
teff
0
冕 冉冕
 d共t兲dt ⫹ 2 ␣
teff
0
0.693teff ␣Tp t2
0
⫺
冊
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
(23)
Biologically effective dose for permanent prostate implants
with  the quadratic coefficient of the linear-quadratic model, ␣ the linear coefficient of the model, the repair rate constant of sublethal damage, Tp the potential tumor doubling time, t2 the time sublethal damage exists, t the time within sublethal damage occurs, teff the effective
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433
treatment time, tm the time of implant evaluation after implantation, d0 the initial dose rate, d(t) the dose rate at time t (Eq. 4), the decay constant of radioactive source, and R(t) as defined in Eq. 3. The optimal timing of implant evaluation is determined when ⑀ is equal 1.
PII S0360-3016(02)02755-4
BIOLOGY CONTRIBUTION
BIOLOGICALLY EFFECTIVE DOSE FOR PERMANENT PROSTATE BRACHYTHERAPY TAKING INTO ACCOUNT POSTIMPLANT EDEMA MARION P.R. VAN GELLEKOM, M.SC., MARINUS A. MOERLAND, PH.D., HENK B. KAL, PH.D, JAN J. BATTERMANN, M.D., PH.D.
AND
Department of Radiotherapy, University Medical Center Utrecht, Utrecht, The Netherlands Purpose: To study the influence of radiobiologic and physical parameters and parameters related to edema on the biologically effective dose (BED) for permanent prostate implants and to determine the optimal timing of seed reconstruction for BED calculation. Methods and Materials: On the basis of the linear-quadratic model, an expression for the BED was derived, including the edema parameters. A set of parameter values was defined, and these parameter values were varied one at a time to examine the effect on the BED and the theoretically effective treatment time (teff). A ratio ⑀ was defined to investigate the optimal timing of seed reconstruction. Results: The maximal BED decreases when the extent of lethal damage is smaller, the potential tumor doubling time is smaller, the half-life time of the seeds is shorter, and the magnitude of prostate volume increase is larger. For 125I, the optimal timing of seed reconstruction is 25 days after implantation. Seed reconstruction 1 day after the implantation results in an underestimation of the BED of at most 43%, depending on the magnitude and half-life of edema. An overestimation of the BED of at most 22% is calculated when seed reconstruction took place at the effective treatment time. Conclusion: The maximal BED depends strongly on the value of ␣, the potential tumor doubling time, and the choice of isotope. If prostate volume increase due to edema is not taken into account, the BED will be underestimated shortly after the implantation and overestimated if the calculations are based on images taken several months after implantation. The optimal timing of BED evaluation for 125I seed implants and typical prostate edema values is 25 days after implantation. © 2002 Elsevier Science Inc. BED, Prostate, Brachytherapy, Edema.
INTRODUCTION
ual shrinkage (8). We derived prostate volume changes from radioactive seed distances on radiographs, originally acquired at 3 days, 1 month, and 3 months after implant to check for any loss of seeds, and found that prostate swelling resolves exponentially in time (data not shown). Waterman et al. (7) acquired series of CT scans and also observed that prostate swelling resolved exponentially in time. Willins and Wallner (9) acquired pre- and postimplant CT scans from 11 patients and found that in all patients, the prostate returned to the preimplant size within 2 months of the procedure and was stable at between 2 and 6 months, indicating that edema is probably the most significant contributor to prostate volume changes. From these observations, which were not specific for resolving edema or radiation-induced shrinkage, we conclude that an exponential model adequately describes the prostate volume changes in the first half-year after implant when the dose is actually administered. Postimplant dosimetry may be based on ultrasonography,
Small tumors of the prostate (Stage T1 and T2) can be treated with brachytherapy. 125I, 103Pd, or 198Au seeds are most commonly used for permanent implantation. The prescription doses for permanent prostate implants with 125I, 103 Pd, or 198Au seeds are, in general, 140, 120, or 60 Gy, respectively (1–3). The smaller prescribed physical doses for 103Pd and 198Au implants compared with 125I implants occur from matching biologic end points for nonedematous prostates when the materials implanted have shorter halflives and therefore higher initial dose rates than 125I. The commonly used technique to introduce the radioactive seeds into the prostate is the transrectal ultrasoundguided transperineal technique (4). The dose actually administered to the prostate is influenced by the postimplant volume changes. A short-term effect is prostate swelling from edema caused by piercing the prostate with needles (5–7); a long-term effect may be radiation-induced continReprint requests to: Marion P.R. Van Gellekom, M.Sc., Department of Radiotherapy, University Medical Center Utrecht, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands. Tel: ⫹31-302509863; Fax: ⫹31-30-2581226; E-mail: m.p.r.vangellekom@ radth.med.uu.nl
Supported by the J. A. Cohen Institute for Radiopathology and Radiation Protection, Leiden, The Netherlands. Received Sep 13, 2001, and in revised form Jan 3, 2002. Accepted for publication Jan 24, 2002. 422
Biologically effective dose for permanent prostate implants
MRI, or CT. On ultrasonography, the seeds give rise to artifacts that hamper evaluation of the prostate volume and the seed distribution. A CT-based evaluation method for permanent prostate implants has been developed by Roy et al. (10) and has become common practice thanks to the widespread availability of CT scanners (1). However, several investigators have remarked that CT images frequently overestimate the size of the prostate in the anterior and lateral areas, because the periprostatic fat pad and venous plexus are not well distinguished from the prostate gland itself (11–13). At our institute, we developed a method for postimplant dosimetry that uses the excellent soft-tissue imaging capabilities of MRI to delineate the prostate and the high resolution of isocentric radiographs to identify the seeds (6). Irrespective of the choice of imaging modality, an optimal time of acquiring the images for postimplant dosimetry exists. Moerland (5) derived an equation for the prostate dose as a function of the magnitude and half-life of the prostate volume increase induced by edema and found an optimal time of seed reconstruction for dose calculations, termed implant evaluation. Before this time, an underestimation of the dose will be calculated and, after the optimal time, the dose will be overestimated. Thus, without taking edema into account, the calculated physical total dose is not a good representation of the real dose in the prostate (14, 15). When the radiobiologic parameters (e.g., potential tumor doubling time and repair of sublethal damage) are taken into account, the biologically effective dose (BED), based on the linear-quadratic model (16), can be calculated. Dale (17, 18) derived an equation for the BED taking into account decaying radioactive sources and a tumor repopulation factor. This model has been extended to include tumor volume changes during treatment due to edema and/or shrinkage (8, 19, 20). In this report, we incorporated a physical model of the edema process in implants and studied its influence on BED calculations. The calculations of the BED are based on images taken a few days or weeks after the implantation. As for the physical dose, there is an optimal time for acquiring those images for relevant BED determinations. In addition to the effects of edema, radiobiologic parameter values may influence the optimal timing of the BED evaluation. The purposes of this study were to examine the influence of the type of isotope, radiobiologic parameter values of prostate tumors, and the effect of edema-induced prostate swelling on the BED; and to determine the optimal time of image acquisition for seed reconstruction, so that the error in the calculated BED will be minimal for the chosen set of radiobiologic parameter values.
● M.P.R. VAN GELLEKOM et al.
dose rate d(t) at a point in the prostate can be expressed as a function of time t: d共t兲 ⫽ d0exp共⫺t兲
Physical dose When decaying radioactive sources (e.g., 125I, 103Pd, 198 Au) are used for the treatment of prostate tumors, the
(1)
where is the decay constant of the radioactive nuclide and d0 is the initial dose rate. A side effect of introducing the needles is edema, which results in swelling of the prostate. If we assume isotropic swelling of the prostate, the distance r of a point in the prostate tissue to any surrounding radioactive seed will change with time according to r共t兲 ⫽ rpre ⫹ 共r0 ⫺ rpre兲exp共⫺edemat兲
(2)
where rpre is the distance preplanned before implantation, r0 is the maximal distance shortly after implantation, and edema is the decay constant of prostate edema. The ratio R of the post- and preimplant distance is R共t兲 ⫽
r共t兲 ⫽ 1 ⫹ 共R0 ⫺ 1兲exp共⫺edemat兲 rpre
(3)
where R0 is the ratio of prostate dimensions shortly after implantation and preimplant dimensions. Because R(t) describes a relative enlargement of the prostate dimensions, the dose rate d(t) in the prostate with edema can be compared with the dose rate when no edema is present (5): d共t兲 ⫽
d0exp共⫺t兲 R2.7共t兲
(4)
where is the decay constant of the radioactive sources and d0 is the initial dose rate. The power 2.7 is explained by the consideration that the radioactive seeds are point sources (d ⬃1/r2) and scatter and absorption g(r) are modeled by g(r) ⬃1/r0.7, which is a valid approximation for 125I, 103Pd, and 198Au seeds. Biologically effective dose The linear-quadratic model (LQ model) of cell inactivation describes the relation between the cell survival fraction S and the dose D (16): S ⫽ exp关⫺共␣D ⫹ D2兲兴
(5)
where ␣ and  are constants. The BED is related to cell survival by (17): S ⫽ exp共⫺␣BED)
METHODS AND MATERIALS
423
BED ⫽ D ⫹
(6)
 2 D ␣
For decaying sources, the BED at time t is expressed by (3):
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BED共t兲 ⫽ D兵1 ⫹ 2共d0兲 共/␣兲/共 ⫺ 兲其 ⫺ 0.693t/共␣Tp兲 (7)
Volume 53, Number 2, 2002
Table 1. Reference set of parameters used for calculation of biologically effective dose Parameter
with
⫽ 关1/共1 ⫺ 兲兴 兵共1 ⫺ 2兲/2 ⫺关1 ⫺ exp共⫺t兲兴/共 ⫹ 兲其 (8) where D is the total dose delivered in time t, d0 is the initial dose rate, is the decay constant of the radioactive source, is the repair rate constant of sublethal damage, Tp is the potential tumor doubling time, and ⫽ exp(⫺t). Equation 7 does not take into account prostate volume changes from edema caused by insertion of the implant needles. For permanent implants, the dose rate in prostate tumors with edema is defined in Eq. 4. Using this equation and the LQ model (Eq. 5), an expression for BED can be derived (Appendix A). The cumulative BED given with permanent implants in the effective treatment time teff and taking into account prostate changes due to edema is expressed by
BEDteff ⫽
冕
exp共⫺t兲 d0 2.7 dt ⫹ R 共t兲
teff
0
冕 冉冕
 2 d02 ␣
teff
0
t2
0
冊
exp共⫺t兲 exp共⫺t2兲 exp共t ⫺ t2兲dt dt2⫺ 2.7 R 共t兲 R2.7共t2兲 0.693teff ␣Tp
(9)
where ␣ is the linear and  the quadratic coefficient of the LQ model, is the repair rate constant of sublethal damage, Tp is the potential tumor doubling time, t2 is the time lapse after induction of sublethal damage, and teff is the effective treatment time. The BED is plotted as a function of the time after the implantation. The BED is a monotonously increasing function until it reaches its maximal value (BEDmax). The point at which the BED is maximum is called the effective treatment time (teff). The terms in Eq. 9 represent two opposite effects contributing to the BED: the first two terms are the rate of tumor cell kill due to irradiation and the third term is the rate of tumor cell repopulation. Those two rates are balanced at a time teff. At times in excess of teff, the effect of repopulation is larger than the effect of cell kill by radiation. Reference parameter set As shown in Eq. 9 many parameters influence the value of teff and the BEDmax achieved at teff. To investigate how the parameters influence BEDmax and teff, the changes in BEDmax and teff are calculated by changing the parameter values one at a time. In Table 1, a reference set of parameters is defined. This set of parameters was used to calculate the BED until the teff.
␣  R0 I125 Pd103 Au198 edema Tp d0(125I)
Value
Reference
0.09 Gy⫺1 0.03 Gy⫺2 0.693 h⫺1 1.09 0.0004798 h⫺1 0.001699 h⫺1 0.010697 h⫺1 0.0032 h⫺1 600 h (25 d) 0.068 Gyh⫺1
Dicker et al. (25) Brenner and Hall (24), Ling (3) Ling (3) Moerland (5), Waterman et al. (7) Nath et al. (21) Nath et al. (21) Ling (3) Moerland (5), Waterman et al. (7) Dicker et al. (25), Ling (3) Moerland (5)
In the case of permanent 125I implants with a half-life of 60 days (21), and the prescribed dose to cover the prostate of 140 Gy in 1 year, the initial dose rate d0 ⫽ 0.068 Gyh⫺1 (5). Other isotopes used for permanent prostate implants are 103 Pd and 198Au with a half-life of 17 and 2.7 days, respectively (3, 21). The d0 values are calculated so that the value of BEDmax for 103Pd and 198Au is equal to the BEDmax for 125 I, in the case of no edema (BEDmax ⫽ 74.9 Gy). Thus, the d0 value for 103Pd and 198Au is 0.16 Gyh⫺1 and 0.634 Gyh⫺1, respectively. These d0 values are somewhat different from values quoted in the literature (3), because we have made the assumption of equal values of BEDmax at teff when no edema is present and with the chosen set of parameters. Part of the radiation-induced damage to the cells will be repaired within 1–3 hours (3, 19, 22, 23). We used for the repair rate constant , the value of 0.693 h⫺1 (T1/2 ⫽ 1 h). In the LQ model, the ␣ and  constants can be related to the induction of lethal and sublethal damage, respectively. In general, for prostate tumors, an ␣/ value of ⱖ10 Gy is applied. For instance, Ling (3) performed BED calculations for an ␣/ value of 10 Gy. However, the results of recent studies have indicated that prostate tumors react like lateresponding tissues (19, 22, 24, 25). An ␣/ value of 3 Gy seems appropriate. To determine the influence of the ␣/ value, ␣/ was varied from 1 to 20 Gy by varying ␣ from 0.03 Gy⫺1 to 0.6 Gy⫺1, while keeping  at 0.03 Gy⫺2 (3, 24). Another radiobiologic parameter is the potential tumor doubling time, Tp. For prostate tumors, Tp may vary from 5 to 67 days (3, 25, 26). In our calculations, the Tp reference value was set to 25 days and the Tp is varied between 10 and 40 days to investigate its influence on the BEDmax and teff. The parameters related to edema, the decay constant of prostate edema edema, and the ratio of post- and preimplant prostate dimensions R0 were also varied to investigate the influence on the BED. Moerland (5) and Waterman et al. (7) found edema-induced prostate volume increases between 0% and 52%, corresponding to R0 variations between 0 and 1.15. The reference value was set to R0 ⫽ 1.09, equivalent to a 30% prostate volume increase, when isotropic enlargement of the prostate is assumed. Postimplant prostate volumes were found to return to preimplant prostate volumes with a mean half-life of about 9 days and variations between 3 and 23 days. This corresponded with a mean edema of
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Fig. 1. BED of the prostate with permanent 125I implants as a function of time for different values of R0 using Eq. 9, and the parameters defined in Table 1. solid line, R0 ⫽ 1.0; large dashes, R0 ⫽ 1.05; short dashes, R0 ⫽ 1.09; and smallest dashes, R0 ⫽ 1.15.
0.0032 h⫺1 and variations between 0.01 and 0.00125 h⫺1, respectively. Optimal timing of implant evaluation In clinical practice, images of the prostate with the implanted radioactive seeds are taken several days or weeks after implantation. Dosimetry calculations are done on the basis of the geometry of the seeds on these images. Because of edema, the volume of the prostate and hence, the distance between the seeds, varies with time. Thus, the moment of taking those images, tm, is important for a correct determination of BED. The optimal timing of implant evaluation is defined as the point after the implantation at which the images are taken and the error in the calculated BED is minimal for the chosen set of radiobiologic parameters values. Moerland (5) has defined, for the physical dose, a parameter ⑀ as the ratio of the estimated cumulative dose and the real cumulative dose as a function of the timing of implant evaluation tm. In analogy, ⑀ can be defined as the ratio of the estimated BED, with a constant volume of the prostate between implantation and teff and the “real” BED, with a variable prostate volume between implantation and teff (for the description of ⑀, see Appendix B). When ⑀ is equal to 1, the optimal time of implant evaluation is achieved for the chosen set of parameters. RESULTS Influence of parameters on BED The BED is calculated as a function of the time after implantation with the parameters in Table 1 (Eq. 9). To investigate the influence of edema on the BED, the extent of edema was varied. The BED was plotted for varying values of R0 in the case of 125I implants in Fig. 1. Without edema (R0 ⫽ 1.0), the BEDmax was equal to 74.9 Gy and teff to 146 days (3504 h). With a prostate volume increase of 30% (R0 ⫽ 1.09), BEDmax decreased to 70.3 Gy, but teff was still 146 days (3504 h). BEDmax decreased further to 67.6 Gy for
a prostate volume increase of 52% (R0 ⫽ 1.15), again the teff was 146 days. The course of prostate swelling is characterized by edema. Varying edema between 0.00125 and 0.01 h⫺1 (equivalent to a half-life of edema between 3 and 23 days), BEDmax varied between 66 and 73.2 Gy and teff remained 146 days (data not shown). In addition to edema, the BED depends on radiobiologic parameter values. The influence of ␣, the parameter of the linear component of the LQ model is shown in Fig. 2. The ␣ values ranged from 0.6 to 0.03 Gy⫺1, with  ⫽ 0.03 Gy⫺2, the ␣/ values ranged from 1 to 20 Gy. The BEDmax ranged from 20 to 120 Gy and teff from 58 days (1392 h) to 310 days (7440 h). Thus, when 125I seeds are implanted, ␣ (or ␣/) influences BEDmax and teff significantly. The  value, correlated with the induction of sublethal damage, on the other hand, hardly changed the BEDmax and teff (data not shown). In Fig. 3, the BED was plotted as a function of time after implantation of 125I seeds for varying values of Tp. A variation in Tp between 10 and 40 days (240 and 960 h) resulted in a variation of BEDmax between 24 and 89 Gy and a variation in teff between 68 and 187 days (1632 and 4488 h). Thus, when the repopulation of prostate tumors is slow, as indicated by Haustermans et al. (26), the BEDmax is large and the t eff is long. The BED also depends on the physical parameters, depending on the choice of the isotope. For instance, 125I has a half-life of 60 days. To cover the prostate with a physical dose of 140 Gy in 1 year, the initial dose rate is equal to 0.068 Gyh⫺1. As shown in Fig. 1, the BEDmax is 74.9 Gy when edema is not taken into account. To relate the BEDmax of 103Pd and 198Au implants to the BEDmax of 125I implants, the initial dose rates are calculated for the implants with the two other isotopes. In the situation without edema, d0 is determined for the BEDmax value of 74.9 Gy. This results in d0 ⫽ 0.16 Gyh⫺1 for 103Pd implants (half-life 17 days) and d0 ⫽ 0.634 Gyh⫺1 for 198Au implants (half-life 2.7 days). With the parameters in Table 1 thus with edema, the BED is calculated as a function of time after implantation for
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Fig. 2. BED of the prostate with permanent 125I implants as a function of time for different values of ␣ using Eq. 9 and the parameters defined in Table 1. 103
Pd and 198Au permanent implants. Figure 4 shows that BEDmax and teff are not the same for those isotopes. For 103 Pd and 198Au, these values are 66.1 Gy and 62 days and 55.6 Gy and 15 days, respectively; without edema, these values are 74.9 Gy and 62 days and 74.9 and 15 days, respectively. This illustrates that edema has more effect on the BED for the short-living isotopes. Optimal timing of implant evaluation Due to edema, the timing of image acquisition is important for seed reconstruction and BED calculations. When implant evaluation takes place at the right time, a realistic BED is calculated. In other words, ⑀ must be equal to 1 Appendix A (Eq. 23). Figures 5 through 7 show ⑀ as a function of the timing of implant evaluation and R0 for 125I, 103 Pd, and 198Au seeds, respectively. The parameters used in Eq. 23 are those defined in Table 1. Implant evaluation right after the implantation always
Fig. 3. BED of the prostate with permanent 125I implants as a function of time for different values of the potential tumor doubling time using Eq. 9 and the parameters defined in Table 1.
results in an ⑀ ⬍1 (for the three isotopes and for all R0 values). For 125I implants, the underestimation of the BED is larger than for 103Pd or 198Au implants. The bigger the R0 value, the larger the underestimation of the dose. Implant evaluation at the effective treatment time results always in an ⑀ ⬎1. The largest overestimation of the BED is calculated when R0 ⫽ 1.15 for all isotopes. Implant evaluation at the effective treatment time results in a larger overestimation of the dose for 198Au implants than for 125I or 103Pd implants. Therefore, there is a point after implantation at which ⑀ ⫽ 1. This optimal timing of implant evaluation depends on the R0 value and the isotope. For small values of R0 and thus small prostate volume increases, ⑀ is close to 1 for the three isotopes at any time after implantation. For a typical prostate volume increase of 30% (R0 ⫽ 1.09) with a half-life of 9 days (edema ⫽ 0.0032 h⫺1), the optimal timing of implant evaluation is 25, 13, and 3 days
Fig. 4. BED of the prostate with permanent implants as a function of time for different isotopes using Eq. 9 and the parameters defined in Table 1.
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Fig. 5. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 125I seeds.
after implantation for 125I, 103Pd, and 198Au implants, respectively. For 125I implants, ⑀ was plotted as a function of R0 and edema for 3 set times after implant: 1 day (Fig. 8A), 25 days (Fig. 8B), and 146 days (Fig. 8C). Implant evaluation 1 day after the seed implantation always results in an underestimation of the BED, because it ignores prostate shrinkage to preimplant dimensions and thus underestimates the dose distribution in the time after implantation. Underestimation depends on the extent of prostate edema and half-life of edema and may amount to 43% for R0 ⫽ 1.15 and edema ⫽
0.0032 h⫺1 (half-life 9 days). When implant evaluation is performed 25 days after seed implantation, a slight overestimation of ⬍1% is found for prostate edema with a half-life of 9 days. At 25 days, the overestimation is at most 3% for prostate edema with a relatively small half-life of 3 days (edema ⫽ 0.01 h⫺1). A relatively large half-life of 23 days (edema ⫽ 0.00125 h⫺1) results in an underestimation of at most 10%. Implant evaluation at teff (146 days after implantation of 125I seeds and the parameter values as defined in Table 1) always results in an ⑀ ⬎1 because of overestimation of the dose contribution delivered in the period when
Fig. 6. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 103Pd seeds.
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Fig. 7. ⑀ as a function of the time of implant evaluation tm and R0 (using Eq. 23 and the parameters in Table 1) for 198Au seeds.
the prostate was swollen. Depending on R0 and edema, the overestimation can be at most 22%. As mentioned before, ␣ and Tp have a strong influence on BEDmax and teff. A smaller ␣ value (0.045 instead of 0.09 Gy⫺1 or ␣/ value of 1.5 instead of 3 Gy) results in a smaller teff (for other values of ␣, see Fig. 2), and the optimal timing of implant evaluation decreases from 25 to 21 days after implantation (data not shown). An increase of the potential tumor doubling time from 25 to 40 days shifts the optimal timing of implant evaluation by only 1 day from 25 to 26 days (data not shown). Figure 9 shows the influence of R0 and edema on the optimal timing of implant evaluation for different isotopes. Optimal timing does not depend on R0, but edema has some influence, especially in case of implants with isotopes with a half-life that is large compared with the half-life of prostate edema, such as 125I implants. For the range of prostate edema half-lives of 3 to 23 days, the optimal timing for implant evaluation ranges from 12 to 37 days, 8 to 16 days, and 3 to 4 days for 125I, 103Pd, and 198Au, respectively. DISCUSSION BED calculations for prostate brachytherapy have been the subject of studies by several authors (3, 25). Dicker et al. (25) calculated BED values for permanent prostate implants and concluded that the BEDmax is equal to 76 Gy (␣ ⫽ 1.0 Gy⫺1, Tp ⫽ 23 days) and 102 Gy (␣ ⫽ 1.0 Gy⫺1, Tp ⫽ 23 days) for 125I and 103Pd implants, respectively. In these studies, postimplant edema was neglected. We derived a theoretical model to calculate the BED in case of a homogeneous physical dose delivery to the prostate taking into account prostate dimension changes due to postimplant edema. The extent of prostate volume increase due to edema
and the edema half-life may vary from patient to patient. For typical edema values (R0 ⫽ 1.09 and edema ⫽ 0.0032 h⫺1) and radiobiologic parameters (␣ ⫽ 0.09 Gy⫺1, Tp ⫽ 25 days), we found BEDmax values of 70.3 Gy and 66.1 Gy for 125 I and 103Pd implants. Variations of R0 and edema showed a further decrease of BED for larger R0 and smaller edema values. The extent of induced lethal damage, represented by ␣ in the LQ equation, and the potential tumor doubling time, Tp, are the only radiobiologic parameters that strongly influence BEDmax. Dicker et al. (25) and Ling (3) found that an increase of Tp results in an increase of BEDmax. They did not take edema effects into account. This is consistent with the results of our calculations: the longer the potential tumor doubling time, the larger the BEDmax and the longer the teff. We varied the Tpot between 10 and 40 days. Recent literature suggests that the Tpot of prostate tumors is around 40 days (19). The teff also depends on ␣. A small ␣ value means a minor extent of lethal damage, and therefore the BEDmax is relatively small and the teff relatively short. Permanent implants deliver 75% of the physical dose in 2 isotope half-lives, which is 5.4, 26, and 120 days for 198Au, 103 Pd, and 125I implants, respectively. Thus, effective treatment times are shorter for implants with short-living isotopes and the effect of postimplant prostate volume changes are larger for these implants. As pointed out by Antipas et al. (19), the absolute teff and BEDmax values depend on the relative biological effectiveness (RBE) of the applied isotope. However, clinical data on RBE values for 125I, 103Pd, and 198Au as a function of the dose rate in prostate brachytherapy are scarce. Therefore, as a reference dose, we applied the physical dose of 140 Gy as applied in clinical prostate 125I brachytherapy and calculated initial dose rates
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Fig. 8. ⑀ as a function of R0 and edema: (a) 1 day after implantation of 125I seeds in the prostate, (b) 25 days after implantation, and (c) 146 days after implantation. Solid line represents edema ⫽ 0.01 h⫺1; dashed line, edema ⫽ 0.0032 h⫺1; and dotted line, edema ⫽ 0.00125 h⫺1.
for 103Pd and 198Au implants to get equivalent BED values in nonedematous prostates, taking into account the radiobiologic parameters and assuming equal RBE. Incorporating different (dose-rate– dependent) RBE values would change the initial dose rates required to get equivalent BEDmax values, but would not change the general conclusions regarding the change of BEDmax as a function of the radiobiologic parameters and edema. In clinical practice, several days or weeks after the implantation, images are taken for
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implant evaluation. Because of induced edema and the decaying dose rate of the seeds, there is an optimal time to take those images. We found an optimal time of 25 days, 13 days, and 3 days after implantation for 125I, 103Pd, and 198 Au implants, respectively. The optimal timing of implant evaluation for the BED is earlier after implantation than the optimal timing for the evaluation of the physical dose. Moerland (5) determined an optimal timing for the physical dose of 28 days after implantation of 125I seeds. However, his calculation of the optimal timing for the physical dose did not take into account repopulation of tumor cells. As shown, the faster the repopulation, the shorter the effective treatment time and so the optimal timing for implant evaluation is earlier after the implantation. Because of logistic reasons, MRI is sometimes done 1 day after seed implantation. Then, the prostate is swollen and therefore an underestimation of the BED of at most 43% is calculated, depending on the extent and the half-life of prostate edema. Yue et al. (14) examined, for a range of half-lives and magnitudes of prostate volume increase, the effect on the optimal time of evaluation for the physical dose. They concluded that the optimal time was highly dependent on the half-life of edema, but the magnitude of prostate volume increase did not change the optimal time significantly for both 125I and 103Pd implants. Our calculations of the BED led to the same conclusion: the optimal timing of image acquisition for the calculation of the BED of 125I, 103Pd, or 198Au implants strongly depends on the half-life of the edema. The larger the half-life of the isotope, the bigger the influence of edema on the optimal timing of the implant evaluation. In addition, the optimal timing is not influenced by the magnitude of the prostate volume increase. For 125I implants and typical prostate edema values (R0 ⫽ 1.09 and edema ⫽ 0.0032 h⫺1), the optimal timing of implant evaluation is 25 days after the implantation. However, in practice, edema will vary from patient to patient. The variations are probably between 0.01 h⫺1 (t1/2 ⫽ 3 days) and 0.00125 h⫺1 (t1/2 ⫽ 23 days), with a mean value of 0.0032 h⫺1 (t1/2 ⫽ 9 days) (7). As a consequence, when the images are taken 25 days after the implantation and edema is either 0.01 h⫺1 or 0.00125 h⫺1, an overestimation of the BED by 3% or an underestimation of the BED by 10% can be calculated, respectively.
CONCLUSION Calculations of the BED after implantation of radioactive seeds in the prostate, based on images taken shortly after implantation, results in an underestimation of the BED of at most 43% for 125I. The BED is overestimated when calculations are based on images taken at teff of at most 9%. Thus, there is an optimal timing of implant evaluation for calculations of the BED, which depends on radiobiologic parameter values and the magnitude and half-life of prostate
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Fig. 9. (a) Optimal timing of implant evaluation as a function of half-life of edema for 125I, 103Pd, and 198Au seeds. (b) Optimal timing of implant evaluation as a function of the extent of edema, R0, for 125I, 103Pd, and 198Au seeds.
volume increase. For the most likely parameter values, we found an optimal time of image acquisition for seed recon-
struction for calculations of the BED of 25 days after seeds implantation.
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REFERENCES 1. Nag S, Beyer D, Friedland J, et al. American Brachytherapy Society (ABS) recommendations for transperineal permanent brachytherapy of prostate cancer. Int J Radiat Oncol Biol Phys 1999;44:789 –799. 2. Nag S, Baird M, Blasko J, et al. American Brachytherapy Society survey of current clinical practice for permanent brachytherapy of prostate cancer. J Brachytherapy Int 1997; 13:243–251. 3. Ling CC. Permanent implants using Au-198, Pd-103 and I-125: Radiobiological considerations based on the linear quadratic model. Int J Radiat Oncol Biol Phys 1992;23:81– 87. 4. Holm HH, Juul N, Pedersen JF, et al. Transperineal iodine125 seed implantation in prostatic cancer guided by transrectal ultrasonography. J Urol 1983;130:283–286. 5. Moerland MA. The effect of edema on postimplant dosimetry of permanent iodine-125 prostate implants: A simulation study. J Brachytherapy Int 1998;14:225–231.
6. Moerland MA, Wijrdeman HK, Beersma R, et al. Evaluation of permanent I-125 prostate implants using radiography and magnetic resonance imaging. Int J Radiat Oncol Biol Phys 1997;37:927–933. 7. Waterman FM, Yue N, Corn BW, et al. Edema associated with I-125 or Pd-103 prostate brachytherapy and its impact on post-implant dosimetry: An analysis based on serial CT acquisition. Int J Radiat Oncol Biol Phys 1998;41:1069 –1077. 8. Dale RG, Jones B, Coles IP. Effect of tumour shrinkage on the biological effectiveness of permanent brachytherapy implants. Br J Radiol 1994;67:639 – 645. 9. Willins J, Wallner K. CT-based dosimetry for transperineal I-125 prostate brachytherapy. Int J Radiat Oncol Biol Phys 1997;39:347–353. 10. Roy JN, Wallner KE, Harrington PJ, et al. A CT-based evaluation method for permanent implants: Application to prostate. Int J Radiat Oncol Biol Phys 1993;26:163–169.
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11. Grimm PD, Blasko JC, Ragde H. Ultrasound-guided transperineal implantation of iodine-125 and palladium-103 for the treatment of early-stage prostate cancer: Technical concepts in planning, operative technique, and evaluation. Atlas Urol Clin North Am 1994;2:113–125. 12. Narayana V, Roberson PL, Pu AT, et al. Impact of differences in ultrasound and computed tomography volumes on treatment planning of permanent prostate implants. Int J Radiat Oncol Biol Phys 1997;37:1181–1185. 13. Narayana V, Roberson PL, Winfield RJ, et al. Impact of ultrasound and computed tomography prostate volume registration on evaluation of permanent prostate implants. Int J Radiat Oncol Biol Phys 1997;39:341–346. 14. Yue N, Chen Z, Peschel R, et al. Optimum timing for imagebased dose evaluation of I-125 and Pd-103 prostate seed implants. Int J Radiat Oncol Biol Phys 1999;45:1063–1072. 15. Chen Z, Yue N, Wang X, et al. Dosimetric effects of edema in permanent prostate seed implants: A rigorous solution. Int J Radiat Oncol Biol Phys 2000;47:1405–1419. 16. Barendsen GW. Dose fractionation, dose rate and iso-effect relationships for normal tissue responses. Int J Radiat Oncol Biol Phys 1982;8:1981–1997. 17. Dale RG. The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy. Br J Radiol 1985;58:515–528. 18. Dale RG. Radiobiological assessment of permanent implants using tumour repopulation factors in the linear-quadratic model. Br J Radiol 1989;62:241–244.
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19. Antipas V, Dale RG, Coles IP. A theoretical investigation into the role of tumour radiosensitivity, clonogen repopulation, tumour shrinkage and radionuclide RBE in permanent brachytherapy implants of I-125 and Pd-103. Phys Med Biol 2001; 46:2557–2569. 20. Dale RG, Jones B. The assessment of RBE effects using the concept of biologically effective dose. Int J Radiat Oncol Biol Phys 1999;43:639 – 645. 21. Nath R, Anderson LL, Luxton G, et al. (GRP) Dosimetry of interstitial brachytherapy sources: Recommendations of the AAPM Radiation Therapy Committee Task Group No. 43. Med Phys 1995;22:209 –234. 22. Fowler J, Chappell R, Ritter M. Is ␣/ for prostate tumors really low? Int J Radiat Oncol Biol Phys 2001;50:1021–1031. 23. King CR. What is the Tpot for prostate cancer? Radiobiological implications of the equivalent outcome with I-125 or Pd-103. Int J Radiat Oncol Biol Phys 2000;47:1165–1167. 24. Brenner DJ, Hall EJ. Fractionation and protraction for radiotherapy of prostate carcinoma. Int J Radiat Oncol Biol Phys 1999;43:1095–1101. 25. Dicker AP, Lin C, Leeper DB, et al. Isotope selection for permanent prostate implants? An evaluation of Pd-103 versus I-125 based on radiobiological effectiviness and dosimetry. Semin Urol Oncol 2000;18:152–159. 26. Haustermans KMG, Hofland I, Van Poppel H, et al. Cell kinetic measurements in prostate cancer. Int J Radiat Oncol Biol Phys 1997;37:1067–1070.
APPENDIX A Dale (17, 18) derived an equation for BED for low-doserate irradiation taking into account repopulation factors. We derived, analogous to Dale’s derivation, an equation for BED, including changing the distance between seeds and the effects of edema. Radiation treatment of tumor with a decaying source The dose rate at any point in the prostate with edema can be compared with the dose rate when no edema is present (see “Physical dose”): d共t兲 ⫽
d0exp共⫺t兲 R2.7共t兲
(10)
with the decay constant of the radioactive sources, d0 the initial dose rate, and R(t) as expressed in Eq. 3. The power 2.7 is explained by the consideration that the radioactive seeds are point sources (d ⬃1/r2), and scatter and absorption g(r) are approximated by g(r) ⬃1/r0.7 (5, 21). Lethal damage (type A damage [17]) induced in the effective treatment time teff after irradiation is expressed as
冕
teff
0
␣d共t兲dt ⫽
冕
teff
0
␣d0exp共⫺t兲 dt R2.7共t兲
(11)
with ␣ the linear constant of the linear-quadratic model. Dale (17) defined lethal damage of type B as the damage resulting from a lethal combination of the damage in the
target pairs induced at two separate sequential irradiations. The probability of sublethal damage in time dt is pd共t兲dt
(12)
with p the probability per unit dose that a sublethal event occurs (i.e., only one target of a pair is hit). Sublethal damage repair can be expressed by e⫺t, with , the sublethal damage repair rate constant. The probability that either target of a pair of targets is hit in one radiation event is equal to 2pd(t)dt. Substituting d(t) (Eq. 10) and taking into account that the damage still exists at time t1, the probability is 2p d0 exp共⫺t兲exp共⫺t1兲 dt R2.7共t兲
(13)
Let t2 ⫽ t ⫹ t1, then t1 ⫽ t2 ⫺ t and then 2p d0 exp共⫺t兲exp关⫺共t2 ⫺ t兲兴 dt R2.7共t兲
(14)
The total probability of sublethal damage in time t2 is presented by
2p d0
冕
t2
0
exp共⫺t ⫺ t2 ⫹ t兲 dt R2.7共t兲
(15)
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The probability that a second undamaged target receives a hit in time dt is pd(t)dt. Therefore, the probability that lethal damage occurs in time dt2 after previous irradiation of duration t2:
冉冕
t2
2 p2 d02
0
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RE ⫽ 1
冕 冉冕 teff
2 d02
0
⫹
冊
exp共⫺t ⫹ t ⫺ t2兲 exp共⫺t2兲 dt dt2 2.7 R 共t兲 R2.7共t2兲
t2
0
冊
exp共⫺t2兲 exp共⫺t ⫹ t ⫺ t2兲 dt dt2 2.7 R 共t兲 R2.7共t2兲
冕
(16)
teff
0
␣ d0 exp共⫺t兲 dt R2.7共t兲 (19)
with the probability that two sublethally damaged targets interact to produce lethal damage. The total probability of lethal damage per target pair in time teff (type B damage) is
This results for the BED dose in
冉冕 冊 teff
BED ⫽ RE
d共t兲dt ⫺
0
冕 冉冕 teff
2d02
0
t2
0
冊
exp共⫺t ⫹ t ⫺ t2兲 exp共⫺t2兲 dt dt2 (17) 2.7 R 共t兲 R2.7共t2兲
with  ⫽ np and n the number of target pairs. According to Dale (17), the average relative effectiveness RE per unit dose (from the linear-quadratic equation BED ⫽ Dtot * RE) is
RE ⫽ 1 ⫹
lethal damage of type B lethal damage of type A
(20)
where the last part of the equation is the result of tumor cell repopulation; Tp is the potential tumour doubling time. Therefore,
冕 冕 冉冕 teff
2
0.693teff ␣Tp
BED ⫽
d0
0
 ⫹ 2 d02 ␣
teff
0
exp共⫺t兲 dt R2.7共t兲 t2
0
冊
exp共⫺t兲 exp共⫺t2兲 exp共t ⫺ t2兲dt dt2 R2.7共t兲 R2.7共t2兲 ⫺
(18)
0.693teff ␣Tp
(21)
APPENDIX B Optimal time of implant evaluation After implantation of the radioactive seeds in the prostate, the prostate is swollen from the induced edema. Because the extent of edema changes with time, the findings of the implant evaluation depend on the timing after implantation. The ratio ⑀ of the calculated BED, with a constant prostate
⑀共tm兲 ⫽
冕
冕 冕 冉冕
exp共⫺t兲 dt R2.7共tm兲
teff
d0
0
teff
0
 d共t兲dt ⫹ 2 ␣
teff
0
t2
0
teff
0
⑀⫽
冕 冉冕
 ␣
teff
0
t2
0
BED共volume constant兲 BED共volume variable兲
(22)
With Eqs. 3, 4 and 9, ⑀ can be expressed by
⫹
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
2
冕
冊
volume, and the “real” BED, with a variable prostate volume, can be expressed by
冊 冊
exp共⫺t兲 exp共⫺t2兲 d0 2.7 exp共t ⫺ t2兲dt d0 2.7 dt2 R 共tm兲 R 共tm兲
冕 冉冕
 d共t兲dt ⫹ 2 ␣
teff
0
t2
0
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
冕
teff
0
冕 冉冕
 d共t兲dt ⫹ 2 ␣
teff
0
0.693teff ␣Tp t2
0
⫺
冊
0.693teff d共t兲exp共t ⫺ t2兲dt d共t2兲dt2 ⫺ ␣Tp
(23)
Biologically effective dose for permanent prostate implants
with  the quadratic coefficient of the linear-quadratic model, ␣ the linear coefficient of the model, the repair rate constant of sublethal damage, Tp the potential tumor doubling time, t2 the time sublethal damage exists, t the time within sublethal damage occurs, teff the effective
● M.P.R. VAN GELLEKOM et al.
433
treatment time, tm the time of implant evaluation after implantation, d0 the initial dose rate, d(t) the dose rate at time t (Eq. 4), the decay constant of radioactive source, and R(t) as defined in Eq. 3. The optimal timing of implant evaluation is determined when ⑀ is equal 1.