INTRODUCTION
Permanent brachytherapy seed implantation to treat prostate cancer is an old technique in which historically radium (226Ra) and gold (198Au) sources were used.Reference Pasteau and Degrais1–Reference Flocks, Kerr, Elkins and Culp3 In the present era of prostate permanent seed brachytherapy, 125I and 103Pd seeds have been used with the help of ultrasound (US) guided transperineal technique. Recently, 131Cs seeds have been implemented in many centres across USA, including UPMC Cancer Centres.Reference Smith, Beriwal, Komanduri, Gibbons and Benoit4–Reference Kehwar, Jones, Huq and Smith6 The design of 131Cs seed is similar to that of 125I and 103Pd, but has a shorter half life of 9.7 days, and slightly higher average photon energy than 125I and 103Pd.Reference Henschke and Lawrence7–Reference Rivard9 The shorter half life of 131Cs can induce differences between planned and delivered doses due to prostatic edema, thus potentially impacting values of dosimetric and radiobiological quantities of interest. The present study investigates the effect of edema on radiobiological quantities, such as tumour cell surviving fraction (SF) and tumour control probability (TCP), for 131Cs seed permanent prostate implants. In prostate implantation, the radioactive seeds are implanted via surgical procedure and edema develops in the prostate immediately after implantation. The edema increases the size of the prostate and consequently changes the relative distances between the radioactive seeds which alters the dosimetric and radiobiological quantities. The effects of edema on dosimetric and radiobiological quantities in 125I and 103Pd implants have been studied by many investigators.Reference Waterman, Yue, Reisinger, Dicker and Corn10–Reference Butler, Stewart and Merrick20 Since 131Cs is a relatively new radioactive source used for prostate permanent seed implants, limited number of studies have been done to investigate the effect of edema for 131Cs permanent seed implants.5,6, 21–24
In a recent study, we derived prostate volume changes after prostate brachytherapy from its original volume using US and CT images acquired at day 0, day 14 and day 28 days after implant, and found that the prostate edema resolves exponentially with post-implant time.Reference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5 Willins and Wallner,Reference Willins and Wallner25 and Waterman et al.Reference Waterman, Yue, Corn and Dicker11 also observed that in 125I seed permanent implants the prostatic edema decayed exponentially with post-implant time. The exponential decay of prostatic edema induces a significant amount of change in dosimetric and radiobiological quantities in 131Cs implants due to its short half life of 9.7 days.Reference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5,Reference Kehwar, Jones, Huq and Smith6 Chen et al.Reference Chen, Deng, Roberts and Nath22 have derived an equation to compute the edema induced dose reduction in 131Cs implants with post-implant time as a function of the magnitude and half-life of the prostatic edema. In this article, the equation of SF (and TCP) is extended to account for the edema induced changes for non-uniform dose distribution within the prostate as a function of the magnitude and edema half life (EHL). In the calculations of SF (and TCP), the equation derived by Chen et al.Reference Chen, Deng, Roberts and Nath22 to compute edema induced dose reduction was used. The calculations of SF are based on the images taken at day 0, day 14 and day 28 after the implantation. The purpose of this study was to examine the influence of edema on the surviving fraction and the tumour control probability prostate tumours.
METHODS AND MATERIALS
Derivation of survival fraction to account for edema effect
Nahum and TaitReference Nahum and Tait26 have derived a double exponential model of the tumour control probability (TCP) for homogeneous dose distributions. This is given by
![{\rm{TCP}} = exp[NS{(D)]}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn1.gif?pub-status=live){kind=small}
where N is the initial number of clonogenic cells in tumour volume (V) and S(D) is the clonogen surviving fraction at dose (D). The number N can be related to the tumour volume (V) using the relationship N = ρV, where ρ is the clonogenic cell density. Equation (1) can then be written as
![{\rm{TCP}} = exp[- \rho V S{(D)}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn2.gif?pub-status=live){kind=small}
The S(D) can be written as
![S(D) = exp[- \alpha {\rm D}({\rm t})^2 + \delta ({\rm t} - {\rm T_k})]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn3.gif?pub-status=live){kind=small}
where α and β represent the single-track and inter-track cellular radiosensitivities. The α is associated with indirect chemical damage, which is independent of dose rate, fraction size and inter-fraction interval, although has partial reparability over a period of time. The β represents cell kill due to interactions between individual particle tracks and is therefore sensitive to fraction size, dose-rate (if small enough), and inter-fraction interval. The γ is the proliferation constant and is given by γ = 0.693/αTp, where Tp is the potential doubling time of tumour cells, and t is the irradiation time of the implant in days, with an account of proliferation kickoff time (Tk). The q(t) is a dose protraction factor to account for the sublethal damage repairReference Zaider and Minerbo27 given by
![{\rm q}({\rm t})=[2(\Lambda {\rm t})^{2}/{\mu ({\rm t})^{2}(1- \Lambda^2/\mu ^2)(1 - {\rm e}^{-\Lambda t})^{2}}][{{\rm e}^{-(\Lambda + \mu)t}} + {\mu {\rm t}(1-{\rm e}^{-2\Lambda t})/(2 \Lambda t)}-{(1+{\rm e}^{-2\Lambda t})/2}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn4.gif?pub-status=live){kind=small}
where λ and μ are the decay constant of radioactive source and repair constant of sublethal damage, respectively. These can be defined by λ=ln(2)/t1/2 and μ=ln(2)/trep, where t1/2 and trep are the half-life of radionuclide used in the implant and half-repair time of sublethal damage, respectively.
In permanent implants, the dose rate at a point ‘P’ by decaying radioactive sources is a simple exponential function of time
and the total dose delivered to point ‘P’ is given by
where Rp(0) is the initial dose rate at point ‘P’ in the tumour.
Effective treatment time teff
Theoretically, the radiation dose is delivered throughout the life time of the patient, i.e., the treatment time ‘t’ approaches to infinity. Hence, the term γ(t − Tk) in equation (3) becomes unrealistic. Antipas et al.Reference Antipas, Dale and Coles28 had shown that in permanent implants the biologically effective dose rate delivered to the tumour cells falls with time due to radionuclide decay and extends to the point at which the biological dose rate falls to the critical dose rate, where it is equal to or less than the tumour cell repopulation. Below this critical dose rate the delivered dose is a waste, because instantaneous tumour cell proliferation rate exceeds the rate of cell killing. The biological dose rate in 131Cs implants approaches the critical dose rate in a shorter period of time due to the short half life of the source. The time interval between the implantation time (day 0) and the time at which the dose rate reaches this critical value is called the effective treatment time (teff).Reference Antipas, Dale and Coles28 The teff provides a measure of the time over which tumour cell kill is ensured and is given by
where Tp, is the tumour cell potential doubling time.
By truncating equation (6), the dose delivered at point ‘P’ in time ‘t’ can be given by
where ‘t’ varies from 0 to teff.
It is clear from the literature that the potential doubling times (Tp) of the prostatic tumour cells are very long.Reference Schmid, McNeal and Stamey29–Reference Hausterman and Fowler32 Different investigators have different views about the use of kick-off time (Tk) in the time factor of the linear quadratic (LQ) model for radiobiological interpretations of the treatment of prostate cancer.18,33–35 One group of the investigatorsReference Wang, Li, Yu and DiBiase33 suggests that the proliferation of the prostate tumour cells speeds up as soon as the tumour shrinks, which can be as early as two or three delivered fractions. Cells in regions of poor nutrition would then become closer to the capillary supplies, so that the growth fraction could increase. Many other researchers have discussed the impact of tumour repopulation on the treatment effectiveness for prostate cancerReference Ling, Roy, Sahoo, Wallner and Anderson36–Reference King, DiPetrillo and Wazer39 and have ignored the “kick-off” time Tk i.e., take Tk = 0 as a common assumption. On the other hand, a second group of investigatorsReference Fowler, Ritter, Fenwick and Chappell34–Reference Ritter35 argued that the values of Tk estimated from clinical data for head-and-neck tumours varied from 21 to 35 days,Reference Roberts and Hendry40,Reference Brenner41 which is six to seven times longer than with short pretreatment median values of Tp of 3.5 daysReference Begg, Hofland, Van Glabekke, Bartelink and Horiot42 or 4.7 days.Reference Zackrisson, Gustafsson, Stenling, Flygare and Wilson43 In another publication, an author of this group,Reference Fowler44 iterated that the value of Tk depends on the pretreatment cell kinetics that determine how rapidly cells are lost from the tumour or tissue, so that spontaneous cell loss can fall close to zero during the later period of treatment. The median Tp of prostate tumours is 42 daysReference Hausterman and Fowler32 which is 10 times the value of head-and-neck tumours. Therefore, it is likely that the value of Tk for prostate be 10 times that of Tk for head-and-neck tumours ranging from 210–350 days.Reference Fowler, Ritter, Fenwick and Chappell34 The scope of this debate is beyond the limits of this article. However, for calculation purposes, we had adopted the arguments of the second group and ignored the time correction factor while calculating the SF and the TCP, considering the fact that even if the Tp of the prostate is as short as 15 days,Reference Hausterman and Fowler32 the Tk would be 150 days which is well in excess of the effective treatment time of 131Cs implants of 61 days. Hence, the factor accounting for proliferation correction in equation (3) can be neglected yielding
![S(D) = exp[- \alpha {\rm D}({\rm t}) - \beta {\rm q}({\rm t}){\rm D}({\rm t})^2]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn9.gif?pub-status=live){kind=small}
Using equation (8) into equation (9), S(D) at ‘P’ can be written as
![S_p(D) =exp [- \alpha {\rm R_p} (0)\int\limits_{0}^{{\rm t}eff} exp(- \lambda {\rm t}){\rm dt}- {\rm R_p} (0)\int\limits_{0}^{{\rm t}eff} exp(- \lambda {\rm t}){\rm dt}^2]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn10.gif?pub-status=live)
Edema-induced dose reduction
In a permanent prostate implant, if there is no implant induced edema observed, the instantaneous dose rate at point ‘P’ and at time ‘t’ can be given by equation (5), which is a simple exponential function of time. With implant induced prostatic edema, the prostate volume and source locations become function of time and thus instantaneous dose rate at point ‘P’ will not be a simple exponential function of time but is given by the following relationReference Chen, Yue, Wang, Roberts, Peschel and Nath45 which accounts for an edema - induced dose reduction at time ‘t’
![{\rm R_p}({\rm t}) = {\rm R_p}(0) [exp(- \lambda {\rm t})/{1 + M_0 exp(-\lambda_e{\rm t})}^ \tau/3]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn11.gif?pub-status=live){kind=small}
and the instantaneous dose at point ‘P’ is given by
![{\rm D_{p,int}}({\rm t}) = {\rm D_p}(0) [exp(- \lambda {\rm t})/{1 + M_0 exp(-\lambda_e{\rm t})}^ \tau/3]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn12.gif?pub-status=live){kind=small}
where Dp,int(t) = Rp(t)/λ and Dp(0) = Rp(0)/λ are doses at times 0 and t, respectively. M0 is the initial magnitude of the edema (is defined by M0 = (V0 − Vp)/Vp, where Vp and V0 are the pre-implant volume and post-implant volume at day 0, respectively), λe is the edema decay constant (defined by λe = ln(2)/te, where te is the edema half life (EHL)), and the exponent τ was determined to have a value of 2.20 by Chen et al.Reference Chen, Deng, Roberts and Nath46 for the Model CS-1 131Cs sources.
By integrating equation (11), the dose delivered at point ‘P’ in time ‘t’is given by
Equation (1) or equation (2) gives the TCP when the dose is spatially uniform throughout the target volume. However, uniform dose distributions are not always achievable in permanent implants. Therefore, it is difficult to deduce the TCP directly from non-uniform dose distributions using above said equations. NiemierkoReference Niemierko47 and Sanchez-Nieto and NahumReference Sanchez-Nieto and Nahum48 and looked into the problem and suggested possible solutions to deduce the outcome in terms of delta-TCP and equivalent uniform dose (EUD), respectively. Webb and NahumReference Webb and Nahum49 modified the TCP model for non-uniform clonogenic cell density and non-uniform dose distributions. In their derivation the tumour volume was divided into n number of infinitesimally small volume voxels and the TCP for such a system can be written as
where n is the number of voxels, TCPReference Pasteau and Degrais1, TCPReference Barringer2, ……,TCPn are the TCPs corresponding to the voxels 1, 2, …….,n. TCPi is the tumour control probability of cells in the i th voxel and can be written as
![{\rm TCP_i} = exp[- \rho_i {\rm V_i} S_i{(D)}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn15.gif?pub-status=live){kind=small}
where ρi ,Vi and Si(D) are the initial clonogenic cell density, voxel volume and surviving fraction of tumour cells in the i th voxel. In the calculations of Si (D), it was assumed that the i th voxel received an average dose rate of Ri(0) at day 0. With help of equation (13), the Si(D) for ith voxel can be written as
![S_i(D)=exp[-\alpha {\rm R_i}(0)\int\limits_{0}^{\rm t}[exp(- \lambda {\rm t})/{1+{\rm M}_0 exp(- \Lambda_{\rm e}{\rm t})}{\tau/3}]{\rm dt}-\beta {\rm q}({\rm t}){{\rm R_i}(0)\int\limits_{0}^{\rm t}exp[(- \lambda {\rm t})/{1+{\rm M}_0 exp(- \Lambda_{\rm e}{\rm t})}{\tau/3}]{\rm dt}^{2}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn16.gif?pub-status=live)
Equation (14), may now be written as
![{\rm TCP}= \sum\limits_{i - 1}^n {exp[- \rho_i {\rm V_i} S_i{(D)}] = exp[- \sum\limits_{i - 1}^n {\rho_i {\rm V_i} S_i{(D)}}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn17.gif?pub-status=live)
If it is assumed that the clonogenic cell density ρ is uniform throughout the tumour volume then equation (17) may have the form given by
![{\rm TCP}= exp[- \sum\limits_{i - 1}^n {\rho {\rm V_i} {\rm S_i}{({\rm D})}}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn18.gif?pub-status=live){kind=small}
By comparing equation (2) and equation (18), S(D) can be written as
![{\rm SD}= (1/{\rm V}) \sum\limits_{i - 1}^{n} {\rm V_i{S_i}}({\rm D})}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn19.gif?pub-status=live){kind=small}
This is the same equation as discussed in earlier articles.Reference Li, Wang, Stewart and DiBiase18,Reference Wang, Mayr, Nag, Montebello, Gupta, Samsami and Kanellitsas19
Many investigators have modeled resolution of post-implant edema.5,11,45,50 Mathematically, the volume of prostate at time ‘t’ with edema resolution can be expressed by
![V_{i}(t)= V_{pi}[1 + M_0 exp(- \lambda_{e}t)]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn20.gif?pub-status=live){kind=small}
where Vi(t) and Vpi are the volumes of the i th voxel at time ‘t’ with edema and at the time before implant (pre-implant volume).
By replacing Vi of equation (19) with Vi(t) of equation (20), the S(D) can be written as
![{\rm SD}= (1/{\rm V}) \sum\limits_{i - 1}^{n} {\rm V_{pi}{1+{\rm M}_0 exp(- \Lambda_{\rm e}{\rm t})}{\rm S_i}({\rm D})}]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn21.gif?pub-status=live){kind=small}
where V is the prostate volume with edema at day 0. In above equations of TCP the prostate volume was used with edema at day 0. It is clear that the total number of clonogenic cells in the prostate remain constant irrespective of the increase of the volume due to increase in the amount of edema. Hence, the TCP for pre-implant prostate volume can be written as
![TCP = exp[- \rho {\rm V_{p}} {\rm S}({\rm D})]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151128095703107-0895:S146039691200009X_eqn22.gif?pub-status=live){kind=small}
where Vp and ρ are the prostate volume and clonogenic cell density before implant procedure, respectively.
Radiobiological parameters
A set of radiobiological parameters is required for the S(D), TCP and teff calculations. The values of these parameters were taken from previously published reports and are as follow: α = 0.15 Gy−1, β = 0.05 Gy−2, α/β = 3.0 Gy, Tp = 42 days, μ = 61.6 d–1 (i.e., μ = ln(2)/trep, here repair half life trep = 0.27 h , ρ = 1×10Reference Kehwar, Jones, Huq and Smith6 (Nath et al. 2009),Reference Nath, Bice, Butler, Chen, Meigooni, Narayana, Rivard and Yu51 and edema decay constant λe = 0.0713 d-1 (i.e., λe = ln(2)/t1/2_edema, where t1/2_edema is the half life of edema decay with a value of 9.72 days).Reference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5
Patients
Thirty-one patients of prostate cancer, who received a prescribed dose of 115 Gy to the prostate by permanent 131Cs seed implants, were analysed in this study. Details of the implant procedure, technique and seed loading were discussed in previous studies.Reference Smith, Beriwal, Komanduri, Gibbons and Benoit4,Reference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5 Briefly, for each patient the transrectal ultrasound (US) was used to obtain images of the prostate prior to the implantation, as well as pre- and post-needle prior to 131Cs seed implantation. The positioning of the needles and seeds in the needles were defined with the guidance of pre- and post-needle US images. The post-implant CT images were also obtained on the day of the implant (day 0) and at day 14 and day 28. Contouring of the prostate on US and CT images, and seed localisation and analysis of the data were performed by the same individual for each patient. The seed locations were generated for US images of pre-needle and pre-seed (but post needle) prostate volumes, and for the CT images of post-implant prostate volumes at days 0, 14 and 28.
RESULTS AND DISCUSSIONS
It was reported in our previous studyReference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5 that for all 31 patients, the initial magnitude of edema (M0) developed immediately after the implantation on day 0 and was determined by comparing the volumes obtained for pre-needle US and post-needle US images, and pre-needle US images and post-seed implant CT images at day 0. The initial M0 was found to be 22.76 ± 5.99 % (ranged from 5.15 to 84.47 %) for pre-needle US and post-needle US images, and 19.81 ± 4.94 % (ranged from 5.36 to 63.23 %) for pre-needle US images and post-seed implant CT images at day 0. The Student’s t-test indicates that there were no statistically significant differences between the magnitude of the edema or between the volumes obtained from post-needle US versus post-seed implant CT images at day 0 (p > 0.05, Student’s t-test). The magnitude of post-implant edema was also determined by comparing the volumes obtained for pre-needle US and post-seed implant CT images at day 14 and 28. These values of M0 were fitted to equation (20) using the method of least square fit to get EHL for these patients, and was found to range from 3.64 days to 34.48 days with a mean of 9.72 ± 8.31 days (mean ± 1SD). In another study, the EHL was reported to vary from 4 days to 30 days.Reference Waterman, Yue, Corn and Dicker11
The increased use of 125I, 103Pd and more recently 131Cs for permanent prostate seed implants poses interesting radiobiological challenges, because during the long period of dose delivery, significant tumour cell repopulation may occur even in slow growing tumours. So, to account for this the teff for 131Cs prostate implants was calculated using equation (7) with an assumption that the prescribed dose of 115 Gy is delivered uniformly to the prostate. The value of teff for the prescribed dose was found to be 60.36 days, which is very close to the 61 days recommended in AAPM TG -137 reportReference Nath, Bice, Butler, Chen, Meigooni, Narayana, Rivard and Yu51 for 131Cs prostate implants. Kehwar et al.Reference Kehwar, Jones, Huq and Smith6 have discussed that in reality the dose distribution within the prostate in permanent implants will never be uniform, hence each of its voxels will have its own effective treatment time. However, it is not clear which value of teff should be used in the calculations of SF or TCP. Should it be the teff of each voxel, or an average value of teff of all voxels within the prostate based on the doses to these voxels, or the value calculated for the prescribed dose? We suggest that the use of teff calculated for the prescribed dose is appropriate due to the fact that most of the tumour cells will receive adequate dose during this period of time. An average value of teff of all voxels within the prostate would also be an appropriate option but will increase the complexity in calculation process to find out a separate teff for each voxel to get an average. The total dose, in absence of edema, delivered during teff is 113.46 Gy, which is 1.33 % lower than the prescribed dose.
Figure 1 shows the relationship between mean surviving fractions and post-implant time. In the figure, the line ‘a’ represents the SF for the prescribed dose calculated using equation (9), where it was assumed that the prescribed dose is uniformly distributed throughout the prostate volume and the effects of edema and cell proliferation were negligible. The line ‘b’ represents for the SF calculated using equation (21) taking into account the initial CT volumes obtained at day 0 and magnitude of the edema with half life of 9.72 days, and the line ‘c’ represents the SF calculated, using equation (9) and (19), for the seed locations generated for individual post-implant CT volumes obtained at days 0, 14 and 28. The values of SF for delivered prescribed dose have single point data, while the SF calculated using either equation (21) or for individual post-implant CT volumes have a range of data for all 31 patients at 14 and 28 days. Hence, to compare the SFs calculated for prescribed dose and for edema decay using equation (21), and for prescribed dose and for individual CT volumes, the Z –test was applied. On the other hand, when comparisons were made for SFs calculated using equation (21) and for individual post-implant CT volumes, the Student t-test was used. These tests indicate that there were statistically significant differences in the SFs at days 14 and 28 for (i) delivered prescribed dose and that calculated using equation (21) (p = 0.02 and 0.01, respectively, Z-test), and (ii) prescribed dose and that calculated using individual post-implant CT volumes (p = 0.02 and 0.03, respectively, Z-test). However, no statistically significant differences were found between the SFs calculated using equation (21) and individual post-implant CT volumes (p = 0.11 and 0.16, respectively, Student’s t-test), hence it is clear that equation (21) calculates the SF accurately. The plots in Figure 1 revealed that the SF calculated using prescribed dose without edema resolution correction overestimates the results than that of actual implants.
Figure 1. Changes in SF with post-implant time. The line ‘a’ represents SF for prescription dose without edema correction, line ‘b’ represents for calculated SF using equation (21) for day 0 CT images and line ‘c’ for individual CT images obtained at day 0, day 14 and day 28.
The TCP values calculated using equation (22) and individual post-implant CT volumes obtained at day 28 have no statistically significant differences (P > 0.05). The calculated values of TCP using equation (22) for the dose delivered in teff, were found to range from 0.55 to 0.99 with a mean and SD of 0.76 ± 0.14. The TCP model described in equation (22) is unique, in that it accounts for the resolution of edema and its magnitude, and a heterogeneous dose distribution throughout the prostate. The patients with low values of TCP represent the poor quality of the implant. Although extremely low TCP values may not be clinically relevant, TCP calculations could be used to identify patients with suboptimal implants who require careful monitoring.
It was mentioned earlier that EHLs calculated for these patients were found to vary from 3.64 days to 34.48 days.Reference Kehwar, Jones, Huq, Beriwal, Benoit and Smith5 Hence, to estimate the effect of different EHL on the SF and the TCP, calculations were performed using equations (21) and (22) for EHLs of 4, 10, 15, 20, 25, and 34 days for all patients.
Figure 2(a) shows the plots of the mean values of SF versus post-implant time for edema half lives of 4, 10, 15, 20, 25, and 34 days. The curves shown in Figure 2(a) were calculated using equation (21) using above mentioned values of EHLs for each patient, and show that the mean SF increased steadily with increasing EHL. Figure 2(b) shows a plot of the dependence of SF on EHL, calculated at Teff. It can be seen from the figure that as EHL is increased from 4 days to 34 days, the mean SF increased by 2.71 log. The changes in TCP resulting from the changes in the SF shown in Figure 2(b)) are shown in Figure 2(c), which shows that as the EHL is increased from 4 days to 34 days, the mean TCP decreases from 0.94 to 0.51.
Figure 2. Change in the SF with post-implant time: (a) plot of SF versus post-implant time for different EHL, (b) plot of SF calculated at teff versus EHL ranging from 4 days to 34 days, and (c) plot of TCP with EHL ranging from 4 days to 34 days corresponding to the SF of (b).
As mentioned earlier, the values of M0 obtained from pre-needle US images and post-seed implant CT images at day 0 were found to range from 5.36 to 63.23 %. Hence, to predict the effect for different amount of edema, calculations of SF and TCP were done for edema magnitudes of 5%, 10%, 20%, 40% and 60%. The pre-implant volumes were calculated by reducing the day 0 CT volumes for corresponding edema size. In the calculations of TCP, the same number of clonogenic cells were used as those calculated for real pre-implant volumes. The plots of SF versus %M0, and TCP versus %M0 are shown in Figures 3(a) and (b) for EHL ranging from 4 days to 34 days. Figures 3(a) and (b) illustrate that the SF increases and the TCP decreases steadily with increasing values of EHL and %M0. At EHLs of 4 days, 10 days, 20 days and 34 days, as %M0 increases from 5 to 60% the SF increases by 1.04 log, 2.44 log, 3.54 log and 4.26 log, respectively, and the TCP decreases by 0.24, 0.67, 0.76 and 0.74, respectively. Similarly, at %M0 of 5%, 10%, 20%, 40% and 60%, as EHL increased from 4 days to 34 days, the SF increased from −18.34 log to −18.02 log, −18.21 log to −17.59 log, −17.97 log to −16.74 log, −17.56 log to −15.15 log, and −17.29 log to −13.77 log, respectively, and the TCP decreased from 0.80 to 0.74, 0.78 to 0.64, 0.73 to 0.39, 0.64 to 0.02, and 0.56 to 0.00, respectively. The change in the SF and TCP is more dramatic when the values of EHL and %M0 are greater than 10 days and 30%, respectively, and is more pronounced for extreme values of EHL and %M0, such as 34 days and 60%, respectively.
Figure 3. Schematic demonstration of role of EHL on SF and TCP calculated post-implant time at Teff. (a) plots of SF as a function of magnitude of edema at different EHL, and (b) plots of TCP as a function of magnitude of edema at different EHL.
Similar results were reported in a study used by Yue et al.Reference Yue, Chen and Nath15 for 103Pd seed implant, as the SF increased from 1.68 log to 4.73 log when values of EHL were increased from 4 days to 30 days for a fixed edema magnitude of 50%, and by 4.75 log when the edema magnitude was increased from 0 to 95% for a fixed EHL of 10 days. In the same study, it was reported that in an 125I permanent seed implant, the SF increased from 0.46 log to 2.29 log when the values of EHL were increased from 4 days to 30 days for a fixed edema magnitude of 50%, and by 1.68 log when the edema magnitude was increased from 0 to 95% for a fixed EHL of 10 days. In the study by Yue et al.,Reference Yue, Chen and Nath15 the SF was calculated using α/β = 10 Gy. The authors mentioned that the general pattern of effects caused by edema are very similar for α/β = 10 Gy and α/β = 2 Gy, and found that the SF decreases with the decrease of the α/β ratio. It was also reported in this study that the edema-induced increase in the SF is independent of tumour cell proliferation rates; it was found to be a function of EHL, amount of edema magnitude, and characteristics of the radioactive source used in the implants.
Figures 4(a)–(d) and Figures 5(a)–(d) illustrate the edema-induced reduction in dose for 131Cs and 125I prostate permanent implants as a function of the edema magnitude and edema half lives, respectively. In these figures, two sets of curves are shown. One is for delivered dose and the other for instantaneous dose with time. The curves which show increasing trend with time represent the delivered dose and the curves with decreasing trend represent the instantaneous dose with implant time. The edema-induced dose reduction calculations were done using the values of τ of 2.20 for the Model CS-1 131Cs sources and 2.35 for the Model 6711 125I sourcesReference Nath, Bice, Butler, Chen, Meigooni, Narayana, Rivard and Yu51 for prescription doses of 115 Gy and 145 Gy, respectively. The values of Teff used for these calculations are 60.36 days and 236 days, respectively, for 131Cs and 125I permanent implants.
Figure 4. Edema induced dose reduction in 131Cs implants as a function of post-implant time at different edema magnitude. (a) EHL of 4 days, (b) EHL of 10 days, (c) EHL of 20 days, and (d) EHL of 34 days.
Figures 4(a)–(d) and Figures 5(a)–(d) show that the dose reductions in 131Cs implants varied from 1.1% (for EHL= 4 days and M0 = 5%) to 32.3% (for EHL= 34 days and M0 = 60%). These values are higher than the corresponding dose reduction for 125I implants, shown in Figures 5(a)–(d), which are found to vary from 0.3% (for EHL= 4 days and M0 = 5%) to 17.5% (for EHL= 34 days and M0 = 60%). The dose reductions in 131Cs and 125I implants as a function of EHL and %M0 are listed in Table 1.
Figure 5. Edema induced dose reduction in 125I implants as a function of post-implant time at different edema magnitude. (a) EHL of 4 days, (b) EHL of 10 days, (c) EHL of 20 days, and (d) EHL of 34 days.
Table 1. Percentage dose reduction for different edema magnitude with edema half lives.
It is seen from both sets of curves and Table 1 that the window of dose reduction increases with increasing %M0, and found to be more pronounced in 131Cs permanent implants compared to the 125I implants and becomes worse for the extreme combination of EHL of 34 days and %M0 of 60%.
Chen et al.Reference Chen, Deng, Roberts and Nath22 reported that the edema-induced dose reduction in preplanned 131Cs implants could exceed 10% of prescription dose with moderate or large edemas. In our study, we did not compute edema-induced dose reduction in pre-planned implants, but calculated the dose reduction in real time implants with the resolution of the edema with post-implant time and found that for an average edema magnitude of 20%, as the EHL is increased from 4 days to 34 days, the dose reduction increased from 4.27 to 11.27% in 131Cs, and 1.03 to 5.98% in 125I implants. In both studies it can be seen that the edema-induced dose reduction is a critical issue and must be addressed to compensate for this effect. In pre-planned 131Cs implants, Chen et al.Reference Chen, Deng, Roberts and Nath22 suggested that to compensate 5%, 10%, 15%, 20%, or 25% edema-induced dose reduction an external beam radiotherapy dose of 2 Gy, 8 Gy, 12 Gy, 14 Gy, or 18 Gy in 1, 4, 6, 7, or 9 fractions, respectively, with 2 Gy per fraction could be delivered. In real time implants, this compensation should be done by redefining the seed positions.
Results of the present study show that the prostatic edema causes significant changes in the SF and TCP due to drastic edema induced dose reduction in 131Cs implants because of the shorter half life of the source. However, it would not be appropriate to conclude that 131Cs prostate seed implants are inferior to 125I prostate seed implants. The effect of edema on the SF and TCP can be minimised by redefining seed positions in real time implantation with the guidance of post-needle plans.
CONCLUSIONS
In this study, the impact of edema and dose non-uniformity on SF and TCP has been discussed. The exponential decaying nature of the edema in prostate implants has been incorporated into the SF equation of the linear quadratic model. The model predicts reasonably accurate SF and TCP with no statistically significant differences (p > 0.05) compared to that calculated for individual post-implant CT images obtained at day 0, day 14 and day 28.
Prostatic edema in 131Cs implants may cause significant increase in the SF and consequently decrease in TCP due to the short half life of radioactive source. The short half life of 131Cs seeds causes drastic edema-induced dose reduction, in the implants, because approximately 80% of the prescribed dose is delivered during first 3 weeks of the implant. As edema magnitude becomes larger and decays more slowly, the dose reduction is more pronounced and consequently more tumour cells survive the treatment of 131Cs seed implants. If this increase in the SF is not compensated for, it may lead to poor local control of these patients. Since the magnitude and resolution time of edema are two major unpredictable factors in the implants, it is difficult to compensate these factors in the implants using preplans. The only way to compensate is using real time implants by properly defining the seed positions with the guidance of post-needle plans, because the edema magnitude of an implant is known by that time.
