International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009) Saratoga Springs, New York, May 3-7, 2009, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2009)
ADJOINT CURRENT-BASED APPROACHES TO PROSTATE BRACHYTHERAPY OPTIMIZATION Jeremy A. Roberts and Douglass L. Henderson University of Wisconsin - Madison 1500 Engineering Drive, Madison, WI 53706
[email protected];
[email protected]
ABSTRACT This paper builds on previous work done at the University of Wisconsin - Madison to employ the adjoint concept of nuclear reactor physics in the so-called greedy heuristic of brachytherapy optimization. Whereas that previous work focused on the adjoint flux, i.e. the importance, this work has included use of the adjoint current to increase the amount of information available in optimizing. Two current-based approaches were developed for 2-D problems, and each was compared to the most recent form of the flux-based methodology. The first method aimed to take a treatment plan from the flux-based greedy heuristic and adjust via application of the current-displacement, or a vector displacement based on a combination of tissue (adjoint) and seed (forward) currents acting as forces on a seed. This method showed promise in improving key urethral and rectal dosimetric quantities. The second method uses the normed current-displacement as the greedy criterion such that seeds are placed in regions of least force. This method, coupled with the dose-update scheme, generated treatment plans with better target irradiation and sparing of the urethra and normal tissues than the flux-based approach. Tables of these parameters are given for both approaches. In summary, these preliminary results indicate adjoint current methods are useful in optimization and further work in 3-D should be performed. Key Words: Brachytherapy, optimization, adjoint methods, greedy method
1. INTRODUCTION Optimizing the placement of permanent brachytherapy seeds is a difficult task. Over the past several years, numerous attempts to apply robust optimization techniques to brachytherapy have been met with ample success; however, even as computational power grows, these routines still require too much time for on-the-fly treatment planning. Therefore, methods that provide adequate plans in short periods of time are desired. Much work has been done at the University of Wisconsin – Madison in applying the adjoint method of reactor physics to brachytherapy optimization. Specifically, the “greedy heuristic” of Yoo [1] [2] and her predecessor Chaswal [3] has led to highly efficient methods and good results. However, it was believed by applying further methods from the adjoint theory that better plans could still be found; the present work aims to do exactly this. 2. BACKGROUND 2.1. The Greedy Heuristic Work has been done to develop a greedy heuristic for brachytherapy optimization based on the adjoint function [1, 2]. Later work refined the general concept by incorporating multi-species seeds
J. A. Roberts and D. L. Henderson
and developing more efficient ways to avoid seed clustering [3–5]. This later work employing efficient seed dispersion [3] is described briefly below and is the standard against which the methods developed in this paper are compared. The adjoint function for a given region of interest (ROI) is defined as + DROI (~r) = C1Su φ+ r)/VROI , ROI (~
(1)
where φ+ ROI is the adjoint flux associated with the ROI, VROI is the volume of the ROI, C1Su is a constant yielding units of Gy/Su , and Su denotes a “source unit”; here, one Su represents a 0.43 mCi 125 I seed. The adjoint function defines the sensitivity of a ROI dose to a 1 Su seed placed at ~r. To capture the sensitivity of all ROIs, the adjoint ratio [6] is defined as ρ(~r) =
wu Du+ (~r) + wr Dr+ (~r) + wn Dn+ (~r) , Dt+ (~r)
(2)
where the subscripts u, r, n, and t denote the urethra, rectum, normal tissue, and target, respectively, and the wi are weighting factors. This ratio is a measure of the dose to the target region relative to overall sparing of sensitive tissues for a seed placed at ~r. Figure 1 depicts a representative prostate slice with annotated regions and correct scale.
Figure 1. Representative prostate slice. Greedy methods aim to solve optimization problems by satisfying some greedy criterion at each step. For brachytherapy, the adjoint ratio is used as the greedy criterion. Essentially, at every step (i.e. every placement), a seed is placed at the position of least ρ. Seeds are placed until the desired target coverage is reached. Because one location of minimum ρ may be in a neighborhood of similarly low ρ values, a mechanism by which seeds are preferentially placed away from current placements is needed. To do this, the “dose-update” scheme was developed [3]. In this scheme, the adjoint ratio is scaled by the dose profile, or ρd (j) = ρ(j)Sdose (j) , 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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where n 1 X D(i, j) , Sdose (j) = Dp
(4)
i=1
and D(i, j) is the dose to a voxel j from a seed at voxel i, and Dp is the prescribed dose. Letting ρd be the modified greedy criterion ensures seed clustering is efficiently avoided.
2.2. Evaluation Parameters To compare the methods developed below to the greedy heuristic just described, a number of evaluation parameters are employed. While more thorough descriptions can be found in the literature, e.g. [3] [7], Table I provides brief descriptions. It should be noted that V100 and CN are the only parameters to be maximized, while the rest are to be minimized.
Table I. Description of Evaluation Parameters. Here, Vt , Vt,Dp , and VDp denote the target volume, target volume receiving at least Dp , and total volume receiving at least Dp = 145 Gy, the prescribed dose. TARGET C OVERAGE V100 V150
percentage of target volume receiving at least Dp percentage of target volume receiving at least 1.5Dp
D OSE Q UALITY DN R CN
“dose non-uniformity ratio”, V150 /V100 “conformation number”, (Vt,Dp /Vt ) × (Vt,Dp /VDp )
U RETHRA V360 D90,u V125,u
percentage of urethral volume receiving 360 Gy minimum dose to 90% of urethral volume percentage of urethral volume receiving at least 1.25Dp
R ECTUM V80,r V90,r D90,r
percentage of rectal volume receiving at least 0.8Dp percentage of rectal volume receiving at least 90 Gy minimum dose to 90% of rectal volume
N ORMAL T ISSUES D90,n
minimum dose to 90% of normal tissue volume
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3. DATA AND MODELING 3.1. Code and Data The transport code DANTSYS was used to generate all fluxes and currents [8]. For this work, a broad three photon group library generated from the lowest three groups of the FENDL 42 group working library was used. The average energy of 125 I, used here for implants, falls in the first energy group, spanning 20-30 keV. The second and third groups span 10-20 and 1-10 keV, respectively [9]. The prostate model used is based on a set of three-dimensional ROIs used by Yoo and Chaswal [1, 3]. The model spans 60 mm × 55 mm in the plane contained in ultrasound images. Additionally, there are 14 such planes, i.e. 14 ultrasound images, each separated by 5 mm. Because this work presents algorithms treated in only two-dimensions, a subset of these images and their associated ROIs are taken as the foundation to the models used herein (e.g. the slice in Figure 1). A DANTSYS input was prepared for each ROI of five ultrasound slices (numbers 3, 5, 7, 9, and 11 of the original 14). The treatment plane is defined by voxels of 1 square mm to coincide with the ultrasound resolution; this serves as the transport mesh. An optional 5 square mm resolution for the purpose of seed location is used to represent treatment grids. Each pixel coinciding with the ROI is set to 1; those outside the ROI are set to 0. The entire treatment plane is modeled as water. 3.2. Flux and Current Generation In generating the forward and adjoint fluxes, it was possible to employ a ray tracing first collision source. Doing so eliminated ray effects, and an SN order of 8 was found to be adequate. To generate the forward and adjoint currents, slightly modified inputs were necessary because the angular flux files are unavailable when a first collision source is used. To help mitigate ray effects, an SN order of 16 was used. For the target and normal tissues, ray effects were largely avoided; however, for the smaller urethra, rectum, and source (described below), some ray effects are present. The extent to which these effects impacted results was not investigated. 3.3. Seed Model: Forward Current and Dose Profiles The previous work on Greedy methods focused primarily on the use of 125 I seeds (Nycomed Amersham model 6711, Arlington Heights, IL). While certainly possible to model in three-dimensions, it was hard to determine how best to model this source in two-dimensions. In the end, it was believed using a small cluster of 1 mm voxels as the forward source definition and then normalizing it to 0.43 mCi would suffice. The source model used throughout is a 1 mm voxel with an additional 1 mm voxel adjacent to each side; thus, five voxels define the seed. Material effects, i.e. the titanium sheath and other components making up the seed, are not included; the seed consists simply of a photon source centered in a water medium. To compute the dose, the ICRP 21 flux-to-dose conversion factors (DF) were used [10]. Table II provides these constants for the energies of interest. Log-log interpolation was used to find the DF 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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Table II. ICRP-21 Flux-to-dose Conversion Factors. Energy (MeV)
(rem/hr) (p/cm2 −s)
DF
0.005 0.010 0.015 0.020 0.025 0.030
1.229e-5 2.780e-6 1.952e-6 5.880e-7 3.719e-7 2.560e-7
Figure 2. Comparison of 2-D Model and TG-43 Radial Dose Profile. values not listed in the report; these are included with the actual tabulated values, denoted via bold font. The DF values at mid-group were used to calculate energy-weighted dose. Previous work has employed the AAPM Task Group 43 recommendations for 125 I dosimetry [11]. A comparison between the midaxial plane of this distribution for a 1 Su seed and the twodimensional 1 Su seed from above was performed. The TG-43 definition has significantly higher dropoff than does the two-dimensional model. This is expected, because the three-dimensional seed incorporates axial falloff in addition to radial falloff; the two-dimensional model has only radial falloff. A plot comparing the midaxial TG-43 and the two-dimensional radial profiles is given in Fig. 2. One may wonder why the TG-43 dose profile was not used. In short, it easily could have been used, but two reasons suggested using the two-dimensional model. First, there is no way to compute the current associated with the TG-43 protocol; because the current is fundamental to this work, it was thought best simply to derive all the data used from one set of models. Second, the dose profile 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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for TG-43 falls off rapidly as noted. Because the problem is confined to a planar solution space, more seeds are needed to cover the volume represented than would be true for the associated threedimensional problem. Hence, excessive over-dosing would arise, and to avoid this, the modeled seed profile is used. 4. Method One: Current-Displacement as an Adjustment In this section, a method is presented that takes the seed placement generated by the flux-based greedy heuristic and applies a displacement based on the seed forward and sensitive tissue adjoint currents. 4.1. Theory Given the adjoint currents for tissues, a summed current is defined for the x- and y- directions as Jxtot (i) = wr Jxr (i) + wu Jxu (i) + wn Jxn (i) + wt Jxt (i)
(5)
Jytot (i) = wr Jyr (i) + wu Jyu (i) + wn Jyn (i) + wt Jyt (i) ,
(6)
and where each weight coefficient wτ is set to 1/Vτ for the sensitive tissue τ and −1/Vt for the target. More succinctly, this summed current can be written J~tot (i) = xˆJxtot (i) + yˆJytot (i) .
(7)
Given initial placement by the flux-based greedy method, the summed current at each seed location ~ can be calculated. This provides everything needed to calculate the current-displacement (C) vector. The essential idea is that the currents in the x- and y- directions from the sensitive tissue may act as a pushing mechanism used to shift the seeds from their initial placement; in other words, the seed placement can be “fine tuned.” However, to avoid seed clustering, the forward current from a seed is included to act on all other seeds. In this way, the current-displacement for a seed at i is defined n X tot ~ ~ (i) + b ~s (i, j), C(i) = a − J J
(8)
j=1 j6=i
where J~s (i, j) denotes the forward current at i from a seed at j, b is a parameter defining the relative importance of the seed currents to the tissue currents, and a is a parameter that converts the summed current magnitudes into a suitable seed displacement value in [mm]. The parameter b is defined b = max J~tot /(k max J~s ) , (9) where max · refers to the highest value of the given function throughout the treatment plane and q p (10) k = 1/ m1 m2 /m1 , m1 = mean((Jxs )2 + (Jys )2 ) , 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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and m2 = mean((Jxtot )2 + (Jytot )2 ) .
(12)
Thus, the relation bears resemblance to the root-mean-square. Finally, the parameter defining the displacement, a, is defined a = k/ max J~tot . (13) These empirical parameter definitions allow for a dynamic current-displacement and provide consistent results for the problems tested. ~ defines how seeds should be displaced. Such adjustment can be done The current-displacement C for any number of seeds at once, and several iterations of adjustments can be performed. This work lets all seeds be displaced and for the total displacement to fall off linearly with each iteration. Doing so effectively stops displacement after some number of iterations and gives the solution. In this study, using three iterations proved adequate for convergence. 4.2. Results For this study, no treatment grid was used for either the baseline greedy heuristic (GY) or GY with the addition of current-displacement (CD1 ). Three cases are presented, one corresponding to each of seed strengths f = 1, 2, and 3 × Su . Table III provides the evaluation parameters averaged over all slices for the given source strength. The right hand column provides percentage changes from the baseline case. Figure 3 provides dose volume histograms (DVH) for each slice and f = 1, comparing GY, CD1 , and CD2 , a method to be discussed below. Additionally, Figure 4 compares the associated seed placements. From Table III, CD1 , on average, maintains the desired target coverage (>98% ). Additionally, V150 is reduced slightly, indicating less over-irradiation of the target. Both CN and DN R are reduced slightly. Recall, CN indicates how well the dose conforms to the target region and DN R quantifies “hot spots”; thus, greater CN and lower DN R indicate better treatments. Given small improvement in two of three variable parameters, CD1 leads to slightly better target irradiation. Additionally, CD1 improves all the parameters for the urethra, and in particular, V125,u is reduced on average by 8.40% compared to GY, marking significant reduction in urethral overdosing. CD1 also reduces V80,r for the rectum. Contrarily, D90,n is slightly increased. However, this increase is smaller than the improvements in the urethra and rectum, and consequently, CD1 improves tissue sparing in an average sense. Note that the number of seeds for CD1 and GY are the same—this is because CD1 merely moves the seeds placed by GY. It should be reiterated that CD1 was not used for problems employing a treatment grid. If, on average, the displacements produced are smaller than typical grid resolution, as seems to be the case, the value of CD1 is questionable. However, it may be the case that treatment methods move toward continuous spatial treatment, for which CD1 could be applied with ease. 5. Method Two: Normed Current-Displacement as Greedy Criterion While the results from the previous section are promising, it was believed applying current-displacement in another way might yield even better results. The idea is essentially this: given the adjoint current 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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Table III. Average evaluation parameters, by f and total. f =1
f =2
f =3
Total
%∆
CD1 GY CD1 GY CD1 GY CD1 GY
98.08 98.42 79.55 80.20 0.51 0.50 0.81 0.81
98.87 99.23 81.50 82.87 0.49 0.50 0.82 0.83
99.39 99.47 85.32 86.89 0.45 0.45 0.86 0.87
98.78 99.04 82.12 83.32 0.48 0.49 0.83 0.84
-0.26
CD1 GY CD1 GY CD1 GY
57.16 67.73 0.00 0.00 167.35 170.95
69.29 75.81 0.00 0.00 177.00 180.90
80.42 82.32 0.00 0.00 195.86 198.33
68.96 75.29 0.00 0.00 180.07 183.39
-8.40
CD1 GY CD1 GY CD1 GY
26.53 28.15 80.87 80.78 86.40 86.47
38.70 41.45 87.73 87.54 90.07 90.41
64.86 64.03 95.68 95.66 102.98 102.21
43.36 44.54 88.09 87.99 93.15 93.03
-2.65
CD1 GY
46.74 45.71
50.79 49.91
53.37 52.82
50.30 49.48
1.65
CD1 GY
19.60 19.60
10.40 10.40
7.60 7.60
12.53 12.53
0.00
TARGET V100 (%) V150 (%) CN DN R
-1.44 -2.30 -1.03
U RETHRA V125,u (%) V360 (%) D90,u (Gy)
0.00 -1.81
R ECTUM V80,r (%) V90,r (%) D90,r (Gy)
0.11 0.13
N ORMAL T ISSUE D90,n (Gy) Q UALITATIVE Seeds
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provides more information straightaway than does the adjoint flux, then perhaps using the current completely in place of the flux might yield nearer-to-optimal plans. 5.1. Theory Let the current-displacement be defined without a seed component as ~ C(i) = xˆJxtot (i) + yˆJytot (i) ,
(14)
where now the summed current from tissues at a voxel i is redefined in the x- and y-directions as Jxtot (i) = wr |Jxt r(i)| + wu |Jxu (i)| + wr |Jxr (i)| + wn |Jxn (i)|
(15)
Jytot (i) = wr |Jyt r(i)| + wu |Jyu (i)| + wr |Jyr (i)| + wn |Jyn (i)| ,
(16)
and where | · | denotes the absolute value. Note the use of the absolute value in Eqs. 15 and 16 is a modification of Eqs. 5 and 6. The normed current-displacement is defined by the L2 norm q ~ ||C(i)|| = Cx2 (i) + Cy2 (i) . (17) 2 ~ 2 is the new greedy criterion. This function ||C|| Defining for a tissue τ the parameter r rmsτ =
τ 2 τ 2 mean (Jx ) + (Jy ) ,
(18)
where rmsτ is simply the root-mean-square of the tissue current, the weighting factor for each tissue τ is defined 1 wi = , τ ∈ (u, r, n) . (19) rmsτ × min rmsτ Note that the target is not considered. In studying these parameters manually, in no case did including the target improve the treatment plan; hence, wt = 0. Using these definitions for the normed current-displacement, a new greedy heuristic is established. In this schematic, the coordinate of least normed current-displacement is chosen for the next seed. However, a method to disperse seeds is needed, and for this purpose, the dose-update scheme defined by Eq. 3 was used. 5.2. Results For this study, GY again refers to the original greedy method, and CD2 refers to the method just described. Four cases are analyzed, one corresponding to each of strengths f = 1 and 2 ×Su both with and without a treatment grid. Table IV provides evaluation parameters averaged over source strengths for the grid and non-grid cases. Every other column provides percentage changes from the baseline (GY). Again, Figure 3 provides the DVHs of all slices for the GY, CD1 , and CD2 methods with f = 1 and Figure 4 compares the associated seed placements. 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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From Table IV, CD2 , on average, slightly increases the target coverage, denoted by V100 . Additionally, V150 is reduced slightly and both CN and DN R are improved. Comparing the grid and no grid cases reveals that the treatment grid has a small impact on the target irradiation. To a relatively small degree, CD2 leads to better target irradiation. Furthermore, CD2 significantly improves all the parameters for the urethra. In particular, V125,u is reduced on average by nearly 13% and 21% in the grid and no grid cases, respectively. This is a substantial improvement in minimizing overdosing of the urethra. The impact of the grid on this sparing is large. Without the grid, the improvement is nearly 1.5 times that of the grid case for V125,u . Additionally, normal tissue sparing is improved, as D90,n is significantly reduced. The difference between the grid and no grid improvements is small compared to the difference for V125,u . Contrarily, CD2 leads to an increase in rectal dose, most evident in the increased V80,r . The increase in dose is impacted greatly by the grid. For the grid case, V80,r increases by only 1.73%. However, for the no grid case, V80,r increases by nearly 22%. A look at the individual cases (i.e. for each f and 2-D slice) reveals that just a few seemingly anomalous cases severely skew the averages. For the case of f = 1, the average over all five slices without the grid, CD2 yields an increase in V80,r of 23.11%. Removing the anamolous case (for which the individual increase is nearly ten-fold), the average increase drops to just 5.30%, a much less troubling increase. Regardless, further work should investigate why such anamolies occur and how they can be avoided. Finally, CD2 requires about one fewer seed on average to provide the treatment coverage. This is important both for tissue sparing and improved economics. 6. CONCLUSION This work has described two novel approaches that employ ROI adjoint current vectors in brachytherapy treatment planning for prostate cancer. The adjoint current measures both the direction and magnitude of the net flow of the importance a seed at any position has on the dose to the ROI. The first approach uses these current vectors to define the current-displacement, which quantifies the “force” exerted on a seed at some position. The force on all seeds is translated into an appropriate displacement vector used to find new seed positions. The adjoint function-driven greedy heuristic is used to generate initial seed placements from which new seed positions are found. The new method applying current-displacement was compared to the original greedy heuristic and was found to improve many of the dosimetric quantities studied, with urethral quantities being most improved. The second approach uses the L2 norm of a modified current-displacement as a new greedy criterion for use in the greedy heuristic. This method was compared to the original greedy heuristic and was found to improve substantially target irradiation and sparing of the urethra and normal tissues. However, increased dose to the rectum was an issue, particularly for the no grid cases in which seemingly anomalous cases arose. A further investigation is needed to solve this issue. Additionally, the new method used fewer seeds on average than the original greedy heuristic. Both uses of adjoint current showed improvement over established methods. Thus, it seems further 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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Table IV. Average evaluation parameters for grid, no grid, and overall. Grid
%∆
No Grid
%∆
Total
%∆
CD2 GY CD2 GY CD2 GY CD2 GY
99.50 99.39 79.64 80.70 0.59 0.56 0.80 0.81
0.11
99.11 98.82 79.87 81.54 0.54 0.52 0.81 0.82
0.29
99.30 99.11 79.76 81.12 0.56 0.54 0.80 0.82
0.20
CD2 GY CD2 GY CD2 GY
72.35 83.13 0.00 0.00 178.86 181.63
-12.98
CD2 GY CD2 GY CD2 GY
42.69 41.97 84.95 84.28 90.73 90.27
1.73
CD2 GY
38.74 44.55
CD2 GY
13.20 14.20
TARGET V100 (%) V150 (%) CN DN R
-1.31 5.40 -1.48
-2.05 4.66 -2.06
-1.68 5.04 -1.77
U RETHRA V125,u (%) V360 (%) D90,u (Gy)
57.03 71.77 0.00 0.00 171.95 175.92
-20.54
42.41 34.80 88.00 84.16 91.73 88.44
21.86
-13.04
40.48 47.81
-7.04
13.90 15.00
0.00 -1.53
64.69 77.45 0.00 0.00 175.41 178.78
-16.48
42.55 38.38 86.48 84.22 91.23 89.35
10.85
-15.13
39.66 46.18
-14.12
-7.33
13.55 14.60
-7.19
0.00 -2.26
0.00 -1.89
R ECTUM V80,r (%) V90,r (%) D90,r (Gy)
0.80 0.52
4.56 3.71
2.68 2.10
N ORMAL T ISSUE D90,n (Gy) Q UALITATIVE Seeds
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(a) Slice 3.
(b) Slice 5.
(c) Slice 7.
(d) Slice 9.
(e) Slice 11.
Figure 3. CD: DVHs for each slice with f = 1, with no grid.
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(a) Slice 3.
(b) Slice 5.
(c) Slice 7.
(d) Slice 9.
(e) Slice 11.
Figure 4. Seed placement for each slice with f = 1, with no grid.
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applications of the adjoint current could also be useful. Chaswal in [3] develops a statisticalbased approach to use the adjoint concept in placement of directional seeds for discrete angles. While this approach does yield good results, it seems that adjoint angular information, e.g. the adjoint angular current, contains exactly the information needed in placing directional seeds in a continuous angular space. Currently, research in that area is being conducted as a next step in this work. As has been noted earlier, there are limitations to working in two-dimensions. The full extent of the success or shortcomings of the two methods described here will only be known upon translation to three-dimensions. It is hoped the methods used throughout are easily adaptable to three-dimensions and furthermore that the same improvements seen here are realized in future work. ACKNOWLEDGEMENTS The first author wishes to thank Marat Seydaliev for several routines that helped with reading binary flux files. REFERENCES [1] S. Yoo, M. E. Kowalok, B. R. Thomadsen, and D. L. Henderson. Treatment planning for prostate brachytherapy using region of interest adjoint functions and a greedy heuristic. Physics in Medicine and Biology, 48(24):4077–4090, DEC 21 2003. [2] S. Yoo, M. Kowalok, D. L. Henderson, and B. R. Thomadsen. Seed- and needle- optimization for brachytherapy using greedy heuristic and adjoint functions. Medical Physics, 30(6):1429– 1430, JUN 2003. [3] V. Chaswal. Adjoint Based Treatment Planning for Brachytherapy: Novel Techniques and Further Developments. PhD thesis, University of Wisconsin - Madison, 2008. [4] V. Chaswal, S. Yoo, B. R. Thomadsen, and D. L. Henderson. Multi-species prostate implant treatment plans incorporating Ir-192 and I-125 using a Greedy Heuristic based 3D optimization algorithm. Medical Physics, 34(2):436–444, FEB 2007. [5] V. Chaswal, S. Yoo, B. R. Thomadsen, and D. L. Henderson. Investigations into the use of multi-species seeds in interstitial prostate implant brachytherapy using the 3-D treatment optimization program based upon the region of interest ajoint functions and greedy heuristic algorithm. Medical Physics, 31(6):1746, JUN 2004. [6] F. C. Difilippo. Forward and adjoint methods for radiotherapy planning. Medical Physics, 25(9):1702–1710, 1998. [7] S. Nag, D. Beyer, J. Friedland, P. Grimm, and R. Nath. American Brachytherapy Society (ABS) recommendations for transperineal permanent brachytherapy of prostate cancer. Int J Radiat Oncol Biol Phys, 44(4):789–99, 1999. [8] DANTSYS. User’s Guide. LA-12969-M(Los Alamos: The Applied Physics (X) Division of Los Alamos National Laboratory, 1997. [9] D. L. Henderson, S. Yoo, M. Kowalok, T. R. Mackie, and B. R. Thomadsen. The Adjoint Method for The Optimization of Brachytherapy and Radiotherapy Patient Treatment Planning Procedures Using Monte Carlo Calculations. Technical report, DOE/ID/13774, University of Wisconsin-Madison (US), 2001. 2009 International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, NY, 2009
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[10] ICRP Committee 3 Task Group, P Grande, and M. C. O’riordant, chairman. Data for Protection Against Ionizing Radiation from External Sources: Supplement to ICRP Publication 15. Technical report, International Commission of Radiological Protection, APR 1971. [11] R. Nath, L.L. Anderson, G. Luxton, K.A. Weaver, J.F. Williamson, and A.S. Meigooni. Dosimetry of interstitial brachytherapy sources: Recommendations of the AAPM Radiation Therapy Committee Task Group No. 43. Medical Physics, 22:209, 1995.
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