- Email: [email protected]
A nomograph for permanent implants of palladium-103 seeds
Inr J. Rodiarmn Oncology Em/ Phys.. Vol. Printed in the U.S.A. All rights reserved.
27, pp.
0360-3016/93 $6.00 + .I0 Copyright 0 1993 Pergamon Press Ltd.
129-135
??Technical Innovations and Notes
A NOMOGRAPH
FOR PERMANENT
LOWELL L. ANDERSON, Memorial
PH.D.,* Sloan-Kettering
IMPLANTS
OF PALLADIUM-103
SEEDS
JANAKI V. MONI, M.D.* AND LOUIS B. HARRISON, Cancer Center,
M.D.+
1275 York Ave., New York, NY 10021
lo3Pd is being substituted for 125Iin permanent implants for which it is desired to deliver a higher initial dose rate while maintaining readily achieved radiation protection. We have constructed a nomograph to assist in determining both the total seed strength required and the appropriate needle spacing for lo3Pd implants. We have calculated the “matched peripheral dose” (MPD), that is, the dose for which the isodose contour volume is equal to the target volume, for 64 125Iand 13 ‘03Pd actual implants as if “‘Pd had been used for all of them, employing a computer lookup table based on single-seed dose distribution measurements in solid water. The calculated data were used to obtain a least-squares fit to a linear relationship between the logarithm of the total seed strength for a given MPD and the logarithm of the average dimension, d. (cm). We found that, for a nominal MPD of 11,500 cGy, total seed strength (in mCi) is given by 3.2 d$” . A lo3Pd nomograph has been constructed on the basis of this power function relationship. Our nomographic guide for planning lo3Pd implants calls for total seed strength to increase significantly faster as a function of target volume average dimension than is the case for 1251.This nomograph will facilitate the application of lo3Pd seeds in permanent implants. lo3Pd seeds, Nomograph, Brachytherapy
planning.
isodose contour volume with estimated target volume, that is, with the volume of an ellipsoid having the same dimensions as the target region (8). Initially, the total ‘03Pd seed strength required to produce a matched peripheral dose (MPD) of 115 Gy was taken to be 3.6 times the total strength specified by the 1251nomograph for the same target average dimension. This factor was based on the results of recalculating MPD’s for recent ‘25I cases as if lo3Pd seeds had been used. It was recognized, of course, that such a factor must, in reality, be volume (or average-dimension) dependent, due to the lower photon energy of lo3Pd. Clearly, a separate nomograph was needed for ‘03Pd.
INTRODUCTION Palladium-
103 decays by electron capture with a half life
of 17 days and emits photons having an average energy of 20.9 keV. Seeds containing lo3Pd were developed by Russell ( 15) and are being used in brachytherapy to treat malignancies at selected sites, notably by Blasko et al. in custom-planned permanent using the ultrasound-guided
implants for prostate cancer, technique that has been de-
scribed for 12’1prostate implants (5). As a result of the difference in half-life, the initial dose rate for a lo3Pd permanent implant with a prescribed dose of 115 Gy is higher than that for an ‘25Ipermanent implant with a prescribed dose of 160 Gy. This higher dose rate provides a theoretical advantage for lo3Pd in eradicating rapidly dividing tumor cells. At our institution, more than 20 patients have received permanent implants of lo3Pd seeds, primarily volume implants for which custom planning was not an option and for which, had ‘25I seeds been used, total seed strength would have been determined using a planning nomograph. For such patients, dose is evaluated by matching
METHODS
AND
MATERIALS
The current version of the “‘1 nomograph has been in use since 1983. For target volumes having an average di-
mension, d,, greater than or equal to 3 cm it gives the total apparent activity*, A, according to the power function, A = 1.34 di.2, with A expressed in mCi and d, in cm. The value of the exponent was determined from cal-
Presented at the Annual Meeting of the American Society for Therapeutic Radiology and Oncology, Washington, DC, 5 November 199 1. * Department of Medical Physics. ’ Brachytherapy Service, Department of Radiation Oncology. Accepted for publication 12 March 1993. * “Apparent activity” is an interim quantity defined (8) as the activity of an unencapsulated point source that would produce the same exposure rate at the same distance in free space
as that produced by the actual source on its transverse axis at a distance large enough that the inverse square law is valid. The currently recommended quantity for source strength is the “airkerma strength”, which is defined (17) as the product of airkerma rate (9) in free space along the transverse axis and the square of distance from the source center, again with the proviso that the distance be large enough to assure that the inverse square law is followed and that, therefore, the defined quantity does not vary with distance. 129
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1. J. Radiation
Oncology
0 Biology 0 Physics
culations for previous implants, which yielded an approximately constant MPD of 160 Gy for a 3-6 cm range of d,. For targets with d, I 3 cm, the original dimensionaveraging rule (7) was applied, that is, the implanted apparent activity in mCi was taken to be five times the average dimension in cm. The dose for these smaller implants is expected to vary roughly as d;‘.2. To obtain a similar, power-function relationship between activity and average dimension for lo3Pd, we make use of MPD data calculated not only for the relatively few lo3Pd implants performed, but also for a number of ‘2.5Iimplants evaluated as if lo3Pd had been used. Planar implants, template implants (mainly prostate) and highly elongated implants were excluded, as were ‘25I cases for which individual seed strength exceeded 0.6 1 mCi (since a lo3Pd seed strength higher by a factor of 3.6 would exceed the maximum strength commercially available). The final data set included 13 lo3Pd and 64 ‘25I implants. In order to assure that a representative set of patient localization data had been selected, the data were first checked for consistency with the current 12’1nomograph. ‘25I MPD’s corresponding to exactly the nomograph value of implanted activity were least-squares fitted to power functions of d,, separately, ford, I 3 cm and d, 2 3 cm, and the fitted exponents were compared with expected values. For six of the implants, d, was 3.0 cm and they were included in both groups. Although the original premise of the ‘25I nomograph (2) was that the seeds would be distributed uniformly, in clinical practice uniformity is only very roughly achieved. A virtue of basing the nomograph on real patient data rather than on idealized configurations is that the resulting recommended total seed strength reflects seed spacing the way it actually occurs. This verification of expected power relationships for “‘1 serves to indicate whether the volume implants selected were consistent, with respect to seed distribution, with those used in the original formulation of the current nomograph. Each MPD determination was performed in the standard way we have been evaluating ‘25I volume implants for many years, which involves first calculating the dose at all mesh points of a 3D rectangular grid extending 2 cm beyond extremum seed positions in each direction. For these calculations, the mesh width was 0.3 cm. For each of 15 uniformly spaced dose levels (e.g., 40 Gy to 320 Gy in steps of 20 Gy). the number of mesh points for which the dose is greater than the dose level is cumulated and multiplied by the cube of the mesh width, to get the corresponding contour volume. By linear interpolation in the resulting table of dose and volume values, the dose is determined that corresponds to the volume of an ellipsoid having the same mutually perpendicular dimensions as those measured (in the operating room) for the target region. The latter volume is the product of the three dimensions and the factor r/6. For the dose calculations as if ‘03Pd had been used, the total apparent activity for each implant was assumed to be 3.6 times the value given by the 12’1 nomograph, and
Volume
27. Number
I. 1993
Table 1. Reference dose vs. distance data for 1 mCi ‘03Pd source Distance (cm)
Dose rate X (distance)2 (cGy h-’ cm2)
0.0 0.2 0.3 0.4 0.5 0.7 1.0 1.5 2.0 2.5 3.0 4.0
0.13 0.91 1.06 1.05 1.oo 0.92 0.80 0.61 0.45 0.33 0.24 1 0.129
Anisotropy factor* 1.oo 1.oo 1.oo 0.94 0.92 0.91 0.91 0.88 0.85 0.83 0.81 0.81
* See text. individual seed strength was obtained by dividing by the number of seeds. In the same way as for the ‘25I implant data, the best-fit slope of log MPD versus log d, was determined for each of the two ranges (I 3 cm and 2 3 cm) of d,, again including in both groups the implants with d, = 3.0 cm. For d, z 3 cm, the desired exponent of d, in the power function specifying ‘03Pd apparent activity was then obtained as the sum of 2.2 and (the negative of) this fitted slope. For d, I 3 cm, the slope was merely compared with that obtained for ‘251. The ‘25I verification calculations made use of singleseed reference data given by Anderson et al. (3) the same reference data that had been used in the original formuFor the lo3Pd calculations, lation of the ‘25I nomograph. the corresponding reference values were those listed in Table 1. The Table 1 data were derived from the twodimensional dose distribution measured for the Model 200 ‘03Pd seed by Chiu-Tsao and Anderson (6) at radial distances up to 4 cm from seed center (similar data were reported earlier by Meigooni et a/. (11). The value at each distance represents the average, over 4 ?r solid angle, of the measured values reported. The “anisotropy factor”, that is, the ratio of this 4 H average to the transverse axis value, is also shown in the table. For distances of 0.1 cm to 0.3 cm, since measured data were available only on the transverse axis, anisotropy factors of 1.O were assumed, in keeping with the trend apparent from the table and since off-axis points so close to source center are either inside or nearly inside the source capsule. The “zero distance” value was obtained by extrapolation on a semilogarithmic plot of dose rate X (distance)2 vs. distance. Dose at distances greater than 4 cm was calculated assuming the same exponential decrease with distance as that exhibited at distances of 2-4 cm.
RESULTS The MPD data for ‘251 volume implants are plotted versus average dimension in Figure 1. The slope of the regression line obtained for d, 2 3 cm is -0.0016, suffi-
Nomograph for “‘Pd implants 0 L. L.
ANDERSON
131
ef al.
the above results would lead us to expect a slope of - 1.56 and we actually observe a slope of - 1.57. By dividing the fitted equation of Figure 2 for d, 2 3 cm, that is, D = 173 .4 dpo.36 a
by the assumption
used for apparent
A = 3.6(1.338)d:.20
t
..I
1
I
3
2
4
. .
..L--.L 5
6
we obtain function
0%. 1) activity,
= 4.82 d:.20
the dose per unit initial
activity
(Eq. 2) as the power
d, (cm)
D/A = 36.0 d,2.56. Fig. 1. 12’1 MPD’s
value that would have been recommended activity had average dimension. Power = 544 d;‘.‘8 for d, 5 3 cm
adjusted in each case to the obtained if exactly the nomographbeen implanted, and plotted versus functions fitted to the data are D and D = 165 d;“.oo’6 for d, 2 3 cm.
that, on the average, the planned dose is indeed independent of target size. Also, for d, I 3 cm, the slope of -1.18 is not greatly different from the - 1.2 value anticipated. The disparity at d, = 3 cm between the separately fitted lines is comparable to the scatter in the data. For the same data set evaluated as lo3Pd implants, if the apparent activity is taken as proportional to that specthe MPD is seen (Fig. 2) to ified by the “‘1 nomograph, decrease as the average dimension increases over 3 cm. As shown below, we can infer from the -0.36 slope of the fitted line that lo3Pd apparent activity should be proportional to the 2.56 power of average dimension in order to compensate for the additional fall-off in MPD beyond that (-2.20 slope) already being compensated for in the “‘1 nomograph. For those implants for which d, I 3 cm, where the apparent activity is directly proportional to d,,
ciently
close
(Eq. 3)
for 77 implants,
to zero
to assure
Solving
(3) for A with D set equal to 115 Gy results in A = 3.2 d;.56.
(Eq. 4)
This power function has been incorporated, for d, 2 3 cm, into the scales on either side of the left-most vertical shown in Figure 3. On the line in the lo3Pd nomograph d, I 3 cm portion of the line, the opposing scales reflect direct proportionality in the relation A = 17.8 d,
(Eq. 5)
which must, of course, agree with (4) for d, = 3 cm. The nomograph in Figure 3 also provides scales for calculating the number of seeds (given the individual seed strength) and for determining how far apart to space the needles through which the seeds are inserted (for a given choice of seed spacing in the needle direction). With respect to overall design, this nomograph is very similar to the one for lz51. In view of the fact, however, that the has not been decurrent version of the ‘25I nomograph scribed in detail, an explanation of its construction has been included here in an Appendix.
DISCUSSION The idea that peripheral dose for interstitial implants is a power function of implant volume (and, by extension, average dimension) is a basic concept of the Manchester system of implant dosimetry (13). Specifically, the formula given for a cylindrical array of radium needles was M = 34 1 V2/3 e0.07(E-I)
2
3
4
5
6
d, (cm)
Fig. 2. lo3Pd MPD’s for 77 implants, adjusted in each case to the value that would have been obtained if exactly 3.6 times the activity specified by the 12’1nomograph had been used. Power functions fitted to the data are D = 642 d;‘.57 ford, 5 3 cm and D = 173 d;“.36 for d, 2 3 cm.
0%. 6)
where M is the number of milligram hours to produce a stated “dose” of 1000 R, V is the implanted volume in cc, and E is the elongation factor, that is, the ratio of longest to shortest dimension. Focussing, for present purposes, only on the exponent, and ignoring elongation effects, the activity per unit dose implied by equation (6) would be proportional to the average dimension squared. To a first approximation, then, the requirement of powers
132
I. J. Radiation Oncology
0 Biology 0 Physics NUMBER OF SEEOS
AVERAGE DIMENSION
RECOMMENOEO
(cm)
(mC1)
Volume
Pd-103
27, Number
1, 1993
PERMANENT
IMPLANTS SPACING BETWEEN
ACTIVITY AVERAGE DIMENSION
SPACING ALONG NEEDLE
km) 140 r
NEEDLES km)
0.5
km)
c
3.5 50 E
4.0 2.5
1.2 1.3 I 1.4t
1.5 6.0
t
L.
1
L. Anderson - 23 - 91
TIE
_INE
Fig. 3. Nomograph for permanent implants of lo3Pd seeds, with lines drawn to illustrate its use to determine the number of seeds required to produce an MPD of about 115 Gy when the target average dimension is 4 cm and the seed apparent activity is 1.5 mCi. Also, needle spacings of 7 mm or 9 mm are indicated for seed spacings of 1.O cm and 0.5 cm, respectively, in the needle direction.
of 2.20 for “‘1 and 2.56 for lo3Pd may be understood as stemming from the lower photon energies of these radionuclides relative to the radium photon spectrum for which equation (6) was derived on the assumption that only geometric (inverse square) attenuation need be accounted for. When attenuation by tissue is appreciable within distances (2-6 cm) comparable to implant dimensions, a greater photon absorption in tissue would be expected to result in a more rapid increase of the activity required as the target volume becomes larger. The procedure followed to determine the power function relationship between source strength and average dimension for lo3Pd (when d, 2 3 cm) was indirect but had the advantage of permitting a comparison of the result with the relationship for ‘251. We were able to observe, in Figure 2, the extent to which the MPD would decrease as a function of d, if we continued to base total seed strength prescription on the “‘1 nomograph. The same result is achieved by calculating by simple proportion, for each patient file, the ‘03Pd activity that would have been required to get an MPD of 115 Gy instead of the MPD actually found for the assumed activity. When these adjusted activities are plotted versus average dimension and fitted with a straight line on the log-log plot, the result (Fig. 4) is seen to verify, within computational accuracy, the previous calculation. Thus, the lo3Pd nomograph, as
developed from this set of actual implants, is completely independent of the “‘1 nomograph. The latter point is important, in view of the fact that the existing 12’1nomograph, to the extent that it is based on outdated dosimetry, is itself in need of revision, that is, relative to values recommended by the Interstitial Collaborative Working
40 b’.
..“‘..,...................d 4
5
6
d, (cm)
Fig. 4. ‘03Pd apparent activity (mCi) that, if used in each of 56 (d, 2 3 cm) implants of Figure 2, would have produced an MPD of 115 Gy. The power function fitted to the data is A = 3.206 d:.555.
Nomograph for ‘03Pdimplants 0 L. L.
Group for the Model 67 11 seed ( 12), dose rates from the nomograph reference data set (also a point-source approximation) were higher by lo%, 14%, and 22% at distances of 1 cm, 2 cm and 3 cm, respectively. However, it is inappropriate to change the 12’1 nomograph until the new data have been accepted and implemented for clinical dose calculations by the medical physics and radiation oncology community, a process that is currently under way via recommendations of Radiation Therapy Task Group No. 43 to the American Association of Physicists in Medicine (AAPM). For the current “‘1 nomograph in the region d, I 3 cm, our brachytherapy staff decided in 1983 to retain the original rule that total seed strength be strictly proportional to average dimension, in order to provide high doses for very small target regions (of presumed greater potential for tumor control, lower propensity for metastasis, and higher normal-tissue tolerance). This policy has been carried forward in the lo3Pd nomograph. Anticipating the forthcoming implementation of AAPM recommendations (1) regarding the quantity for source strength, we have prepared an alternative nomograph (Fig. 5) that specifies, as a function of average dimension the lo3Pd air-kerma strength required to achieve an MPD’of 115 Gy. The conversion from activity in mCi to air-kerma strength in U involved merely adjusting the total strength and seed strength scales by a factor
NUMBER OF SEEOS AVERAGE
DIMENSION km1 \
RECOMMENDED AIR-KERMA STRENGTH
( u 1
/
ANDERSON
133
et al.
equal to the air-kerma rate constant (16), that is, by 1.30 U/mCi. It is important to recognize that the MPD targeted by the nomographs described here is based on matching the volume of the treatment isodose contour to a target volume estimated as the volume of an ellipsoid having the same mutually orthogonal dimensions as the target. Only in the unlikely event that both the target volume and the treatment isodose contour actually were ellipsoidal in shape would the MPD be the same as the true minimum peripheral dose. In general, the shapes are different and the fact that the volumes are equal means that the two surfaces, ideally, are “interwoven.” The extent to which the target volume protrudes from the treatment isodose contour determines the extent to which the minimum peripheral dose is less than the MPD determined by this method. Although the matched peripheral dose is considered a useful measure for comparisons with other implants evaluated by the same method, it should be used with caution in comparisons with implants planned or evaluated, for example, by the dose contour method (4). The latter method relies on 3D imaging to define the target volume and the locations of implanted sources, and defines the treatment dose as that for which the isodose contour encompasses the target volume. For the same prescription dose, the total source strength required to encompass the target with the corresponding isodose contour
Pd-103
PERMANENT
IMPLANTS
280 260 240 220 200 160 160
AVERAGE DIMENSION
SPACING ALONG NEEDLE
(cm)
140
km)
6.0
SPACING BETWEEN NEEOLES km)
I
0.5 t
5.5 2.5
6o 3.0
SEED STRENGTH
2.0 -
5.0
0.6 1.5 4.5
(U
1 0.7 -
L. L. Anderson 1 - 23 - 91
10
TIE
LINE
Fig. 5. Nomograph of Figure 3, modified to show (total and individual seed) air-kerma strengths (U) rather than apparent activities (mCi).
134
I. J. Radiation Oncology 0 Biology 0 Physics
will, in general, be larger than the nomograph-specified value. When 3D imaging is available and custom planning is possible, the dose contour method of evaluation is obviously preferred. In other situations (for example, when implanting a non-resectable tumor intra-operatively), a nomographic guide is very useful, and basing it on prior implants evaluated by the matched-peripheral-dose method has the already-mentioned advantage of being realistic with respect to seed spacing. It must be recognized, however, that it is an approximate method, that it takes no account of possible geographic miss, and that it generally calls for a higher prescription dose because of the shape-related tendency to overestimate minimum dose. The prescription dose of 160 Gy associated with the “‘1 nomograph is one that, on the basis of clinical ex-
Volume 27. Number I, 1993
perience, has been considered a “full-treatment” dose for permanent implants. The 115 Gy dose level used in generating the ‘03Pd nomograph is the dose estimated to have the same “time-dose-factor” (TDF) (14) as that corresponding to 160 Gy from a permanent of ‘251( 15). Using linear quadratic formulas to compare the relative cellkilling effectiveness of the two radionuclides for these doses in permanent implants, Ling (10) has shown that ‘03Pd may be more effective for tumor doubling times of a few days and that 125I may be more effective for longer tumor doubling times. It should be noted that, since the MPD for a given seed configuration is directly proportional to seed strength (assumed uniform), a dose different from 115 Gy is readily prescribed by scaling the total seed strength given by the nomograph up or down proportionately.
REFERENCES 1. AAPM Radiation Therapy Committee Task Group 32. Specification of brachytherapy source strength, AAPM Report No. 21. New York: American Institute of Physics; 1987. 2. Anderson, L. L. Spacing nomograph for interstitial implants of “‘1 seeds. Med. Phys. 3:48-5 1;1976. 3. Anderson, L. L.; Kuan, H. M.; Ding, I. Y. Clinical dosimetry with ‘*% In: George, F. W., ed. Modern interstitial and intracavitary radiation management. New York: Masson Publishing USA; 1981:9-15. 4. Anderson, L. L.; Weaver, K. A.; Nath, R.; Phillips, T. L.; Nori, D.; Son, Y. H. Dose contour evaluation and approximate methods. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. 5. Blasko, J. C.; Ragde, H.; Schumacher, D. Transperineal percutaneous iodine- 125 implantation for prostatic carcinoma using transrectal ultrasound and template guidance. Endocuriether. Hyperther. Oncol. 3: 13 l-l 39; 1987. 6. Chiu-Tsao, S. T.; Anderson, L. L. Thermoluminescent dosimetry for lo3Pd seeds (model 200) in solid water phantom. Med. Phys. 18:449-452; 199 1. 7. Henschke, U. K.; Cevc, P. Dimension averaging a simple method for dosimetry of interstitial implants. Radiobiol. Radiother. 9:287-298; 1968. 8. Hilaris, B. S.; Nori, D.; Anderson, L. L. Atlas of brachytherapy. New York: Macmillan Publishing Company: 1988. 9. International Commission on Radiation Units and Measurements. Radiation quantities and units, ICRU Report
10.
1 I.
12.
13. 14. 15.
16.
17.
33. Washington, D.C.: International Commission on Radiation Units and Measurements; 1980. Ling, C. C. Permanent implants using Au- 198, Pd- 103 and I- 125: Radiobiological considerations based on the linear quadratic model. Int. J. Radiat. Oncol. Biol. Phys. 23:8187; 1992. Meigooni, A. S.; Sabnis, S.; Nath, R. Dosimetry of palladium- 103 brachytherapy sources for permanent implants. Endocuriether. Hyperther. Oncol. 6: 107- 117; 1990. Meli, J. A.; Anderson, L. L.; Weaver, K. A. Dose distribution. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. Meredith, W. J., ed. Radiation dosage, the Manchester System, 2nd edition. Edinburgh: E. & S. Livingstone; 1967. Orton, C. G. Time-dose factors (TDF’s) in brachytherapy. Br. J. Radiol. 47:603-607; 1974. Russell, J. L., Jr. Calculated dose from TheraSeed implants. Theragenics Corporation Internal Report TM- 100 14C, 4 November 1984. See also, Theragenics 1988 product insert, PI-06-2 l-88, Theragenics Corporation, Norcross, GA 30093. Weaver, K. A.; Anderson, L. L.; Meli, J. A. Source characteristics. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. Williamson, J. F.; Nath, R. Clinical implementation of AAPM Task Group 32 recommendations on brachytherapy source strength specification. Med. Phys. 18:439-448; 199 1.
APPENDIX
Nomograph construction The format of the current 125I nomograph for permanent implants, which is here adapted for lo3Pd implants, is based on seven equally spaced vertical lines, each with a logarithmic scale, to represent the variables involved in deciding: ( 1) How many seeds to implant for a given target average dimension; and (2) How to space needles for a selected (along the needle) seed spacing. Among the average dimension, d,, the total (apparent) activity, A, the seed strength, s, and the number of seeds, N, the relationships represented are
A = k, d:
(Fq. la)
log(A) = log(k,) + a log(d,)
(Fq. lb)
or, equivalently,
and N = A/s.
(Fq. 2)
Equation (1 b) is represented by juxtaposed log scales on either side of the leftmost vertical line, with d, on the left
Nomograph for ‘03Pdimplants 0 L. L. ANDERSON
of the line and A on the right. The scale factor, that is, the distance (cm) per unit logarithm (base 10) of the scale variable, is a times for the d, scale what it is for the A scale. For ‘03Pd, when d, I 3 cm, a = 1.0 and k, = 17.76, whereas when d, 2 3 cm, a = 2.56 and k, = 3.2. Thus, the upper (< 3 cm) part of the d scale is drawn to the same scale as the A scale, but the lower part of the d, scale is “stretched” relative to the A scale by a factor of 2.56. By convention, scale factors are positive if the variable increases in the “up” direction on the paper and negative if the opposite is true (as for the d,, A and s scales). The scale factor for the N scale is here chosen to be equal and opposite to that of the A scale, so the scale factor for the s scale must have the same sign as that of the A scale and one-half the magnitude (to offset the factor-of-two geometric amplification that occurs between N and s scales for a line hinged at the A scale). It is necessary to arrange the scales on the page in such a manner that no scale-variable range of interest extends beyond a desired rectangular border. To this end, the longest scale is identified (here, the N scale); it is the scale having the largest product of relative scale-factor and logarithm of scale-variable maximum/minimum ratio. It is assigned a scale factor that produces a comfortable fit within the margins. For the N scale, the maximum of the range is simply the greatest number of seeds one would ever want to implant; together with the maximum target size (d,) one would ever want to implant, it determines the minimum value of seed strength (s) that must be represented. The maximum available seed strength and the minimum average dimension together determine the minimum number of seeds. Other scales are approximately centered vertically by placing mid-range values of the variables on the same horizontal line. Scales can be made longer or shorter by adjusting the scale factor, as long as the necessary relationships among them are preserved. The spacing part of the nomograph for lo3Pd is conceptually unchanged from that of the lz51 nomograph. The target volume is assumed to have an elongation of 1.5 and to be ellipsoidal. Since elongation is the ratio, r, of maximum to minimum dimension and the third dimension could be anywhere in between, a volume formula is used that averages those of the extreme shapes, that is, those of a prolate spheroid and an oblate spheroid. Thus, if a is the minimum diameter and fg is an average-dimension correction factor that preserves volume, the volume is given by (7r/6)ra3 = (a/6)fi
di
(Eq. 3)
* This may be readily shown for cubical seed-arrays within spherical targets, for example, an array of 8 seeds, where the fractional distance is 0.39, and an array of 27 seeds, where the average fractional distance of peripheral seeds is 0.33. For a 64 seed cube, reduced to 56 seeds by eliminating corner seeds in
135
etal.
for an prolate spheroid, it is
and fp = 0.98 1 for r = 1.5, whereas
(a/6)?a3
= (rr/6)fh dz
(Eq. 4)
for an oblate spheroid, and fp = 0.983 for r = 1.5. Substituting the average value, fp = 0.982, the volume becomes V = 0.496 d:.
(Eq. 5)
The cell volume allocated to each seed within the target volume is u, where u, is the cell dimension along the needle path and u, is the mesh width of a rectangular array of needle crossing points in a plane perpendicular to the needle direction. This convention results in peripheral seeds that are at a distance less than half the mesh width, on the average, from the target surface.* Setting the total volume equal to the volume from (5)
uf,
V = Nu,uf, = 0.496 d: and the needle spacing u
”
(Eq. 6)
is given by
= 0 704 d3/2N-+-i/2 s . a
.
(Eq. 7)
The nomograph assumes a seed applicator+ with discrete choices of seed spacing along the needle and that the spacing selected will be uniformly applied throughout the implant. The nomograph solution to equation (7) starts with the N scale already in place and adds a new (unsegmented) d, scale on the next vertical line in order to evaluate the intermediate product, t = 0.704 d;/2N-i’2
(Eq. 8)
on the following vertical line. This scale, which need not be labelled, is referred to simply as the “tie line.” The final multiplication is performed by drawing a line between the tie line result and the value selected from the u, scale to arrive at a u,-scale value on the right-most vertical line, that is, u, = tu,“2.
(Eq. 9)
From the exponents in (8) and (9), together with the “elemental nomograph” equation (2) it may be inferred that the scale factors for the d, scale, the u, scale and the u, scale must be, in the same order, 2, i, and -2 times that of the N scale.
order to meet the volume-matching requirement, the average fractional distance of the 48 “peripheral” seeds is 0.49. ’ Mick Applicator, Mick Radio-Nuclear Instruments, Inc., 1470 Outlook Ave., Bronx, NY 10465.
27, pp.
0360-3016/93 $6.00 + .I0 Copyright 0 1993 Pergamon Press Ltd.
129-135
??Technical Innovations and Notes
A NOMOGRAPH
FOR PERMANENT
LOWELL L. ANDERSON, Memorial
PH.D.,* Sloan-Kettering
IMPLANTS
OF PALLADIUM-103
SEEDS
JANAKI V. MONI, M.D.* AND LOUIS B. HARRISON, Cancer Center,
M.D.+
1275 York Ave., New York, NY 10021
lo3Pd is being substituted for 125Iin permanent implants for which it is desired to deliver a higher initial dose rate while maintaining readily achieved radiation protection. We have constructed a nomograph to assist in determining both the total seed strength required and the appropriate needle spacing for lo3Pd implants. We have calculated the “matched peripheral dose” (MPD), that is, the dose for which the isodose contour volume is equal to the target volume, for 64 125Iand 13 ‘03Pd actual implants as if “‘Pd had been used for all of them, employing a computer lookup table based on single-seed dose distribution measurements in solid water. The calculated data were used to obtain a least-squares fit to a linear relationship between the logarithm of the total seed strength for a given MPD and the logarithm of the average dimension, d. (cm). We found that, for a nominal MPD of 11,500 cGy, total seed strength (in mCi) is given by 3.2 d$” . A lo3Pd nomograph has been constructed on the basis of this power function relationship. Our nomographic guide for planning lo3Pd implants calls for total seed strength to increase significantly faster as a function of target volume average dimension than is the case for 1251.This nomograph will facilitate the application of lo3Pd seeds in permanent implants. lo3Pd seeds, Nomograph, Brachytherapy
planning.
isodose contour volume with estimated target volume, that is, with the volume of an ellipsoid having the same dimensions as the target region (8). Initially, the total ‘03Pd seed strength required to produce a matched peripheral dose (MPD) of 115 Gy was taken to be 3.6 times the total strength specified by the 1251nomograph for the same target average dimension. This factor was based on the results of recalculating MPD’s for recent ‘25I cases as if lo3Pd seeds had been used. It was recognized, of course, that such a factor must, in reality, be volume (or average-dimension) dependent, due to the lower photon energy of lo3Pd. Clearly, a separate nomograph was needed for ‘03Pd.
INTRODUCTION Palladium-
103 decays by electron capture with a half life
of 17 days and emits photons having an average energy of 20.9 keV. Seeds containing lo3Pd were developed by Russell ( 15) and are being used in brachytherapy to treat malignancies at selected sites, notably by Blasko et al. in custom-planned permanent using the ultrasound-guided
implants for prostate cancer, technique that has been de-
scribed for 12’1prostate implants (5). As a result of the difference in half-life, the initial dose rate for a lo3Pd permanent implant with a prescribed dose of 115 Gy is higher than that for an ‘25Ipermanent implant with a prescribed dose of 160 Gy. This higher dose rate provides a theoretical advantage for lo3Pd in eradicating rapidly dividing tumor cells. At our institution, more than 20 patients have received permanent implants of lo3Pd seeds, primarily volume implants for which custom planning was not an option and for which, had ‘25I seeds been used, total seed strength would have been determined using a planning nomograph. For such patients, dose is evaluated by matching
METHODS
AND
MATERIALS
The current version of the “‘1 nomograph has been in use since 1983. For target volumes having an average di-
mension, d,, greater than or equal to 3 cm it gives the total apparent activity*, A, according to the power function, A = 1.34 di.2, with A expressed in mCi and d, in cm. The value of the exponent was determined from cal-
Presented at the Annual Meeting of the American Society for Therapeutic Radiology and Oncology, Washington, DC, 5 November 199 1. * Department of Medical Physics. ’ Brachytherapy Service, Department of Radiation Oncology. Accepted for publication 12 March 1993. * “Apparent activity” is an interim quantity defined (8) as the activity of an unencapsulated point source that would produce the same exposure rate at the same distance in free space
as that produced by the actual source on its transverse axis at a distance large enough that the inverse square law is valid. The currently recommended quantity for source strength is the “airkerma strength”, which is defined (17) as the product of airkerma rate (9) in free space along the transverse axis and the square of distance from the source center, again with the proviso that the distance be large enough to assure that the inverse square law is followed and that, therefore, the defined quantity does not vary with distance. 129
130
1. J. Radiation
Oncology
0 Biology 0 Physics
culations for previous implants, which yielded an approximately constant MPD of 160 Gy for a 3-6 cm range of d,. For targets with d, I 3 cm, the original dimensionaveraging rule (7) was applied, that is, the implanted apparent activity in mCi was taken to be five times the average dimension in cm. The dose for these smaller implants is expected to vary roughly as d;‘.2. To obtain a similar, power-function relationship between activity and average dimension for lo3Pd, we make use of MPD data calculated not only for the relatively few lo3Pd implants performed, but also for a number of ‘2.5Iimplants evaluated as if lo3Pd had been used. Planar implants, template implants (mainly prostate) and highly elongated implants were excluded, as were ‘25I cases for which individual seed strength exceeded 0.6 1 mCi (since a lo3Pd seed strength higher by a factor of 3.6 would exceed the maximum strength commercially available). The final data set included 13 lo3Pd and 64 ‘25I implants. In order to assure that a representative set of patient localization data had been selected, the data were first checked for consistency with the current 12’1nomograph. ‘25I MPD’s corresponding to exactly the nomograph value of implanted activity were least-squares fitted to power functions of d,, separately, ford, I 3 cm and d, 2 3 cm, and the fitted exponents were compared with expected values. For six of the implants, d, was 3.0 cm and they were included in both groups. Although the original premise of the ‘25I nomograph (2) was that the seeds would be distributed uniformly, in clinical practice uniformity is only very roughly achieved. A virtue of basing the nomograph on real patient data rather than on idealized configurations is that the resulting recommended total seed strength reflects seed spacing the way it actually occurs. This verification of expected power relationships for “‘1 serves to indicate whether the volume implants selected were consistent, with respect to seed distribution, with those used in the original formulation of the current nomograph. Each MPD determination was performed in the standard way we have been evaluating ‘25I volume implants for many years, which involves first calculating the dose at all mesh points of a 3D rectangular grid extending 2 cm beyond extremum seed positions in each direction. For these calculations, the mesh width was 0.3 cm. For each of 15 uniformly spaced dose levels (e.g., 40 Gy to 320 Gy in steps of 20 Gy). the number of mesh points for which the dose is greater than the dose level is cumulated and multiplied by the cube of the mesh width, to get the corresponding contour volume. By linear interpolation in the resulting table of dose and volume values, the dose is determined that corresponds to the volume of an ellipsoid having the same mutually perpendicular dimensions as those measured (in the operating room) for the target region. The latter volume is the product of the three dimensions and the factor r/6. For the dose calculations as if ‘03Pd had been used, the total apparent activity for each implant was assumed to be 3.6 times the value given by the 12’1 nomograph, and
Volume
27. Number
I. 1993
Table 1. Reference dose vs. distance data for 1 mCi ‘03Pd source Distance (cm)
Dose rate X (distance)2 (cGy h-’ cm2)
0.0 0.2 0.3 0.4 0.5 0.7 1.0 1.5 2.0 2.5 3.0 4.0
0.13 0.91 1.06 1.05 1.oo 0.92 0.80 0.61 0.45 0.33 0.24 1 0.129
Anisotropy factor* 1.oo 1.oo 1.oo 0.94 0.92 0.91 0.91 0.88 0.85 0.83 0.81 0.81
* See text. individual seed strength was obtained by dividing by the number of seeds. In the same way as for the ‘25I implant data, the best-fit slope of log MPD versus log d, was determined for each of the two ranges (I 3 cm and 2 3 cm) of d,, again including in both groups the implants with d, = 3.0 cm. For d, z 3 cm, the desired exponent of d, in the power function specifying ‘03Pd apparent activity was then obtained as the sum of 2.2 and (the negative of) this fitted slope. For d, I 3 cm, the slope was merely compared with that obtained for ‘251. The ‘25I verification calculations made use of singleseed reference data given by Anderson et al. (3) the same reference data that had been used in the original formuFor the lo3Pd calculations, lation of the ‘25I nomograph. the corresponding reference values were those listed in Table 1. The Table 1 data were derived from the twodimensional dose distribution measured for the Model 200 ‘03Pd seed by Chiu-Tsao and Anderson (6) at radial distances up to 4 cm from seed center (similar data were reported earlier by Meigooni et a/. (11). The value at each distance represents the average, over 4 ?r solid angle, of the measured values reported. The “anisotropy factor”, that is, the ratio of this 4 H average to the transverse axis value, is also shown in the table. For distances of 0.1 cm to 0.3 cm, since measured data were available only on the transverse axis, anisotropy factors of 1.O were assumed, in keeping with the trend apparent from the table and since off-axis points so close to source center are either inside or nearly inside the source capsule. The “zero distance” value was obtained by extrapolation on a semilogarithmic plot of dose rate X (distance)2 vs. distance. Dose at distances greater than 4 cm was calculated assuming the same exponential decrease with distance as that exhibited at distances of 2-4 cm.
RESULTS The MPD data for ‘251 volume implants are plotted versus average dimension in Figure 1. The slope of the regression line obtained for d, 2 3 cm is -0.0016, suffi-
Nomograph for “‘Pd implants 0 L. L.
ANDERSON
131
ef al.
the above results would lead us to expect a slope of - 1.56 and we actually observe a slope of - 1.57. By dividing the fitted equation of Figure 2 for d, 2 3 cm, that is, D = 173 .4 dpo.36 a
by the assumption
used for apparent
A = 3.6(1.338)d:.20
t
..I
1
I
3
2
4
. .
..L--.L 5
6
we obtain function
0%. 1) activity,
= 4.82 d:.20
the dose per unit initial
activity
(Eq. 2) as the power
d, (cm)
D/A = 36.0 d,2.56. Fig. 1. 12’1 MPD’s
value that would have been recommended activity had average dimension. Power = 544 d;‘.‘8 for d, 5 3 cm
adjusted in each case to the obtained if exactly the nomographbeen implanted, and plotted versus functions fitted to the data are D and D = 165 d;“.oo’6 for d, 2 3 cm.
that, on the average, the planned dose is indeed independent of target size. Also, for d, I 3 cm, the slope of -1.18 is not greatly different from the - 1.2 value anticipated. The disparity at d, = 3 cm between the separately fitted lines is comparable to the scatter in the data. For the same data set evaluated as lo3Pd implants, if the apparent activity is taken as proportional to that specthe MPD is seen (Fig. 2) to ified by the “‘1 nomograph, decrease as the average dimension increases over 3 cm. As shown below, we can infer from the -0.36 slope of the fitted line that lo3Pd apparent activity should be proportional to the 2.56 power of average dimension in order to compensate for the additional fall-off in MPD beyond that (-2.20 slope) already being compensated for in the “‘1 nomograph. For those implants for which d, I 3 cm, where the apparent activity is directly proportional to d,,
ciently
close
(Eq. 3)
for 77 implants,
to zero
to assure
Solving
(3) for A with D set equal to 115 Gy results in A = 3.2 d;.56.
(Eq. 4)
This power function has been incorporated, for d, 2 3 cm, into the scales on either side of the left-most vertical shown in Figure 3. On the line in the lo3Pd nomograph d, I 3 cm portion of the line, the opposing scales reflect direct proportionality in the relation A = 17.8 d,
(Eq. 5)
which must, of course, agree with (4) for d, = 3 cm. The nomograph in Figure 3 also provides scales for calculating the number of seeds (given the individual seed strength) and for determining how far apart to space the needles through which the seeds are inserted (for a given choice of seed spacing in the needle direction). With respect to overall design, this nomograph is very similar to the one for lz51. In view of the fact, however, that the has not been decurrent version of the ‘25I nomograph scribed in detail, an explanation of its construction has been included here in an Appendix.
DISCUSSION The idea that peripheral dose for interstitial implants is a power function of implant volume (and, by extension, average dimension) is a basic concept of the Manchester system of implant dosimetry (13). Specifically, the formula given for a cylindrical array of radium needles was M = 34 1 V2/3 e0.07(E-I)
2
3
4
5
6
d, (cm)
Fig. 2. lo3Pd MPD’s for 77 implants, adjusted in each case to the value that would have been obtained if exactly 3.6 times the activity specified by the 12’1nomograph had been used. Power functions fitted to the data are D = 642 d;‘.57 ford, 5 3 cm and D = 173 d;“.36 for d, 2 3 cm.
0%. 6)
where M is the number of milligram hours to produce a stated “dose” of 1000 R, V is the implanted volume in cc, and E is the elongation factor, that is, the ratio of longest to shortest dimension. Focussing, for present purposes, only on the exponent, and ignoring elongation effects, the activity per unit dose implied by equation (6) would be proportional to the average dimension squared. To a first approximation, then, the requirement of powers
132
I. J. Radiation Oncology
0 Biology 0 Physics NUMBER OF SEEOS
AVERAGE DIMENSION
RECOMMENOEO
(cm)
(mC1)
Volume
Pd-103
27, Number
1, 1993
PERMANENT
IMPLANTS SPACING BETWEEN
ACTIVITY AVERAGE DIMENSION
SPACING ALONG NEEDLE
km) 140 r
NEEDLES km)
0.5
km)
c
3.5 50 E
4.0 2.5
1.2 1.3 I 1.4t
1.5 6.0
t
L.
1
L. Anderson - 23 - 91
TIE
_INE
Fig. 3. Nomograph for permanent implants of lo3Pd seeds, with lines drawn to illustrate its use to determine the number of seeds required to produce an MPD of about 115 Gy when the target average dimension is 4 cm and the seed apparent activity is 1.5 mCi. Also, needle spacings of 7 mm or 9 mm are indicated for seed spacings of 1.O cm and 0.5 cm, respectively, in the needle direction.
of 2.20 for “‘1 and 2.56 for lo3Pd may be understood as stemming from the lower photon energies of these radionuclides relative to the radium photon spectrum for which equation (6) was derived on the assumption that only geometric (inverse square) attenuation need be accounted for. When attenuation by tissue is appreciable within distances (2-6 cm) comparable to implant dimensions, a greater photon absorption in tissue would be expected to result in a more rapid increase of the activity required as the target volume becomes larger. The procedure followed to determine the power function relationship between source strength and average dimension for lo3Pd (when d, 2 3 cm) was indirect but had the advantage of permitting a comparison of the result with the relationship for ‘251. We were able to observe, in Figure 2, the extent to which the MPD would decrease as a function of d, if we continued to base total seed strength prescription on the “‘1 nomograph. The same result is achieved by calculating by simple proportion, for each patient file, the ‘03Pd activity that would have been required to get an MPD of 115 Gy instead of the MPD actually found for the assumed activity. When these adjusted activities are plotted versus average dimension and fitted with a straight line on the log-log plot, the result (Fig. 4) is seen to verify, within computational accuracy, the previous calculation. Thus, the lo3Pd nomograph, as
developed from this set of actual implants, is completely independent of the “‘1 nomograph. The latter point is important, in view of the fact that the existing 12’1nomograph, to the extent that it is based on outdated dosimetry, is itself in need of revision, that is, relative to values recommended by the Interstitial Collaborative Working
40 b’.
..“‘..,...................d 4
5
6
d, (cm)
Fig. 4. ‘03Pd apparent activity (mCi) that, if used in each of 56 (d, 2 3 cm) implants of Figure 2, would have produced an MPD of 115 Gy. The power function fitted to the data is A = 3.206 d:.555.
Nomograph for ‘03Pdimplants 0 L. L.
Group for the Model 67 11 seed ( 12), dose rates from the nomograph reference data set (also a point-source approximation) were higher by lo%, 14%, and 22% at distances of 1 cm, 2 cm and 3 cm, respectively. However, it is inappropriate to change the 12’1 nomograph until the new data have been accepted and implemented for clinical dose calculations by the medical physics and radiation oncology community, a process that is currently under way via recommendations of Radiation Therapy Task Group No. 43 to the American Association of Physicists in Medicine (AAPM). For the current “‘1 nomograph in the region d, I 3 cm, our brachytherapy staff decided in 1983 to retain the original rule that total seed strength be strictly proportional to average dimension, in order to provide high doses for very small target regions (of presumed greater potential for tumor control, lower propensity for metastasis, and higher normal-tissue tolerance). This policy has been carried forward in the lo3Pd nomograph. Anticipating the forthcoming implementation of AAPM recommendations (1) regarding the quantity for source strength, we have prepared an alternative nomograph (Fig. 5) that specifies, as a function of average dimension the lo3Pd air-kerma strength required to achieve an MPD’of 115 Gy. The conversion from activity in mCi to air-kerma strength in U involved merely adjusting the total strength and seed strength scales by a factor
NUMBER OF SEEOS AVERAGE
DIMENSION km1 \
RECOMMENDED AIR-KERMA STRENGTH
( u 1
/
ANDERSON
133
et al.
equal to the air-kerma rate constant (16), that is, by 1.30 U/mCi. It is important to recognize that the MPD targeted by the nomographs described here is based on matching the volume of the treatment isodose contour to a target volume estimated as the volume of an ellipsoid having the same mutually orthogonal dimensions as the target. Only in the unlikely event that both the target volume and the treatment isodose contour actually were ellipsoidal in shape would the MPD be the same as the true minimum peripheral dose. In general, the shapes are different and the fact that the volumes are equal means that the two surfaces, ideally, are “interwoven.” The extent to which the target volume protrudes from the treatment isodose contour determines the extent to which the minimum peripheral dose is less than the MPD determined by this method. Although the matched peripheral dose is considered a useful measure for comparisons with other implants evaluated by the same method, it should be used with caution in comparisons with implants planned or evaluated, for example, by the dose contour method (4). The latter method relies on 3D imaging to define the target volume and the locations of implanted sources, and defines the treatment dose as that for which the isodose contour encompasses the target volume. For the same prescription dose, the total source strength required to encompass the target with the corresponding isodose contour
Pd-103
PERMANENT
IMPLANTS
280 260 240 220 200 160 160
AVERAGE DIMENSION
SPACING ALONG NEEDLE
(cm)
140
km)
6.0
SPACING BETWEEN NEEOLES km)
I
0.5 t
5.5 2.5
6o 3.0
SEED STRENGTH
2.0 -
5.0
0.6 1.5 4.5
(U
1 0.7 -
L. L. Anderson 1 - 23 - 91
10
TIE
LINE
Fig. 5. Nomograph of Figure 3, modified to show (total and individual seed) air-kerma strengths (U) rather than apparent activities (mCi).
134
I. J. Radiation Oncology 0 Biology 0 Physics
will, in general, be larger than the nomograph-specified value. When 3D imaging is available and custom planning is possible, the dose contour method of evaluation is obviously preferred. In other situations (for example, when implanting a non-resectable tumor intra-operatively), a nomographic guide is very useful, and basing it on prior implants evaluated by the matched-peripheral-dose method has the already-mentioned advantage of being realistic with respect to seed spacing. It must be recognized, however, that it is an approximate method, that it takes no account of possible geographic miss, and that it generally calls for a higher prescription dose because of the shape-related tendency to overestimate minimum dose. The prescription dose of 160 Gy associated with the “‘1 nomograph is one that, on the basis of clinical ex-
Volume 27. Number I, 1993
perience, has been considered a “full-treatment” dose for permanent implants. The 115 Gy dose level used in generating the ‘03Pd nomograph is the dose estimated to have the same “time-dose-factor” (TDF) (14) as that corresponding to 160 Gy from a permanent of ‘251( 15). Using linear quadratic formulas to compare the relative cellkilling effectiveness of the two radionuclides for these doses in permanent implants, Ling (10) has shown that ‘03Pd may be more effective for tumor doubling times of a few days and that 125I may be more effective for longer tumor doubling times. It should be noted that, since the MPD for a given seed configuration is directly proportional to seed strength (assumed uniform), a dose different from 115 Gy is readily prescribed by scaling the total seed strength given by the nomograph up or down proportionately.
REFERENCES 1. AAPM Radiation Therapy Committee Task Group 32. Specification of brachytherapy source strength, AAPM Report No. 21. New York: American Institute of Physics; 1987. 2. Anderson, L. L. Spacing nomograph for interstitial implants of “‘1 seeds. Med. Phys. 3:48-5 1;1976. 3. Anderson, L. L.; Kuan, H. M.; Ding, I. Y. Clinical dosimetry with ‘*% In: George, F. W., ed. Modern interstitial and intracavitary radiation management. New York: Masson Publishing USA; 1981:9-15. 4. Anderson, L. L.; Weaver, K. A.; Nath, R.; Phillips, T. L.; Nori, D.; Son, Y. H. Dose contour evaluation and approximate methods. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. 5. Blasko, J. C.; Ragde, H.; Schumacher, D. Transperineal percutaneous iodine- 125 implantation for prostatic carcinoma using transrectal ultrasound and template guidance. Endocuriether. Hyperther. Oncol. 3: 13 l-l 39; 1987. 6. Chiu-Tsao, S. T.; Anderson, L. L. Thermoluminescent dosimetry for lo3Pd seeds (model 200) in solid water phantom. Med. Phys. 18:449-452; 199 1. 7. Henschke, U. K.; Cevc, P. Dimension averaging a simple method for dosimetry of interstitial implants. Radiobiol. Radiother. 9:287-298; 1968. 8. Hilaris, B. S.; Nori, D.; Anderson, L. L. Atlas of brachytherapy. New York: Macmillan Publishing Company: 1988. 9. International Commission on Radiation Units and Measurements. Radiation quantities and units, ICRU Report
10.
1 I.
12.
13. 14. 15.
16.
17.
33. Washington, D.C.: International Commission on Radiation Units and Measurements; 1980. Ling, C. C. Permanent implants using Au- 198, Pd- 103 and I- 125: Radiobiological considerations based on the linear quadratic model. Int. J. Radiat. Oncol. Biol. Phys. 23:8187; 1992. Meigooni, A. S.; Sabnis, S.; Nath, R. Dosimetry of palladium- 103 brachytherapy sources for permanent implants. Endocuriether. Hyperther. Oncol. 6: 107- 117; 1990. Meli, J. A.; Anderson, L. L.; Weaver, K. A. Dose distribution. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. Meredith, W. J., ed. Radiation dosage, the Manchester System, 2nd edition. Edinburgh: E. & S. Livingstone; 1967. Orton, C. G. Time-dose factors (TDF’s) in brachytherapy. Br. J. Radiol. 47:603-607; 1974. Russell, J. L., Jr. Calculated dose from TheraSeed implants. Theragenics Corporation Internal Report TM- 100 14C, 4 November 1984. See also, Theragenics 1988 product insert, PI-06-2 l-88, Theragenics Corporation, Norcross, GA 30093. Weaver, K. A.; Anderson, L. L.; Meli, J. A. Source characteristics. In: Interstitial Collaborative Working Group: Interstitial brachytherapy-Physical, biological and clinical considerations. New York: Raven Press; 1990. Williamson, J. F.; Nath, R. Clinical implementation of AAPM Task Group 32 recommendations on brachytherapy source strength specification. Med. Phys. 18:439-448; 199 1.
APPENDIX
Nomograph construction The format of the current 125I nomograph for permanent implants, which is here adapted for lo3Pd implants, is based on seven equally spaced vertical lines, each with a logarithmic scale, to represent the variables involved in deciding: ( 1) How many seeds to implant for a given target average dimension; and (2) How to space needles for a selected (along the needle) seed spacing. Among the average dimension, d,, the total (apparent) activity, A, the seed strength, s, and the number of seeds, N, the relationships represented are
A = k, d:
(Fq. la)
log(A) = log(k,) + a log(d,)
(Fq. lb)
or, equivalently,
and N = A/s.
(Fq. 2)
Equation (1 b) is represented by juxtaposed log scales on either side of the leftmost vertical line, with d, on the left
Nomograph for ‘03Pdimplants 0 L. L. ANDERSON
of the line and A on the right. The scale factor, that is, the distance (cm) per unit logarithm (base 10) of the scale variable, is a times for the d, scale what it is for the A scale. For ‘03Pd, when d, I 3 cm, a = 1.0 and k, = 17.76, whereas when d, 2 3 cm, a = 2.56 and k, = 3.2. Thus, the upper (< 3 cm) part of the d scale is drawn to the same scale as the A scale, but the lower part of the d, scale is “stretched” relative to the A scale by a factor of 2.56. By convention, scale factors are positive if the variable increases in the “up” direction on the paper and negative if the opposite is true (as for the d,, A and s scales). The scale factor for the N scale is here chosen to be equal and opposite to that of the A scale, so the scale factor for the s scale must have the same sign as that of the A scale and one-half the magnitude (to offset the factor-of-two geometric amplification that occurs between N and s scales for a line hinged at the A scale). It is necessary to arrange the scales on the page in such a manner that no scale-variable range of interest extends beyond a desired rectangular border. To this end, the longest scale is identified (here, the N scale); it is the scale having the largest product of relative scale-factor and logarithm of scale-variable maximum/minimum ratio. It is assigned a scale factor that produces a comfortable fit within the margins. For the N scale, the maximum of the range is simply the greatest number of seeds one would ever want to implant; together with the maximum target size (d,) one would ever want to implant, it determines the minimum value of seed strength (s) that must be represented. The maximum available seed strength and the minimum average dimension together determine the minimum number of seeds. Other scales are approximately centered vertically by placing mid-range values of the variables on the same horizontal line. Scales can be made longer or shorter by adjusting the scale factor, as long as the necessary relationships among them are preserved. The spacing part of the nomograph for lo3Pd is conceptually unchanged from that of the lz51 nomograph. The target volume is assumed to have an elongation of 1.5 and to be ellipsoidal. Since elongation is the ratio, r, of maximum to minimum dimension and the third dimension could be anywhere in between, a volume formula is used that averages those of the extreme shapes, that is, those of a prolate spheroid and an oblate spheroid. Thus, if a is the minimum diameter and fg is an average-dimension correction factor that preserves volume, the volume is given by (7r/6)ra3 = (a/6)fi
di
(Eq. 3)
* This may be readily shown for cubical seed-arrays within spherical targets, for example, an array of 8 seeds, where the fractional distance is 0.39, and an array of 27 seeds, where the average fractional distance of peripheral seeds is 0.33. For a 64 seed cube, reduced to 56 seeds by eliminating corner seeds in
135
etal.
for an prolate spheroid, it is
and fp = 0.98 1 for r = 1.5, whereas
(a/6)?a3
= (rr/6)fh dz
(Eq. 4)
for an oblate spheroid, and fp = 0.983 for r = 1.5. Substituting the average value, fp = 0.982, the volume becomes V = 0.496 d:.
(Eq. 5)
The cell volume allocated to each seed within the target volume is u, where u, is the cell dimension along the needle path and u, is the mesh width of a rectangular array of needle crossing points in a plane perpendicular to the needle direction. This convention results in peripheral seeds that are at a distance less than half the mesh width, on the average, from the target surface.* Setting the total volume equal to the volume from (5)
uf,
V = Nu,uf, = 0.496 d: and the needle spacing u
”
(Eq. 6)
is given by
= 0 704 d3/2N-+-i/2 s . a
.
(Eq. 7)
The nomograph assumes a seed applicator+ with discrete choices of seed spacing along the needle and that the spacing selected will be uniformly applied throughout the implant. The nomograph solution to equation (7) starts with the N scale already in place and adds a new (unsegmented) d, scale on the next vertical line in order to evaluate the intermediate product, t = 0.704 d;/2N-i’2
(Eq. 8)
on the following vertical line. This scale, which need not be labelled, is referred to simply as the “tie line.” The final multiplication is performed by drawing a line between the tie line result and the value selected from the u, scale to arrive at a u,-scale value on the right-most vertical line, that is, u, = tu,“2.
(Eq. 9)
From the exponents in (8) and (9), together with the “elemental nomograph” equation (2) it may be inferred that the scale factors for the d, scale, the u, scale and the u, scale must be, in the same order, 2, i, and -2 times that of the N scale.
order to meet the volume-matching requirement, the average fractional distance of the 48 “peripheral” seeds is 0.49. ’ Mick Applicator, Mick Radio-Nuclear Instruments, Inc., 1470 Outlook Ave., Bronx, NY 10465.