0360-3016/92
Int. I Radramn Oncology Biol. Phys., Vol. 24, pp. 795-801 Printed in the U.S.A. All rights resewed.
t .oO
$5.00 Press Ltd.
Copynsht B 1992Per@mon
??Technical Innovations and Notes
DOSE DETERMINATION
IN HIGH DOSE-RATE BRACHYTHERAPY
PAVEL V. HOUDEK, PH.D., JAMES G. SCHWADE, M.D., XIAODONG Wu, M.S., VINCENT PISCIOTTA, M.S., JEFFREY A. FIEDLER, M.S., CHRISTOPHER F. SERAGO, PH.D., ARNOLD M. MARKOE, M.D., ANDRE A. ABITBOL, M.D., ALAN A. LEWIN, M.D., PAUL G. BRAUNSCHWEIGER, PH.D. AND MARSHALL D. SKLAR, M.D. Department
of Radiation Oncology, University of Miami School of Medicine, Miami, FL
Although high dose-rate brachytherapy with a single, rapidly moving radiation source is becoming a common treatment modality, a suitable formalism for determinationof the dose deliveredby a movingradiationsource has not yet been developed. At present, brachytherapy software simulates high dose-rate treatments using only a series of stationary sources, and consequently fails to account for the dose component delivered while the source is in motion. We now describe a practical model for determination of the true, total dose administered. The algorithm calculatesboth the dose delivered while the source is in motion within and outside of the implanted volume (dynamic component), and the dose delivered while the source is stationary at a series of fixed dwell points. It is shown that the dynamic dose element cannot be ignored because it always increases the dose at the prescription points and, in addition, distorts the dose distribution within and outside of the irradiated volume. Failure to account for the dynamic dose component results in dosimetric errors that range from significant (> 10%) to negligible (< I%), depending on the prescribed dose, source activity, and source speed as defined by the implant geometry. Dosimetry, High dose-rate brachytherapy,
Moving source, Algorithm.
INTRODUCTION Modern, computer controlled, high dose-rate (HDR) afterloading machines are usually equipped with a single, high activity, miniaturized, Iridium-192 source that can be moved through several channels between the storage/ safe unit and patient (3). The treatment proceeds in the following manner: the source is rapidly moved from the afterloader to the first implant dwell point; it is held stationary there for a pre-programmed period of time (dwell time); then, somewhat less rapidly, the source moves to occupy the next dwell point. The process is repeated at all dwell points in a given channel; the source is then retracted into the afterloader and the process repeated in subsequent channels. Commonly used brachytherapy software simulates a prescribed treatment by using a series of stationary sources only and, consequently, fails to account for the component of dose delivered when the source is in motion. Since exclusion of this dynamic dose element should lead to underestimation of dose at the prescription points, and concomitant distortion of the dose distribution in the irradiated volume, we have developed a formalism that de-
Reprints requests to: Pave1 V. Houdek, Ph.D., Department of Radiation Oncology (D-31), University of Miami School of Medicine, P.O. Box 016960, Miami, FL 33101, U.S.A.
scribes both the dynamic as well as stationary components of HDR treatments. The results are presented herein. METHODS
AND MATERIALS
General considerations It is assumed that a point radiation source S moves in
space along curve c with velocity o, and that source position at any time t is defined by the three functions: X = f(t),
y = g(t),
2 = h(t).
The dose D delivered to point P when the source S, Figure 1, moves from point A to point B during time interval t = t2 - tl is: (2
D=C
s 11
F(t)dt,
ok. 1)
where constant C specifies the radiation emission parameters of S, and function F(t) describes the attenuation of radiation between S and P at instant t. For the purposes
Accepted for publication
11 May 1992.
796
1. J. Radiation Oncology 0 Biology0 Physics
Volume 24, Number 4, 1992
the x axis, source position at any time t is: x = f(t) = vt, y = a, z = 0, and the magnitude of the position vector r is: Ir I = (x2 + a2)“2. From the definition of F(t) andf(t) it follows that: F(t) = l/ )r (2 = 1/(x2 + a’) and dt = dx/v. Therefore, the dose delivered to point P by moving source S as it travels from A to B is:
s 12
D,,,,=C
11
F(t)dt = (C/v)
X2( l/(x2 + a2))dx s x1
= (C/av)[arctg(x/a)]$;.
(Eq. 2)
Fig. 1. Radiation source S moving in space along path c.
of brachytherapy dosimetry, the constant C is equal to the product of the specific dose rate constant and source strength (C = 4.33* lo4 cGy cm2 h-* was used for the data presented here) (9) and its numerical value is available in the literature (4, 8, 9). Assuming that none other than inverse-square attenuation is involved, function F(t) is defined by:
F(t) = l/lrl’,
2. Assume that source S, Figure 2 (bottom), moves in a gynecologic ring applicator ( 1) with constant angular velocity Q = dol/dtin a circular path from A to B. The source position at any time t is: x = rcos(a), y = rsin(a), z = 0, where angular displacement a! = fit. The magnitude of the position vector r is constant at any time t, Ir I = (x2 + y2)‘j2 = r, and hence F(t) = l/r2. Since dt = da/Q, the dose delivered to point P by moving source S is: D,,,.,= (C/(r2s2)) se2 dcu = (C/(r2Q))[a]::. al
where r is the position vector of source S. If i, j, and k are unit vectors along the x, y, and z axes, then r is given by:
3. Suppose that source S is stationary at a dwell point
Y
r = ix + jy + kz.
A
The magnitude of r, for example (r 1, is the distance between the source S and origin P:
C
A
S
B
Irl = r = (x2 + y2 + 22)1’2. The velocity vector v of the moving source Sis obtained by differentiation of position vector r with respect to time: v = drfdt = i(dx/dt) + j(dy/dt) + k(dz/dt).
Its magnitude
1v ) is the source instantaneous
velocity o:
(VI = u = ((dx/dt)2 + (dy/dt)2 + (dz/dt)2)“2.
Velocity is positive (negative) when the source is moving towards increasing (decreasing) values of r. Brachytherapy dosirnetry
Herein. three solutions of equation (l), relevant to HDR dosimetry, are discussed: 1. Suppose that in a linear implant the point source S, Figure 2 (top), moves along path c from the point A(x, , a, 0) to the point B(x2, a, 0) with constant velocity 2) = dx/dt. Since the source trajectory is parallel with
Fig. 2. Trajectory of source S in a linear implant (top, PA = AB = 1.0 cm) and in a gynecologic ring applicator (bottom, PA = 1.O cm, (Y*- LY,= [180/r] degrees).
Dose determination in high dose-rate brachytherapy 0 P. V. HOUDEK
having coordinates x, y, and z. The magnitude of the position vector is constant in time, (r ( = (x2 + y* + z2)‘O, and hence F(t) = l/r*. Then the dose delivered to the point Pin time interval t = t2 - tl is: D, = (C/r*) j-” dt = (C/r*)[t]$
11
Table 1. Source
speed* as a function of the distance travelled+
Distance (mm)
Speed (cm/set)
2.5 5.0 (Eq. 3)
191
ef ~1.
10.0
20.0
22.1 27.1 30.3 32.3
Distance (mm) 30.0 50.0 100.0
995.0
Speed (cm/set) 32.7 33.1 34.5 49.9
* Average speed for dwell times of 0.1 set, 0.3 set, and 0.5 sec. t Microselectron, Nucletron Corp., Columbia, MD 2 1045.
RESULTS Dose computation Total dose Dr, delivered to the point P during a HDR
treatment session is:
Implant geometry DT=DS+DM,
where Ds and DM are the dose components delivered by the stationary and moving source respectively. For example, if the source in Figure 2 (top) dwells for 1 second at point A(0, 1, 0),then moves with constant speed o = 1 cm/s to point B( 1, 1, 0) and dwells there for 1 second, then using equations (2) and (3), the dose delivered to point P is: DT
=
&pant
A +
DS,point =
B +
DM
C + (C/2) + 0.78X
= 2.28X.
If dose is computed using a standard algorithm that ignores the dynamic dose component, then the dose to point P is underestimated by more than 50% since DT/DS = 1.523. If, however, the source travels between points A and B with a speed of 100 cm/s, the dosimetric error would only be 0.5%. Thus, in general, the simulation of HDR treatments using only a stationary dose formalism is always incorrect, but the magnitude of the resulting error may range from significant to negligible depending on the source speed (determined by the implant geometry), source activity and prescribed dose. Source speed The average speed of the source between dwell points within an implant was measured using an oscilloscope linked to the pulse generating circuit of the stepping motor that moves the source. The time needed to transfer the source from one dwell point to another was determined from oscilloscope readings, and the source speed then computed. The average source speed outside of the implanted volume, that is the maximum source speed, was measured using a photo-diode circuit. The time required for the source to travel a distance of 199 cm (afterloaderpatient-afterloader) was measured using an electronic timer that was triggered by the circuit when the light beam was interrupted by the moving source. The speed was then calculated. The results are presented in Table 1.
The speed was found to increase with distance travelled from a minimum of 23 cm/s for the shortest allowable interval between dwell points (0.25 cm), to a maximum of 50 cm/s outside of the patient. Consequently, implant geometry, specifically the inter-dwell point distances, determines the total time that the source is in motion, and hence the magnitude of the dynamic dose component DM that is delivered to the prescription point by the moving radiation source. For instance, the dynamic dose component delivered to point P would increase by 33% if an additional three dwell points, 0.25 cm apart, would be inserted between points A and B in the example shown in Figure 2 (bottom). This is a consequence of a speed change from 30.3 cm/s, when the source moves directly from A to B (1 .O cm for r = 1 cm), to 22.7 cm/s (Table 1) when the source travels the same distance in 0.25 cm steps. Source activity Similarly, if the source activity would double and the prescribed dose and source speed would remain constant, the dynamic dose component D,,.,, would increase accordingly ( 100%). Prescribed dose
Currently used dose determination formalisms consider only the stationary dose component Ds, which is always taken as being numerically equal to the prescribed dose. Consequently, the ratio DT/Ds, (total true dose administered to the prescription point/dose prescribed at the dose prescription point), which determines the magnitude of the dosimetric error resulting from the failure to account for the dynamic dose component DM, is inversely proportional to the prescribed dose. Clinical significance
To assess the significance of our findings, a clinical example, schematically shown in Figure 3, was considered. A gynecologic implant was simulated using a vaginal cylinder 2.6 cm in diameter having six catheters with hexagonal geometry on the periphery of a 2.0 cm diameter circle.
798
1. J. Radiation Oncology 0 Biology0 Physics
Volume 24, Number 4, 1992
Fig. 3. Vaginal cylinder with six catheters arranged in hexagonal geometry. Sagittal (top) and transverse (bottom) views. Open circles indicate the dose prescription points; full circles show the source dwell positions; location of plane F (7.0 cm from the origin) is illustrated in the insert.
The doses prescribed at the dose prescription points 1 through 6, Figure 3, were identical in all planes A through E 1 cm apart. The prescribed dose values were as follows: 100 cGy at point 1, 120 cGy at points 2 and 6, 150 cGy at points 3 and 5, and 200 cGy at point 4. In each catheter, there were 17 dwell positions 2.5 mm apart. The geometrically-based dose distribution optimization (2, 7) for the prescribed doses yielded dwell times that are summarized in Table 2. The resulting dose distribution, in sag&al and transverse planes, is shown in Figure 4. Figure 5 demonstrates the deterioration of dose distribution homogeneity with increasing step size, and accounts for the choice of step size used in the example above.
The true total dose delivered to each dose prescription point, that is DT, was then calculated and the ratio DT/ Ds computed. The results, Table 3, show that the dosimetric error for this particular HDR treatment ranges from 4% to 13% within the implanted volume. Outside of this region, for example in plane F (Fig. 3, insert), the error is significantly higher. It is noted that if the prescribed dose at point 4 would be 1000 cGy, and the dose at point 1 would stay 100 cGy, the maximum error would remain 13%. The variation of this error due to source activity, speed (step size), and prescribed dose is summarized in Table 4. Since there are a variety of HDR afterloading machines available, the value of Dr/Ds for afterloaders
Dose
determination in high dose-rate brachytherapy 0
799
P. V. HOUDEK et al.
Table 2. Dwell times at 17 dwell points in 6 catheters* Dwell time (set)
Dwell position(s)
Catheter(s)
1 and 2 and 3 and 4 and 5 and 6 and 7 and 8 and 9
17 16 15 14 13 12 11 10
1
2 and 6
3 and 5
4
0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.3 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.1
0.4 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2
0.6 0.5 0.4 0.4 0.3 0.3 0.3 0.3 0.3
* For 370 GBq ‘921rsource.
with (i) a 20 mCi (740 GBq) source and (ii) a minimum source speed of 15.9 cm/set have been included in this table. The dynamic dose component D,,,,, which is constant for all dose prescription points in a given plane and independent of the prescribed dose, is also listed in Table 3. The distribution of the dynamic dose component, FigFig. 5. Sag&al dose distribution obtained for the same applicator using source step size of 0.5 cm (top, 54 dwell positions) and 1.O cm (bottom, 30 dwell positions).
ure 6, clearly demonstrates that DM is not constant throughout the implanted volume. Consequently, II,,., always distorts the dose distribution derived by the standard algorithm which only recognizes the stationary dose component Ds. The distribution of DM outside of the target is shown in the insert of Figure 6. Summary
The total dose DT delivered to the prescription points during an HDR treatment session consists of dynamic Table 3. Dose delivered/dose prescribed (&/OS) at 30 dose points Dose point (prescribed dose in cGy)
Fig. 4. Optimized dose distribution (50%, lOO%, 120%, 150%, and 200%) for the applicator shown in Figure 3, in sag&al (top) and transverse (bottom) planes. Full circles show the dose prescription points, open (cross-hatched) squares indicate the source dwell positions in (outside) of the planes of interest. There is a total of 102 dwell positions (sources).
Plane
1 (100)
2 and 6 (120)
3 and 5 (150)
4 (200)
&f* (cGy)
A B C D
1.074 1.120 1.129 1.131
1.062 1.100 1.108 1.109
1.049 1.080 1.086 1.087
1.037 1.060 1.065 1.066
7.4 12.0 12.9 13.1
;
1.970 1.116
1.970 1.097
1.882 1.077
1.882 1.058
11.6 9.7 _
* C = 4.33’104 cGycm2h-r. + The points Fl through F6, on plane F, 7.0 cm from the origin, are outside of the treatment volume (Fig. 3, insert). No dose is prescribed to these points; however, they receive a calculated Ds dose of 10 cGy.
800
1. J. Radiation Oncology 0 Biology 0 Physics
Volume 24, Number 4, 1992
14
12
10
.......................*........................... i ..................................................... I i ........
. . . . . .. . . .
........
..........
5..
b..
6 ,..
........ . . . . . . . . /
.. .... 6
i i 1-7
I..,
795 I
,
.A
,..
1
101.5
!
595
395
I I
99.5
1
I
97.5
I
95.5
I
I
93.5
a
1
91.5
19.5
-0.5
I
I
b..,
89.5
SOURCE POSITION (cm) Fig. 6. Dose distribution
of the dynamic dose component
(DM) and stationary (Ds) dose components. DM is determined by the speed of the source (implant geometry) and its activity. Ds is identical to the prescribed dose. The dynamic dose element cannot be ignored because it always escalates the dose at the prescription points. The magnitude of dosimetric error is determined by the value of DT/Ds. The underestimation of dose is more significant for low prescribed doses, high source activities, and low source velocities (Table 4). The value of DT/Ds is not constant throughout the implanted volume. ConTable 4. Variation of the ratio D=/D,* Source step activity size
Or/OS for prescribed dose
GBq
cm
Source speed cm/s
740 370 185 370 370 370
0.25 0.25 0.25 0.50 1.oo 0.25
22.7 22.7 22.7 27.1 30.3 15.9
in cGy 100
200
500
1.262 1.131 1.066 1.118 1.111 1.186
1.131 1.066 1.033 1.059 1.056 1.093
1.052 1.026 1.013 1.024 1.022 1.037
* Dose point D, .
within and outside (insert) of the implanted volume.
sequently, the failure to account for dose delivered by a moving radiation source also results in distortion of the relative dose distribution both within and outside of the irradiated volume.
DISCUSSION
AND CONCLUSION
The major advantage of high dose-rate brachytherapy techniques is that the treatment of each individual patient can be optimized (3,5,6). This is achieved by appropriate selection of dwell points and times. Usually, the best results, for example those compatible with the objectives of HDR conformal brachytherapy, are obtained by using a large number of dwell points (small inter-dwell distances) that in return permits fine variation of dwell times. Using current dose determination algorithms, however, then results in underestimation of dose delivered to the prescription points, as well as incorrect computation of the dose distribution within and outside of the implanted volume. Thus, we conclude that to fully use the advantages of the HDR treatment modality, it is essential that a new formalism for the determination of administered dose be adopted.
REFERENCES 1. Abitbol, A.; Houdek, P.; Schwade, J. G.; Lewin, A. A.; Ser-
ago, C.; Brandon, A. Ring applicator with rectal retractor:
applicability to high dose rate brachytherapy cancer. Activity 4:68-69;1990.
of cervical
Dose determination in high dose-rate brachytherapy 0 P. V. 2. Edmundson, G. K. Geometry based optimization for stepping source implants. In: Martinez, A. A., Orton, C. G., Mould, R. F., ed. Brachytherpy HDR and LDR. Columbia, MD: Nucletron; 1990: 184- 192. 3. Houdek, P. V.; Schwade, J. G.; Abitbol, A. A.; Pisciotta, V. J.; Wu, X.; Serago, C. F.; Markoe, A. M.; Lewin, A. A.; Averette, H. E.; Sevin, B.; Brandon, A. H.; Fiedler, J. A.; Cohen, A. M.; Fayad, F. Optimization of high dose rate cervix brachytherapy; Part I: Dose distribution. Int. J. Radiat. Oncol. Biol. Phys. 21:1621-1625;1991. 4. Nath, R.; Meigooni, A. S.; Meli, J. A. Dosimetry on transverse axes of “‘1 and 19*Irinterstitial brachytherapy sources. Med. Phys. 17:1032-1040;1990.
HOUDEK et al.
801
5. Orton, C. G. HDR: Forget not “time” and “distance.” Int. J. Radiat. Oncol. Biol. Phys. 20: 113 1- 1132; 199 1. 6. Speiser, B. Advantages of high dose rate remote afterloading systems: Physics or biology. Int. J. Radiat. Oncol. Biol. Phys. 20:1133-l 135;1991. 7. van der Laarse, R.; de Boer, R. W. Optimization of high dose rate brachytherapy. Activity 2: 14- 15; 1989. 8. Weaver, K. A.; Smith, V.; Huang, D.; Barnett, C.; Schell, M. C.; Ling, C. Dose parameters of “‘1 and ‘921rseed sources. Med. Phys. 16:636-643; 1989. 9. 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.
Int. I Radramn Oncology Biol. Phys., Vol. 24, pp. 795-801 Printed in the U.S.A. All rights resewed.
t .oO
$5.00 Press Ltd.
Copynsht B 1992Per@mon
??Technical Innovations and Notes
DOSE DETERMINATION
IN HIGH DOSE-RATE BRACHYTHERAPY
PAVEL V. HOUDEK, PH.D., JAMES G. SCHWADE, M.D., XIAODONG Wu, M.S., VINCENT PISCIOTTA, M.S., JEFFREY A. FIEDLER, M.S., CHRISTOPHER F. SERAGO, PH.D., ARNOLD M. MARKOE, M.D., ANDRE A. ABITBOL, M.D., ALAN A. LEWIN, M.D., PAUL G. BRAUNSCHWEIGER, PH.D. AND MARSHALL D. SKLAR, M.D. Department
of Radiation Oncology, University of Miami School of Medicine, Miami, FL
Although high dose-rate brachytherapy with a single, rapidly moving radiation source is becoming a common treatment modality, a suitable formalism for determinationof the dose deliveredby a movingradiationsource has not yet been developed. At present, brachytherapy software simulates high dose-rate treatments using only a series of stationary sources, and consequently fails to account for the dose component delivered while the source is in motion. We now describe a practical model for determination of the true, total dose administered. The algorithm calculatesboth the dose delivered while the source is in motion within and outside of the implanted volume (dynamic component), and the dose delivered while the source is stationary at a series of fixed dwell points. It is shown that the dynamic dose element cannot be ignored because it always increases the dose at the prescription points and, in addition, distorts the dose distribution within and outside of the irradiated volume. Failure to account for the dynamic dose component results in dosimetric errors that range from significant (> 10%) to negligible (< I%), depending on the prescribed dose, source activity, and source speed as defined by the implant geometry. Dosimetry, High dose-rate brachytherapy,
Moving source, Algorithm.
INTRODUCTION Modern, computer controlled, high dose-rate (HDR) afterloading machines are usually equipped with a single, high activity, miniaturized, Iridium-192 source that can be moved through several channels between the storage/ safe unit and patient (3). The treatment proceeds in the following manner: the source is rapidly moved from the afterloader to the first implant dwell point; it is held stationary there for a pre-programmed period of time (dwell time); then, somewhat less rapidly, the source moves to occupy the next dwell point. The process is repeated at all dwell points in a given channel; the source is then retracted into the afterloader and the process repeated in subsequent channels. Commonly used brachytherapy software simulates a prescribed treatment by using a series of stationary sources only and, consequently, fails to account for the component of dose delivered when the source is in motion. Since exclusion of this dynamic dose element should lead to underestimation of dose at the prescription points, and concomitant distortion of the dose distribution in the irradiated volume, we have developed a formalism that de-
Reprints requests to: Pave1 V. Houdek, Ph.D., Department of Radiation Oncology (D-31), University of Miami School of Medicine, P.O. Box 016960, Miami, FL 33101, U.S.A.
scribes both the dynamic as well as stationary components of HDR treatments. The results are presented herein. METHODS
AND MATERIALS
General considerations It is assumed that a point radiation source S moves in
space along curve c with velocity o, and that source position at any time t is defined by the three functions: X = f(t),
y = g(t),
2 = h(t).
The dose D delivered to point P when the source S, Figure 1, moves from point A to point B during time interval t = t2 - tl is: (2
D=C
s 11
F(t)dt,
ok. 1)
where constant C specifies the radiation emission parameters of S, and function F(t) describes the attenuation of radiation between S and P at instant t. For the purposes
Accepted for publication
11 May 1992.
796
1. J. Radiation Oncology 0 Biology0 Physics
Volume 24, Number 4, 1992
the x axis, source position at any time t is: x = f(t) = vt, y = a, z = 0, and the magnitude of the position vector r is: Ir I = (x2 + a2)“2. From the definition of F(t) andf(t) it follows that: F(t) = l/ )r (2 = 1/(x2 + a’) and dt = dx/v. Therefore, the dose delivered to point P by moving source S as it travels from A to B is:
s 12
D,,,,=C
11
F(t)dt = (C/v)
X2( l/(x2 + a2))dx s x1
= (C/av)[arctg(x/a)]$;.
(Eq. 2)
Fig. 1. Radiation source S moving in space along path c.
of brachytherapy dosimetry, the constant C is equal to the product of the specific dose rate constant and source strength (C = 4.33* lo4 cGy cm2 h-* was used for the data presented here) (9) and its numerical value is available in the literature (4, 8, 9). Assuming that none other than inverse-square attenuation is involved, function F(t) is defined by:
F(t) = l/lrl’,
2. Assume that source S, Figure 2 (bottom), moves in a gynecologic ring applicator ( 1) with constant angular velocity Q = dol/dtin a circular path from A to B. The source position at any time t is: x = rcos(a), y = rsin(a), z = 0, where angular displacement a! = fit. The magnitude of the position vector r is constant at any time t, Ir I = (x2 + y2)‘j2 = r, and hence F(t) = l/r2. Since dt = da/Q, the dose delivered to point P by moving source S is: D,,,.,= (C/(r2s2)) se2 dcu = (C/(r2Q))[a]::. al
where r is the position vector of source S. If i, j, and k are unit vectors along the x, y, and z axes, then r is given by:
3. Suppose that source S is stationary at a dwell point
Y
r = ix + jy + kz.
A
The magnitude of r, for example (r 1, is the distance between the source S and origin P:
C
A
S
B
Irl = r = (x2 + y2 + 22)1’2. The velocity vector v of the moving source Sis obtained by differentiation of position vector r with respect to time: v = drfdt = i(dx/dt) + j(dy/dt) + k(dz/dt).
Its magnitude
1v ) is the source instantaneous
velocity o:
(VI = u = ((dx/dt)2 + (dy/dt)2 + (dz/dt)2)“2.
Velocity is positive (negative) when the source is moving towards increasing (decreasing) values of r. Brachytherapy dosirnetry
Herein. three solutions of equation (l), relevant to HDR dosimetry, are discussed: 1. Suppose that in a linear implant the point source S, Figure 2 (top), moves along path c from the point A(x, , a, 0) to the point B(x2, a, 0) with constant velocity 2) = dx/dt. Since the source trajectory is parallel with
Fig. 2. Trajectory of source S in a linear implant (top, PA = AB = 1.0 cm) and in a gynecologic ring applicator (bottom, PA = 1.O cm, (Y*- LY,= [180/r] degrees).
Dose determination in high dose-rate brachytherapy 0 P. V. HOUDEK
having coordinates x, y, and z. The magnitude of the position vector is constant in time, (r ( = (x2 + y* + z2)‘O, and hence F(t) = l/r*. Then the dose delivered to the point Pin time interval t = t2 - tl is: D, = (C/r*) j-” dt = (C/r*)[t]$
11
Table 1. Source
speed* as a function of the distance travelled+
Distance (mm)
Speed (cm/set)
2.5 5.0 (Eq. 3)
191
ef ~1.
10.0
20.0
22.1 27.1 30.3 32.3
Distance (mm) 30.0 50.0 100.0
995.0
Speed (cm/set) 32.7 33.1 34.5 49.9
* Average speed for dwell times of 0.1 set, 0.3 set, and 0.5 sec. t Microselectron, Nucletron Corp., Columbia, MD 2 1045.
RESULTS Dose computation Total dose Dr, delivered to the point P during a HDR
treatment session is:
Implant geometry DT=DS+DM,
where Ds and DM are the dose components delivered by the stationary and moving source respectively. For example, if the source in Figure 2 (top) dwells for 1 second at point A(0, 1, 0),then moves with constant speed o = 1 cm/s to point B( 1, 1, 0) and dwells there for 1 second, then using equations (2) and (3), the dose delivered to point P is: DT
=
&pant
A +
DS,point =
B +
DM
C + (C/2) + 0.78X
= 2.28X.
If dose is computed using a standard algorithm that ignores the dynamic dose component, then the dose to point P is underestimated by more than 50% since DT/DS = 1.523. If, however, the source travels between points A and B with a speed of 100 cm/s, the dosimetric error would only be 0.5%. Thus, in general, the simulation of HDR treatments using only a stationary dose formalism is always incorrect, but the magnitude of the resulting error may range from significant to negligible depending on the source speed (determined by the implant geometry), source activity and prescribed dose. Source speed The average speed of the source between dwell points within an implant was measured using an oscilloscope linked to the pulse generating circuit of the stepping motor that moves the source. The time needed to transfer the source from one dwell point to another was determined from oscilloscope readings, and the source speed then computed. The average source speed outside of the implanted volume, that is the maximum source speed, was measured using a photo-diode circuit. The time required for the source to travel a distance of 199 cm (afterloaderpatient-afterloader) was measured using an electronic timer that was triggered by the circuit when the light beam was interrupted by the moving source. The speed was then calculated. The results are presented in Table 1.
The speed was found to increase with distance travelled from a minimum of 23 cm/s for the shortest allowable interval between dwell points (0.25 cm), to a maximum of 50 cm/s outside of the patient. Consequently, implant geometry, specifically the inter-dwell point distances, determines the total time that the source is in motion, and hence the magnitude of the dynamic dose component DM that is delivered to the prescription point by the moving radiation source. For instance, the dynamic dose component delivered to point P would increase by 33% if an additional three dwell points, 0.25 cm apart, would be inserted between points A and B in the example shown in Figure 2 (bottom). This is a consequence of a speed change from 30.3 cm/s, when the source moves directly from A to B (1 .O cm for r = 1 cm), to 22.7 cm/s (Table 1) when the source travels the same distance in 0.25 cm steps. Source activity Similarly, if the source activity would double and the prescribed dose and source speed would remain constant, the dynamic dose component D,,.,, would increase accordingly ( 100%). Prescribed dose
Currently used dose determination formalisms consider only the stationary dose component Ds, which is always taken as being numerically equal to the prescribed dose. Consequently, the ratio DT/Ds, (total true dose administered to the prescription point/dose prescribed at the dose prescription point), which determines the magnitude of the dosimetric error resulting from the failure to account for the dynamic dose component DM, is inversely proportional to the prescribed dose. Clinical significance
To assess the significance of our findings, a clinical example, schematically shown in Figure 3, was considered. A gynecologic implant was simulated using a vaginal cylinder 2.6 cm in diameter having six catheters with hexagonal geometry on the periphery of a 2.0 cm diameter circle.
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Volume 24, Number 4, 1992
Fig. 3. Vaginal cylinder with six catheters arranged in hexagonal geometry. Sagittal (top) and transverse (bottom) views. Open circles indicate the dose prescription points; full circles show the source dwell positions; location of plane F (7.0 cm from the origin) is illustrated in the insert.
The doses prescribed at the dose prescription points 1 through 6, Figure 3, were identical in all planes A through E 1 cm apart. The prescribed dose values were as follows: 100 cGy at point 1, 120 cGy at points 2 and 6, 150 cGy at points 3 and 5, and 200 cGy at point 4. In each catheter, there were 17 dwell positions 2.5 mm apart. The geometrically-based dose distribution optimization (2, 7) for the prescribed doses yielded dwell times that are summarized in Table 2. The resulting dose distribution, in sag&al and transverse planes, is shown in Figure 4. Figure 5 demonstrates the deterioration of dose distribution homogeneity with increasing step size, and accounts for the choice of step size used in the example above.
The true total dose delivered to each dose prescription point, that is DT, was then calculated and the ratio DT/ Ds computed. The results, Table 3, show that the dosimetric error for this particular HDR treatment ranges from 4% to 13% within the implanted volume. Outside of this region, for example in plane F (Fig. 3, insert), the error is significantly higher. It is noted that if the prescribed dose at point 4 would be 1000 cGy, and the dose at point 1 would stay 100 cGy, the maximum error would remain 13%. The variation of this error due to source activity, speed (step size), and prescribed dose is summarized in Table 4. Since there are a variety of HDR afterloading machines available, the value of Dr/Ds for afterloaders
Dose
determination in high dose-rate brachytherapy 0
799
P. V. HOUDEK et al.
Table 2. Dwell times at 17 dwell points in 6 catheters* Dwell time (set)
Dwell position(s)
Catheter(s)
1 and 2 and 3 and 4 and 5 and 6 and 7 and 8 and 9
17 16 15 14 13 12 11 10
1
2 and 6
3 and 5
4
0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.3 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.1
0.4 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2
0.6 0.5 0.4 0.4 0.3 0.3 0.3 0.3 0.3
* For 370 GBq ‘921rsource.
with (i) a 20 mCi (740 GBq) source and (ii) a minimum source speed of 15.9 cm/set have been included in this table. The dynamic dose component D,,,,, which is constant for all dose prescription points in a given plane and independent of the prescribed dose, is also listed in Table 3. The distribution of the dynamic dose component, FigFig. 5. Sag&al dose distribution obtained for the same applicator using source step size of 0.5 cm (top, 54 dwell positions) and 1.O cm (bottom, 30 dwell positions).
ure 6, clearly demonstrates that DM is not constant throughout the implanted volume. Consequently, II,,., always distorts the dose distribution derived by the standard algorithm which only recognizes the stationary dose component Ds. The distribution of DM outside of the target is shown in the insert of Figure 6. Summary
The total dose DT delivered to the prescription points during an HDR treatment session consists of dynamic Table 3. Dose delivered/dose prescribed (&/OS) at 30 dose points Dose point (prescribed dose in cGy)
Fig. 4. Optimized dose distribution (50%, lOO%, 120%, 150%, and 200%) for the applicator shown in Figure 3, in sag&al (top) and transverse (bottom) planes. Full circles show the dose prescription points, open (cross-hatched) squares indicate the source dwell positions in (outside) of the planes of interest. There is a total of 102 dwell positions (sources).
Plane
1 (100)
2 and 6 (120)
3 and 5 (150)
4 (200)
&f* (cGy)
A B C D
1.074 1.120 1.129 1.131
1.062 1.100 1.108 1.109
1.049 1.080 1.086 1.087
1.037 1.060 1.065 1.066
7.4 12.0 12.9 13.1
;
1.970 1.116
1.970 1.097
1.882 1.077
1.882 1.058
11.6 9.7 _
* C = 4.33’104 cGycm2h-r. + The points Fl through F6, on plane F, 7.0 cm from the origin, are outside of the treatment volume (Fig. 3, insert). No dose is prescribed to these points; however, they receive a calculated Ds dose of 10 cGy.
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1. J. Radiation Oncology 0 Biology 0 Physics
Volume 24, Number 4, 1992
14
12
10
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SOURCE POSITION (cm) Fig. 6. Dose distribution
of the dynamic dose component
(DM) and stationary (Ds) dose components. DM is determined by the speed of the source (implant geometry) and its activity. Ds is identical to the prescribed dose. The dynamic dose element cannot be ignored because it always escalates the dose at the prescription points. The magnitude of dosimetric error is determined by the value of DT/Ds. The underestimation of dose is more significant for low prescribed doses, high source activities, and low source velocities (Table 4). The value of DT/Ds is not constant throughout the implanted volume. ConTable 4. Variation of the ratio D=/D,* Source step activity size
Or/OS for prescribed dose
GBq
cm
Source speed cm/s
740 370 185 370 370 370
0.25 0.25 0.25 0.50 1.oo 0.25
22.7 22.7 22.7 27.1 30.3 15.9
in cGy 100
200
500
1.262 1.131 1.066 1.118 1.111 1.186
1.131 1.066 1.033 1.059 1.056 1.093
1.052 1.026 1.013 1.024 1.022 1.037
* Dose point D, .
within and outside (insert) of the implanted volume.
sequently, the failure to account for dose delivered by a moving radiation source also results in distortion of the relative dose distribution both within and outside of the irradiated volume.
DISCUSSION
AND CONCLUSION
The major advantage of high dose-rate brachytherapy techniques is that the treatment of each individual patient can be optimized (3,5,6). This is achieved by appropriate selection of dwell points and times. Usually, the best results, for example those compatible with the objectives of HDR conformal brachytherapy, are obtained by using a large number of dwell points (small inter-dwell distances) that in return permits fine variation of dwell times. Using current dose determination algorithms, however, then results in underestimation of dose delivered to the prescription points, as well as incorrect computation of the dose distribution within and outside of the implanted volume. Thus, we conclude that to fully use the advantages of the HDR treatment modality, it is essential that a new formalism for the determination of administered dose be adopted.
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ago, C.; Brandon, A. Ring applicator with rectal retractor:
applicability to high dose rate brachytherapy cancer. Activity 4:68-69;1990.
of cervical
Dose determination in high dose-rate brachytherapy 0 P. V. 2. Edmundson, G. K. Geometry based optimization for stepping source implants. In: Martinez, A. A., Orton, C. G., Mould, R. F., ed. Brachytherpy HDR and LDR. Columbia, MD: Nucletron; 1990: 184- 192. 3. Houdek, P. V.; Schwade, J. G.; Abitbol, A. A.; Pisciotta, V. J.; Wu, X.; Serago, C. F.; Markoe, A. M.; Lewin, A. A.; Averette, H. E.; Sevin, B.; Brandon, A. H.; Fiedler, J. A.; Cohen, A. M.; Fayad, F. Optimization of high dose rate cervix brachytherapy; Part I: Dose distribution. Int. J. Radiat. Oncol. Biol. Phys. 21:1621-1625;1991. 4. Nath, R.; Meigooni, A. S.; Meli, J. A. Dosimetry on transverse axes of “‘1 and 19*Irinterstitial brachytherapy sources. Med. Phys. 17:1032-1040;1990.
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5. Orton, C. G. HDR: Forget not “time” and “distance.” Int. J. Radiat. Oncol. Biol. Phys. 20: 113 1- 1132; 199 1. 6. Speiser, B. Advantages of high dose rate remote afterloading systems: Physics or biology. Int. J. Radiat. Oncol. Biol. Phys. 20:1133-l 135;1991. 7. van der Laarse, R.; de Boer, R. W. Optimization of high dose rate brachytherapy. Activity 2: 14- 15; 1989. 8. Weaver, K. A.; Smith, V.; Huang, D.; Barnett, C.; Schell, M. C.; Ling, C. Dose parameters of “‘1 and ‘921rseed sources. Med. Phys. 16:636-643; 1989. 9. 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.
