Download a simple analytic derivation suggests that prostate cancer / ratio is

Int. J. Radiation Oncology Biol. Phys., Vol. 51, No. 1, pp. 213–214, 2001
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BIOLOGY CONTRIBUTION
A SIMPLE ANALYTIC DERIVATION SUGGESTS THAT PROSTATE CANCER
␣/␤ RATIO IS LOW
CHRISTOPHER R. KING, PH.D., M.D.,*
AND
JACK F. FOWLER, D.SC., PH.D.†
*Department of Radiation Oncology, Stanford University School of Medicine, Stanford, CA; †Department of Human Oncology,
University of Wisconsin Hospital, Madison, WI
Prostate cancer, Radiotherapy, Radiobiology, ␣/␤ ratio.
Since the publication of Brenner and Hall’s recent paper (1)
suggesting that the ␣/␤ ratio for prostate cancer is 1.5 Gy, a
debate has been propagated through the radiotherapy community (2–5). This surprisingly low value carries implications of more efficient (and perhaps less costly) regimens of
external beam radiotherapy (i.e., hypofractionated schedules) and optimal forms of brachytherapy (i.e., high dose
rate). Should it prove to be true, such a low ratio would
negate the therapeutic advantage of conventional fractionation regimens and call out for clinical trials. There is
certainly indirect evidence for a low ␣/␤ ratio. The very
small proportion of proliferating cells (6, 7), the very long
observed doubling times (8, 9), the slow response to irradiation, and the recent interim results of high-dose-rate
brachytherapy of varying fractionation (10) are all are consistent with such a low value. The surprise is that this value
is as low as, if not even lower than, normal tissue response
to irradiation.
Can one apply the linear-quadratic (LQ) equation to
outcomes of a population of clinically localized prostate
cancer after radiotherapy? The LQ equation, although
purely phenomenological, has certainly proven to be of
great use, both in the study of cultured tumor cells irradiated
in vitro, in comparing regimens of different fractionation,
and in the study of normal tissue response to radiation.
Although issues of inter- and intratumor heterogeneity are
not directly accounted for in the standard LQ formalism,
these issues might not necessarily affect the ␣/␤ ratio when
applied to changes in fraction size (4). Whereas the ␣ term,
a direct measure of radiosensitivity, does depend upon heterogeneity (3) and is intimately tied to the number of
clonogens, the ratio ␣/␤ might not.
We present a simple analytical derivation of ␣/␤ that is a
logical conclusion from the observation that, for clinically
localized prostate cancer, external beam and permanent
brachytherapy appear to achieve equivalent outcomes. This
derivation makes no assumptions and does not attempt to fit
models to specific clinical data.
For fractionated external-beam irradiation to a total dose
De, with dose per fraction d, the LQ survival fraction of
clonogens can be written as follows:
Reprint requests to: Christopher R. King, Ph.D., M.D., Department of Radiation Oncology, Stanford University School of
Medicine, 300 Pasteur Drive, Stanford, CA 94305. Tel: (650)
723-1420; Fax: (650) 498-6922; E-mail: christopher@reyes.
stanford.edu
Received Dec 5, 2000. Accepted for publication Feb 15, 2001.
S e ⫽ exp(⫺␣ D e关1 ⫹ d/共 ␣ / ␤ 兲兴)
(1)
For permanent low-dose-rate brachytherapy, the extension
of the LQ equation from Dale (11) can be written as follows:
S b ⫽ exp(⫺␣ D b关1 ⫹ GD b/共 ␣ / ␤ 兲兴)
(2)
where Db is the minimal peripheral dose to decay and G
(sometimes called the “Lea-Catcheside” term) accounts for
source decay and sublethal damage repair kinetics and is
given by G ⫽ ␭/共␭ ⫹ ␮兲, with ␭ the decay constant and
␮ the repair constant. Values of 59.7 days for the half-life of
125
I and typically 1–2 hours for repair half-life result in
G⬇10⫺3. Thus Eq. 2 may be simplified to:
S b ⫽ exp(⫺␣ D b)
(3)
Because external beam and permanent brachytherapy are
indeed nearly equivalent therapies, Se ⫽ Sb. Simple algebraic manipulation of Eqs. 1 and 3 yields:
␣ / ␤ ⫽ d/共D b/D e ⫺ 1兲
(4)
For standard values (i.e., d ⫽ 1.8 –2 Gy, De ⫽ 70 –74 Gy,
and Db ⫽ 145 Gy [TG-43]), this yields: ␣/␤⬇2 Gy.
If one wishes to include source decay and repair kinetics,
then Eq. 4 becomes:
␣ / ␤ ⫽ 共d ⫺ GD b2/D e兲/共D b/D e ⫺ 1兲
(5)
For typical values this yields: ␣/␤ ⬃ 1.8 Gy.
213
214
I. J. Radiation Oncology
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Biology
●
Physics
It is interesting that this derivation is not dependent upon
␣ or estimates of clonogen number, or specific clinical
outcome data. It simply follows from the LQ equations and
the clinical observation that external beam and permanent
brachytherapy achieve similar outcomes with current regimens. Because clinical outcomes for 125I or 103Pd are equivalent (12), similar values are expected if 103Pd data are used
(i.e., half-life of 17 days and prescribed dose of 124 Gy
Volume 51, Number 1, 2001
[NIST-99]). These values are ␣/␤ ⬃ 2.8 (or ⬃ 2 if decay
and repair are included, as in Eq. 5). The values obtained
here are consistent with those of Brenner and Hall (1) and
Fowler et al. (5), who fit the LQ model to clinical series and
continue to provide interesting insights into the clinical
radiobiology of prostate cancer. They also provoke thought
that perhaps prostate cancer might be optimally treated with
hypofractionated regimens.
REFERENCES
1. Brenner DJ, and Hall EJ. Fractionation and protraction for
radiotherapy of prostate carcinoma. Int J Radiat Oncol Biol
Phys 1999;43:1095–1101.
2. Duschesne GM, Peters LJ. What is the ␣/␤ ratio for prostate
cancer? Rationale for hypofractionated high-dose-rate brachytherapy. Int J Radiat Oncol Biol Phys 1999;44:747–748 (editorial).
3. King CR, and Mayo CS. Is the prostate ␣/␤ Ŕatio of 1.5 from
Brenner and Hall a modeling artifact? Int J Radiat Oncol Biol
Phys 2000;47:536 –538 (letter).
4. Brenner DJ, Hall EJ. Low ␣/␤ values for prostate cancer
appear to be independent of modeling details. Int J Radiat
Oncol Biol Phys 2000;47:538 –539 (letter).
5. Fowler JF, Chappell R, Ritter MA. Is ␣/␤ for prostate really
low? Int J Radiat Oncol Biol Phys 2001;50:1021–1031.
6. Haustermans KMG, Hofland I, Van Poppel H, Oyen R, Van
der Voorde W, Begg AC, and Fowler JF. Cell kinetic measurements in prostate cancer. Int J Radiat Oncol Biol Phys
1997;37:1067–1070.
7. Haustermans KMG, and Fowler JF. A comment on prolifer-
8.
9.
10.
11.
12.
ation rates in human prostate cancer. Int J Radiat Oncol Biol
Phys 2000;48:303 (letter).
Lee WR, Hanks GE, Corn BW, Schultheiss TE. Observations
of pretreatment prostate-specific antigen doubling time in 107
patients referred for definitive radiotherapy. Int J Radiat Oncol Biol Phys 1995;31:21–24.
Schmid HP. Tumour markers in patients on deferred treatment: Prostate specific antigen doubling times. Cancer Surv
1995;23:157–167.
Martinez AA, Kestin LL, Stromberg JS, et al. Interim report of
image-guided conformal high-dose-rate brachytherapy for patients with unfavorable prostate cancer: The William Beaumont phase II dose-escalation trial. Int J Radiat Oncol Biol
Phys 2000;47:343–352.
Dale RG. The application of the linear-quadratic dose-effect
equation to fractionated and protracted radiotherapy. Br J
Radiol 1985;58:515–528.
Cha CM, Potters L, Ashley R, Freeman K, Wang X-H, Waldbaum R, and Leibel S. Isotope selection for patients undergoing prostate brachytherapy. Int J Radiat Oncol Biol Phys
1999;45:391–395.