An analytical approximation of the Bragg curve for therapeutic proton beams

Medical Physics - Tập 24 Số 12 - Trang 2024-2033 - 1997
Thomas Bortfeld1,2
1Deutsches Krebsforschungszentrum (DKFZ), Abteilung Medizinische Physik
2Universität Heidelberg, Fakultät für Physik und Astronomie, Heidelberg, Germany

Tóm tắt

The knowledge of proton depth‐dose curves, or “Bragg curves,” is a fundamental prerequisite for dose calculations in radiotherapy planning, among other applications. In various cases it is desirable to have an analytical representation of the Bragg curve, rather than using measured or numerically calculated data. This work provides an analytical approximation of the Bragg curve in closed form. The underlying model is valid for proton energies between about 10 and 200 MeV. Its main four constituents are: (i) a power‐law relationship describing the range‐energy dependency; (ii) a linear model for the fluence reduction due to nonelastic nuclear interactions, assuming local deposition of a fraction of the released energy; (iii) a Gaussian approximation of the range straggling distribution; and (iv) a representation of the energy spectrum of poly‐energetic beams by a Gaussian with a linear “tail.” Based on these assumptions the Bragg curve can be described in closed form using a simple combination of Gaussians and parabolic cylinder functions. The resulting expression can be fitted to measurements within the measurement error. Very good agreement is also found with numerically calculated Bragg curves.

Từ khóa


Tài liệu tham khảo

10.1088/0031-9155/41/8/005

10.1088/0031-9155/38/7/009

10.1118/1.596887

S. Scheib “Spot‐Scanning mit Protonen: Experimentelle Resultate und Therapieplanung ” Ph.D. thesis ETH Zürich 1993.

K.‐U. Gardey “A Pencil Beam Model for Proton Therapy—Treatment Planning and Experimental Results ” Ph.D. thesis Universität Heidelberg 1996.

10.1016/0168-583X(93)95944-Z

C. V. Levin R. Gonin and S. Wynchank “Computed limitations of spot scanning for therapeutic proton beams ” Int. J. Biomed. Comput.23 33–41 (1988).

10.1002/ima.1850060102

J. O. Deasy D. M. Shepard and T. R. Mackie “Distal edge tracking: A proposed delivery method for conformal proton therapy using intensity modulation ”XIIth International Conference on the Use of Computers in Radiation Therapy edited by D. D. Leavitt and G. Starkschall (Medical Physics Publishing Madison 1997) pp. 406–409.

A. Ahnesjö P. Andreo and A. Brahme “Calculation and application of point spread functions for treatment planning with high energy photon beams ” Acta Oncolog.26 49–56 (1987).

H. A. Bethe and J. Askhin “Passage of radiations through matter ” inExperimental Nuclear Physics Vol. 1 edited by E. Segrè (Wiley New York 1953).

10.1016/0092-640X(82)90004-3

M. J. Berger “Penetration of proton beams through water I. Depth‐dose distribution spectra and LET distribution ” Report NISTIR 5226 National institute of standards and technology physics laboratory Gaithersburg MD 20899 July 1993.

R. D. Evans The Atomic Nucleus(Robert E. Krieger Malabar FL 1982). Reprint.

M. R. Raju Heavy Particle Radiotherapy(Academic New York 1980).

ICRU Report 49: Stopping powers and ranges for protons and alpha particles. International Commission on Radiation Units and Measurements Bethesda MD 1993.

H. Bichsel and T. Hiraoka “Energy spectra and depth‐dose curves for 70 MeV protons ” Int. J. Quantum Chem. Quantum Chem. Symp.23 565–574 (1989).

M. Abramowitz and I. A. Stegun Eds. Handbook of Mathematical Functions(Dover New York 1972).

I. S. Gradshteyn and I. M. Ryzhik Table of Integrals Series and Products(Academic San Diego 1980). Corrected and enlarged edition.

10.1063/1.1143946

10.1118/1.597535

10.1259/0007-1285-34-399-143

10.1016/0029-554X(75)90430-9

U. Oelfke K. ‐U. Gardey E. W. Blackmore and G. K. Y. Lam “Proton dosimetry at TRIUMF: Experimental profiles and PTRAN MC calculations ” in PTCOG XXI San Francisco April 25 1995.

10.1103/PhysRev.71.385

Thông tin
Thông tin xuất bản

Medical Physics

Tập 24 Số 12

2024-2033

Thông tin tác giả