Abstract
A numerical integration method is proposed to evaluate a very computationally expensive integration encountered in the analysis of the optimal dose grid size for the intensity modulated proton therapy (IMPT) fluence map optimization (FMO). The resolution analysis consists of obtaining the Fourier transform of the 3-dimensional (3D) dose function and then performing the inverse transform numerically. When the proton beam is at an angle with the dose grid, the Fourier transform of the 3D dose function contains integrals involving oscillatory sine and cosine functions and oscillates in all of its three dimensions. Because of the oscillatory behavior, it takes about 300 hours to compute the integration of the inverse Fourier transform to achieve a relative accuracy of 0.1 percent with a 2 GHz Intel PC and using an iterative division algorithm. The proposed method (subsequently referred to as table method) solves integration problems with a partially separated integrand by integrating the inner integral for a number of points of the outer integrand and finding the values of other evaluation points by interpolation. The table method reduces the computational time to less than one percent for the integration of the inverse Fourier transform. This method can also be used for other integration problems that fit the method application conditions.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17, 437–451 (1991)
Bortfeld, T.: An analytical approximation of the Bragg curve for therapeutic proton beams. Med. Phys. 24, 2024–2033 (1997)
Bracewell, R.N.: The Fourier Transform and Its Application. McGraw-Hill, New York (1978)
de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of loop integrals using extrapolation. Computer Physics Communications 159, 145–156 (2004)
de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F., Cucos, L., Van Voorst, J.: Loop integration results using numerical extrapolation for a non-scalar integral. Nuclear Instruments and Methods in Physics Research Section A 539, 269–273 (2004); hep-ph/0405098
Dempsey, J.F., et al.: A Fourier analysis of the dose grid resolution required for accuracte IMRT fluence map optimization. Med. Phys. 32, 380–388 (2005)
Genz, A., Malik, A.: An adaptive algorithm for numerical integration over an n-dimensional rectangular region. Journal of Computational and Applied Mathematics 6, 295–302 (1980)
Genz, A., Malik, A.: An imbedded family of multidimensional integration rules. SIAM J. Numer. Anal. 20, 580–588 (1983)
Li, H.S., Dempsey, J.F., Romeijin, H.E.: A Fourier analysis on the optimal grid size for discrete proton beam dose calculation. Submitted to Medical Physics
Li, S., de Doncker, E., Kaugars, K.: On iterated numerical integration. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3514, pp. 123–130. Springer, Heidelberg (2005)
Piessens, R., de Doncker, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK, A Subroutine Package for Automatic Integration. Springer Series in Computational Mathematics. Springer, Heidelberg (1983)
Szymanowski, H., Mazal, A., et al.: Experimental determination and verification of the parameters used in a proton pencil beam algorithm. Med. Phys. 28, 975–987 (2001)
ParInt. ParInt web site, http://www.cs.wmich.edu/parint
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, S., de Doncker, E., Kaugars, K., Li, H.S. (2006). A Fast Integration Method and Its Application in a Medical Physics Problem. In: Gavrilova, M.L., et al. Computational Science and Its Applications - ICCSA 2006. ICCSA 2006. Lecture Notes in Computer Science, vol 3984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11751649_87
Download citation
DOI: https://doi.org/10.1007/11751649_87
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34079-9
Online ISBN: 978-3-540-34080-5
eBook Packages: Computer ScienceComputer Science (R0)