A Fast Integration Method and Its Application in a Medical Physics Problem
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.
KeywordsInverse Fourier Transform Adaptive Method Table Method Integration Problem Dose Function
Unable to display preview. Download preview PDF.
- 1.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)Google Scholar
- 2.Bortfeld, T.: An analytical approximation of the Bragg curve for therapeutic proton beams. Med. Phys. 24, 2024–2033 (1997)Google Scholar
- 3.Bracewell, R.N.: The Fourier Transform and Its Application. McGraw-Hill, New York (1978)Google Scholar
- 4.de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of loop integrals using extrapolation. Computer Physics Communications 159, 145–156 (2004)Google Scholar
- 5.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/0405098Google Scholar
- 6.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)Google Scholar
- 7.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)Google Scholar
- 8.Genz, A., Malik, A.: An imbedded family of multidimensional integration rules. SIAM J. Numer. Anal. 20, 580–588 (1983)Google Scholar
- 9.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 PhysicsGoogle Scholar
- 12.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)Google Scholar
- 13.ParInt. ParInt web site, http://www.cs.wmich.edu/parint