A Fast Integration Method and Its Application in a Medical Physics Problem

  • Shujun Li
  • Elise de Doncker
  • Karlis Kaugars
  • Haisen S. Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3984)


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.


Inverse Fourier Transform Adaptive Method Table Method Integration Problem Dose Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Shujun Li
    • 1
  • Elise de Doncker
    • 1
  • Karlis Kaugars
    • 1
  • Haisen S. Li
    • 2
  1. 1.Computer ScienceWestern Michigan University 
  2. 2.Department of Radiation OncologyUniversity of Florida 

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