Journal of Mathematical Chemistry

, Volume 54, Issue 10, pp 2048–2061 | Cite as

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

  • Hiroyuki Fujii
  • Shinpei Okawa
  • Yukio Yamada
  • Yoko Hoshi
  • Masao Watanabe
Original Paper


Numerical calculation of photon migration in biological tissue using the radiative transfer equation (RTE) has attracted great interests in biomedical optics and imaging. Because biological tissue is a highly forward-peaked scattering medium, renormalization of the phase function in numerical calculation of the RTE is crucial. This paper proposes a simple approach of renormalizing the phase function by the double exponential formula, which was heuristically modified from the original one. Firstly, the validity of the proposed approach was tested by comparing numerical results for an average cosine of the polar scattering angle calculated by the proposed approach with those by the conventional approach in highly forward-peaked scattering. The results show that calculation of the average cosine converged faster using the proposed approach than using the conventional one as a total number of discrete angular directions increases. Next, the accuracy of the numerical solutions of the RTE using the proposed approach was examined by comparing the numerical solutions with the analytical solutions of the RTE in a homogeneous medium with highly forward-peaked scattering. It was found that the proposed approach reduced the errors of the numerical solutions from those using the conventional one especially at a small value of the total number of the directions. This result suggests that the proposed approach has a possibility to improve the accuracy for the numerical results of the RTE in the highly scattering medium.


Radiative transfer equation Renormalization approach of the phase function Highly forward-peaked scattering media Double-exponential formula 

Mathematics Subject Classification

85A25 92C55 65D30 74E10 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Hiroyuki Fujii
    • 1
  • Shinpei Okawa
    • 2
  • Yukio Yamada
    • 3
  • Yoko Hoshi
    • 4
  • Masao Watanabe
    • 1
  1. 1.Division of Mechanical and Space Engineering, Faculty of EngineeringHokkaido UniversitySapporoJapan
  2. 2.National Defense Medical CollegeTokorozawaJapan
  3. 3.Brain Science Inspired Life Support Research CenterUniversity of Electro-CommunicationsChofuJapan
  4. 4.Preeminent Medical Photonics Education and Reseach CenterHamamatsu University School of MedicineHamamatsuJapan

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