A comparison between transport and diffusion calculations using a ﬁnite element-spherical harmonics radiation transport method E. D. Aydin a) Canakkale Onsekiz Mart University, Faculty of Arts and Sciences, Department of Physics, Terzioglu Campus, 17100, Canakkale, Turkey C. R. E. de Oliveira and A. J. H. Goddard Computational Physics and Geophysics Group, Department of Earth Sciences and Engineering, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BP, United Kingdom Received 4 September 2001; accepted for publication 14 June 2002; published 12 August 2002 Most researchers choose the diffusion approximation to the transport equation as the model to describe photon migration in biological tissues. However, the applicability of this approximation is limited and, in certain cases, invalid. In this paper we introduce a two-dimensional, ﬁnite element- spherical harmonics FE-P N  radiation transport method for the simulation of light propagation in tissue. The propagation of light is investigated ﬁrst in a layered cylinder, which can be seen as a very simplistic approximation of a human head. Effects of the anisotropy factor g on the photon migration is then examined in homogeneous and heterogeneous media for different values of g and  s . The inﬂuence of void-like heterogeneities and channels in which absorption and scattering are very small compared with the surrounding medium on the transport of photons is also investigated. Signiﬁcant differences between transport and diffusion calculations are shown to occur in all cases. © 2002 American Association of Physicists in Medicine. DOI: 10.1118/1.1500404 Key words: photon migration, Boltzmann transport equation, ﬁnite elements, spherical harmonics, anisotropic scattering I. INTRODUCTION The propagation of light in tissue is a question of growing concern in many medical applications such as disease diag- nosis and medical imaging using optical techniques. 1,2 The diffusion approximation to the transport equation has be- come a widely used model for the migration of photons in turbid media, especially for the problems with large propa- gation distances. 3,4 Furthermore, solutions for the inverse problems of photon propagation in the literature are almost exclusively based on the diffusion theory approximations. 5–7 However, it is well known that, in order to apply the diffu- sion approximation, photon scattering must be the dominant process in the material. 8,9 This can be expressed mathemati- cally as  a  s where  a is the absorption coefﬁcient and  s is the scattering coefﬁcient. This condition is generally true for tissue, where the typical parameter ranges are  a =0.01– 0.1 mm -1 and  s =10– 100 mm -1 , when the near- infrared NIR wavelength that is used to probe the tissue is between 600 and 900 nm. The diffusion theory, therefore, becomes an appropriate approximation for many biomedical applications at these wavelengths. 10 Although media such as skin, bone, brain matter, and breast tissue satisfy the above-given condition, there also ex- ist low-scattering almost clear regions, such as cerebrospinal- ﬂuid-ﬁlled spaces in the brain or the synovial-ﬂuid-ﬁlled space in joints. It has been shown in the literature that the diffusion theory fails to accurately describe the light propa- gation in these regions. 11,9 Moreover, the problems involving very thin tissue layers cannot correctly be described by this approximation due to the fact that they may not be scatter dominated. 12,13 Another problem with diffusion theory oc- curs at strong discontinuities regions where the optical prop- erties change dramatically from the surrounding region. 14 In addition, description of an anisotropic source and anisotropic photon scattering cannot directly be taken into account within this theory. 15,16 To model anisotropic scattering sev- eral researchers have applied Monte Carlo MC methods. 17–20 Although MC simulations provide essentially correct solutions, these simulations are too slow to be used for realistic geometries large and optically dense volumes, such as the breast and brain. Therefore, they have normally been limited to small tissue volumes. Among the researchers, Boas 21 employed the P 3 approxi- mation for an inﬁnite homogeneous media. He determined optical properties of highly absorbing media solving the frequency-domain diffusion equation and transport equation within P 3 approximation. He found that, in most cases, the P 3 approximation permits a more accurate determination of the optical properties of highly absorbing media. Hielscher et al. 22,23 employed a ﬁnite-difference discrete-ordinate transport method for the calculation of photon migration in biological tissues. They also reported the differences be- tween transport and diffusion solution for various homoge- neous and heterogeneous tissue-like media. A number of studies have been performed to investigate the propagation of light especially in various models of the adult head including nonscattering region CSF. Okada et al. 24 employed models consisting of three- or four-layered 2013 2013 Med. Phys. 29 „9…, September 2002 0094-2405Õ2002Õ29„9…Õ2013Õ11Õ$19.00 © 2002 Am. Assoc. Phys. Med.