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Light transport in biological tissue based on the simplified spherical harmonics equations. (English) Zbl 1122.78015
Light propagation models in biomedical optics are essential for tomographic imaging of biological tissues using visible and infrared light. These models predict light intensities which can be compared to actually measured light intensities on the tissue boundary. Combinations of predicted and measured data are the basis of image reconstruction algorithms. Light transport in scattering media is modeled in terms of suitable radiative transfer equations whose solvability is a major endavour. Discrete ordinates and spherical harmonics equations are among the reliable approximation methods to this end.
The main goal of the paper is to transfer from the neutron transport theory to the biomedical optics, the so-called simplified spherical harmonics approximation method. Suitable equations are formulated with partially reflective boundary conditions in anisotropically scattering media. Solutions are found to give better results than the standard diffusion model methods. The respective equations are numerically solved for the two-dimensional media mimicking typical small tissue properties. There is a significant computational time gain incurred, in comparison with the standard transport calculations.

78A70 Biological applications of optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
62P10 Applications of statistics to biology and medical sciences; meta analysis
92C55 Biomedical imaging and signal processing
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
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[1] Gibson, A.P.; Hebden, J.C.; Arridge, S.R., Recent advances in diffuse optical imaging, Phys. med. biol., 50, R1-R43, (2005)
[2] Bluestone, A.Y.; Abdoulaev, G.; Schmitz, C.; Barbour, R.L.; Hielscher, A.H., Three-dimensional optical tomography of hemodynamics in the human head, Opt. exp., 9, 6, 272-286, (2001)
[3] Hielscher, A.H.; Klose, A.D.; Scheel, A.K.; Moa-Anderson, B.; Backhaus, M.; Netz, U.; Beuthan, J., Sagittal laser optical tomography for imaging of rheumatoid finger joints, Phys. med. biol., 49, 1147-1163, (2004)
[4] Hielscher, A.H., Optical tomographic imaging of small animals, Curr. opin. biotechnol., 16, 79-88, (2005)
[5] Weissleder, R.; Mahmood, U., Molecular imaging, Radiology, 219, 316-333, (2001)
[6] Welch, A.J.; van Gemert, M.J.C., Optical-thermal response of laser-irradiated tissue, (1995), Plenum Press New York
[7] Cheong, W.F.; Prahl, S.A.; Welch, A.J., A review of the optical properties of biological tissue, IEEE J. quantum electron., 26, 2166-2185, (1990)
[8] Tuchin, V., Tissue optics, (2000), SPIE Bellingham, WA
[9] Hielscher, A.H.; Alcouffe, R.E.; Barbour, R.L., Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues, Phys. med. biol., 43, 1285-1302, (1998)
[10] Aydin, E.D.; de Oliveira, C.R.E.; Goddard, A.J.H., A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method, Med. phys., 29, 9, 2013-2023, (2002)
[11] Rice, B.W.; Cable, M.D.; Nelson, M.B., In vivo imaging of light-emitting probes, J. biomed. opt., 6, 4, 432-440, (2001)
[12] Troy, T.; Jekic-McMullen, D.; Sambucetti, L.; Rice, B., Quantitative comparison of the sensitivity of detection of fluorescent and bioluminescent reporters in animal models, Mol. imaging, 3, 1, 9-23, (2004)
[13] Viviani, V.R.; Oehlmeyer, T.L.; Arnoldi, F.G.C.; Brochetto-Braga, M.R., A new firefly luciferase with bimodal spectrum: identification of structural determinants of spectral ph-sensitivity in firefly luciferases, Photochem. photobiol., 81, 843-848, (2005)
[14] Zhao, H.; Doyle, T.C.; Coquoz, O.; Kalish, F.; Rice, B.; Contag, C.H., Emission spectra of bioluminescent reporters and interaction with Mammalian tissue determine the sensitivity of detection in vivo, J. biomed. opt., 10, 4, 041210-1-041210-9, (2005)
[15] Case, K.M.; Zweifel, P.F., Linear transport theory, (1967), Addison-Wesley Reading, MA · Zbl 0132.44902
[16] Adams, M.L.; Larsen, E.W., Fast iterative methods for discrete-ordinates particle transport calculations, Prog. nucl. energy, 40, 1, 3-159, (2002)
[17] Fletcher, J.K., The solution of the multigroup neutron transport equation using spherical harmonics, Nucl. sci. eng., 84, 33-46, (1983) · Zbl 0514.65096
[18] Kobayashi, K.; Oigawa, H.; Yamagata, H., The spherical harmonics method for the multigroup transport equations in x-y geometry, Ann. nucl. energy, 13, 12, 663-678, (1986)
[19] Klose, A.D.; Ntziachristos, V.; Hielscher, A.H., The inverse source problem based on the radiative transfer equation in optical molecular imaging, J. comput. phys., 202, 323-345, (2005) · Zbl 1061.65143
[20] E.M. Gelbard, Application of spherical harmonics methods to reactor problems, WAPD-BT-20, Bettis Atomic Power Laboratory, 1960.
[21] Gelbard, E.M.; Davis, J.; Pearson, J., Iterative solutions to the Pl and double pl equations, Nucl. sci. eng., 5, 36-44, (1959)
[22] Tomasevic, D.I.; Larsen, E.W., The simplified P2 approximation, Nucl. sci. eng., 122, 309-325, (1996)
[23] Larsen, E.W.; Morel, J.E.; McGhee, J.M., Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering, Nucl. sci. eng., 123, 328, (1996)
[24] Ackroyd, R.T.; de Oliveira, C.R.E.; Zolfaghari, A.; Goddard, A.J.H., On a rigorous resolution of the transport equation into a system of diffusion-like equations, Prog. nucl. eng., 35, 1, 1-64, (1999)
[25] Brantley, P.S.; Larsen, E.W., The simplified P3 approximation, Nucl. sci. eng., 134, 1-21, (2000)
[26] Morel, J.E.; McGhee, J.M.; Larsen, E.W., A three-dimensional time-dependent unstructured tetrahedral-mesh SPN method, Nucl. sci. eng., 123, 319-327, (1996)
[27] Kotiluoto, P., Fast tree multigrid transport application for the simplified P3 approximation, Nucl. sci. eng., 138, 269-278, (2001)
[28] Kotiluoto, P.; Hiismáki, P., Application of the new multitrans SP3 radiation transport code in BNCT dose planning, Med. phys., 28, 9, 1905-1910, (2001)
[29] Ciolini, R.; Coppa, G.G.M.; Montagnini, B.; Ravetto, P., Simplified PN and AN methods in neutron transport, Prog. nucl. energy, 40, 2, 237-264, (2002)
[30] Josef, J.A.; Morel, J.E., Simplified spherical harmonic method for coupled electron – photon transport calculations, Phys. rev. E, 57, 5, 6161-6171, (1998)
[31] Larsen, E.W.; Thoemmes, G.; Klar, A.; Deaid, M.; Goetz, T., Simplified PN approximations to the equations of radiative heat transfer and applications, J. comput. phys., 183, 652-675, (2002)
[32] Lemanska, M., On the simplified Pn method in the 2-D diffusion code EXTERMINATOR, Atomkernenergie, 37, 173-175, (1981)
[33] Lewis, E.E.; Palmiotti, G., Simplified spherical harmonics in the variational nodal method, Nucl. sci. eng., 126, 48-58, (1997)
[34] Kienle, A.; Forster, F.K.; Hibst, R., Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance, Opt. lett., 26, 20, 1571-1573, (2001)
[35] Sharma, S.K.; Banerjee, S., Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues, J. opt. A, 5, 294-302, (2003)
[36] Marshak, R.E., Note on the spherical harmonic method as applied to the Milne problem for a sphere, Phys. rev., 71, 7, 443-446, (1947) · Zbl 0032.37504
[37] Keijzer, M.; Star, W.M.; Storchi, P.R.M., Optical diffusion in layered media, Appl. opt., 27, 9, 1820-1824, (1988)
[38] Haskell, R.C.; Svaasand, L.O.; Tsay, T.-T.; Feng, T.-C.; McAdams, M.S.; Tromberg, B.J., Boundary conditions for the diffusion equation in radiative transfer, J. opt. soc. am. A, 11, 10, 2727-2741, (1994)
[39] Spott, T.; Svaasand, L.O., Collimated light sources in the diffusion approximation, Appl. opt., 39, 34, 6453-6465, (2000)
[40] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in C: the art of scientific computing, (1994), Cambridge University Press Cambridge
[41] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations, (1994), Cambridge University Press Cambridge · Zbl 0811.65063
[42] Lathrop, K.D., Spatial differencing of the transport equation: positivity vs. accuracy, J. comput. phys., 4, 475-498, (1969) · Zbl 0199.50703
[43] Duderstadt, J.J.; Martin, W.R., Transport theory, (1979), Wiley New York · Zbl 0407.76001
[44] Lacroix, D.; Berour, N.; Boulet, P.; Jeandel, G., Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions, Heat mass transfer, 42, 322-337, (2006)
[45] Bluestone, A.Y.; Stewart, M.; Lei, B.; Kass, I.S.; Lasker, J.; Abdoulaev, G.S.; Hielscher, A.H., Three-dimensional optical tomographic brain imaging in small animals, part I: hypercapnia, J. biomed. opt., 9, 5, 1046-1062, (2004)
[46] Bluestone, A.Y.; Stewart, M.; Lasker, J.; Abdoulaev, G.S.; Hielscher, A.H., Three-dimensional optical tomographic brain imaging in small animals, part II: unilateral carotid occlusion, J. biomed. opt., 9, 5, 1063-1073, (2004)
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