Abstract
This chapter discusses diffuse optical tomography. We present the origins of this method in terms of spectroscopic analysis of tissue using near-infrared light and its extension to an imaging modality. Models for light propagation at the macroscopic and mesoscopic scale are developed from the radiative transfer equation (RTE). Both time and frequency domain systems are discussed. Some formal results based on Green’s function models are presented, and numerical methods are described based on discrete finite element method (FEM) models and a Bayesian framework for image reconstruction. Finally, some open questions are discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References and Further Reading
Amaldi E (1959) The production and slowing down of neutrons. In Flügge S (ed) Encyclopedia of physics, vol 38/2. Springer, Berlin, pp 1–659
Aronson R (1995) Boundary conditions for diffusion of light. J Opt Soc Am A 12:2532–2539
Aydin ED (2007) Three-dimensional photon migration through voidlike regions and channels. Appl Opt 46(34):8272–8277
Aydin ED, de Oliveira CRE, Goddard AJH (2004) A finite element-spherical harmonics radiation transport model for photon migration in turbid media. J Quant Spectrosc Radiat Transf 84: 247–260
Bal G (2002) Transport through diffusive and nondiffusive regions, embedded objects, and clear layers. SIAM J Appl Math 62(5):1677–1697
Bal G (2006) Radiative transfer equation with varying refractive index: a mathematical perspective. J Opt Soc Am A 23:1639–1644
Bal G (2009) Inverse transport theory and applications. Inv Probl 25:053001 (48pp)
Bal G, Maday Y (2002) Coupling of transport and diffusion models in linear transport theory. Math Model Numer Anal 36(1):69–86
Bluestone AV, Abdoulaev G, Schmitz CH, Barbour RL, Hielscher AH (2001) Three-dimensionalopticaltomographyofhemodynamicsinthehumanhead.OptExpress 9(6):272–286
Contini D, Martelli F, Zaccanti G (1997) Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory Appl Opt 36(19):4587–4599
Dehghani H, Arridge SR, Schweiger M, Delpy DT (2000) Optical tomography in the presence of void regions. J Opt Soc Am A 17(9): 1659–1670
Fantini S, Franceschini MA, Gratton E (1997) Effective source term in the diffusion equation for photon transport in turbid media. Appl Opt 36(1):156–163
Ferwerda HA (1999) The radiative transfer equation for scattering media with a spatially varying refractive index. J Opt A Pure Appl Opt 1(3): L1–L2
Furutsu K (1980) Diffusion equation derived from space-time transport equation. J Opt Soc Am 70(4):360–366
Groenhuis RAJ, Ferwerda HA, Ten Bosch JJ (1983) Scattering and absorption of turbid materials determined from reflection measurements. Part 1: Theory Appl Opt 22(16):2456–2462
Hebden JC, Gibson A, Md Yusof R, Everdell N, Hillman EMC, Delpy DT, Arridge SR, Austin T, Meek JH, Wyatt JS (2002) Three-dimensional optical tomography of the premature infant brain. Phys Med Biol 47:4155–4166
Khan T, Jiang H (2003) A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices. J Opt A Pure Appl Opt 5:137–141
Kim AD, Ishimaru A (1998) Optical diffusion of continuos-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer. Appl Opt 37(22):5313–5319
Klose AD, Larsen EW (2006) Light transport in biological tissue based on the simplified spherical harmonics equations. J Comput Phys 220: 441–470
Kolehmainen V, Arridge SR, Vauhkonen M, Kaipio JP (2000) Simultaneous reconstruction of internal tissue region boundaries and coefficients in optical diffusion tomography. Phys Med Biol 45:3267–3283
Marti-Lopez L, Bouza-Dominguez J, Hebden JC, Arridge SR, Martinez-Celorio RA (2003) Validity conditions for the radiative transfer equation. J Opt Soc Am A 20(11):2046–2056
Wang LV (1998) Rapid modeling of diffuse reflectance of light in turbid slabs. J Opt Soc Am A 15(4):936–944
Wright S, Schweiger M, Arridge SR (2007) Reconstruction in optical tomography using the PN approximations. Meas Sci Technol 18:79–86
Ackroyd RT (1997) Finite element methods for particle transport : applications to reactor and radiation physics. Research Studies, Taunton
Anderson BDO, Moore JB (1979) Optimal filtering. Prentice Hall, Englewood Cliffs
Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15(2):R41–R93
Arridge SR, Cope M, Delpy DT (1992) Theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis. Phys Med Biol 37:1531–1560
Arridge SR, Dehghani H, Schweiger M, Okada E (2000) The finite element model for the propagation of light in scattering media: a direct method for domains with non-scattering regions. Med Phys 27(1):252–264
Arridge SR, Kaipio JP, Kolehmainen V, Schweiger M, Somersalo E, Tarvainen T, Vauhkonen M (2006) Approximation errors and model reduction with an application in optical diffusion tomography. Inverse Probl 22(1):175–196
Arridge SR, Lionheart WRB (1998) Non-uniqueness in diffusion-based optical tomography. Opt Lett 23:882–884
Arridge SR, Schotland JC (2009) Optical tomography: forward and inverse problems. Inverse Prob 25(12):123010 (59pp)
Arridge SR, Schweiger M, Hiraoka M, Delpy DT (1993) A finite element approach for modeling photon transport in tissue. Med Phys 20(2): 299–309
Arridge SR, Kaipio JP, Kolehmainen V, Schweiger M, Somersalo E, Tarvainen T, Vauhkonen M (2006) Approximation errors and model reduction with an application in optical diffusion tomography. Inverse Probl 22:175–195
Benaron DA, Stevenson DK (1993) Optical time-of-flight and absorbance imaging of biological media. Science 259:1463–1466
Berg R, Svanberg S, Jarlman O (1993) Medical transillumination imaging using short-pulse laser diodes. Appl Opt 32:574–579
Berger JO (2006) Statistical decision theory and Bayesian analysis. Springer, New York
Calvetti D, Kaipio JP, Somersalo E (2006) Aristotelian prior boundary conditions. Int J Math 1:63–81
Case MC, Zweifel PF (1967) Linear transport theory. Addison-Wesley, New York
Cope M, Delpy DT (1988) System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near infra- red transillumination. Med Biol Eng Comput 26:289–294
Cutler M (1929) Transillumination as an aid in the diagnosis of breast lesions. Surg Gynecol Obstet 48:721–729
Delpy DT, Cope M, van der Zee P, Arridge SR, Wray S, Wyatt J (1988) Estimation of optical pathlength through tissue from direct time of flight measurement. Phys Med Biol 33:1433–1442
Diamond SG, Huppert TJ, Kolehmainen V, Franceschini MA, Kaipio JP, Arridge SR, Boas DA (2006) Dynamic physiological modeling for functional diffuse optical tomography. Neuroimage 30:88–101
Dorn O (1997) Das inverse Transportproblem in der Lasertomographie. PhD thesis, University of MĂĽnster
Doucet A, de Freitas N, Gordon N (2001) Sequential Monte Carlo methods in practice. Springer, New York
Duderstadt JJ, Martin WR (1979) Transport theory. Wiley, New York
Durbin J, Koopman J (2001) Time series analysis by state space methods. Oxford University Press, Oxford
Firbank M, Arridge SR, Schweiger M, Delpy DT (1996) An investigation of light transport through scattering bodies with non-scattering regions. Phys Med Biol 41:767–783
Haskell RC, Svaasand LO, Tsay T-T, Feng T-C, McAdams MS, Tromberg BJ (1994) Boundary conditions for the diffusion equation in radiative transfer. J Opt Soc Am A 11(10):2727–2741
Hayashi T, Kashio Y, Okada E (2003) Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region. Appl Opt 42(16):2888–2896
Hebden JC, Kruger RA, Wong KS (1991) Time resolved imaging through a highly scattering medium. Appl Opt 30(7):788–794
Heino J, Somersalo E (2002) Estimation of optical absorption in anisotropic background. Inverse Prob 18:559–573
Heino J, Somersalo E (2004) A modelling error approach for the estimation of optical absorption in the presence of anisotropies. Phys Med Biol 49:4785–4798
Heino J, Somersalo E, Kaipio JP (2005) Compensation for geometric mismodelling by anisotropies in optical tomography. Opt Express 13(1):296–308
Henyey LG, Greenstein JL (1941) Diffuse radiation in the galaxy. AstroPhys J 93:70–83
Hielscher AH, Alcouffe RE, Barbour RL (1998) Comparison of finitedifference transport and diffusion calculations for photon migration in homogeneous and hetergeneous tissue. Phys Med Biol 43:1285–1302
Ho PP, Baldeck P, Wong KS, Yoo KM, Lee D, Alfano RR (1989) Time dynamics of photon migration in semiopaque random media. Appl Opt 28:2304–2310
Huttunen JMJ, Kaipio JP (2007) Approximation error analysis in nonlinear state estimation with an application to state-space identification. Inverse Prob 23:2141–2157
Huttunen JMJ, Kaipio JP (2007) Approximation errors in nostationary inverse problems. Inverse Prob Imaging 1(1):77–93
Huttunen JMJ, Kaipio JP (2009) Model reduction in state identification problems with an application to determination of thermal parameters. Appl Numer Math 59: 877–890
Huttunen JMJ, Lehikoinen A, Hämäläinen J, Kaipio JP (2009) Importance filtering approach for the nonstationary approximation error method. Inverse Prob in review
Ishimaru A (1978) Wave propagation and scattering in random media, vol 1. Academic, New York
Jarry G, Ghesquiere S, Maarek JM, Debray S, Bui M-H, Laurent HD (1984) Imaging mammalian tissues and organs using laser collimated transillumination. J Biomed Eng 6:70–74
Jöbsis FF (1977) Noninvasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters. Science 198: 1264–1267
Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New York
Kaipio J, Somersalo E (2007) Statistical and computational inverse problems. J Comput Appl Math 198:493–504
Kaipio JP, Kolehmainen V, Vauhkonen M, Somersalo E (1999) Inverse problems with structural prior information. Inverse Probl 15:713–729
Kak AC, Slaney M (1987) Principles of computerized tomographic imaging. IEEE, New York
Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME. J Basic Eng 82D(1):35–45
Kolehmainen V, Prince S, Arridge SR, Kaipio JP (2000) A state estimation approach to non-stationary optical tomography problem. J Opt Soc Am A 20:876–884
Kolehmainen V, Schweoger M, Nissilä I, Tarvainen T, Arridge SR, Kaipio JP (2009) Approximation errors and model reduction in three-dimensional optical tomography. J Optical Soc Amer A 26:2257–2268
Kolehmainen V, Tarvainen T, Arridge SR, Kaipio JP (2010) Marginalization of uninteresting distributed parameters in inverse problems – application to diffuse optical tomography. Int J Uncertainty Quantification, In press
Lakowicz JR, Berndt K (1990) Frequency domain measurement of photon migration in tissues. Chem Phys Lett 166(3):246–252
Lehikoinen A, Finsterle S, Voutilainen A, Heikkinen LM, Vauhkonen M, Kaipio JP (2007) Approximation errors and truncation of computational domains with application to geophysical tomography. Inverse Probl Imaging 1: 371–389
Lehikoinen A, Huttunen JMJ, Finsterle S, Kowalsky MB, Kaipio JP: Dynamic inversion for hydrological process monitoring with electrical resistance tomography under model uncertainties, Water Resour Res 46: W04513, doi:10.1029/2009WR008470, 2010
Mitic G, Kolzer J, Otto J, Plies E, Solkner G, Zinth W (1994) Timegated transillumination of biological tissue and tissuelike phantoms. Opt Lett 33:6699–6710
Natterer F, WĂĽbbeling F (2001) Mathematical methods in image reconstruction. SIAM, Philadelphia
Nissilä I, Noponen T, Kotilahti K, Tarvainen T, Schweiger M, Lipiänen L, Arridge SR, Katila T (2005) Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography. Rev Sci Instrum 76(4):004302
Nissinen A, Heikkinen LM, Kolehmainen V, Kaipio JP (2009) Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography. Meas Sci Technol 20, doi: 10.1088/0957–0233/20/10/105504
Nissinen A, Kolehmainen V, Kaipio JP: Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography, IEEE Trans Med Imaging, in review, 2010.
Nissinen A, Heikkinen LM, Kaipio JP (2008) Approximation errors in electrical impedance tomography – an experimental study. Meas Sci Technol 19, doi: 10.1088/0957-0233/19/1/015501
Ntziachristos V, Ma X, Chance B (1998) Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography. Rev Sci Instrum 69:4221–4233
Okada E, Schweiger M, Arridge SR, Firbank M, Delpy DT (1996) Experimental validation of Monte Carlo and Finite-Element methods for the estimation of the optical path length in inhomogeneous tissue. Appl Opt 35(19):3362–3371
Prince S, Kolehmainen V, Kaipio JP, Franceschini MA, Boas D, Arridge SR (2003) Time series estimation of biological factors in optical diffusion tomography. Phys Med Biol 48(11): 1491–1504
Schmidt A, Corey R, Saulnier P (1995) Imaging through random media by use of low-coherence optical heterodyning. Opt Lett 20:404–406Â
Schmidt FEW, Fry ME, Hillman EMC, Hebden JC, Delpy DT (2000) A 32-channel time-resolved instrument for medical optical tomography. Rev Sci Instrum 71(1):256–265
Schmitt JM, Gandbjbakhche AH, Bonner RF (1992) Use of polarized light to discriminate short-path photons in a multiply scattering medium. Appl Opt 31:6535–6546
Schotland JC, Markel V (2001) Inverse scattering with diffusing waves. J Opt Soc Am A 18: 2767–2777
Schweiger M, Arridge SR (1997) The finite element model for the propagation of light in scattering media: frequency domain case. Med Phys 24(6):895–902
Schweiger M, Arridge SR, Hiraoka M, Delpy DT (1995) The finite element model for the propagation of light in scattering media: boundary and source conditions. Med Phys 22(11): 1779–1792
Schweiger M, Arridge SR, Nissilä I (2005) Gauss–Newton method for image reconstruction in diffuse optical tomography. Phys Med Biol 50:2365–2386
Schweiger M, Nissilä I, Boas DA, Arridge SR (2007) Image reconstruction in optical tomography in the presence of coupling errors. Appl Opt 46(14):2743–2756
Spears KG, Serafin J, Abramson NH, Zhu X, Bjelkhagen H (1989) Chronocoherent imaging for medicine. IEEE Trans Biomed Eng 36: 1210–1221
Sylvester J, Uhlmann G (1987) A global uniquness theorem for an inverse boundary value problem. Ann Math 125:153–169
Tarvainen T, Kolehmainen V, Pulkkinen A, Vauhkonen M, Schweiger M, Arridge SR, Kaipio JP (2010) Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography. Inverse Probl 26, doi: 10.1088/0266–5611/ 26/1/015005
Tarvainen T, Vauhkonen M, Kolehmainen V, Arridge SR, Kaipio JP (2005) Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions. Phys Med Biol 50:4913–4930
Tarvainen T, Vauhkonen M, Kolehmainen V, Kaipio JP (2005) A hybrid radiative transfer – diffusion model for optical tomography. Appl Opt 44(6):876–886
Tarvainen T, Vauhkonen M, Kolehmainen V, Kaipio JP (2006) Finite element model for the coupled radiative transfer equation and diffusion approximation. Int J Numer Meth Engng 65(3):383–405
Tervo J, Kolmonen P, Vauhkonen M, Heikkinen LM, Kaipio JP (1999) A finite-element model of electron transport in radiation therapy and a related inverse problem. Inverse Prob 15: 1345–1362
Wang L, Ho PP, Liu C, Zhang G, Alfano RR (1991) Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate. Science 253:769–771
Wang L, Jacques SL (1993) Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media. J Opt Soc Am A 10(8):1746–1752
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this entry
Cite this entry
Arridge, S.R., Kaipio, J.P., Kolehmainen, V., Tarvainen, T. (2011). Optical Imaging. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_17
Download citation
DOI: https://doi.org/10.1007/978-0-387-92920-0_17
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-92919-4
Online ISBN: 978-0-387-92920-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering