Abstract

Light propagation in tissue is known to be favored in the Near Infrared spectral range. Capitalizing on this fact, new classes of molecular contrast agents are engineered to fluoresce in the Near Infrared. The potential of these new agents is vast as it allows tracking non-invasively and quantitatively specific molecular events in-vivo. However, to monitor the bio-distribution of such compounds in thick tissue proper physical models of light propagation are necessary. To recover 3D concentrations of the compound distribution, it is necessary to perform a model based inverse problem: Diffuse Optical Tomography. In this work, we focus on Fluorescent Diffuse Optical Tomography expressed within the normalized Born approach. More precisely, we investigate the performance of Fluorescent Diffuse Optical Tomography in the case of time resolved measurements. The different moments of the time point spread function were analytically derived to construct the forward model. The derivation was performed from the zero order moment to the second order moment. This new forward model approach was validated with simulations based on relevant configurations. Enhanced performance of Fluorescent Diffuse Optical Tomography was achieved using these new analytical solutions when compared to the current formulations.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. A Yodh and B. Chance, �??Spectroscopy and imaging with diffusing light,�?? Phys. Today 48, 34-40 (1995).
    [CrossRef]
  2. X. Intes and B. Chance, �??Non-PET Functional Imaging Techniques Optical,�?? Clin. No. Am. 43, 221-234 (2005).
  3. F. Jobsis, �??Noninvasive infrared monitoring of cerebral and myocardial sufficiency and circulatory parameters,�?? Science 198, 1264-1267 (1977).
    [CrossRef] [PubMed]
  4. Y. Lin, G. Lech, S. Nioka, X. Intes and B. Chance, �??Noninvasive, low-noise, fast imaging of blood volume and deoxygenation changes in muscles using light-emitting diode continuous-wave imager,�?? Rev. Sci. Instrum. 73, 3065-3074 (2002).
    [CrossRef]
  5. Y. Chen, C. Mu, X. Intes, D. Blessington and B. Chance, �??Frequency domain phase cancellation instrument for fast and accurate localization of fluorescent heterogeneity,�?? Rev. Sci. Instrum. 74, 3466-3473 (2003).
    [CrossRef]
  6. B. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, et al., �??Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,�?? Neoplasia 2, 26-40 (2000).
    [CrossRef] [PubMed]
  7. D. Grosenick, H. Wabnitz, K. Moesta, J. Mucke, M. Moller, C. Stroszczunski, J. Stobel, B. Wassermann, P. Schlag and H. Rinnerberg, �??Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography,�?? Phys Med. Biol. 49, 1165-1181 (2004).
    [CrossRef] [PubMed]
  8. H. Jiang, N. Iftimia, J. Eggert, L. Fajardo and K. Klove, �??Near-infrared optical imaging of the breast with model-based reconstruction,�?? Acad. Radiol. 9, 186-194 (2002).
    [CrossRef] [PubMed]
  9. M. Franceschini, K. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, et al., �??Frequency-domain techniques enhance optical mammography: Initial clinical results,�?? Proc. Nat. Acad. Sci. Am. 94, 6468-6473 (1997).
    [CrossRef]
  10. S. Colak, M. van der Mark, G. Hooft, J. Hoogenraad, E. van der Linden, F. Kuijpers, �??Clinical optical tomography and NIR spectroscopy for breast cancer detection,�?? IEEE J. Sel. Top. Quatum Electron. 5, 1143-1158 (1999).
    [CrossRef]
  11. X. Intes, S. Djeziri, Z. Ichalalene, N. Mincu, Y. Wang, P. St.-Jean, F. Lesage, D. Hall, D. A. Boas, M. Polyzos, �??Time-Domain Optical Mammography Softscan®: Initial Results on Detection and Characterization of Breast Tumors�??, Proc. SPIE 5578, 188-197 (2004).
    [CrossRef]
  12. D. B. Jakubowski, A. E. Cerussi, F. Bevilacqua, N. Shah, D. Hsiang, J. Butler, B. J. Tromberg, �??Monitoring neoadjuvant chemotherapy in breast cancer using quantitative diffuse optical spectroscopy: a case study,�?? J. Biomed Opt. 9, 230-238 (2004).
    [CrossRef] [PubMed]
  13. G. Strangman, D. A. Boas, J. Sutton, �??Non-invasive neuroimaging using Near-Infrared light,�?? Biol. Psychiatry 52, 679-693 (2002).
    [CrossRef] [PubMed]
  14. Y. Chen, D. Tailor, X. Intes and B. Chance, �??Quantitative correlation between Near-Infrared spectroscopy (NIRS) and magnetic resonance imaging (MRI) on rat brain oxygenation modulation,�?? Phys. Med. Biol. 48, 417-427 (2003).
    [CrossRef] [PubMed]
  15. M. Stankovic, D. Maulik, W. Rosenfeld, P. Stubblefield, A. Kofinas, E. Gratton, et al., �??Role of frequency domain optical spectroscopy in the detection of neonatal brain hemorrhage- a newborn piglet study,�?? J. Matern. Fetal Med. 9, 142-149 (2000).
    [CrossRef] [PubMed]
  16. J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, �??Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,�?? Phys Med Biol. 49, 1117-1130 (2004).
    [CrossRef] [PubMed]
  17. V. Quaresima, R. Lepanto and M. Ferrari, �??The use of near infrared spectroscopy in sports medicine,�?? J. Sports Med. Phys. Fitness 43, 1-13 (2003).
    [PubMed]
  18. X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh and B. Chance, �??In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,�?? Med. Phys. 30, 1039-1047 (2003).
    [CrossRef] [PubMed]
  19. R. Weissleder and U. Mahmood, �??Molecular imaging,�?? Radiology 219, 316-333 (2001).
    [PubMed]
  20. J. V. Frangioni, �??In vivo near-infrared fluorescence imaging,�?? Curr. Opin. Chem. Biol. 7, 626�??634 (2003).
    [CrossRef] [PubMed]
  21. K. Licha, �??Contrast agents for optical imaging,�?? Topics in Current Chemistry 222, 1-29 (2002).
    [CrossRef]
  22. G. Zheng, Y. Chen, X. Intes, B. Chance and J. Glickson, �??Contrast-Enhanced NIR Optical Imaging for subsurface cancer detection,�?? J. Porphyrin and Phthalocyanines 8, 1106- 1118 (2004).
    [CrossRef]
  23. S. Achilefu, R. Dorshow, J. Bugaj and R. Rajagopalan, �??Novel receptor-targeted fluorescent contrast agents for in-vivo tumor imaging,�?? Invest. Radiol. 35, 479-485 (2000).
    [CrossRef] [PubMed]
  24. Y. Chen, G. Zheng, Z. Zhang, D. Blessington, M. Zhang, H. Li, et al., �??Metabolism Enhanced Tumor Localization by Fluorescence Imaging: In Vivo Animal Studies,�?? Opt. Lett. 28, 2070-2072 (2003).
    [CrossRef] [PubMed]
  25. R. Weissleder, C. H. Tung, U. Mahmood, A. Bogdanov, �??In vivo imaging with protease-activated near-infrared fluorescent probes,�?? Nat. Biotech. 17, 375-378 (1999).
    [CrossRef]
  26. R. Weinberg, �??How Does Cancer Arise,�?? 275, 62-71 (1996).
    [CrossRef] [PubMed]
  27. X. Intes, Y. Chen, X. Li and B. Chance, �??Detection limit enhancement of fluorescent heterogeneities in turbid media by dual-interfering excitation,�?? Appl. Opt. 41, 3999-4007 (2002).
    [CrossRef] [PubMed]
  28. J. Lewis, S. Achilefu, J. R. Garbow, R. Laforest, M. J. Welch, �??Small animal imaging: current technology and perspectives for oncological imaging,�?? Eur. Jour. Cancer 38, 2173�??88 (2002).
    [CrossRef]
  29. V. Ntziachristos and R. Weissleder, �??Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,�?? Opt. Lett. 26, 893-895 (2001).
    [CrossRef]
  30. M. J. Eppstein, D. J. Hawrysz, A. Godavarty and E. M. Sevick-Muraca, �??Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: Near-infrared fluorescence tomography,�?? Proc. Nat. Acad. Sci. Am. 99, 9619-9624 (2002).
    [CrossRef]
  31. A. B. Milstein, J.J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, �??Fluorescence optical diffusion tomography using multiple-frequency data,�?? J. Opt. Soc. Am. A 21, 1035-1049 (2004).
    [CrossRef]
  32. X. Li, �??Fluorescence and diffusive wave diffraction tomographic probes in turbid media,�?? PhD University of Pennsylvania (1996).
  33. M. O�??Leary, �??Imaging with diffuse photon density waves,�?? PhD University of Pennsylvania (1996).
  34. E. Hillman, �??Experimental and theoretical investigations of near infrared tomographic imaging methods and clinical applications,�?? PhD University College London (2002).
  35. A. Liebert, H. Wabnitz, D. Grosenick, M. Moller, R. Macdonald and H. Rinnerberg, �??Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons,�?? Appl. Opt. 42, 5785-5792 (2003).
    [CrossRef] [PubMed]
  36. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams and B. J. Tromberg, �??Boundary conditions for the diffusion equation in radiative transfer,�?? J. Opt. Soc. Am A 11, 2727-41 (1994).
    [CrossRef]
  37. R. Gordon, R. Bender and G. Herman, �??Algebraic reconstruction techniques (ART) for the three dimensional electron microscopy and X-Ray photography,�?? J. Theoret. Biol. 69, 471-482 (1970).
    [CrossRef]
  38. A. Kak and M. Slaney, �??Computerized tomographic Imaging�??, IEEE Press, N.Y. (1987).
  39. D. Ros, C. Falcon, I. Juvells and J. Pavia, �??The influence of a relaxation parameter on SPECT iterative reconstruction algorithms,�?? Phys. Med. Biol. 41, 925-937 (1996).
    [CrossRef] [PubMed]
  40. X. Intes, V. Ntziachristos, J. Culver, A. G. Yodh and B. Chance, �??Projection access order in Algebraic Reconstruction Techniques for Diffuse Optical Tomography,�?? Phys. Med. Biol. 47, N1-N10 (2002).
    [CrossRef] [PubMed]
  41. E. Graves, J. Culver, J. Ripoll, R. Weissleder and V. Ntziachristos, �??Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,�?? J. Opt. Soc. Am. A 21, 231-241 (2004).
    [CrossRef]

Acad. Radiol. (1)

H. Jiang, N. Iftimia, J. Eggert, L. Fajardo and K. Klove, �??Near-infrared optical imaging of the breast with model-based reconstruction,�?? Acad. Radiol. 9, 186-194 (2002).
[CrossRef] [PubMed]

Appl. Opt. (2)

Biol. Psychiatry (1)

G. Strangman, D. A. Boas, J. Sutton, �??Non-invasive neuroimaging using Near-Infrared light,�?? Biol. Psychiatry 52, 679-693 (2002).
[CrossRef] [PubMed]

Clin. No. Am. (1)

X. Intes and B. Chance, �??Non-PET Functional Imaging Techniques Optical,�?? Clin. No. Am. 43, 221-234 (2005).

Curr. Opin. Chem. Biol. (1)

J. V. Frangioni, �??In vivo near-infrared fluorescence imaging,�?? Curr. Opin. Chem. Biol. 7, 626�??634 (2003).
[CrossRef] [PubMed]

Eur. Jour. Cancer 38, 2173???88 (2002). (1)

J. Lewis, S. Achilefu, J. R. Garbow, R. Laforest, M. J. Welch, �??Small animal imaging: current technology and perspectives for oncological imaging,�?? Eur. Jour. Cancer 38, 2173�??88 (2002).
[CrossRef]

IEEE J. Sel. Top. Quatum Electron. (1)

S. Colak, M. van der Mark, G. Hooft, J. Hoogenraad, E. van der Linden, F. Kuijpers, �??Clinical optical tomography and NIR spectroscopy for breast cancer detection,�?? IEEE J. Sel. Top. Quatum Electron. 5, 1143-1158 (1999).
[CrossRef]

Invest. Radiol. (1)

S. Achilefu, R. Dorshow, J. Bugaj and R. Rajagopalan, �??Novel receptor-targeted fluorescent contrast agents for in-vivo tumor imaging,�?? Invest. Radiol. 35, 479-485 (2000).
[CrossRef] [PubMed]

J. Biomed Opt. (1)

D. B. Jakubowski, A. E. Cerussi, F. Bevilacqua, N. Shah, D. Hsiang, J. Butler, B. J. Tromberg, �??Monitoring neoadjuvant chemotherapy in breast cancer using quantitative diffuse optical spectroscopy: a case study,�?? J. Biomed Opt. 9, 230-238 (2004).
[CrossRef] [PubMed]

J. Matern. Fetal Med. (1)

M. Stankovic, D. Maulik, W. Rosenfeld, P. Stubblefield, A. Kofinas, E. Gratton, et al., �??Role of frequency domain optical spectroscopy in the detection of neonatal brain hemorrhage- a newborn piglet study,�?? J. Matern. Fetal Med. 9, 142-149 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am A (1)

R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams and B. J. Tromberg, �??Boundary conditions for the diffusion equation in radiative transfer,�?? J. Opt. Soc. Am A 11, 2727-41 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Porphyrin and Phthalocyanines (1)

G. Zheng, Y. Chen, X. Intes, B. Chance and J. Glickson, �??Contrast-Enhanced NIR Optical Imaging for subsurface cancer detection,�?? J. Porphyrin and Phthalocyanines 8, 1106- 1118 (2004).
[CrossRef]

J. Sports Med. Phys. Fitness (1)

V. Quaresima, R. Lepanto and M. Ferrari, �??The use of near infrared spectroscopy in sports medicine,�?? J. Sports Med. Phys. Fitness 43, 1-13 (2003).
[PubMed]

J. Theoret. Biol. (1)

R. Gordon, R. Bender and G. Herman, �??Algebraic reconstruction techniques (ART) for the three dimensional electron microscopy and X-Ray photography,�?? J. Theoret. Biol. 69, 471-482 (1970).
[CrossRef]

Med. Phys. (1)

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh and B. Chance, �??In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,�?? Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

Nat. Biotech. (1)

R. Weissleder, C. H. Tung, U. Mahmood, A. Bogdanov, �??In vivo imaging with protease-activated near-infrared fluorescent probes,�?? Nat. Biotech. 17, 375-378 (1999).
[CrossRef]

Neoplasia (1)

B. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, et al., �??Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,�?? Neoplasia 2, 26-40 (2000).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys Med Biol. (1)

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek and J. S. Wyatt, �??Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography,�?? Phys Med Biol. 49, 1117-1130 (2004).
[CrossRef] [PubMed]

Phys Med. Biol. (1)

D. Grosenick, H. Wabnitz, K. Moesta, J. Mucke, M. Moller, C. Stroszczunski, J. Stobel, B. Wassermann, P. Schlag and H. Rinnerberg, �??Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography,�?? Phys Med. Biol. 49, 1165-1181 (2004).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

Y. Chen, D. Tailor, X. Intes and B. Chance, �??Quantitative correlation between Near-Infrared spectroscopy (NIRS) and magnetic resonance imaging (MRI) on rat brain oxygenation modulation,�?? Phys. Med. Biol. 48, 417-427 (2003).
[CrossRef] [PubMed]

D. Ros, C. Falcon, I. Juvells and J. Pavia, �??The influence of a relaxation parameter on SPECT iterative reconstruction algorithms,�?? Phys. Med. Biol. 41, 925-937 (1996).
[CrossRef] [PubMed]

X. Intes, V. Ntziachristos, J. Culver, A. G. Yodh and B. Chance, �??Projection access order in Algebraic Reconstruction Techniques for Diffuse Optical Tomography,�?? Phys. Med. Biol. 47, N1-N10 (2002).
[CrossRef] [PubMed]

Phys. Today (1)

A Yodh and B. Chance, �??Spectroscopy and imaging with diffusing light,�?? Phys. Today 48, 34-40 (1995).
[CrossRef]

Proc. Nat. Acad. Sci. Am. (2)

M. Franceschini, K. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, et al., �??Frequency-domain techniques enhance optical mammography: Initial clinical results,�?? Proc. Nat. Acad. Sci. Am. 94, 6468-6473 (1997).
[CrossRef]

M. J. Eppstein, D. J. Hawrysz, A. Godavarty and E. M. Sevick-Muraca, �??Three-dimensional, Bayesian image reconstruction from sparse and noisy data sets: Near-infrared fluorescence tomography,�?? Proc. Nat. Acad. Sci. Am. 99, 9619-9624 (2002).
[CrossRef]

Proc. SPIE (1)

X. Intes, S. Djeziri, Z. Ichalalene, N. Mincu, Y. Wang, P. St.-Jean, F. Lesage, D. Hall, D. A. Boas, M. Polyzos, �??Time-Domain Optical Mammography Softscan®: Initial Results on Detection and Characterization of Breast Tumors�??, Proc. SPIE 5578, 188-197 (2004).
[CrossRef]

Radiology (1)

R. Weissleder and U. Mahmood, �??Molecular imaging,�?? Radiology 219, 316-333 (2001).
[PubMed]

Rev. Sci. Instrum. (2)

Y. Lin, G. Lech, S. Nioka, X. Intes and B. Chance, �??Noninvasive, low-noise, fast imaging of blood volume and deoxygenation changes in muscles using light-emitting diode continuous-wave imager,�?? Rev. Sci. Instrum. 73, 3065-3074 (2002).
[CrossRef]

Y. Chen, C. Mu, X. Intes, D. Blessington and B. Chance, �??Frequency domain phase cancellation instrument for fast and accurate localization of fluorescent heterogeneity,�?? Rev. Sci. Instrum. 74, 3466-3473 (2003).
[CrossRef]

Sci. Am. (1)

R. Weinberg, �??How Does Cancer Arise,�?? 275, 62-71 (1996).
[CrossRef] [PubMed]

Science (1)

F. Jobsis, �??Noninvasive infrared monitoring of cerebral and myocardial sufficiency and circulatory parameters,�?? Science 198, 1264-1267 (1977).
[CrossRef] [PubMed]

Topics in Current Chemistry (1)

K. Licha, �??Contrast agents for optical imaging,�?? Topics in Current Chemistry 222, 1-29 (2002).
[CrossRef]

Other (4)

X. Li, �??Fluorescence and diffusive wave diffraction tomographic probes in turbid media,�?? PhD University of Pennsylvania (1996).

M. O�??Leary, �??Imaging with diffuse photon density waves,�?? PhD University of Pennsylvania (1996).

E. Hillman, �??Experimental and theoretical investigations of near infrared tomographic imaging methods and clinical applications,�?? PhD University College London (2002).

A. Kak and M. Slaney, �??Computerized tomographic Imaging�??, IEEE Press, N.Y. (1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Typical TPSF and respective moments. The IFS curve corresponds to a typical instrument response function.

Fig. 2.
Fig. 2.

Configuration used for the simulations herein. The source (detectors) locations are depicted by red (blue) dots

Fig. 3.
Fig. 3.

Results of repartition of energy, meantimes and variance of 1,000 randomly generated noised TPSF.

Fig. 4.
Fig. 4.

Example of sensitivity matrices. a) and b) correspond respectively to m0λ2(r s,r d) and m0λ2(r s,r dm2λ2(r s,r d) for a 6cm thick slab with source-detector facing each other and a 0.1 µM background of Cy 7. c) and d) correspond to the same parameters for a 0.1 µM background of Cy 5.5. Last, e) and f) correspond to the same parameters for a 0.1 µM background of Cy 3B.

Fig. 5.
Fig. 5.

Reconstruction from synthetic data for Cy 7: a) 0th moment only, b) 0th, 1st and 2nd moments; Cy 5.5 : c) 0th moment only, d) 0th, 1st and 2nd moments; and Cy 3B: e) 0th moment only, f) 0th, 1st and 2nd moments. The quantitative values are expressed in µM.

Fig. 6.
Fig. 6.

Reconstruction from synthetic data for Cy 5.5 using all three moments noisy data.

Tables (3)

Tables Icon

Table 1. Parameters used in the simulations

Tables Icon

Table 2. Fluorochrome investigated herein.

Tables Icon

Table 3. Noise model used. The standard deviations are expressed in percent of the mean.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

1 v t Φ ( r , t ) D Δ 2 Φ ( r , t ) + μ a Φ ( r , t ) = S ( r , t )
t N ex ( r , t ) = 1 τ N ex ( r , t ) + σ · Φ λ 1 ( r , t ) [ N tot ( r , t ) 2 N ex ( r , t ) ]
N ex ( r , ω ) τ = σ · N tot ( r ) 1 i ω τ · Φ λ 1 ( r , ω )
Φ λ 2 ( r s , r d , ω ) = η volume N ex ( r , ω ) · Φ λ 2 ( r , r d , ω ) · d 3 r
Φ λ 2 ( r s , r d , ω ) = volume Φ λ 1 ( r s , r , ω ) · Q eff · N tot ( r ) 1 i ω τ · Φ λ 2 ( r , r d , ω ) · d 3 r
Φ λ 2 ( r s , r d , ω ) Φ 0 λ 1 ( r s , r d , ω ) = 1 Φ 0 λ 1 ( r s , r d , ω ) volume Φ 0 λ 1 ( r s , r , ω ) · Q eff · C tot ( r ) 1 i ω τ · Φ 0 λ 2 ( r , r d , ω ) · d 3 r
Φ λ 2 ( r s , r d , ω ) Φ 0 λ 1 ( r s , r d , ω ) = D λ 1 G λ 1 ( r s , r d , ω ) voxels 1 D λ 1 G λ 1 ( r s , r v , ω ) · Q eff · C tot ( r v ) 1 i ω τ · 1 D λ 2 G λ 2 ( r v , r d , ω ) · h 3
m k = t k = + t k · p ( t ) dt + p ( t ) dt
m 0 λ 2 ( r s , r d ) = Φ N λ 2 ( r s , r d , ω = 0 ) = voxels G λ 1 ( r s , r v , ω = 0 ) · G λ 2 ( r v , r d , ω = 0 ) G λ 1 ( r s , r d , ω = 0 ) × Q eff h 3 D λ 2 × C tot ( r v )
m 0 λ 2 ( r s , r d ) × m 1 λ 2 ( r s , r d ) = voxels { ( τ + r s r v + r s r d 2 . v μ a λ D λ r v r d 2 . v μ a D λ ) × G λ ( r s , r v , ω = 0 ) · G λ ( r v , r d , ω = 0 ) G λ ( r s , r d , ω = 0 ) × Q eff h 3 D λ × C tot ( r v ) }
m 0 λ 2 ( r s , r d ) · m 2 λ 2 ( r 2 , r d ) = voxels { ( τ 2 + r s r v + r s r d 4 . v 2 μ a λ μ a λ D λ + r v r d 4 . v 2 μ a λ μ a λ D λ + { τ + r s r v 2 . v μ a λ D λ + r v r d 2 . v μ a λ D λ } 2 t λ 2 ( r s , r d ) · { τ + r s r v + r v r d 2 . v μ a λ D λ r v r d 2 . v μ a λ D λ } ) × G λ ( r s , r v , ω = 0 ) · G λ ( r v , r d , ω = 0 ) G λ ( r s , r d , ω = 0 ) × Q eff h 2 D λ × C tot ( r v ) }
m 0 λ 2 ( r s 1 , r d 1 ) m 0 λ 2 ( r sm , r dm ) m 0 λ 2 ( r s 1 , r d 1 ) · m 1 λ 2 ( r s 1 , r d 1 ) m 0 λ 2 ( r sm , r dm ) · m 1 λ 2 ( r sm , r dm ) m 0 λ 2 ( r s 1 , r d 1 ) · m 2 λ 2 ( r s 1 , r d 1 ) m 0 λ 2 ( r sm , r dm ) · m 2 λ 2 ( r sm , r dm ) = W 11 m 0 λ 2 W ln m 0 λ 2 W ml m 0 λ 2 W mn m 0 λ 2 W 11 m 0 λ 2 · m 1 λ 2 W ln m 0 λ 2 · m 1 λ 2 W m 1 m 0 λ 2 · m 1 λ 2 W mn m 0 λ 2 · m 1 λ 2 W 11 m 0 λ 2 · m 2 λ 2 W ln m 0 λ 2 · m 2 λ 2 W m 1 m 0 λ 2 · m 2 λ 2 W mn m 0 λ 2 · m 2 λ 2 · C tot ( r vl ) C tot ( r vn )
b = A · x
x j ( k + 1 ) = x j ( k ) + ξ b i i a ij x j ( k ) i a ij a ij i a ij

Metrics