Abstract

Two efficient Monte Carlo models are described, facilitating predictions of complete time-resolved fluorescence spectra from a light-scattering and light-absorbing medium. These are compared with a third, conventional fluorescence Monte Carlo model in terms of accuracy, signal-to-noise statistics, and simulation time. The improved computation efficiency is achieved by means of a convolution technique, justified by the symmetry of the problem. Furthermore, the reciprocity principle for photon paths, employed in one of the accelerated models, is shown to simplify the computations of the distribution of the emitted fluorescence drastically. A so-called white Monte Carlo approach is finally suggested for efficient simulations of one excitation wavelength combined with a wide range of emission wavelengths. The fluorescence is simulated in a purely scattering medium, and the absorption properties are instead taken into account analytically afterward. This approach is applicable to the conventional model as well as to the two accelerated models. Essentially the same absolute values for the fluorescence integrated over the emitting surface and time are obtained for the three models within the accuracy of the simulations. The time-resolved and spatially resolved fluorescence exhibits a slight overestimation at short delay times close to the source corresponding to approximately two grid elements for the accelerated models, as a result of the discretization and the convolution. The improved efficiency is most prominent for the reverse-emission accelerated model, for which the simulation time can be reduced by up to two orders of magnitude.

© 2003 Optical Society of America

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2002 (1)

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

2001 (4)

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

M. G. Muller, I. Georgakoudi, Q. Zhang, J. Wu, M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. 40, 4633–4646 (2001).
[CrossRef]

2000 (1)

1999 (1)

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

1998 (3)

1997 (5)

1996 (3)

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

1995 (4)

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

M. S. Patterson, S. Andersson-Engels, B. C. Wilson, E. K. Osei, “Absorption spectroscopy in tissue-simulating materials: a theoretical and experimental study of photon paths,” Appl. Opt. 34, 22–30 (1995).
[CrossRef] [PubMed]

1994 (3)

1993 (2)

1989 (2)

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Andersson-Engels, S

A Pifferi, R Berg, P Taroni, S Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Andersson-Engels, S.

A. Pifferi, P. Taroni, G. Valentini, S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
[CrossRef]

M. S. Patterson, S. Andersson-Engels, B. C. Wilson, E. K. Osei, “Absorption spectroscopy in tissue-simulating materials: a theoretical and experimental study of photon paths,” Appl. Opt. 34, 22–30 (1995).
[CrossRef] [PubMed]

S. Andersson-Engels, A. M. K. Enejder, J. Swartling, A. Pifferi, “Accelerated Monte Carlo models to simulate fluorescence of layered tissue,” in Photon Migration, Diffuse Spectroscopy, and Optical Coherence Tomography: Imaging and Functional Assessment, S. Andersson-Engels, J. G. Fujimoto, eds., Proc. SPIE4160, 14–15 (2000).
[CrossRef]

Arridge, S. R.

Avrillier, S.

Berg, R

A Pifferi, R Berg, P Taroni, S Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

R Berg, “Laser-based cancer diagnostics and therapy—tissue optics considerations,” Ph.D. thesis (Lund Institute of Technology, Lund, Sweden, 1995).

Bigio, I. J.

I. J. Bigio, J. R. Mourant, “Ultraviolet and visible spectroscopies for tissue diagnostics: fluorescence spectroscopy and elastic-scattering spectroscopy,” Phys. Med. Biol. 42, 803–814 (1997).
[CrossRef] [PubMed]

Bogdanov, A.

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

Bonner, R. F.

Caltabiano, T.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Canpolat, M.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Cecchi, G.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Chan, E.

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

Chance, B.

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

Chen, A. U.

Chen, J.

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

Cheong, W. F.

Condarelli, D.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Cothren, R.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Crilly, R. J.

Criswell, G.

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

Davis, M. J.

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Ediger, M. N.

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

Edner, H.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Enejder, A. M. K.

S. Andersson-Engels, A. M. K. Enejder, J. Swartling, A. Pifferi, “Accelerated Monte Carlo models to simulate fluorescence of layered tissue,” in Photon Migration, Diffuse Spectroscopy, and Optical Coherence Tomography: Imaging and Functional Assessment, S. Andersson-Engels, J. G. Fujimoto, eds., Proc. SPIE4160, 14–15 (2000).
[CrossRef]

Ettori, D.

Feld, M. S.

M. G. Muller, I. Georgakoudi, Q. Zhang, J. Wu, M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. 40, 4633–4646 (2001).
[CrossRef]

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Fitzmaurice, M.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Frank, R.

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

Gandjbakhche, A. H.

Gardner, C. M.

Gélébart, B.

Georgakoudi, I.

Gordon, H. R.

Gossage, K.

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

Holtz, J. Z.

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Island, T. C.

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Jacques, S. L.

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Johnston, A. L.

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Kienle, A.

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

Kittrell, C.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Klavuhn, K. G.

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Kramers, J. R.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Law, E. J.

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Mahmood, U.

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

Metha, A.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Motamedi, M.

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

Mourant, J. R.

M. Canpolat, J. R. Mourant, “Monitoring photosensitizer concentration by use of a fiber-optic probe with a small source-detector separation,” Appl. Opt. 39, 6508–6514 (2000).
[CrossRef]

I. J. Bigio, J. R. Mourant, “Ultraviolet and visible spectroscopies for tissue diagnostics: fluorescence spectroscopy and elastic-scattering spectroscopy,” Phys. Med. Biol. 42, 803–814 (1997).
[CrossRef] [PubMed]

Muller, M. G.

Nishioka, N. S.

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

Nossal, R.

Osei, E. K.

Paithankar, D. Y.

Pantani, L.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Patterson, M. S.

Pfefer, J.

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

Pfefer, T. J.

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

Pifferi, A

A Pifferi, R Berg, P Taroni, S Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Pifferi, A.

A. Pifferi, P. Taroni, G. Valentini, S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
[CrossRef]

S. Andersson-Engels, A. M. K. Enejder, J. Swartling, A. Pifferi, “Accelerated Monte Carlo models to simulate fluorescence of layered tissue,” in Photon Migration, Diffuse Spectroscopy, and Optical Coherence Tomography: Imaging and Functional Assessment, S. Andersson-Engels, J. G. Fujimoto, eds., Proc. SPIE4160, 14–15 (2000).
[CrossRef]

Pogue, B. W.

Pope, K.

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Prahl, S. A.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

Ramanujam, N.

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

Ratliff, N. B.

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Rava, R. P.

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Richards-Kortum, R

A. J. Welch, R Richards-Kortum, “Monte Carlo simulation of the propagation of fluorescent light,” in Laser-induced Interstitial Thermotherapy, G. Müller, A. Roggan, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1995), pp. 174–189.

Richards-Kortum, R.

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Saarnak, A. E.

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

Schomacker, K. T.

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

Schweiger, M.

Sevick-Muraca, E. M.

Sinaasappel, M.

Spears, J. R.

Star, W. M.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef]

Sterenborg, H. J. C. M.

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

M. Sinaasappel, H. J. C. M. Sterenborg, “Quantification of the hematoporphyrin derivative by fluorescence measurement using dual-wavelength excitation and dual-wavelength detection,” Appl. Opt. 32, 541–548 (1993).
[CrossRef] [PubMed]

Still, J. M.

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Svanberg, S.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Swartling, J.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

S. Andersson-Engels, A. M. K. Enejder, J. Swartling, A. Pifferi, “Accelerated Monte Carlo models to simulate fluorescence of layered tissue,” in Photon Migration, Diffuse Spectroscopy, and Optical Coherence Tomography: Imaging and Functional Assessment, S. Andersson-Engels, J. G. Fujimoto, eds., Proc. SPIE4160, 14–15 (2000).
[CrossRef]

Taroni, P

A Pifferi, R Berg, P Taroni, S Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Taroni, P.

Thomsen, S.

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Tinet, E.

Tualle, J.-M.

Tung, C.-H.

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

Valentini, G.

van Gemert, M. J. C.

A. J. Welch, M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, New York, 1995).

Wagnières, G. A.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef]

Wang, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Warren, S.

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Weibring, P.

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Weiss, G. H.

Weissleder, R.

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

Welch, A. J.

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

C. M. Gardner, A. J. Welch, “Monte Carlo simulation of light transport in tissue: unscattered absorption events,” Appl. Opt. 33, 2743–2745 (1994).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

A. J. Welch, R Richards-Kortum, “Monte Carlo simulation of the propagation of fluorescent light,” in Laser-induced Interstitial Thermotherapy, G. Müller, A. Roggan, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1995), pp. 174–189.

A. J. Welch, M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, New York, 1995).

Wilson, B.

Wilson, B. C.

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef]

M. S. Patterson, S. Andersson-Engels, B. C. Wilson, E. K. Osei, “Absorption spectroscopy in tissue-simulating materials: a theoretical and experimental study of photon paths,” Appl. Opt. 34, 22–30 (1995).
[CrossRef] [PubMed]

Wu, J.

Yazdi, Y.

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

Zhang, Q.

Zheng, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (15)

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

M. Canpolat, J. R. Mourant, “Monitoring photosensitizer concentration by use of a fiber-optic probe with a small source-detector separation,” Appl. Opt. 39, 6508–6514 (2000).
[CrossRef]

M. Sinaasappel, H. J. C. M. Sterenborg, “Quantification of the hematoporphyrin derivative by fluorescence measurement using dual-wavelength excitation and dual-wavelength detection,” Appl. Opt. 32, 541–548 (1993).
[CrossRef] [PubMed]

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

C. M. Gardner, S. L. Jacques, A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

M. G. Muller, I. Georgakoudi, Q. Zhang, J. Wu, M. S. Feld, “Intrinsic fluorescence spectroscopy in turbid media: disentangling effects of scattering and absorption,” Appl. Opt. 40, 4633–4646 (2001).
[CrossRef]

M. S. Patterson, S. Andersson-Engels, B. C. Wilson, E. K. Osei, “Absorption spectroscopy in tissue-simulating materials: a theoretical and experimental study of photon paths,” Appl. Opt. 34, 22–30 (1995).
[CrossRef] [PubMed]

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, G. H. Weiss, “Effects of multiple-passage probabilities on fluorescent signals from biological media,” Appl. Opt. 36, 4613–4619 (1997).
[CrossRef] [PubMed]

C. M. Gardner, A. J. Welch, “Monte Carlo simulation of light transport in tissue: unscattered absorption events,” Appl. Opt. 33, 2743–2745 (1994).
[CrossRef] [PubMed]

R. J. Crilly, W. F. Cheong, B. Wilson, J. R. Spears, “Forward-adjoint fluorescence model: Monte Carlo integration and experimental validation,” Appl. Opt. 36, 6513–6519 (1997).
[CrossRef]

S. Avrillier, E. Tinet, D. Ettori, J.-M. Tualle, B. Gélébart, “Influence of the emission-reception geometry in laser-induced fluorescence spectra from turbid media,” Appl. Opt. 37, 2781–2787 (1998).
[CrossRef]

A. Pifferi, P. Taroni, G. Valentini, S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
[CrossRef]

H. R. Gordon, “Equivalence of the point and beam spread functions of scattering media: a formal demonstration,” Appl. Opt. 33, 1120–1122 (1994).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

Burns (1)

J. M. Still, E. J. Law, K. G. Klavuhn, T. C. Island, J. Z. Holtz, “Diagnosis of burn depth using laser-induced indocyanine green fluorescence: a preliminary clinical trial,” Burns 27, 364–371 (2001).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

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

T. J. Pfefer, K. T. Schomacker, M. N. Ediger, N. S. Nishioka, “Light propagation in tissue during fluorescence spectroscopy with single-fiber probes,” IEEE J. Sel. Top. Quantum Electron. 7, 1004–1012 (2001).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

N. Ramanujam, J. Chen, K. Gossage, R. Richards-Kortum, B. Chance, “Fast and noninvasive fluorescence imaging of biological tissues in vivo using a flying-spot scanner,” IEEE Trans. Biomed. Eng. 48, 1034–1041 (2001).
[CrossRef] [PubMed]

S. Warren, K. Pope, Y. Yazdi, A. J. Welch, S. Thomsen, A. L. Johnston, M. J. Davis, R. Richards-Kortum, “Combined ultrasound and fluorescence spectroscopy for physico-chemical imaging of atherosclerosis,” IEEE Trans. Biomed. Eng. 42, 121–132 (1995).
[CrossRef] [PubMed]

J. Photochem. Photobiol. B (1)

H. J. C. M. Sterenborg, A. E. Saarnak, R. Frank, M. Motamedi, “Evaluation of spectral correction techniques for fluorescence measurements on pigmented lesions in vivo,” J. Photochem. Photobiol. B 35, 159–165 (1996).
[CrossRef] [PubMed]

Lasers Surg. Med. (2)

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distribution in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef]

A. J. Welch, C. M. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, S. Warren, “Propagation of fluorescence light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef]

Nat. Biotechnol. (1)

R. Weissleder, C.-H. Tung, U. Mahmood, A. Bogdanov, “In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,” Nat. Biotechnol. 17, 375–378 (1999).
[CrossRef] [PubMed]

Opt. Lasers Eng. (1)

P. Weibring, J. Swartling, H. Edner, S. Svanberg, T. Caltabiano, D. Condarelli, G. Cecchi, L. Pantani, “Optical monitoring of volcanic sulphur dioxide emissions—comparison between four different remote-sensing spectroscopic techniques,” Opt. Lasers Eng. 37, 267–284 (2002).
[CrossRef]

Photochem. Photobiol. (1)

G. A. Wagnières, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef]

Phys. Med. Biol. (2)

I. J. Bigio, J. R. Mourant, “Ultraviolet and visible spectroscopies for tissue diagnostics: fluorescence spectroscopy and elastic-scattering spectroscopy,” Phys. Med. Biol. 42, 803–814 (1997).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

Spectrochim. Acta Part A (1)

R. Richards-Kortum, R. P. Rava, R. Cothren, A. Metha, M. Fitzmaurice, N. B. Ratliff, J. R. Kramers, C. Kittrell, M. S. Feld, “A model for extraction of diagnostic information from laser induced fluorescence spectra of human artery wall,” Spectrochim. Acta Part A 45, 87–93 (1989).
[CrossRef]

Other (6)

A. J. Welch, R Richards-Kortum, “Monte Carlo simulation of the propagation of fluorescent light,” in Laser-induced Interstitial Thermotherapy, G. Müller, A. Roggan, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1995), pp. 174–189.

R Berg, “Laser-based cancer diagnostics and therapy—tissue optics considerations,” Ph.D. thesis (Lund Institute of Technology, Lund, Sweden, 1995).

A. J. Welch, M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, New York, 1995).

A Pifferi, R Berg, P Taroni, S Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

S. Andersson-Engels, A. M. K. Enejder, J. Swartling, A. Pifferi, “Accelerated Monte Carlo models to simulate fluorescence of layered tissue,” in Photon Migration, Diffuse Spectroscopy, and Optical Coherence Tomography: Imaging and Functional Assessment, S. Andersson-Engels, J. G. Fujimoto, eds., Proc. SPIE4160, 14–15 (2000).
[CrossRef]

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

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Figures (9)

Fig. 1
Fig. 1

The process of fluorescence emission from a multilayered semi-infinite turbid medium is schematically illustrated.

Fig. 2
Fig. 2

Geometry used for the simulations. (a) The medium is divided into volume elements using a grid along the r and z axes. Similarly, time is divided into d t intervals. (b) View of the coordinate system used to calculate the convolution for a slab of thickness d z .

Fig. 3
Fig. 3

The concepts of (a) forward Monte Carlo (FMC) and (b) reverse Monte Carlo (RMC) methods are illustrated. The tissue surface is along the r axis, and n is the normal vector. In (a), an isotropic point source at S 1 is considered, corresponding to fluorescence emission from this point. The radiant flux across the surface boundary is measured at D 1 . In (b), a point source at S 2 is assumed. The fluence rate is measured at D 2 . The gray line represents one possible photon trajectory between the two points. The cone ΔΩ at D 1 and at S 2 represents the solid angle of acceptance for emission at the surface, defined by the condition for total reflection. Furthermore, r S 2 = r D 1 and r D 2 = r S 1 .

Fig. 4
Fig. 4

Examples of results obtained from simulations using the reference parameters listed in Table 1 when employing the (a) SMC, (b) FMC, and (c) RMC methods. A total number of 10 6 photons was used for the SMC method, yielding a computation time of 13 min on a Pentium III 933-MHz processor. For the FMC and RMC methods, 10 5 photons were used, and the corresponding computation times were 77 and 75 s, respectively.

Fig. 5
Fig. 5

Plots of F CW ( r ) are shown, obtained from SMC (diamonds), RMC (squares), and FMC (triangles) simulations. The simulation conditions are the reference values listed in Table 1 if not otherwise stated.

Fig. 6
Fig. 6

F r ( t ) is shown for (a)–(c) r = 0.05   cm and (d)–(f ) r = 0.5   cm plotted versus time. The results are obtained from SMC (diamonds), RMC (squares), and FMC (triangles) simulations. The simulation conditions are again the reference values listed in Table 1 if not otherwise stated.

Fig. 7
Fig. 7

Data derived from SMC simulations are compared with results obtained from (a) RMC and (b) FMC simulations, with respect to the size of the time grid element ( r = 0.05   cm , and optical properties are as defined in Table 1).

Fig. 8
Fig. 8

Simulated fluorescence spectra for PpIX for two different measurement geometries: imaging and direct optical fiber contact. The case of no photobleaching is shown. The intrinsic spectrum used as input is also shown.

Fig. 9
Fig. 9

Relative fluorescence intensity for three different bleaching states and two geometries, imaging (diamonds) and direct optical fiber contact (squares). State 1 corresponds to no bleaching (a 3-mm-thick layer of PpIX) and was used as the reference point. State 2 corresponds to a 2-mm layer of PpIX, starting 1 mm below the surface. In state 3, a 1-mm layer of PpIX located 2 mm below the surface was simulated. The result is shown for the 700-nm simulation.

Tables (4)

Tables Icon

Table 1 Standard Values of Input Parameters Used in the Simulations

Tables Icon

Table 2 Values of F TOT Obtained for Different Optical Propertiesa

Tables Icon

Table 3 Computation Time for the Different Models

Tables Icon

Table 4 Coefficient of Variation (CV) for the Three Models

Equations (21)

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ϕ eff ( λ m ) = ϕ   μ a xf μ a x   η ( λ m ) Δ λ 0 η ( λ ) d λ ,
D ( t ) = 1 τ   exp - t τ
F z inst ( r ,   t ,   z ) d z = d z 0 r d r 0 2 π d φ 0 ϕ eff A ( r ,   t ,   z ) × E ( d ,   t - t ,   z ) d t ,
F inst ( r ,   t ) = 0 F z inst ( r ,   t ,   z ) d z .
F ( r ,   t ) = 0 F inst ( r ,   t ) D ( t - t ) d t .
FWD ( r D 1 ) = Δ Ω L ( r D 1 ,   - s ) [ 1 - R ( | - s n | ) ] × ( - s n ) d Ω ( s ) .
R ( θ 2 ) = 1 2   sin 2 ( θ 2 - θ 1 ) sin 2 ( θ 2 + θ 1 ) + tan 2 ( θ 2 - θ 1 ) tan 2 ( θ 2 + θ 1 ) ,
θ 2 = cos - 1 ( | s n | ) , n 1   sin   θ 1 = n 2   sin   θ 2 .
REV ( r D 2 ) = 4 π L ( r D 2 ,   s ) d Ω ( s ) .
4 π   L 2 ( r ,   s ) Q 1 ( r ,   - s ) d Ω ( s ) d V = 4 π   L 1 ( r ,   - s ) Q 2 ( r ,   s ) d Ω ( s ) d V .
Q 1 ( r ,   s ) = P fwd 4 π   δ ( r - r S 1 ) ,
Q 2 ( r ,   s ) = P rev Δ Ω   [ 1 - R ( | s n | ) ] ( s - n ) δ ( r - r S 2 ) if   s   is inside Δ Ω 0 if   s   is not inside Δ Ω ,
LHS = 4 π   L 2 ( r ,   s )   P fwd 4 π   δ ( r - r S 1 ) d Ω ( s ) d V = P fwd 4 π   4 π L 2 ( r S 1 ,   s ) d Ω ( s ) .
RHS = 4 π   L 1 ( r ,   - s )   P rev Δ Ω   [ 1 - R ( | s n | ) ] ( s - n ) × δ ( r - r S 2 ) s Δ Ω d Ω ( s ) d V = P rev Δ Ω   Δ Ω L 1 ( r S 2 ,   - s ) [ 1 - R ( | s   n | ) ] × ( s - n ) d Ω ( s ) .
REV ( r D 2 ) = P rev P fwd   4 π Δ Ω   FWD ( r D 1 ) .
P rev = Δ Ω 4 π   P fwd
w = i = 1 nl   exp - μ a m , j ct i n i ,
F TOT = 0 2 π r d r 0 F ( r ,   t ) d t
F CW ( r ) = 0 F ( r ,   t ) d t .
F r ¯ ( t ) = F ( r ¯ ,   t ) ,
CV = 1 N   i     | σ i ( r i ,   t i ) | F ( r i ,   t i ) ,

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