Abstract

Photon diffusion theory was used to model photobleaching and tissue necrosis resulting from broad-beam therapeutic light irradiation of tissue containing a photosensitizer. The photosensitizer fluorescence signal at the tissue surface was simulated with both broad-beam and pencil-beam excitation. The relationship between the decreasing fluorescence signal and the increasing depth of tissue photodynamic damage during treatment was examined. By analyzing spatially resolved fluorescence measured at the tissue surface in terms of an equivalent virtual point or planar source of fluorescence within the tissue, predictions of necrosis depth that are insensitive to a range of initial treatment parameters were shown to be possible. Preliminary measurements in tissue-simulating phantoms supported the main theoretical findings. The potential value and feasibility of this technique for photodynamic therapy dosimetry are discussed.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  5. T. H. Foster, M. G. Nichols, “Oxygen sensitivity of PDT determined from time-dependent electrode measurements in spheroids,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy IV, T. J. Dougherty, ed., Proc. SPIE2392, 141–151 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

1997

B. C. Wilson, M. S. Patterson, L. Lilge, “Implicit and explicit dosimetry in photodynamic therapy: a new paradigm,” Lasers Med. Science 12, 182–199 (1997).
[CrossRef]

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Non-invasive in vivo measurements of photosensitizer uptake using diffuse reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

B. W. Pogue, T. Hasan, “A theoretical study of light fractionation and dose rate effects in photodynamic therapy,” Radiat. Res. 147, 551–559 (1997).
[CrossRef] [PubMed]

A. J. L. Jongen, H. J. C. M. Sterenborg, “Mathematical description of photobleaching in vivo describing the influence of tissue optics on measured fluorescence signals,” Phys. Med. Biol. 42, 1701–1716 (1997).
[CrossRef] [PubMed]

1996

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

1993

H. I. Pass, “Photodynamic therapy in oncology: mechanisms and clinical use,” J. Natl. Cancer Inst. 85, 443–456 (1993).
[CrossRef] [PubMed]

1992

T. H. Foster, L. Gao, “Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density,” Radiat. Res. 130, 379–383 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1990

M. S. Patterson, B. C. Wilson, R. Graff, “In vivo tests of the concept of photodynamic threshold dose in normal rat liver photosensitized by aluminum chlorosulphonated phthalocyanine,” Photochem. Photobiol. 51, 343–349 (1990).
[CrossRef] [PubMed]

W. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1988

1987

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

W. R. Potter, T. S. Mang, T. J. Dougherty, “The theory of photodynamic therapy dosimetry: consequences of photodestruction of sensitizer,” Photochem. Photobiol. 46, 97–101 (1987).
[CrossRef] [PubMed]

1986

J. Moan, “Effect of bleaching on porphyrin sensitizers during photodynamic therapy,” Cancer Lett. 33, 45–53 (1986).
[CrossRef] [PubMed]

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

1983

Akdemir, D.

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

Barr, H.

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

Beck, E. R.

T. J. Farrell, M. S. Patterson, J. E. Hayward, B. C. Wilson, E. R. Beck, “A CCD and neural network based instrument for the non-invasive determinations of tissue optical properties in vivo,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 117–128 (1994).
[CrossRef]

Berns, M. W.

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1992).

Bown, S. G.

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

Cheong, W.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Coleridge-Smith, P. D.

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

Diamond, K. R.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Non-invasive in vivo measurements of photosensitizer uptake using diffuse reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Dougherty, T. J.

W. R. Potter, T. S. Mang, T. J. Dougherty, “The theory of photodynamic therapy dosimetry: consequences of photodestruction of sensitizer,” Photochem. Photobiol. 46, 97–101 (1987).
[CrossRef] [PubMed]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, J. E. Hayward, B. C. Wilson, E. R. Beck, “A CCD and neural network based instrument for the non-invasive determinations of tissue optical properties in vivo,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 117–128 (1994).
[CrossRef]

Ferwerda, H. A.

Foster, T. H.

T. H. Foster, L. Gao, “Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density,” Radiat. Res. 130, 379–383 (1992).
[CrossRef] [PubMed]

T. H. Foster, M. G. Nichols, “Oxygen sensitivity of PDT determined from time-dependent electrode measurements in spheroids,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy IV, T. J. Dougherty, ed., Proc. SPIE2392, 141–151 (1995).
[CrossRef]

Gao, L.

T. H. Foster, L. Gao, “Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density,” Radiat. Res. 130, 379–383 (1992).
[CrossRef] [PubMed]

Gerald, C. F.

C. F. Gerald, P. O. Wheatley, Applied Numerical Analysis (Addison-Wesley, Don Mills, Ontario, 1970).

Gofstein, G.

S. L. Jacques, R. Joseph, G. Gofstein, “How photobleaching affects dosimetry and fluorescence monitoring of PDT in turbid media,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy II, T. J. Dougherty, ed., Proc. SPIE1881, 168–179 (1993).
[CrossRef]

Graff, R.

M. S. Patterson, B. C. Wilson, R. Graff, “In vivo tests of the concept of photodynamic threshold dose in normal rat liver photosensitized by aluminum chlorosulphonated phthalocyanine,” Photochem. Photobiol. 51, 343–349 (1990).
[CrossRef] [PubMed]

Groenhuis, R. A.

Hasan, T.

B. W. Pogue, T. Hasan, “A theoretical study of light fractionation and dose rate effects in photodynamic therapy,” Radiat. Res. 147, 551–559 (1997).
[CrossRef] [PubMed]

Hawkes, R. P.

R. P. Hawkes, “Photodynamic therapy dosimetry through measurement of fluorescence decrease due to photobleaching,” M.S. thesis (McMaster University, Hamilton, Ontario, Canada, 1997).

Hayward, J. E.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Non-invasive in vivo measurements of photosensitizer uptake using diffuse reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, J. E. Hayward, B. C. Wilson, E. R. Beck, “A CCD and neural network based instrument for the non-invasive determinations of tissue optical properties in vivo,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 117–128 (1994).
[CrossRef]

Jacques, S. L.

S. L. Jacques, R. Joseph, G. Gofstein, “How photobleaching affects dosimetry and fluorescence monitoring of PDT in turbid media,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy II, T. J. Dougherty, ed., Proc. SPIE1881, 168–179 (1993).
[CrossRef]

Jongen, A. J. L.

A. J. L. Jongen, H. J. C. M. Sterenborg, “Mathematical description of photobleaching in vivo describing the influence of tissue optics on measured fluorescence signals,” Phys. Med. Biol. 42, 1701–1716 (1997).
[CrossRef] [PubMed]

Joseph, R.

S. L. Jacques, R. Joseph, G. Gofstein, “How photobleaching affects dosimetry and fluorescence monitoring of PDT in turbid media,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy II, T. J. Dougherty, ed., Proc. SPIE1881, 168–179 (1993).
[CrossRef]

Keijzer, M.

Lilge, L.

B. C. Wilson, M. S. Patterson, L. Lilge, “Implicit and explicit dosimetry in photodynamic therapy: a new paradigm,” Lasers Med. Science 12, 182–199 (1997).
[CrossRef]

MacRobert, A. J.

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

Mang, T. S.

W. R. Potter, T. S. Mang, T. J. Dougherty, “The theory of photodynamic therapy dosimetry: consequences of photodestruction of sensitizer,” Photochem. Photobiol. 46, 97–101 (1987).
[CrossRef] [PubMed]

Mathews, J.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, Menlo Park, Calif., 1970).

Moan, J.

J. Moan, “Effect of bleaching on porphyrin sensitizers during photodynamic therapy,” Cancer Lett. 33, 45–53 (1986).
[CrossRef] [PubMed]

Nichols, M. G.

T. H. Foster, M. G. Nichols, “Oxygen sensitivity of PDT determined from time-dependent electrode measurements in spheroids,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy IV, T. J. Dougherty, ed., Proc. SPIE2392, 141–151 (1995).
[CrossRef]

Pass, H. I.

H. I. Pass, “Photodynamic therapy in oncology: mechanisms and clinical use,” J. Natl. Cancer Inst. 85, 443–456 (1993).
[CrossRef] [PubMed]

Patterson, M. S.

B. C. Wilson, M. S. Patterson, L. Lilge, “Implicit and explicit dosimetry in photodynamic therapy: a new paradigm,” Lasers Med. Science 12, 182–199 (1997).
[CrossRef]

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Non-invasive in vivo measurements of photosensitizer uptake using diffuse reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. C. Wilson, R. Graff, “In vivo tests of the concept of photodynamic threshold dose in normal rat liver photosensitized by aluminum chlorosulphonated phthalocyanine,” Photochem. Photobiol. 51, 343–349 (1990).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, J. E. Hayward, B. C. Wilson, E. R. Beck, “A CCD and neural network based instrument for the non-invasive determinations of tissue optical properties in vivo,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 117–128 (1994).
[CrossRef]

W. M. Star, B. C. Wilson, M. S. Patterson, “Light delivery and optical dosimetry in photodynamic therapy of solid tumors,” in Photodynamic Therapy, B. W. Henderson, T. J. Dougherty, eds. (Marcel Dekker, New York, 1992), pp. 335–368.

M. S. Patterson, B. C. Wilson, “A theoretical study of the influence of sensitizer photobleaching on depth of necrosis in photodynamic therapy,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy III, T. J. Dougherty, ed., Proc. SPIE2133, 208–219 (1994).
[CrossRef]

Pogue, B. W.

B. W. Pogue, T. Hasan, “A theoretical study of light fractionation and dose rate effects in photodynamic therapy,” Radiat. Res. 147, 551–559 (1997).
[CrossRef] [PubMed]

Potter, W. R.

W. R. Potter, T. S. Mang, T. J. Dougherty, “The theory of photodynamic therapy dosimetry: consequences of photodestruction of sensitizer,” Photochem. Photobiol. 46, 97–101 (1987).
[CrossRef] [PubMed]

W. R. Potter, “PDT dosimetry and response,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed. (SPIE, Bellingham, Wash., 1989), pp. 88–99.
[CrossRef]

Prahl, S. A.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1992).

Star, W. M.

M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

W. M. Star, B. C. Wilson, M. S. Patterson, “Light delivery and optical dosimetry in photodynamic therapy of solid tumors,” in Photodynamic Therapy, B. W. Henderson, T. J. Dougherty, eds. (Marcel Dekker, New York, 1992), pp. 335–368.

Sterenborg, H. J. C. M.

A. J. L. Jongen, H. J. C. M. Sterenborg, “Mathematical description of photobleaching in vivo describing the influence of tissue optics on measured fluorescence signals,” Phys. Med. Biol. 42, 1701–1716 (1997).
[CrossRef] [PubMed]

Storchi, P. R. M.

Svaasand, L. O.

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

Tadir,

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

Ten Bosch, J. J.

Tralau, C. J.

C. J. Tralau, A. J. MacRobert, P. D. Coleridge-Smith, H. Barr, S. G. Bown, “Photodynamic therapy with phthalocyanine sensitization: quantitative studies in a transplantable rat fibrosarcoma,” Br. J. Cancer 55, 399–395 (1987).
[CrossRef]

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

Tromberg, B. J.

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

Walker, R. L.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, Menlo Park, Calif., 1970).

Weersink, R. A.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Non-invasive in vivo measurements of photosensitizer uptake using diffuse reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Welch, A. J.

W. Cheong, S. A. Prahl, A. J. Welch, “A review of optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wheatley, P. O.

C. F. Gerald, P. O. Wheatley, Applied Numerical Analysis (Addison-Wesley, Don Mills, Ontario, 1970).

Wieman, T. J.

S. G. Bown, C. J. Tralau, P. D. Coleridge-Smith, D. Akdemir, T. J. Wieman, “Photodynamic therapy with porphyrin and phthalocyanine sensitization: quantitative studies in normal rat liver,” Br. J. Cancer 54, 43–52 (1986).
[CrossRef] [PubMed]

Wilson, B. C.

B. C. Wilson, M. S. Patterson, L. Lilge, “Implicit and explicit dosimetry in photodynamic therapy: a new paradigm,” Lasers Med. Science 12, 182–199 (1997).
[CrossRef]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. C. Wilson, R. Graff, “In vivo tests of the concept of photodynamic threshold dose in normal rat liver photosensitized by aluminum chlorosulphonated phthalocyanine,” Photochem. Photobiol. 51, 343–349 (1990).
[CrossRef] [PubMed]

B. C. Wilson, “Photodynamic therapy: light delivery and dosage for second-generation photosensitizers,” in Photosensitizing Compounds: Their Chemistry, Biology and Clinical Use, G. Bock, S. Harnett, eds. (Wiley, Chichester, 1989), pp. 60–77.

T. J. Farrell, M. S. Patterson, J. E. Hayward, B. C. Wilson, E. R. Beck, “A CCD and neural network based instrument for the non-invasive determinations of tissue optical properties in vivo,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 117–128 (1994).
[CrossRef]

W. M. Star, B. C. Wilson, M. S. Patterson, “Light delivery and optical dosimetry in photodynamic therapy of solid tumors,” in Photodynamic Therapy, B. W. Henderson, T. J. Dougherty, eds. (Marcel Dekker, New York, 1992), pp. 335–368.

M. S. Patterson, B. C. Wilson, “A theoretical study of the influence of sensitizer photobleaching on depth of necrosis in photodynamic therapy,” in Optical Methods for Tumor Treatment and Detection: Mechanisms and Techniques in Photodynamic Therapy III, T. J. Dougherty, ed., Proc. SPIE2133, 208–219 (1994).
[CrossRef]

Wyss, M.

L. O. Svaasand, P. Wyss, M. Wyss, Tadir, B. J. Tromberg, M. W. Berns, “Dosimetry model for photodynamic therapy with topically administered photosensitizers,” Lasers Surg. Med. 18, 139–149 (1996).
[CrossRef] [PubMed]

Wyss, P.

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Figures (10)

Fig. 1
Fig. 1

Schematic illustration of the treatment geometry and the excitation and the emission geometries for the different fluorescence measurement schemes. (a) Broad-beam treatment geometry; isofluence lines are parallel with tissue surface and are depth dependent, and fluorescence emitted from the surface is spatially uniform. (b) Pencil-beam excitation, highly scattering case; excitation fluence has both radial and depth dependence, and fluorescence emitted from the surface is radially dependent. (c) Pencil-beam excitation, highly absorbing case; excitation fluence is confined to axis of pencil beam and is depth dependent, and fluorescence emitted from the surface is radially dependent.

Fig. 2
Fig. 2

Changes in (a) fluence rate versus depth in tissue and (b) PS absorption coefficient (∝ concentration) versus depth as photobleaching proceeds for broad-beam light irradiation and standard tissue and PS parameters: μ a0 = 0.04 mm-1, μ s ′ = 1.0 mm-1, μ aPS = 0.02 mm-1, and β = 0.05 cm2 J-1. In (a) the fluence rate is normalized to the incident fluence rate, and the dotted curve is for μ aPS = 0, i.e., complete photobleaching. In (b) the incident light fluence is indicated by the increasing values of Ψ, and the dotted curve indicates a specific (arbitrary) necrosis threshold level.

Fig. 3
Fig. 3

Fluorescence signal as photobleaching progresses for broad-beam fluorescence excitation: (a) d n versus Ψ, (b) relative fluorescence versus Ψ, (c) d n versus relative fluorescence. Initial conditions as in Fig. 2.

Fig. 4
Fig. 4

Dependence of the depth of necrosis versus relative broad-beam fluorescence signal for varying initial conditions: (a) varying tissue absorption, (b) varying tissue scatter, (c) varying initial PS absorption, (d) same curves as (a) and (b) combined with d n plotted in units of penetration depth. In each case the other three parameters are fixed; the fixed values are the same as those in Fig. 2.

Fig. 5
Fig. 5

Fluorescence signal for different radial distances as photobleaching progresses for pencil-beam fluorescence excitation in highly scattering tissue: (a) F(ρ) versus ψ and (b) d n versus F(ρ). In each graph the minimum ρ value is 0.33 mm, with increments of 0.33 mm. Initial conditions μ a0,x = 0.04 mm-1, μ s,x ′ = 1.0 mm-1, μ aPS,x = 0.02 mm-1, β = 0.05 cm2 J-1, μ a0,m = 0.04 mm-1, μ s,m ′ = 1.0 mm-1, and μ aPS,m = 0.00 mm-1.

Fig. 6
Fig. 6

Spatially resolved fluorescence F(ρ) as a function of ρ for pencil-beam fluorescence excitation, normalized to the value at ρ = 1 mm. In (a) the curves correspond to Ψ of 0, 0.5, 1, 1.5, 2, 3, 5, 15, and 50 J cm-2, whereas in (b) the curves correspond to virtual plane depth of 0, 0.5, 1, 1.5, 2, 3, 4, 5, 15, and 30 mm. Initial conditions are the same as those in Fig. 5.

Fig. 7
Fig. 7

(a) Virtual plane depth versus Ψ and (b) d n versus virtual plane depth for pencil-beam fluorescence excitation and highly scattering tissue. Initial conditions are the same as those in Fig. 5.

Fig. 8
Fig. 8

Necrosis depth as a function of virtual plane depth for pencil-beam fluorescence excitation in highly scattering tissue for varying initial conditions: (a) varying tissue absorption, (b) varying tissue scatter, (c) varying initial PS absorption. In each case the other parameters are fixed; the fixed values are the same as those in Fig. 5.

Fig. 9
Fig. 9

Necrosis depth as a function of virtual point source depth for pencil-beam fluorescence excitation in highly absorbing tissue for varying initial conditions: (a) tissue absorption varied, (b) tissue scatter varied, (c) initial PS concentration varied. Corresponding fixed values for the other parameters: μ a0,x = 0.40 mm-1, μ s,x ′ = 1.0 mm-1, μ aPS,x = 0.02 mm-1, β = 0.01 cm2 J-1, μ a0,m = 0.04 mm-1, μ s,m ′ = 1.0 mm-1, and μ aPS,m = 0.00 mm-1.

Fig. 10
Fig. 10

(a) Raw fluorescence emission spectra at ρ = 0.33 mm on the surface of a tissue-simulating phantom-containing Photofrin with 514-nm pencil-beam excitation (peak shown). Successive curves correspond to the increasing fluence values, ψ. The 630- and the 690-nm fluorescence peaks are bleached as treatment progresses. A secondary photoproduct peak near 660 nm increased initially and then was also photobleached. (b) F(ρ) as a function of source–detector distance ρ, normalized to the value at ρ = 0.33 mm at the start (ψ = 0 J cm-2) and at the end of treatment (Ψ = 125 J cm-2).

Tables (1)

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Table 1 Summary of Standard Optical Properties Used in Simulations

Equations (45)

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D 2 Φ r - μ a Φ r = - S r ,
D = 3 μ a + μ s - 1 .
D z 2 z 2   Φ z - μ a z Φ z = - S z .
S z = S 0 μ s   exp - 0 z   μ t w d w ,
2 z 2   Φ z - μ eff 2 z Φ z = - 3 S 0 μ s μ t z ×   exp - 0 z   μ t w d w ,
Φ z = S 0 b μ eff z 1 / 2 exp - 0 z   μ eff w d w + C z exp - 0 z   μ t w d w
d μ a PS z ,   t d t = - β Φ z ,   t μ a PS z ,   t ,
μ a PS z ,   t + Δ t = μ a PS z ,   t - β Φ z ,   t μ a PS z ,   t Δ t .
f z = γ Φ z μ a PS , x z ,
E 0 ;   z s = 1 2 exp - μ eff , m z s 1 + exp - 4 A μ eff , m 3 μ t , m ,
F 0 = γ   0   E 0 ;   z s Φ z s μ a PS , x z s d z s .
S r = S 0 μ s , x   exp - μ t , x z δ ρ ,
S r = S 0 μ s , x μ t , x   δ z - 1 μ t , x δ ρ .
Φ x ρ s ,   z s ;   z 0 = S 0 μ s , x μ t , x 1 4 π D x exp - μ eff , x r 1 r 1 - exp - μ eff , x r 2 r 2 ,
r 1 = z s - z 0 2 + ρ s 2 1 / 2 ,
r 2 = z 0 + 2 z b + z s 2 + ρ s 2 1 / 2 .
μ eff , x = 3 μ s , x + μ a 0 , x μ a 0 , x 1 / 2 , D x = 3 μ s , x + μ a 0 , x - 1 .
f ρ s ,   z s = γ Φ x ρ s ,   z s μ a PS , x z s .
E ρ ,   0 ;   ρ s ,   z s = 1 4 π D m exp - μ eff , m r 1 r 1 - exp - μ eff , m r 2 r 2 ,
r 1 = z s 2 + ρ - ρ s 2 1 / 2 ,
r 2 = z s + 2 z b 2 + ρ - ρ s 2 1 / 2 .
r 1 = z s 2 + ρ s 2 + ρ 2 - 2 ρ s ρ   cos   θ 1 / 2 , r 2 = z s + 2 z b 2 + ρ s 2 + ρ 2 - 2 ρ s ρ   cos   θ 1 / 2 .
F ρ ,   0 ;   z s =   0 0 2 π   E ρ ,   0 ;   ρ s ,   z s f ρ s ,   z s ρ s d θ d ρ s .
F ρ ,   0 = 0   F ρ ,   0 ;   z s d z s .
Φ r = Φ 0   exp - 0 z   μ a , x w d w δ ρ .
f 0 ,   z s = γ μ a PS , x z s exp - 0 z   μ a , x w d w .
E ρ ,   0 ;   0 ,   z s = 1 4 π z s μ eff , m + 1 r 1 exp - μ eff , m r 1 r 1 2 + z s + 2 z b μ eff , m + 1 r 2 ×   exp - μ eff , m r 2 r 2 2 ,
r 1 = z s 2 + ρ 2 1 / 2 ,     r 2 = z s + 2 z b 2 + ρ 2 1 / 2 .
F ρ ,   0 = γ   0   μ a PS , x z s exp - 0 z s   μ a , x z d z × E ρ ,   0 ;   0 ,   z s d z 0 .
d D d t = γ 0 Φ z ,   t μ a PS z ,   t ,
d D d t = - γ 0 β d μ a PS d t .
D z ,   t = γ 0 β μ a PS z ,   0 - μ a PS z ,   t .
2 z 2   Φ z - μ eff 2 z Φ z = - 3 μ s μ t z ×   exp - 0 z   μ t w d w .
Φ z 1 μ eff z 1 / 2 a   exp + 0 z   μ eff w d w + b   exp - 0 z   μ eff w d w .
Φ z b μ eff z exp - 0 z   μ eff w d w + P z exp - 0 z   μ t w d w ,
Φ 0 - 2 D z z   Φ z | z = 0 = 0 .
Φ 0 - 2 AD z z   Φ z | z = 0 = 0 .
A = 1 + r d 1 - r d ,
r d = - 1.440 n rel - 2 + 0.710 n rel - 1 + 0.668 + 0.0636 n rel ,
b = - 3 μ t μ eff 3 / 2 3 μ t μ eff + 2 A μ eff 2 + A   z   μ eff 1 + 2 3   A P z | z = 0 - 2 A 3 μ t z   P z | z = 0 ,
Φ z = S 0 b μ eff z 1 / 2 exp - 0 z   μ eff w d w + C z exp - 0 z   μ t w d w ,
Φ z = G   exp - μ eff z + H   exp - μ t z ,
G = S 0 1 - 3 μ a μ t μ s 3 + 2 A μ t 1 + 2 A μ a 3 μ t 1 / 2 , H = - 2 S 0 1 - 3 μ a μ t .
2 z 2   P z - 2 μ t z z   P z + μ t z 2 - μ eff z 2 - z   μ t z P z + 3 μ s μ t z = 0 .
P z z = - 3 μ s μ t 0 μ t 0 2 - μ eff 0 2 ,

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