Abstract

We estimated the absorption and reduced scattering coefficients of tissue-simulating phantoms from interstitial measurements of the phase difference and relative amplitude signals at two distances from a sinusoidally modulated isotropic source. It was found that absorption and reduced scattering coefficients can be recovered within 10% and slightly over 10% respectively, using either the data collected by two detectors 3mm apart or by two detectors 5mm apart with light collected by one detector attenuated by a neutral density filter. This accuracy was achieved over a wide range of optical properties, µa=0.008 to 0.17mm-1 and µ s ’=0.3 to 1.8mm-1. Additional factors affecting accuracy including source anisotropy, uncertainty in fiber placement, phase amplitude crosstalk, and the forward light propagation model (the combined isotropic similarity model and standard diffusion approximation versus the modified spherical harmonics method) were studied by Monte Carlo simulations (first two factors) and experiments (last two factors).

© 2006 Optical Society of America

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    [CrossRef]

2006

H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006).
[CrossRef] [PubMed]

G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006).
[CrossRef]

2005

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys Med Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

Z. Huang, "A review of progress in clinical photodynamic therapy," Technol. Cancer. Res. Treat. 4,283-293 (2005).
[PubMed]

R. A. Weersink, A. Bogaards, M, Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, "Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities," J. Photochem. Photobiol B: Biology. 79. 211-222 (2005).
[CrossRef] [PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, "Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in vivo in human prostate during during motexafin lutetium mediated photodynamic therapy," J. Photochem. Photobiol B: Biology. 79, 231-241 (2005).
[CrossRef] [PubMed]

2004

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

V. A. Markel, "Modified spherical harmonics method for solving the radiative transport equation," Waves Random Media 14, L13-L19 (2004).
[CrossRef]

2003

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

R. Weissleder and V. Ntziachristos, "Shedding light onto live molecule targets," Nature. Med. 9,123-128 (2003).
[CrossRef] [PubMed]

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

2002

M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002).
[CrossRef]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002).
[CrossRef]

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
[CrossRef] [PubMed]

2000

1999

1998

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
[CrossRef]

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998).
[CrossRef]

1997

1996

R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 47,555-606 (1996).
[CrossRef] [PubMed]

1995

1993

J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A. 10,127-140 (1993).
[CrossRef] [PubMed]

1992

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

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

1981

1941

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93,70-83, (1941).
[CrossRef]

Alford, K.

K. Alford and Y. Wickramasinghe, "Phase-amplitude crosstalk in intensity modulated near infrared spectroscopy," Rev. Sci. Instrum. 71,2191-2195 (2000).
[CrossRef]

Baker, K.S.

Beckers, J.

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Berger, A. J.

Bevilacqua, F.

Bogaards, A.

R. A. Weersink, A. Bogaards, M, Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, "Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities," J. Photochem. Photobiol B: Biology. 79. 211-222 (2005).
[CrossRef] [PubMed]

Brun, P.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Bush, T.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Cerussi, A. E.

Cerussi, A.E.

Chance, B.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
[CrossRef]

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998).
[CrossRef]

Chen, Q.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Cope, M.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
[CrossRef]

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys Med Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Douplik, A.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Du, C.

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998).
[CrossRef]

Fallstrom, K.

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002).
[CrossRef]

Fantini, S.

Faris, G. W.

Farrell, T. J.

H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006).
[CrossRef] [PubMed]

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
[CrossRef] [PubMed]

Finlay, F. C.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys Med Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, "Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in vivo in human prostate during during motexafin lutetium mediated photodynamic therapy," J. Photochem. Photobiol B: Biology. 79, 231-241 (2005).
[CrossRef] [PubMed]

Fishkin, J. B.

J. B. Fishkin, P. T. C. So, A.E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, "Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom," Appl. Opt. 34,1143-1155 (1995).
[CrossRef] [PubMed]

J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A. 10,127-140 (1993).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

Foster, T. H.

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

Franceschini, M. A.

Gao, L.

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

Gerken, M.

Gilead, A.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Glatstein, E.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Gratton, E.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
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J. B. Fishkin, P. T. C. So, A.E. Cerussi, S. Fantini, M. A. Franceschini, and E. Gratton, "Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom," Appl. Opt. 34,1143-1155 (1995).
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J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A. 10,127-140 (1993).
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Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93,70-83, (1941).
[CrossRef]

Gross, S.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Hahn, S. M.

T. C. Zhu, J. C. Finlay, and S. M. Hahn, "Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in vivo in human prostate during during motexafin lutetium mediated photodynamic therapy," J. Photochem. Photobiol B: Biology. 79, 231-241 (2005).
[CrossRef] [PubMed]

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Hayward, J. E.

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93,70-83, (1941).
[CrossRef]

Hetzel, F. W.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

His, R. A.

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
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Huang, Z.

Z. Huang, "A review of progress in clinical photodynamic therapy," Technol. Cancer. Res. Treat. 4,283-293 (2005).
[PubMed]

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Hunter, R. J.

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
[CrossRef] [PubMed]

Jacques, S. L.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

Jakubowski, D.

Jankun, J.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Kapatkin, A. S.

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Katila, T.

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002).
[CrossRef]

Keck, R. W.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Kotilahti, K.

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002).
[CrossRef]

LaRue, S. M.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Lilge, L.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Luck, D.

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Ma, H. Y.

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998).
[CrossRef]

Madsen, S. J.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

Malkowicz, S. B.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Markel, V. A.

G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006).
[CrossRef]

V. A. Markel, "Modified spherical harmonics method for solving the radiative transport equation," Waves Random Media 14, L13-L19 (2004).
[CrossRef]

Miles, J.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Moes, C. J. M.

Neeman, M.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Niedre, M.

M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002).
[CrossRef]

Nissila, I.

I. Nissila, K. Kotilahti, K. Fallstrom, and T. Katila, "Instrumentation for the accurate measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 73,3306-3312 (2002).
[CrossRef]

Ntziachristos, V.

R. Weissleder and V. Ntziachristos, "Shedding light onto live molecule targets," Nature. Med. 9,123-128 (2003).
[CrossRef] [PubMed]

Okamoto, A.

S. Yokoyama and A. Okamoto, "Examination to eliminate undesirable phase delay of an avalanche photodiode (APD) for intensity-modulated light," Rev. Sci. Instrum. 66,5331-5336 (1995).
[CrossRef]

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006).
[CrossRef]

Patterson, M. S.

H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006).
[CrossRef] [PubMed]

R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
[CrossRef] [PubMed]

M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002).
[CrossRef]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

Pestka, M.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Pomerleau-Dalcourt, N.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Prahl, S. A.

Ramanujam, N.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
[CrossRef]

N. Ramanujam, C. Du, H. Y. Ma, and B. Chance, "Sources of phase noise in homodyne phase modulation devices used for tissue oximetry studies," Rev. Sci. Instrum. 69,3042-3054 (1998).
[CrossRef]

Richards-Kortum, R.

R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 47,555-606 (1996).
[CrossRef] [PubMed]

Rodriguez, C. E.

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Salomon, Y.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Scherz, A.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

Schotland, J. C.

G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006).
[CrossRef]

Selman, S. H.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Sevick-Muraca, E.

R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 47,555-606 (1996).
[CrossRef] [PubMed]

Shapiro, H.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Smith, R. C.

So, P. T. C.

Star, W. M.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

Stripp, D.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Szkudlarek, M.

L. Lilge, N. Pomerleau-Dalcourt, A. Douplik, S. H. Selman, R. W. Keck, M. Szkudlarek, M. Pestka, and J. Jankun, "Transperineal in vivo fluence-rate dosimetry in the canine prostate during SnET2-mediated PDT," Phys. Med. Biol. 49,3209-3225 (2004).
[CrossRef] [PubMed]

Tochner, Z.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Trncic, N.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Tromberg, B.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, "Phase measurement of light absorption and scatter in human tissue," Rev. Sci. Instrum. 69,3457-3481 (1998).
[CrossRef]

Tromberg, B. J.

Van Gemert, M. J. C.

H. J. van Staveren, C. J. M. Moes, J. Van Marle, S. A. Prahl, and M. J. C. Van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 30,4507-4514 (1997).
[CrossRef]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

Van Marle, J.

van Staveren, H. J.

Vulcan, T. G.

T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

Weersink, R. A.

R. A. Weersink, A. Bogaards, M, Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, "Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities," J. Photochem. Photobiol B: Biology. 79. 211-222 (2005).
[CrossRef] [PubMed]

Weissleder, R.

R. Weissleder and V. Ntziachristos, "Shedding light onto live molecule targets," Nature. Med. 9,123-128 (2003).
[CrossRef] [PubMed]

Whittington, R.

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Wickramasinghe, Y.

K. Alford and Y. Wickramasinghe, "Phase-amplitude crosstalk in intensity modulated near infrared spectroscopy," Rev. Sci. Instrum. 71,2191-2195 (2000).
[CrossRef]

Wilson, B. C.

Z. Huang, Q. Chen, N. Trncic, S. M. LaRue, P. Brun, B. C. Wilson, H. Shapiro, and F. W. Hetzel, "Effects of Pd-bacteriopheophorbide (Tookad) mediated photodynamic therapy on canine prostate pretreated with ionizing radiation," Radiat. Res. 161, 723-731, (2004).
[CrossRef] [PubMed]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
[CrossRef]

M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002).
[CrossRef]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

Xu, H.

H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006).
[CrossRef] [PubMed]

Yokoyama, S.

S. Yokoyama and A. Okamoto, "Examination to eliminate undesirable phase delay of an avalanche photodiode (APD) for intensity-modulated light," Rev. Sci. Instrum. 66,5331-5336 (1995).
[CrossRef]

Zhu, T. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys Med Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

T. C. Zhu, J. C. Finlay, and S. M. Hahn, "Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in vivo in human prostate during during motexafin lutetium mediated photodynamic therapy," J. Photochem. Photobiol B: Biology. 79, 231-241 (2005).
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T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
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Annu. Rev. Phys. Chem.

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Appl. Opt.

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[CrossRef]

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H. Xu, T. J. Farrell, and M. S. Patterson, "Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements," J. Biomed. Opt.(to be published2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A.

J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A. 10,127-140 (1993).
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R. A. Weersink, A. Bogaards, M, Gertner, S. R. H. Davidson, K. Zhang, G. Netchev, J. Trachtenberg, and B. C. Wilson, "Techniques for delivery and monitoring of TOOKAD (WST09)-mediated photodynamic therapy of the prostate: clinical experience and practicalities," J. Photochem. Photobiol B: Biology. 79. 211-222 (2005).
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T. C. Zhu, J. C. Finlay, and S. M. Hahn, "Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in vivo in human prostate during during motexafin lutetium mediated photodynamic therapy," J. Photochem. Photobiol B: Biology. 79, 231-241 (2005).
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J. Phys. A

G. Panasyuk, J. C. Schotland, and V. A. Markel, "Radiative transport equation in rotated reference frames," J. Phys. A,  39,115-137 (2006).
[CrossRef]

Lasers Surg. Med.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. Van Gemert, "Optical properties of Liposyn: a phantom medium for light propagation studies," Lasers Surg. Med. 12,510-519 (1992).
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Nat. Med.

S. Gross, A. Gilead, A. Scherz, M. Neeman, and Y. Salomon, "Monitoring photodynamic therapy of solid tumors on-line by BOLD-contrast MRI," Nat. Med. 9, 1327-1331 (2003).
[CrossRef] [PubMed]

Nature. Med.

R. Weissleder and V. Ntziachristos, "Shedding light onto live molecule targets," Nature. Med. 9,123-128 (2003).
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Opt. Lett.

Photochem. Photobio.

M. Niedre, M. S. Patterson, and B. C. Wilson, "Direct near-infrared luminescence detection of singlet oxygen generated by photodynamic therapy in cells in vitro and tissues in vivo," Photochem. Photobio. 75,382-391 (2002).
[CrossRef]

Q. Chen, Z. Huang, D. Luck, J. Beckers, P. Brun, B. C. Wilson, A. Scherz, Y. Salomon, and F. W. Hetzel, "Preclinical studies in normal canine prostate of a novel palladium-bacteriopheophorbide (WST09) photosensitizer for photodynamic therapy of prostate cancer," Photochem. Photobio. 76,438-445, (2002).
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T. C. Zhu, S. M. Hahn, A. S. Kapatkin, A. Dimofte, C. E. Rodriguez, T. G. Vulcan, E. Glatstein, and R. A. His, "In-vivo optical properties of normal canine prostate at 732nm using motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 77,81-88 (2003).
[CrossRef] [PubMed]

T. C. Zhu, A. Dimofte, F. C. Finlay, D. Stripp, T. Bush, J. Miles, R. Whittington, S. B. Malkowicz, Z. Tochner, E. Glatstein, and S. M. Hahn, "Optical properties of human prostate at 732nm measured in-vivo during motexafin lutetium mediated photodynamic therapy," Photochem. Photobiol. 81,96-105 (2005).
[CrossRef]

Phys Med Biol.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys Med Biol. 50,2291-2311 (2005).
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R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, "Haemoglobin oxygenation of a two-layer tissue-simulating phantom from time-resolved reflectance: effect of top layer thickness," Phys Med Biol. 47,193-208 (2002).
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Phys. Med. Biol.

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Other

M. S. Patterson and B. C. Wilson, "Photodynamic therapy," in The Modern Technology of Radiation Oncology, J. Van Dyk, eds. (Medical Physics Publishing, Madison, Wisconsin 1999), Chap. 23.

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Figures (8)

Fig. 1.
Fig. 1.

The geometry of the two-distance detection scheme. Two detector fibers are placed parallel to each other as well as to the source fiber. For these experiments, a single detector fiber was translated between positions.

Fig. 2.
Fig. 2.

Phase shift vs. amplitude at high voltage of 700V on PMT. The light was attenuated using a variable optical attenuator. Error bars show the standard deviations.

Fig. 3.
Fig. 3.

Phase shift due to the attenuation through four neutral density filters (O.D.=0.38, 0.43, 0.92, and 0.99). Separation between the two collimating lenses was 3.0cm. The output current at the PMT was maintained at 0.35uA. Error bars show the standard deviations.

Fig. 4.
Fig. 4.

Phase shift caused by the insertion of 0.92 OD neutral density filter at four different separations (0.0, 3.1, 5.5, and 8.0cm) between the two collimating lenses. Error bars show the standard deviations.

Fig. 5.
Fig. 5.

Fluence rate (mm-2s-1) versus polar angle defined in the insert. Fluence rate was calculated at 3mm from the source. In the insert, the bold line represents the fiber, and the angle denoted by θ varies from 0 (positive z direction) to 180 (negative z direction).

Fig. 6.
Fig. 6.

Percent errors in estimated OPs using ISM/SDA model for nine phantoms at locations 1, 2, and 3 that are shown by sub-Fig (a), (b) and (c) respectively. Phantom identification numbers can be found in Table 5.

Fig. 7.
Fig. 7.

Percent errors in estimated OPs using MSH model for nine phantoms at locations 1, 2, and 3 that are shown by sub-Figs. (a), (b) and (c) respectively. Phantom identification numbers can be found in Table 5.

Fig. 8.
Fig. 8.

Percent errors in estimated OPs using ISM/SDA [sub-Fig. (a)] and MSH [sub-Fig. (b)] models for nine phantoms at location (5, 10)mm. Phantom identification numbers can be found in Table 5.

Tables (6)

Tables Icon

Table. 1. Recovered OPs using ISM/SDA and MSH for three pairs of positions. The expected OPs: µs’=0.4mm-1, µa=0.08mm-1

Tables Icon

Table. 2. Experimental comparison of MSH, and ISM/SDA. Detection position pair was (5, 8). The expected µs’=0.2mm-1

Tables Icon

Table 3. Monte Carlo study of relative difference of recovered OPs using ISM/SDA model in µa (left) and µs’ (right).

Tables Icon

Table 4. Monte Carlo study of the effect of the shift of two detectors on recovered OPs using ISM/SDA model in µa and µs’.

Tables Icon

Table 5. OPs for the phantoms used in Figs. 4, 5 (top two rows), and 6 (bottom two rows).

Tables Icon

Table 6. The average of root-mean-square percent errors and standard deviations (in bracket) in recovered OPs for nine phantoms (for OPs, see text) using ISM/SDA and MSH.

Equations (14)

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

Re ( φ ) = μ t ' 4 π r [ k 0 2 c exp ( a 0 μ t ' r ) + 0 1 g ( c , ξ ) exp ( μ t ' r ξ ) d ξ ξ 2
Im ( φ ) = 1 2 π r . ω v . 1 2 π i + k ' k ' 2 + 1 . exp [ i k ' ( μ a + μ s ' ) r ] [ 1 ( c k ' ) · tan 1 k ' ] 2 d k '
μ a = ( 3.28 ± 0.15 ) · ( % c ) + ( 0.0038 ± 0.0008 ) ( m m 1 )
ϕ ( z ) = A ( μ s ' , μ a ) · exp ( μ eff z )
μ eff 2 = 3 μ s ' μ a ink + 3 μ s ' μ a w
μ ' s = ( 0.97 ± 0.08 ) · ( % c ) ( m m 1 )
θ = 3 r 1 r 2 2 v μ s ' μ a ω 2 π
μ a = ω 2 v · ln ( r 2 A 1 r 1 A 2 ) θ
Δ μ a μ a = Δθ θ + ( Δ A 1 A 1 + Δ A 2 A 2 ) ln ( A 1 A 2 )
μ s = 4 v 2 3 d 2 ω 2 μ a θ 2
Δ μ s μ s = 2 Δ θ θ + Δ μ a μ a
R l l = b l δ l , l 1 + b l + 1 δ l , l + 1
S ll = μ l δ ll
φ ( r ) = 2 μ 0 r n 0 | y n y n | 0 λ n 2 exp ( r λ n )

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