Abstract

Interstitial determination of the tissue optical properties is important in biomedicine, especially for interstitial laser therapies. Continuous wave (CW) radiance techniques which examine light from multiple directions have been proposed as minimally invasive methods for determining the optical properties under an interstitial probe arrangement. However, both the fitting algorithm based on the P3 approximation and the analytical method based on the diffusion approximation (DA), which are currently used recovery algorithms, cannot extract the optical properties of tissue with low transport albedos accurately from radiance measurements. In this paper, we proposed an incomplete P3 approximation for the radiance, the P3in for short, which is the asymptotic part of the solution for the P3 approximation. The relative differences between the P3in and the P3 were within 0.48% over a wide range of clinically relevant optical properties for measurements at source detector separations (SDS) from 5 mm to 10 mm and angles from 0° to 160°. Based on the P3in, we developed an analytical method for extracting the optical properties directly using simple expressions constructed from the radiance measurements at only two SDSs and four angles. The developed recovery algorithm was verified by simulated and experimental radiance data. The results show that both the absorption and reduced scattering coefficients were recovered accurately with relative errors within 5.28% and 3.86%, respectively, from the simulated data and with relative errors within 10.82% and 10.67%, respectively, from the experimental data over a wide range of albedos from 0.5 to 0.99. Since the developed P3in-based radiance technique can obtain the optical properties rapidly from the measurements at only two SDSs and four angles, it is expected to be used for in vivo and in situ determination of the optical properties in online treatment planning during laser therapies.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2017 (1)

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

2016 (3)

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

S.-Y. Tzeng, J.-Y. Guo, C.-C. Yang, C.-K. Hsu, H.-J. Huang, S.-J. Chou, C.-H. Hwang, and S.-H. Tseng, “Portable handheld diffuse reflectance spectroscopy system for clinical evaluation of skin: a pilot study in psoriasis patients,” Biomed. Opt. Express 7(2), 616–628 (2016).
[PubMed]

2015 (1)

2014 (1)

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

2013 (1)

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

2012 (4)

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Y. Pu, W. Wang, M. Al-Rubaiee, S. K. Gayen, and M. Xu, “Determination of optical coefficients and fractal dimensional parameters of cancerous and normal prostate tissues,” Appl. Spectrosc. 66(7), 828–834 (2012).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

2011 (2)

2010 (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

2008 (2)

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

2007 (1)

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

2006 (2)

2005 (4)

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

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(10), 2291–2311 (2005).
[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 motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

2001 (2)

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).

1997 (1)

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

1991 (1)

Aernouts, B.

Alerstam, E.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Al-Rubaiee, M.

Andersson-Engels, S.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Ballangrud, Å. M.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Barajas, O.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Baran, T. M.

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Bargo, P. R.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Bartels, K. E.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Beeson, K.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Bellnier, D.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Blair, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Bunting, C. F.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Chin, L. C. L.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Chou, S.-J.

Dickey, D. J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

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(10), 2291–2311 (2005).
[PubMed]

Einarsdóttír, M.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Evers, D. J.

Fenn, M. C.

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

Finlay, J. C.

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 motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

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(10), 2291–2311 (2005).
[PubMed]

Foschum, F.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Foster, T. H.

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).

Gayen, S. K.

Goodell, T. T.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Grabtchak, S.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Guo, J.-Y.

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 motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Hamilton, S.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Hendriks, B. H.

Holyoak, G. R.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Hsu, C.-K.

Huang, H.-J.

Hull, E. L.

Hwang, C.-H.

Jacques, S. L.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Jiang, Z.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Khan, T.

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Kienle, A.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

Koval, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Li, J.

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

Liemert, A.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

Lucassen, G. W.

Messing, E. M.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Miller, G. G.

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Mitra, S.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Moes, C. J.

Montgomery, L. G.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

Moore, R. B.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Nachabé, R.

Nagarajan, V. K.

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Oakley, E.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Palmer, T. J.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

Parilov, E.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Patterson, M. S.

Piao, D.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Potasek, M.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Prahl, S. A.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991).
[PubMed]

Pu, Y.

Rayner, D. C.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

Ritchey, J. W.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Ruers, T. J.

Saeys, W.

Shafirstein, G.

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

Sleven, R. A.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

Slobodov, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Svanberg, K.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Svensson, T.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

Thomas, A.

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Tseng, S.-H.

Tulip, J.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

Tzeng, S.-Y.

Van Beers, R.

van der Voort, M.

van Gemert, M. J.

van Marie, J.

van Staveren, H. J.

Vitkin, I. A.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Waldman, D. L.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Wang, W.

Watté, R.

Wesseling, J.

Whelan, W. M.

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

S. Grabtchak and W. M. Whelan, “Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements,” Biomed. Opt. Express 3(10), 2371–2380 (2012).
[PubMed]

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006).
[PubMed]

Wilson, J. D.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Worthington, A. E.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

Xu, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

Xu, H.

Xu, M.

Yang, C.-C.

Yao, J. L.

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

Yu, B.

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Zhu, T. C.

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[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 motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

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(10), 2291–2311 (2005).
[PubMed]

Appl. Opt. (2)

Appl. Spectrosc. (1)

Biomed. Opt. Express (3)

Cancers (Basel) (1)

G. Shafirstein, D. Bellnier, E. Oakley, S. Hamilton, M. Potasek, K. Beeson, and E. Parilov, “Interstitial photodynamic therapy-a focused review,” Cancers (Basel) 9(2), 12 (2017).
[PubMed]

IEEE. J. Sel. Top. Quant. (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE. J. Sel. Top. Quant. 16(4), 715–729 (2010).

J. Biomed. Opt. (7)

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012).
[PubMed]

T. M. Baran, “Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers,” J. Biomed. Opt. 21(7), 77001 (2016).
[PubMed]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[PubMed]

S. Grabtchak, L. G. Montgomery, and W. M. Whelan, “Feasibility of interstitial near-infrared radiance spectroscopy platform for ex vivo canine prostate studies: optical properties extraction, hemoglobin and water concentration, and gold nanoparticles detection,” J. Biomed. Opt. 19(5), 057003 (2014).
[PubMed]

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[PubMed]

T. M. Baran, M. C. Fenn, and T. H. Foster, “Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe,” J. Biomed. Opt. 18(10), 107007 (2013).
[PubMed]

T. M. Baran, J. D. Wilson, S. Mitra, J. L. Yao, E. M. Messing, D. L. Waldman, and T. H. Foster, “Optical property measurements establish the feasibility of photodynamic therapy as a minimally invasive intervention for tumors of the kidney,” J. Biomed. Opt. 17(9), 98002 (2012).
[PubMed]

J. Biophotonics (1)

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J. Biophotonics 1(3), 200–203 (2008).
[PubMed]

J. Opt. Soc. Am. A (1)

J. Photochem. Photobiol. B (1)

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 motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005).
[PubMed]

Lasers Surg. Med. (1)

V. K. Nagarajan and B. Yu, “Monitoring of tissue optical properties during thermal coagulation of ex vivo tissues,” Lasers Surg. Med. 48(7), 686–694 (2016).
[PubMed]

Opt. Commun. (1)

T. Khan and A. Thomas, “Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,” Opt. Commun. 255(1), 130–166 (2005).

Opt. Express (2)

Phys. Med. Biol. (4)

J. Li and T. C. Zhu, “Determination of in vivo light fluence distribution in a heterogeneous prostate during photodynamic therapy,” Phys. Med. Biol. 53(8), 2103–2114 (2008).
[PubMed]

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001).
[PubMed]

O. Barajas, Å. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997).
[PubMed]

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(10), 2291–2311 (2005).
[PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3 Pt 2), 036605 (2011).
[PubMed]

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Figures (8)

Fig. 1
Fig. 1 The maximum relative differences between the P3in and the P3, ε max , in the optical property range of 0.01 mm 1 μ a 1.0 mm 1 and 0.5 mm 1 μ s 4.5 mm 1 at SDSs from 5 mm to 10 mm and angles from 0° to 180°.
Fig. 2
Fig. 2 Geometry for measurements of the radiances, L m (r,θ), at SDSs of r and r and angles of θ 0 , θ 1 , θ 2 and θ 3 .
Fig. 3
Fig. 3 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of typical scattering. (a) is for μ a and (b) is for μ s . The insets show the relative errors (REs) of the recovered optical properties.
Fig. 4
Fig. 4 Recovered optical properties based on the DA and the P3in from the simulated radiance data calculated by the P25 in the case of low scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 5
Fig. 5 A schematic of measuring the radiances from an isotropic point source at SDS of r and angle of θ.
Fig. 6
Fig. 6 Recovered optical properties based on the DA and the P3in from the phantom measurements in the case of typical scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 7
Fig. 7 Recovered optical properties obtained from the phantom measurements in the case of low scattering. (a) is for μ a and (b) is for μ s . The insets show the REs of the recovered optical properties.
Fig. 8
Fig. 8 REs of the recovered optical properties for 0.01 mm 1 μ a 1.0 mm 1 and 0.5 mm 1 μ s 4.5 mm 1 . (a) and (b) are based on the P3in. (c) and (d) are based on the P5in. (a) and (c) are for μ a . (b) and (d) are for μ s .

Equations (17)

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L(r,θ)= S 0 4π l=0 N (2l+1) φ l (r) P l (cosθ) ,
l+1 2l+1 [ d dr + l+2 r ] φ l+1 (r)+ l 2l+1 [ d dr l1 r ] φ l1 (r)+ σ l φ l (r)= S 0 δ l0 ,l=0,...,3,
L(r,θ)= S 0 4π l=0 3 (2l+1)[ C h l ( ν ) Q l ( ν r)+ D h l ( ν + ) Q l ( ν + r)] P l (cosθ) ,
C = ( ν ) 5 12π μ a 2 σ 1 (3 μ a σ 1 ( ν + ) 2 ) ( ( ν ) 2 ( ν + ) 2 ) , D = ( ν + ) 5 12π μ a 2 σ 1 (3 μ a σ 1 ( ν ) 2 ) ( ( ν + ) 2 ( ν ) 2 ) .
h 0 =1, h 1 ( ν ± )= μ a ν ± , h 2 ( ν ± )= 1 2 + 3 μ a σ 1 2 ( ν ± ) 2 , h 3 ( ν ± )= 9 μ a σ 1 14 σ 3 ν ± 3 ν ± 14 σ 3 .
L in (r,θ)= S 0 C 4π l=0 3 (2l+1) h l (ν) Q l (νr) P l (cosθ) .
l=0 3 S 0 C 4π (2l+1) h l (ν) Q l (νr) P l (cos θ i ) = L m (r, θ i ),i=0,...,3.
S 0 C h 0 (ν) Q 0 (νr)= 4π( α 3 (r, θ 0 , θ 1 , θ 2 ) β 13 ( θ 0 , θ 1 , θ 3 ) α 3 (r, θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 2 )) ( β 03 ( θ 0 , θ 1 , θ 2 ) β 13 ( θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 2 )) ,
S 0 C h 1 (ν) Q 1 (νr)= 4π( α 3 (r, θ 0 , θ 1 , θ 2 ) β 03 ( θ 0 , θ 1 , θ 3 ) α 3 (r, θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 2 )) 3( β 13 ( θ 0 , θ 1 , θ 2 ) β 03 ( θ 0 , θ 1 , θ 3 ) β 13 ( θ 0 , θ 1 , θ 3 ) β 03 ( θ 0 , θ 1 , θ 2 )) ,
S 0 C h 2 (ν) Q 2 (νr)= 4π( α 1 (r, θ 0 , θ 1 , θ 2 ) β 31 ( θ 0 , θ 1 , θ 3 ) α 1 (r, θ 0 , θ 1 , θ 3 ) β 31 ( θ 0 , θ 1 , θ 2 )) 5( β 21 ( θ 0 , θ 1 , θ 2 ) β 31 ( θ 0 , θ 1 , θ 3 ) β 21 ( θ 0 , θ 1 , θ 3 ) β 31 ( θ 0 , θ 1 , θ 2 )) ,
α i (r, θ 0 , θ 1 , θ 2 )= ( L m (r, θ 0 ) P i (cos θ 1 ) L m (r, θ 1 ) P i (cos θ 0 ))( P i1 (cos θ 0 ) P i (cos θ 2 ) P i1 (cos θ 2 ) P i (cos θ 0 )), ( L m (r, θ 0 ) P i (cos θ 2 ) L m (r, θ 2 ) P i (cos θ 0 ))( P i1 (cos θ 0 ) P i (cos θ 1 ) P i1 (cos θ 1 ) P i (cos θ 0 ))
β ij ( θ 0 , θ 1 , θ 2 )= ( P i (cos θ 0 ) P j (cos θ 1 ) P i (cos θ 1 ) P j (cos θ 0 ))( P j1 (cos θ 0 ) P j (cos θ 1 ) P j1 (cos θ 1 ) P j (cos θ 0 )). ( P i (cos θ 0 ) P j (cos θ 2 ) P i (cos θ 2 ) P j (cos θ 0 ))( P j1 (cos θ 0 ) P j (cos θ 2 ) P j1 (cos θ 2 ) P j (cos θ 0 ))
h 0 (ν) Q 0 (νr) h 0 (ν) Q 0 (ν r ) = f 0 (r, θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) .
ν= ln[ f 0 (r, θ 0 , θ 1 , θ 2 , θ 3 ) f 0 ( r , θ 0 , θ 1 , θ 2 , θ 3 ) r r ]/ ( r r) .
h l (ν) Q l (νr) h 0 (ν) Q 0 (νr) = f l (r, θ 0 , θ 1 , θ 2 , θ 3 ) f 0 (r, θ 0 , θ 1 , θ 2 , θ 3 ) ,l=1,2.
h l (ν)= Q 0 (νr) Q l (νr) f l (r, θ 0 , θ 1 , θ 2 , θ 3 ) f 0 (r, θ 0 , θ 1 , θ 2 , θ 3 ) ,l=1,2.
μ a = h 1 (ν)ν, μ s = (2 h 2 (ν)+1)ν 3 h 2 (ν) h 1 (ν)ν.

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