Abstract

This is Part II of the work that examines photon diffusion in a homogenous medium enclosed by a concave circular cylindrical applicator or enclosing a convex circular cylindrical applicator. Part I of this work [J. Opt. Soc. Am. A 27, 648 (2010)] analytically examined the steady-state photon diffusion between a source and a detector for two specific cases: (1) the detector is placed only azimuthally with respect to the source, and (2) the detector is placed only longitudinally with respect to the source, in the infinitely long concave and convex applicator geometries. For the first case, it was predicted that the decay rate of photon fluence would become smaller in the concave geometry and greater in the convex geometry than that in the semi-infinite geometry for the same source–detector distance. For the second case, it was projected that the decay rate of photon fluence would be greater in the concave geometry and smaller in the convex geometry than that in the semi-infinite geometry for the same source–detector distance. This Part II of the work quantitatively examines these predictions from Part I through several approaches, including (a) the finite-element method, (b) the Monte Carlo simulation, and (c) experimental measurement. Despite that the quantitative examinations have to be conducted for finite cylinder applicators with large length-to-radius ratio to approximate the infinite-length condition modeled in Part I, the results obtained by these quantitative methods for two concave and three convex applicator dimensions validated the qualitative trend predicted by Part I and verified the quantitative accuracy of the analytic treatment of Part I in the diffusion regime of the measurement, at a given set of absorption and reduced scattering coefficients of the medium.

© 2011 Optical Society of America

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References

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2010 (2)

2009 (1)

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

2008 (2)

X. Zhou and T. C. Zhu, “Interstitial diffuse optical tomography using an adjoint model with linear sources,” Proc. SPIE 6845, 68450C (2008).
[CrossRef]

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907-5925 (2008).
[CrossRef] [PubMed]

2007 (1)

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

2005 (1)

2003 (1)

1997 (2)

1994 (2)

1992 (1)

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

1989 (1)

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Alfano, R. R.

Bunting, C. F.

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648-662 (2010).
[CrossRef]

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

Carpenter, C. M.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Chen, N.

Contini, D.

Davis, S. C.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Dawson, J. B.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Dehghani, H.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

Dolne, J.

Driver, I.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Eames, M. E.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Fantini, S.

Farrell, T. J.

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

Feather, J. W.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Feng, T.

Foschum, F.

Franceschini, M. A.

Gratton, E.

Haidekker, M. A.

Haskell, R. C.

Huang, M.

Kienle, A.

King, P. R.

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Liemert, A.

Liu, F.

Martelli, F.

McAdams, M. S.

Michels, R.

Musgrove, C.

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

Patterson, M. S.

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

Paulsen, K. D.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Piao, D.

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648-662 (2010).
[CrossRef]

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

Pogue, B. W.

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648-662 (2010).
[CrossRef]

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

Polishchuk, A. Ya.

Srinivasan, S.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

Wilson, B. C.

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

Xie, T.

Yalavarthy, P. K.

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

Yao, G.

Zaccanti, G.

Zhang, A.

Zhou, X.

X. Zhou and T. C. Zhu, “Interstitial diffuse optical tomography using an adjoint model with linear sources,” Proc. SPIE 6845, 68450C (2008).
[CrossRef]

Zhu, Q.

Zhu, T. C.

X. Zhou and T. C. Zhu, “Interstitial diffuse optical tomography using an adjoint model with linear sources,” Proc. SPIE 6845, 68450C (2008).
[CrossRef]

Appl. Opt. (3)

Commun. Numer. Methods Eng. (1)

.H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711-732 (2009).
[CrossRef]

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

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

Med. Phys. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

I. Driver, J. W. Feather, P. R. King, and J. B. Dawson, “The optical properties of aqueous suspensions of Intralipid, a fat emulsion,” Phys. Med. Biol. 34 (12), 1927-1930 (1989).
[CrossRef]

Proc. SPIE (2)

X. Zhou and T. C. Zhu, “Interstitial diffuse optical tomography using an adjoint model with linear sources,” Proc. SPIE 6845, 68450C (2008).
[CrossRef]

C. Musgrove, C. F. Bunting, H. Dehghani, B. W. Pogue, and D. Piao, “Computational aspects of endoscopic near-infrared optical tomography: initial investigations,” Proc. SPIE 6434, 643409 (2007).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Three-dimensional rendering of the finite-element mesh of the cylindrical applicator: (a) concave geometry for the case-azi configuration shown with denser mesh along the azimuth direction at the outer surface of the cylinder domain, (b) concave geometry for the case-longi configuration shown with denser mesh along the longitudinal direction at the outer surface of the cylinder domain, (c) convex geometry for both case-azi and case-longi configurations, (d) discretization of the convex imaging volume for the case-azi configuration shown with denser mesh along the azimuth direction at the inner surface of the cylinder domain.

Fig. 2
Fig. 2

Geometry of the MC simulation: (a) concave geometry for case-azi configuration, (b) concave geometry for case-longi configuration, (c) convex geometry for case-azi configuration, (d) convex geometry for case-longi configuration..

Fig. 3
Fig. 3

Five cylindrical applicators made from the same black acetal material, shown by the (a) sketch and (b) photograph. The lower front two with radii of 0.95 cm and 2.53 cm were used for the concave geometry, and the upper rear three with radii of 1.27 cm , 2.41 cm , and 5.07 cm were used for the convex geometry..

Fig. 4
Fig. 4

Photographs of the experimental setup for the case-azi configuration in the convex geometry. The tank that housed the Intralipid solution for immersing the setup is not shown..

Fig. 5
Fig. 5

Illustration of the experimental setup for the case-azi configurations: (a) concave geometry with the source and the detector placed azimuthally in proximity to the inner surface of the cylinder applicator, (b) convex geometry with the source and the detector placed azimuthally in proximity to the outer surface of the cylinder applicator.

Fig. 6
Fig. 6

Illustration of the experimental setup for the case-longi configurations: (a) concave geometry with the detector penetrating the cylinder wall and the source placed longitudinally in proximity to the inner surface of the cylinder applicator, (b) convex geometry with the source and the detector placed longitudinally in proximity to the outer surface of the cylinder applicator.

Fig. 7
Fig. 7

Details of ρ r < and ρ r > in the azimuth plane: (a) concave geometry, (b) convex geometry.

Fig. 8
Fig. 8

Effect of positioning error of the source or the detector in case-azi configurations: (a) source is fixed, but the detector is shifted radially by 0 mm , 0.5 mm and 1 mm , (b) detector is fixed, but the source is shifted radially by 0.5 mm , 0 mm , and 0.5 mm . Conditions for (a) and (b) are not identical due to the requirement of an isotropic source placed 1 / μ s distance into the medium, whereas the detector is ideally located on the surface..

Fig. 9
Fig. 9

Effect of positioning error of the source or the detector in case-longi configurations: (a) source is fixed, but the detector is shifted radially by 0 mm , 0.5 mm , and 1 mm , (b) detector is fixed, but the source is shifted radially by 0.5 mm , 0 mm , and 0.5 mm . Conditions for (a) and (b) are not identical due to the requirement of an isotropic source placed 1 / μ s distance into the medium, whereas the detector is ideally located on the surface..

Fig. 10
Fig. 10

Effect of the measurement error of the initial source–detector distance. Initial source–detector distance is changed + 0.5 mm , 0.5 mm , + 1 mm , 1 mm , + 2 mm , and 2 mm for (a) case-azi configuration and (b) case-longi configuration..

Fig. 11
Fig. 11

Comparisons of analytic prediction, the FEM simulation, the MC simulation, and experimental results for both concave and convex geometries: (a) case-azi configuration, (b) case-longi configuration. Optical properties are μ a = 0.025 cm 1 , μ s = 5 cm 1 , A = 1.86 , and S = 1 ..

Equations (18)

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

Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) · 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] · 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } .
Ψ = S 2 π 2 D 0 d k { m = 0 ε m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { m = 0 ε m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m [ k eff ( R 0 R a ) ] K m ( k eff R 0 ) 1 I m ( k eff R 0 ) K m ( k eff R 0 ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff R 0 ) K m [ k eff ( R 0 + R a ) ] 1 K m ( k eff R 0 ) I m ( k eff R 0 ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] } .
ln ( Ψ · d ) = μ a D d + ln ( S 4 π D ) ,
Ψ = S 4 π D l real e k 0 l real S 4 π D l imag e k 0 l imag ,
l real = d 2 + R a 2 , R a = 1 / μ s ,
l imag = d 2 + ( 2 R b + R a ) 2 , R b = 2 A D .
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) · 1 I m ( k eff ρ r > ) K m ( k eff ρ r > ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) · 1 K m ( k eff ρ r < ) I m ( k eff ρ r < ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } ,
ρ r < = R 0 R a ξ | source , ρ r > = R 0 ξ | detector
ρ r < = R 0 + ξ | detector , ρ r > = R 0 + R a + ξ | source
Ψ = S 2 π 2 D 0 d k { m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) 1 I m ( k eff ρ r > ) K m ( k eff ρ r > ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) 1 K m ( k eff ρ r < ) I m ( k eff ρ r < ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] cos [ m ( φ φ ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) 1 I m ( k eff ρ r > ) K m ( k eff ρ r > ) K m [ k eff ( R 0 + R b ) ] I m [ k eff ( R 0 + R b ) ] } ,
Ψ = S 2 π 2 D 0 d k { cos [ k ( z z ) ] m = 0 ε m I m ( k eff ρ r < ) K m ( k eff ρ r > ) 1 K m ( k eff ρ r < ) I m ( k eff ρ r < ) I m [ k eff ( R 0 R b ) ] K m [ k eff ( R 0 R b ) ] } .

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