Abstract

We have modified the existing convolution method of the Monte Carlo simulation for finite photon beams with both translational and rotational invariance. The modified convolution method was applied to simulate the optical fluence distribution in tissue in dark-field confocal photoacoustic microscopy. We studied the influence of the size of the dark field and the illumination incident angle on the depth position of the effective optical focus (the region with the highest fluence) and the fluence ratio (the ratio of the optical fluence at the effective optical focus inside the tissue to the optical fluence on the tissue surface along the ultrasonic axis). Within the reach of diffuse photons, the depth position of the effective optical focus increases with the size of the dark field and is much less sensitive to the incident angle. The findings show that, while the fluence at the effective optical focus decreases, the fluence ratio increases with the size of the dark field. The incident angle has a weaker influence on the fluence ratio than the size of the dark field does. An incident angle between 30 and 50 degrees gives the highest fluence at the effective optical focus.

© 2009 Optical Society of America

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References

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  1. M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-1-041101-22 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (2)

L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 171-179 (2008).
[CrossRef]

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758-5767 (2008).
[CrossRef]

2007 (5)

Q. Liu and N. Ramanujam, “Scaling method for fast Monte Carlo simulation of diffuse reflectance spectra from multilayered turbid media,” J. Opt. Soc. Am. A 24, 1011-1025(2007).
[CrossRef]

M. C. Skala, G. M. Palmer, K. M. Vrotsos, A. Gendron-Fitzpatrick, and N. Ramanujam, “Comparison of a physical model and principal component analysis for the diagnosis of epithelial neoplasias in vivo using diffuse reflectance spectroscopy,” Opt. Express 15, 7863-7875 (2007).
[CrossRef] [PubMed]

A. Wang, V. Nammalvar, and R. Drezek, “Targeting spectral signatures of progressively dysplastic stratified epithelia using angularly variable fiber geometry in reflectance Monte Carlo simulations,” J. Biomed. Opt. 12, 044012-1-044012-14 (2007).
[CrossRef]

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

H. F. Zhang, K. Maslov, and L. V. Wang, “In vivo imaging of subcutaneous structures using functional photoacoustic microscopy,” Nat. Protocols 2, 797-804 (2007).
[CrossRef]

2006 (4)

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, M. Li, G. Stoica, and L. V. Wang, “In vivo volumetric imaging of subcutaneous microvasculature by photoacoustic microscopy,” Opt. Express 14, 9317-9323 (2006).
[CrossRef] [PubMed]

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-1-041101-22 (2006).
[CrossRef]

2005 (1)

2002 (1)

R. K. Wang, “Signal degradation by multiple scattering in optical coherence tomography of dense tissue: a Monte Carlo study,” Phys. Med. Biol. 47, 2281-2299 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (2)

1997 (1)

L. Wang, S. L. Jacques, and L. Zheng, “CONV--convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

1995 (1)

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

1989 (3)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 2. Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Arifler, D.

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

Dong, K.

Drezek, R.

A. Wang, V. Nammalvar, and R. Drezek, “Targeting spectral signatures of progressively dysplastic stratified epithelia using angularly variable fiber geometry in reflectance Monte Carlo simulations,” J. Biomed. Opt. 12, 044012-1-044012-14 (2007).
[CrossRef]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 2. Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

Follen, M.

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

Gendron-Fitzpatrick, A.

Hu, X. H.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “CONV--convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

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

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Li, M.

Liang, G.

Liu, Q.

Lu, J. Q.

MacAulay, C.

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

Maslov, K.

H. F. Zhang, K. Maslov, and L. V. Wang, “In vivo imaging of subcutaneous structures using functional photoacoustic microscopy,” Nat. Protocols 2, 797-804 (2007).
[CrossRef]

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

H. F. Zhang, K. Maslov, M. Li, G. Stoica, and L. V. Wang, “In vivo volumetric imaging of subcutaneous microvasculature by photoacoustic microscopy,” Opt. Express 14, 9317-9323 (2006).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

K. Maslov, G. Stoica, and L. V. Wang, “In vivo dark-field reflection-mode photoacoustic microscopy,” Opt. Lett. 30, 625-627 (2005).
[CrossRef] [PubMed]

Nammalvar, V.

A. Wang, V. Nammalvar, and R. Drezek, “Targeting spectral signatures of progressively dysplastic stratified epithelia using angularly variable fiber geometry in reflectance Monte Carlo simulations,” J. Biomed. Opt. 12, 044012-1-044012-14 (2007).
[CrossRef]

Palmer, G. M.

Patterson, M. S.

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 2. Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Ramanujam, N.

Richards-Kortum, R.

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

Sivaramakrishnan, M.

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

Skala, M. C.

Song, Z.

Stoica, G.

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, M. Li, G. Stoica, and L. V. Wang, “In vivo volumetric imaging of subcutaneous microvasculature by photoacoustic microscopy,” Opt. Express 14, 9317-9323 (2006).
[CrossRef] [PubMed]

K. Maslov, G. Stoica, and L. V. Wang, “In vivo dark-field reflection-mode photoacoustic microscopy,” Opt. Lett. 30, 625-627 (2005).
[CrossRef] [PubMed]

Vrotsos, K. M.

Wang, A.

A. Wang, V. Nammalvar, and R. Drezek, “Targeting spectral signatures of progressively dysplastic stratified epithelia using angularly variable fiber geometry in reflectance Monte Carlo simulations,” J. Biomed. Opt. 12, 044012-1-044012-14 (2007).
[CrossRef]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “CONV--convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

Wang, L. V.

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758-5767 (2008).
[CrossRef]

L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 171-179 (2008).
[CrossRef]

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

H. F. Zhang, K. Maslov, and L. V. Wang, “In vivo imaging of subcutaneous structures using functional photoacoustic microscopy,” Nat. Protocols 2, 797-804 (2007).
[CrossRef]

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-1-041101-22 (2006).
[CrossRef]

H. F. Zhang, K. Maslov, M. Li, G. Stoica, and L. V. Wang, “In vivo volumetric imaging of subcutaneous microvasculature by photoacoustic microscopy,” Opt. Express 14, 9317-9323 (2006).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

K. Maslov, G. Stoica, and L. V. Wang, “In vivo dark-field reflection-mode photoacoustic microscopy,” Opt. Lett. 30, 625-627 (2005).
[CrossRef] [PubMed]

L. V. Wang and G. Liang, “Absorption distribution of an optical beam focused into a turbid medium,” Appl. Opt. 38, 4951-4958 (1999).
[CrossRef]

Wang, L.-H.

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

Wang, R. K.

R. K. Wang, “Signal degradation by multiple scattering in optical coherence tomography of dense tissue: a Monte Carlo study,” Phys. Med. Biol. 47, 2281-2299 (2002).
[CrossRef] [PubMed]

Welch, A. J.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 2. Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

Xu, M.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-1-041101-22 (2006).
[CrossRef]

Zhang, H. F.

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

H. F. Zhang, K. Maslov, and L. V. Wang, “In vivo imaging of subcutaneous structures using functional photoacoustic microscopy,” Nat. Protocols 2, 797-804 (2007).
[CrossRef]

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, M. Li, G. Stoica, and L. V. Wang, “In vivo volumetric imaging of subcutaneous microvasculature by photoacoustic microscopy,” Opt. Express 14, 9317-9323 (2006).
[CrossRef] [PubMed]

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “CONV--convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

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

Appl. Opt. (3)

Appl. Phys. Lett. (1)

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90, 053901-1-053901-3 (2007).

Comput. Methods Programs Biomed. (2)

L. Wang, S. L. Jacques, and L. Zheng, “CONV--convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

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

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

L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 171-179 (2008).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 1. Model prediction and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162-1168(1989).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, and M. S. Patterson, “Monte-Carlo modeling of light-propagation in highly scattering tissue. 2. Comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

J. Biomed. Opt. (2)

A. Wang, V. Nammalvar, and R. Drezek, “Targeting spectral signatures of progressively dysplastic stratified epithelia using angularly variable fiber geometry in reflectance Monte Carlo simulations,” J. Biomed. Opt. 12, 044012-1-044012-14 (2007).
[CrossRef]

D. Arifler, C. MacAulay, M. Follen, and R. Richards-Kortum, “Spatially resolved reflectance spectroscopy for diagnosis of cervical precancer: Monte Carlo modeling and comparison to clinical measurements,” J. Biomed. Opt. 11, 064027-1-064027-16 (2006).
[CrossRef]

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

Med. Phys. (1)

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758-5767 (2008).
[CrossRef]

Nat. Biotechnol. (1)

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24, 848-851(2006).
[CrossRef] [PubMed]

Nat. Protocols (1)

H. F. Zhang, K. Maslov, and L. V. Wang, “In vivo imaging of subcutaneous structures using functional photoacoustic microscopy,” Nat. Protocols 2, 797-804 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

R. K. Wang, “Signal degradation by multiple scattering in optical coherence tomography of dense tissue: a Monte Carlo study,” Phys. Med. Biol. 47, 2281-2299 (2002).
[CrossRef] [PubMed]

Proc. SPIE (1)

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE IS 5, 102-111 (1989).

Rev. Sci. Instrum. (1)

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101-1-041101-22 (2006).
[CrossRef]

Other (1)

“American national standard for safe use of lasers,” ANSI Standard Z136.1-2007 (Laser Institute of America, 2007).

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Figures (6)

Fig. 1
Fig. 1

Geometries of pencil beams: (a) rotation of a pencil beam; (b) rotation and translation of a pencil beam. PB i , incident pencil beam; PB r , pencil beam after rotation by θ from PB i ; PB r s , pencil beam after translation by r from PB r .

Fig. 2
Fig. 2

Schematic of the finite-size laser beams: (a) laser beam with only parallel translation between decomposed pencil beams; (b) dark-field illumination laser beam with both rotation and translation between decomposed pencil beams.

Fig. 3
Fig. 3

Optical fluence distributions along the r z plane for (a) bright-field and (b) dark-field optical illumination. (c), (d) Contour plots of (a) and (b), respectively.

Fig. 4
Fig. 4

Fluence distribution for an obliquely incident pencil beam with a 45-degree incident angle: (a) depth- resolved fluence distribution; fluence values on each plane that is parallel to the x y were summed; (b) fluence distribution along the x y plane at the depth z = 0.2 cm .

Fig. 5
Fig. 5

Simulation results with a constant incident angle: (a) changes of optical fluence at the effective optical focus and on the skin surface as a function of the inner radius; (b) changes of the fluence ratio as a function of the inner radius; (c) changes of the depth position of the effective optical focus as a function of the inner radius; (d) changes of optical fluence at the effective optical focus when the depth position of the effective optical focus increases with the inner radius.

Fig. 6
Fig. 6

Simulation results as a function of the incident angle with a fixed size of the illumination ring: (a) changes of optical fluence at the effective optical focus and on the skin surface; (b) changes of the fluence ratio; (c) changes of the depth position of the effective optical focus.

Tables (1)

Tables Icon

Table 1 Optical Properties of the Two-Layer Tissue Model in the MC Simulation a

Equations (12)

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

G d ( x , y , z ) = G p ( x 2 + y 2 , arctan 2 ( x , y ) , z ) , G p ( r , θ , z ) = G d ( cos θ , r sin θ , z ) ,
G p ( r , θ , z : θ ) = G p ( r , θ θ , z ) = G d ( r cos ( θ θ ) , r sin ( θ θ ) , z ) = G d ( x , y , z ) ,
G d ( x , y , z : r , θ ) = G d ( x , y , z : r ) = G d ( x r , y , z ) = G d ( r cos ( θ θ ) r , r sin ( θ θ ) , z ) .
G p ( r , θ , z : r , θ ) = G p ( r 2 + r 2 2 r r cos ( θ θ ) , arctan 2 ( r cos ( θ θ ) r , r sin ( θ θ ) ) , z ) .
C p ( r , θ , z ) = S G p ( r , θ , z : r , θ ) s ( r , θ ) d S .
G d ( x , y , z : x , y ) = G d ( x x , y y , z ) ,
G p ( r , θ , z : r , θ ) = G p ( r 2 + r 2 2 r r cos ( θ θ ) , arctan 2 ( r cos θ r cos θ , r sin θ r sin θ ) , z ) .
C p ( r , θ , z ) = S G p ( r 2 + r 2 2 r r cos ( θ θ ) , arctan 2 ( r cos θ r cos θ , r sin θ r sin θ ) , z ) s ( r , θ ) r d θ d r .
C p ( r , θ , z ) = S G p ( r 2 + r 2 2 r r cos ( θ θ ) , z ) s ( r , θ ) r d θ d r ,
C d ( x , y , z ) = S G d ( x x + y y ( x 2 + y 2 ) x 2 + y 2 , y x x y x 2 + y 2 , z ) s ( x , y ) d x d y , C p ( r , θ , z ) = S G p ( r 2 + r 2 2 r r cos ( θ θ ) , arctan 2 ( r cos ( θ θ ) r , r sin ( θ θ ) ) , z ) s ( r , θ ) r d θ d r .
C p ( r , z ) = C d ( r , 0 , z ) ,
C p ( r , z ) = S G d ( r cos θ r , r sin θ , z ) s ( r , θ ) r d θ d r .

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