Abstract

Using the principle of energy conservation and laws of geometrical optics, we derive the photon transport equation for turbid biological media with spatially varying isotropic refractive index. We show that when the refractive index is constant, our result reduces to the standard radiative transfer equation and when the medium is lossless and free of scattering to the well known geometrical optics equations in refractive media.

© 2005 Optical Society of America

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References

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Am. Assoc. Phys. Med.

E. D. Aydin, C. R. E. De Oliveira and A. J. H. Goddard, �??A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,�?? Am. Assoc. Phys. Med. 29, 2013�??2023 (2002).

Appl. Opt.

Comm. Pure and Appl. Maths.

M. Kline, �??A note on the expansion coefficient of geometrical optics,�?? Comm. Pure and Appl. Maths. 14, 473�??479 (1961).
[CrossRef]

Comp. Meth. and Prog. in Biomed.

L-H.Wang, S. L. Jacques and L-Q Zheng, �??MCML - Monte Carlo modeling of photon transport in multi-layered tissues,�?? Comp. Meth. and Prog. in Biomed. 47, 131�??146 (1995).
[CrossRef]

Devel. Biol.

S. A. Boppart, M. E. Brezinski, B. E. Bouma, G. J. Tearney and J. G. Fujimoto, �??Investigation of developing embryonic morphology using optical coherence tomography,�?? Devel. Biol. 177, 54�??63 (1996).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

H. A. Ferwerda, �??The radiative transfer equation for scattering media with a spatially varying refractive index,�?? J. Opt. A: Pure Appl. Opt. 1, L1 �?? L2 (1999).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

T. Khan and H. Jiang, �??A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,�?? J. Opt. A: Pure Appl. Opt. 5, 137 �?? 141 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Lasers in Surgery and Medicine

A. J. Welch, G. Yoon and M. J. C. Van Gemert, �??Practical models for light distribution in laser-irradiated tissue,�?? Lasers in Surgery and Medicine 6, 488 �?? 493 (1987).
[CrossRef] [PubMed]

Opt. Express

Phys. med. Biol.

J. C. Hebden, A. Gibson, R. M Yusof, N. Everdell, E. M. C. Hillman, D. T. Deply, S. R. Arridge, T. Austin, J. H. Meek and J. S. Wyatt, �??Three-dimensional optical tomography of the premature infant brain,�?? Phys. Med. Biol. 47, 4155�??4166 (2002).
[CrossRef] [PubMed]

SPIE Milestone Series MS

B. R. Masters (ed.) Selected Papers on Optical Low-Coherence Reflectometry & Tomography, SPIE Milestone Series MS 165 (SPIE Optical Engineering Press, Bellingham, 2001).

Other

A. J. Welch and M. J. C. Van-Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Lasers, Photonics and Electro-Optics), (Plenum Publishing Corporation, New York, 1995).

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, (SPIE Press, Bellingham, 2000).

M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Third, Revised Edition, (Springer, New Jersey, 2004).

M. Born and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, (Wiley Interscience Publishers, New Jersey, 1965).

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978).

S. Chandrasekhar, Radiative Transfer, (Dover Publications, New York, 1960).

S. I. Grossman, Calculus , Fifth Edition, (Harcourt Brace College Publishers, Philadelphia, 1991).

G. Yankovsky, Higher Algebra , (Mir Publishers, Moscow, 1980).

D. J. Griffiths, Introduction to Electrodynamics, Third Edition, (Prentice Hall, New Jersey, 1999).

W. L. Burke, Applied Differential Geometry, (Cambridge University Press, Cambridge, 1985).

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Figures (2)

Fig. 1.
Fig. 1.

Transport of an infinitesimally small photon packet with phase space volume, V S 0 , along an infinitesimally small ray tube surrounding a central light ray.

Fig. 2.
Fig. 2.

Details of the stationary and moving coordinate systems

Equations (54)

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Ω ̂ = d r d s = 1 μ 2 cos ( φ ) x ̂ + 1 μ 2 sin ( φ ) y ̂ + μ z ̂
V 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s ( μ a + μ s ) 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
+ Δ s δ V s μ s ( 1 1 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ
+ Δ s δ V s ε ( x , y , z , μ , φ , t ) d x d y d z μ d φ + O ( Δ s 2 )
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= δ V s 0 [ ( 𝓛 J ) s + Δ s , t + Δ t ( 𝓛 J ) s , t ] d x 0 d y 0 d z 0 d μ 0 d φ 0
( 𝓛 J ) s + Δ s , t + Δ t ( 𝓛 J ) s , t = J n c 𝓛 t Δ s + J 𝓛 s Δ s + L J s Δ s + O ( Δ s 2 )
J = x x 0 x y 0 x z 0 x μ 0 x φ 0 y x 0 y y 0 y z 0 y μ 0 y φ 0 z x 0 z y 0 z z 0 z μ 0 z φ 0 μ x 0 μ y 0 μ z 0 μ μ 0 μ φ 0 φ x 0 φ y 0 φ z 0 φ μ 0 φ φ 0
J = σ sgn ( σ ) J 1 , σ ( 1 ) J 2 , σ ( 2 ) J 3 , σ ( 3 ) J 4 , σ ( 4 ) J 5 , σ ( 5 )
J = σ sgn ( σ ) j = 1 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
J s = v { x 0 , y 0 , z 0 } σ sgn ( σ ) ( k = 1 5 Ω v ϑ ( k ) ϑ ( k ) v ) j = 1 ϑ 0 ( j ) v 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
+ v { μ 0 , φ 0 } σ sgn ( σ ) ( k = 1 5 ϑ ( k ) v ϑ ( k ) ( v s ) ) j = 1 ϑ 0 ( j ) v 5 ϑ ( j ) ϑ 0 ( σ ( j ) )
J s = ( Ω x x + Ω y y + Ω z z + μ ( μ s ) + φ ( φ s ) ) J
r · V = V x x + V y y + V z z
J s = J r · Ω ̂ + ( μ ( μ s ) + φ ( φ s ) ) J
r · Ω ̂ = 1 R 1 ( s ) + 1 R 2 ( s )
( n Ω ̂ ) s = r n
Ω ̂ s = r n n 1 n d n d s Ω ̂
μ ̂ = μ cos ( φ ) x ̂ μ sin ( φ ) y ̂ + 1 μ 2 z ̂
φ ̂ = sin ( φ ) x ̂ + cos ( φ ) y ̂
Ω ̂ s = 1 1 μ 2 μ s μ ̂ + 1 μ 2 φ s φ ̂
μ s = 1 μ 2 r n · μ ̂ n
φ s = r n · φ ̂ n 1 μ 2
μ ̂ μ = Ω ̂ 1 μ 2
φ ̂ φ = μ μ ̂ 1 μ 2 Ω ̂
μ ( μ s ) = μ 1 μ 2 r n · μ ̂ n r n · Ω ̂ n
φ ( φ s ) = μ 1 μ 2 r n · μ ̂ n r n · Ω ̂ n
J s = ( 1 R 1 ( s ) + 1 R 2 ( s ) 2 r n · Ω ̂ n ) J
J s = ( 1 R 1 ( s ) + 1 R 2 ( s ) 2 n d n d s ) J
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s 0 [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] J d x 0 d y 0 d z 0 d μ 0 d φ 0 + O ( Δ s 2 )
δ V s + Δ s 𝓛 ( x , y , z , μ , φ , t + Δ t ) d x d y d z d μ d φ δ V s 𝓛 ( x , y , z , μ , φ , t ) d x d y d z d μ d φ
= Δ s δ V s [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] d x d y d z d μ d φ + O ( Δ s 2 )
δ V s [ n c 𝓛 t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 + n 2 s ( 𝓛 n 2 ) ] d x d y d z d μ d φ
δ V s ( ( μ a + μ s ) 𝓛 + ε ) d x d y d z d μ d φ
δ V s μ s ( 1 1 0 2 π f ( μ , φ , μ ̀ , φ ̀ ) 𝓛 ( x , y , z , μ ̀ , φ ̀ , t ) d μ ̀ d φ ̀ ) d x d y d z d μ d φ = 0
n ( r ) c 𝓛 ( r , Ω ̂ , t ) t + ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) + n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
r 𝒮 = n ( r ) Ω ̂
n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= n 2 ( r ) r s · r ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + n 2 ( r ) Ω ̂ s · Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
Ω ̂ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) = μ ̂ 1 μ 2 μ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) ) + φ ̂ 1 μ 2 φ ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
n 2 ( r ) s ( 𝓛 ( r , Ω ̂ , t ) n 2 ( r ) )
= Ω ̂ · r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) r n ( r ) · Ω ̂ 𝓛 ( r , Ω ̂ , t ) 2 n ( r ) Ω ̂ · r n ( r )
n ( r ) c 𝓛 ( r , Ω ̂ , t ) t + Ω ̂ · r 𝓛 ( r , Ω ̂ , t ) + 1 n ( r ) r n ( r ) · Ω ̂ 𝓛 ( r , Ω ̂ , t )
+ ( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ , t ) 2 n ( r ) Ω ̂ · r n ( r )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
n 0 c 𝓛 ( r , Ω ̂ , t ) t + Ω ̂ · r 𝓛 ( r , Ω ̂ , t )
= ( μ a ( r ) + μ s ( r ) ) 𝓛 ( r , Ω ̂ , t ) + ε ( r , Ω ̂ , t ) + μ s ( r ) 4 π f ( Ω ̂ , Ω ̀ ) 𝓛 ( r , Ω ̀ , t ) d Ω ̀
( 1 R 1 ( s ) + 1 R 2 ( s ) ) 𝓛 ( r , Ω ̂ ) + n 2 ( r ) d d s ( 𝓛 ( r , Ω ̂ ) n 2 ( r ) ) = 0
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0 exp ( s 0 s ( 1 R 1 ( s ) + 1 R 2 ( s ) ) d s )
𝓛 ( r , Ω ̂ ) = n 2 ( r ) n 2 ( r 0 ) 𝓛 0

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