Abstract

We develop a modified method to simplify the analytical solutions to the simplified spherical harmonics equations (SPN). The scheme decouples the SPN partial differential equations into independent equations using eigen decompositions and calculates the Green’s function of the photon migrations based on the eigenvectors and eigenvalues. In contrast to the established solutions that are based on the original coupled equations, the proposed derivation is theoretically concise and universally extendable to other regular geometries. We validate the proposed method in comparison with Monte-Carlo simulations for an infinite scattering medium and a circular geometry as an example of the boundary value problems.

© 2013 Optical Society of America

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  1. Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
    [CrossRef]
  2. A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
    [CrossRef]
  3. J. B. Domínguez and Y. Bérubé-Lauzière, Biomed. Opt. Express 2, 817 (2011).
    [CrossRef]
  4. M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
    [CrossRef]
  5. M. Chu and H. Dehghani, Opt. Express 17, 24208 (2009).
    [CrossRef]
  6. A. Liemert and A. Kienle, Opt. Lett. 35, 3507 (2010).
    [CrossRef]
  7. A. Liemert and A. Kienle, Opt. Lett. 36, 4041 (2011).
    [CrossRef]
  8. S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
    [CrossRef]
  9. A. Doronin and I. Meglinski, J. Biomed. Opt. 17, 090504 (2012).
    [CrossRef]
  10. E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
    [CrossRef]

2012

A. Doronin and I. Meglinski, J. Biomed. Opt. 17, 090504 (2012).
[CrossRef]

2011

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

J. B. Domínguez and Y. Bérubé-Lauzière, Biomed. Opt. Express 2, 817 (2011).
[CrossRef]

A. Liemert and A. Kienle, Opt. Lett. 36, 4041 (2011).
[CrossRef]

2010

2009

M. Chu and H. Dehghani, Opt. Express 17, 24208 (2009).
[CrossRef]

Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
[CrossRef]

2007

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

2006

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

1992

S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef]

Bérubé-Lauzière, Y.

Chu, M.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef]

Dehghani, H.

Deply, D. T.

S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef]

Domínguez, J. B.

Doronin, A.

A. Doronin and I. Meglinski, J. Biomed. Opt. 17, 090504 (2012).
[CrossRef]

Frank, M.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Guo, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Hanlon, E. B.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Hu, X.

Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
[CrossRef]

Itzkan, I.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Jiang, H.

Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
[CrossRef]

Kienle, A.

Klar, A.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Klose, A. D.

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

Larsen, E. W.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

Liemert, A.

Meglinski, I.

A. Doronin and I. Meglinski, J. Biomed. Opt. 17, 090504 (2012).
[CrossRef]

Perelman, L. T.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Qiu, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Turzhitsky, V.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Vitkin, E.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Yasuda, S.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Yuan, Z.

Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
[CrossRef]

Biomed. Opt. Express

J. Biomed. Opt.

A. Doronin and I. Meglinski, J. Biomed. Opt. 17, 090504 (2012).
[CrossRef]

J. Comput. Phys.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

Nat. Commun.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, Nat. Commun. 2, 587 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

S. R. Arridge, M. Cope, and D. T. Deply, Phys. Med. Biol. 37, 1531 (1992).
[CrossRef]

Z. Yuan, X. Hu, and H. Jiang, Phys. Med. Biol. 54, 67 (2009).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Schematics of source-detector configurations for (a) infinite and (b) circular scattering geometries.

Fig. 2.
Fig. 2.

(a) Fluence rates calculated from the SP3, SP5, DA, and MC, respectively, for an infinite geometry with μa=0.01mm1, and (b) the model errors, both as functions of the SDS.

Fig. 3.
Fig. 3.

(a) Fluence rates calculated from the SP3, SP5, DA, and MC, respectively, for an infinite geometry with μa=1mm1, and (b) the model errors, both as functions of the SDS.

Fig. 4.
Fig. 4.

(a) Exiting flux from two-dimensional circle boundary versus the detection angle, with absorption coefficients of μa=0.02mm1, calculated from the SPN, DA and MC, and (b) model errors.

Fig. 5.
Fig. 5.

(a) Exiting flux from two-dimensional circle boundary versus the detection angle θ, with absorption coefficients of μa=0.2mm1, calculated from the SPN, DA and MC, and (b) the model errors.

Tables (2)

Tables Icon

Table 1. Standard Deviations of the Fluence Rates Calculated by the DA and SPN for the Different Absorption Coefficients

Tables Icon

Table 2. Standard Deviations of the Exiting Flux Calculated by the DA and SPN for the Different Absorption Coefficients

Equations (16)

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·13μa1ϕ1(r)+μa0ϕ1(r)=Q(r)+23μa0ϕ2(r)815μa0ϕ3(r)·17μa3ϕ2(r)+(49μa0+59μa2)ϕ2(r)=23Q(r)+(23μa0)ϕ1(r)+(1645μa0+49μa2)ϕ3(r)·111μa5ϕ3(r)+(64225μa0+1645μa2+925μa4)ϕ3(r)=815Q(r)(815μa0)ϕ1(r)+(1645μa0+49μa2)ϕ2(r),
(2IA)Φ(r)=εQ(r),
A=[3μa0μa12μa0μa18μa0μa1514μa0μa33(28μa09+35μa29)μa3(112μa04528μa29)μa388μa0μa515(176μa04544μa29)μa5(704μa0225+176μa245+99μa425)μa5]
(2IB1AB)B1Φ(r)=B1εQ(r),
[2Idiag(λ12,λ22,λ32)]Φ(r)=εQ(r),
(2λi2)ϕi(r)=εiQ(r)(i=1,2,3).
(12+A1)ϕ1(r)+(1+B13μa1)n^·ϕ1(r)=(18+C1)ϕ2(r)+(D1μa3)n^·ϕ2(r)+(116+E1)ϕ3(r)+(F1μa5)n^·ϕ3(r),(724+A2)ϕ2(r)+(1+B27μa3)n^·ϕ2(r)=(18+C2)ϕ1(r)+(D2μa1)n^·ϕ1(r)+(41384+E2)ϕ3(r)+(F2μa5)n^·ϕ3(r),(4071920+A3)ϕ3(r)+(1+B311μa5)n^·ϕ3(r)=(116+C3)ϕ1(r)+(D3μa1)n^·ϕ1(r)+(41384+E3)ϕ2(r)+(F3μa3)n^·ϕ2(r),
ϕ(r)=ϕ1(r)23ϕ2(r)+815ϕ3(r),
J=(14+J0)(ϕ123ϕ2+815ϕ3)(0.5+J13μa1)n^·ϕ1+(516+J2)(13ϕ2415ϕ3)(J37μa3)n^·ϕ2+15ϕ3(332+J4)(J511μa5)n^·ϕ3,
ϕi(r)=εieλir4πr(i=13).
ϕiinf(r)=j=13Bijεjeλjr4πr(i=13),
ϕi(r)=Hi(r)+Gi(r),
Hi(r)=n=an(i)In(rλi)cos(nφ)Gi(r)=εi2πn=[In(rλi)Kn(rλi)]cos(nφ)r>r(i=13),
ϕicirc(r)=j=13Bijn=[an(j)In(rλj)+εj2πIn(rλj)Kn(rλj)]cos(nθ)(i=13).
εϕ=ϕϕMCϕMC×100%;εJ=JJMCJMC×100%.
σ=1Nd=1N(χ(d)χMC(d)χMC(d))2×100%,

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