Abstract

The analytical solutions for the coupled diffusion equations that are encountered in diffuse fluorescence spectroscopy/imaging for regular geometries were compared with the well-established numerical models, which are based on the finite element method. Comparison among the analytical solutions obtained using zero boundary conditions and extrapolated boundary conditions (EBCs) was also performed. The results reveal that the analytical solutions are in close agreement with the numerical solutions, and solutions obtained using EBCs are more accurate in obtaining the mean time of flight data compared to their counterpart. The analytical solutions were also shown to be capable of providing bulk optical properties through a numerical experiment using a realistic breast model.

© 2013 Optical Society of America

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References

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  1. K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging of biological tissues in regular geometries. Part I: zero and extrapolated boundary conditions,” J. Opt. Soc. Am. A30, 537–552(2013).
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    [CrossRef]
  5. S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
    [CrossRef]
  6. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).
  7. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues” Opt. Lett. 26, 1335–1337 (2001).
    [CrossRef]
  8. T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
    [CrossRef]
  9. E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
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2009

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

2008

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef]

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

2005

2001

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues” Opt. Lett. 26, 1335–1337 (2001).
[CrossRef]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

1989

Achilefu, S.

Ayyalasomayajula, K. R.

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging of biological tissues in regular geometries. Part I: zero and extrapolated boundary conditions,” J. Opt. Soc. Am. A30, 537–552(2013).

Bevilacqua, F.

Bloch, S.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

Chance, B.

Culver, J.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Dunn, A. K.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Hayakawa, C. K.

Jacques, S. L.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef]

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

Jiang, S.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Kumar, S.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

McBride, T. O.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Osterberg, U. L.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Patterson, M. S.

Patwardhan, S.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Pogue, B. W.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Rasmussen, J. C.

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

Riseborough, P.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Sadoqi, M.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Sevick-Muraca, E. M.

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

Spanier, J.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

Tromberg, B. J.

Venugopalan, V.

Wilson, B. C.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging of biological tissues in regular geometries. Part I: zero and extrapolated boundary conditions,” J. Opt. Soc. Am. A30, 537–552(2013).

You, J. S.

Appl. Opt.

Commun. Numer. Methods Eng.

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: algorithms for numerical model and image reconstruction algorithms,” Commun. Numer. Methods Eng. 25, 711–732 (2009).
[CrossRef]

J. Biomed. Opt.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef]

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Rev. Sci. Instrum.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo,” Rev. Sci. Instrum. 72, 1817–1824 (2001).
[CrossRef]

Other

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging of biological tissues in regular geometries. Part I: zero and extrapolated boundary conditions,” J. Opt. Soc. Am. A30, 537–552(2013).

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Figures (5)

Fig. 1.
Fig. 1.

Geometries indicating the source detectors arrangement used in the numerical models. (a) Slab (transmittance), (b) slab (reflectance), (c) circle, (d) sphere, and (e) cylinder.

Fig. 2.
Fig. 2.

Comparison of numerical (FEM) and analytical solutions subjected to ZBC for the geometries shown in Fig. 1. (a) and (c) Give the logarithm of amplitude. (b) and (d) Shows comparison of computed phase as a function of source/detector distance.

Fig. 3.
Fig. 3.

(a) Shows % difference (or error) between numerical and analytical solutions given in Figs. 2(a) and 2(c) for the case of amplitude data. (b) Same effort as (a) except for phase data. (c) and (d) Same effort as Figs. 2(a) and 2(b) except the boundary condition is changed to EBC for the case of infinite slab [Figs. 1(a) and 1(b)].

Fig. 4.
Fig. 4.

(a) and (c) Same effort as Figs. 2(a) and 2(c) with ZBC in the case of analytical solution for the MTOF data. (b) and (d) Same effort as (a) and (c) except analytical solution is computed using EBC.

Fig. 5.
Fig. 5.

(a) Cube geometry showing the source/detector arrangement that was used for numerical solution (FEM) computation. (b) Comparison of analytical solution [Eq. (3)] and FEM solution as a function of source/detector distance. (c) Breast geometry that was used for generating numerical experimental data. (d) The comparison of computed data that were obtained using the best fits obtained using the analytical models (given in the legend).

Tables (3)

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Table 1. Specification of the Finite Element Meshes that were Used in this Worka

Tables Icon

Table 2. Optical Property Values for all Cases that were Discussed in this Work

Tables Icon

Table 3. Comparison of Computational Time Required for Computing Amplitude Using Numerical (FEM) and Analytical Models Using ZBC and EBCa

Equations (5)

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

grpϕp(r,r,t,t)=8abcl=1m=1n=1sinlπxasinlπx0asinmπybsinmπy0bsinnπzcsinnπz0ceγp2π2(tt)[l2a2+m2b2+n2c2],
gcubeϕp(r,r,t,t)=8a3l=1m=1n=1sinlπxasinlπx0asinmπyasinmπy0asinnπzasinnπz0aeγp2π2(tt)[l2+m2+n2a2].
gcubeϕfl(r,r,t,t)=nζ2γm2γm2γx2[gcubeϕx(r,r,t,t)gcubeϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]].
Φfl0=Φfl2,x=x1,t>0;Φfl0x=Φfl2x,x=x1,t>0,
Φfl0|Ω=Φfl2|Ω,t>0;Φfl0n|Ω=Φfl2n|Ω,t>0,

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