Abstract

We compare two fundamentally different ways to evaluate the time dependence in Monte Carlo simulations of photon migration: estimating the pulse response in time versus evaluating the transfer function at discrete points in the frequency domain. We show that these two methods differ in accuracy owing to quantization and sampling errors, whereas the statistical error is essentially the same for both methods. From our analysis we also derive alternative methods to sample the time-domain pulse response with reduced quantization and sampling error. Simulation results are included to illustrate our theoretical analysis.

© 1999 Optical Society of America

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References

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  1. M. H. Kalos, P. A. Whitlock, Monte Carlo Methods: Basics (Wiley, New York, 1986), Vol. 1.
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 7, pp. 148–167 and Chap. 9, pp. 175–190.
  3. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  4. H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]
  5. H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).
  6. Brian W. Pogue, Markus Testorf, Troy McBride, Ulf Osterberg, Keith Paulsen, “Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection,” Opt. Express 1, 391–403 (1997).
    [CrossRef] [PubMed]
  7. L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C (University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992–1993) ( http://ece.ogi.edu/omlc/science/software/mc/index.html ).
  8. S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
    [CrossRef]
  9. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  10. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
    [CrossRef]
  11. I. V. Yaroslavsky, A. N. Yaroslavski, V. V. Tuchin, H.-J. Schwarzmaier, “Effect of the scattering delay on time-dependent photon migration in turbid media,” Appl. Opt. 36, 6529–6538 (1997).
    [CrossRef]
  12. S. M. Ross, Introduction to Probability Models (Academic, New York, 1997), Chap. 2.2, pp. 25–30 and Chap. 5.3, pp. 249–276.
  13. R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Boston, 1986), Chap. 10, pp. 183–218.
  14. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]

1998

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

1997

1996

1994

1993

1992

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Akchurin, G. G.

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Boston, 1986), Chap. 10, pp. 183–218.

Chandrasekhar, S.

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

Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Fantini, S.

Fishkin, J. B.

Franceschini, M. A.

Gratton, E.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 7, pp. 148–167 and Chap. 9, pp. 175–190.

Jacques, S. L.

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C (University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992–1993) ( http://ece.ogi.edu/omlc/science/software/mc/index.html ).

Jiang, H.

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

Kalos, M. H.

M. H. Kalos, P. A. Whitlock, Monte Carlo Methods: Basics (Wiley, New York, 1986), Vol. 1.
[CrossRef]

McBride, Troy

Osterberg, Ulf

Österberg, U. L.

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

Paulsen, K. D.

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

Paulsen, Keith

Pogue, B. W.

Pogue, Brian W.

Ross, S. M.

S. M. Ross, Introduction to Probability Models (Academic, New York, 1997), Chap. 2.2, pp. 25–30 and Chap. 5.3, pp. 249–276.

Schwarzmaier, H.-J.

I. V. Yaroslavsky, A. N. Yaroslavski, V. V. Tuchin, H.-J. Schwarzmaier, “Effect of the scattering delay on time-dependent photon migration in turbid media,” Appl. Opt. 36, 6529–6538 (1997).
[CrossRef]

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Testorf, Markus

Tuchin, V. V.

I. V. Yaroslavsky, A. N. Yaroslavski, V. V. Tuchin, H.-J. Schwarzmaier, “Effect of the scattering delay on time-dependent photon migration in turbid media,” Appl. Opt. 36, 6529–6538 (1997).
[CrossRef]

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Wang, L.

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C (University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992–1993) ( http://ece.ogi.edu/omlc/science/software/mc/index.html ).

Whitlock, P. A.

M. H. Kalos, P. A. Whitlock, Monte Carlo Methods: Basics (Wiley, New York, 1986), Vol. 1.
[CrossRef]

Yaroslavski, A. N.

Yaroslavsky, A. N.

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Yaroslavsky, I. V.

Yaroslavsky, V.

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Med. Phys.

H. Jiang, K. D. Paulsen, U. L. Österberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multi-target tissue-like phantoms,” Med. Phys. 25(2), 183–193 (1998).

Opt. Express

Phys. Med. Biol.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlength in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Other

S. M. Ross, Introduction to Probability Models (Academic, New York, 1997), Chap. 2.2, pp. 25–30 and Chap. 5.3, pp. 249–276.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed. (McGraw-Hill, Boston, 1986), Chap. 10, pp. 183–218.

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-Layered Tissues in Standard C (University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992–1993) ( http://ece.ogi.edu/omlc/science/software/mc/index.html ).

V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency domain,” in Photon Propagation in Tissue, B. Chance, D. T. Delpy, G. Muller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

M. H. Kalos, P. A. Whitlock, Monte Carlo Methods: Basics (Wiley, New York, 1986), Vol. 1.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 7, pp. 148–167 and Chap. 9, pp. 175–190.

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

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Figures (7)

Fig. 1
Fig. 1

Interrelation of the frequency-domain signal and the absolute errors of amplitude and phase in the Gaussian plane.

Fig. 2
Fig. 2

Sampling of the time response function within a window of size t w . The sampling interval δt is distinguished from the interval of time quantization, Δt.

Fig. 3
Fig. 3

(a) Modulation depth and (b) phase of the frequency-domain signal are evaluated with different scoring techniques. The simulation results are compared with the predictions of diffusion theory.

Fig. 4
Fig. 4

Independence of the frequency-domain signal on the local accuracy of the time histogram. The poor accuracy in time (a) of 103 simulated photon packets still provides an ac signal (b) without notable error.

Fig. 5
Fig. 5

Impact of the sampling distance δt on the ac signal. The (a) response function and the (b) modulation depth are altered only for large sampling distances, whereas the (c) phase information is severely affected even for small δt.

Fig. 6
Fig. 6

Effect of (a), (b) a finite width of the time window t w and (c), (d) the time interval Δt on the ac signal.

Fig. 7
Fig. 7

Influence of averaging on the time response: (a) averaging of the theoretical response function, (b) simulation of 106- and 104-photon packets, (c) moving average of the simulations in Fig. 7(b), (d) difference between the estimates in Fig. 7(c) compared with the performance of a sinc function as an averaging window.

Equations (24)

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1ct Ir, s, t+sIr, s, t+μtIr, s, t=μs4π4π ps, sIr, s, tdΩ+r, s, t.
s=-lnξμt,
Wk=Wk-1μsμt,
mΔttd<m+1Δt;  m=0, 1, 2,.
ũthω=m nm expiωmΔt.
sĨr, s, ω-1+iωt¯μtĨr, s, ω=μs4π4π ps, sĨr, s, ωdΩ+˜r, s, ω,
|Zk|=|Zk-1| μsμt11+ωt¯1/2, tan φk=sin φk-1-ωt¯ cos φk-1cos φk-1+ωt¯ sin φk-1.
ũscω0=d Wd expiω0td.
μm=Enm=Npm; σm2=Varnm=Npm,
m=0M pm=1.
Δnmnm=1Npm1/2.
Eũreω=m Enmcosωtm=N m pm cosωtmNp˜reω,
Eũimω=m Enmsinωtm=N m pm sinωtmNp˜imω.
ϕξ=EexpξX,
ϕcos,mξ=exp(Npmexpξ cosωtm-1), ϕsin,mξ=exp(Npmexpξ sinωtm-1),
Varũreω=N m=0M pm cos2ωtm, Varũimω=N m=0M pm sin2ωtm.
Var|ũ|=Eũre2+ũim2-E2ũre2+ũim21/2=Eũre2+Eũim2-|ũ|2.
Δ|ũω||ũω|=Var|ũ|1/2E|ũ|=1N1|p˜ω|,
tanΔϑ=Δ|ũω||ũω|=1N1|p˜ω|.
pt=gt * recttΔtm δt-mδt,
pt=1Δtt-Δt/2t+Δt/2gt+ġtt+12 g¨tt2dt=gt+124 g¨tΔt2.
Δptpt=124g¨tgt Δt2.
p˜ω=g˜ωsincΔtω * n δω-n 2πδt,
pt=gt * sinctΔtm δt-mδt.

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