## Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. III. Frequency-domain and time-domain results

JOSA A, Vol. 27, Issue 7, pp. 1723-1742 (2010)

http://dx.doi.org/10.1364/JOSAA.27.001723

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### Abstract

We study the performance of a previously proposed perturbation theory for the diffusion equation in frequency and time domains as they are known in the field of near infrared spectroscopy and diffuse optical tomography. We have derived approximate formulas for calculating higher order self- and mixed path length moments, up to the fourth order, which can be used in general diffusive media regardless of geometry and initial distribution of the optical properties, for studying the effect of absorbing defects. The method of Padé approximants is used to extend the validity of the theory to a wider range of absorption contrasts between defects and background. By using Monte Carlo simulations, we have tested these formulas in the semi-infinite and slab geometries for the cases of single and multiple absorbing defects having sizes of interest (d=4–10 mm, where d is the diameter of the defect). In frequency domain, the discrepancy between the two methods of calculation (Padé approximants and Monte Carlo simulations) was within 10% for absorption contrasts Δμ_{a}≤0.2 mm^{−1} for alternating current data, and usually to within 1° for Δμ_{a}≤0.1 mm^{−1} for phase data. In time domain, the average discrepancy in the temporal range of interest (a few nanoseconds) was 2%–3% for Δμ_{a}≤0.06 mm^{−1}. The proposed method is an effective fast forward problem solver: all the time-domain results presented in this work were obtained with a computational time of less than about 15 s with a Pentium IV 1.66 GHz personal computer.

© 2010 Optical Society of America

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.5280) Medical optics and biotechnology : Photon migration

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 2, 2010

Manuscript Accepted: May 26, 2010

Published: June 25, 2010

**Virtual Issues**

Vol. 5, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini, "Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. III. Frequency-domain and time-domain results," J. Opt. Soc. Am. A **27**, 1723-1742 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=josaa-27-7-1723

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### References

- N. Shah, A. Cerussi, C. Eker, J. Espinoza, J. Butler, J. Fishkin, R. Hornung, and B. Tromberg, “Noninvasive functional optical spectroscopy of human breast tissue,” Proc. Natl. Acad. Sci. U.S.A. 98, 4420–4425 (2001). [CrossRef]
- T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002). [CrossRef]
- Y. Yu, N. Liu, A. Sassaroli, and S. Fantini, “Near-infrared spectral imaging of the female breast for quantitative oximetry in optical tomography,” Appl. Opt. 48, D225–D235 (2009). [CrossRef]
- G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005). [CrossRef]
- M. A. Franceschini, D. K. Joseph, T. J. Huppert, S. G. Diamond, and D. A. Boas, “Diffuse optical imaging of the whole head,” J. Biomed. Opt. 11, 054007 (2006). [CrossRef]
- A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001). [CrossRef]
- V. Toronov, A. Webb, J. H. Choi, M. Wolf, U. Wolf, and E. Gratton, “Study of local cerebral hemodynamics by frequency-domain near-infrared spectroscopy and correlation with simultaneously acquired functional magnetic resonance imaging,” Opt. Express 9, 417–427 (2001). [CrossRef]
- T. Durduran, C. Zhou, B. L. Edlow, G. Yu, R. Choe, M. N. Kim, B. L. Cucchiara, M. E. Putt, Q. Shah, S. E. Kasner, J. H. Greenberg, A. G. Yodh, and J. A. Detre, “Transcranial optical monitoring of cerebrovascular hemodynamics in acute stroke patients,” Opt. Express 17, 3884–3902 (2009). [CrossRef]
- B. Chance, E. Anday, S. Nioka, S. Zhou, H. Long, K. Worden, C. Li, T. Turray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2, 411–423 (1998). [CrossRef]
- J. Steinbrink, A. Villringer, F. Kempf, D. Haux, S. Boden, and H. Obrig, “Illuminating the BOLD signal: combined fMRI-fNIRS studies,” Magn. Reson. Imaging 24, 495–505 (2006). [CrossRef]
- M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). [CrossRef]
- K. Ren, G. Abdoulaev, G. Bal, and A. H. Hielscher, “Frequency-domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. (USA) 28, 1463–1489 (2006). [CrossRef]
- E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3 approximation,” J. Opt. Soc. Am. A 18, 584–599 (2001). [CrossRef]
- K. G. Phillips and S. L. Jacques, “Solution of transport equations in layered media with refractive index mismatch using the PN-method,” J. Opt. Soc. Am. A 26, 2147–2162 (2009). [CrossRef]
- B. Wassermann, “Limits of high-order perturbation theory in time-domain optical mammography,” Phys. Rev. E 74, 031908 (2006). [CrossRef]
- F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E 67, 056623 (2003). [CrossRef]
- V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001). [CrossRef]
- D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: Analytical solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994). [CrossRef]
- A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999). [CrossRef]
- S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997). [CrossRef]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modelling photon transport in tissue,” Med. Phys. 20, 299–309 (1993). [CrossRef]
- W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24, 034011 (2008). [CrossRef]
- 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: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25, 711–732 (2009). [CrossRef]
- Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009). [CrossRef]
- V. Chernomordik, D. W. Hattery, D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, P. M. Schlag, and A. Gandjbakhche, “Quantification of optical properties of a breast tumor using random walk theory,” J. Biomed. Opt. 7, 80–87 (2002). [CrossRef]
- D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404–413 (1997). [CrossRef]
- M. R. Ostermeyer and S. L. Jacques, “Perturbation theory for diffuse light transport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997). [CrossRef]
- D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg, “Evaluation of higher-order time-domain perturbation theory of photon diffusion on breast equivalent phantoms and optical mammograms,” Phys. Rev. E 76, 061908 (2007). [CrossRef]
- A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory,” J. Opt. Soc. Am. A 23, 2105–2118 (2006). [CrossRef]
- A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. II. Continuous-wave results,” J. Opt. Soc. Am. A 23, 2119–2131 (2006). [CrossRef]
- A. Sassaroli, F. Martelli, S. Fantini, “Fourth order perturbation theory for the diffusion equation: continuous wave results for absorbing defects,” Proc. SPIE 7174, 717402 (2009). [CrossRef]
- A. Sassaroli, F. Martelli, and S. Fantini, “Higher order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries,” Appl. Opt. 48, D62–D73 (2009). [CrossRef]
- R. Graaff and K. Rinzema, “Practical improvements on photon diffusion theory: application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001). [CrossRef]
- J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, 1992).
- D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997). [CrossRef]
- A. Torricelli, L. Spinelli, A. Pifferi, P. Taroni, and R. Cubeddu, “Use of a nonlinear perturbation approach for in vivo breast lesion characterization by multi-wavelength time-resolved optical mammography,” Opt. Express 11, 853–867 (2003). [CrossRef]
- G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge U. Press, 1996).

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