## Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source

JOSA A, Vol. 16, Issue 4, pp. 837-844 (1999)

http://dx.doi.org/10.1364/JOSAA.16.000837

Acrobat PDF (286 KB)

### Abstract

The images method is widely used to solve the diffusion equation for the propagation of light within a semi-infinite medium (*z*>0) in which both scattering and absorption occur. The results are widely used both to analyze data and to illustrate the failings of the diffusion approximation. We emphasize that the images method does not properly obey the extrapolation boundary conditions of diffusion theory, namely, (1− *z*_{e} d/d*z*)φ(*z*)|_{z=0}=0; instead, it approximates these by φ(−*z*_{e})=0. The images method also does not describe the source of diffusing photons produced by the exponential attenuation of a collimated incident beam. By correcting these defects and comparing with a simulation, we show that most of the error in the images solution is due to the source and boundary treatment rather than to the diffusion approximation. Our prediction for backscattered intensity versus distance from the source point should be useful for improved analysis of experimental data, especially in cases of nonzero wall reflectivity.

© 1999 Optical Society of America

**OCIS Codes**

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

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**Citation**

D. J. Durian and J. Rudnick, "Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source," J. Opt. Soc. Am. A **16**, 837-844 (1999)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-4-837

Sort: Year | Journal | Reset

### References

- B. Chance and R. R. Alfano, eds., Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389 (1995).
- A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
- A. G. Yodh, B. Tromberg, E. Sevick-Muraca, and D. J. Pine, eds., feature issue on diffusing photons in turbid media, J. Opt. Soc. Am. A 14, 136–342 (1997); Appl. Opt. 36, 5–231 (1997).
- K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967).
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
- J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, New York, 1979).
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
- J. Masoliver, J. M. Porra, and G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
- D. J. Durian, “Two-stream theory of diffusing-light spectroscopies,” Physica A 229, 218–235 (1996).
- J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
- R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
- R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
- G. C. Pomraning and B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy 22, 787–817 (1995).
- D. J. Durian, “The influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (1994).
- M. U. Vera and D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
- M. U. Vera, P.-A. Lemieux, and D. J. Durian, “The angular distribution of diffusely backscattered light,” J. Opt. Soc. Am. A 14, 2800–2808 (1997).
- G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
- R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements,” Appl. Opt. 22, 2456–2467 (1983).
- B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, and R. Boretsky, “Comparison of time-resolved and-unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
- T. J. Farrell, M. J. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance from the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
- H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, and B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
- 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).
- M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
- J. M. Schmitt, A. Knuttel, and R. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
- J. B. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, UK, 1959).
- M. Lax, V. Nayaranamurti, and R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, Leningrad, June 1987, J. L. Birman, H. Z. Cummins, and A. A. Kaplyanskii, eds. (Plenum, New York, 1987).
- K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
- D. J. Durian and J. Rudnick, “Photon migration at short times and distances, and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
- D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 33, 1502–504 (1998).
- I. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989).
- A. Y. Polishchuk, S. Gutman, M. Lax, and R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
- J. M. Porrà, J. Masoliver, and G. H. Weiss, “When the te-legrapher’s equation furnishes a better approximation to the transport equation than the diffusion approximation,” Phys. Rev. E 55, 7771–7774 (1997).
- R. Aronson and N. Corngold, “The photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A (to be published).
- K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
- M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, “Independence of the diffusion coefficient from absorption: ex-perimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
- T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
- P.-A. Lemieux, M. U. Vera, and D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.