## Heuristic Green’s function of the time dependent radiative transfer equation for a semi-infinite medium

Optics Express, Vol. 15, Issue 26, pp. 18168-18175 (2007)

http://dx.doi.org/10.1364/OE.15.018168

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### Abstract

The Green’s function of the time dependent radiative transfer equation for the semi-infinite medium is derived for the first time by a heuristic approach based on the extrapolated boundary condition and on an almost exact solution for the infinite medium. Monte Carlo simulations performed both in the simple case of isotropic scattering and of an isotropic point-like source, and in the more realistic case of anisotropic scattering and pencil beam source, are used to validate the heuristic Green’s function. Except for the very early times, the proposed solution has an excellent accuracy (>98% for the isotropic case, and >97% for the anisotropic case) significantly better than the diffusion equation. The use of this solution could be extremely useful in the biomedical optics field where it can be directly employed in conditions where the use of the diffusion equation is limited, e.g. small volume samples, high absorption and/or low scattering media, short source-receiver distances and early times. Also it represents a first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and diffuse optical imaging. Moreover the proposed solution can be useful to several research fields where the study of a transport process is fundamental.

© 2007 Optical Society of America

## 1. Introduction

2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–R93 (1999). [CrossRef]

3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2000). [CrossRef]

4. T. Feng, P. Edström, and M. Gulliksson, “Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation,” Inverse Probl. **23**, 879–891 (2007). [CrossRef]

5. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E **51**, 6134–6141 (1995). [CrossRef]

6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: Numerical and experimental investigation,” Phys. Med. Biol. **45**, 1359–1373 (2000). [CrossRef] [PubMed]

2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–R93 (1999). [CrossRef]

3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2000). [CrossRef]

7. For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol.49, N. 7 (2004). [PubMed]

*ρ*, typically in the range 20–40 mm, because photons received at shorter distances are not well described by the DE [6

6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: Numerical and experimental investigation,” Phys. Med. Biol. **45**, 1359–1373 (2000). [CrossRef] [PubMed]

8. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. **95**, 078101 (2005). [CrossRef] [PubMed]

9. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. **43**, 1285–1302 (1998). [CrossRef] [PubMed]

10. H. Dehghani, S. R. Arridge, and M. Schweiger, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A **17**,1659–1670 (2000). [CrossRef]

11. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**, 313–320 (2005). [CrossRef] [PubMed]

8. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. **95**, 078101 (2005). [CrossRef] [PubMed]

12. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E **56**, 1135–1141 (1997). [CrossRef]

12. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E **56**, 1135–1141 (1997). [CrossRef]

## 2. Theory

12. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E **56**, 1135–1141 (1997). [CrossRef]

*(*

_{i}*r, t*), emitted by a pulsed point-like isotropic source (i.e.

*source*=

*δ*(

**r**)

*δ*(

*t*)) in an infinite non-absorbing medium

*x*) is the step function (zero for

*x*<0 and 1 for

*x*>0),

*µ*,

_{s}*µ′*s and v are respectively the scattering coefficient, the reduced scattering coefficient and the speed of light in the medium, while the expression for the function

_{s}*G*is provided as the following

*x*) the first expression of Eq. (2) has to be preferred to the second one because it is more accurate, while for late times they tend to give the same values. The first term of Eq. (1) represents the ballistic peak, while the second term is the scattered component. The accuracy of Eq. (1) is better than 2% [12

**56**, 1135–1141 (1997). [CrossRef]

**56**, 1135–1141 (1997). [CrossRef]

*µ′*instead of

_{s}*µ*. Since

_{s}*µ′*=

_{s}*µ*(1-

_{s}*g*), where the symbol,

*g*, denotes the anisotropy factor, for isotropic scattering (

*g*=0) the two formalisms are identical. Furthermore, our approach would be useful to treat anisotropic scattering (

*g*≠0) for which Eq. (2) can still be heuristically used. The effect of absorption can be introduced, according to the RTE, by multiplying the above time-domain solution obtained for the case of zero absorption by exp(-

*µ*).

_{a}vt*z*

_{0}would involve the radiance,

*I*(

**r**,

**ŝ**,

*t*), emerging from the medium along any direction

**ŝ**. If the refractive indexes of both media are the same (as it is assumed in our derivation), the boundary condition results:

*I*(

**r**,

**ŝ**,

*t*)=0, for every point

**r**≡(

*x*,

*y*,

*z*) at the external boundary and for all directions

**ŝ**inwardly directed. Although rigorous, this approach cannot be implemented to solve the RTE with an analytical method. In order to overcome this problem the same boundary condition used for the DE [13

13. 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] [PubMed]

**r**

^{+}and

**r**

^{-}, of the real source (positive) and of the image source (negative) are shown in Fig. 1. In this way the fluence rate will vanish at

*z*=-

*z*, which identifies the extrapolated boundary, and it is composed by two terms of fluence from an infinite medium, Φ

_{e}*: one positive from the source at*

_{i}**r**

^{+}and the other negative from the source at

**r**

^{-}. This heuristic approach leads to the following expression for the fluence rate in a semi-infinite medium, Φ

*(*

_{si}*x*,

*y*,

*z*,

*z*

_{0},

*t*), with refractive index matched at the boundary

13. 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] [PubMed]

*z*=2/3

_{e}*µ′*. Note that it differs in about 6%from the

_{s}*extrapolated end point*,

*d*, of the Milne problem [1] (

_{e}*d*=0.7104/

_{e}*µ′*) that is accounted as the distance from the interface at which the asymptotic component of the density of energy extrapolates to zero. Although this fact does not represent a ground for the heuristic boundary condition employed, it anyway suggests that for a source deeply sinked inside the medium the EBC tends within few per cents toward the boundary conditions of the classical Milne problem of photon transport. It can be also noticed that the differences between these two solutions become negligible for a source far from the boundary and for the cases analyzed in this work. For instance, for

_{s}*z*

_{0}=1/

*µ′*the differences observed are within 2% and the accuracy of these two boundary conditions is thus similar. The time-resolved reflectance emerging from the medium is derived by Fick’s law [15

_{s}15. M. H. Lee, “Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion,” Phys. Rev. Lett. **85**, 2422–2425 (2000). [CrossRef] [PubMed]

*t*≫1/(

*µ′*).

_{s}v**56**, 1135–1141 (1997). [CrossRef]

*t*

_{1}and

*t*

_{2}we have the singularities due to the ballistic peak of the two sources. These effects arise from the delta source considered in the solution and are confined to very early times (e.g.

*t*

_{1}=4.7 ps and

*t*

_{2}=8.7 ps for

*ρ*=1 mm,

*µ′*=1 mm

_{s}^{-1}and

*v*=0.3 mm/ps). We want to stress that Eq. (4) is zero for

*t*<

*t*

_{1}and, with the exception of the singularities at

*t*

_{1}and

*t*

_{2}, it is a continuous function.

## 3. Results

16. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. **30**, 4600–4612 (1997). [CrossRef]

17. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. **3**, 897–905 (1994). [CrossRef]

*µ′*=1 mm

_{s}^{-1}. Since both Eq. (4) (denoted RTE in the figures) and the MC use the same dependence on

*µ*the comparison is done for a non absorbing medium without any loss of generality. The results obtained with the DE theory (Eq. (36) of Ref. [13

_{a}13. 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] [PubMed]

*z*

_{0}=1/

*µ′*and

_{s}*v*is assumed to be 0.3 mm/ps in the medium. In Fig. 2 (a) and (b) the reflectance from a semi-infinite medium is shown at

*ρ*=1 mm and 5 mm, respectively. In Fig. 2 (c) and (d) the plots show the ratio of the MC results to those of RTE and DE. In Fig. 2 (a) and (b) the error bars on the MC data are omitted since they have similar size as the symbols. In Fig. 2 (c) and (d) the error bars are obtained from the statistical standard deviations on the MC data. Similar results have been obtained for

*ρ*in the range 0.1–20 mm and for

*z*

_{0}>0.2 mm. Both for RTE and for MC response the contribution of ballistic photons has not been included.

*t*≤12 ps for

*ρ*=1 mm or 16.9≤

*t*≤26 ps for

*ρ*=5 mm), the RTE solution is within 2% in agreement with the results of the MC simulations. Approximately, for

*ρ*=0.1, 1, 5, 10 and 20 mm the error is less than 2% just 8 ps after

*t*

_{1}(that corresponds to 2.4 mean free paths). This accuracy is similar to that of Eq. (1). Thus even at very short distances the accuracy of Eq. (4) is similar to that of Eq. (1), assuring that the introduction of the heuristic boundary condition does not diminish the accuracy of the Paasschens’ solution. These results also show for the DE an accuracy significantly worse than the proposed heuristic solution. Moreover the solution from the DE is nonzero also for

*t*<

*t*

_{1}, due to the non-causality hypothesis assumed in this approach. The good agreement obtained at early times suggests that the CW reflectance obtained from an integration of Eq. (4) works well even at short distances and for high values of

*µ*.

_{a}*g*=0.9, typical of biological tissue, was assumed. In Fig. 3 (a) and (b) the reflectance is shown at

*ρ*=1 mm and 5 mm respectively. The ratio of the MC results to those of RTE and DE is plotted in figures (c) and (d). Equation (4) and the DE are evaluated with the source at

*z*

_{0}=1/

*µ′*. The distance

_{s}*ρ*=1 mm could be an actual value for measurements at null source-detector distance. These results show an agreement between the heuristic solution and the MC results similar to that of Fig. 2 and we still can see clear advantages in using this solution instead of the DE. These results support the use of Eq. (4) for analyzing measurements on tissue at short source-receiver distances and at early times.

## 4. Discussion and conclusions

*t*

_{1}<

*t*≤

*t*

_{1}+8 ps for

*v*=0.3 mm/ps and

*µ′*=1 mm

_{s}^{-1}) the proposed solution shows an error less than 2% even in regions where the DE shows errors larger than 10%. Also the comparisons with MC results for a medium with anisotropic scattering (

*g*=0.9) illuminated by a pencil light beam (Fig. 3) showed an excellent agreement. This is surprising, since the proposed solution is based on the response for the infinite medium that is exact only for isotropic scatterers. These surprising good results are probably due to a compensation effect between the different approximations introduced to obtain the heuristic solution (boundary conditions, pencil beam modelled with a point-like isotropic source, isotropic scattering). Anyway, the excellent agreement shown by the comparisons in Fig. 3 for the temporal response even at short distances and at early times, suggests that also the CW reflectance obtained integrating Eq. (4) gives accurate results even at short distances and for high values of absorption for diffusive media with anisotropic scattering.

18. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model for light propagation through diffusive layered media,” Phys. Med. Biol. **50**, 2159–2166 (2005). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | J. J. Duderstadt and W. R. Martin, |

2. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

3. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

4. | T. Feng, P. Edström, and M. Gulliksson, “Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation,” Inverse Probl. |

5. | L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E |

6. | F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: Numerical and experimental investigation,” Phys. Med. Biol. |

7. | For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol.49, N. 7 (2004). [PubMed] |

8. | A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. |

9. | A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. |

10. | H. Dehghani, S. R. Arridge, and M. Schweiger, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A |

11. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

12. | J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E |

13. | 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. |

14. | E. Zauderer, |

15. | M. H. Lee, “Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion,” Phys. Rev. Lett. |

16. | F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results,” Appl. Opt. |

17. | G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. |

18. | F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model for light propagation through diffusive layered media,” Phys. Med. Biol. |

**OCIS Codes**

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

(170.5280) Medical optics and biotechnology : Photon migration

(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: October 31, 2007

Revised Manuscript: December 7, 2007

Manuscript Accepted: December 17, 2007

Published: December 19, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Fabrizio Martelli, Angelo Sassaroli, Antonio Pifferi, Alessandro Torricelli, Lorenzo Spinelli, and Giovanni Zaccanti, "Heuristic Green’s function of the time dependent radiative transfer equation for a semi-infinite medium," Opt. Express **15**, 18168-18175 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-18168

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### References

- J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley&Sons, New York, 1979).
- S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999). [CrossRef]
- A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000). [CrossRef]
- T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007). [CrossRef]
- L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995). [CrossRef]
- F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: Numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000). [CrossRef] [PubMed]
- For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol. 49, N. 7 (2004). [PubMed]
- A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005). [CrossRef] [PubMed]
- A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998). [CrossRef] [PubMed]
- H. Dehghani, S. R. Arridge, and M. Schweiger, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17,1659-1670 (2000). [CrossRef]
- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005). [CrossRef] [PubMed]
- J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997). [CrossRef]
- 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] [PubMed]
- E. Zauderer, Partial Differential Equations of Applied Mathematics, (John Wiley&Sons, New York, 1989) Sec. 7.5, p. 484.
- M. H. Lee, "Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion," Phys. Rev. Lett. 85, 2422-2425 (2000). [CrossRef] [PubMed]
- F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results," Appl. Opt. 30, 4600-4612 (1997). [CrossRef]
- G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994). [CrossRef]
- F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005). [CrossRef] [PubMed]

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