## An improved Monte Carlo diffusion hybrid model for light reflectance by turbid media

Optics Express, Vol. 15, Issue 10, pp. 5905-5918 (2007)

http://dx.doi.org/10.1364/OE.15.005905

Acrobat PDF (243 KB)

### Abstract

This paper introduces an improved diffusion model which is accurate and fast in
computation for the cases of
μ_{a}/μ’_{s} <
007 as good as the conventional diffusion model for the cases of
μ_{a}/μ’_{s} <
0.007 for surface measurement, hence more suitable than the conventional model
to be the forward model used in the image reconstruction in the diffuse optical
tomography. Deviation of the diffusion approximation (DA) on the medium surface
is first studied in the Monte Carlo (MC) diffusion hybrid model for reflectance
setup. A modification of DA and an improved MC diffusion hybrid model based on
this modified DA are introduced. Numerical tests show that for media with
relatively strong absorption the present modified diffusion approach can reduce
the surface deviation significantly in both the hybrid and pure diffusion model,
and consumes nearly no more computation time than the conventional diffusion
approach.

© 2007 Optical Society of America

## 1. Introduction

1. A. Yodh and B. Chance, “Spectroscopy and imaging with
diffusing light,” Phys. Today **48**, 34–40
(1995). [CrossRef]

3. K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, “Constrast features of breast cancer
in frequency-domain laser scanning mammography,”
J. Biomed. Opt. **3**, 129–136
(1998) [CrossRef]

4. A. Villringer and B. Chance, “Non-invasive optical spectroscopy
and imaging of human brain function,”
Trends Neurosci. **20**, 435–442
(1997). [CrossRef] [PubMed]

2. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process Mag. 57–75 (2001). [CrossRef]

7. B. C. Wilson and G. Adam, “A Monte Carlo model for the
absorption and flux distributions of light in
tissue,” Med. Phys. **10**, 824–830
(1983). [CrossRef] [PubMed]

*μ*

_{a}/

*μ*’

_{s}≤ 0.005, the deviation of the DE solution from the RTE solution is usually smaller than 1%). Another limitation of DE is that it can not correctly model light propagation in the “near-source region” (the region near the incident point of the collimated light source) [10–13

10. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid
materials determined from reflection measurements. 1.
Theory,” Appl. Opt. **22**, 2456–2462
(1983). [CrossRef] [PubMed]

14. L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo
simulation and diffusion theory for light reflectance by turbid
media,” J. Opt. Soc. Am. A **10**, 1746–1752
(1993). [CrossRef]

17. G. Bal and Y. Maday, “Coupling of transport and diffusion
models in linear transport theory,” Math.
Model Numer. Anal. **36**, 69–86
(2002). [CrossRef]

*μ*

_{a}/

*μ*’

_{s}< 0.007 is not satisfied. For example, in human brain near-infrared imaging,

*μ*

_{a}/

*μ*’

_{s}of the scalp is about 0.0095, and

*μ*

_{a}/

*μ*’

_{s}of the gray matter is about 0.0164 [5

5. E. Okada and D. T. Delpy, “Near-infrared light propagation in
an adult head model. II,” Appl. Opt. **42**, 2915–2922
(2003). [CrossRef] [PubMed]

*μ*

_{a}/

*μ*’

_{s}≥ 0.05, the deviation of the diffusion modeled light intensity at the detector from the RTE modeled value is usually greater than 7%. Figure 2 shows a numerical example of the comparison between the conventional MC diffusion hybrid model and the pure MC model. From Fig. 2 one sees that the deviation of the surface diffuse intensity is greater than 7% when the distance from the incident point is larger than 1.2 cm.

*μ*

_{a}/

*μ*’

_{s}< 0.07 as good as the conventional diffusion model for the cases of

*μ*

_{a}/

*μ*’

_{s}< 0.007 for surface measurement, hence more suitable than the conventional model to be the forward model used in the DOT image reconstruction. The idea is presented in the following two paragraphs.

19. 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). [CrossRef]

*μ*

_{a}/

*μ*’

_{s}< 0.07 is as good as the conventional diffusion model for the cases of

*μ*

_{a}/

*μ*’

_{s}< 0.007. Meanwhile, this improved model (as a forward model) consumes nearly no more computation time than the conventional diffusion model. Therefore, this model is more suitable than the conventional model to be the forward model for the image reconstruction in DOT.

*μ*

_{a}/

*μ*’

_{s}< 0.07, we first study in Section 2 the accuracy behavior of the basic equations (listed in Section 2.1) of the conventional DA when the conventional model is applied to the cases of 0.05 ≤

*μ*

_{a}/

*μ*’

_{s}< 0.07. This study shows which equations listed in Section 2.1 are more inaccurate and should be modified to correctly model the light propagation in the near-surface region. In Section 3.1 several empirical formulae are proposed to modify the basic equations of the conventional DA. These modified basic equations represent the “modified DA”. A modified MC diffusion hybrid model (for light reflectance by turbid media) based on the modified DA is introduced in Section 3.2 and numerical tests for the modified model are provided in Section 3.3. Supported by the numerical experiments including that shown in Section 3.3, we give our conclusion in Section 4.

## 2. Surface deviation of the conventional diffusion approximation

### 2.1. The conventional diffusion approximation

*μ’*=

_{t}*μ’*+

_{s}*μ*is the reduced transport coefficient,

_{a}*J*(

_{d}**r**,

**s**) is the diffuse radiance (Wsr

^{-1}cm

^{-2}) at point r in the direction of unit vector

**s**, and Ω represents the far-from-source region of the turbid medium. Note that Eq. (1) (called the “conventional DE” in the present paper) is derived from the following two equations [6]:

**F**

_{d}(

**r**) ≜ ∫

_{4π}

*J*(

_{d}**r**,

**s**)

**s**d

*ω*is the diffuse flux.

21. M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layered
media,” Appl. Opt. **27**, 1820–1824
(1988). [CrossRef] [PubMed]

*S*represents the surface of the medium,

**n**is the outward normal to

*S*, and

*A*is calculated with the following expression [22

22. 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*(

*θ*) is the Fresnel reflection coefficient for the diffuse light hitting the medium surface (from inside) at an angle of

*θ*(with respect to

**n**). Coefficient

*A*is a function of the relative refractive index

*n*of the medium and can be approximated with a polynomial fit of

*n*in practical calculation [22

22. 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]

### 2.2. Three implementations of the conventional MC diffusion hybrid model

_{i}(

*i*= 0,1,2,3) are defined for the convenience of describing the various boundary conditions to be used in the present paper.

*V*

_{2}and Γ

_{2}). Then, the finite element method (FEM) [23–25] based on the conventional DE [Eq. (1)] is applied only in the far-from-source region Ω (The size of

*V*

_{2}is chosen large enough so that the light in Ω is completely diffusive.) The diffuse intensity at point

**r**in Ω decreases rapidly when the distance between

**r**and O (the incident point) increases. Volume

*V*

_{1}is chosen large enough so that the diffuse intensity on Γ

_{0}is small enough and zero boundary condition can be applied at Γ

_{0}for the DE solution.

_{2}. In all these three implementations, the FEM is used to solve Eq. (1) (combined with different boundary conditions).

21. M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layered
media,” Appl. Opt. **27**, 1820–1824
(1988). [CrossRef] [PubMed]

_{0}, Γ

_{1}and Γ

_{3}as in Conventional Hybrid I. For the boundary condition on Γ

_{2}, we use in Conventional Hybrid II

*α*(

**r**∈ Γ

_{2}) can be obtained from the MC simulation. One way to evaluate

*α*(

**r**∈ Γ

_{2}) with MC simulation is to evaluate

*J*(

_{d}**r**∈ Γ

_{2},

**s**) at first, then calculate

*α*(

**r**∈ Γ

_{2}) with the definition of

*α*(

**r**∈ Γ

_{2}) for

*U*(

_{d}**r**∈ Γ

_{2}), and hence improves the accuracy of

*U*(

_{d}**r**) for

**r**near Γ

_{2}. Note that the square ÉF́ǴH́ does not need to be much bigger than square EFGH, and square EFGH should be set a proper size. In the MC diffusion hybrid models,

*U*on Γ

_{d}_{2}is evaluated with MC simulation. The computation time for the simulation to evaluate

*U*on Γ

_{d}_{2}under a required precision increases exponentially with the distance between Γ

_{2}and the source. Empirically, this distance is set about 5/

*μ*́, i.e., the size of the “near-source region” is about 10/

_{t}*μ*́. Outside this near-source region, the near-field deviation is quite small and it is suitable to use the diffusion model instead of the MC model to save the computation time. Setting a larger size (more than 10

_{t}*μ*́) of the near-source region will consume extra time for the computation. For Conventional Hybrids I and III, one can set ÉF́ǴH́ almost identical to EFGH (i.e., Γ

_{t}_{2}reduces to the four edge lines of square EFGH).

*μ*/

_{a}*μ*́ < 0.007 . But in the following subsection, we apply Conventional Hybrids I-III to a case of

_{s}*μ*/

_{a}*μ*́ = 0.06 and compare their solutions to study the behavior of the conventional DA in the cases of

_{s}*μ*/

_{a}*μ*́ ≥ 0.05 .

_{s}### 2.3. Numerical study for the surface deviation

*x*-axis obtained by Conventional Hybrids I-III. The relative deviations are defined by

*sfUd*

_{MC}(

*x*),

*sfUd*

_{Hybrid I, II, III}(

*x*) are the surface diffuse intensities at position (

*x*,0,0) obtained by the pure MC simulation and Conventional Hybrid (I, II or III), respectively.

_{2}. Therefore, as shown in Fig. 4, the significant difference between

*DEV*

_{hybrid II}(

*x*) and

*DEV*

_{hybrid III}(

*x*) in region 0.4380cm ≤

*x*≤ 0.4517cm (i.e.,

*sfUd*

_{hybrid II}(

**r**∈ Γ

_{2}) is much greater than

*sfUd*

_{MC}(

**r**∈ Γ

_{2}) while

*sfUd*

_{hybrid III}(

**r**∈ Γ

_{2}) is close to

*sfUd*

_{MC}(

**r**∈ Γ

_{2})) implies that Eq. (8) is not quite accurate, i.e., Eq. (3) for

**r**near the surface is not quite accurate in this example. Also note that

*DEV*

_{hybrid II}(

*x*) is a bit worse than

*DEV*

_{hybrid I}(

*x*). This indicates that Eq. (8) is less accurate than the second equation in system (7) [derived from Eqs. (3) and (4)]. This difference between

*DEV*

_{hybrid II}(

*x*) and

*DEV*

_{hybrid I}(

*x*) implies that Eq. (4) is not accurate. The underlying reason for the inaccuracy of Eqs. (3) and (4) is the relatively strong anisotropy of the diffuse light near the surface of the medium.

## 3. The modified DA and the improved diffusion (hybrid or pure) models

### 3.1. The modified diffusion approximation

*η*

_{1}(

**r**) and

*η*

_{2}(

**r**) are modification parameter functions introduced to reduce the surface deviation. They are usually selected empirically. For the setup and the coordinate system shown in Fig. 3 (the source beam incidents along

*z*axis, and

*x*-

*y*plane is defined on the surface of the medium), we propose that

*η*

_{1}(

**r**) and

*η*

_{2}(

**r**) be approximated as:

*c*

_{1}(

*n*) and

*c*

_{2}(

*n*) are constants determined only by the refractive index

*n*of the medium.

*η*

_{1}(

**r**) and

*η*

_{2}(

**r**) in Eq. (11). 1. For light reflectance setup shown in Fig. 3, the deviation of the surface measurement evaluated by the conventional diffusion model from that by the MC model increases almost proportionally with

*μ*/

_{a}*μ*́ [22

_{s}22. 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]

*μ*/

_{a}*μ*́. 2. The anisotropic light becomes nearly isotropic when it travels a path length in the scattering dominated medium. Therefore, the modification should decrease rapidly with the depth from the surface. It’s natural to choose

_{s}*η*

_{1}(

**r**) to decrease exponentially with the dimensionless path length. 3. The deviation of the DE on the surface increases with the mismatch of the refractive index at the interface [19-20

19. 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). [CrossRef]

*c*

_{1}(

*n*) and

*c*

_{2}(

*n*) should be determined by the refractive index

*n*. For a fixed

*n*,

*c*

_{1}and

*c*

_{2}are two constants. Usually their values are unknown, however they are not arbitrary. The values of

*c*

_{1}and

*c*

_{2}for a fixed

*n*can be estimated from large numbers of numerical examples. In this paper, we use

*n*= 1.35. We have done hundreds of numerical comparisons between the modified DA and the pure MC simulation for various optical properties of turbid media. In each numerical experiment, we choose various pairs of values of

*c*

_{1}and

*c*

_{2}to find a pair of values which makes the modified DA consistent with the pure MC simulation best. We found that for the case of

*n*= 1.35, when we use the values

*c*

_{1}= 3.14 and

*c*

_{2}= 2.2, the modified DA is consistent with the pure MC simulation very good in most cases. Though the best pair values of

*c*

_{1}and

*c*

_{2}are unknown, they should be near the pair values of

*c*

_{1}= 3.14 and

*c*

_{2}= 2.2 which are used for the present modified DA in this paper.

### 3.2. The improved MC diffusion hybrid model (the present hybrid model

*α*(

**r**∈ Γ

_{2}) and

*D*(

**r**) is the diffusion coefficient that is defined in the present paper by

*η*

_{1}(

**r**) = 0, The definition of

*D*(

**r**) by Eq. (14) reduces to the conventional definition of the diffusion coefficient

*D*(

**r**) = [3

*μ*́(

_{t}**r**)]

^{-1}(Ref. [6]) and at the same time the modified DE [Eq. (12)] reduces to the conventional DE [Eq. (1)]. Absorption-independent diffusion coefficient

*D*(

**r**) = [3

*μ*́(

_{t}**r**)]

^{-1}(corresponding to

*η*

_{1}(

**r**) =

*μ*(

_{a}**r**)/

*μ*́(

_{s}**r**)) was suggested in Refs. [22

**36**, 4587–4599
(1997). [CrossRef] [PubMed]

26. 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). [CrossRef]

*D*(

**r**) by using the

*η*

_{1}(

**r**) function described in Eq. (11). The partial current boundary condition in the second equation of system (13) is modified by the

*η*

_{2}(

**r**) in Eq. (11). And in the present paper, we use the values of

*c*

_{1}= 3.14 and

*c*

_{2}= 2.2 , i.e., for light reflectance setups shown in Figs. 1 and 3, we propose the following diffusion coefficient and partial current boundary condition: (The refractive index of the medium

*n*= 1.35)

*D*(

**r**) = [3

*μ*́(

_{s}**r**)]

^{-1}in Refs. [22

**36**, 4587–4599
(1997). [CrossRef] [PubMed]

26. 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). [CrossRef]

13. T. Spott and L. O. Svaasand, “Collimated light sources in the
diffusion approximation,” Appl. Opt. **39**, 6453–6465
(2000). [CrossRef]

*U*(

_{d}**r**):

*ϕ*(

**r**) is the test function satisfying

*ϕ*(

**r**∈ Γ

_{0}∪ Γ

_{2}) = 0 , and

*U*(

_{d}**r**) is imposed with

*U*(

_{d}**r**∈ Γ

_{0}) = 0 and

*U*(

_{d}**r**∈ Γ

_{2}) =

*α*(

**r**∈ Γ

_{2}).

*η*

_{1}(

**r**) ≡ 0 and

*η*

_{2}(

**r**) = 0 , Eq. (16) reduces to the Galerkin variation equation in the conventional model. This implies that the computation complexity of the forward problem in the modified model is the same as in the conventional model. Therefore, the modified DA approach consumes nearly no more computation time than the conventional DA approach.

### 3.3. Numerical tests for the modified DAs

*η*

_{1}(

**r**) =

*μ*(

_{a}**r**)/

*μ*́ (

_{s}**r**) and

*η*

_{2}(

**r**) = 0. A modification between Modified DA I and II is

*η*

_{1}(

**r**)=

*μ*(

_{a}**r**)/

*μ*́(

_{s}**r**) and

*η*

_{2}(

**r**) = 2.2

*μ*(

_{a}**r**)/

*μ*́(

_{t}**r**).

*V*

_{1}=

*V*

_{2}∪ Ω is the whole domain of the medium,

*S*(

**r**) is the diffuse source which is derived from the reduced incident radiance [6, 10

10. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid
materials determined from reflection measurements. 1.
Theory,” Appl. Opt. **22**, 2456–2462
(1983). [CrossRef] [PubMed]

21. M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layered
media,” Appl. Opt. **27**, 1820–1824
(1988). [CrossRef] [PubMed]

*S*(

**r**) is very small and can be considered as zero, i.e.

*S*(

**r**∈ Ω) ≈ 0. And the boundary conditions system (13) is replaced by

*z*axis, and

*x*-

*y*plane is defined on the surface of the medium. The incident point O is at (0, 0, 0).

*μ*/

_{a}*μ*́ (0.005) used in this example.

_{t}*z*axis (the axis of the incident beam) and described by

*r*= √

*x*

^{2}+

*y*

^{2}+

*z*

^{2},(

*z*≥ 0). As shown in Fig. 6, Present Hybrid model is the most accurate (the deviation of Present Hybrid from the pure MC model is the smallest). The present pure diffusion model (denoted by “Present Pure” in Fig. 6) is much more accurate than the conventional pure diffusion model (denoted by “Conventional Pure” in Fig. 6), and in the far-from-source region, Present Pure is also more accurate than Conventional Hybrid III. The large values of

*DEV*

_{Conventional, Present Pure}(

*x*≥ 0.1cm) are due to the near-field deviation of the diffusion model. Figure 6 shows that the present modified DA improves the accuracy significantly not only for the hybrid model but also for the pure diffusion model.

**36**, 4587–4599
(1997). [CrossRef] [PubMed]

26. 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). [CrossRef]

14. L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo
simulation and diffusion theory for light reflectance by turbid
media,” J. Opt. Soc. Am. A **10**, 1746–1752
(1993). [CrossRef]

*μ*/

_{a}*μ*́ → 0; and Fig. 7 shows that the present modified DA is also better than other modified DAs reported before (e.g., Modified DA II which was suggested in Refs. [22

_{t}**36**, 4587–4599
(1997). [CrossRef] [PubMed]

**14**, 3358–3365
(1997). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Yodh and B. Chance, “Spectroscopy and imaging with
diffusing light,” Phys. Today |

2. | D. A. Boas, D. H. Brooks, E. L. Miller, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process Mag. 57–75 (2001). [CrossRef] |

3. | K. T. Moesta, S. Fantini, H. Jess, S. Totkas, M. A. Franceschini, M. Kaschke, and P. M. Schlag, “Constrast features of breast cancer
in frequency-domain laser scanning mammography,”
J. Biomed. Opt. |

4. | A. Villringer and B. Chance, “Non-invasive optical spectroscopy
and imaging of human brain function,”
Trends Neurosci. |

5. | E. Okada and D. T. Delpy, “Near-infrared light propagation in
an adult head model. II,” Appl. Opt. |

6. | A. Ishimaru, |

7. | B. C. Wilson and G. Adam, “A Monte Carlo model for the
absorption and flux distributions of light in
tissue,” Med. Phys. |

8. | L. Wang, S. L. Jacques, and L. Zhen, “MCML - Monte Carlo modeling of light
transport in multi-layered tissues,”
Comput. Methods Programs Biomed. |

9. | D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code
for photon migration through complex heterogeneous media including the adult
human head,” Opt. Express |

10. | R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid
materials determined from reflection measurements. 1.
Theory,” Appl. Opt. |

11. | 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. |

12. | S. R. Arridge, “Optical tomography in medical
imaging,” Inv. Probl. |

13. | T. Spott and L. O. Svaasand, “Collimated light sources in the
diffusion approximation,” Appl. Opt. |

14. | L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo
simulation and diffusion theory for light reflectance by turbid
media,” J. Opt. Soc. Am. A |

15. | G. Alexandrakis, T. J. Farrrell, and M. S. Patterson, “Monte Carlo diffusion hybrid model
for photon migration in a two-layer turbid medium in the frequency
domain,” Appl. Opt. |

16. | T. Hayashi, Y. Kashio, and E. Okada, “Hybrid Monte Carlo-diffusion method
for light propagation in tissue with a low-scattering
region,” Appl. Opt. |

17. | G. Bal and Y. Maday, “Coupling of transport and diffusion
models in linear transport theory,” Math.
Model Numer. Anal. |

18. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, “Finite element model for the coupled
radiative transfer equation and diffusion
approximation,” Int. J. Numer. Methods
Eng. , |

19. | 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 |

20. | B. Q. Chen, K. Stamnes, and J. J. Stamnes, “Validity of the diffusion
approximation in bio-optical imaging”,
Appl. Opt. |

21. | M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layered
media,” Appl. Opt. |

22. | 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. |

23. | S. C. Brenner and L. R. Scott, |

24. | P. G. Cialet, |

25. | J. M. Jin, |

26. | T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, “Does the photon-diffusion
coefficient depend on absorption,” J.
Opt. Soc. Am. A |

**OCIS Codes**

(170.5270) Medical optics and biotechnology : Photon density waves

(170.6960) Medical optics and biotechnology : Tomography

(170.7050) Medical optics and biotechnology : Turbid media

(290.1990) Scattering : Diffusion

**ToC Category:**

Scattering

**History**

Original Manuscript: January 8, 2007

Revised Manuscript: March 14, 2007

Manuscript Accepted: April 25, 2007

Published: April 30, 2007

**Virtual Issues**

Vol. 2, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Bin Luo and Sailing He, "An improved Monte Carlo diffusion hybrid model for light reflectance by turbid media," Opt. Express **15**, 5905-5918 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-5905

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### References

- A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48,34-40 (1995). [CrossRef]
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