## Light diffusion in a turbid cylinder. II. Layered case

Optics Express, Vol. 18, Issue 9, pp. 9266-9279 (2010)

http://dx.doi.org/10.1364/OE.18.009266

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### Abstract

This paper is the second of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for an N-layered finite cylinder. Solutions are given in the steady-state, frequency, and time domains for a point beam incident at an arbitrary position of the first layer and for a circular flat beam incident at the middle of the cylinder top. For special cases the solutions were compared to other solutions of the diffusion equation showing excellent agreement. In addition, the derived solutions were validated by comparison with Monte Carlo simulations. In the time domain we also derived a fast solution (≈ 10ms) for the case of equal reduced scattering coefficients and refractive indices in all layers.

© 2010 Optical Society of America

## 1. Introduction

*et al*. presented approximative solutions of the two-layered diffusion equation [3

3. I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. **39**, 1567–1582 (1992). [CrossRef]

4. A. Kienle, M.S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. **37**, 779–791 (1998). [CrossRef]

5. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. **37**, 6852–6862 (1998). [CrossRef]

6. A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. **44**, 2689–2702 (1999). [CrossRef] [PubMed]

8. X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media **16**, 121–135 (2006). [CrossRef]

7. J.M. Tualle, H.M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, “Asymptotic Behavior and Inverse Problem in Layered Scattering Media,” J. Opt. Soc. Am. A **21**, 24–34 (2004). [CrossRef]

9. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted. [PubMed]

10. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted. [PubMed]

11. G. Alexandrakis, T.J. Farrell, and M.S. Patterson, “Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium,” Appl. Opt. **37**, 7401–7409 (1998). [CrossRef]

12. S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. **23**, 3165–3167 (2005). [CrossRef]

13. G. Alexandrakis, T.J. Farrell, and M.S. Patterson, “Monte Carlo Diffusion Hybrid Model for Photon Migration in a Two-Layer Turbid Medium in the Frequency Domain,” Appl. Opt. **39**, 2235–2244 (2000). [CrossRef]

14. M. Das, C. Xu, and Q. Zhu, “Analytical Solution for Light Propagation in a Two-Layer Tissue Structure with a Tilted Interface for Breast Imaging,” Appl. Opt. **45**, 5027–5036 (2006). [CrossRef] [PubMed]

15. A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. **201**, 771–797 (2004). [CrossRef]

16. C. Donner and H.W. Jensen, “Rapid Simulations of Steady-State Spatially Resolved Reflectance and Transmittance Profiles of Multilayered Turbid Materials,” J. Opt. Soc. Am. A **23**, 1382–1390 (2006). [CrossRef]

*et al*. for the case of two and three layers [17

17. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, 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]

18. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. **52**, 2827–2843 (2007). [CrossRef] [PubMed]

19. 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]

## 2. Theory

### 2.1. Diffusion Theory

*l*,

_{k}*n*,

_{k}*μ*′

*, and*

_{sk}*μ*, respectively. The radius of the cylinder is indicated with

_{ak}*a*. As usual, it is assumed that the incident beam can be represented by an isotropic source at a distance of 1/(

*μ*′

_{s1}+

*μ*

_{a1}) from the location of incidence at the boundary of the cylinder. The point source is located in the first layer of the cylinder.

*R*

_{eff,k}is the angle-averaged probability for reflection at the boundary between layer k and the surrounding medium and

*D*= 1/(3

_{k}*μ*′

_{sk}) is the diffusion coefficient of layer k. For the boundary at the top and the bottom of the cylinder k equals 1 and N, respectively. For the boundary at the cylinder barrel k is assumed to be the layer at which the remitted or transmitted light is calculated, because it is more involved to solve the diffusion equation for N layers using a different boundary lengths at the cylinder barrel for each layer.

#### 2.1.1. Solution in the frequency domain

*c*, and

*ω*denote the fluence rate, the speed of light, and the angular frequency of the intensity modulated light, respectively. The source function for a point source in cylindrical coordinates is

*s*are determined from the following equation

_{n}*a*′ =

*a*+

*z*, and

_{bk}*J*is the first kind Bessel function of order m. By applying Eq. 4 to Eq. 3 the resulting ordinary differential equation for Φ = Φ(

_{m}*s*,

_{n}*ν*,

*m*,

*z*,

*ω*) is

*G*

_{1}

^{(p)}(

*s*,

_{n}*z*,

*ω*) we use the Fourier transform to find the one-dimensional Green’s function for an unbounded region in

*z*-direction. By applying

*k*= 1

*A*und

_{k}*B*are determined by using the following boundary conditions in z-direction

_{k}*z*<

*l*

_{1}) is given by

^{th}layer (

*L*

_{N-1}≤

*z*<

*L*) by

_{N}*β*

_{3}and

*γ*

_{3}are obtained by recurrence relations. The start values are

*β*

_{3}= sinh[

*α*

_{2}(

*l*

_{2}+

*z*

_{b2})] und

*γ*

_{3}= cosh[

*α*

_{2}(

*l*

_{2}+

*z*

_{b2})] have to be applied. For a three-layered cylinder (N = 3) only Eq. 17 is required. We note that we derived the above presented solutions already for the case of a laterally infinitely extended medium [9

9. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted. [PubMed]

*k*is obtained using Eq. 8 and the following recurrence relation

*ρ*incident onto the middle of the cylinder top we refer to the accompanying paper [1] where we gave the corresponding solution for a homogeneous cylinder. Accordingly, the solution of an N-layered cylinder is

_{w}*ω*= 0.

#### 2.1.2. Solution in the time domain

*α*in the frequency domain have the form

_{k}*s*=

*σ*+

*iω*is

*σ*=

*μ*+

_{ak}c*Dcs*

^{2}

*one obtains*

_{n}*x*(

*t*)

*e*

^{-Dcsn2t}. The region of convergence for this case is Re{

*s*} > -

*μ*. The imaginary axis is within the region of convergence, so that we can calculate the inverse transform of

_{ak}c*X*(

*μ*+

_{a}c*iω*) by means of the Fourier transform, in particular the FFT algorithm. For this special case we obtain the solution as

*z*-direction (flux-term) are used, whereas for all other solutions only the flux term is applied [9

9. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted. [PubMed]

10. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted. [PubMed]

### 2.2. Monte Carlo simulations

^{7}photons were used in Monte Carlo simulations.

## 3. Results

### 3.1. Comparison with other solutions of the diffusion theory

17. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, 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]

18. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. **52**, 2827–2843 (2007). [CrossRef] [PubMed]

*ρ*= 17mm.

^{-5}and can normally be neglected for applications in the field of biophotonics. If necessary, they can be decreased by putting more effort into the numerical evaluation.

*a*→ ∞) and compared it to solutions derived for N-layered turbid media that are laterally infinitely extended [9

10. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted. [PubMed]

*ρ*= 50, 60, 70mm). The point source is incident at the center of the cylinder top. The optical properties and thicknesses of the seven layers used for the calculations shown in Fig. 4 are listed in Table 1.

layer |
μ′_{s}[mm^{−1}] |
μ[mm_{a}^{−1}] | n |
l
_{1}[mm] |
---|---|---|---|---|

1 | 1.2 | 0.01 | 1.4 | 3 |

2 | 1.1 | 0.03 | 1.3 | 2 |

3 | 1.3 | 0.02 | 1.5 | 2 |

4 | 1.6 | 0.015 | 1.6 | 2 |

5 | 1.5 | 0.008 | 1.1 | 2 |

6 | 1.4 | 0.035 | 1.7 | 2 |

7 | 1.7 | 0.025 | 1.4 | 3 |

^{-10}and 10

^{-8}. Finally, we compare the derived solutions of the two-layered diffusion equation using the same optical properties in both layers to the solution of a homogeneous cylinder [1]. The point source is incident at the center of the cylinder and time resolved transmittance from the cylinder barrel at a depth of

*z*= 7mm is calculated, see Fig. 6. The results are given for two radii of the cylinder (

*a*= 18mm (red curve),

*a*= 40mm (green curve)). As in the other comparisons the relative difference between the curves calculated with the two models depend on the absolute reflectance data. In general, the larger the reflectance the smaller the relative differences.

*a*= 18mm and the green curve those for

*a*= 40mm.

21. A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A **22**, 1883–1888 (2005). [CrossRef]

*a*= 80mm in

*x*- and

*y*-direction and a length of 10mm in

*z*-direction are also shown in Fig. 6 (dashed curves). Similar as for the cylinder the parallelepiped is illuminated at the center and the transmittance is calculated at a depth of

*z*= 7mm in the middle of a lateral side, so that the distance between the source and the detector is the same as for the cylinder. For small times the solutions are close, for larger times they again diverge due to the different geometries.

### 3.2. Comparison with Monte Carlo simulations

*n*= 1.0 for all layers and the surrounding. The transmittance from the center of the cylinder bottom is calculated for three different radii

*a*= 4mm (red curve),

*a*= 5mm (green curve) and

*a*= 10mm (blue curve). Fig. 8 shows the results obtained from the solution of the diffusion equation (solid curves) and those obtained from the Monte Carlo simulations (points).

*r⃗*

_{0}= (

*x*,

*y*,

*z*) = (4√2mm, 4√2mm, 1/(

*μ*

_{a1}+

*μ*′

_{s1})mm). Fig. 9 shows the spatially resolved reflectance along the

*x*-axis (blue curve), along the negative

*y*-axis (green curve), and along the bisecting line between the

*x*-axis and

*y*-axis (red line), see inset. The optical and geometrical properties are those used for the calculations shown in Fig. 8 besides the radius of the cylinder, which is changed to

*a*= 12mm.

## 4. Discussion

18. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. **52**, 2827–2843 (2007). [CrossRef] [PubMed]

## Acknowledgement

## References and links

1. | A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder. I Homogeneous Case,” submitted. |

2. | A. Ishimaru, “Wave Propagation and Scattering in Random Media,” Academic Press, New York (1978). |

3. | I. Dayan, S. Havlin, and G.H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. |

4. | A. Kienle, M.S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. |

5. | A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. |

6. | A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. |

7. | J.M. Tualle, H.M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, “Asymptotic Behavior and Inverse Problem in Layered Scattering Media,” J. Opt. Soc. Am. A |

8. | X.C. Wang and S.M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,” Waves Rand. Compl. Media |

9. | A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt., accepted. [PubMed] |

10. | A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt., accepted. [PubMed] |

11. | G. Alexandrakis, T.J. Farrell, and M.S. Patterson, “Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium,” Appl. Opt. |

12. | S-H. Tseng, C. Hayakawa, B.J. Tromberg, J. Spanier, and A.J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. |

13. | G. Alexandrakis, T.J. Farrell, and M.S. Patterson, “Monte Carlo Diffusion Hybrid Model for Photon Migration in a Two-Layer Turbid Medium in the Frequency Domain,” Appl. Opt. |

14. | M. Das, C. Xu, and Q. Zhu, “Analytical Solution for Light Propagation in a Two-Layer Tissue Structure with a Tilted Interface for Breast Imaging,” Appl. Opt. |

15. | A.H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. |

16. | C. Donner and H.W. Jensen, “Rapid Simulations of Steady-State Spatially Resolved Reflectance and Transmittance Profiles of Multilayered Turbid Materials,” J. Opt. Soc. Am. A |

17. | F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for Layered Diffusive Media by the Eigenfunction Method,” Phys. Rev. E |

18. | F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. |

19. | F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation Model for Light Propagation through Diffusive Layered Media,” Phys. Med. Biol. |

20. | F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, “Light Propagation through Biological Tissue and other Diffusive Media,” SPIE Press, Bellingham (2010). |

21. | A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A |

22. |

**OCIS Codes**

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

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

(170.5280) Medical optics and biotechnology : Photon migration

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: February 12, 2010

Revised Manuscript: April 7, 2010

Manuscript Accepted: April 14, 2010

Published: April 19, 2010

**Virtual Issues**

Vol. 5, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

André Liemert and Alwin Kienle, "Light diffusion in a turbid cylinder. II. Layered case," Opt. Express **18**, 9266-9279 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-9-9266

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### References

- A. Liemert and A. Kienle, “Light Diffusion in a N-layered Turbid Cylinder.I Homogeneous Case,” submitted.
- A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).
- I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-Layer Turbid Medium. A Diffusion Analysis,” J. Mod. Opt. 39, 1567–1582 (1992). [CrossRef]
- A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagniàres, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. 37, 779–791 (1998). [CrossRef]
- A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of Two-Layered Turbid Media with Time-Resolved Reflectance,” Appl. Opt. 37, 6852–6862 (1998). [CrossRef]
- A. Kienle and T. Glanzmann, “In Vivo Determination of the Optical Properties of Muscle Using a Layered-Model,” Phys. Med. Biol. 44, 2689–2702 (1999). [CrossRef] [PubMed]
- J. M. Tualle, H. M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, “Asymptotic Behavior and Inverse Problem in Layered Scattering Media,” J. Opt. Soc. Am. A 21, 24–34 (2004). [CrossRef]
- X. C. Wang and S. M. Wang, “Light Transport Modell in a N-Layered Mismatched Tissue,”Waves Rand. Compl. Media 16, 121–135 (2006). [CrossRef]
- A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Steady-State Domain,” J. Biomed. Opt.accepted. [PubMed]
- A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt.accepted. [PubMed]
- G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium,” Appl. Opt. 37, 7401–7409 (1998). [CrossRef]
- S-H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative Spectroscopy of Superficial Turbid Media,” Opt. Lett. 23, 3165–3167 (2005). [CrossRef]
- G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Monte Carlo Diffusion Hybrid Model for Photon Migration in a Two-Layer Turbid Medium in the Frequency Domain,” Appl. Opt. 39, 2235–2244 (2000). [CrossRef]
- M. Das, C. Xu, and Q. Zhu, “Analytical Solution for Light Propagation in a Two-Layer Tissue Structure with a Tilted Interface for Breast Imaging,” Appl. Opt. 45, 5027–5036 (2006). [CrossRef] [PubMed]
- A. H. Barnett, “A Fast Numerical Method for Time-Resolved Photon Diffusion in General Stratified Turbid Media,” J. Comp. Phys. 201, 771–797 (2004). [CrossRef]
- C. Donner and H. W. Jensen, “Rapid Simulations of Steady-State Spatially Resolved Reflectance and Transmittance Profiles of Multilayered Turbid Materials,” J. Opt. Soc. Am. A 23, 1382–1390 (2006). [CrossRef]
- F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada Y, 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]
- F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the Time-Dependent Diffusion Equation for a Three-Layer Medium: Application to Study Photon Migration through a Simplified Adult Head Model,” Phys. Med. Biol. 52, 2827–2843 (2007). [CrossRef] [PubMed]
- 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]
- F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and other Diffusive Media (SPIE Press, Bellingham, 2010).
- A. Kienle, “Light Diffusion Through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005). [CrossRef]
- http://www.uni-ulm.de/ilm/index.php?id=10020200.

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