## Light diffusion in a turbid cylinder. I. Homogeneous case

Optics Express, Vol. 18, Issue 9, pp. 9456-9473 (2010)

http://dx.doi.org/10.1364/OE.18.009456

Acrobat PDF (1101 KB)

### Abstract

This paper is the first of two dealing with light
diffusion in a turbid cylinder. The diffusion equation was solved for a homogeneous finite cylinder that is illuminated at an arbitrary location. Three solutions were derived for an incident *δ*-light source in the steady-state, frequency, and time domains, respectively, applying different integral transformations. The performance of these solutions was compared with respect to accuracy and speed. Excellent agreement between the solutions, of which some are very fast (< 10ms), was found. Six of the nine solutions were extended to a circular flat beam which is incident onto the top side. Furthermore, the validity of the solutions was tested against Monte Carlo simulations.

© 2010 Optical Society of America

## 1. Introduction

1. F. Voit, J. Schäfer, and A. Kienle, “Light Scattering by Multiple Spheres: Comparison between Maxwell Theory and Radiative-Transfer-Theory Calculations,” Opt. Lett. **34**, 2593–2595 (2009). [CrossRef] [PubMed]

3. 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] [PubMed]

6. S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Pathlength in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. **37**, 1531–1560 (1992). [CrossRef] [PubMed]

7. B.W. Pogue and M.S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. **39**, 1157–1180 (1994). [CrossRef] [PubMed]

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

6. S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Pathlength in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. **37**, 1531–1560 (1992). [CrossRef] [PubMed]

7. B.W. Pogue and M.S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. **39**, 1157–1180 (1994). [CrossRef] [PubMed]

9. A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. **44**, 2747–2763 (1999). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Diffusion Theory

*z*-coordinate, see Fig. 1. The boundary conditions in

*z*-direction are then implemented applying two different approaches in the frequency domain: use of an infinite series of point sources (version A) and use of hyperbolic sine functions (version B), and applying two approaches in the time domain: use of the finite sine transform (version A) and again use of an infinite series of point sources (version B). In addition, for these six cases we give also the solutions for a flat beam incident onto the cylinder top. In the second part we use the finite sine transform in z-direction and then solve the resulting modified Bessel differential equation both in the frequency and time domains. We note that the results in the steady-state domain can be easily obtained from those derived in the frequency domain by setting the modulation frequency

*ω*= 0.

*et al*. presented a two-dimensional solution of the diffusion equation for a cylinder in the frequency domain solving the modified Bessel differential equation [6

6. S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Pathlength in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. **37**, 1531–1560 (1992). [CrossRef] [PubMed]

*et al*. extended this solution to locations outside the incident plane and for the extrapolated boundary condition [9

9. A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. **44**, 2747–2763 (1999). [CrossRef] [PubMed]

7. B.W. Pogue and M.S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. **39**, 1157–1180 (1994). [CrossRef] [PubMed]

*a*and

*l*, respectively. The incident light beam position can be chosen arbitrarily, e.g. it can be incident onto the top (or bottom) and onto the barrel. It is assumed that the incident light beam can be represented by a point source in the turbid medium in direction of the beam at a distance of l/(

_{z}*μ*′

_{s}+

*μ*) from the location of (perpendicular) incidence at the cylinder barrel, see Fig. 1, where

_{a}*μ*and

_{a}*μ*′

_{s}are the absorption and reduced scattering coefficients of the homogeneous cylinder. The position of the point source is given in cylindrical coordinates

*r*⃗

_{0}= (

*ρ*

_{0},

*ϕ*

_{0},

*z*

_{0}). In case of an oblique incident beam the Snel’s law has to be applied to determine the position of the point source. Extrapolated boundary conditions are used to describe the influence of the border between the turbid medium and the surrounding, characterized by the refractive indices,

*n*

_{1}and

*n*

_{0}, respectively. As in earlier papers we use the fluence rate and the flux term for calculation of the reflectance in the steady-state domain [12

12. A. Kienle and M. S. Patterson, “Improved Solutions of the Steady-State and the Time-Resolved Diffusion Equations for Reflectance from a Semi-Infinite Turbid Medium,” J. Opt. Soc. Am. A **14**, 246–254 (1997). [CrossRef]

12. A. Kienle and M. S. Patterson, “Improved Solutions of the Steady-State and the Time-Resolved Diffusion Equations for Reflectance from a Semi-Infinite Turbid Medium,” J. Opt. Soc. Am. A **14**, 246–254 (1997). [CrossRef]

#### 2.1.1. Solutions derived via the finite Hankel transform

*D*= 1/(3

*μ*′

_{s}),

*c*, and

*ω*denote the fluence rate, the diffusion coefficient, the speed of light in the turbid medium and the angular frequency of the intensity modulated light, respectively. Equation (1) can be rewritten by using cylindrical coordinates as follows

*r*⃗,

*ω*). First, a cosine transform of the form

*ϕ*derivative. Thus, we obtain

*m*order

^{th}*s*are the positive roots of the Bessel functions of first kind and m

_{n}^{th}order divided by

*a*′, i.e.

*J*(

_{m}*a*′

*s*) = 0. The roots of the Bessel functions are determined using Halley’s method and applying the start values described by Abramowitz and Stegun [17]. The upper limit of the integral

_{n}*a*′ results from the extrapolated boundary condition Φ(

*ρ*=

*a*′) = 0, where we use

*a*′ =

*a*+

*z*. The extrapolation length

_{b}*z*is calculated with

_{b}*R*represents the fraction of photons that is internally diffusely reflected at the boundary to the non-scattering surrounding.

_{eff}*R*was calculated by solving the integrals presented by Haskell

_{eff}*et al*[13

13. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A **11**, 2727–2741 (1994). [CrossRef]

*ρ*=

*a*′, we obtain the following ordinary differential equation where Φ = Φ(

*s*,

_{n}*ϕ*,

*m*,

*z*,

*ω*)

*G*(

*s*,

_{n}*z*,

*ω*as the solution of

*α*

^{2}=

*s*

^{2}

_{n}+

*μ*/

_{a}*D*+

*iω*/(

*Dc*). This equation is similar to the equation we derived solving the diffusion equation with lateral infinitely large extensions [15

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

*m*≥ 0, the following equation is obtained

*r*⃗ = (0,0,0), Eq. (20) becomes

*δ*-source. For the derivation of the solution of a finite flat beam incident onto the center of the cylinder top we use the diffusion equation for the rotationally symmetric case (no

*ϕ*-dependence)

*S*(

*r*⃗,

*ω*) is a rotationally symmetric source. For a flat beam we have

*ρ*is the radius of the flat beam and circ is defined as

_{w}*G*, see Eq. (9). The boundary conditions in

*z*-direction for the homogeneous cylinder are given by

#### Version A

#### Version B

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

*z*<

*l*as

_{z}15. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, “Nonin-vasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. **37**, 779–791 (1998). [CrossRef]

*l*is small.

_{z}#### Version A

#### Version B

*α*is now

#### 2.1.2. Solutions derived via modified Bessel differential equation

*z*-dependency in Eq. (2)

*ρ*=

*a*′,

*ω*) = 0. The solution can be derived by using a formula given in reference [6

**37**, 1531–1560 (1992). [CrossRef] [PubMed]

*ρ*derivative for the modified Bessel functions, see Eq. (22). The result is

*r*⃗,

*ω*), is used and split into real and imaginary parts [17]. Then, the inverse Fourier transform is calculated by using the FFT algorithm.

*z*-coordinate [18

18. A. Liemert and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. **15**, 025002 (2010). [CrossRef] [PubMed]

*I*′

_{m}(

*μ*) =

_{eff}a*∂I*(

_{m}*μ*)/

_{eff}ρ*∂ρ*∣

_{ρ=a}=

*μ*

_{eff}*I*

_{m+1}(

*μ*)+

_{eff}a*m*/

*aI*(

_{m}*μ*).

_{eff}a### 2.2. Monte Carlo Simulations

*n*

_{0}=

*n*

_{1}= 1.0. The Henyey-Greenstein phase function is used for calculating the scattering angles using an anisotropy factor of

*g*= 0.8. 10

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

## 3. Results

### 3.1. Comparison of the different solutions of the diffusion equation

*z*= 8mm, see inset of Fig. 2. The steady-state transmittance is calculated around the cylinder barrel at a depth of

*z*= 10mm. The height and radius of the cylinder are

*l*= 20mm and

_{z}*a*= 5mm, respectively. The reduced scattering coefficient is

*μ*′

_{s}= 0.9mm

^{-1}, whereas the absorption coefficient is varied (

*μ*= 0.005,0.01,0.015mm

_{a}^{-1}). For the calculations shown in Fig. 2 we used Eq. (54) for

*ω*= 0. As expected the transmitted light decreases up to the opposite point relative to the incident beam (at

*ϕ*=

*π*) and the transmittance is smaller for increasing absorption coefficients.

^{-10}, thus, the agreement is excellent. The relative differences between the two steady-state solutions calculated via the Hankel transform (version A and version B) are even smaller (data not shown).

*a*= 9mm and a height of

*l*= 12mm. The cylinder barrel is illuminated at

_{z}*z*

_{0}= 9mm and the transmittance is detected at

*z*= 9mm on the opposite site of the cylinder (

*ϕ*=

*π*) (black curve) and at the middle of the cylinder bottom (red curve).

^{-10}(data not shown). For comparison, in Fig. 4 also the time resolved transmittance from a rectangular parallelepiped is shown for both detection positions (dashed curves) [8

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

*x*-,

*y*-, and

*z*-directions are 9mm, 9mm, and 12mm, respectively. As expected, the results diverge for longer times when the detected photons have experienced the region where the two geometries (cylinder and parallelepiped) are different.

*ω*

_{0}= 2

*π*· 500MHz) for a pencil beam that is incident onto the cylinder barrel computed with the solution obtained by solving the modified Bessel differential equation. The light transmitted from the barrel at an angle of

*ϕ*= 3

*π*/4 at the same height is shown. The transmittance is computed for different cylinder radii

*a*= 5mm (black curve),

*a*= 6mm (green curve),

*a*= 7mm (red curve).

*a*= 6mm:

^{-10}.

### 3.2. Comparison with Monte Carlo simulations

*a*= 14mm, whereas the height of the cylinders is

*l*= 5mm (blue curves) and

_{z}*l*= 10mm (red curves). The optical properties are

_{z}*μ*′

_{s}= 1.3mm

^{-1}and

*μ*= 0.008mm

_{a}^{-1}.

*ρ*= 0mm (blue curve),

_{w}*ρ*= 5mm (green curve), and

_{w}*ρ*= 10mm (red curve). We note that the beam with

_{w}*ρ*= 0mm corresponds to a pencil beam. The optical properties of the cylinder are

_{w}*μ*′

_{s}= 1.2mm

^{-1}and

*μ*= 0.01mm

_{a}^{-1}. The radius and height are

*a*= 20mm and

*l*= 5mm, respectively.

_{z}*ρ*= 0mm. For large beam diameters the agreement is even good for small distances from the source.

_{w}## 4. Discussion

## Acknowledgement

## References and links

1. | F. Voit, J. Schäfer, and A. Kienle, “Light Scattering by Multiple Spheres: Comparison between Maxwell Theory and Radiative-Transfer-Theory Calculations,” Opt. Lett. |

2. | A. Ishimaru, |

3. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time Resolved Reflectance and Transmittance for the Noninvasive Measurement of Tissue Optical Properties,” Appl. Opt. |

4. | T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties |

5. | 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. |

6. | S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Pathlength in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. |

7. | B.W. Pogue and M.S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. |

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

9. | A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. |

10. | A. Liemert and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” accepted (2010). |

11. | |

12. | A. Kienle and M. S. Patterson, “Improved Solutions of the Steady-State and the Time-Resolved Diffusion Equations for Reflectance from a Semi-Infinite Turbid Medium,” J. Opt. Soc. Am. A |

13. | R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A |

14. | E. Meissel and G.B. Mathews, |

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

16. | H. S. Carslaw and J. C. Jaeger, |

17. | M. Abramowitz and I. A. Stegun, |

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

**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 21, 2010

**Virtual Issues**

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

**Citation**

André Liemert and Alwin Kienle, "Light diffusion in a turbid cylinder. I. Homogeneous case," Opt. Express **18**, 9456-9473 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-9-9456

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### References

- F. Voit, J. Schäfer, and A. Kienle, “Light Scattering by Multiple Spheres: Comparison between Maxwell Theory and Radiative-Transfer-Theory Calculations,” Opt. Lett. 34, 2593–2595 (2009). [CrossRef] [PubMed]
- A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic Press, New York, 1978).
- 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] [PubMed]
- T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A Diffusion Theory Model of Spatially Resolved, Steady-State Diffuse Reflectance for the Noninvasive Determination of Tissue Optical Properties in Vivo,” Med. Phys. 19, 879–888 (1992). [CrossRef] [PubMed]
- 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]
- S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis for the Determination of Optical Path length in Tissue: Temporal and Frequency Analysis,” Phys. Med. Biol. 37, 1531–1560 (1992). [CrossRef] [PubMed]
- B. W. Pogue, and M. S. Patterson, “Frequency Domain Optical Absorption Spectroscopy of Finite Tissue Volumes Using Diffusion Theory,” Phys. Med. Biol. 39, 1157–1180 (1994). [CrossRef] [PubMed]
- A. Kienle, “Light Diffusion through a Turbid Parallelepiped,” J. Opt. Soc. Am. A 22, 1883–1888 (2005). [CrossRef]
- A. Sassaroli, F. Martelli, D. Imai, and Y. Yamada, “Study on the Propagation of Ultra-Short Pulse Light in Cylindrical Optical Phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999). [CrossRef] [PubMed]
- A. Liemert, and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” accepted (2010).
- http://www.uni-ulm.de/ilm/index.php?id=10020200.
- A. Kienle, and M. S. Patterson, “Improved Solutions of the Steady-State and the Time-Resolved Diffusion Equations for Reflectance from a Semi-Infinite Turbid Medium,” J. Opt. Soc. Am. A 14, 246–254 (1997). [CrossRef]
- R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994). [CrossRef]
- E. Meissel, and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics, (Bibliobazaar, 2008).
- 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]
- H. S. Carslaw, and J. C. Jaeger, Conduction of Heat in Solids, (Clarendon, Oxford, 1959).
- M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1971).
- A. Liemert, and A. Kienle, “Light Diffusion in N-layered Turbid Media: Frequency and Time Domains,” J. Biomed. Opt. 15, 025002 (2010). [CrossRef] [PubMed]

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