## Effect of a clear layer at the surface of a diffusive medium on measurements of transmittance and reflectance

Optics Express, Vol. 12, Issue 22, pp. 5510-5517 (2004)

http://dx.doi.org/10.1364/OPEX.12.005510

Acrobat PDF (134 KB)

### Abstract

The effect of a clear layer at the surface of a diffusive medium on measurements of reflectance and transmittance has been investigated with Monte Carlo simulations. To quantify the effect of the clear layer Monte Carlo results have been fitted with the solution of the diffusion equation for the homogeneous medium in order to reconstruct the optical properties of the diffusive medium. The results showed that the clear layer has a small effect on measurements of transmittance. On the contrary measurements of reflectance are greatly perturbed and the accurate reconstruction of the optical properties of the diffusive medium becomes almost impossible.

© 2004 Optical Society of America

## 1. Introduction

1. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. **37**, 1531–60 (1992). [CrossRef] [PubMed]

2. 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**, 4587–99 (1997). [CrossRef] [PubMed]

*et al*. [3

3. M. Firbank, S. R. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. **41**, 767–783 (1996). [CrossRef] [PubMed]

*et al*. [4

4. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A **17**, 1671–1681 (2000). [CrossRef]

3. M. Firbank, S. R. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. **41**, 767–783 (1996). [CrossRef] [PubMed]

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

6. H. Kawaguchi, T. Hayashi, T. Kato, and E. Okada, “Theoretical evaluation of accuracy in position and size of brain activity obtained by near-infrared topography,” Phys. Med. Biol. **49**, 2753–2765 (2004). [CrossRef] [PubMed]

3. M. Firbank, S. R. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. **41**, 767–783 (1996). [CrossRef] [PubMed]

*et al*. [4

4. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A **17**, 1671–1681 (2000). [CrossRef]

7. 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**, 1359–1374 (2000). [CrossRef] [PubMed]

## 2. Description of the MC code and of the fitting procedure

8. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A **19**, 71–80 (2002). [CrossRef]

*n*, and

_{d}, n_{cl}*n*the refractive index of the diffusive medium, of the clear layer and of the external medium respectively. Numerical simulations provide the temporal point spread function, i.e., the probability to detect the emitted photon per unit time and per unit area of the receiver. The statistical error has been obtained from the number of photons received within each temporal interval. To reduce the computation time we used a scattering function with the asymmetry factor

_{e}*g*=0. Since in practical applications measurements of transmittance are usually carried out for optical mammography, and measurements of reflectance are carried out on muscular tissues, we used different optical properties for simulations of transmittance and reflectance. For simulations of transmittance we used

*µ*′

*=1 mm*

_{s}^{-1}for the reduced scattering coefficient and

*µ*=0.003 mm

_{a}^{-1}for the absorption coefficient to mimic the optical properties of the breast; for reflectance we used

*µ*′

*=1 and 0.5 mm*

_{s}^{-1}and

*µ*=0.01 mm

_{a}^{-1}, values representative of many biological tissues.

2. 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**, 4587–99 (1997). [CrossRef] [PubMed]

*z*, which also accounts for the effect of reflections at the interface between the diffusive medium and the exterior, has been evaluated as [9

_{e}9. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection and diffusive light in random media,” Phys. Rev. A **44**, 3948–59 (1991). [CrossRef] [PubMed]

2. 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**, 4587–99 (1997). [CrossRef] [PubMed]

*R*(

*µ*) is the reflection coefficient and cos

^{-1}(

*µ*) is the incidence angle. When a clear layer is present at the surface of the diffusive medium reflections occur both at the inner and outer surface of the clear layer and the total boundary reflectivity for a photon striking the inner boundary at an angle cos

^{-1}(

*µ*) is given by [10

10. D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E **50**, 857–66 (1994). [CrossRef]

*R*is the polarization-averaged Fresnel reflectivity of the

_{ij}*ij*interface. Durian [10

10. D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E **50**, 857–66 (1994). [CrossRef]

## 3. Numerical results

### 3.1 Diffuse transmittance

*s*, of the transparent walls that enclose the medium, has been investigated with reference to a diffusive slab of thickness

_{cl}*s*=40 mm between two layers of Plexiglas in air (

_{d}*n*=1.49,

_{cl}*n*=1). Examples of numerical results are reported in Fig. 1 for a diffusive medium having

_{e}*µ*′

*=1 mm*

_{s}^{-1},

*µ*=0.003 mm

_{a}^{-1}, and

*n*=1.33. These values are representative for a compressed breast at near infrared wavelengths. The results have been reported for different values of the thickness of the clear layers in the range between

_{d}*s*=0 (no layers) and

_{cl}*s*=10 mm. The figure reports the time resolved transmittance,

_{cl}*T*(

*t*), for two values of the distance

*ρ*of the receiver from the pencil light beam:

*ρ*=0 (coaxial receiver) and

*ρ*=20 mm. The coaxial receiver had a radius of 3 mm, that at

*ρ*=20 mm was a ring 2 mm thick. The acceptance angle was 90 degrees. These results show that the two layers slightly affect measurements of transmittance. In particular, the temporal point spread functions for

*s*≤1 mm are almost indistinguishable, indicating that multiple reflections inside the thin clear layers have a small effect on the total time of flight and that the clear layers do not significantly affect the effective reflection coefficient of the boundary. This result is in agreement with predictions of Eqs. (1) and (2) that give very similar results for the extrapolation length in the two cases:

_{cl}*z*=1.677/µ

_{e}*′*for water-air, and

_{s}*z*=1.753/µ

_{e}*′*for water-Plexiglas-air. When

_{s}*s*increases we observe an appreciable decrease of

_{cl}*T*(

*t*) for short times and a shift of the maximum toward longer times. The perturbation decreases as the distance from the light beam increases. For the cw transmittance variations remain within 45% for

*ρ*=0 and within 30% for

*ρ*=20 mm.

*z*=1.753/µ

_{e}*′*obtained with the reflectivity of the water-Plexiglas-air boundary. Four parameters have been fitted:

_{s}*µ*,

_{afit}*µ*′

*, an amplitude factor*

_{sfit}*F*, and a temporal shift

*t*. For the fit we used only the response within the range delimited by data corresponding to 1% of the maximum value. Furthermore, since the DE is not very accurate for early-received photons [2

_{0}**36**, 4587–99 (1997). [CrossRef] [PubMed]

*t*<2.5

*t*where

_{b}*t*is the time of flight for ballistic photons. The temporal shift would account for the time spent by received photons inside the walls if this parameter were independent of the overall time of flight. The results obtained for

_{b}*µa*and

_{fit}*µ*′

*are reported in Fig. 2 for two receivers at*

_{sfit}*ρ*=0 and 20 mm. For a better readability of the results we chose a non-linear scale for the x-axis. The reduced

*χ*

^{2}was already smaller than about 1 indicating an excellent agreement between the curves retrieved by the fit and the MC results. These results show that the distortion on the time resolved transmittance due to the presence of the Plexiglas causes a small error on the optical properties of the diffusive medium. The shift of the maximum of

*T*(

*t*) toward longer times observed when

*s*increases, mainly causes an increase of the reduced scattering coefficient. The effect is larger for the coaxial receiver. The variation remains within 15% even for

_{cl}*s*=10 mm.

_{cl}### 3.2 Diffuse reflectance

*R*(

*t*) for a semi-infinite medium having

*µ*′

*=1 mm*

_{s}^{-1},

*µ*=0.01 mm

_{a}^{-1},

*n*=1.33,

_{d}*n*=1.49, and

_{cl}*n*=1. These values of

_{e}*µ*′

*and*

_{s}*µ*are typical of biological tissue at near infrared wavelengths. The thickness of the clear layer ranges between

_{a}*s*=0 (no layer) and

_{cl}*s*=10 mm. The figure reports the results for:

_{cl}*ρ*=10, 20, 30, and 50mm. The receivers were rings with thicknesses 0.2, 0.6, 1.6, and 4 mm respectively, and acceptance angle of 90 degrees. The results show that a thin clear layer at the surface of the diffusive medium does not appreciably perturb light propagation: even at

*ρ*=50 mm the responses for

*s*=0 (no layer) 0.01, and 0.1 mm are indistinguishable. On the contrary, a thick clear layer provokes a strong distortion: The reflectance at short times strongly increases and the maximum of the curves, especially for large distances, strongly shifts toward shorter times. Nevertheless, the slope at long times remains almost unchanged. Since in an inversion procedure based on the solution of the DE for the homogeneous medium the retrieved reduced scattering coefficient mainly depends on the position of the maximum, and the absorption coefficient on the slope at long times, we expect large errors especially on

_{cl}*µ*′

*. This is shown in Fig. 4 where the results of the fit have been reported as a function of the source-receiver distance for different values of*

_{s}*s*. Also for this figure, for a better readability of the results we chose a non-linear scale for the x-axis. As for transmittance, four parameters have been fitted and the value

_{cl}*z*=1.753/

_{e}*µ*′

*has been used. For*

_{s}*s*≤0.1 mm the results of the fit are in excellent agreement with the actual values, the standard error is small (smaller than the marks) and the values of the reduced

_{cl}*χ*

^{2}smaller than about 1. As

*s*increases discrepancies rapidly increase. Also the values of the reduced

_{cl}*χ*

^{2}rapidly increase, indicating that the solution of the DE is not suitable to fit the time resolved reflectance when a thick clear layer is present. As expected there are larger discrepancies for the reduced scattering coefficient, that is strongly underestimated, while the values of the absorption coefficient remain reasonably close to the actual values: apart from the datum for

*ρ*=10 mm and

*s*=10 mm discrepancies are within 50%.

_{cl}*R*(

*ρ,s*)/

_{cl}*R*(

*ρ,s*=0)=1.03, 1.5, 2.3, 1.6, and 2.0 for

_{cl}*ρ*=10 mm and

*s*=0.1, 1, 3, 5, and 10 mm respectively, and becomes 1.05, 2.8, 59.2, 251, and 960 for

_{cl}*ρ*=50 mm.

## 4. Discussion and conclusions

*s*>1 mm the information on

_{cl}*µ*′

*is completely lost and an error of about 50% can be made on*

_{s}*µ*in the experimental situation typical for measurements on biological tissue. We point out that for MC simulations the clear layer was assumed perfectly transparent (

_{a}*µ*′

*s*=µ

_{cl}*=0) with perfectly smooth plane surfaces. Small differences with respect to this idealized situation can greatly perturb propagation through the clear layer and significantly different results can be obtained. This is shown in the example of Fig. 5 in which the comparison among the time resolved reflectance simulated for three clear layers having slightly different optical properties has been reported. We also point out that, since*

_{acl}*n*>

_{cl}*n*>

_{d}*n*, there are the conditions for a guided propagation through the clear layer that, in the idealized situation of perfectly smooth plane surfaces, cannot be established since light cannot penetrate into the clear layer with angles greater than the limit angle. However, recent experiments carried out to develop a phantom for studying light propagation through layered media [11

_{e}11. S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express **12**, 2102–2111 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2102. [CrossRef] [PubMed]

*α*: We repeated simulations for

*α*=10, 30, and 90 degrees. Simulations showed that the shape of the temporal response is slightly affected by the acceptance angle. Consequently, the values of

*µ*and

_{afit}*µ*′

*retrieved by the fit have a weak dependence on*

_{sfit}*α*. As an example, from the time resolved reflectance referring to a clear layer with

*s*=5 mm and to a receiver with

_{cl}*α*=10 degrees (collimated detection) we obtained:

*µ*=0.0064, 0.0083, 0.0088, and 0.0103 mm

_{afit}^{-1}and

*µ*

_{s}*′*=0.35, 0.13, 0.09, and 0.06 mm

_{fit}^{-1}for

*ρ*=10, 20, 30, and 50 mm respectively. The corresponding results for

*α*=90 degrees (open detector) were:

*µ*=0.0057, 0.0136, 0.0098, and 0.0104 mm

_{afit}^{-1}and

*µ*′

_{s}*=0.25, 0.04, 0.09, and 0.05 mm*

_{fit}^{-1}.

*µ*′

*=0.5 mm*

_{s}^{-1}and

*µ*=0.01 mm

_{a}^{-1}. The results of the fit on reflectance data are summarized in Fig. 6. Also these results show the strong effect of the clear layer on the retrieved optical properties. Although we observe an overall behaviour similar to the one of Fig. 4 referring to

*µ*′

*=1 mm*

_{s}^{-1}, it is not possible to describe the effect of the clear layer on the fitting results as a systematic error. Therefore, even if a reference measurement on a reference medium could be available, a correction procedure of the fitting results seems unrealistic.

11. S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express **12**, 2102–2111 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2102. [CrossRef] [PubMed]

## References and Links

1. | S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. |

2. | 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. |

3. | M. Firbank, S. R. Arridge, M. Schweiger, and D. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. |

4. | J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J Opt Soc Am A |

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

6. | H. Kawaguchi, T. Hayashi, T. Kato, and E. Okada, “Theoretical evaluation of accuracy in position and size of brain activity obtained by near-infrared topography,” Phys. Med. Biol. |

7. | 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. |

8. | F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximate solutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc. Am. A |

9. | J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection and diffusive light in random media,” Phys. Rev. A |

10. | D. J. Durian, “Influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E |

11. | S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Liquid phantom for investigating light propagation through layered diffusive media,” Opt. Express |

**OCIS Codes**

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

(170.3830) Medical optics and biotechnology : Mammography

(170.5280) Medical optics and biotechnology : Photon migration

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 22, 2004

Revised Manuscript: October 20, 2004

Published: November 1, 2004

**Citation**

Samuele Del Bianco, Fabrizio Martelli, and Giovanni Zaccanti, "Effect of a clear layer at the surface of a diffusive medium on measurements of transmittance and reflectance," Opt. Express **12**, 5510-5517 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5510

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### References

- S. R. Arridge, M. Cope, and D. T. Delpy, �??The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,�?? Phys. Med. Biol. 37, 1531-60 (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, 4587-99 (1997). [CrossRef] [PubMed]
- M. Firbank, S. R. Arridge, M. Schweiger, and D. Delpy, �??An investigation of light transport through scattering bodies with non-scattering regions,�?? Phys. Med. Biol. 41, 767-783 (1996). [CrossRef] [PubMed]
- J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, �??Boundary conditions for light propagation in diffusive media with nonscattering regions,�?? J Opt Soc Am A 17, 1671-1681 (2000). [CrossRef]
- H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, �??Optical tomography in the presence of void regions,�?? J Opt Soc Am A 17, 1659-1670 (2000). [CrossRef]
- H. Kawaguchi, T. Hayashi, T. Kato, and E. Okada, �??Theoretical evaluation of accuracy in position and size of brain activity obtained by near-infrared topography,�?? Phys. Med. Biol. 49, 2753-2765 (2004). [CrossRef] [PubMed]
- 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, 1359-1374 (2000). [CrossRef] [PubMed]
- F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, �??Analytical approximate solutions of the time-domain diffusion equation in layered slabs,�?? J. Opt. Soc. Am. A 19, 71-80 (2002). [CrossRef]
- J. X. Zhu, D. J. Pine, and D. A. Weitz, �??Internal reflection and diffusive light in random media,�?? Phys. Rev. A 44, 3948-59 (1991). [CrossRef] [PubMed]
- D. J. Durian, �??Influence of boundary reflection and refraction on diffusive photon transport,�?? Phys. Rev. E 50, 857-66 (1994). [CrossRef]
- S. Del Bianco, F. Martelli, F. Cignini, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, �??Liquid phantom for investigating light propagation through layered diffusive media,�?? Opt. Express 12, 2102-2111 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2102">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2102</a.. [CrossRef] [PubMed]

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