## Retrieving the optical parameters of biological tissues using diffuse reflectance spectroscopy and Fourier series expansions. I. theory and application |

Biomedical Optics Express, Vol. 3, Issue 10, pp. 2395-2404 (2012)

http://dx.doi.org/10.1364/BOE.3.002395

Acrobat PDF (2385 KB)

### Abstract

The determination of optical parameters of biological tissues is essential for the application of optical techniques in the diagnosis and treatment of diseases. Diffuse Reflection Spectroscopy is a widely used technique to analyze the optical characteristics of biological tissues. In this paper we show that by using diffuse reflectance spectra and a new mathematical model we can retrieve the optical parameters by applying an adjustment of the data with nonlinear least squares. In our model we represent the spectra using a Fourier series expansion finding mathematical relations between the polynomial coefficients and the optical parameters. In this first paper we use spectra generated by the Monte Carlo Multilayered Technique to simulate the propagation of photons in turbid media. Using these spectra we determine the behavior of Fourier series coefficients when varying the optical parameters of the medium under study. With this procedure we find mathematical relations between Fourier series coefficients and optical parameters. Finally, the results show that our method can retrieve the optical parameters of biological tissues with accuracy that is adequate for medical applications.

© 2012 OSA

## 1. Introduction

1. T. J. Farrell, M.S Patterson, and B. 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**(4), 879–896 (1992). [CrossRef] [PubMed]

2. E. Vitkin, V. Turzhitsky, L. Qiu, L. Gou, I. Itzkan, E. B. Hanlon, and L. T. Perelman “Photon diffusion near the point of entry in anisotropically scattering turbid media,” Nat. Commun. **2**, 587 (2011). [CrossRef] [PubMed]

3. V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman “A predictive model of backscattering at subdiffusion length scale,” Biomed. Opt. Express **1**, 1034–1046 (2010). [CrossRef]

4. I. Seo, C. K. Hayakawa, and V. Venugopalan “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. **35**(2), 681–693 (2008). [CrossRef] [PubMed]

5. E. L. Hull and T. H. Foster “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A **18**(3), 584–599 (2001). [CrossRef]

7. R. Reif, O. A. Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. **46**(32) 7317–7328 (2007). [CrossRef] [PubMed]

## 2. Theoretical foundations

**Fourier Series Expansion:**the basic idea of these series is that any periodical function (

*T*) can be expressed as a trigonometric sum of sines and cosines where their frequencies are multiples of the fundamental frequency

*ω*

_{0}, but at the same intervals in which the function is defined. In our particular case, we do not have a periodical function. However, it could be considered a periodical function by parts, being defined as a range of period

*T*equivalent to the radial distance to be studied. The idea of this proposal is to represent the reflectance as a linear combination of sines and cosines, called an order

*n*trigonometric polynomial [8], Eq (1), where the expansion coefficients are determined through the adjustment of data. where

*r*is the radial distance from the incident point to the point of exit of the light this distance is due to the multiple point of absorption and scattering within the sample before exiting.

*ω*

_{0}is the natural frequency defined for the period

*T*(

*ω*

_{0}= 2

*π*/

*T*) and

*a*,

_{n}*b*are the Fourier coefficients.

_{n}*R*(

_{n}*r*) with experimental data for to establish the relation of the Fourier series coefficients

*a*and

_{n}*b*with the optical parameters

_{n}*a*(

_{n}*μ*,

_{a}*μ*,

_{s}*n*,

*g*) and

*b*(

_{n}*μ*,

_{a}*μ*,

_{s}*n*,

*g*) must be found.

## 3. Methodology

9. L. H. 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]

*d*.

*Matlab Version 7.9.0.525*adjusting tools, the expansion of the Fourier series expansion was defined order 8. Using the results of the simulations obtained by Monte Carlo method as experimental data, we studied the behavior of Fourier series coefficients with the variation of the optical parameters and proceeded to propose an analytical relation (between the Fourier coefficients and the optical parameters) to facilitate the future retrieval of the optical parameters.

## 4. Results

**Simulation of radial diffuse reflection:**to evaluate the change of the diffuse reflection radial distribution using optical parameters, we first vary the scattering coefficient keeping the other parameters constant. Second, we vary the absorption coefficient without varying the other parameters. Third, we vary the refraction index, keeping the other optical parameters constant. All simulations are carried out with, 1,000,000 photons and with a sample thickness

*d*= 200

*cm*.

*μ*= 0.1

_{a}*mm*

^{−1}), the anisotropy factor (

*g*= 0.8), the refraction index (

*n*= 1.4), varying the scattering coefficients over a range (

*μ*= 10 – 200

_{s}*mm*

^{−1}), In Fig 2 (A), as you can see, the diffuse reflectance intensity rises due to the increase of the scattering centers.

*μ*= 0.1 to 1

_{a}*mm*

^{−1}) for this simulation, the input parameters are the scattering coefficient (

*μ*= 75

_{s}*mm*

^{−1}), the anisotropy factor (

*g*= 0.8) and the refraction index (

*n*= 1.4), Note that as the absorption coefficient increases, the reflection curve decreases progressively (see Fig 2 (B)). This was expected due to the increase of the absorption centers. Finally, the third case is for the refraction index. The variation was made in the interval 1.4 to 1.6 with the follows optical parameters: the scattering coefficient (

*μ*= 75

_{s}*mm*

^{−1}), the anisotropy factor (g=0. 8) and absorption coefficients (

*μ*= 0.1

_{a}*mm*

^{−1}). Note that as that parameter increases, the reflection curve decreases (see Fig 2 (C)).

**Adjustment of the Reflection Curve with Trigonometric Functions:**Once the simulations of the diffuse reflection resolution radial have been obtained, we do the trigonometric fit with least squares using Fourier series expansion. Fig 3 shows the radial reflection curve using the trigonometric polynomial of order 8, with an

*ω*

_{0}= 2.137(1/

*cm*) and the root mean square of 0.9997.

**FS**) and Farrell model [1

1. T. J. Farrell, M.S Patterson, and B. 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**(4), 879–896 (1992). [CrossRef] [PubMed]

**DA**). We observed that

**FS**presents a good adjustment

**DA**, but outside diffusion approximation, the adjustment by

**FS**continued being effective; therefore, fitting the curve by diffuse reflectance Fourier series is more versatile and can be applied under different conditions.

*a*. Note that when increasing the scattering coefficient, the coefficients

_{n}*a*

_{1},

*a*

_{6}and

*a*

_{8}decrease but coefficients

*a*

_{2},

*a*

_{5}and

*a*

_{7}increase (see Fig 6). Then, two types of fit were made (linear and cubic) with the help of the program MATLAB, Version 7.9.0.525 computing tool

*CurveFitting*.

*b*were studied. The results are shows in Fig 8. Note that coefficients

_{n}*b*

_{2},

*b*

_{4}and

*b*

_{6}, increase when the coefficient of scattering increases.

*a*

_{0}has been discarded from the study because its only contribution in the series is the displacement of the curve in the abscissa axis as seen in Fig 9.

*a*and

_{n}*b*of the Fourier series expansion have been analyzed. The other terms were eliminated for their limited contribution in the series, though the adjustment made was of order 8 obtaining 16 Fourier coefficients (8 for

_{n}*a*and 8 for

_{n}*b*). Fig 10 shows the first five Fourier series coefficients. The nominal values are inappreciable in relation to the first terms.

_{n}*a*

_{1}is shown. Note an increase in

*a*

_{1}where the absorption coefficient increases. In Table 2, the adjustments made to the curve obtained in Fig 11 are shown in order to find the analytical relation between the coefficient

*a*

_{1}and the absorption coefficient, obtaning an analytical relation between both coefficients of an order 5 polynomial.

*a*of the Fourier series is shown, obtaining the same polynomial as the previous case in addition, a study of the variations of the coefficients

_{n}*b*of the Fourier series expansion was carried out.

_{n}*a*is shown where one can observe the curve made by the method of least squares for a order 2 polynomial in the case of the refraction index (see Table 3).

_{n}*a*and

_{n}*b*can be attributed to the stochastic nature of Monte Carlo method.

_{n}## 5. Conclusion

*a*and

_{n}*b*of the Fourier series expansion. For the absorption coefficient there is a mathematical relationship of an order 5 polynomial and for the refraction index, the relationship is of an order 2 polynomial. Finally, no relation was found between expansion coefficients for the anisotropy factor, at least not the first ten expansions.

_{n}## References and links

1. | T. J. Farrell, M.S Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. |

2. | E. Vitkin, V. Turzhitsky, L. Qiu, L. Gou, I. Itzkan, E. B. Hanlon, and L. T. Perelman “Photon diffusion near the point of entry in anisotropically scattering turbid media,” Nat. Commun. |

3. | V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman “A predictive model of backscattering at subdiffusion length scale,” Biomed. Opt. Express |

4. | I. Seo, C. K. Hayakawa, and V. Venugopalan “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. |

5. | E. L. Hull and T. H. Foster “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A |

6. | B. Morales Cruzado and S. Vázquez y Montiel, “Obtención de los parámetros ópticos de la piel usando algoritmos genéticos y MCML,” Rev. Mex. Fis. |

7. | R. Reif, O. A. Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt. |

8. | J. S. Walker, |

9. | L. H. 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. | J. Qin and R. Lu, “Monte Carlo simulation for quantification of light transport features in apples,” Comput. Electron. Agri. |

11. | I. V. Meglinski and S. J. Matcher, “Computer simulation of the skin reflectance spectra,” Comput. Meth. Programs Bio. |

**OCIS Codes**

(170.7050) Medical optics and biotechnology : Turbid media

(290.3200) Scattering : Inverse scattering

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: May 24, 2012

Revised Manuscript: June 23, 2012

Manuscript Accepted: June 30, 2012

Published: September 5, 2012

**Citation**

Aarón A. Muñoz Morales and Sergio Vázquez y Montiel, "Retrieving the optical parameters of biological tissues using diffuse reflectance spectroscopy and Fourier series expansions. I. theory and application," Biomed. Opt. Express **3**, 2395-2404 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-10-2395

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### References

- T. J. Farrell, M.S Patterson, and B. 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(4), 879–896 (1992). [CrossRef] [PubMed]
- E. Vitkin, V. Turzhitsky, L. Qiu, L. Gou, I. Itzkan, E. B. Hanlon, and L. T. Perelman “Photon diffusion near the point of entry in anisotropically scattering turbid media,” Nat. Commun.2, 587 (2011). [CrossRef] [PubMed]
- V. Turzhitsky, A. Radosevich, J. D. Rogers, A. Taflove, and V. Backman “A predictive model of backscattering at subdiffusion length scale,” Biomed. Opt. Express1, 1034–1046 (2010). [CrossRef]
- I. Seo, C. K. Hayakawa, and V. Venugopalan “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys.35(2), 681–693 (2008). [CrossRef] [PubMed]
- E. L. Hull and T. H. Foster “Steady-state reflectance spectroscopy in the P-3 approximation,” J. Opt. Soc. Am. A18(3), 584–599 (2001). [CrossRef]
- B. Morales Cruzado and S. Vázquez y Montiel, “Obtención de los parámetros ópticos de la piel usando algoritmos genéticos y MCML,” Rev. Mex. Fis.57, 375–381. (2011).
- R. Reif, O. A. Amar, and I. J. Bigio, “Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media,” Appl. Opt.46(32) 7317–7328 (2007). [CrossRef] [PubMed]
- J. S. Walker, Fourier Analysis (Oxford University Press, 1988), pp. 5–28.
- L. H. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A10, 1746–1752. (1993). [CrossRef]
- J. Qin and R. Lu, “Monte Carlo simulation for quantification of light transport features in apples,” Comput. Electron. Agri.68, 44–51 (2009). [CrossRef]
- I. V. Meglinski and S. J. Matcher, “Computer simulation of the skin reflectance spectra,” Comput. Meth. Programs Bio.70, 179–186 (2003). [CrossRef]

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