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Noninvasive determination of tissue optical properties based on radiative transfer theory
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Optics & Laser Technology 36 (2004) 353 – 359 www.elsevier.com/locate/optlastec
Noninvasive determination of tissue optical properties based on radiative transfer theory Fujun Zhanga; b , Bingquan Chenb; c;∗ , Shengzhi Zhaoa , Shangming Yangb , Ruiping Chenb , Dongcao Songb a School
of Information Science and Engineering, Shandong University, Jinan, Shandong 250100, China b Institute of Biomedical Optics, Yantai University, Yantai, Shandong 264005, China c Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA Received 18 March 2003; received in revised form 12 September 2003; accepted 20 October 2003
Abstract We present a feasibility study of a new method for determining the tissue optical properties, including the absorption and scattering coe8cients and the scattering asymmetry factor. A state-of-the-art radiative transfer model for the coupled air/tissue system, based on rigorous radiative transfer theory, is used in our forward modeling simulations. The concept of the e;ective photon penetration depth is introduced and used to help determine the depth below, which information about the tissue will not be available through noninvasive imaging of a biological tissue using re
1. Introduction Any imaging or spectroscopic method for bio-optical research must deal with both the absorption by the primary component of tissue, i.e. water, and the scattering by various types of tissue particles. Knowledge of the optical properties of biological tissue is a prerequisite for carrying out studies in bio-optical imaging, as well as for developing instruments for medical diagnosis [1–3]. The physiological state of biological tissue can be obtained from its optical properties, if an accurate bio-optical model for the tissue is available. Thus, it is desirable to develop (i) adequate bio-optical models that relate tissue optical properties to its physiological state, and (ii) accurate and reliable methods for determining optical properties of tissue. Because techniques that are based on the use of coherent light usually cannot provide information about the optical properties of a biological tissue, much ongoing research [4– 15] is aimed at developing methods for interpreting images ∗
Corresponding author. Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA. Tel.: +1-201-216-5587; fax: +1-201-216-5638. E-mail address: bingquan [email protected] (B. Chen). 0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.10.001
based on the use of incoherent or di;use light. In imaging with di;use light, solutions of the radiative transfer problem are frequently based on the di;usion approximation to the radiative transfer equation. However, it has been argued that the di;usion approximation holds only for regions deep inside the medium far from its boundaries and far from the source [16]. Di;usion approximations work well for large optical depths, single-scattering albedo values near unity, and asymmetry factors near zero [3,17,18]. But for many types of tissue, the scattering asymmetry factor is in the range between 0.6 and 0.95, and it can be as large as 0.990 – 0.999, for example, for blood [3,19,20]. Moreover, our recent study on the validity of di;usion approximation [21] indicated that: (i) For light incident on a nonabsorbing tissue (i.e. the single scattering albedo equals unity) with an index-matched boundary, the di;usion approximation provides accurate re
354
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
di;usion approximation does not work for a tissue with a nonindex-matched boundary. (iv) A validity criterion for the di;usion approximation has been established using the single scattering albedo a and the asymmetry factor g. Therefore, new techniques aimed at reliable and accurate determination of the optical properties of tissue should be based on rigorous radiative transfer theory rather than di;usion approximations. In vivo determination of tissue optical properties have been carried out by many groups around the world, e.g. by Bevilacqua and co-authors as well as in other references in the paper by Bevilacqua et al. [22]. In those studies, little attention has been paid to the e;ects of g in noninvasive optical imaging. Methods using the absorption coe8cient and the so-called reduced scattering coe8cient to represent the optical properties of a tissue [1,2,9,10,23] have limited validity because they are based on di;usion approximations [21]. The purpose of this study, is to investigate the feasibility of developing a reliable and accurate method based on rigorous radiative transfer theory rather than di;usion approximations for determining the optical properties of a biological tissue, and to develop an inversion algorithm for accurate noninvasive determination of tissue optical properties using radiative transfer theory in conjunction with measurements of re
optical fiber spatial filter
I 0 (θ 0 , φ 0 )
detector
θ0
to laser
collector lens
θ
collimating lens
I r (θ, φ)
Air, n 0 = 1.0
Tissue: α, σ, g n = 1.34
Fig. 1. Schematic diagram of experimental setup for acquiring re
where S( ; ; ) a( ) = 4
2
0
d
1
−1
d p( ; ; ; ; )I ( ; ; )
+ Q( ; ; ); Q( ; ; ) =
(2)
a( )F0 0 Tb (− 0 ; mrel ) 4 0m × p( ; − 0m; 0 ; ; )e− = 0m ;
(3)
and 1=2 0m = 0m ( 0 ; mrel ) = [1 − (1 − 02 )m−2 rel ] :
(4)
Here I is the intensity or radiance measured in the direction of ( ; ’), and 0 = cos , where 0 is the polar angle and ’0 is the azimuthal angle of the incident collimated beam. mrel is the real part of the index of refraction of the tissue relative to the air, Tb (− 0 ; mrel ) ≡ Tb (− 0 ; 0; − u0m ; 0 ; mrel ) is the beam transmittance through the air/tissue interface. The optical properties of the medium areas are as follows: is the optical depth, ( ) is the absorption coe8cient, ( ) is the scattering coe8cient, a( ) ≡ ( )=[( ) + ( )] is the single-scattering albedo. F0 is the
(6)
where g is the scattering asymmetry factor which is the Nrst moment of the phase function, = cos , where is for the polar angle prior to scattering; = cos 0 , where is for the polar angle after scattering, is the azimuthal angle prior to scattering and ’ is the azimuthal angle after scattering. The internal source Q( ; u; ’) depends also on the
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
355
direction ( 0 ; ’0 ) of the incident light (see Fig. 1), the collimated beam intensity and the refractive index of the tissue relative to the adjacent air. We employ the discrete-ordinate method to solve the radiative transfer equation pertaining to a slab of biological tissue (see Fig. 1). The re
n = 1:34, and the absorption and scattering coe8cients of the tissue layer are denoted by and , respectively. The scattering asymmetry factor of the tissue layer is denoted by g, corresponding to phase functions p as indicated in Fig. 1. In this study the incident
3. Forward modeling
3.1. Dependence of the re
In our forward simulations, we assume that the refractive indices of the air and the tissue are taken to be n0 = 1:0 and
Fig. 2 shows the dependence of the re
0.100 0.25 0.090 0.080 Reflectance
Reflectance
0.20 0.15 0.10
0.070 0.060 0.050
0.05
0.040
0.00
0.030 0
(a)
20
40 60 Viewing Angle
80 (b)
0 10 20 30 40 50 Scattering coefficient (1/mm)
0.25
Reflectance
0.20 0.15 0.10 0.05 0.00 0 (c)
20
40 60 Viewing Angle
80
Fig. 2. Dependence of the re
356
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
0 = 0◦ , but di;erent viewing direction . In Fig. 2(a) the single-scattering albedo a is allowed to vary, while g is kept constant. We see from Fig. 2(a) that for a Nxed illumination direction 0 = 0◦ , the re
In the simulations of Fig. 3, we use the following values for the tissue optical properties: (a) = 0:001–0:5 mm−1 ; = 15 mm−1 ; g = 0:95. (b) = 0:01 mm−1 ; = 2–50 mm−1 ; g = 0:95. (c) = 0:01 mm−1 ; = 15 mm−1 ; g = 0:3– 0.99. As shown in Fig. 3, there is no any change in the re
4. Inverse problems From the forward simulations presented above, we see that the re
0.40
0.40
0.30
0.30
Reflected Radiance
Reflected Radiance
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
0.20
0.10
0.20
0.10 0.00
0.00 (a)
357
0 10 20 30 40 Thickness of Tissue Layer (mm)
(b)
0 10 20 30 40 Thickness of Tissue Layer (mm)
Reflected Radiance
0.40 0.30 0.20 0.10 0.00 (c)
0 10 20 30 40 Thickness of Tissue Layer (mm)
Fig. 3. Dependence of the e;ective photon penetration depth on the tissue optical properties. 0 = 0, = 8◦ . The solid curves are used to illustrate the variation of the e;ective photon penetration depth. (a) = 15 mm−1 ; g = 0:95. (b) = 0:01 mm−1 ; g = 0:95. (c) = 0:01 mm−1 , = 15 mm−1 .
4.1. Accurate retrieval of single scattering albedo using an asymmetry factor model
0.95 a (retrieved)
Based on the discussions in Section 3, we can employ the radiative transfer theory to accurately retrieve the tissue single-scattering albedo using the measured re
1.00
0.90 0.85 0.80 0.75 0.75 0.80 0.85 0.90 0.95 1.00 a (exact)
Fig. 4. Comparison between retrieved and exact values of single-scattering albedo with a model of asymmetry factor (gm ). The exact value of g is 0.9.
358
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
Measurement I
Measurement II
Determine γ
Determine g
No
γ =1+g
∆g < ε Yes
Determine α, σ’
Determine a
Output: α,σ’, a, g Fig. 5. Flowchart of inversion algorithm that combines our theory and method and the technique developed by Bevilacqua et al. [22] (within dashed-line box). is the small value determined by error criterion.
albedo and the asymmetry factor, we present an inversion algorithm in conjunction with other existing methods discussed below. 4.2. Inversion algorithm in conjunction with other existing techniques Bevilacqua et al. [22] developed a technique for in vivo local determination of tissue optical properties , and , where = 1 − g, ! = (1 − g2 )=(1 − g), and g2 is the second moment of the phase function (note that a and s were used in the paper by Bevilacqua et al. instead of and used here). As we discussed in Section 2 above, if we use the synthetic Henyey–Greenstein phase function to describe the angular light scattering pattern due to particles in the tissue, g2 = g2 , and the parameter = 1 + g. Bevilacqua et al. [22] concluded that can be determined within ±0:2, and the accuracies of retrieved and depend on the accuracy of . According to the discussions in Section 4.1, an accuracy of ±0:2 for or g will lead to signiNcant error of the retrieved single-scattering albedo that is related to the absorption and scattering coe8cients. We now present an inversion algorithm that combines our theory and method and the technique developed by Bevilacqua et al. [22], as indicated by the
with improved accuracy using measurement I and the algorithm developed by Bevilacqua et al. [22]. Finally, we can compare the retrieved a from measurement I and calculated a from retrieved , and g using measurement I. The performance of this inversion algorithm has been preliminarily validated by our experiments. First, we have repeated the experiment and data processing as described by Bevilacqua et al. [22], which can be considered as Measurement I as indicated in Fig. 5. We then used our experimental setup (see Fig. 1) to perform Measurement II. Our algorithm program for processing Measurement II data is based on a radiative transfer computation software, named as DISORT [25]. Experimental results show that the di;erences between the method by Bevilacqua et al. [22] and our algorithm described in this paper for the retrieved values of and are between 15 and 43%. As discussed above, we believe our approach provides more accurate results because our algorithm can determine g more accurately. For now, we call this inversion algorithm (see Fig. 5) as a hybrid inversion algorithm. 5. Conclusions We have discussed a new technique for determining the optical properties, i.e. the absorption and scattering coe8cients and the scattering asymmetry factor. A comprehensive radiative transfer model for the coupled air/tissue system is used for forward modeling simulations, in which the change of the refractive index across the air/tissue interface is taken into account. We have found that the re
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359 [6] Chance B, Anday E, Nioka S, Zhou S, Hong L, Worden K, Li C, Murray T, Ovetsky Y, Pidikiti D, Thomas R. A novel method for fast imaging of brain function, non-invasively, with light. Opt Express 1998;2:411–23. [7] Chen B, Stamnes JJ, Stamnes K. Reconstruction algorithm for di;raction tomography of di;use photon density waves in a random medium. Pure Appl Opt 1998;7:1161–80. [8] Cheng X, Boas DA. Di;use optical re
359
[16] Ishimaru A. Wave propagation and scattering in random media. New York: IEEE Press; 1997. [17] Kim AD, Ishimaru A. Optical di;usion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer. Appl Opt 1998;37:5313–9. [18] Durduran T, Yodh AG, Chance B, Boas DA. Does the photon-di;usion coe8cient depend on absorption? J Opt Soc Am 1997;A 14:3358–65. [19] Cheong W-F, Prahl SA, Welch AJ. A review of the optical properties of biological tissue. IEEE J Quantum Electron 1990;26:2166–85. [20] Roggan A, Friebel M, Dorschel K, Hahn A, Mueller G. Optical properties of circulating human blood in the wavelength range 400 –2500 nm. J Biomed Opt 1999;4:36–46. [21] Chen B, Stamnes K, Stamnes JJ. Validity of di;usion approximation in bio-optical imaging. Appl Opt 2001;40:6356–66. [22] Bevilacqua F, Piguet D, Marquet P, Gross JD, Tromberg BJ, Depeursinge C. In vivo local determination of tissue optical properties: applications to human brain. Appl Opt 1999;38:4939–50. [23] Patterson MS, Chance B, Wilson BC. Time resolved re
Optics & Laser Technology 36 (2004) 353 – 359 www.elsevier.com/locate/optlastec
Noninvasive determination of tissue optical properties based on radiative transfer theory Fujun Zhanga; b , Bingquan Chenb; c;∗ , Shengzhi Zhaoa , Shangming Yangb , Ruiping Chenb , Dongcao Songb a School
of Information Science and Engineering, Shandong University, Jinan, Shandong 250100, China b Institute of Biomedical Optics, Yantai University, Yantai, Shandong 264005, China c Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA Received 18 March 2003; received in revised form 12 September 2003; accepted 20 October 2003
Abstract We present a feasibility study of a new method for determining the tissue optical properties, including the absorption and scattering coe8cients and the scattering asymmetry factor. A state-of-the-art radiative transfer model for the coupled air/tissue system, based on rigorous radiative transfer theory, is used in our forward modeling simulations. The concept of the e;ective photon penetration depth is introduced and used to help determine the depth below, which information about the tissue will not be available through noninvasive imaging of a biological tissue using re
1. Introduction Any imaging or spectroscopic method for bio-optical research must deal with both the absorption by the primary component of tissue, i.e. water, and the scattering by various types of tissue particles. Knowledge of the optical properties of biological tissue is a prerequisite for carrying out studies in bio-optical imaging, as well as for developing instruments for medical diagnosis [1–3]. The physiological state of biological tissue can be obtained from its optical properties, if an accurate bio-optical model for the tissue is available. Thus, it is desirable to develop (i) adequate bio-optical models that relate tissue optical properties to its physiological state, and (ii) accurate and reliable methods for determining optical properties of tissue. Because techniques that are based on the use of coherent light usually cannot provide information about the optical properties of a biological tissue, much ongoing research [4– 15] is aimed at developing methods for interpreting images ∗
Corresponding author. Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA. Tel.: +1-201-216-5587; fax: +1-201-216-5638. E-mail address: bingquan [email protected] (B. Chen). 0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.10.001
based on the use of incoherent or di;use light. In imaging with di;use light, solutions of the radiative transfer problem are frequently based on the di;usion approximation to the radiative transfer equation. However, it has been argued that the di;usion approximation holds only for regions deep inside the medium far from its boundaries and far from the source [16]. Di;usion approximations work well for large optical depths, single-scattering albedo values near unity, and asymmetry factors near zero [3,17,18]. But for many types of tissue, the scattering asymmetry factor is in the range between 0.6 and 0.95, and it can be as large as 0.990 – 0.999, for example, for blood [3,19,20]. Moreover, our recent study on the validity of di;usion approximation [21] indicated that: (i) For light incident on a nonabsorbing tissue (i.e. the single scattering albedo equals unity) with an index-matched boundary, the di;usion approximation provides accurate re
354
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
di;usion approximation does not work for a tissue with a nonindex-matched boundary. (iv) A validity criterion for the di;usion approximation has been established using the single scattering albedo a and the asymmetry factor g. Therefore, new techniques aimed at reliable and accurate determination of the optical properties of tissue should be based on rigorous radiative transfer theory rather than di;usion approximations. In vivo determination of tissue optical properties have been carried out by many groups around the world, e.g. by Bevilacqua and co-authors as well as in other references in the paper by Bevilacqua et al. [22]. In those studies, little attention has been paid to the e;ects of g in noninvasive optical imaging. Methods using the absorption coe8cient and the so-called reduced scattering coe8cient to represent the optical properties of a tissue [1,2,9,10,23] have limited validity because they are based on di;usion approximations [21]. The purpose of this study, is to investigate the feasibility of developing a reliable and accurate method based on rigorous radiative transfer theory rather than di;usion approximations for determining the optical properties of a biological tissue, and to develop an inversion algorithm for accurate noninvasive determination of tissue optical properties using radiative transfer theory in conjunction with measurements of re
optical fiber spatial filter
I 0 (θ 0 , φ 0 )
detector
θ0
to laser
collector lens
θ
collimating lens
I r (θ, φ)
Air, n 0 = 1.0
Tissue: α, σ, g n = 1.34
Fig. 1. Schematic diagram of experimental setup for acquiring re
where S( ; ; ) a( ) = 4
2
0
d
1
−1
d p( ; ; ; ; )I ( ; ; )
+ Q( ; ; ); Q( ; ; ) =
(2)
a( )F0 0 Tb (− 0 ; mrel ) 4 0m × p( ; − 0m; 0 ; ; )e− = 0m ;
(3)
and 1=2 0m = 0m ( 0 ; mrel ) = [1 − (1 − 02 )m−2 rel ] :
(4)
Here I is the intensity or radiance measured in the direction of ( ; ’), and 0 = cos , where 0 is the polar angle and ’0 is the azimuthal angle of the incident collimated beam. mrel is the real part of the index of refraction of the tissue relative to the air, Tb (− 0 ; mrel ) ≡ Tb (− 0 ; 0; − u0m ; 0 ; mrel ) is the beam transmittance through the air/tissue interface. The optical properties of the medium areas are as follows: is the optical depth, ( ) is the absorption coe8cient, ( ) is the scattering coe8cient, a( ) ≡ ( )=[( ) + ( )] is the single-scattering albedo. F0 is the
(6)
where g is the scattering asymmetry factor which is the Nrst moment of the phase function, = cos , where is for the polar angle prior to scattering; = cos 0 , where is for the polar angle after scattering, is the azimuthal angle prior to scattering and ’ is the azimuthal angle after scattering. The internal source Q( ; u; ’) depends also on the
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
355
direction ( 0 ; ’0 ) of the incident light (see Fig. 1), the collimated beam intensity and the refractive index of the tissue relative to the adjacent air. We employ the discrete-ordinate method to solve the radiative transfer equation pertaining to a slab of biological tissue (see Fig. 1). The re
n = 1:34, and the absorption and scattering coe8cients of the tissue layer are denoted by and , respectively. The scattering asymmetry factor of the tissue layer is denoted by g, corresponding to phase functions p as indicated in Fig. 1. In this study the incident
3. Forward modeling
3.1. Dependence of the re
In our forward simulations, we assume that the refractive indices of the air and the tissue are taken to be n0 = 1:0 and
Fig. 2 shows the dependence of the re
0.100 0.25 0.090 0.080 Reflectance
Reflectance
0.20 0.15 0.10
0.070 0.060 0.050
0.05
0.040
0.00
0.030 0
(a)
20
40 60 Viewing Angle
80 (b)
0 10 20 30 40 50 Scattering coefficient (1/mm)
0.25
Reflectance
0.20 0.15 0.10 0.05 0.00 0 (c)
20
40 60 Viewing Angle
80
Fig. 2. Dependence of the re
356
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
0 = 0◦ , but di;erent viewing direction . In Fig. 2(a) the single-scattering albedo a is allowed to vary, while g is kept constant. We see from Fig. 2(a) that for a Nxed illumination direction 0 = 0◦ , the re
In the simulations of Fig. 3, we use the following values for the tissue optical properties: (a) = 0:001–0:5 mm−1 ; = 15 mm−1 ; g = 0:95. (b) = 0:01 mm−1 ; = 2–50 mm−1 ; g = 0:95. (c) = 0:01 mm−1 ; = 15 mm−1 ; g = 0:3– 0.99. As shown in Fig. 3, there is no any change in the re
4. Inverse problems From the forward simulations presented above, we see that the re
0.40
0.40
0.30
0.30
Reflected Radiance
Reflected Radiance
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
0.20
0.10
0.20
0.10 0.00
0.00 (a)
357
0 10 20 30 40 Thickness of Tissue Layer (mm)
(b)
0 10 20 30 40 Thickness of Tissue Layer (mm)
Reflected Radiance
0.40 0.30 0.20 0.10 0.00 (c)
0 10 20 30 40 Thickness of Tissue Layer (mm)
Fig. 3. Dependence of the e;ective photon penetration depth on the tissue optical properties. 0 = 0, = 8◦ . The solid curves are used to illustrate the variation of the e;ective photon penetration depth. (a) = 15 mm−1 ; g = 0:95. (b) = 0:01 mm−1 ; g = 0:95. (c) = 0:01 mm−1 , = 15 mm−1 .
4.1. Accurate retrieval of single scattering albedo using an asymmetry factor model
0.95 a (retrieved)
Based on the discussions in Section 3, we can employ the radiative transfer theory to accurately retrieve the tissue single-scattering albedo using the measured re
1.00
0.90 0.85 0.80 0.75 0.75 0.80 0.85 0.90 0.95 1.00 a (exact)
Fig. 4. Comparison between retrieved and exact values of single-scattering albedo with a model of asymmetry factor (gm ). The exact value of g is 0.9.
358
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359
Measurement I
Measurement II
Determine γ
Determine g
No
γ =1+g
∆g < ε Yes
Determine α, σ’
Determine a
Output: α,σ’, a, g Fig. 5. Flowchart of inversion algorithm that combines our theory and method and the technique developed by Bevilacqua et al. [22] (within dashed-line box). is the small value determined by error criterion.
albedo and the asymmetry factor, we present an inversion algorithm in conjunction with other existing methods discussed below. 4.2. Inversion algorithm in conjunction with other existing techniques Bevilacqua et al. [22] developed a technique for in vivo local determination of tissue optical properties , and , where = 1 − g, ! = (1 − g2 )=(1 − g), and g2 is the second moment of the phase function (note that a and s were used in the paper by Bevilacqua et al. instead of and used here). As we discussed in Section 2 above, if we use the synthetic Henyey–Greenstein phase function to describe the angular light scattering pattern due to particles in the tissue, g2 = g2 , and the parameter = 1 + g. Bevilacqua et al. [22] concluded that can be determined within ±0:2, and the accuracies of retrieved and depend on the accuracy of . According to the discussions in Section 4.1, an accuracy of ±0:2 for or g will lead to signiNcant error of the retrieved single-scattering albedo that is related to the absorption and scattering coe8cients. We now present an inversion algorithm that combines our theory and method and the technique developed by Bevilacqua et al. [22], as indicated by the
with improved accuracy using measurement I and the algorithm developed by Bevilacqua et al. [22]. Finally, we can compare the retrieved a from measurement I and calculated a from retrieved , and g using measurement I. The performance of this inversion algorithm has been preliminarily validated by our experiments. First, we have repeated the experiment and data processing as described by Bevilacqua et al. [22], which can be considered as Measurement I as indicated in Fig. 5. We then used our experimental setup (see Fig. 1) to perform Measurement II. Our algorithm program for processing Measurement II data is based on a radiative transfer computation software, named as DISORT [25]. Experimental results show that the di;erences between the method by Bevilacqua et al. [22] and our algorithm described in this paper for the retrieved values of and are between 15 and 43%. As discussed above, we believe our approach provides more accurate results because our algorithm can determine g more accurately. For now, we call this inversion algorithm (see Fig. 5) as a hybrid inversion algorithm. 5. Conclusions We have discussed a new technique for determining the optical properties, i.e. the absorption and scattering coe8cients and the scattering asymmetry factor. A comprehensive radiative transfer model for the coupled air/tissue system is used for forward modeling simulations, in which the change of the refractive index across the air/tissue interface is taken into account. We have found that the re
F. Zhang et al. / Optics & Laser Technology 36 (2004) 353 – 359 [6] Chance B, Anday E, Nioka S, Zhou S, Hong L, Worden K, Li C, Murray T, Ovetsky Y, Pidikiti D, Thomas R. A novel method for fast imaging of brain function, non-invasively, with light. Opt Express 1998;2:411–23. [7] Chen B, Stamnes JJ, Stamnes K. Reconstruction algorithm for di;raction tomography of di;use photon density waves in a random medium. Pure Appl Opt 1998;7:1161–80. [8] Cheng X, Boas DA. Di;use optical re
359
[16] Ishimaru A. Wave propagation and scattering in random media. New York: IEEE Press; 1997. [17] Kim AD, Ishimaru A. Optical di;usion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer. Appl Opt 1998;37:5313–9. [18] Durduran T, Yodh AG, Chance B, Boas DA. Does the photon-di;usion coe8cient depend on absorption? J Opt Soc Am 1997;A 14:3358–65. [19] Cheong W-F, Prahl SA, Welch AJ. A review of the optical properties of biological tissue. IEEE J Quantum Electron 1990;26:2166–85. [20] Roggan A, Friebel M, Dorschel K, Hahn A, Mueller G. Optical properties of circulating human blood in the wavelength range 400 –2500 nm. J Biomed Opt 1999;4:36–46. [21] Chen B, Stamnes K, Stamnes JJ. Validity of di;usion approximation in bio-optical imaging. Appl Opt 2001;40:6356–66. [22] Bevilacqua F, Piguet D, Marquet P, Gross JD, Tromberg BJ, Depeursinge C. In vivo local determination of tissue optical properties: applications to human brain. Appl Opt 1999;38:4939–50. [23] Patterson MS, Chance B, Wilson BC. Time resolved re