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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 3, Iss. 10 — Oct. 1, 2012
  • pp: 2371–2380
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Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements

Serge Grabtchak and William M. Whelan  »View Author Affiliations


Biomedical Optics Express, Vol. 3, Issue 10, pp. 2371-2380 (2012)
http://dx.doi.org/10.1364/BOE.3.002371


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Abstract

We present a new method for extracting the effective attenuation coefficient and the diffusion coefficient from relative spectrally resolved cw radiance measurements using the diffusion approximation. The method is validated on both simulated and experimental radiance data sets using Intralipid-1% as a test platform. The effective attenuation coefficient is determined from a simple algebraic expression constructed from a ratio of two radiance measurements at two different source–detector separations and the same 90° angle. The diffusion coefficient is determined from another ratio constructed from two radiance measurements at two angles (0° and 180°) and the same source–detector separation. The conditions of the validity of the method as well as possible practical applications are discussed.

© 2012 OSA

1. Introduction

Extraction of optical parameters of biological tissues is of great interest for biomedical optics for tissue diagnostics, treatment planning or monitoring [1

1. W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42(5), 763–787 (1997). [CrossRef] [PubMed]

3

3. M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6(4), 379–390 (1991). [CrossRef]

]. For example, the in vivo determination of optical properties of cancerous tissue is needed to optimize the light dose distribution during photodynamic therapy [4

4. S. R. H. Davidson, R. A. Weersink, M. A. Haider, M. R. Gertner, A. Bogaards, D. Giewercer, A. Scherz, M. D. Sherar, M. Elhilali, J. L. Chin, J. Trachtenberg, and B. C. Wilson, “Treatment planning and dose analysis for interstitial photodynamic therapy of prostate cancer,” Phys. Med. Biol. 54(8), 2293–2313 (2009). [CrossRef] [PubMed]

6

6. C. M. Moore, C. A. Mosse, C. Allen, H. Payne, M. Emberton, and S. G. Bown, “Light penetration in the human prostate: a whole prostate clinical study at 763 nm,” J. Biomed. Opt. 16(1), 015003 (2011). [CrossRef] [PubMed]

]. When light propagates in turbid media, both absorption and scattering by tissue contribute to the total attenuation of light, characterized by two optical parameters, the effective attenuation coefficient (μeff(λ)) and the diffusion coefficient (D(λ)) when the diffusion approximation is valid [7

7. D. T. Delpy and M. Cope, ““Quantification in tissue near-infrared spectroscopy,” Philos. Trans. R. Soc. London Ser. B 352(1354), 649–659 (1997). [CrossRef]

]. A detailed coverage of various aspects of light interaction with turbid media is given in several excellent books on this subject [8

8. A. J. Welch and M. J. C. van Gemert, eds., Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011).

10

10. F. Martelli, S. del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and software (SPIE, 2010).

].The effective attenuation coefficient quantifies the combined effect due to absorption and scattering and is of great value by itself. Both μeff(λ) and D(λ) can be determined by fitting measured optical data to a diffusion model. Further, the absorption coefficient (μa(λ)) and the reduced scattering coefficient (μs(λ)) can be determined using the following relations: D(λ)=1/3(μs(λ)+μa(λ)) and μeff(λ)=μa(λ)/D(λ) [8

8. A. J. Welch and M. J. C. van Gemert, eds., Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011).

11

11. 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(5), 1359–1373 (2000). [CrossRef] [PubMed]

]. Typically, all techniques can be classified based on a type of light sources used to interrogate tissue and are divided into three categories: continuous wave (cw), intensity modulated and time resolved [2

2. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010). [CrossRef]

,7

7. D. T. Delpy and M. Cope, ““Quantification in tissue near-infrared spectroscopy,” Philos. Trans. R. Soc. London Ser. B 352(1354), 649–659 (1997). [CrossRef]

,12

12. B. C. Wilson and S. L. Jacques, “Optical reflectance and transmittance of tissues - principles and applications,” IEEE J. Quantum Electron. 26(12), 2186–2199 (1990). [CrossRef]

] (see also Chap. 8 in Ref. [8

8. A. J. Welch and M. J. C. van Gemert, eds., Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011).

]). Figure 1
Fig. 1 Schematic representation of responses of the turbid media to external stimuli used in various measurement approaches.
shows a typical schematic representation of responses of a turbid medium to external stimuli used in various measurement approaches. An external stimulus is shown on a left from the block representing a turbid medium, while a medium response produces a modified signal shown on a right from the block. Since all considered measurement approaches involve some sort of intensity measurements, Y-axis is labeled as “intensity”. X-axes are linked to a domain where measurements are performed. The techniques are listed in the order of increased information content that can be extracted from a single source–detector measurement pair (i.e., optode).

The continuous wave (or intensity) techniques are usually associated with fluence-type measurements where isotropic detectors collect photons from 4π solid angle. An external stimulus is usually a constant intensity laser or a multispectral light source. Tissue response results in attenuation of the input signal. One of first measurements of spatially resolved steady-state reflectance for determination of μa and μs of the tissue was reported by Reynolds et al. [13

13. L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15(9), 2059–2067 (1976). [CrossRef] [PubMed]

]. Since then, reflectance has been used frequently for determination of optical properties of turbid media [14

14. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt. 22(16), 2456–2462 (1983). [CrossRef] [PubMed]

21

21. P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005). [CrossRef] [PubMed]

]. Performing relative fluence measurements at multiple source–detector separations enables the determination of μeff(λ) from a value of the slope [22

22. G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42(19), 4023–4030 (2003). [CrossRef] [PubMed]

,23

23. A. Johansson, J. Axelsson, S. Andersson-Engels, and J. Swartling, “Realtime light dosimetry software tools for interstitial photodynamic therapy of the human prostate,” Med. Phys. 34(11), 4309–4321 (2007). [CrossRef] [PubMed]

]. A variation of multi-distance fluence measurements, called the added absorber technique allows obtaining both μa and μs by changing the absorption coefficient by small known quantities [24

24. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef] [PubMed]

,25

25. B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1(4), 235–244 (1986). [CrossRef]

]. Otherwise, to obtain D(λ) absolute fluence measurements with complex calibrations are required [26

26. J. S. Dam, C. B. Pedersen, T. Dalgaard, P. E. Fabricius, P. Aruna, and S. Andersson-Engels, “Fiber-optic probe for noninvasive real-time determination of tissue optical properties at multiple wavelengths,” Appl. Opt. 40(7), 1155–1164 (2001). [CrossRef] [PubMed]

28

28. T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005). [CrossRef] [PubMed]

]. Such calibrations can be difficult to implement in vivo.

Intensity modulated techniques rely on exploring the diffusive wave in tissue that is produced by modulating the light source with a frequency in a range of 100 MHz −1 GHz [2

2. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010). [CrossRef]

,42

42. J. Fishkin, E. Gratton, M. J. Vandeven, and W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” Proc. SPIE 1431, 122–135 (1991). [CrossRef]

48

48. S. J. Madsen, E. R. Anderson, R. C. Haskell, and B. J. Tromberg, “Portable, high-bandwidth frequency-domain photon migration instrument for tissue spectroscopy,” Opt. Lett. 19(23), 1934–1936 (1994). [CrossRef] [PubMed]

]. In response to an external intensity modulated stimulus, the medium attenuates the signal and introduces a phase shift relative to the input signal. Obtaining the change in modulation and the phase allows for both μa(λ) and μs(λ) determination. In time resolved techniques a short pulse of light (<100 ps) is launched to the medium at a selected point and time. As it propagates through tissue, the pulse undergoes attenuation and broadening that can be used to extract μa(λ) and μs(λ) [49

49. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989). [CrossRef] [PubMed]

53

53. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J Biophotonics 1(3), 200–203 (2008). [CrossRef] [PubMed]

]. It was found that pulse shape (rather than the fluence rate levels) analysis makes the technique insensitive to local bleeding at fiber tips [52

52. T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007). [CrossRef] [PubMed]

,53

53. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J Biophotonics 1(3), 200–203 (2008). [CrossRef] [PubMed]

]. One can also notice certain parallels between transformations of initially confined light excitation in the angular and time domains.

In order for techniques to be used in clinical applications in vivo they should ideally be minimally invasive (limited fiber translations), provide complete tissue characterization without requiring complex absolute calibrations, enable the detection and characterization of tissue inhomogeneities and be cost effective. Frequency-domain and time-domain techniques enable the extraction of optical absorption and scattering from a single optode pair (i.e., minimally invasive) without absolute calibrations, but they require more complex and costly equipment than cw-based methods.

The goal of this article is to demonstrate that our cw-based radiance approach doesn’t have limitations inherent to cw-based fluence techniques, and still allows for accurate determination of both μa(λ) and μs(λ) from relative measurements requiring only a rotation and a single translation in tissue with a potential of completely eliminating the translation. It may bring cw-based radiance closer to frequency and time domain techniques in terms of being able to provide a complete tissue characterization from a single optode measurement.

2. Methods and materials

A detailed description of the experimental setup for radiance measurements has been published elsewhere [39

39. S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt. 16(7), 077003 (2011). [CrossRef] [PubMed]

,41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

]. Therefore, only a brief description is given here. A schematic of the experimental setup is shown in Fig. 2
Fig. 2 A schematic of the experimental setup for radiance measurements.
. A phantom Lucite box (with blackened 18-cm walls) was filled with Intralipid-1% solution and accommodated a fiber with a 800-μm spherical diffuser (connected to a white light source) and a 600-μm side firing fiber (the radiance detector). A size of the box insured an infinite medium geometry. The side firing fiber had a well-defined angular acceptance window of ~10 degrees in water. Both fibers were threaded through ~1.1-mm stainless steel tubes for mechanical stability. Source–detector separation was varied from 6.5 mm to 30.5 mm in various measurements. The side firing fiber was mounted on a computer-controlled rotation stage. The radiance profiles were acquired by rotating the side firing fiber over a 360 degree range with a 2 degree step. The side firing fiber was connected to a computer-controlled USB 4000 spectrometer (Ocean Optics) that collected spectra at every angular step.

A method of extracting of optical properties of the turbid medium from radiance measurements was introduced earlier [41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

]. Briefly, we evaluated the ability to extract μeff(λ) and D(λ) for Intralipid-1% (450–950 nm range) using the diffusion approximation from both experimental and simulated distance-dependent radiance data sets. Simulated radiance data sets have been obtained using an analytical solution of the RTE (radiative transfer equation) for the radiance caused by an isotropic light source placed inside an infinitely extended anisotropic scattering medium [54

54. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3), 036605 (2011). [CrossRef] [PubMed]

]. Radiance was expressed as a function of optical properties of Intralipid-1% (μa(λ),μs(λ)and anisotropy factor, g(λ)) that were obtained directly from (non-radiance) collimated transmission and spatially resolved reflectance measurements (called basic characterization measurements). Thus, known optical properties of Intralipid were input parameters for simulated radiance data sets. Then both simulated and experimentally obtained radiance data sets were fit to a formula of relative radiance derived from the diffusion-based expression for radiance to recover μeff(λ) and D(λ). Using multiple distance measurements within a 6.5–30.5 mm range (with detector facing the source, i.e. 0° angle) and normalizing the radiance data to the value obtained at the shortest source–detector separation (6.5 mm, that is approximately 6 optical penetration depths away from the light source), we demonstrated a successful recovery of μeff(λ) but failed to recover D(λ). This may be a result of the use of relative radiance, in that it didn’t provide enough resolving power to extract both parameters from the fit data. Recovery of optical parameters from relative radiance built in this way likely suffered from non-uniqueness due to optical similarity. The current article proposes a better solution that recovers both parameters.

A starting point for the current approach is also the expression for the radiance, I under the diffusion approximation:

I(r,θ,λ)=P0(4π)2D(λ)[1+3(D(λ)r+μeff(λ)D(λ))cosθ]exp(μeff(λ)r)r
(1)

where r is a distance between the source and the detector, θ is the angle between the direction of propagation and the direction of scattering, λ is a wavelength, P0 is the source power. In order to exclude P0 and D(λ) and obtain an expression that depends solely on μeff(λ), a ratio of two radiance values measured at two different distances, r but the same 90° angle was constructed:

I(r,90°,λ)/I(r0,90°,λ)=exp(μeff(λ)r)exp(μeff(λ)r0)r0/r,
(2)

where r0 was chosen to correspond to the shortest source–detector separation. Equation (2) can be arranged to solve for μeff(λ) as follows:

μeff(λ)=ln(I(r,90°,λ)r/I(r0,90°,λ)r0)/(r0r).
(3)

To determine D(λ) another ratio was constructed using two different angles, 180° and 0° and the same distance, r:

I(r,180°,λ)I(r,0°,λ)=3D(λ)+r(13μeff(λ)D(λ))3D(λ)+r(1+3μeff(λ)D(λ)).
(4)

With knowledge of μeff(λ) from Eq. (3), Eq. (4) can be solved for D(λ) as follows:

D(λ)=1I(r,180°,λ)/I(r,0°,λ)3(μeff(λ)+1/r)(1+I(r,180°,λ)/I(r,0°,λ)).
(5)

The advantage of this approach (Eqs. (3) and (5)) is that data fitting, multiple and absolute measurements are not required, rather both μeff(λ) and D(λ) can be determined directly from simple algebraic expressions that contain only measured quantities.

3. Results and discussion

This approach was tested first with simulated radiance using P approximation as described by Grabtchak et al. [41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

]. Results of applying Eq. (3) to different distance-pairs are shown in Fig. 3(a)
Fig. 3 A comparison of optical parameters extracted from simulated data (lines) with actual parameters obtained from basic characterization experiments (symbols): a) extracted μeff(λ) based on Eq. (3) for different pairs of source–detector separations, b) extracted D(λ) based on Eq. (5) for different source–detector separations.
along with the actual values of μeff(λ) measured independently [41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

]. All distance-pair data produced excellent agreement with actual μeff(λ) experimental values. Figure 3(b) shows results of applying Eq. (5) to determine D(λ) at different distances. Actual values of D(λ) are also shown in the plot. Again, the agreement between the extracted and actual values is excellent. The plot also shows that spectrally resolved radiance obtained at any distance produces identical results. This represents a strong contrast to the inability of retrieving correct values of D(λ) from simulated results using the earlier approach from our group [41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

].

This demonstrates the potential of this radiance approach to recover both optical parameters of the turbid medium with a few-percent error provided that the finite detector size and geometry are taken into account. In addition, there is no need to perform more than two distance dependent measurements and the distances can be separated by any values. It may be beneficial to perform measurements close enough to the source that would minimize noise which becomes more noticeable at higher source–detector separations above 850 nm as seen in Fig. 4(b).

Once μeff(λ) and D(λ) have been obtained, μa(λ)and μs(λ)can be determined from them using two relations listed in the introduction part as follows: μa(λ)=μeff2(λ)D(λ), μs(λ)=(13μeff2(λ)D2(λ))/3D(λ). Figures 5(a)
Fig. 5 A comparison of (a) μa(λ) and (b) μs(λ) expressed from non-corrected μeff(λ) and D(λ)obtained from experimental radiance data (lines) with those expressed from actual values of μeff(λ) and D(λ) obtained from basic characterization measurements (symbols).
and 5(b) demonstrate the spectral behavior of μa(λ) and μs(λ) extracted from values of D(λ) and μeff(λ) obtained from the radiance experiments at 12.5 mm source–detector separation with no correction applied. For comparison, the same parameters obtained from separate (non-radiance) basic characterization measurements are shown with symbols. One can see that the systematic error observed in D(λ) translates to almost equivalent errors in bothμa(λ)andμs(λ). However, when experimental values of radiance data are corrected for backscatter, the error doesn’t exceed the value of a random error in D(λ) (i.e., 4%).

We’d like to discuss the origin of a successful application of the diffusion approximation for radiance (Eq. (1)) even though the discrepancies for the angular radiance obtained from P1 and P3 on one hand and P and Monte Carlo simulations on the other hand are well known and documented [11

11. 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(5), 1359–1373 (2000). [CrossRef] [PubMed]

,36

36. L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006). [CrossRef] [PubMed]

,54

54. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3), 036605 (2011). [CrossRef] [PubMed]

]. In order for the diffusion approximation to be valid two conditions must be fulfilled: μaμs and μsr>>1. A simple check of the first condition using our published data for Intralipid-1% [41

41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

] indicates that even in the range of high water absorption, the first condition is satisfied at λ = 840 nm: μa=3×103mm1, μs=0.88 mm1. The second condition fails at short and even medium distances used in the experiments giving 5.7 ≫ 1 at 6.5 mm and 8 ≫ 1 at 10 mm which are not quite correct. Therefore, a discrepancy between the diffusion and P radiance profiles is expected, and Fig. 6
Fig. 6 Normalized to maximum at 0° angular radiance from analytical solution of RTE (solid red line) and from diffusion approximation (dashed blue line) for 850 nm and 10-mm source–detector separation.
demonstrates it for a 10 mm source–detector separation.

Other wavelengths produce somewhat similar results. Note that both profiles in Fig. 6 are normalized to the value of the forward scattered radiance at 0° that hides the existing discrepancy in absolute values but still shows a different shape. Another important observation is that the ratios of the backscattered to the forward scattered radiance have the same value for both approximations. Since Eq. (4) was developed around this ratio, it explains why the use of the diffusion-based Eq. (1) produced accurate results. It becomes also evident that as long as any ratios formed from Eq. (1) involve the same angle (i.e., 90° and two distances) or probe the angular extremes of radiance (i.e., 0° and 180° and a single distance), the exact shape of the angular dependent radiance is not considered in the calculations. For example, our attempt to replicate the results forming a ratio between radiance values at 90° and 0° failed when testing with simulated data. Indeed, this ratio relies on a certain intermediate value of the angular radiance that has to be correct in order to produce valid results. Therefore, the approach described here can be applied beyond the formal conditions for the diffusion approximation. The exact range for the validity of this approach has to be tested with extensive modeling with P approximation or Monte Carlo simulations. However, comparing normalized backscattered radiance for the diffusion approximation and Monte Carlo simulation using existing published data indicated that the condition of μaμs is more important than μsr>>1 in order for the approach to work. For example, when a medium has optical properties not satisfying the first condition, μs/μa=5.7 (μs=1 mm1, μa=0.175 mm1) [36

36. L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006). [CrossRef] [PubMed]

] measurements close to the source (r = 5 mm) and far from the source (r = 13 mm) produce, correspondingly 21% and 7% mismatch between normalized backscattered radiance values under two approximations. Treating this mismatch as errors in the backscattered radiance indicates that the error in the recovered D(λ) would exceed 22%. However, the expression used to recover μeff(λ) can still produce accurate values because the value of the backscattered radiance is not used in this calculation.

We would like to put current findings in a perspective of a potential applicability to prostate measurements. Optical properties of prostate show a wide range of values indicating intra-organ and inter-patient variability. Still, using some reported optical properties we estimated the ratioμs/μato produce values from10 to 20 at NIR wavelengths [52

52. T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007). [CrossRef] [PubMed]

]. This is a range where the first condition of the diffusion approximation for radiance is valid therefore we expect the demonstrated approach to retain its utility when applied to prostate.

As oppose to Intralipid, real tissues are more complex exhibiting various degrees of heterogeneity. Due to multiple scattering, radiance detects photons that travel through different regions in a sample. Thus, radiance measurements produce some averaged properties of the medium. Changes in these properties as a result of a diseased tissue state (e.g., cancer) or due to treatment can also be measured. Bleeding around interstitially inserted detector poses the same problem for any interstitial techniques, fluence or radiance. Blood around the detector would introduce unwanted increased optical absorption thus filtering the measured radiance signal coming from deeper tissues. When radiance measurements are performed on the prostate in a minimally invasive way (for example, illumination via rectum and detection via urethra), there should be no bleeding. According to the proposed method, this single point measurement produces enough data to determine the effective attenuation coefficient of the prostate. To determine the diffusion coefficient, only one additional interstitial point measurement is required, and this measurement may result in bleeding indeed. However, a need to puncture the prostate only once minimizes the degree of invasiveness in our opinion.

In order to eliminate a translation in tissue and extract all optical properties from angular measurements only, the need for the correct description of the angular radiance limits the use of the diffusion approximation. If a closed formed expression for radiance can be generated from Monte Carlo simulations or P approximation, that would permit a separation of variables (P0,μs,μa, g), a typical radiance data set can provide a large number (up to 180 when taking data every 2° in 0°–360° range) of independent equations to find the unknowns. Otherwise, all angular radiance data sets would increase a robustness of parameter recovery from simultaneous multiple curve fittings similar to the one used in a multi-spectral approach for DOT [2

2. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010). [CrossRef]

].

4. Conclusion

We have introduced a new approach that can be used to recover both μeff(λ) and D(λ) of a turbid medium from relative spectrally resolved radiance measurements using the expression from the diffusion approximation. The approach requires probe rotation and only one translation in the medium. The explicit form of the angular dependence of the radiance is not involved in calculations but the accurate value of the ratio of the backscattered to the forward scattered radiance is required. It has been shown that when μaμs both the diffusion approximation and more accurate descriptions like Monte Carlo or P approximation produce the same value for the ratio that ensures a success of the approach even when the second condition for the diffusion approximation, μsr>>1 is not valid.

Acknowledgments

The authors (S. G. and W. W.) acknowledge financial support from the Natural Sciences and Engineering Research Council, and Canadian Institutes of Health Research, Atlantic Innovation Fund and Canada Research Chair program (W. W.)

References

1.

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42(5), 763–787 (1997). [CrossRef] [PubMed]

2.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010). [CrossRef]

3.

M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6(4), 379–390 (1991). [CrossRef]

4.

S. R. H. Davidson, R. A. Weersink, M. A. Haider, M. R. Gertner, A. Bogaards, D. Giewercer, A. Scherz, M. D. Sherar, M. Elhilali, J. L. Chin, J. Trachtenberg, and B. C. Wilson, “Treatment planning and dose analysis for interstitial photodynamic therapy of prostate cancer,” Phys. Med. Biol. 54(8), 2293–2313 (2009). [CrossRef] [PubMed]

5.

M. L. Pantelides, C. Whitehurst, J. V. Moore, T. A. King, and N. J. Blacklock, “Photodynamic therapy for localised prostatic cancer: light penetration in the human prostate gland,” J. Urol. 143(2), 398–401 (1990). [PubMed]

6.

C. M. Moore, C. A. Mosse, C. Allen, H. Payne, M. Emberton, and S. G. Bown, “Light penetration in the human prostate: a whole prostate clinical study at 763 nm,” J. Biomed. Opt. 16(1), 015003 (2011). [CrossRef] [PubMed]

7.

D. T. Delpy and M. Cope, ““Quantification in tissue near-infrared spectroscopy,” Philos. Trans. R. Soc. London Ser. B 352(1354), 649–659 (1997). [CrossRef]

8.

A. J. Welch and M. J. C. van Gemert, eds., Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011).

9.

L. H. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

10.

F. Martelli, S. del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and software (SPIE, 2010).

11.

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(5), 1359–1373 (2000). [CrossRef] [PubMed]

12.

B. C. Wilson and S. L. Jacques, “Optical reflectance and transmittance of tissues - principles and applications,” IEEE J. Quantum Electron. 26(12), 2186–2199 (1990). [CrossRef]

13.

L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15(9), 2059–2067 (1976). [CrossRef] [PubMed]

14.

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt. 22(16), 2456–2462 (1983). [CrossRef] [PubMed]

15.

R. A. J. Groenhuis, J. J. Ten Bosch, and H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 2: measuring method and calibration,” Appl. Opt. 22(16), 2463–2467 (1983). [CrossRef] [PubMed]

16.

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), 879–888 (1992). [CrossRef] [PubMed]

17.

R. Bays, G. Wagnières, D. Robert, D. Braichotte, J. F. Savary, P. Monnier, and H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35(10), 1756–1766 (1996). [CrossRef] [PubMed]

18.

J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, and H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source-detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36(22), 5655–5661 (1997). [CrossRef] [PubMed]

19.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, and H. J. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44(4), 967–981 (1999). [CrossRef] [PubMed]

20.

W. Ko, Y. Kwak, and S. Kim, “Measurement of optical coefficients of tissue-like solutions using a combination method of infinite and semi-infinite geometries with continuous near infrared light,” Jpn. J. Appl. Phys. 45(9A), 7158–7162 (2006). [CrossRef]

21.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005). [CrossRef] [PubMed]

22.

G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42(19), 4023–4030 (2003). [CrossRef] [PubMed]

23.

A. Johansson, J. Axelsson, S. Andersson-Engels, and J. Swartling, “Realtime light dosimetry software tools for interstitial photodynamic therapy of the human prostate,” Med. Phys. 34(11), 4309–4321 (2007). [CrossRef] [PubMed]

24.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef] [PubMed]

25.

B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. 1(4), 235–244 (1986). [CrossRef]

26.

J. S. Dam, C. B. Pedersen, T. Dalgaard, P. E. Fabricius, P. Aruna, and S. Andersson-Engels, “Fiber-optic probe for noninvasive real-time determination of tissue optical properties at multiple wavelengths,” Appl. Opt. 40(7), 1155–1164 (2001). [CrossRef] [PubMed]

27.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005). [CrossRef] [PubMed]

28.

T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B 79(3), 231–241 (2005). [CrossRef] [PubMed]

29.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983). [CrossRef] [PubMed]

30.

A. M. Ballangrud, P. J. Wilson, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Light distribution and optical coefficients in prostate tumor,” Proc. SPIE 2371, 148–152 (1995).

31.

A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med. 19(4), 471–479 (1996). [CrossRef] [PubMed]

32.

O. Barajas, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol. 42(9), 1675–1687 (1997). [CrossRef] [PubMed]

33.

D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol. 46(9), 2359–2370 (2001). [CrossRef] [PubMed]

34.

T. Xu, C. P. Zhang, X. Y. Wang, L. S. Zhang, and J. G. Tian, “Measurement and analysis of light distribution in intralipid-10% at 650 nm,” Appl. Opt. 42(28), 5777–5784 (2003). [CrossRef] [PubMed]

35.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14(9), 1813–1820 (2005). [CrossRef]

36.

L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt. 45(9), 2101–2114 (2006). [CrossRef] [PubMed]

37.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007). [CrossRef] [PubMed]

38.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys. 105(10), 102025 (2009). [CrossRef]

39.

S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt. 16(7), 077003 (2011). [CrossRef] [PubMed]

40.

S. Grabtchak, T. J. Palmer, and W. Whelan, “Radiance spectroscopy tool box for characterizing Au nanoparticles in tissue mimicking phantoms as applied to prostate,” J. Cancer Sci. Ther. S1, 8 (2011).

41.

S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt. 17(6), 067007 (2012). [CrossRef] [PubMed]

42.

J. Fishkin, E. Gratton, M. J. Vandeven, and W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” Proc. SPIE 1431, 122–135 (1991). [CrossRef]

43.

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. 30(31), 4474–4476 (1991). [CrossRef] [PubMed]

44.

J. M. Schmitt, A. Knüttel, and J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9(10), 1832–1843 (1992). [CrossRef] [PubMed]

45.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32(4), 607–616 (1993). [CrossRef] [PubMed]

46.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69(10), 3457–3481 (1998). [CrossRef]

47.

S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34(1), 32–42 (1995). [CrossRef]

48.

S. J. Madsen, E. R. Anderson, R. C. Haskell, and B. J. Tromberg, “Portable, high-bandwidth frequency-domain photon migration instrument for tissue spectroscopy,” Opt. Lett. 19(23), 1934–1936 (1994). [CrossRef] [PubMed]

49.

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989). [CrossRef] [PubMed]

50.

S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28(12), 2223–2229 (1989). [CrossRef] [PubMed]

51.

S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, “Time-resolved transillumination for medical diagnostics,” Opt. Lett. 15(21), 1179–1181 (1990). [CrossRef] [PubMed]

52.

T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt. 12(1), 014022 (2007). [CrossRef] [PubMed]

53.

T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J Biophotonics 1(3), 200–203 (2008). [CrossRef] [PubMed]

54.

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(3), 036605 (2011). [CrossRef] [PubMed]

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics
(170.7050) Medical optics and biotechnology : Turbid media
(290.4210) Scattering : Multiple scattering
(170.6935) Medical optics and biotechnology : Tissue characterization

ToC Category:
Optics of Tissue and Turbid Media

History
Original Manuscript: June 12, 2012
Revised Manuscript: August 27, 2012
Manuscript Accepted: August 27, 2012
Published: September 4, 2012

Citation
Serge Grabtchak and William M. Whelan, "Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements," Biomed. Opt. Express 3, 2371-2380 (2012)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-10-2371


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References

  1. W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol.42(5), 763–787 (1997). [CrossRef] [PubMed]
  2. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys.73(7), 076701 (2010). [CrossRef]
  3. M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci.6(4), 379–390 (1991). [CrossRef]
  4. S. R. H. Davidson, R. A. Weersink, M. A. Haider, M. R. Gertner, A. Bogaards, D. Giewercer, A. Scherz, M. D. Sherar, M. Elhilali, J. L. Chin, J. Trachtenberg, and B. C. Wilson, “Treatment planning and dose analysis for interstitial photodynamic therapy of prostate cancer,” Phys. Med. Biol.54(8), 2293–2313 (2009). [CrossRef] [PubMed]
  5. M. L. Pantelides, C. Whitehurst, J. V. Moore, T. A. King, and N. J. Blacklock, “Photodynamic therapy for localised prostatic cancer: light penetration in the human prostate gland,” J. Urol.143(2), 398–401 (1990). [PubMed]
  6. C. M. Moore, C. A. Mosse, C. Allen, H. Payne, M. Emberton, and S. G. Bown, “Light penetration in the human prostate: a whole prostate clinical study at 763 nm,” J. Biomed. Opt.16(1), 015003 (2011). [CrossRef] [PubMed]
  7. D. T. Delpy and M. Cope, ““Quantification in tissue near-infrared spectroscopy,” Philos. Trans. R. Soc. London Ser. B352(1354), 649–659 (1997). [CrossRef]
  8. A. J. Welch and M. J. C. van Gemert, eds., Optical-Thermal Response of Laser-Irradiated Tissue, 2nd ed. (Springer, 2011).
  9. L. H. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).
  10. F. Martelli, S. del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions and software (SPIE, 2010).
  11. 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(5), 1359–1373 (2000). [CrossRef] [PubMed]
  12. B. C. Wilson and S. L. Jacques, “Optical reflectance and transmittance of tissues - principles and applications,” IEEE J. Quantum Electron.26(12), 2186–2199 (1990). [CrossRef]
  13. L. Reynolds, C. Johnson, and A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt.15(9), 2059–2067 (1976). [CrossRef] [PubMed]
  14. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: theory,” Appl. Opt.22(16), 2456–2462 (1983). [CrossRef] [PubMed]
  15. R. A. J. Groenhuis, J. J. Ten Bosch, and H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 2: measuring method and calibration,” Appl. Opt.22(16), 2463–2467 (1983). [CrossRef] [PubMed]
  16. 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), 879–888 (1992). [CrossRef] [PubMed]
  17. R. Bays, G. Wagnières, D. Robert, D. Braichotte, J. F. Savary, P. Monnier, and H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt.35(10), 1756–1766 (1996). [CrossRef] [PubMed]
  18. J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, and H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source-detector separations for which path lengths do not depend on scattering properties,” Appl. Opt.36(22), 5655–5661 (1997). [CrossRef] [PubMed]
  19. R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, and H. J. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol.44(4), 967–981 (1999). [CrossRef] [PubMed]
  20. W. Ko, Y. Kwak, and S. Kim, “Measurement of optical coefficients of tissue-like solutions using a combination method of infinite and semi-infinite geometries with continuous near infrared light,” Jpn. J. Appl. Phys.45(9A), 7158–7162 (2006). [CrossRef]
  21. P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt.10(3), 034018 (2005). [CrossRef] [PubMed]
  22. G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt.42(19), 4023–4030 (2003). [CrossRef] [PubMed]
  23. A. Johansson, J. Axelsson, S. Andersson-Engels, and J. Swartling, “Realtime light dosimetry software tools for interstitial photodynamic therapy of the human prostate,” Med. Phys.34(11), 4309–4321 (2007). [CrossRef] [PubMed]
  24. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt.30(31), 4507–4514 (1991). [CrossRef] [PubMed]
  25. B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci.1(4), 235–244 (1986). [CrossRef]
  26. J. S. Dam, C. B. Pedersen, T. Dalgaard, P. E. Fabricius, P. Aruna, and S. Andersson-Engels, “Fiber-optic probe for noninvasive real-time determination of tissue optical properties at multiple wavelengths,” Appl. Opt.40(7), 1155–1164 (2001). [CrossRef] [PubMed]
  27. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol.50(10), 2291–2311 (2005). [CrossRef] [PubMed]
  28. T. C. Zhu, J. C. Finlay, and S. M. Hahn, “Determination of the distribution of light, optical properties, drug concentration, and tissue oxygenation in-vivo in human prostate during motexafin lutetium-mediated photodynamic therapy,” J. Photochem. Photobiol. B79(3), 231–241 (2005). [CrossRef] [PubMed]
  29. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys.10(6), 824–830 (1983). [CrossRef] [PubMed]
  30. A. M. Ballangrud, P. J. Wilson, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Light distribution and optical coefficients in prostate tumor,” Proc. SPIE2371, 148–152 (1995).
  31. A. M. Ballangrud, P. J. Wilson, K. Brown, G. G. Miller, R. B. Moore, M. S. McPhee, and J. Tulip, “Anisotropy of radiance in tissue phantoms and Dunning R3327 rat tumors: radiance measurements with flat cleaved fiber probes,” Lasers Surg. Med.19(4), 471–479 (1996). [CrossRef] [PubMed]
  32. O. Barajas, A. M. Ballangrud, G. G. Miller, R. B. Moore, and J. Tulip, “Monte Carlo modelling of angular radiance in tissue phantoms and human prostate: PDT light dosimetry,” Phys. Med. Biol.42(9), 1675–1687 (1997). [CrossRef] [PubMed]
  33. D. J. Dickey, R. B. Moore, D. C. Rayner, and J. Tulip, “Light dosimetry using the P3 approximation,” Phys. Med. Biol.46(9), 2359–2370 (2001). [CrossRef] [PubMed]
  34. T. Xu, C. P. Zhang, X. Y. Wang, L. S. Zhang, and J. G. Tian, “Measurement and analysis of light distribution in intralipid-10% at 650 nm,” Appl. Opt.42(28), 5777–5784 (2003). [CrossRef] [PubMed]
  35. T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys.14(9), 1813–1820 (2005). [CrossRef]
  36. L. C. L. Chin, W. M. Whelan, and I. A. Vitkin, “Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification,” Appl. Opt.45(9), 2101–2114 (2006). [CrossRef] [PubMed]
  37. L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt.12(6), 064027 (2007). [CrossRef] [PubMed]
  38. L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105(10), 102025 (2009). [CrossRef]
  39. S. Grabtchak, T. J. Palmer, and W. M. Whelan, “Detection of localized inclusions of gold nanoparticles in Intralipid-1% by point-radiance spectroscopy,” J. Biomed. Opt.16(7), 077003 (2011). [CrossRef] [PubMed]
  40. S. Grabtchak, T. J. Palmer, and W. Whelan, “Radiance spectroscopy tool box for characterizing Au nanoparticles in tissue mimicking phantoms as applied to prostate,” J. Cancer Sci. Ther.S1, 8 (2011).
  41. S. Grabtchak, T. J. Palmer, F. Foschum, A. Liemert, A. Kienle, and W. M. Whelan, “Experimental spectro-angular mapping of light distribution in turbid media,” J. Biomed. Opt.17(6), 067007 (2012). [CrossRef] [PubMed]
  42. J. Fishkin, E. Gratton, M. J. Vandeven, and W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” Proc. SPIE1431, 122–135 (1991). [CrossRef]
  43. M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt.30(31), 4474–4476 (1991). [CrossRef] [PubMed]
  44. J. M. Schmitt, A. Knüttel, and J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A9(10), 1832–1843 (1992). [CrossRef] [PubMed]
  45. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt.32(4), 607–616 (1993). [CrossRef] [PubMed]
  46. B. Chance, M. Cope, E. Gratton, N. Ramanujam, and B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum.69(10), 3457–3481 (1998). [CrossRef]
  47. S. Fantini, M. A. Franceschinifantini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain mutlichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng.34(1), 32–42 (1995). [CrossRef]
  48. S. J. Madsen, E. R. Anderson, R. C. Haskell, and B. J. Tromberg, “Portable, high-bandwidth frequency-domain photon migration instrument for tissue spectroscopy,” Opt. Lett.19(23), 1934–1936 (1994). [CrossRef] [PubMed]
  49. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.28(12), 2331–2336 (1989). [CrossRef] [PubMed]
  50. S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt.28(12), 2223–2229 (1989). [CrossRef] [PubMed]
  51. S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, “Time-resolved transillumination for medical diagnostics,” Opt. Lett.15(21), 1179–1181 (1990). [CrossRef] [PubMed]
  52. T. Svensson, S. Andersson-Engels, M. Einarsdóttír, and K. Svanberg, “In vivo optical characterization of human prostate tissue using near-infrared time-resolved spectroscopy,” J. Biomed. Opt.12(1), 014022 (2007). [CrossRef] [PubMed]
  53. T. Svensson, E. Alerstam, M. Einarsdóttír, K. Svanberg, and S. Andersson-Engels, “Towards accurate in vivo spectroscopy of the human prostate,” J Biophotonics1(3), 200–203 (2008). [CrossRef] [PubMed]
  54. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(3), 036605 (2011). [CrossRef] [PubMed]

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