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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 4 — Apr. 1, 2009
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Spectral determination of a two-parametric phase function for polydispersive scattering liquids

Tobias Lindbergh, Ingemar Fredriksson, Marcus Larsson, and Tomas Strömberg  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1610-1621 (2009)
http://dx.doi.org/10.1364/OE.17.001610


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Abstract

A method for determining a two-parametric Gegenbauer-kernel phase function that accurately describes the diffuse reflectance from a polydispersive scattering media at small source-detector separations (0.23 to 1.2 mm), is presented. The method involves spectral collimated transmission measurements, spatially resolved spectral diffuse reflectance (SRDR) measurements, and inverse Monte Carlo technique. Both absolute calibration (using a monodispersive polystyrene microsphere suspension) and relative calibration (eliminating differences between fibers) of SRDR spectra yielded comparable results. When applied to water dilutions of milk, simulated and measured spectra deviated less than 6.5% and 2.5% for the absolute and relative calibration case, respectively, even for the closest fiber separation. Corresponding values for milk including ink as an absorber, were 13.4% and 7.3%.

© 2009 Optical Society of America

1. Introduction

To accurately model how photons propagate in tissue-like optical phantoms consisting of scatterers, absorbers, and bulk media [1

1. B.W. Pogue and M.S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11, 041102-1 – 041102-16 (2006). [CrossRef] [PubMed]

], it is essential to gain knowledge on the optical properties, i.e. the refractive index, the scattering and absorption coefficients (μ s and μ a), and the scattering phase function (p(θ)). At distances of several d mfp′ =1(μ a+μ s′) away from the source, where diffusion models are valid, only the reduced scattering coefficient μ s′ = μ s (1-g) is needed to describe light scattering [2

2. 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, 879–888 (1992). [CrossRef] [PubMed]

]. For an accurate description close to the source using Monte Carlo (MC) simulations, the anisotropy factor g = g 1 = 〈cos(θ)〉 (first moment) and higher order of moments (gn,n≥2) of the phase function need to be taken into account [3–5

3. F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999). [CrossRef]

].

Polystyrene microsphere suspensions are frequently used as the scattering component in optical phantoms. When using microspheres, Mie theory can be employed to accurately describe both the scattering coefficient and the phase function of a monodispersive suspension [6

6. G.M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45, 1062“1071 (2006). [CrossRef] [PubMed]

] or combinations of spheres with different, known sizes [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

]. Michels et al. [7

7. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [CrossRef] [PubMed]

] used Mie theory, a sophisticated goniometry setup, spatially resolved reflectance, and collimated transmission to estimate the scattering properties of six different nutritional fat emulsions including their scattering size distribution of the scattering particles. However, in the latter study, the scatterer size distribution could not be freely fitted but had to be restricted to a linear decay in the logarithmic size scale.

The scattering properties of milk depend on the size distribution of fat globules and casein micelles, which varies greatly between homogenized and unhomogenized milk [8

8. C.L. Crofcheck, F.A. Payne, and M.P. Mengüc, “Characterization of milk properties with a radiative transfer model,” Appl. Opt. 41, 2028–2037 (2002). [CrossRef] [PubMed]

], and the refractive index differences between fat globules and casein micelles [9

9. M. C. Ambrose Griffin and W.G. Griffin, “A simple turbidimetric method for the determination of the refractive index of large colloidal particles applied to casein micelles,” J. Colloid Interface Sci. 104, 409–415 (1985). [CrossRef]

]. Hence, Mie theory cannot readily be employed to describe its scattering properties. Thueler et al. [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

] proposed an adaptive light scattering model that includes a parameter γ = (1 - g 2)/ (1 - g 1) taking into account not only the anisotropy factor, but also the second moment of the phase function. Estimations of the scattering properties were achieved by matching the model with measurements from a fiber optic probe with multiple fiber separations (0.3 – 1.35 mm). Their model did not match the measurements for the closest two fibers (< 0.4 mm distance), when measuring on phantoms consisting of combinations of five polystyrene spheres, ranging from 0.05 μm to 1 μm in diameter. This indicates the need to include higher order of moments of the phase function to accurately model how photons propagate in microsphere suspensions.

The aim of this study was to present a method for determining the scattering properties of non-absorbing polydispersive optical phantoms in sufficient detail to allow accurate modeling of light intensities close to the source. The basis of the proposed method has some similarities with those presented by Thueler et al. [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

], Palmer et al. [6

6. G.M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45, 1062“1071 (2006). [CrossRef] [PubMed]

], and Michels et al [7

7. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [CrossRef] [PubMed]

], such as: 1) The phase function estimation is improved by first measuring the scattering coefficient μ s with a spectral collimated transmission (SCT) setup; 2) Measured and simulated spatially resolved diffuse reflectance (SRDR) with small source-detector separations are matched using non-linear fitting.

On the other hand, the proposed method differs from the methods presented in the above mentioned studies, in two important ways: 1) Mie theory is not used but the phase function is approximated with a two-parametric (α Gk and g, see below) Gegenbauer kernel (Gk) phase function [10

10. L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering.,” J. Opt. Soc. Am. 70, 1206–1212 (1980). [CrossRef]

], enabling application to various polydispersive non-absorbing media with unknown size distributions, covering the γ range used by Thueler et al [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

]; 2) Intensity calibration (simulations vs. measurements) is done either using a polystyrene sphere suspension (absolute calibration; no fitting) or in the fitting procedure (relative calibration). The latter does not, in contrast to Thueler et al. [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

] and Palmer et al. [6

6. G.M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45, 1062“1071 (2006). [CrossRef] [PubMed]

], require measurements on reference microsphere suspensions. The robustness is further increased by measurements on multiple phantoms with different concentrations of scatterers. Together, this improves the accuracy in the two-parametric phase function determination in a way that enables accurate light modeling close to the source.

2. Material and methods

2.1 SCT setup

Collimated transmission is a widely used and accepted method for measuring the scattering coefficient (μ s), absorption coefficient (μ a) or the total attenuation coefficient (μ t) of optical phantoms. The method is based on the Beer-Lambert law which describes collimated light attenuation as a function of optical pathlength through a medium. The essentials of the method can be found elsewhere [11

11. L. Wang and S.L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using collimated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994). [CrossRef] [PubMed]

]. A schematic view of the SCT setup is shown in Fig. 1 left panel. The two fiber collimators (Edmund Optics Inc., UK) were connected to a broadband tungsten halogen lamp (HL-2000, Ocean Optics Inc., USA) and a spectrometer (400–900 nm AvaSpec 2048-5-RM, Avantes BV, The Netherlands, grating: VB 600 lines/mm), respectively, using glass fibers with core/cladding sizes of 200/230 μm and a numerical aperture (NA) of 0.37. The two collimators, aligned and placed 250 mm apart facing each other, ensured a uniform and collimated illumination and detection of the light passing through the sample cuvette. The divergence of the collimators was 0.37 degrees. For each sample, 10 consecutive measurements where conducted at different sample thicknesses, achieved using a submerged glass cylinder connected to a high resolution translational micro stage. For each sample, the step size was adjusted to ensure that no measurement was conducted on optical thicknesses greater than 3 mfp (mean free path; mfp = 1/(μ a+μ s)).

2.2 SRDR setup

For the SRDR measurements, the radial decay in backscattered light intensity was measured using a fiber probe with several light collecting fibers positioned at different distances away from a light emitting fiber. The SRDR profile (magnitude and spatial shape) depends on the optical properties and the inverse problem can be solved using either the diffusion approximation [12

12. A. Kienle, L. Lilge, M.S. Patterson, R. Hibst, R. Steiner, and B.C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304”2314 (1996). [CrossRef] [PubMed]

] or Monte Carlo simulations [4

4. P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

].

A schematic view of the SRDR probe tip is shown in Fig. 1, right panel. The fiber-optic probe used in the SRDR setup consisted of 5 linearly aligned fibers, with one of the outermost fibers as the illuminating source fiber (S in Fig. 1 right panel) and the remaining 4 fibers as detector fibers (D 1–4 in Fig. 1 right panel). The center-to-center source-detector separations were 230 μm, 690 μ, 1150 μm, and 1610 μm, respectively. The spectrometer, light source and optical fiber specifications were the same as used in the SCT setup. The dependency between emission angle and light intensity of the emitting fiber was quantified in a goniometric setup. This angular dependency was found to correspond well with the specified NA, and was taken into account in the analysis of the simulation data by weighting each photon according to its emission angle. Similarly, the angle sensitivity of the backscattered light, collected by the fiber-optic probe when attached to the spectroscope, was characterized and taken into account in the analysis of the simulation data. This sensitivity, however, was found limited by the spectroscope to about ±6 degrees. When performing the measurements, the SRDR probe was handheld and vertically submerged (approx. 1 cm) in a 100 mL beaker. No movement artifacts due to inhomogeneous illumination were observed during the measurements.

Fig. 1. Left Panel: The SCT setup. Right Panel: The tip of the probe used for SRDR measurements, with source fiber (S) and detector fibers (D 1–4). Note: The SRDR probe is not drawn to scale, except for fiber separations. The distance from the edge of the probe to the edge of the closest fiber (S) was 2 mm.

2.3 Spectral data preprocessing

Since the sensitivity of the spectrometer was insufficient below 500 nm and above 800 nm, only the 500–800 nm range was used in the analysis. For both the SCT and the SRDR measurements, dark spectra for every channel were recorded after each measurement and subtracted from the signal spectra before further processing. To compensate for any changes in lamp intensity and color during the SRDR measurements, all signal spectra were normalized with simultaneously recorded lamp spectra. Additionally, all measurements were normalized with a lamp-normalized white reference spectra recorded on a smooth surface covered with BaSO4 with a reflectivity higher than 98% in the 500-800 nm range [13

13. E. Häggblad, T. Lindbergh, M.G.D. Karlsson, M. Larsson, H. Casimir-Ahn, E.G. Salerud, and T. Strömberg, “Myocardial tissue oxygenation estimated with calibrated diffuse reflectance spectroscopy during coronary artery bypass grafting,” J. Biomed. Opt. 13, 054030-1 ” 054030-9 (2008). [CrossRef] [PubMed]

].

2.4 Preparation and scattering coefficient characterization of milk dilutions

Homogenized ultra-high temperature (UHT) treated milk (Arla Foods, Sweden), with a fat content of 1.5% and protein content of 3.4%, was used as a scattering basis for the optical phantoms. By diluting the milk with water, using a precision scale and assuming a density of 1.0 g/cm3 for the phantom components, four aqueous dilutions with different fractional milk contents were prepared. The dilutions were named OP 1 through OP 4 (OP = Optical Phantom), and the fractional milk contents were set to 1/8, 1/4, 1/2 and 1, respectively. The spectral scattering coefficient was determined by SCT measurements, assuming,μ a = 0.0 mm-1. The water absorption, with a maximum μ a of below 0.003 mm-1 in the wavelength region of interest [14

14. H. Buiteveld, J.M.H. Hakvoort, and M. Donze, “The optical properties of pure water,” in Ocean Optics XII -Proceedings of SPIE, 174–183 (1994).

], was found to have minimal or no impact on the detected intensity and was thus neglected.

2.5 Calibration of the SRDR system

To individually calibrate the detector fibers of the SRDR system, simulations and measurements of the diffuse reflectance spectra for an aqueous monodispersive polystyrene microsphere suspension (Bangs Laboratories, Inc., USA, radius 155 μm) were compared. The Monte Carlo (MC) setup consisted of a semi-infinite slab representing the suspension, with,μ a = 0.0 mm-1, μ s as determined from a SCT measurement, and a Mie phase function calculated using a wavelength dependent relative refractive index based on water [15

15. G.M. Hale and M.R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef] [PubMed]

] and polystyrene [16

16. I.D. Nikolov and I.D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067–2070 (2000). [CrossRef]

]. The absolute calibration factors (A abs) for each detecting fiber were calculated as the ratio between the mean intensity in the wavelength range 500–800 nm of the measured and the simulated spectra.

The relative calibration factors (A rel) between the detecting fibers were determined from a measurement where all fibers were illuminated uniformly and equally. For this measurement, the probe was vertically submerged in a 100 mL beaker containing a high-scattering polystyrene sphere suspension, which was illuminated from below by a white halogen lamp (Flexilux 150 HL, Schölly Fiberoptik GmbH, Denzlingen, Germany).

2.6 Two-parametric phase function determination

The idea for determining the two-parametric phase function was to solve the inverse problem by comparing measured SRDR spectra with pre-simulated MC spectra. The MC setup used the two parametric Gk phase function (parameters α Gk and α Gk):

PGk(θ)=αGkgGk(1gGk2)2αGkπ[(1+gGk)2αGk(1gGk)2αGk][1+gGk22gGkcos(θ)]αGk+1.
(1)

The anisotropy factor g depends on both α Gk and g Gk; g Gk only equals g when α Gk = 0.5 (i.e. the Henyey-Greenstein phase function) [10

10. L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering.,” J. Opt. Soc. Am. 70, 1206–1212 (1980). [CrossRef]

]. Simulations were performed for 7 values of α Gk (0.025, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5) and 11 values of the anisotropy factor g (0.7, 0.72, …, 0.9). The g Gk parameter needed for simulation input was numerically calculated using the relationship between g Gk, α Gk and g described in [10

10. L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering.,” J. Opt. Soc. Am. 70, 1206–1212 (1980). [CrossRef]

]. Each of these 7 × 11 = 77 simulations were performed by injecting photons in a single point. The direction of each photon was randomly chosen based on a uniformly illuminated solid angle (limited by NA = 0.37) assumption. Only a single scattering coefficient, denoted μ s,sim =3.72 mm-1 (average μ s for OP 2 in the wavelength range 500–800 nm), was simulated. Additional μ s levels were added in the post-processing by rescaling the original simulation data [17

17. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304-1 – 041304-10 (2008). [CrossRef] [PubMed]

]. In short, the rescaling is performed by multiplying the optical pathlength and the photon detection point from the original simulation by the factor μ ssim/μ s . For each of the 77 combinations of α Gk and g, recalculations were done for the SCT measured μ s levels at wavelengths λ = 500, 510, …, 800 nm for each of the four phantoms (OP 1–4). The effect of the source distribution (radius 100 μm) was added after rescaling by randomly offsetting each detection position. The emission and detection angle dependency of the SRDR system was taken into account by adjusting the weight of each photon accordingly, as described earlier. Finally, the detected intensity at the 4 source-detector separations (D 1–4) was calculated, based on at least 1.0×105 detected photons.

The error functions between MC generated and measured intensity (Eq. 2: absolute calibration method and Eq. 3: relative calibration method) were minimized in a least-square sense using the Levenberg-Marquardt algorithm with αGk and g as fitting parameters. The initial values of these parameters were chosen as the mid-points in the two vectors α Gk and g (0.2625 and 0.8), respectively. Evaluating other initial values yielded identical results. Both α Gk and g are assumed to depend only on λ, and not OP and D .

Eabs(αGk,g)=IMC(αGk,g,λ,OP,D)Imean(λ,OP,D)/Aabs(D)1
(2)
Erel(αGk,g)=IMC(αGk,g,λ,OP,D)Imeas(λ,OP,D)/Arel(D)(IMC(αGk,g,λ,OP,D)Imeas(λ,OP,D)/Arel(D)λ,OP,D)11.
(3)

The minimizers were denoted α Gk, abs * (α Gk, rel *) and g * abs(g * rel), giving the optimal MC generated intensity, I * MC,abs (I * MC,rel), for the absolute (relative) calibrated case. Note that the minimizers are wavelength dependent and that the optimization based on the relative error function needs to be performed for all wavelengths simultaneously.

For the phase function approximation, OP 4 was omitted from the analysis, since a high concentration of scatterers likely affects the phase function (see Discussion). In addition, for some optical phantoms and wavelengths, the signal level for D 4 was below two standard deviations of the detector dark noise and was therefore excluded from the analysis.

The robustness of the method was evaluated by excluding data from the optimization procedure. This was achieved by limiting the measured data utilized for the phase function approximation to either a single fiber separation ([OP 1–3, Dj], j = 1, 2, 3) or a single optical phantom ([OPi, D 1–3], i = 1, 2, 3), resulting in six different α Gk,abs * and g * abs combinations, and six different α Gk,rel * and g * abscombinations. For each of these six combinations, I * MC,abs and I * MC,rel were evaluated for all nine OPi and Dj combinations and compared to the corresponding I meas. The deviation between simulated and measured spectra was quantified as the root-mean-square (rms) errors

Eabsrms=(Eabs2(αGk,abs*,gabs*)λ)1/2
(4)

and

Erelrms=(Erel2(αGk,rel*,grel*)λ)1/2.
(5)

For comparison, the optimal anisotropy factor g * HG,abs for a Henyey-Greenstein (HG) phase function approximation (α Gk = 0.5) was calculated using [OP 1–3, D 1–3].

2.7 Optical phantoms including absorption

To investigate how the two-parametric phase function describe light propagation in optical phantoms containing both absorption and scattering components, SRDR measurements were performed on three scattering and absorbing optical phantoms (OP 5–7), and were compared to simulated spectra. As absorption component, water soluble, oil-free blue ink (Coloris, Germany) was used. The ink was diluted with water to 1/500 of the original concentration, and the spectral absorption coefficient was determined by SCT measurements. The fractional content of the diluted ink in OP 5–7 was set to 1/20, 1/7 and 1/2, respectively. The fractional milk concentration was set to 1/4 (equal to the scattering component concentration of OP 2) in all three optical phantoms.

The simulated spectra for OP 5–7 were calculated based on both (α Gk,abs *, g * abs) and (α Gk,rel *, g * rel), determined from [OP 1–3, D 1–3]. By logging the optical pathlength of every detected photon in the original simulation, inclusion of the absorption effect was done after the μ s,sim/μ s rescaling by reweighting every detected photon by a factor of exp(-μ a·PL), where PL is the rescaled optical pathlength [17

17. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304-1 – 041304-10 (2008). [CrossRef] [PubMed]

]. The μ a values used in the simulated spectra were obtained by multiplying the fractional content described above by the μ a spectra from the SCT measurements of the blue ink dilution.

3. Results

3.1 Calibration of the SRDR system

The fit between the measured and simulated spectra for the polystyrene microsphere suspension is shown in Fig. 2 left panel. The measured spectra are multiplied with the absolute calibration factors A abs. The right panel shows a comparison between A abs and A rel. In this figure, both factors were normalized with its mean over D. The quotients (A rel/〈A relD)/(A abs/(〈A absD) were -2.2%, -0.6%, 1.3%, and 1.0% for D 1, D 2, D 3 and D 4, respectively.

Fig. 2. Left Panel: Calibration measurements using a polystyrene microsphere suspension. Measured (black) and simulated (gray) spectra for the source-detector separations D 1–4. Measured spectra are normalized by A abs (see Eq. 2). Note: The intensities are unitless, since the simulated spectra are normalized with the incident light intensity. Right Panel: A abs for D 1–4 from the microsphere suspension measurement (*) and A rel from the uniform illumination measurement (Δ). Both series are normalized by their mean value.

3.2 Scattering coefficient

The maximal optical path lengths during the measurements were in the interval 0.15 mm (OP 4) and 0.55 mm (OP 1), giving an mfp < 3 (for OP 3 and λ =500 nm). The small acceptance angle of the fiber collimator (0.37 degrees), makes detection of scattered light negligible. Calculations according to Wang et al. 11 or OP 3 and the Gk phase function with α Gk,abs * and g * abs (Fig. 4), gives a μ s -error less than 0.3%. In Fig. 3 left panel, μ s, as measured by SCT, is shown as a function of wavelength for OP 1–4. The scattering coefficient of the optical phantoms displayed a linear response to concentration, except for OP 4, which displayed lower values, as seen in the quotients μs,OPi μs,OP1 OP 1, i = 2, 3, 4, Fig. 3 right panel. This non-linear response between μ s vs. concentration has been observed by others [18–20

18. A. Giusto, R. Saija, M.A. Iati, P. Denti, F. Borghese, and O.I. Sindoni, “Optical properties of high-density dispersions of particles: Application to intralipid solutions,” Appl. Opt. 42, 4375–4380 (2003). [CrossRef] [PubMed]

], and is further commented in the discussion.

Fig. 3. Left Panel: μ s for OP 1–4 as a function of wavelength. Right Panel: μs,OPi/μs,OP1, i = 2,3,4 . Dotted lines: Expected quotients assuming linear response between μ s and concentration. Solid lines: experimental quotients.

3.3 Two-parametric phase function determination

In Fig. 4 left panel, α *Gk,abs and α* Gk,rel, utilizing [OP 1–3, D 1–3], are shown, and g * abs and g * rel are displayed in Fig. 4 right panel.

Fig. 4. The wavelength-resolved parameters for the two-parametric phase function. Left Panel: α Gk,abs *, (oe-17-03-1610-i001) and α Gk,rel * (oe-17-03-1610-i002). Right Panel: g * abs (oe-17-03-1610-i003), g * rel(oe-17-03-1610-i004) and g HG,abs (O).

For both the absolute calibrated and the relative calibrated case, the anisotropy factor displayed decreasing values with increasing wavelength. The anisotropy factor with the HG phase function was lower than those with the Gk phase function. The difference between g * rel and g * abs was in the interval -1.2% to -0.8%.

The range and the average of the rms-errors, as given by Eq. 4 and 5, for each of the six Gk phase functions estimated from the subsets [OP 1–3, D j], j = 1, 2, 3, and [OPi, D 1–3], i = 1, 2, 3, are presented in table 1.

Table 1:. Deviations (E rms abs and E rms rel) between simulated and measured spectra evaluated over OP 1–3, D 1–3, using different data-sets a) in the phase function estimation. Deviations are given as mean and min-max range.

table-icon
View This Table

When utilizing all available data [OP 1–3, D 1–3] in the phase function approximation, the worst case rms-errors in Eq. 4 and 5, were 6.5% and 2.5% for the absolute and relative calibrated cases, respectively. For the absolute calibrated case, the corresponding value when utilizing the HG phase function and [OP 1–3, D 1–3], were 8.5%, with a min - max range of 3.3% to 16.8%, displaying substantially larger errors compared to the cases utilizing the Gk phase function.

For phase function approximations utilizing only subsets of OP and D, Eq. 4 displayed worst case rms-errors above 10% except when calibrating using [OP 2, D 1–3]. In general, the relative calibration case displayed lower rms-errors, even when the determination was based on a sub-set of data while evaluation was performed on the complete data-set. Fig. 5 shows measured intensity, I meas, together with I MC,abs * and I MC,rel * based on the phase function approximation utilizing [OP 1–3, D 1–3]. There was no relationship between the rms-error and either of the level of scattering (OPi, i = 1, 2, 3) or the source-detector distances.

Fig. 5: Measured (black) and simulated (solid gray : I MC,abs *, dashed black : I MC,rel *) intensities, with phase function approximation based on [OP 1–3, D 1–3]. Left Panel: D 1, Right Panel: D 3.

3.4 Optical phantoms including absorption

In Fig. 6 left panel, the absorption coefficient of the diluted (1/500) blue ink as a function of wavelength is shown. Fig. 6 right panel displays the comparison between measured and simulated spectra in D 3 for OP 5 and OP 7. D 3 was chosen as an illustrative example in favor of D 1 and D 2, since the effect of absorption is most noticeable in D 3.

Fig. 6: Left Panel: μ a for diluted blue ink (1/500) as function of wavelength. Right Panel: Measured (black) and simulated (solid gray : I MC,abs *, dashed black : I MC,rel *) intensities for D 3.

For OP 5–7 the rms-errors were below 13.4% and 7.3% for the absolute and relative calibrated cases, respectively. There was no relationship between the rms-error and the source-detector distance, while the error increased with the level of absorption (in OPi, i = 5, 6, 7). The only rms-errors above 10% were for OP 7 and the absolute calibrated cases.

4. Discussion and conclusions

We have presented a method for a two-parametric phase function determination at small source-detector separations that can be applied to polydispersive scattering liquids. The method involves spectral scattering coefficient determination, SRDR measurements, and MC modeling and simulation of light transport. The phase function parameters are determined by nonlinear fitting minimizing the difference between measured and simulated SRDR data. To speed-up the inverse Monte Carlo algorithm, pre-simulated multi-dimensional data-sets, based on a two-parametric phase function, and scaling for μ s, is utilized. The inverse algorithm employs spectral scattering coefficients that are separately determined using collimated transmission measurements.

Evaluations were performed on four different concentrations of UHT-treated milk with 1.5% fat content and 3.4% protein content. Two different intensity calibration methods were presented; one using a monodispersive polystyrene sphere suspension for an absolute calibration, and another using a relative calibration scheme where the difference in sensitivity between fibers is equalized.

4.1 Spectral scattering coefficient

The measured spectral scattering coefficients for OP 1–3 were found to be multiples of each other which is expected since the fractional milk content in OP 1–4 was set to 1/8, 1/4, 1/2, and 1, respectively. Assuming a linear relationship between concentration and μ s, the measured μ s for OP 4 was smaller than expected. However, several studies [19

19. G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, and W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994). [CrossRef] [PubMed]

, 20

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

] have reported that for sufficiently high volume fraction of scatterers in liquid phantoms, the scattering processes are not independent of each other. Zaccanti et al. [20

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

] show that deviations in μ s are larger than for μ s′, indicating changes in both μ s and the anisotropy factor. In their work, this nonlinear response was observed for volume fractions larger than 0.01. In our work, the volume fraction of scattering particles (fat globules and casein micelles) in OP 4 (undiluted milk) was calculated to be 0.042, assuming 80% of the protein to be casein micelles [21

21. P. Walstra and R. Jenness, Dairy Chemistry and Physics (Wiley, New York,1984).

]. The milk concentration for OP 4 clearly was not in this independent-scattering concentration region, which was the reason for exclusion in the phase function approximation. For OP 1–3 the measured μ s displayed a mean (over wavelength) deviation of less than 1 % from the expected μ s when assuming independent scattering events.

4.2 Calibration of the SRDR system

The proposed method involves intensity calibration on a single known optical phantom (absolute calibration method) or the calibration is done in the fitting procedure (relative calibration method), in contrast to several other calibrated SRDR methods [22–26

22. 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, 034018-1 034018-15 (2005). [CrossRef] [PubMed]

], which relies on an extensive number of phantoms. Most of those methods are also based on adaptations of analytical light transport models, which have shown to fail when dealing with small source-detector separations or large ranges in optical properties.

In the absolute calibration method, an alteration in the calibration factors will affect the absolute level of both α Gk,abs * and g * abs. However, repeated calibration measurements displayed intensity variations not greater than 5%. Tests have shown that a corresponding alteration in the intensity of the measured spectra gave a change in calculated anisotropy within the interval 0.005 to 0.01 (varying somewhat with wavelength), indicating a robust calibration process.

In the relative calibration method, the amplification factors A rel only compensated for inter-channel differences in fiber transmittance, fiber-spectroscope coupling, and detector efficiencies. The scaling factor between the measured and MC simulated SRDR intensities is compensated for by the normalization described in Eq. 3. As a measure of how well the relative calibration method performs compared to the absolute calibrated method, we calculated the scaling factor k between I meas and I MC,rel *. When comparing 〈k A relOP,D to 〈A absOP,D, the difference was -5.8%. This difference was only slightly larger than the intensity variations for the repeated calibration measurements in the absolute calibration method, suggesting that the relative calibration performed equally well as the absolute calibration method, given that [OP 1–3, D 1–3] is used.

4.3 Two-parametric phase function determination

The Gk phase function was determined utilizing seven different combinations of OP and D, data for both the absolute and relative calibrated method. In addition, the anisotropy factor was calculated based on a HG phase function approximation. Other authors [24

24. G. Zonios, L.T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M.S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo,” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

, 27

27. A. Amelink, H.J.C.M. Sterenborg, M.P.L. Bard, and S.A. Burgers, “In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

] have used Mie theory when describing light scattering in liquid optical phantoms. However, if the Mie phase function would be employed in the MC simulations used in our phase function approximation algorithm, every particle size would correspond to one dimension in I MC. In liquids containing many different scattering particle sizes, such as milk, the amount of data in this array would be huge, and the overall MC simulation time becomes unrealistic. This could be solved by using a semi-analytic Monte Carlo approach where the simulation data is rescaled in each optimization iteration [28

28. E. Tinet, S. Avrillier, and J.M. Tualle, “Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996). [CrossRef]

]. However, it is likely that multiple solutions to the inverse problems are rapidly introduced as the number of particle sizes increases. The difficulty of finding a scatterer size distribution has been observed by others [7

7. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [CrossRef] [PubMed]

].

When utilizing [OP 1–3, D 1–3] for the Gk phase function determination, the results for α Gk and g were very similar in both spectral behavior and absolute levels when comparing the absolute and relative calibrated methods. The I MC * vs. I meas rms-errors were lower using the relative calibrated method, except when utilizing [OP 3, D 1–3] for the phase function approximation. For phase function approximations utilizing only subsets of OP and D, the I MC * vs. I meas rms-errors were larger compared to the case when utilizing [OP 1–3, D 1–3]. In general, utilization of only D 1 in the phase function approximation resulted in I MC * < I meas, whereas I MC * > I meas when only D 3 was utilized for the phase function approximation. In addition, the D 3 intensity variation in I MC as a function of α Gk was very small. For example, inspection of I MC(α Gk, g = 0.80, λ = 500nm, OP 1, D) showed an intensity variation of 31 % for D 1, 4.6 % for D 2 and 3.3 % for D 3. It is, therefore, reasonable to assume that for the scattering components and the concentrations used in this study, determination of the two-parametric phase function has to be performed at small source-detector separations (0.23 mm) and not larger than 1.2 mm.

The comparison between simulated (both absolute and relative calibrated) spectra and measured spectra for the optical phantoms including absorption (OP 5–7), gave only slightly larger average rms-errors than in the OP 1-3 cases. These results are an additional indicator that the two-parametric phase function as determined by the proposed method is able to predict the light propagation also in liquid optical phantoms including absorption. The absorption level in OP 7 is close to 1 mm-1, which is, in terms of biological tissue, regarded as very high.

Acknowledgements

The study was financed by grants from the Swedish Research Council 2002–5204, 2003–4751, 2005–3934, and the Competence Centre NIMED (Non-invasive Medical Measurements).

References and links

1.

B.W. Pogue and M.S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11, 041102-1 – 041102-16 (2006). [CrossRef] [PubMed]

2.

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, 879–888 (1992). [CrossRef] [PubMed]

3.

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999). [CrossRef]

4.

P. Thueler, I. Charvet, F. Bevilacqua, M.S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, “In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties,” J. Biomed. Opt. 8, 495–503 (2003). [CrossRef] [PubMed]

5.

A. Kienle, F.K. Forster, and R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26, 1571–1573 (2001). [CrossRef]

6.

G.M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45, 1062“1071 (2006). [CrossRef] [PubMed]

7.

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [CrossRef] [PubMed]

8.

C.L. Crofcheck, F.A. Payne, and M.P. Mengüc, “Characterization of milk properties with a radiative transfer model,” Appl. Opt. 41, 2028–2037 (2002). [CrossRef] [PubMed]

9.

M. C. Ambrose Griffin and W.G. Griffin, “A simple turbidimetric method for the determination of the refractive index of large colloidal particles applied to casein micelles,” J. Colloid Interface Sci. 104, 409–415 (1985). [CrossRef]

10.

L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering.,” J. Opt. Soc. Am. 70, 1206–1212 (1980). [CrossRef]

11.

L. Wang and S.L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using collimated light transmission,” Phys. Med. Biol. 39, 2349–2354 (1994). [CrossRef] [PubMed]

12.

A. Kienle, L. Lilge, M.S. Patterson, R. Hibst, R. Steiner, and B.C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304”2314 (1996). [CrossRef] [PubMed]

13.

E. Häggblad, T. Lindbergh, M.G.D. Karlsson, M. Larsson, H. Casimir-Ahn, E.G. Salerud, and T. Strömberg, “Myocardial tissue oxygenation estimated with calibrated diffuse reflectance spectroscopy during coronary artery bypass grafting,” J. Biomed. Opt. 13, 054030-1 ” 054030-9 (2008). [CrossRef] [PubMed]

14.

H. Buiteveld, J.M.H. Hakvoort, and M. Donze, “The optical properties of pure water,” in Ocean Optics XII -Proceedings of SPIE, 174–183 (1994).

15.

G.M. Hale and M.R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef] [PubMed]

16.

I.D. Nikolov and I.D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. 39, 2067–2070 (2000). [CrossRef]

17.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304-1 – 041304-10 (2008). [CrossRef] [PubMed]

18.

A. Giusto, R. Saija, M.A. Iati, P. Denti, F. Borghese, and O.I. Sindoni, “Optical properties of high-density dispersions of particles: Application to intralipid solutions,” Appl. Opt. 42, 4375–4380 (2003). [CrossRef] [PubMed]

19.

G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, and W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994). [CrossRef] [PubMed]

20.

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

21.

P. Walstra and R. Jenness, Dairy Chemistry and Physics (Wiley, New York,1984).

22.

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, 034018-1 034018-15 (2005). [CrossRef] [PubMed]

23.

S.L. Jacques, Optical fiber reflectance spectroscopy, http://omlc.ogi.edu/news/oct03/saratov/index.htm.

24.

G. Zonios, L.T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M.S. Feld, “Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo,” Appl. Opt. 38, 6628–6637 (1999). [CrossRef]

25.

T.J. Pfefer, L.S. Matchette, C.L. Bennett, J.A. Gall, J.N. Wilke, A.J. Durkin, and M.N. Ediger, “Reflectance-based determination of optical properties in highly attenuating tissue,” J. Biomed. Opt. 8, 206–215 (2003). [CrossRef] [PubMed]

26.

J.C. Finlay and T.H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation,” Med. Phys. 31, 1949–1959 (2004). [CrossRef] [PubMed]

27.

A. Amelink, H.J.C.M. Sterenborg, M.P.L. Bard, and S.A. Burgers, “In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy,” Opt. Lett. 29, 1087–1089 (2004). [CrossRef] [PubMed]

28.

E. Tinet, S. Avrillier, and J.M. Tualle, “Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996). [CrossRef]

29.

T. Lindbergh, M. Larsson, I. Fredriksson, and T. Strömberg, “Reduced scattering coefficient determination by non-contact oblique angle illumination: Methodological considerations,” in Progress in Biomedical Optics and Imaging - Proceedings of SPIE, 64350I-1 – 64350I-12 (2007).

30.

S.A. Ramakrishna and K.D. Rao, “Estimation of light transport parameters in biological media using coherent backscattering,” Pramana J. Phys. 54, 255–267 (2000). [CrossRef]

31.

I.V. Yaroslavsky, A.N. Yaroslavsky, T. Goldbach, and H.-J. Schwarzmaier, “Inverse hybrid technique for determining the optical properties of turbid media from integrating-sphere measurements,” Appl. Opt. 35, 6797–6809 (1996). [CrossRef] [PubMed]

OCIS Codes
(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics
(290.5820) Scattering : Scattering measurements
(290.7050) Scattering : Turbid media
(300.6550) Spectroscopy : Spectroscopy, visible
(170.6935) Medical optics and biotechnology : Tissue characterization

ToC Category:
Scattering

History
Original Manuscript: November 18, 2008
Revised Manuscript: January 16, 2009
Manuscript Accepted: January 19, 2009
Published: January 27, 2009

Virtual Issues
Vol. 4, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Tobias Lindbergh, Ingemar Fredriksson, Marcus Larsson, and Tomas Strömberg, "Spectral determination of a two-parametric phase function for polydispersive scattering liquids," Opt. Express 17, 1610-1621 (2009)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-3-1610


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References

  1. B. W. Pogue and M. S. Patterson, "Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry," J. Biomed. Opt. 11, 041102-1 - 041102-16 (2006). [CrossRef] [PubMed]
  2. 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, 879-888 (1992). [CrossRef] [PubMed]
  3. F. Bevilacqua and C. Depeursinge, "Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path," J. Opt. Soc. Am. A 16, 2935-2945 (1999). [CrossRef]
  4. P. Thueler, I. Charvet, F. Bevilacqua, M. S. Ghislain, G. Ory, P. Marquet, P. Meda, B. Vermeulen, and C. Depeursinge, "In vivo endoscopic tissue diagnostics based on spectroscopic absorption, scattering, and phase function properties," J. Biomed. Opt. 8, 495-503 (2003). [CrossRef] [PubMed]
  5. A. Kienle, F. K. Forster, and R. Hibst, "Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance," Opt. Lett. 26, 1571-1573 (2001). [CrossRef]
  6. G. M. Palmer and N. Ramanujam, "Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms," Appl. Opt. 45, 1062-1071 (2006). [CrossRef] [PubMed]
  7. R. Michels, F. Foschum, and A. Kienle, "Optical properties of fat emulsions," Opt. Express 16, 5907-5925 (2008). [CrossRef] [PubMed]
  8. C. L. Crofcheck, F. A. Payne, and M. P. Mengüc, "Characterization of milk properties with a radiative transfer model," Appl. Opt. 41, 2028-2037 (2002). [CrossRef] [PubMed]
  9. M. C. Ambrose Griffin and W. G. Griffin, "A simple turbidimetric method for the determination of the refractive index of large colloidal particles applied to casein micelles," J. Colloid Interface Sci. 104, 409-415 (1985). [CrossRef]
  10. L. O. Reynolds and N. J. McCormick, "Approximate two-parameter phase function for light scattering," J. Opt. Soc. Am. 70, 1206-1212 (1980). [CrossRef]
  11. L. Wang and S. L. Jacques, "Error estimation of measuring total interaction coefficients of turbid media using collimated light transmission," Phys. Med. Biol. 39, 2349-2354 (1994). [CrossRef] [PubMed]
  12. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, "Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue," Appl. Opt. 35, 2304-2314 (1996). [CrossRef] [PubMed]
  13. E. Häggblad, T. Lindbergh, M. G. D. Karlsson, M. Larsson, H. Casimir-Ahn, E. G. Salerud, and T. Strömberg, "Myocardial tissue oxygenation estimated with calibrated diffuse reflectance spectroscopy during coronary artery bypass grafting," J. Biomed. Opt. 13, 054030-1 - 054030-9 (2008). [CrossRef] [PubMed]
  14. H. Buiteveld, J. M. H. Hakvoort, and M. Donze, "The optical properties of pure water," in Ocean Optics XII - Proc. SPIE 2258, 174-183 (1994).
  15. G. M. Hale and M. R. Querry, "Optical constants of water in the 200-nm to 200-µm wavelength region," Appl. Opt. 12, 555-563 (1973). [CrossRef] [PubMed]
  16. I. D. Nikolov and C. D. Ivanov, "Optical plastic refractive measurements in the visible and the near-infrared regions," Appl. Opt. 39, 2067-2070 (2000). [CrossRef]
  17. E. Alerstam, S. Andersson-Engels, and T. Svensson, "White Monte Carlo for time-resolved photon migration," J. Biomed. Opt. 13, 041304-1 - 041304-10 (2008). [CrossRef] [PubMed]
  18. A. Giusto, R. Saija, M. A. Iati, P. Denti, F. Borghese, and O. I. Sindoni, "Optical properties of high-density dispersions of particles: Application to intralipid solutions," Appl. Opt. 42, 4375-4380 (2003). [CrossRef] [PubMed]
  19. G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, and W. Zinth, "Time-gated transillumination of biological tissues and tissuelike phantoms," Appl. Opt. 33, 6699-6710 (1994). [CrossRef] [PubMed]
  20. G. Zaccanti, S. Del Bianco, and F. Martelli, "Measurements of optical properties of high-density media," Appl. Opt. 42, 4023-4030 (2003). [CrossRef] [PubMed]
  21. P. Walstra and R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).
  22. 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, 034018-1034018-15 (2005). [CrossRef] [PubMed]
  23. S. L. Jacques, Optical fiber reflectance spectroscopy, http://omlc.ogi.edu/news/oct03/saratov/index.htm.
  24. G. Zonios, L. T. Perelman, V. Backman, R. Manoharan, M. Fitzmaurice, J. Van Dam, and M. S. Feld, "Diffuse reflectance spectroscopy of human adenomatous colon polyps in vivo," Appl. Opt. 38, 6628-6637 (1999). [CrossRef]
  25. T. J. Pfefer, L. S. Matchette, C. L. Bennett, J. A. Gall, J. N. Wilke, A. J. Durkin, and M. N. Ediger, "Reflectance-based determination of optical properties in highly attenuating tissue," J. Biomed. Opt. 8, 206-215 (2003). [CrossRef] [PubMed]
  26. J. C. Finlay and T. H. Foster, "Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation," Med. Phys. 31, 1949-1959 (2004). [CrossRef] [PubMed]
  27. A. Amelink, H. J. C. M. Sterenborg, M. P. L. Bard, and S. A. Burgers, "In vivo measurement of the local optical properties of tissue by use of differential path-length spectroscopy," Opt. Lett. 29, 1087-1089 (2004). [CrossRef] [PubMed]
  28. E. Tinet, S. Avrillier, and J. M. Tualle, "Fast semianalytical Monte Carlo simulation for time-resolved light propagation in turbid media," J. Opt. Soc. Am. A 13, 1903-1915 (1996). [CrossRef]
  29. T. Lindbergh, M. Larsson, I. Fredriksson, and T. Strömberg, "Reduced scattering coefficient determination by non-contact oblique angle illumination: Methodological considerations," Proc. SPIE, 64350I-1 - 64350I-12 (2007).
  30. S. A. Ramakrishna and K. D. Rao, "Estimation of light transport parameters in biological media using coherent backscattering," Pramana J. Phys. 54, 255-267 (2000). [CrossRef]
  31. I. V. Yaroslavsky, A. N. Yaroslavsky, T. Goldbach, and H.-J. Schwarzmaier, "Inverse hybrid technique for determining the optical properties of turbid media from integrating-sphere measurements," Appl. Opt. 35, 6797-6809 (1996). [CrossRef] [PubMed]

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