## Spectral determination of a two-parametric phase function for polydispersive scattering liquids

Optics Express, Vol. 17, Issue 3, pp. 1610-1621 (2009)

http://dx.doi.org/10.1364/OE.17.001610

Acrobat PDF (247 KB)

### 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

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]

*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]

*g*=

*g*

_{1}= 〈cos(

*θ*)〉 (first moment) and higher order of moments (

*g*,

_{n}*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]

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]

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]

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]

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]

*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.

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]

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]

*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.

*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]

*γ*range used by Thueler et al [4

**8**, 495–503 (2003). [CrossRef] [PubMed]

**8**, 495–503 (2003). [CrossRef] [PubMed]

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]

## 2. Material and methods

### 2.1 SCT setup

*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]

*μ*

_{a}+

*μ*

_{s})).

### 2.2 SRDR setup

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]

**8**, 495–503 (2003). [CrossRef] [PubMed]

*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.

### 2.3 Spectral data preprocessing

_{4}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

^{3}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], was found to have minimal or no impact on the detected intensity and was thus neglected.

### 2.5 Calibration of the SRDR system

*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-*μ*m 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]

*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.

*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

*α*

_{Gk}and

*α*

_{Gk}):

*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]

*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]

*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]

*μ*

_{s}sim/

*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

*μ*m) 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 1.0×10

^{5}detected photons.

*α*

_{Gk}(7),

*g*(11),

*λ*(31),

*OP*

_{1–4}(4), and

*D*

_{1–4}(4). Finally, the intensity array was smoothed in the

*α*

_{Gk}and

*g*dimensions using cubic polynomials to eliminate local minima caused by small stochastic variations in the simulation data. No trends in the residuals between smoothed and original data could be seen, and the maximum deviation was below 2%. The final intensity array was denoted

*I*

_{MC}. Similarly, the array containing the measured intensities was denoted

*I*

_{meas}, with the dimensions (lengths)

*λ*(31),

*OP*

_{1–4}(4), and

*D*

_{1–4}(4).

_{Gk}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*.

*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.

*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.

*OP*

_{1–3},

*D*],

_{j}*j*= 1, 2, 3) or a single optical phantom ([

*OP*,

_{i}*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*

^{*}

_{abs}combinations. For each of these six combinations,

*I*

^{*}

_{MC,abs}and

*I*

^{*}

_{MC,rel}were evaluated for all nine

*OP*and

_{i}*D*combinations and compared to the corresponding

_{j}*I*

_{meas}. The deviation between simulated and measured spectra was quantified as the root-mean-square (rms) errors

*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

*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.

*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]

*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

*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*

_{rel}〉

_{D})/(

*A*

_{abs}/(〈

*A*

_{abs}〉

_{D}) were -2.2%, -0.6%, 1.3%, and 1.0% for

*D*

_{1},

*D*

_{2},

*D*

_{3}and

*D*

_{4}, respectively.

### 3.2 Scattering coefficient

*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

*μ*

_{s,OPi}*μ*OP 1,

_{s,OP1}*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]

### 3.3 Two-parametric phase function determination

*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.

*g*

^{*}

_{rel}and

*g*

^{*}

_{abs}was in the interval -1.2% to -0.8%.

*OP*

_{1–3},

*D*

_{j}],

*j*= 1, 2, 3, and [

*OP*,

_{i}*D*

_{1–3}],

*i*= 1, 2, 3, are presented in table 1.

*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.

*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 (

*OP*,

_{i}*i*= 1, 2, 3) or the source-detector distances.

### 3.4 Optical phantoms including absorption

*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}.

*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

*OP*,

_{i}*i*= 5, 6, 7). The only rms-errors above 10% were for

*OP*

_{7}and the absolute calibrated cases.

## 4. Discussion and conclusions

*μ*

_{s}, is utilized. The inverse algorithm employs spectral scattering coefficients that are separately determined using collimated transmission measurements.

## 4.1 Spectral scattering coefficient

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

*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]. 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

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]

*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.

*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*

_{rel}〉

_{OP,D}to 〈

*A*

_{abs}〉

_{OP,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

*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.

*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.

*μ*

_{s}values derived from SCT, was used to simulate a multi-dimensional array of SRDR spectra. The two phase function parameters were estimated from minimizing the rms-error between simulated and measured SRDR spectra using a non-linear search algorithm. Intensity calibration of the SRDR measurements was done either by a separate measurement on a monodispersive polystyrene sphere aqueous suspension (compensating for measured-simulated amplification differences) or by uniformly illuminating the fiber probe (only compensating for between-fiber amplification differences). An additional overall amplification factor, calculated as the average ratio between fitted and measured spectra, was added to the relative calibration case when comparing intensities. Both the absolute and the relative calibrations yielded similar phase function estimations when measuring on several non-absorbing phantoms containing different concentration levels of the same scattering substance. The deviation between fitted and measured spectra was low, even when adding ink to the samples, which strongly supports that the estimated phase functions are accurate.

## Acknowledgements

## 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. |

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. |

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 |

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. |

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. |

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. |

7. | R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express |

8. | C.L. Crofcheck, F.A. Payne, and M.P. Mengüc, “Characterization of milk properties with a radiative transfer model,” Appl. Opt. |

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. |

10. | L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering.,” J. Opt. Soc. Am. |

11. | L. Wang and S.L. Jacques, “Error estimation of measuring total interaction coefficients of turbid media using collimated light transmission,” Phys. Med. Biol. |

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. |

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. |

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- |

16. | I.D. Nikolov and I.D. Ivanov, “Optical plastic refractive measurements in the visible and the near-infrared regions,” Appl. Opt. |

17. | E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. |

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. |

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. |

20. | G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. |

21. | P. Walstra and R. Jenness, |

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. |

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. |

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. |

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. |

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. |

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 |

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. |

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. |

**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

- 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]
- 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]
- 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]
- 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. 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]
- 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]
- R. Michels, F. Foschum, and A. Kienle, "Optical properties of fat emulsions," Opt. Express 16, 5907-5925 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- L. O. Reynolds and N. J. McCormick, "Approximate two-parameter phase function for light scattering," J. Opt. Soc. Am. 70, 1206-1212 (1980). [CrossRef]
- 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. 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- G. Zaccanti, S. Del Bianco, and F. Martelli, "Measurements of optical properties of high-density media," Appl. Opt. 42, 4023-4030 (2003). [CrossRef] [PubMed]
- P. Walstra and R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).
- 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]
- S. L. Jacques, Optical fiber reflectance spectroscopy, http://omlc.ogi.edu/news/oct03/saratov/index.htm.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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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