## Dependent scattering in
^{®} |

Optics Express, Vol. 22, Issue 5, pp. 6086-6098 (2014)

http://dx.doi.org/10.1364/OE.22.006086

Acrobat PDF (1884 KB)

### Abstract

The effect of dependent scattering on the bulk scattering properties of intralipid phantoms in the 600-1850 nm wavelength range has been investigated. A set of 57 liquid optical phantoms, covering a wide range of intralipid concentrations (1-100% *v/v*), was prepared and the bulk optical properties were accurately determined. The bulk scattering coefficient as a function of the particle density could be well described with Twersky’s packing factor (R^{2} > 0.990). A general model was elaborated taking into account the wavelength dependency and the effect of the concentration of scattering particles (R^{2} = 0.999). Additionally, an empirical approach was followed to characterize the effect of dense packing of scattering particles on the anisotropy factor (R^{2} = 0.992) and the reduced scattering coefficient (R^{2} = 0.999) of the phantoms. The derived equations can be consulted in future research for the calculation of the bulk scattering properties of intralipid dilutions in the 600-1850 nm range, or for the validation of theories that describe the effects of dependent scattering on the scattering properties of intralipid-like systems.

© 2014 Optical Society of America

## 1. Introduction

*µ*, bulk scattering coefficient

_{a}*µ*, reduced scattering coefficient

_{s}*µ*, and scattering phase function

_{s}’*p(θ)*or derived anisotropy factor

*g*. In biomedical optics and spectroscopic applications, knowledge about the product’s BOP is, therefore, essential for a correct understanding of measured optical signals [1,2]. Accordingly, accurate calibration, validation and comparison of measurement setups with optical phantoms, for which the BOP’s are known accurately, is crucial. Optical phantoms with variable BOP’s can be constructed by mixing an absorbing dye, a scattering agent and a (neutral) matrix in different ratios.

*Intralipid*(IL) is a fat-in-water emulsion which is frequently used in biomedical optics as scattering component for the preparation of liquid optical phantoms [3

^{®}3. B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. **11**(4), 041102 (2006). [CrossRef] [PubMed]

4. B. Cletus, R. Künnemeyer, P. Martinsen, and V. A. McGlone, “Temperature-dependent optical properties of Intralipid measured with frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. **15**(1), 017003 (2010). [CrossRef] [PubMed]

12. B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express **21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

4. B. Cletus, R. Künnemeyer, P. Martinsen, and V. A. McGlone, “Temperature-dependent optical properties of Intralipid measured with frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. **15**(1), 017003 (2010). [CrossRef] [PubMed]

5. P. D. Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. **56**(2), N21–N28 (2011). [CrossRef] [PubMed]

13. P. I. Rowe, R. Künnemeyer, A. McGlone, S. Talele, P. Martinsen, and R. Oliver, “Thermal stability of intralipid optical phantoms,” Appl. Spectrosc. **67**(8), 993–996 (2013). [CrossRef] [PubMed]

*T*-matrix computations have been derived [16–21

21. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A **13**(11), 2266–2278 (1996). [CrossRef]

23. M. I. Mishchenko, “Directional radiometry and radiative transfer: A new paradigm,” J. Quantum Spectrosc. Radiat. Transfer **112**(13), 2079–2094 (2011). [CrossRef]

24. M. I. Mishchenko, D. H. Goldstein, J. Chowdhary, and A. Lompado, “Radiative transfer theory verified by controlled laboratory experiments,” Opt. Lett. **38**(18), 3522–3525 (2013). [CrossRef] [PubMed]

*µ*,

_{s}*µ*,

_{s}’*g*and

*p(θ)*). For this reason,

*µ*and

_{s}*µ*deviate from the linear relation with the volume concentration of scattering particles

_{s}’*Φ*for a specific scattering agent. This deviation typically increases with increasing

_{p}*Φ*. Additionally, the normalized

_{p}*p(θ)*and

*g*become dependent on

*Φ*[11

_{p}11. 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]

*W*decreases monotonously from unity to zero as

_{p}*Φ*increases from zero to unity, representing the deviation from the linear relation between

_{p}*µ*and

_{s}*Φ*[29

_{p}29. V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. **64**(6), 1710–1719 (1978). [CrossRef]

*Φ*, Twersky’s packing factor only depends on the packing dimension

_{p}*p*[29

29. V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. **64**(6), 1710–1719 (1978). [CrossRef]

31. J. M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. **37**(13), 2788–2797 (1998). [CrossRef] [PubMed]

31. J. M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. **37**(13), 2788–2797 (1998). [CrossRef] [PubMed]

35. M. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quantum Spectrosc. Radiat. Transfer **52**(1), 95–110 (1994). [CrossRef]

*T*-matrix computations to investigate the effect of the compaction state of monodisperse scattering spheres on the anisotropy factor

*g*[18

18. D. W. Mackowski and M. I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quantum Spectrosc. Radiat. Transfer **123**, 103–112 (2013). [CrossRef]

*g*with an increase of

*Φ*, especially if the particle radii were in the range of the wavelength.

_{p}6. P. Di Ninni, F. Martelli, and G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express **2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

11. 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]

14. A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. **42**(21), 4375–4380 (2003). [CrossRef] [PubMed]

*et al.*measured the reduced scattering coefficient (

*µ*) at 751 and 833 nm for IL dilutions with a

_{s}’*Φ*up to 0.032 ml/ml. The relation between

_{p}*Φ*and

_{p}*µ*could be well explained by second-order polynomials (through the origin) and showed a deviation within 2% from the linear relation (through the origin) for

_{s}’*Φ*< 0.023 ml/ml [6

_{p}6. P. Di Ninni, F. Martelli, and G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express **2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

*et al.*and Giusto

*et al.*studied the bulk scattering properties (

*µ*,

_{s}*µ*and

_{s}’*g*) at 633 nm for IL dilutions with a

*Φ*up to 0.227 ml/ml (pure IL) [11

_{p}11. 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]

14. A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. **42**(21), 4375–4380 (2003). [CrossRef] [PubMed]

*µ*and

_{s}*µ*followed a second-order polynomial (through the origin) as a function of

_{s}’*Φ*, while

_{p}*g*followed a nearly linear decrease. Unfortunately, no information is available about the effect of dependent scattering on the scattering properties of IL dilutions for wavelengths above 850 nm. However, for spectroscopic applications, the longer wavelength range is more valuable because of the presence of many important molecular overtones and combination vibrations in the NIR [36

36. S. N. Thennadil, H. Martens, and A. Kohler, “Physics-based multiplicative scatter correction approaches for improving the performance of calibration models,” Appl. Spectrosc. **60**(3), 315–321 (2006). [CrossRef] [PubMed]

37. E. Zamora-Rojas, B. Aernouts, A. Garrido-Varo, D. Pérez-Marín, J. E. Guerrero-Ginel, and W. Saeys, “Double integrating sphere measurements for estimating optical properties of pig subcutaneous adipose tissue,” Innov. Food Sci. Emerg. Technol. **19**, 218–226 (2013). [CrossRef]

*µ*and/or

_{s}*µ*of biological tissues or other turbid samples. Additionally, the effect of dependent scattering is even more dominant for longer wavelengths [6

_{s}’6. P. Di Ninni, F. Martelli, and G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express **2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

12. B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express **21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

14. A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. **42**(21), 4375–4380 (2003). [CrossRef] [PubMed]

27. A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. **72**(10), 1317–1320 (1982). [CrossRef]

31. J. M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. **37**(13), 2788–2797 (1998). [CrossRef] [PubMed]

38. R. Watté, N. N. Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. M. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express **21**(26), 32630–32642 (2013). [CrossRef] [PubMed]

*µ*,

_{s}*µ*and

_{s}’*g*) as a function of the wavelength

*λ*and the volume concentration of scattering IL particles

*Φ*.

_{p}## 2. Materials and methods

### 2.1 Liquid optical phantoms

12. B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express **21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*v/v*). Methylene Blue (M9140, Sigma-Aldrich, Missouri, USA),

*Intralipid*(batch 10FH1726, expiring date 07/2014, Fresenius Kabi, Germany) and deionized water were respectively used as absorbing, scattering and dilution agent and mixed in different ratios. Methylene Blue (MB) was chosen as absorber since it has a sharp absorption peak in the 550-750 nm range where absorption by water and IL is minimal (both

^{®}20%*µ*< 0.03 cm

_{a}^{−1}) [8

8. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**(8), 5907–5925 (2008). [CrossRef] [PubMed]

*v/v*) scattering particles (soybean oil + egg lecithin) [8

8. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**(8), 5907–5925 (2008). [CrossRef] [PubMed]

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

*v*/

*v*) or a

*Φ*of respectively 0.00227, 0.00454, 0.00908, 0.01816, 0.03632, 0.07264 and 0.14528 ml/ml. Additionally, pure IL (

_{p}*Φ*= 0.227 ml/ml) was also measured.

_{p}### 2.2 Optical characterization of phantoms

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*d*): 0.55 and 1.1 mm. Measurement of the same sample at two thicknesses should finally result in the same estimated BOP values. Therefore, it provides an interesting means to evaluate the measurement setup and the BOP estimation routine [39

39. S. A. Prahl, “Everything I think you should know about Inverse Adding-Doubling,” http://omlc.ogi.edu/software/iad/iad-3-9-10.zip.

39. S. A. Prahl, “Everything I think you should know about Inverse Adding-Doubling,” http://omlc.ogi.edu/software/iad/iad-3-9-10.zip.

*n*) of the sample has to be provided to the algorithm and was therefore calculated based on the volume concentration of scattering particles (

*Φ*) in the sample:

_{p}*n*=

_{sample}*n*+ 0.14

_{water}*Φ*[8

_{p}8. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express **16**(8), 5907–5925 (2008). [CrossRef] [PubMed]

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

40. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. **12**(3), 555–563 (1973). [CrossRef] [PubMed]

*µ*) and reduced scattering coefficient (

_{a}*µ*) could be estimated for the entire range. However, separation between

_{s}’*µ*and

_{s}*g*could not be established when the scattering depth (

*µ**

_{s}*d)*was very high. The reason for this was the presence of a significant amount of scattered photons in the measurement of the unscattered transmittance [12

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*and

_{s}*g*were obtained for

*λ*≥ 560 + 2200

*Φ*in the case of 0.55 mm sample thickness and

_{p}*λ*≥ 600 + 3000

*Φ*for 1.1 mm sample thickness (

_{p}*λ*in nm and

*Φ*in ml/ml). Moreover, because of low signal levels in the diffuse reflectance measurements for wavelengths above 1200 nm (high absorption by water) for phantoms with very low scattering levels (

_{p}*Φ*≤ 0.02 ml/ml), no accurate estimates for the anisotropy factor could be established for that specific range [12

_{p}**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*,

_{s}*g*and

*µ*).

_{s}’### 2.3 Equation fitting to the bulk scattering properties

*µ*and

_{s}*g*were available for phantoms with

*Φ*≥ 0.018 ml/ml since the unscattered transmittance measurements were incorrect. Above 1850 nm, the estimated BOP’s for low scattering levels (

_{p}*Φ*≤ 0.03 ml/ml) showed large variability. This can be explained by the low signal levels in the diffuse reflectance measurements, due to combination of the high absorption by water and the low scattering for low IL concentrations [12

_{p}**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*and

_{s}*Φ*. In order to describe the direct relation between

_{p}*µ*and

_{s}*Φ*, including dependent scattering, the Twersky equation needs to be multiplied with this linear relation. An accurate equation for the

_{p}*µ*spectrum of pure IL in the 500-2250 nm range, in absence of dependent scattering, was derived in [12

_{s}**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*in cm

_{s}^{−1}and

*λ*in nm:

*µ*and

_{s}*Φ*for independent scattering can be obtained by correcting Eq. (2) with 0.227 ml/ml (

_{p}*Φ*for pure IL) and multiplying with the actual

_{p}*Φ*. The final equation, describing

_{p}*µ*as a function of

_{s}*λ*and

*Φ*, taking into account dependent scattering by including the packing factor, can be written as follows:

_{p}*p*is unknown for IL in the Vis-NIR range, Eq. (3) was evaluated by fitting a

*p*-parameter at each wavelength. The fitted

*p*-parameters were then used in the further analysis.

*Φ*→ 0 ml/ml), independent scattering is valid and the

_{p}*g*-value only depends on the wavelength (

*λ*). In [12

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*λ*in nm:

*g*(

*λ*,

*Φ*) in the 600-1850 nm range, also valid for the case of dependent scattering (

_{p}*Φ*> 0.02 ml/ml), will be determined empirically after visual inspection of the data, starting from the formula for

_{p}*g*(

^{indep}*λ*) [Eq. (4)].

## 3. Results and discussion

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*) can be observed, which confirms the conclusion presented in [6

_{a}**2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

### 3.1 Bulk scattering coefficient

*µ*) values are presented as a function of the volume concentration of scattering particles

_{s}*Φ*for respectively 3 wavelengths below and 3 wavelengths above 1000 nm.

_{p}*µ*and

_{s}*g*could be established. Therefore,

*µ*data for short wavelengths were only available for low IL concentrations [Fig. 1(a)]. Equation (3), with a variable packing dimension

_{s}*p*, was fitted successfully (all R

^{2}> 0.996) to the

*µ*data for all 6 wavelengths [Figs. 1(a) and 1(b), solid lines]. In Fig. 1(a), the second-order polynomial derived for IL at 633 nm by Zaccanti

_{s}*et al.*[11

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

*p*- and R

^{2}-values demonstrated in respectively Figs. 2(a) and 2(b). Overall, Eq. (3) fitted very well at every wavelength (all R

^{2}> 0.990). The dip in R

^{2}around 1450 nm is probably caused by a minor water absorption bump in the bulk scattering spectra for almost all phantoms [12

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

*µ*and

_{a}*µ*(or

_{s}*µ*). The baseline in the

_{s}’*µ*values for this wavelength range introduces an underestimation of the actual

_{s}*p*-values as the Twersky equation is forced to go through the origin, resulting in lower R

^{2}values. Taking this into consideration, the packing dimension follows a nearly linear increase with increasing wavelength above 850 nm [Fig. 2(a)]. However, for the wavelengths below 850 nm, the fitted

*p*-values are highly unstable and the relation with

*λ*is far from linear. This is because no

*µ*data is available in this wavelength range for the phantoms with high IL concentrations [Fig. 1(a)]. Hence, the relation between

_{s}*µ*and

_{s}*Φ*is fairly linear for the remaining points (

_{p}*Φ*< 0.08 ml/ml) and a neighboring

_{p}*p*-value will fit almost equally good. Accordingly, it was assumed that the linear relation between

*p*and

*λ*could be extended for wavelengths below 850 nm. The fitted linear function (R

^{2}= 0.827) is given by the equation and solid line in Fig. 2(a). Substitution of this linear function and Eq. (2) in Eq. (3) provides a general equation for

*µ*, taking into account the effect of both

_{s}*λ*and

*Φ*. This general equation can be represented by a surface in a 3D space, as shown in Fig. 3. Next to the surface, also the measured

_{p}*µ*values are presented (grey dots) and a very good agreement was found between the data and the model (R

_{s}^{2}= 0.999).

29. V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. **64**(6), 1710–1719 (1978). [CrossRef]

30. P. A. Bascom and R. S. Cobbold, “On a fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. **98**(6), 3040–3049 (1995). [CrossRef] [PubMed]

*µ*and

_{s}*Φ*. This decreasing effect with decreasing wavelength was also noticed in other studies [6

_{p}**2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

**21**(26), 32450–32467 (2013). [CrossRef] [PubMed]

**42**(21), 4375–4380 (2003). [CrossRef] [PubMed]

27. A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. **72**(10), 1317–1320 (1982). [CrossRef]

**37**(13), 2788–2797 (1998). [CrossRef] [PubMed]

**37**(13), 2788–2797 (1998). [CrossRef] [PubMed]

30. P. A. Bascom and R. S. Cobbold, “On a fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. **98**(6), 3040–3049 (1995). [CrossRef] [PubMed]

32. N. E. Berger, R. J. Lucas, and V. Twersky, “Polydisperse scattering theory and comparisons with data for red blood cells,” J. Acoust. Soc. Am. **89**(3), 1394–1401 (1991). [CrossRef] [PubMed]

33. V. Twersky, “Low-frequency scattering by mixtures of correlated nonspherical particles,” J. Acoust. Soc. Am. **84**(1), 409–415 (1988). [CrossRef]

*p*at 1800-1850 nm) and 3.

### 3.2 Anisotropy factor

*k*was fitted to

*g*as a function of

*Φ*. The results for 3 wavelengths below and 3 wavelengths above 1000 nm are presented in respectively Figs. 4(a) and 4(b). It can be concluded that the fitted linear equations describe the effect of dependent scattering on

_{p}*g*well. Comparison of our results at 635 nm with the results reported by Zaccanti

*et al.*[11

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

*g*as a function of

*Φ*at 633 nm shows a good agreement. Nevertheless, one might argue that for

_{p}*Φ*going to zero, a convergence to a stable value would be more appropriate. Moreover, following these linear equations for

_{p}*Φ*above 0.227 ml/ml (pure IL),

_{p}*g*would become negative which is very rare for biological tissues or fluids [35

35. M. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quantum Spectrosc. Radiat. Transfer **52**(1), 95–110 (1994). [CrossRef]

^{2}for each fit is illustrated in Fig. 5(b). The R

^{2}values corresponding to the linear equations for wavelengths outside the 800-1400 nm range indicate a rather poor fit. This can be explained by the relatively large variation in

*g*for a specific

*Φ*compared to the variability of the average

_{p}*g*between different levels of

*Φ*, characterized by a slope close to 0. However, the linear function tends to describe the slightly decreasing trend fairly good. The slope, and with that the effect of dependent scattering on

_{p}*g*, is more negative for wavelengths in the 800-1400 nm window with a minimum around 1000 nm. It was noticed by Mishchenko [35

35. M. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quantum Spectrosc. Radiat. Transfer **52**(1), 95–110 (1994). [CrossRef]

*g*is maximum if the wavelength is between 0.5 and 20 times larger than the radii of the scattering particles. As most of the fat globules in IL have a diameter smaller than 500 nm, with a mean particle radius around 50 nm [9

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

*g*as a function of both

*λ*and

*Φ*, an equation for the slope parameter

_{p}*k*had to be derived, while the wavelength dependency of

*k*cannot be described by a simple function. From Fig. 5(a), it can be observed that the

*k*-parameter decreases with increasing wavelength until it reaches a minimum around 1000 nm. For longer wavelengths it increases again until it tends to converge to a constant value. A fifth order polynomial was fitted (R

^{2}= 0.934) to the

*k*-data and plotted as a solid line in Fig. 5(a). The combination of the equation for

*k*and Eqs. (4) and (6) provides a general equation for

*g*as a function of both

*λ*and

*Φ*. This general model is presented as a surface in Fig. 6 and compared (R

_{p}^{2}= 0.992) with the measured

*g*-data (grey dots). It should, however, be emphasized that the general equations for

*µ*and

_{s}*g*were only validated against data for which

*λ*≥ 560 + 2200

*Φ*,

_{p}*λ*≤ 1850 nm and

*Φ*≤ 0.227 ml/ml, and that extrapolation of the models is not advised.

_{p}### 3.3 Reduced scattering coefficient

*µ*and

_{s}*g*are combined with Eq. (5), a general model is obtained describing the reduced scattering coefficient (

*µ*) as a function of

_{s}’*λ*and

*Φ*. In Figs. 7(a) and 7(b), the effect of dependent scattering on the measured

_{p}*µ*values (cyan markers) for respectively 3 wavelengths below and 3 wavelengths above 1000 nm is presented. Moreover, the results of the general equation for

_{s}’*µ*at the 6 discrete wavelengths are plotted as solid lines. A very good agreement was found between the measured values and the model (all R

_{s}’^{2}> 0.993). At 635 nm, the model was in better agreement with the measured data than the results at 633 nm presented in [11

**42**(19), 4023–4030 (2003). [CrossRef] [PubMed]

*et al.*[6

**2**(8), 2265–2278 (2011). [CrossRef] [PubMed]

*µ*(surface) is plotted together with the measured

_{s}’*µ*data (grey dots). A very good correspondence (R

_{s}’^{2}= 0.999) can be observed. The model even performs well in the region for which no

*µ*and

_{s}*g*data could be extracted from

*µ*because of erroneous unscattered transmittance measurements (

_{s}’*λ*< 560 + 2200

*Φ*,

_{p}*λ*≥ 600 nm and

*Φ*≤ 0.227 ml/ml). This result strengthens the general models for

_{p}*µ*and

_{s}*g*, even in the regions where only limited data was available.

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. V. Tuchin, |

2. | A. Ishimaru, |

3. | B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. |

4. | B. Cletus, R. Künnemeyer, P. Martinsen, and V. A. McGlone, “Temperature-dependent optical properties of Intralipid measured with frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. |

5. | P. D. Ninni, F. Martelli, and G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. |

6. | P. Di Ninni, F. Martelli, and G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express |

7. | S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. |

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

9. | H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. |

10. | X. Wen, V. V. Tuchin, Q. Luo, and D. Zhu, “Controling the scattering of intralipid by using optical clearing agents,” Phys. Med. Biol. |

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

12. | B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express |

13. | P. I. Rowe, R. Künnemeyer, A. McGlone, S. Talele, P. Martinsen, and R. Oliver, “Thermal stability of intralipid optical phantoms,” Appl. Spectrosc. |

14. | A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. |

15. | G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media |

16. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

17. | M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express |

18. | D. W. Mackowski and M. I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quantum Spectrosc. Radiat. Transfer |

19. | M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. |

20. | V. P. Dick, “Applicability limits of Beer’s law for dispersion media with a high concentration of particles,” Appl. Opt. |

21. | D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A |

22. | A. Ishimaru, |

23. | M. I. Mishchenko, “Directional radiometry and radiative transfer: A new paradigm,” J. Quantum Spectrosc. Radiat. Transfer |

24. | M. I. Mishchenko, D. H. Goldstein, J. Chowdhary, and A. Lompado, “Radiative transfer theory verified by controlled laboratory experiments,” Opt. Lett. |

25. | R. West, D. Gibbs, L. Tsang, and K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A |

26. | L. Tsang, J. Kong, and T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. |

27. | A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. |

28. | V. Twersky, “Low-frequency scattering by correlated distributions of randomly oriented particles,” J. Acoust. Soc. Am. |

29. | V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. |

30. | P. A. Bascom and R. S. Cobbold, “On a fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. |

31. | J. M. Schmitt and G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. |

32. | N. E. Berger, R. J. Lucas, and V. Twersky, “Polydisperse scattering theory and comparisons with data for red blood cells,” J. Acoust. Soc. Am. |

33. | V. Twersky, “Low-frequency scattering by mixtures of correlated nonspherical particles,” J. Acoust. Soc. Am. |

34. | A. Bashkatov, E. Genina, V. I. Kochubey, and V. Tuchin, “Effects of scattering particles concentration on light propagation through turbid media,” Proc. SPIE |

35. | M. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quantum Spectrosc. Radiat. Transfer |

36. | S. N. Thennadil, H. Martens, and A. Kohler, “Physics-based multiplicative scatter correction approaches for improving the performance of calibration models,” Appl. Spectrosc. |

37. | E. Zamora-Rojas, B. Aernouts, A. Garrido-Varo, D. Pérez-Marín, J. E. Guerrero-Ginel, and W. Saeys, “Double integrating sphere measurements for estimating optical properties of pig subcutaneous adipose tissue,” Innov. Food Sci. Emerg. Technol. |

38. | R. Watté, N. N. Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. M. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express |

39. | S. A. Prahl, “Everything I think you should know about Inverse Adding-Doubling,” http://omlc.ogi.edu/software/iad/iad-3-9-10.zip. |

40. | G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. |

41. | H. C. van de Hulst, |

**OCIS Codes**

(290.5820) Scattering : Scattering measurements

(290.5850) Scattering : Scattering, particles

(290.7050) Scattering : Turbid media

(170.6935) Medical optics and biotechnology : Tissue characterization

**ToC Category:**

Scattering

**History**

Original Manuscript: January 31, 2014

Revised Manuscript: February 20, 2014

Manuscript Accepted: February 21, 2014

Published: March 6, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Ben Aernouts, Robbe Van Beers, Rodrigo Watté, Jeroen Lammertyn, and Wouter Saeys, "Dependent scattering in Intralipid® phantoms in the 600-1850 nm range," Opt. Express **22**, 6086-6098 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-6086

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### References

- V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2nd ed. (SPIE Press, 2007), p. 840.
- A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 1: Single Scattering and Transport Theory, 1st ed. (Academic Press, 1978), p. 250.
- B. W. Pogue, M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006). [CrossRef] [PubMed]
- B. Cletus, R. Künnemeyer, P. Martinsen, V. A. McGlone, “Temperature-dependent optical properties of Intralipid measured with frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. 15(1), 017003 (2010). [CrossRef] [PubMed]
- P. D. Ninni, F. Martelli, G. Zaccanti, “Intralipid: towards a diffusive reference standard for optical tissue phantoms,” Phys. Med. Biol. 56(2), N21–N28 (2011). [CrossRef] [PubMed]
- P. Di Ninni, F. Martelli, G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express 2(8), 2265–2278 (2011). [CrossRef] [PubMed]
- S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, M. J. van Gemert, “Optical properties of Intralipid: a phantom medium for light propagation studies,” Lasers Surg. Med. 12(5), 510–519 (1992). [CrossRef] [PubMed]
- R. Michels, F. Foschum, A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 5907–5925 (2008). [CrossRef] [PubMed]
- H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef] [PubMed]
- X. Wen, V. V. Tuchin, Q. Luo, D. Zhu, “Controling the scattering of intralipid by using optical clearing agents,” Phys. Med. Biol. 54(22), 6917–6930 (2009). [CrossRef] [PubMed]
- G. Zaccanti, S. Del Bianco, F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42(19), 4023–4030 (2003). [CrossRef] [PubMed]
- B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013). [CrossRef] [PubMed]
- P. I. Rowe, R. Künnemeyer, A. McGlone, S. Talele, P. Martinsen, R. Oliver, “Thermal stability of intralipid optical phantoms,” Appl. Spectrosc. 67(8), 993–996 (2013). [CrossRef] [PubMed]
- A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. 42(21), 4375–4380 (2003). [CrossRef] [PubMed]
- G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995). [CrossRef]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 3rd ed. (Cambridge University, 2006).
- M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15(6), 2822–2836 (2007). [CrossRef] [PubMed]
- D. W. Mackowski, M. I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quantum Spectrosc. Radiat. Transfer 123, 103–112 (2013). [CrossRef]
- M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85(4), 621–629 (1952). [CrossRef]
- V. P. Dick, “Applicability limits of Beer’s law for dispersion media with a high concentration of particles,” Appl. Opt. 37(21), 4998–5004 (1998). [CrossRef] [PubMed]
- D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996). [CrossRef]
- A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing, 1st ed. (Academic, 1978), p. 319.
- M. I. Mishchenko, “Directional radiometry and radiative transfer: A new paradigm,” J. Quantum Spectrosc. Radiat. Transfer 112(13), 2079–2094 (2011). [CrossRef]
- M. I. Mishchenko, D. H. Goldstein, J. Chowdhary, A. Lompado, “Radiative transfer theory verified by controlled laboratory experiments,” Opt. Lett. 38(18), 3522–3525 (2013). [CrossRef] [PubMed]
- R. West, D. Gibbs, L. Tsang, K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11(6), 1854–1858 (1994). [CrossRef]
- L. Tsang, J. Kong, T. Habashy, “Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus–Yevick approximation,” J. Acoust. Soc. Am. 71(3), 552–558 (1982). [CrossRef]
- A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72(10), 1317–1320 (1982). [CrossRef]
- V. Twersky, “Low-frequency scattering by correlated distributions of randomly oriented particles,” J. Acoust. Soc. Am. 81(5), 1609–1618 (1987). [CrossRef]
- V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64(6), 1710–1719 (1978). [CrossRef]
- P. A. Bascom, R. S. Cobbold, “On a fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. 98(6), 3040–3049 (1995). [CrossRef] [PubMed]
- J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37(13), 2788–2797 (1998). [CrossRef] [PubMed]
- N. E. Berger, R. J. Lucas, V. Twersky, “Polydisperse scattering theory and comparisons with data for red blood cells,” J. Acoust. Soc. Am. 89(3), 1394–1401 (1991). [CrossRef] [PubMed]
- V. Twersky, “Low-frequency scattering by mixtures of correlated nonspherical particles,” J. Acoust. Soc. Am. 84(1), 409–415 (1988). [CrossRef]
- A. Bashkatov, E. Genina, V. I. Kochubey, V. Tuchin, “Effects of scattering particles concentration on light propagation through turbid media,” Proc. SPIE 3917, 256–263 (2000). [CrossRef]
- M. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quantum Spectrosc. Radiat. Transfer 52(1), 95–110 (1994). [CrossRef]
- S. N. Thennadil, H. Martens, A. Kohler, “Physics-based multiplicative scatter correction approaches for improving the performance of calibration models,” Appl. Spectrosc. 60(3), 315–321 (2006). [CrossRef] [PubMed]
- E. Zamora-Rojas, B. Aernouts, A. Garrido-Varo, D. Pérez-Marín, J. E. Guerrero-Ginel, W. Saeys, “Double integrating sphere measurements for estimating optical properties of pig subcutaneous adipose tissue,” Innov. Food Sci. Emerg. Technol. 19, 218–226 (2013). [CrossRef]
- R. Watté, N. N. Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. M. Nicolaï, W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 21(26), 32630–32642 (2013). [CrossRef] [PubMed]
- S. A. Prahl, “Everything I think you should know about Inverse Adding-Doubling,” http://omlc.ogi.edu/software/iad/iad-3-9-10.zip .
- G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12(3), 555–563 (1973). [CrossRef] [PubMed]
- H. C. van de Hulst, Light Scattering by Small Particles (John Wiley, 1957), p. 470.

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