## Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method

Optics Express, Vol. 15, Issue 2, pp. 486-500 (2007)

http://dx.doi.org/10.1364/OE.15.000486

Acrobat PDF (235 KB)

### Abstract

In spite of many progresses achieved both with theories and with experiments in studying light propagation through diffusive media, a reliable method for accurate measurements of the optical properties of diffusive media at NIR wavelengths is, in our opinion, still missing. It is therefore difficult to create a diffusive medium with well known optical properties to be used as a reference. In this paper we describe a method to calibrate the reduced scattering coefficient, *μ*́*
_{s}
*, of a liquid diffusive medium and the absorption coefficient,

*μ*, of an absorbing medium with a standard error smaller than 2% both on

_{a}*μ*́

*and on*

_{s}*μ*. The method is based on multidistance measurements of fluence into an infinite medium illuminated by a CW source. The optical properties are retrieved with simple inversion procedures (linear fits) exploiting the knowledge of the absorption coefficient of the liquid into which the diffuser and the absorber are dispersed. In this study Intralipid diluted in water has been used as diffusive medium and Indian ink as absorber. For a full characterization of these media measurements of collimated transmittance have also been carried out, from which the asymmetry factor of the scattering function of Intralipid and the single scattering albedo of Indian ink have been determined.

_{a}© 2007 Optical Society of America

## 1. Introduction

*μ*, and the reduced scattering coefficient,

_{a}*μ*́

*) remains a difficult task. Many methodologies based on measurements both in the time, frequency, and CW domain, have been presented that, in principle, can give accurate measurements of*

_{s}*μ*and

_{a}*μ*́

*, and many phantoms have been proposed to assess the performance of NIR instrumentation (see Ref. [1] for a review). However, looking at recently published papers it seems difficult to find measurements with accuracy better than 10% even when measurements are carried out on homogeneous media in the most favorable experimental condition.*

_{s}2. A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. **44**,2104–2114 (2005). [CrossRef] [PubMed]

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

4. H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express **14**,6485–6501 (2006),http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]

5. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm,” Opt. Express **14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

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

4. H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express **14**,6485–6501 (2006),http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]

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

*μ*́

*of Liposyn was obtained with a standard deviation of 16% and absorption of ink with a standard deviation of 8%. In [4*

_{s}4. H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express **14**,6485–6501 (2006),http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]

*μ*́

*was 8%, while the absorption of Intralipid was assumed equal to that of water and the absorption coefficient of Indian ink equal to the extinction coefficient obtained from measurements of transmittance. In [5*

_{s}5. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm,” Opt. Express **14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

## 2. The proposed method

*μ*and

_{a}*μ*́

*is easily obtained diluting and mixing them in known concentrations. In this section we describe the procedures to calibrate a diffusive and an absorbing medium and we discuss the intrinsic approximations of the proposed method.*

_{s}### 2.1. Calibration of the diffusive medium

*μ*, obtained from multidistance measurements of fluence rate, Φ(

_{eff}*r*), inside an infinite medium illuminated by a CW source. The optical properties are retrieved making use of the solution of the diffusion equation (DE). For the infinite medium illuminated by a CW pointlike isotropic source emitting a unitary power the fluence at distance

*r*from the source is given by [6

6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. **45**,1359–1373 (2000). [CrossRef] [PubMed]

*μ*=(3

_{eff}*μ*

_{a}*μ*́

*)*

_{s}^{1/2}is the effective attenuation coefficient and

*μ*and

_{a}*μ*́

*are the absorption and the reduced scattering coefficients of the medium. From Eq. 1 results*

_{s}*μ*as the absolute value of the slope of ln[

_{eff}*r*Φ(

*r*)] as a function of

*r*. A single measure of

*μ*is not sufficient to reconstruct

_{eff}*μ*and

_{a}*μ*́

*. The reconstruction becomes possible if measurements can be repeated after controlled changes of the optical properties of the diffusive medium. This is feasible when measurements are carried out on liquid diffusive media in which the optical properties can be easily changed in a well known way varying the concentration of scatterers or of absorber. If an absorber with calibrated absorption coefficient is available, the so called method of adding absorption can be used [7*

_{s}7. B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. **1**,235–244 (1986). [CrossRef]

8. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. **31**,4507–1514 (1991). [CrossRef]

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

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

^{−1}over a wide range of values for

*μ*́

*(0.5 <*

_{s}*μ*́

*< 18mm*

_{s}^{−1}).

10. J. C. Hebden, R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I.Experimental techniques,” Phys. Med. Biol. **42**,825–840 (1997). [CrossRef] [PubMed]

11. R. Cubeddu, C. D andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dovar, D. Johnson, M. Ruiz-Altisent, and C. Valero, “Nondestructive quantification of chemical and physical properties of fruits by time-resolved reflectance spectroscopy in the wavelength range 650-1000 nm,” Appl. Opt. **40**,538–543 (2001). [CrossRef]

*μ*as a function of the volume concentration of the diffusive medium. The diffuser we used is the Intralipid-20% (Fresenius Kabi, Uppsala, Sweden), a highly scattering medium with small absorption. The optical properties are retrieved exploiting the knowledge of the absorption coefficient of the liquid (water) into which the diffusive medium is dispersed. In fact, if we denote with

_{eff}*ε*

_{ail}and

*ε*́

_{sil}the intrinsic absorption and the reduced scattering coefficient of Intralipid-20%, and with

*ε*

_{aH2O}the absorption coefficient of pure water, the optical properties of the aqueous dilution result

*ρ*

_{il}is the volume concentration of Intralipid, i.e. the ratio between the volume of Intralipid-20% and the total volume of Intralipid and water. If

*ε*

_{aH2O}is known, by using Eq. 5 the absorption and reduced scattering coefficient of the non diluted diffusive medium can be obtained from the coefficients of a non-linear fit with a second degree polynomial. Alternately, Eq. 5 can be linearized dividing by

*ρ*

_{il}

*ε*and

_{a}*ε*́

*can be obtained with a more simple linear fit as*

_{s}*I*

_{il}and

*S*

_{il}are respectively the intercept and the slope of the straight line that best fits

*μ*

^{2}

*(*

_{eff}*ρ*

_{il})/

*ρ*

_{il}as a function of

*ρ*

_{il}.

*ε*

_{aH2O}with high accuracy. Absorption spectra of pure water have been reported in many papers (see the web page of the Oregon Medical Laser Center for a useful compendium [12

12. S. Prahl, “Optical absorption of water,” http://omlc.ogi.edu/spectra/water/index.html.

12. S. Prahl, “Optical absorption of water,” http://omlc.ogi.edu/spectra/water/index.html.

^{−1}(a significantly lower value, 0.000219 mm

^{−1}is also present [13]) and a discrepancy of ≃ 5% remains (0.00286 and 0.00273 mm

^{−1}) even between the two more recently published data [14

14. H. Buiteveld, J. M. H. Hakvoort, and M. Donze, “The optical properties of pure water,” in *Ocean Optics XII*,
J. S. Jaffe ed., Proc. SPIE **2258**,174–183 (1994). [CrossRef]

15. L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65-2.5 μm spectral range,” Appl. Opt. **32**,3531–3540 (1993). [CrossRef] [PubMed]

9. 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*. The knowledge of the absorption, reduced scattering, and extinction coefficients enabled us to determine the asymmetry factor,

_{ext}*g*, of the medium.

### 2.2. Calibration of the absorbing medium

*μ*on a diffusive medium with calibrated reduced scattering coefficient as a function of the volume concentration,

_{eff}*ρ*

_{ink}, of Indian ink. If we denote with

*ε*ink the absorption coefficient of the non diluted Indian ink and we assume that the added absorber does not significantly alter the reduced scattering coefficient of the medium we obtain

_{a}*μ*

_{a0}and

*μ*́

_{s0}are the absorption and the reduced scattering coefficient of the diffusive medium before the addition of ink. If

*μ*́

_{s0}is known, it is possible to retrieve

*ε*ink with a linear fit as

_{a}*S*

_{ink}is the slope of the straight line that best fits

*μ*

^{2}

*(*

_{eff}*ρ*ink) as a function of

*ρ*ink. The intercept,

*I*

_{ink}, can be used to retrieve

*μ*

_{a0}as

_{ink}.

### 2.3. Accuracy of the proposed method: Discussion of the intrinsic approximations

*μ*, comes from the solution of the DE that is an approximate solution of the more rigorous radiative transfer equation. Furthermore, the solution refers to a pointlike source and a pointlike receiver inside an infinite medium, whereas measurements are carried out with source and receiver with finite dimension inside a finite medium. Also Eq. 6, used to retrieve

_{eff}*ε*

_{ail}and

*ε*́

_{sil}, is subjected to restrictions, since it comes from Eqs. 3 and 4 that are based on the independent scattering approximation. It is therefore necessary to investigate 1) the range of applicability of the DE, 2) the effect of the finite size of source and receiver, 3) the effect of the perturbation on light propagation due to the presence of the source and of the receiver in the medium, 4) the effect of the finite medium, and 5) the validity of the independent scattering approximation.

#### Applicability of the diffusion equation.

6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. **45**,1359–1373 (2000). [CrossRef] [PubMed]

*r*> 2/

*μ*́

*.*

_{s}#### Effect of the finite size of source and receiver.

16. S. Fantini, M. A. Franceschini, and E. Gratton, “Effective source term in the diffusion equation for photon transport in turbid media,” Appl. Opt. **36**,156–163 (1997). [CrossRef] [PubMed]

*F*= sinh(

_{S}*μ*

_{eff}*a*)/(

*μ*

_{eff}*a*) where

*a*is the radius of the sphere. It has been also shown, integrating Eq. 1 over a spherical receiver [17

17. M. Bassani ,*Limits of validity of the diffusion equation and methodologies for measuring optical properties of
highly scattering media*, (in Italian) M. S. Thesis , University of Florence, Italy (1997). [PubMed]

*F*=

_{R}*F*. For the values of

_{S}*a*and

*μ*

_{eff}*a*used in our experiments the correction factors remain between 1 and 1.01, but the most important point is that they do not depend on

*r*, so that the slope of ln[

*r*Φ(

*r*)] as a function of

*r*remains unchanged. Therefore, the finite size of the source and of the receiver does not perturb the measure of

*μ*.

_{eff}#### Perturbation introduced by the source and by the receiver.

*r*

_{1}and

*r*

_{2}are the distances of the volume element of the inhomogeneity, d

*V*, from the source and from the receiver respectively. To evaluate the perturbation due to the diffusive sphere that illuminates the medium we integrated Eq. 12 making the assumptions

_{i}*r*

_{1}+

*r*

_{2}≅

*r*

_{2}≅

*r*With these assumptions we obtain

*r*

_{2}>>

*r*

_{1}two volumes elements symmetric with respect to the center of the source bring perturbations of equal intensity but with opposite sign. We therefore expect a negligible scattering perturbation from the source fibre. An identical consideration can be applied for the receiving fibre.

*μ*́

*independent of the source-receiver distance that does not affect the measure of*

_{s}*μ*. This conclusion has been supported by the results of an experimental investigation: We repeated multidistance measurements of fluence with and without a third fiber, identical to the others, in contact with the source fibre. As expected, we observed that the third fibre reduces the measured signal by a small factor, proportional to

_{eff}*μ*́

*, independent of the source-receiver distance. The reduction was of 1% for*

_{s}*μ*́

*= 0.4mm*

_{s}^{−1}.

#### Effect of the finite medium.

*V*, 1 ≤

*V*≤ 3L. The source and the receiver were centered inside the cylinder and measurements have been carried out for 10 ≤

*r*≤ 35mm. The perturbation due to the finite volume decreases when the reduced scattering coefficient or the absorption coefficient of the medium increases. At λ = 750nm, the wavelength used in our experiments at which the absorption coefficient of water is 0.00278 mm

^{−1}, we did not observe any difference between measurements with

*V*= 1.5 and 3L when

*μ*́

*≥ 0.3mm*

_{s}^{−1}. With

*μ*́

*≥ 0.8mm*

_{s}^{−1}differences have not been observed even when

*V*= 1L. At NIR wavelengths, where the absorption coefficient of water is ≳ 0.002mm

^{−1}, a quite small volume is therefore sufficient to mimic an infinite medium.

#### Validity of the independent scattering approximation.

*ρ*́

_{il}< 0.025 (

*ρ*

_{il}< 0.11), deviations from the linearity remain within 4%. We have simulated the effect of these deviations on the inversion procedure to reconstruct the intrinsic absorption and the reduced scattering coefficient,

*ε*

_{ail}and

*ε*́

_{sil}. For the values of

*ρ*

_{il}we used, the error introduced is insignificant (< 0.01%) for

*ε*́

_{sil}, whereas it leads to an underestimate ≅ 0.001mm

^{−1}for

*ε*

_{ail}.

## 3. Experimental setup

*μ*is basically the same described in Ref. [9

_{eff}**42**,4023–4030 (2003). [CrossRef] [PubMed]

*λ*= 750nm. As described in Sect. 2 two identical plastic fibers with a 3 mm diameter diffusive sphere have been used to illuminate the medium and to collect received photons. The center-to-center distance

*r*has been varied with a computer controlled translation stage between 10 ≤

*r*≤ 35mm (1 mm step). The error on the relative displacement was very small (0.002 mm), but the uncertainty in positioning the fibres can lead to a systematic error of 0.1 mm on

*r*. The fluence was measured with a photomultiplier and a lock-in amplifier with a standard error ≲ 1% . The volume concentration has been obtained with an error smaller than 0.1% measuring the weight of water (with an error of 1 g) and of Intralipid (with an error of 1 mg) and using for the specific weight of constituents the values reported in Ref. [8

8. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. **31**,4507–1514 (1991). [CrossRef]

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

^{2}. The extinction coefficient has been obtained measuring the transmittance through a scattering cell 30mm thick as a function of the concentration of the diffusing or absorbing medium.

^{2}) immersed in water we measured the received power

*P*

_{0}as a function of the depth. The photodiode was tilted in order to avoid multiple reflections and attention was paid to avoid air bubbles. An accurate calibration has been obtained from the slope of the line that best fits ln

*P*

_{0}as a function of the depth.

## 4. Experimental results

### 4.1. Measurements of effective attenuation coefficient

*V*≃ 3L and

*μ*́

*≳ 0.3mm*

_{s}^{−1}. Examples of multidistance measurements are reported in Fig. 1. The figure reports ln[

*r*Φ(

*r*)] for three values of Intralipid concentration:

*ρ*

_{il}= 0.0204, 0.0401, and 0.0857. The standard error on ln[

*r*Φ(

*r*)] is almost independent both of

*r*and of

*ρ*

_{il}. The error bars have not been shown since smaller than the marks. The lines that best fit the results are also reported. The corresponding values of

*μ*are 0.0585±0.0001, 0.08197±0.00007, and 0.11938±0.00005 mm

_{eff}^{−1}. The standard deviation has been evaluated only considering the statistical errors (the systematic error on

*r*has been disregarded).

### 4.2. Absorption coefficient of water

*ε*

_{aH2O}we measured, with a photodiode immersed in water, the power

*P*

_{0}received as a function of the depth. The absolute value of the slope of the line that best fits ln

*P*

_{0}as a function of the depth represents the absorption coefficient of water (at NIR wavelengths the scattering of pure water is negligible with respect to absorption). An example of results is shown in Fig. 2. The error bars are smaller than the marks. Almost identical results have been obtained from measurements repeated on different samples of water or on the same sample but at different times (after 24 hours). The value we obtained at λ = 750nm is

*ε*

_{aH2O}= (2.78 ± 0.01)10

^{−3}mm

^{−1}.

### 4.3. Calibration of Intralipid

*ε*

_{ail}and

*ε*

_{sil}

^{́}, is obtained from the slope and the intercept of the straight line that best fits

*μ*

^{2}

*(*

_{eff}*ρ*

_{il})/

*ρ*

_{il}as a function of

*ρ*

_{il}. An example of experimental results is reported in Fig. 3. The straight line that best fits the results, for which

*I*

_{il}= 0.1692±0.0004mm

^{−2}and

*S*

_{il}= −0.0325±0.005mm

^{−2}, is also shown. The corresponding values of

*ε*

_{ail}and

*ε*́

_{sil}are:

*ε*

_{ail}=(2.25±0.09)10

^{−3}mm

^{−1}and

*ε*́

_{sil}=20.3±0.1mm

^{−1}. The results of the fit slightly change (

*ε*

_{ail}= (2.17±0.09)10

^{−3}mm

^{−1}and

*ε*́

_{sil}= 20.3±0.1mm

^{−1}) if the point at

*ρ*

_{il}= 0.0204 is excluded.

*ε*

_{ail}and

*ε*́

_{sil}have been evaluated considering only the statistical errors on

*I*

_{il},

*S*

_{il}, and

*ε*

_{aH2O}. To evaluate the error due to the uncertainty in positioning the fibres, measurements have been analyzed after introducing a shift of ±0.1 mm on the values of

*r*. The shift gives an error≃±1% on

*ε*́

_{sil}and ≃∓7% on

*ε*

_{ail}. If we include this source of error we obtain for the calibration of Intralipid-20%:

*ε*́

_{sil}= 20.3±0.3mm

^{−1}and

*ε*

_{ail}= (2.25±0.26)10

^{−3}mm

^{−1}.

*ε*́

_{sil}, while the value of

*ε*

_{ail}may be underestimated of ≅0.001mm

^{−1}.

### 4.4. Calibration of Indian ink

*μ*́

_{s0}. Absorption was varied adding small quantities of ink previously diluted (0.722 g of ink in 231 g of water). An example of experimental results is reported in Fig. 4. The figure reports

*μ*

^{2}

*(*

_{eff}*ρ*

_{ink}) as a function of

*ρ*

_{ink}for measurements carried out with

*ρ*

_{il}= 0.0857 corresponding to

*μ*́

_{s0}= 1.74±0.03mm

^{−1}and

*μ*

_{a0}= (2.73 ± 0.02)10

^{−3}mm

^{−1}. Also in this case the straight line that best fits the results, for which

*I*

_{ink}= (1.427 ± 0.001)10

^{−2}mm

^{−2}and

*S*

_{ink}= 3412±7mm

^{−2}, is reported. From

*S*

_{ink}and

*I*

_{ink}we obtained the following values for the absorption coefficient of the non diluted Indian ink and the absorption coefficient of the diffusive medium:

*ε*

_{aink}= 654±11mm

^{−1}and

*μ*

_{a0}= (2.73 ± 0.04)10

^{−3}mm

^{−1}. The errors are mainly due to the error on

*μ*́

_{s0}. We notice that the value of

*μ*

_{a0}retrieved by the fit is, within the standard deviation, equal to the value expected from the calibration of Intralipid.

### 4.5. Extinction coefficient of Intralipid and Indian ink

*ε*

_{extil}= 69.0±0.2mm

^{−1}for Intralipid-20%, and

*ε*

_{extink}= 747±4mm

^{−1}for non diluted Indian ink. From these values we obtained for the asymmetry factor of Intralipid

*g*= 0.706±0.006, and for the single scattering albedo of Indian ink Λ

_{ink}= 0.125±0.02.

## 5. Discussion and conclusions

### 5.1. Summary of the results

*ε*

_{ail}may be systematically underestimated of ≅ 0.001mm

^{−1}.

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

*r*. Measurements at larger distances would be also useful to better fulfill the independent scattering approximation, since lower concentrations of diffusive medium would be necessary to meet the condition

*r*>2/

*μ*́

*for the validity of the diffusion approximation. The disadvantage of measurements at larger distances is the larger volume of diffusive medium necessary to mimic an infinite medium.*

_{s}*ε*

_{ail}= 0.00225mm

^{−1}, is in good agreement with the value 0.00238mm

^{−1}expected from absorption of water, 0.00278mm

^{−1}, and lipids, 0.001mm

^{−1}[18] (value also reported in the web page of the Oregon Medical Laser Center http://omlc.ogi.edu/spectra/). In spite of the large error, the important information we obtained is that at λ = 750 nm

*ε*

_{ail}is not significantly different from absorption of water. Therefore, even though the error on

*ε*

_{ail}is quite large (≃ 50%), the error on the absorption of a dilution remains ≲ 2% for volume concentrations

*ρ*

_{il}≲0.05 necessary to obtain values of

*μ*́

*≲ 1mm*

_{s}^{−1}as commonly used for phantoms with optical properties similar to biological tissue.

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

*ρ*

_{il}. Errors are ≲ 1% for

*ρ*

_{il}< 0.035, corresponding to

*μ*́

*≃ 0.7mm*

_{s}^{−1}, but may become 4% for

*ρ*

_{il}= 0.1 corresponding to

*μ*́

*≃ 2mm*

_{s}^{−1}. We notice that the failure of the independent scattering approximation for high concentrations of scatterers is often disregarded. Comparisons between experimental results on the optical properties of Intralipid are commonly made without paying attention to the different concentrations at which measurements have been carried out. As an example, in Refs. [19

19. S. Jacques, “Optical properties of Intralipid, an aqueous suspension of lipid droplets,” http://omlc.ogi.edu/spectra/intralipid/index.html

5. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm,” Opt. Express **14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

8. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. **31**,4507–1514 (1991). [CrossRef]

### 5.2. Validation of the experimental results

*ε*́

_{sil}and for

*ε*

_{aink}the accuracy was similar to that obtained from CW measurements. The methodology is fully described in Ref. [20]. The comparison showed an excellent agreement: Discrepancies were within the standard deviation

### 5.3. Comparison with published data

**31**,4507–1514 (1991). [CrossRef]

19. S. Jacques, “Optical properties of Intralipid, an aqueous suspension of lipid droplets,” http://omlc.ogi.edu/spectra/intralipid/index.html

*et al*. [8

**31**,4507–1514 (1991). [CrossRef]

**31**,4507–1514 (1991). [CrossRef]

*ε*

_{extil}= 63.62mm

^{−1},

*g*= 0.664, and

*ε*

_{́sil}= 21.35mm

^{−1}. For the absorption coefficient it is difficult to find data at the same wavelength: At 750 nm we only know the result reported by Chen

*et al*. [5

**14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

**14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

_{ink}. Comparisons with other experimental data have not been made since we did not find any results at the wavelength we used. The value we obtained, Λ

_{ink}= 0.125±0.02, is very close to the value, 0.115, expected from Mie theory for small particles (size 0.1

*μ*m) of carbon suspended in water [21

21. S. J. Madsen, M. S. Patterson, and B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. **37**,985–993 (1992). [CrossRef] [PubMed]

### 5.4. Comparison with other CW methodologies

*et al*. [3

**50**,2291–2311 (2005). [CrossRef] [PubMed]

**14**,6485–6501 (2006),http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]

*et al*. used suspensions of Liposyn and Indian ink to test the performance of an interstitial setup for quick measurements of both absorption and reduced scattering coefficient of tissue

*in vivo*. The setup is based on absolute multidistance measurements of fluence in the geometry of infinite medium. The detector has been calibrated using an integrating sphere. The reduced scattering coefficient of Liposyn was reconstructed with a standard deviation of 16% and absorption of ink with a standard deviation of 8%.

**14**,6485–6501 (2006),http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]

*μ*, but to invert the results they made the assumption that absorption of Intralipid is equal to absorption of water and that Λ

_{eff}_{ink}=0, i.e., they ignored the scattering component of Indian ink. Furthermore, they used for absorption of water the value

*ε*

_{aH2O}= 0.00261mm

^{−1}taken from published data. They obtained

*μ*́

_{sil}(

*ρ*

_{il}=0.05) = 0.97mm

^{−1}with an error of 8%, a value in good agreement with our result.

*et al*. [5

**14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

### 5.5. Advantages and disadvantages of the proposed method

6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. **45**,1359–1373 (2000). [CrossRef] [PubMed]

*l*(

*r*)⟩ =

*r*√3

*μ*́

*/4*

_{s}*μa*. As an example, for an aqueous diffusive medium with

*μ*́

*=1mm*

_{s}^{−1},

*μ*= 0.0025mm

_{a}^{−1}, and

*r*= 35mm results ⟨

*l*(

*r*)⟩ = 606mm, corresponding to a mean time of about 2.7 ns. Photons with a so long mean time inside the diffusive medium are difficult to detect with time resolved measurements (the temporal range is usually limited to ≃ 10ns by the pulse repetition rate, typically ≃ 100 MHz) or even with CW measurements carried out with different geometry. As an example, the method described by Cheng

*et al*. [5

**14**,
7420–7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]

*μ*́

*= 15mm*

_{s}^{−1}, we obtain from the DE [22

22. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I) Theory,” Appl. Opt. **36**,4587–4599 (1997). [CrossRef] [PubMed]

*l*⟩ ≃ 0.5mm for diffusely transmitted photons and ⟨

*l*⟩ ≃ 0.35mm for diffusely reflected photons. Due to the small mean pathlength the effect of absorption is negligible: Only 0.1% of the emitted photons is absorbed when μ

*= 0.0025mm*

_{a}^{−1}, a value close to that expected for Intralipid-20%.

*r*> 2/

*μ*́

*it is almost coincident with the solution of the radiative transfer equation. Furthermore, the simplicity of the solution leads to simple inversion procedures: The optical properties are obtained using only linear fits. On the contrary, as an example, the lack of an accurate analytical forward model makes the inversion difficult of CW measurements on thin slabs of diffusive media: As mentioned in Sect. 5.4 complicated numerical Monte Carlo simulations are necessary to invert measurements of total reflectance and transmittance. To invert measurements of diffuse transmittance or diffuse reflectance an analytical forward model exists, but more approximated and complicated with respect to that for the infinite medium, since solutions of the DE for finite geometry are usually obtained with approximate boundary conditions. As an example, for time resolved measurements the solution of the DE commonly used as forward model to invert experimental results is inaccurate at short times even at large source-receiver distances. Furthermore, the complexity of the solution makes necessary the use of non-linear fits. The fit is further complicated because measurements are commonly in arbitrary units and they may be affected by a systematic error on the time scale. Therefore, in addition to*

_{s}*μ*and

_{a}*μ*́

*it is often necessary to fit an amplitude factor and a temporal shift. The strong correlation between the fitted parameters makes the inversion procedure unstable and sensitive to the initial guess. The inversion is also complicated by the temporal instrument response function.*

_{s}### 5.6. Conclusions

23. L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, “Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue,” Rev. Sci. Instrum. **75**,5315–5325 (2004). [CrossRef]

2. A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. **44**,2104–2114 (2005). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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

2. | A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. |

3. | A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. |

4. | H. Xu and M. S. Patterson, “Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference,” Opt. Express |

5. | C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm,” Opt. Express |

6. | F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. |

7. | B. C. Wilson, M. S. Patterson, and D. M. Burns, “Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light,” Lasers Med. Sci. |

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

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

10. | J. C. Hebden, R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I.Experimental techniques,” Phys. Med. Biol. |

11. | R. Cubeddu, C. D andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dovar, D. Johnson, M. Ruiz-Altisent, and C. Valero, “Nondestructive quantification of chemical and physical properties of fruits by time-resolved reflectance spectroscopy in the wavelength range 650-1000 nm,” Appl. Opt. |

12. | S. Prahl, “Optical absorption of water,” http://omlc.ogi.edu/spectra/water/index.html. |

13. | R. M. Pope, “Optical absorption of pure water and sea water using the integrating cavity absorption meter,” Phd Thesis, 1993, Texas A&M University. |

14. | H. Buiteveld, J. M. H. Hakvoort, and M. Donze, “The optical properties of pure water,” in |

15. | L. Kou, D. Labrie, and P. Chylek, “Refractive indices of water and ice in the 0.65-2.5 μm spectral range,” Appl. Opt. |

16. | S. Fantini, M. A. Franceschini, and E. Gratton, “Effective source term in the diffusion equation for photon transport in turbid media,” Appl. Opt. |

17. | M. Bassani , |

18. | R.L.P. van Veen, H.J.C.M. Sterenborg, A. Pifferi, A. Torricelli, and R. Cubeddu, “Determination of VIS- NIR absorption coefficients of mammalian fat, with time-and spatially resolved diffuse reflectance and transmission spectroscopy,” OSA Annual BIOMED Topical Meeting, 2004. |

19. | S. Jacques, “Optical properties of Intralipid, an aqueous suspension of lipid droplets,” http://omlc.ogi.edu/spectra/intralipid/index.html |

20. | L. Spinelli, F. Martelli, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time resolved method,” In preparation. |

21. | S. J. Madsen, M. S. Patterson, and B. C. Wilson, “The use of India ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. |

22. | D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I) Theory,” Appl. Opt. |

23. | L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, “Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue,” Rev. Sci. Instrum. |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.5280) Medical optics and biotechnology : Photon migration

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: December 12, 2006

Revised Manuscript: January 11, 2007

Manuscript Accepted: January 13, 2007

Published: January 22, 2007

**Virtual Issues**

Vol. 2, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Fabrizio Martelli and Giovanni Zaccanti, "Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. CW method," Opt. Express **15**, 486-500 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-486

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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 (2006).
- A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, "Performance assessment of photon migration instruments: the MEDPHOT protocol," Appl. Opt. 44,2104-2114 (2005). [CrossRef] [PubMed]
- A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005). [CrossRef] [PubMed]
- H. Xu and M. S. Patterson, "Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference," Opt. Express 14,6485-6501 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485. [CrossRef] [PubMed]
- C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, "A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm, " Opt. Express 14,7420-7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420. [CrossRef] [PubMed]
- F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000). [CrossRef] [PubMed]
- B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986). [CrossRef]
- H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991). [CrossRef]
- G. Zaccanti, S. Del Bianco, and F. Martelli, "Measurements of optical properties of high density media," Appl. Opt. 42,4023-4030 (2003). [CrossRef] [PubMed]
- J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997). [CrossRef] [PubMed]
- R. Cubeddu, C. D’andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dovar, D. Johnson, M. Ruiz-Altisent, and C. Valero, "Nondestructive quantification of chemical and physical properties of fruits by time-resolved reflectance spectroscopy in the wavelength range 650-1000 nm," Appl. Opt. 40,538-543 (2001). [CrossRef]
- S. Prahl, "Optical absorption of water," http://omlc.ogi.edu/spectra/water/index.html.
- R. M. Pope, "Optical absorption of pure water and sea water using the integrating cavity absorption meter," Phd Thesis, 1993, Texas A&M University.
- H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994). [CrossRef]
- L. Kou, D. Labrie, and P. Chylek, "Refractive indices of water and ice in the 0.65-2.5 m spectral range," Appl. Opt. 32,3531-3540 (1993). [CrossRef] [PubMed]
- S. Fantini, M. A. Franceschini, and E. Gratton, "Effective source term in the diffusion equation for photon transport in turbid media," Appl. Opt. 36,156-163 (1997). [CrossRef] [PubMed]
- M. Bassani, Limits of validity of the diffusion equation and methodologies for measuring optical properties of highly scattering media, (in Italian) M. S. Thesis, University of Florence, Italy (1997). [PubMed]
- R.L.P. van Veen and H.J.C.M. Sterenborg, A. Pifferi, A. Torricelli, and R. Cubeddu, "Determination of VIS- NIR absorption coefficients of mammalian fat, with time-and spatially resolved diffuse reflectance and transmission spectroscopy," OSA Annual BIOMED Topical Meeting, 2004.
- S. Jacques, "Optical properties of Intralipid, an aqueous suspension of lipid droplets," http://omlc.ogi.edu/spectra/intralipid/index.html
- L. Spinelli, F. Martelli, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, "Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time resolved method," In preparation.
- S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992). [CrossRef] [PubMed]
- D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I) Theory," Appl. Opt. 36,4587-4599 (1997). [CrossRef] [PubMed]
- L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004). [CrossRef]

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