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, μ́s, of a liquid diffusive medium and the absorption coefficient, μa, of an absorbing medium with a standard error smaller than 2% both on μ́s and on μa. 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.
© 2007 Optical Society of America
The study of light propagation through diffusing media has been deeply investigated over the past two decades especially with reference to measurements on biological tissues at near infrared wavelengths (NIR). From a physical point of view one of the main problems in therapeutic and diagnostic applications of light in medicine is the determination of optical properties of tissue. However, in spite of many progress both with theories and with instrumentation, the accurate measurement of optical properties of diffusive media (the absorption coefficient, μa, and the reduced scattering coefficient, μ́s) 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 μa and μ́s, and many phantoms have been proposed to assess the performance of NIR instrumentation (see Ref.  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.
As a first example we refer to Ref. , in which a protocol to asses the performance of photon migration instrumentation has been proposed. The paper reports comparisons among the optical properties of solid phantoms obtained by 8 different instruments from 5 European countries. There are significant differences in the results from different instruments, and comparisons are complicated since the ’true value’ of the optical properties of the used phantoms remains unknown.
As a second example we refer to Refs. , , , three recently published papers in which calibrations of the optical properties of Intralipid have been reported. In  and  the optical properties of aqueous suspensions of Intralipid or Liposyn and Indian ink were obtained from CW multidistance measurements of fluence rate. In  μ́s of Liposyn was obtained with a standard deviation of 16% and absorption of ink with a standard deviation of 8%. In  the standard deviation for μ́s 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  a method has been presented based on measurements of diffuse reflectance and diffuse transmittance with an integrating sphere. The experimental results at λ = 633 nm show an error larger than 20% for the reduced scattering coefficient of Intralipid while the absorption coefficient, probably due to the small thickness of samples used, was overestimated by at least two orders of magnitude. Looking at these recent results, even the calibration of Intralipid and Indian Ink, diffusive and absorbing media widely used to prepare phantoms for NIR measurements, seems to be an open problem.
In this paper we present a method, based on CW relative multidistance measurements of fluence rate carried out in an infinite medium, to calibrate liquid diffusive media with high accuracy. We calibrated the reduced scattering of Intralipid and the absorption coefficient of Indian ink with a standard deviation smaller than 2%. The optical properties have been retrieved exploiting the knowledge of the absorption coefficient of the liquid into which the diffusive medium is dispersed. 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.
The experimental results have been validated by a comparison with the results from time resolved measurements carried out on samples of Intralipid and Indian ink from the same batch used in our experiments. The comparison showed an excellent agreement.
The proposed method is presented in Sect. 2 together with a discussion of errors due to the intrinsic approximations. The experimental setup and examples of experimental results are presented in Sects. 3 and 4. Discussion and conclusions are in Sect. 5.
2. The proposed method
To prepare a medium with controlled scattering and absorption properties we need both a calibrated diffuser and a calibrated absorber. With a calibrated diffuser and a calibrated absorber a phantom with known μa and μ́s 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.
2.1. Calibration of the diffusive medium
The method we propose to calibrate the optical properties of a liquid diffusive medium is based on measurements of effective attenuation coefficient, μeff, obtained from multidistance measurements of fluence rate, Φ(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 
where μeff =(3μa μ́s)1/2 is the effective attenuation coefficient and μa and μ́s are the absorption and the reduced scattering coefficients of the medium. From Eq. 1 results
From accurate relative multidistance measurements of fluence it is therefore possible, with a linear fit, to retrieve with an excellent accuracy μeff as the absolute value of the slope of ln[rΦ(r)] as a function of r. A single measure of μeff is not sufficient to reconstruct μa and μ́s. 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 , , . At visible wavelengths it is possible to use dyes, purely absorbing media, whose absorption coefficient can be easily and precisely calibrated with measurements of collimated transmittance. As an example, from measurements on aqueous suspensions of Intralipid by using an He-Ne laser at 632.8 nm and the Cresil blue dye as absorber , the reduced scattering coefficient has been reconstructed with an error smaller than 2% and the absorption coefficient with an error smaller than 0.00005 mm−1 over a wide range of values for μ́s (0.5 < μ́s < 18mm−1).
Unfortunately at NIR wavelengths, where measurements are carried out for many important applications (tissue optics , characterization of agricultural products ) suitable dyes are, to our knowledge, not easily available and the method of adding absorption is inapplicable. The method we propose is based on measurements of μeff 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 ε ail and έsil 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
where 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
and εa and έs can be obtained with a more simple linear fit as
where 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.
Equations 7 and 8 show that to get accurate measurements of the optical properties of the diffusive medium it is necessary to know 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 ). Published data at NIR wavelengths are in reasonable agreement, but an appreciable spread of values remains also when more recently published data are compared. At λ = 750nm, the wavelength used in our experiments, almost all data reported in Ref.  are between 0.00247 and 0.00286 mm−1 (a significantly lower value, 0.000219 mm−1 is also present ) and a discrepancy of ≃ 5% remains (0.00286 and 0.00273 mm−1) even between the two more recently published data , . Discrepancies may be due both to the experimental procedure and to the quality of water. To obtain accurate measurements of optical properties of diffusing media by using the proposed method, an accurate calibration of the absorption coefficient of water is therefore necessary.
To better characterize the diffuser we also carried out measurements of collimated transmittance  to determine the extinction coefficient, μext. The knowledge of the absorption, reduced scattering, and extinction coefficients enabled us to determine the asymmetry factor, g, of the medium.
2.2. Calibration of the absorbing medium
The absorption coefficient of a material with negligible scattering properties can be easily obtained with high accuracy from measurements of collimated transmittance. The absorber we used is the the Indian ink (Rotring waterproof), a medium with high absorption but with not negligible scattering. A measure of collimated transmittance is therefore insufficient to characterize it. To determine the absorption coefficient we carried out measurements of μeff on a diffusive medium with calibrated reduced scattering coefficient as a function of the volume concentration, ρ ink, of Indian ink. If we denote with εaink 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
where μ a0 and μ́s0 are the absorption and the reduced scattering coefficient of the diffusive medium before the addition of ink. If μ́s0 is known, it is possible to retrieve εaink with a linear fit as
where S ink is the slope of the straight line that best fits μ 2 eff (ρink) as a function of ρink. The intercept, I ink, can be used to retrieve μ a0 as
The calibrated diffusive medium we used is a suspension of Intralipid previously calibrated with the procedure described in Sect. 2.1. The assumption that the added absorber does not significantly alter the reduced scattering coefficient is well verified in our experiments since we added few milliliters of absorber to ≃ 3L of diffusive medium.
For a better characterization of the absorber we also measured the extinction coefficient. The knowledge of absorption and extinction coefficients enabled us to determine the single scattering albedo, Λink.
2.3. Accuracy of the proposed method: Discussion of the intrinsic approximations
The optical properties both of the diffusive and of the absorbing medium are obtained with linear fits starting from multidistance measurements of fluence rate. In principle, from accurate measurements of fluence it is therefore possible to retrieve the optical properties with high accuracy. However, the equations we used are subjected to some approximations and their range of applicability should be discussed.
First of all Eq. 2, used to retrieve μeff, 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 ε ail and έsil, 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.
The accuracy of the DE to describe light propagation through an infinite medium has been investigated in depth in Ref.  both with numerical simulations and with experiments. In particular it has been shown that for moderate values of absorption Eq. 1 is almost exact (discrepancies with respect to the solution of the radiative transfer equation smaller than 1%) provided r > 2/μ́s.
Effect of the finite size of source and receiver.
In our experimental setup two identical diffusive sphere (3-mm diameter) attached to the tip of two plastic fibers have been used to illuminate the medium and to collect received photons. It has been shown  that the solution of the DE for a spherical source with constant strength differs from the solution for a pointlike source only by an amplitude factor FS = sinh(μeff a)/(μeff a) where a is the radius of the sphere. It has been also shown, integrating Eq. 1 over a spherical receiver , that the finite size of the receiver leads to an amplitude factor FR =FS. For the values of a and μeff 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[rΦ(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 μeff.
Perturbation introduced by the source and by the receiver.
The measurements of fluence in an infinite medium are disturbed by the presence of the source and of the receiver. In fact the optical properties of the fibres used to illuminate the medium and to collect the fluence are unavoidably different with respect to the diffusive medium. Furthermore, part of the light impinging the tips leaves the diffusive medium. For our experiments it is mainly important to understand if the perturbation due to the fibres changes the dependence of ln[rΦ(r)] on r. This point has been investigated solving the DE for the infinite medium with a perturbation approach. For an absorbing inhomogeneity the perturbation of the fluence results
where r 1 and r 2 are the distances of the volume element of the inhomogeneity, dVi, 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 r 1 + r 2 ≅r 2 ≅ r With these assumptions we obtain
An identical expression can be obtained for the perturbation due to the spherical receiver.
For a scattering inhomogeneity the perturbation results
where ϑ is the angle between the source-inhomogeneity vector and the receiver-inhomogeneity vector. To evaluate the perturbation due to the spherical source we observe that if 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.
In conclusion we expect that the presence of the fibres cause only a perturbation proportional to μ́s independent of the source-receiver distance that does not affect the measure of μeff. 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 μ́s, independent of the source-receiver distance. The reduction was of 1% for μ́s = 0.4mm−1.
Effect of the finite medium.
To investigate the effect of the finite medium we compared measurements repeated on different volumes of the same diffusive medium. We used cylindrical containers with diameter ≃ to the height with volumes, 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 μ́s ≥ 0.3mm−1. With μ́s ≥ 0.8mm−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.
Equation 6, used to retrieve the optical properties of the diffusive medium, comes from Eqs. 3 and 4 that are based on the independent scattering approximation, i.e. on the assumption that the average distance between scatterers is sufficiently large with respect to the wavelength so that they act as independent and uncorrelated scatterers. To fulfil this assumption the fractional volume occupied by scatterers, ρ̃il, must be sufficiently small (ρ̃il ≲ 0.01). We notice that, taking into account the specific weight of different components , for Intralipid-20% the volume concentration of scatterers is related to the volume concentration of Intralipid, ρil, by ρ̃il = 0.227ρ il. The dependence of μa and μ́s on ρ̃il has been investigated in Ref.  with measurements on Intralipid-20% at λ = 632.8nm carried out up to ρ̃il = 0.227, i.e. on non diluted Intralipid-20%. The results reported in  showed that μa increases linearly with ρ̃il as expected when the independent scattering approximation holds. Significantly different results have been observed for μ́s for which the following expression has been obtained 
Equation 4 can be therefore used only for moderate concentrations. From Eq. 15 it results that for the concentrations used in our experiments, ρ́il < 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 έsil. For the values of ρ il we used, the error introduced is insignificant (< 0.01%) for έsil, whereas it leads to an underestimate ≅ 0.001mm−1 for ε ail.
3. Experimental setup
The experimental setup we used to measure μeff is basically the same described in Ref. . The light source was a 5 mW laser diode emitting at 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. .
Also the experimental setup we used to measure the extinction coefficient of Intralipid and of Indian ink is basically the same described in Ref. . For these measurements the light emitted by the laser diode was collimated to obtain a beam with a cross-section ≃ 3mm2. 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.
The same experimental setup has been also used to measure the absorption coefficient of pure water. We used a cell 430 mm long filled with water purified with a purification system (Elix 3 Millipore). With a photodiode (active area 100 mm2) 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
Both the calibration of the diffusive and of the absorbing medium are based on measurements of effective attenuation coefficient. To avoid errors due to limitations of the DE and to the finite volume of the medium all measurements have been carried out with V ≃ 3L and μ́s ≳ 0.3mm−1. Examples of multidistance measurements are reported in Fig. 1. The figure reports ln[rΦ(r)] for three values of Intralipid concentration: ρ il= 0.0204, 0.0401, and 0.0857. The standard error on ln[rΦ(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 μeff are 0.0585±0.0001, 0.08197±0.00007, and 0.11938±0.00005 mm−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
To get accurate measurements of the optical properties of the diffusive medium it is necessary to know the absorption coefficient of water with high accuracy. As explained in Sect. 3, to determine 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)10−3mm−1.
4.3. Calibration of Intralipid
The calibration of the intrinsic absorption and reduced scattering coefficient of Intralipid-20%, ε ail and ε siĺ, 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 έsil are: ε ail =(2.25±0.09)10−3mm−1 and έsil =20.3±0.1mm−1. The results of the fit slightly change (ε ail = (2.17±0.09)10−3mm−1 and έ sil = 20.3±0.1mm−1) if the point at ρ il= 0.0204 is excluded.
The errors on ε ail and έsil 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 έsil and ≃∓7% on ε ail. If we include this source of error we obtain for the calibration of Intralipid-20%: έsil = 20.3±0.3mm−1 and ε ail = (2.25±0.26)10−3mm−1.
A further, systematic error, may come from the assumption of the independent scattering approximation. As discussed in Sect. 2, for measurements on Intralipid-20% at the volume concentrations we used, we expect an insignificant effect on έ sil , while the value of ε ail may be underestimated of ≅0.001mm−1.
4.4. Calibration of Indian ink
To calibrate the Indian ink we used the previously calibrated Intralipid-20% at a known volume concentration as the medium with known μ́s0. 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 μ́s0 = 1.74±0.03mm−1 and μ a0 = (2.73 ± 0.02)10−3mm−1. Also in this case the straight line that best fits the results, for which I ink = (1.427 ± 0.001)10−2mm−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 μa0 = (2.73 ± 0.04)10−3mm−1. The errors are mainly due to the error on μ́s0. 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
As mentioned in Sect. 2, we also carried out measurements of extinction coefficient to better characterize both Intralipid and Indian ink. For the intrinsic extinction coefficient we obtained 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 Λink = 0.125±0.02.
5. Discussion and conclusions
5.1. Summary of the results
We have presented a methodology to calibrate the optical properties of liquid diffusive and absorbing media at NIR wavelengths. The methodology requires a simple experimental setup, based on a CW laser diode, a computer controlled translation stage, a photomultiplier and a lock-in amplifier for measurements of CW intensity. The methodology has been used to calibrate the optical properties of Intralipid-20% as a diffuser and of Indian ink as an absorber. The results we obtained are summarized in Tab. I. Each value is reported with its standard deviation. We remind that, due to the assumption of the independent scattering approximation, the value of ε ail may be systematically underestimated of ≅ 0.001mm−1.
Both the reduced scattering coefficient of Intralipid-20% and the absorption coefficient of Indian ink have been obtained with an error smaller than 2%, similar to the error obtained at visible wavelength with the method of adding absorption . The error is mainly due to the uncertainty, ±0.1mm, in positioning the fibers. This uncertainty is unlikely to be lowered, but its effect might be reduced if multidistance measurements were carried out for larger values of 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/μ́s 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.
A significantly larger percent error has been obtained for the intrinsic absorption coefficient of Intralipid-20%. The error is mainly due to the assumption of the independent scattering approximation and to the error in positioning the fibres. The value we obtained, 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  (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 μ́s ≲ 1mm−1 as commonly used for phantoms with optical properties similar to biological tissue.
The results of measurements repeated on samples from the same batch showed that the optical properties both of Intralipid and of Indian ink are very stable (the results we obtained were within the errors reported in Table I) over a period of at least one year (for Intralipid even after the date of expiry of the drug).
Mixing suitably diluted quantities of the calibrated Intralipid-20% and of the calibrated Indian ink it is possible to obtain a medium whose optical properties can be predicted, with an error smaller than 2%, by Eqs. 3 and 4. A similar accuracy is, in our knowledge, unlikely to be obtained with other methodologies commonly used at NIR wavelengths. However, we point out that Eqs. 3 and 4 should be used carefully. The results reported in  indicate that Eq. 3 predicts the absorption coefficient for all concentrations, but Eq. 4 can be used only for moderate values of ρ il. Errors are ≲ 1% for ρ il < 0.035, corresponding to μ́s≃ 0.7mm−1, but may become 4% for ρ il = 0.1 corresponding to μ́s ≃ 2mm−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.  and  comparisons are made between results obtained from measurements on non-diluted Intralipid-10% and -20% and from measurements on a suspension of 3% Intralipid-10% . In our opinion, to compare experimental results it is important to specify the concentration at which measurements have been carried out.
5.2. Validation of the experimental results
The experimental results have been validated by a comparison with the results from time resolved measurements carried out on samples of Intralipid and Indian ink from the same batch used in our experiments. The absorption coefficient of Indian ink was obtained with a linear fit from measurements of time resolved transmittance carried out on a suspension of Intralipid before and after the addition of a weighed quantity of ink. Again with a linear fit, the reduced scattering coefficient of Intralipid was obtained from multidistance measurements of time resolved transmittance. Both for έsil and for ε aink the accuracy was similar to that obtained from CW measurements. The methodology is fully described in Ref. . The comparison showed an excellent agreement: Discrepancies were within the standard deviation
5.3. Comparison with published data
To compare the results we obtained for Intralipid with previously published data, we should notice that 1000 mL of the Intralipid-20% we used contain 200 g of soybean oil, 22.5 g of glycerin, 12 g of egg phospholipid, and water. The composition is slightly different with respect to the one reported in Refs. [8 ] and , where instead of 12 g of egg phospholipid there are 12 g of lecithin. This difference may explain some of the discrepancies observed. The results we obtained for the scattering properties can be compared with data reported by van Stavereen et al.  that carried out measurements for concentrations similar to ours. From data reported in Ref.  we expect, at 750 nm, 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.  that is about two orders of magnitude higher with respect to our results. As will be discussed later, a possible explanation for this discrepancy may be the poor sensitivity of the methodology used in  to variations of absorption coefficient when measurements are carried out on low absorbing media.
For Indian ink the most relevant information is Λink. 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, Λink = 0.125±0.02, is very close to the value, 0.115, expected from Mie theory for small particles (size 0.1 μm) of carbon suspended in water [21 ].
5.4. Comparison with other CW methodologies
Two methods based on multidistance measurements of fluence to measure the optical properties of Intralipid or Liposyn and Indian ink have been recently used by Dimofte et al.  and by Xu and Patterson .
Dimofte 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%.
Xu and Patterson  carried out measurements at 750 nm on a semiinfinite medium irradiated by a broad beam, varying the concentrations of Intralipid and Indian ink. The reduced scattering coefficient of Intralipid was obtained from the values of μeff, but to invert the results they made the assumption that absorption of Intralipid is equal to absorption of water and that Λink=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 μ́sil(ρ il =0.05) = 0.97mm−1 with an error of 8%, a value in good agreement with our result.
A method based on measurements of diffuse reflectance and diffuse transmittance with an integrating sphere has been recently presented by Cheng et al.  as a primary method to calibrate diffusive media. They carried out measurements on layers of non diluted Intralipid-20% a few tenths of millimeter thick. The algorithm used to invert the measurements was based on complicated Monte Carlo simulations in which propagation inside the integrating sphere was included. As an example, the experimental results at λ = 633 nm show an error larger than 20% for the reduced scattering coefficient of Intralipid while the absorption coefficient, probably due to the small thickness of the samples used (see Sect. 5.5 for a deeper discussion), was overestimated by at least two orders of magnitude.
5.5. Advantages and disadvantages of the proposed method
The accuracy we obtained to calibrate the optical properties both of the diffuser and of the absorber are, in our opinion, unlikely to be obtained with other methodologies commonly used for NIR measurements. In our opinion there are two main reasons for this better accuracy. The first is that measurements are carried out with the geometry of infinite medium for which, due to the long paths that received photons can follow before reaching the receiver, they have a high sensitivity to absorption. For the infinite medium the mean pathlength is given by  ⟨l(r)⟩ =r√3μ́s/4μa. As an example, for an aqueous diffusive medium with μ́s =1mm−1, μa = 0.0025mm−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.  is based on measurements on slabs ≃ 0.2mm thick. Propagation through so a thin layer is almost unaffected by the small absorption of Intralipid, since the mean pathlength followed by photons before emerging from the slab is very small: For a non-absorbing slab 0.2mm thick, with the same refractive index as the external medium and μ́s = 15mm−1, we obtain from the DE  ⟨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 μa = 0.0025mm−1, a value close to that expected for Intralipid-20%.
The second reason for the good results is the accuracy and the simplicity of the model used to retrieve the optical properties. The CW solution of the DE for the infinite medium is obtained with exact boundary conditions, and for r > 2/μ́s 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 μa and μ́s 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.
We have presented a method, based on a simple experimental setup, to calibrate a diffusive and an absorbing medium with high accuracy. We calibrated the reduced scattering coefficient of Intralipid and the absorption coefficient of Indian ink with errors smaller than 2%. Intralipid and Indian ink are widely used to prepare phantoms for biological tissue. Calibrated phantoms with known optical properties can be useful 1) to calibrate instrumentation for absolute measurements of diffused light , 2) to establish the accuracy of methodologies and instrumentation for measuring the optical properties of diffusive media, and 3) for a deeper comprehension of photon migration through diffusive media. We also point out that the availability of a calibrated absorber enables us to use the method of adding absorption (the method of adding absorption is simpler with respect to the proposed method, and can be more widely used) also at NIR wavelengths.
The calibration method has been discussed with reference to aqueous suspensions, but it is applicable to any liquid suspension provided the absorption coefficient of the liquid is known with high accuracy.
This research was partially supported by MIUR under the project PRIN2005 (prot. 2005025333).
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