1. Introduction

The study of light propagation through diffusive layered media is of interest for tissue optics since tissues often have a layered structure. It is for instance the case of measurements of muscle oxygenation in which data interpretation is complicated by the subcutaneous fat layer, or of brain activation complicated by the layered structure of the adult head. For a proper analysis of measurements a clear understanding of photon migration through layered media is highly desirable. Both analytical and numerical models have been proposed to describe photon migration through diffusive layered media. Analytical models are usually based on the diffusion equation (DE). Solutions are available for the reflectance from a two-layered medium [1,2]. Numerical models have been developed mainly to obtain solutions of the DE with the finite element method [3,4] or to reconstruct solutions of the radiative transfer equation with Monte Carlo (MC) simulations [2,5,6]; however, the DE provides only approximate solutions of the physical problem and MC simulations are time consuming. Experiments on layered media are therefore useful both for a better comprehension of light propagation and to check the reliability of approximate solutions.

For this purpose phantoms with layered structure are desirable. They should have well known geometrical and optical properties that mimic the actual tissue, with the potentiality to easily change the optical properties of each layer. It should also be possible to insert easily an inhomogeneity. Many phantoms have been used to investigate light propagation through tissue. Phantoms are usually solid [7–10], liquid-solid [11–13] or liquid [14–16]. With respect to solid or liquid-solid phantoms a liquid phantom has many advantages, because the optical properties of each layer can be easily varied by adding small quantities of calibrated scattering or absorbing materials. With this phantom it is simple to study how the measured response changes even when small variations of optical properties are involved. Since nothing else changes during the experiment apart from local variations of optical properties, it is possible to obtain reliable experimental results even for small changes of optical properties in a small-localized volume. This is for instance the case of measurements for investigating brain activation. Different approaches have been proposed for producing inhomogeneous structures with a liquid phantom, such as use of glass [14] or Mylar [15] tubes or of a thin, optically transparent sheet of plastic film to separate two diffusive layers [16]. Although some influence of glass [14] and polystyrene tubes [17] has already been observed, there are no systematic studies on the design of inhomogeneous liquid phantoms and on the perturbation produced by different materials.

The phantom we have developed is an aqueous suspension with known absorption and reduced scattering coefficients, μ_a and μ_s ' , obtained diluting previously calibrated scatterers and absorbers. A thin membrane separates layers with different optical properties. The membrane can be also used to enclose small volumes of diffusive medium to obtain small inhomogeneities with well known optical properties that can be introduced and easily moved into each layer of the liquid phantom. The phantom is described in Section 2. A significant part of this section has been devoted to the search and characterisation of a suitable material for the membrane that does not introduce appreciable perturbation on light propagation. In Section 3 examples of applications of the liquid phantom are reported with reference to a three-layered medium.

2. Proposed phantom

2.1 Description of the phantom

The phantom we developed and used is illustrated in Fig. 1. It is a scattering cell whose lateral and bottom walls are obtained joining U-shaped spacers of different thickness, while plain rectangular sheets provide the front and back wall, respectively. The cell is made of black polyvinyl chloride (refractive index 1.54). A thin membrane can be inserted between the U-shaped spacers to separate aqueous suspensions (refractive index 1.33) with different optical properties. The U-shaped spacers are sealed with a thin film of silicone and kept together by screws. Small transparent windows (diameter 3 mm) symmetric with respect to the centre of the cell (the small circles in the front wall in Fig. 1) are used to illuminate the diffusive medium and to collect the diffuse reflectance at different distances from the light source. The windows are made of Plexiglas and have polished surfaces. The black walls provide well-known boundary conditions, since the impinging photons are either reflected or absorbed. For this study we used Intralipid as a scattering medium and India ink or Cresyl blue as absorbers, and the phantom was 115 mm high, 100 mm wide, and 60 mm thick.

 

Fig. 1. View of the proposed phantom.

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Great attention has been paid to choose the thickness and the material for the membrane. In fact the membrane should separate volumes with different optical properties without introducing appreciable perturbations on light propagation. We firstly considered the possibility of using a thin transparent material, since simple considerations suggest that the perturbation should vanish as the thickness of the layer approaches zero. This was confirmed by MC simulations. Although the MC code [2] was developed to use scattering functions generated with Mie theory, in order to reduce the computation time almost all numerical results were obtained using the scattering function for Rayleigh scatterers, having the asymmetry factor g = 0. For all the source-receiver distances considered, comparisons with simulations carried out using the scattering function for Intralipid, for which g ≅ 0.8, showed results that are indistinguishable from those obtained with g = 0, provided the depth of the transparent layer (or slightly scattering, as is the case of results reported in Fig. 5) is >1/ μ_s ' . When this condition is met the layer is illuminated by diffused light, and propagation is sensitive to μ_s ' and not to the single scattering properties μ_s and g . Reflection and refraction due to the refractive index mismatch between the diffusive medium and the membrane and the cell walls were also taken into account by the code. Plane, perfectly smooth surfaces were assumed for both the membrane and the walls. Examples of results are shown in Fig. 2. The figure reports the time resolved reflectance, R(ρ, t), for a homogeneous medium with μ_s ' = 1 mm^-1, μ_a = 0.01 mm^-1 and refractive index n ₁ = 1.33, and for the same medium with a transparent, non-absorbing membrane of different thickness, s_cl, at a depth of 5 mm. The refractive index of the membrane is n ₂ = 1.5. The results are shown for two values of the source-receiver separation: ρ = 20 and 50 mm. The curves corresponding to the homogeneous medium and to s_cl = 0.1 and 0.021 mm are indistinguishable, indicating that the perturbation introduced by the transparent layer is negligible, within the statistical fluctuations on MC results, provided the thickness of the layer is s_cl ≤ 0.1 mm. These results also show that the refractive index mismatch between the membrane and the diffusive medium does not appreciably perturb propagation. We point out that these results were obtained assuming a perfectly smooth plane surface, for which reflection and refraction laws are applicable. Measurements have shown substantial deviations from these expected results.

 

Fig. 2. MC time resolved reflectance for a homogeneous medium and for the same medium with a transparent membrane of different thickness, s_cl, at a depth of 5 mm. The results are shown for two values of the source-receiver separation: ρ = 20 and 50 mm. The diffusive medium has μ_s ' = 1 mm^-1, μ_a =0.01 mm^-1, and refractive index n ₁ = 1.33. The refractive index of the membrane is n ₂ = 1.5.

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2.2 Perturbation due to the clear layer: time resolved measurements

Based on the MC results we first tested a phantom in which a 100 μm thick transparency film for overhead projectors was used as a separator. To measure the perturbation due to the clear layer we repeated multidistance time resolved measurements of reflectance on a homogeneous medium with and without the membrane. The diffusive medium was an aqueous suspension of Intralipid and India ink with μ_s ' = 1.00 mm^-1 and μ_a = 0.01 mm^-1. The measurement system [18] was based on a tuneable mode-locked Ti:Sapphire laser source for illumination and on a micro-channel plate photomultiplier tube (R1564U with S1 photocathode, Hamamatsu, Japan) and a PC board for time correlated single-photon counting (SPC130, Becker & Hickl, Germany) for the detection of time distributions. Multimode fibres were used both to lead the laser beam to the phantom and to collect the diffuse reflectance. Measurements were carried out at the wavelength of 750 nm. A variable neutral density circular filter allowed us to optimise the illumination power. Measurements were performed with the scattering cell described in Fig. 1. First we carried out measurements with the membrane and then we repeated the experiment after having cut and removed it, without changing anything else. The absorption and the reduced scattering coefficients of the diffusive medium had been calibrated with accuracy better than 5%.

The results we obtained for the transparency film were unexpectedly bad: A significantly higher fraction of photons was received, especially at short times and for large source-receiver distances, when the transparent membrane was present. Examples of results are reported in Fig. 3. Similar results were obtained also with a significantly thinner layer (31 μm) of cellophane. The large perturbations at short times for large values of ρ suggest that a significant fraction of photons reaches the receiver due to a guided propagation through the membrane. Due to the refractive index mismatch, the condition for a guided propagation through the membrane exists. If plane and smooth surfaces are assumed, the membrane acts like a lossless guide, but photons cannot penetrate from the diffusive medium into the membrane with angles greater than the limit angle for which guided propagation occurs. However, due to the roughness of the surfaces, or to the scattering of sedimented particles, the membrane acts as a lossy guide and some photons can penetrate inside with angles greater than the limit angle and travel a long distance inside the membrane. In principle it is possible to include this effect of guided propagation in the MC code but, even omitting considerations about the complexity of this change, it would require detailed information on the optical properties of the membrane that was not available for the materials we used. The undesired propagation through the transparent membrane can be avoided by using a material with scattering properties. Among the materials we tested the most appropriate was a 23 μm thick film of Mylar, for which measurements of time resolved reflectance, with and without the layer, showed indistinguishable results. These measurements are also shown in Fig. 3.

 

Fig. 3. Examples of perturbation due to three different transparent membranes on multidistance (ρ = 10, 20, 30, 40 mm) measurements of time resolved reflectance. The figure reports the results for the homogeneous medium (μ_s ' =1.00±0.03 mm^-1 and μ_a = 0.01±0.0005 mm^-1) with (red curves) and without (black curves) the membrane. For all measurements the membrane was at a depth of 4.5 mm.

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2.3 Perturbation due to the membrane: CW measurements

To deeply investigate the effect of the membrane used as a separator we carried out measurements of CW reflectance at λ = 632.8 nm by using the set-up reported in [12,19]. We used a larger scattering cell, 175 mm high, 175 mm wide, and 105 mm thick, and we measured the reflectance, R(ρ), as a function of the distance, d , of the membrane from the input surface. The membrane was stuck on a frame moved by a step motor. Examples of experimental results are reported in Fig. 4. The figure displays the relative perturbation (R - R _hom) / R _hom (R _hom is the reflectance for the homogeneous medium without the membrane) as a function of d . Three thin membranes of similar thickness, but different material were used: cellophane (31 μm), white plastic film (38 μm), and Mylar (23 μm). The reduced scattering coefficient of the diffusive medium was 1.0 mm^-1. Measurements were carried out for different values of the absorption coefficient. Measurements with the transparent membrane of cellophane show a small perturbation (within 2%) when absorption is small, but the perturbation strongly increases as absorption increases: The thin membrane causes a perturbation that becomes larger than 100% for ρ = 40 mm and μ_a = 0.025 mm^-1, when d > 4 mm. These results also suggest that the perturbation is mainly due to photons that, propagating through the clear layer, are not significantly affected by the absorption of the diffusive medium. When the absorption is small, and the CW reflectance is dominated by photons that travel long paths inside the diffusive medium, these photons are a small fraction of the measured signal. On the contrary, they become a significant fraction of the total as the absorption increases, since photons with long paths are strongly attenuated. As an example, the signal measured at ρ = 40 mm decreased by a factor ~ 3200 as the absorption changed from μ_a = 0.0003 to 0.025 mm^-1.

 

Fig. 4. Examples of perturbation due to three different membranes of similar thickness but of different material, on measurements of CW reflectance. The figure reports the relative perturbation (R - R _hom)/ R _hom as a function of the depth of the membrane. Measurements have been repeated for three values of absorption for a diffusive medium with μ_s' =1.0±0.05mm^-1. The blue, green, red, and black curves refer to source-receiver distances ρ = 10, 20, 30, and 40 mm, respectively.

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Measurements with the white plastic film showed significantly different results. The perturbation, almost independent of μ_a , was larger for small values of ρ, but always smaller than 20%. These results suggest that the effect of guided propagation is insignificant for this material. In this case the perturbation is probably dominated by the scattering coefficient of the white plastic that is significantly higher with respect to the diffusive medium. Finally, the results for the Mylar show perturbations always smaller than 8%, and almost independent of μ_a . Also in this case the contribution of guided propagation is insignificant. The small perturbation on measurements for μ_a = 0.0003 mm^-1 was ascribed to a small effect of absorption. This was supported by comparisons with MC results: An excellent agreement between experimental and numerical results (discrepancies within the standard error) was obtained when an absorption coefficient μ_a = 0.15 mm^-1 was assumed for the Mylar. MC simulations without including the effect of absorption showed perturbations within 1%, of the same order of statistical fluctuations on MC results. Simulations also included the effect of reflection at the water-Mylar interface due to the refractive index mismatch (n = 1.65 for Mylar). The large noise on data referring to large values of ρ and high absorption is due to the strong attenuation of the signal.

Both time resolved and CW measurements show that the Mylar we used is a suitable material for the phantom. To better characterize this material we carried out measurements for the estimate of the extinction coefficient μ_ext [19]. With the CW experimental setup we measured the collimated transmittance P ₀/ P_e through two optical windows enclosing some layers of material (P_e is the power measured without layers inside the optical windows) and we plotted ln(P_e / P ₀) as a function of the thickness of the enclosed layers. From the slope of the straight line that best fits the results we evaluated the extinction coefficient μ_ext. After subtracting the effect of reflections we obtained for the Mylar μ_ext ≅ μ_s ≅ 30 mm^-1.

3. Examples of results obtained with the liquid phantom

To demonstrate the versatility of the liquid phantom we report results both of time resolved and of CW measurements. Time resolved measurements were carried out on a three-layered medium in which the first and the third layer (thickness 4.5 and 51 mm respectively) were filled with the same aqueous suspension of Intralipid and India ink having μ _s1 ' = μ _s3 '= 1.0 mm^-1 and μ _a1 = μ _a3 = 0.01 mm^-1. Multidistance measurements of reflectance were repeated for different values of the optical properties, μ _s2 ' and μ _s2, of the second layer (thickness 4.5 mm). Examples of results are reported in Figs. 5 and 6. To show the effect of a clear layer on measurements of reflectance, Fig. 5 reports the comparison between measurements for the homogeneous medium and for the layered medium with μ _s2 ' = 0.1 mm^-1 and μ _a2 = 0.003 mm^-1. The figure reports the experimental results for ρ = 20 and 40 mm together with the predictions of MC simulations. For the comparison MC results were convolved with the temporal response of the experimental setup. Furthermore, each curve of the experimental results, that are available in arbitrary units, has been multiplied by a constant factor to overlap the maximum of the experimental and of the MC curve. The marked difference between the results for the homogeneous and for the layered medium points out the strong effect that a clear layer has both on the shape and on the amplitude of the time resolved reflectance. For ρ = 40 mm the total reflectance from the layered medium was about 40 times larger than from the homogeneous medium. The agreement between numerical and experimental results is excellent, showing both the reliability of the MC code, and the adequacy of the phantom we used, for investigating photon migration through layered media.

 

Fig. 5. Effect of a clear layer on measurements of time resolved reflectance. The three-layered medium has the first and the third layer (thickness 4.5 and 51 mm, respectively) with the same optical properties: μ _s1 ' = μ _s3 ' = 1.0 mm^-1 and μ _a1 = μ _a3 = 0.01 mm^-1. The figure shows the comparison between measurements for the homogeneous medium and for the layered medium with μ _s2 ' = 0.1 mm^-1 and μ _a2 = 0.003 mm^-1. The thickness of the second layer was 4.5 mm. The results for ρ = 20 and 40 mm are reported together with predictions of MC simulations.

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From time resolved measurements repeated after having varied only the absorption coefficient μ _a2 by a small quantity, ∆μ _a2, the time resolved mean path, 〈l ₂〉(t), followed inside the second layer by detected photons was obtained as [12]

Examples of results are shown in Fig. 6 for a layered medium having: thickness of the first, second, and third layer 4.5, 4.5, and 51 mm respectively; μ _s1 ' = μ _s3 ' = 100 mm^-1, μ _s2 ' = 1.05 mm^-1, μ _a1 = μ _a3 = 0.01 mm^-1, and μ _a2 = 0.003 mm^-1. The absorption coefficient of the second layer was varied by ∆μ _a2 = 0.0034 mm^-1. The figure reports the results of measurements at distances ρ = 10, 20, 30, and 40 mm together with the prediction of the DE [2]. The results from the DE show that 〈l ₂〉 is larger for photons that have spent a longer time inside the diffusive medium, but 〈l ₂〉 is almost independent of the source-receiver distance. The agreement of experimental results with predictions of the DE is rather good. These results indicate that the sensitivity to a small variation of μ _a2 is larger at longer times, but is almost independent of the source-receiver distance. In fact, the perturbation on measurements of time resolved reflectance is proportional to exp(-∆μ _a2 〈l ₂〉(t)).

 

Fig. 6. Time resolved mean path followed by received photons inside the second layer of a medium having: thickness of the first, second, and third layer 4.5, 4.5, and 51 mm respectively; μ _s1 ' = μ _s3 ' = 1.00 mm^-1, μ _s2 ' = 1.05 mm^-1, μ _a1 = μ _a3 = 0.01 mm^-1 and μ _a2 = 0003 mm^-1. The results are reported for measurements at ρ = 10, 20, 30, and 40 mm together with the prediction of the diffusion equation.

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As an example of CW measurements Fig. 7 reports the perturbation due to a small absorbing inhomogeneity immersed in the third layer of a medium having: thickness of the first, second, and third layer 11, 3, and 46 mm respectively; μ _s1 ' = 1.7 mm^-1, μ _s3 ' = 5.7 mm^-1, μ _a1 = μ _a2 = μ _a3 = 0.0003 mm^-1. Measurements have been repeated for four values of μ _s2 ': 0, 0.18, 0.40, and 1.5 mm^-1. The source and the receiver were at x = -20 and +20 mm respectively. The inhomogeneity (volume 1.1 ml, roughly spherical) was a small quantity of liquid diffusive medium with μ_si ' = μ _s3 ' = 5.7 mm^-1 and μ_ai = 0.012 mm^-1 enclosed with the film of Mylar. The Mylar film was first put into a small cylinder, filled with the liquid, and then sealed with a thin fishing line. The inhomogeneity was glued to a thin, white-painted steel wire (diameter 0.2 mm) moved by two computer-controlled translators. We checked that the perturbation due to the wire was negligible. Measurements of reflectance were repeated moving the inhomogeneity inside the third layer in the plane containing the x -axis and the direction of the laser beam. The figure reports the perturbation as a function of the x -coordinate of the centre of the inhomogeneity for two values of the depth within the third layer. At the depth of 22 mm the inhomogeneity skimmed the surface of the second layer. The scattering properties of the medium were chosen as a rough model of an adult head: the three layers represent the scattering properties of the scalp and skull, of the cerebrospinal fluid (CSF), and of the grey and white matter respectively [20,21]. The results of Fig. 7 give us information on the influence of the scattering properties of the CSF on measurements of CW perturbation due to localized variations of absorption inside the brain. However, since the absorption coefficient of the phantom is significantly smaller with respect to values expected for the adult head (μ_a of the order of 0.15-0.36 mm^-1 [20,21]) the results of Fig. 7 can be only used as indicative.

 

Fig. 7. Perturbation due to an absorbing inhomogeneity (volume = 1.1 ml, μ_si ' = 5.7 mm^-1 and μ_ai = 0.012 mm^-1) immersed into the third layer of a medium having: thickness of the first, second, and third layer 11, 3, and 46 mm respectively; μ _s1 ' = 1.7 mm^-1, μ _s3 ' = 5.7 mm^-1, μ _a1 = μ _a2 = μ _a3 = 0.0003 mm^-1. Results are shown for four values of μ _s2 ' . The source and the receiver are at x = -20 and +20 mm respectively. The perturbation is reported as a function of the x-coordinate of the centre of the inhomogeneity for two values of the depth: 22 mm (red curve) and 25 mm (black curve) within the third layer.

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4. Conclusions

We have described a liquid phantom for studying light propagation through diffusive layered media. From MC simulations we expected that any optically transparent material with thickness smaller than about 0.1 mm would have been suitable for separating volumes of liquid with different optical properties. On the contrary, the experimental results of Sect. 2 showed that a very thin transparent layer of cellophane greatly perturbs both the amplitude and the shape of the time resolved reflectance, especially when measurements are carried out at large source-receiver distances on media with high absorption. The unexpected large perturbation was explained with a guided propagation that, due to the refractive index mismatch, can be established through the transparent membrane: Photons that, due to the roughness of the surface or to scattering of particles sedimented on the surface, penetrate into the membrane with angles for which total internal reflection occurs can easily reach the receiver propagating through the membrane. Therefore, a great attention should be paid to the choice of the membrane. Experimental results showed that a material with scattering properties should be used to avoid the undesired guided propagation. Among the materials we tested, the most suitable is the Mylar. The Mylar film looks like a transparent material, but measurements of extinction showed a high scattering coefficient. The Mylar film does not perturb the shape of the temporal response, and the perturbation on CW measurements always remains smaller than 8% for all experimental situations we investigated. However, experimental results showed that any thin film of material with scattering properties could be acceptable. Also the highly scattering white plastic film, although causing a 20% change in the CW reflectance, does not significantly affect the shape of the temporal response.

The examples of results reported in Figs. 5–7 demonstrate the versatility of the proposed phantom. The liquid phantom has many advantages with respect to solid or solid-liquid phantoms because the optical properties of each layer can be easily varied by adding small quantities of calibrated scattering or absorbing materials. Furthermore, small inhomogeneities with well known optical properties can be obtained enclosing small quantities of calibrated liquid inside a thin Mylar film, and easily moved into each layer of the phantom. Therefore, it is easy to study how the measured response changes when even small variations of optical properties are involved. Since nothing else changes during the experiment, apart from local variations of optical properties, it is possible to obtain reliable and accurate experimental results also for small variations of optical properties in a small localized volume.