1. Introduction

Light propagation through diffusive media is well described by the diffusion equation (DE). For homogeneous media accurate analytical solutions are available for many geometries [1,2]. These solutions are commonly used to invert measurements of reflectance or transmittance in order to determine the optical properties of the medium. Analytical solutions are also available for a diffusive medium having a layered structure and for a homogeneous medium with small inhomogeneities inside. However, there are situations of practical interest in which a clear layer is present inside, or at the surface of the diffusive medium, for which the DE is not suitable to provide an accurate analytical solution. It is the case, for instance, of the head, where an almost transparent layer of cerebrospinal fluid is present between the skull and the brain. The effect of the cerebrospinal fluid on light propagation has been widely investigated with reference to brain imaging. To study this problem Firbank et al. [3] used the hybrid radiosity-diffusion theory, whereas Ripoll et al. [4] obtained a general expression for the boundary conditions when a void region is embedded in a diffusive medium. The expression is rigorous within the diffusion approximation. By using these approaches numerical solutions of the DE have been obtained using the finite element method [3,5], whereas numerical solutions of the radiative transfer equation have been obtained with Monte Carlo simulations (see for instance Ref. [6]). It has been shown that measurements of time resolved reflectance are strongly affected by the light piping through the clear layer.

The case of a clear layer at the surface of a diffusive medium is also of practical interest, since measurements on phantoms are often carried out on liquid suspensions of scatterers enclosed into cells with transparent walls (e.g., glass, Plexiglas). Also measurements on biological tissues are sometimes carried out with a clear layer at the surface. It is particularly relevant the case of optical mammography, in which the breast is compressed between two parallel transparent plates and measurements of transmittance are carried out with a collinear source-receiver system. Also of interest are measurements of reflectance carried out on tissues slightly compressed with a plate to get a more regular geometry.

In this paper we focus on the effect of a clear layer at the surface of the diffusive medium illuminated by an external source. The problem has been investigated with Monte Carlo (MC) simulations: We evaluated both the time resolved reflectance and the time resolved transmittance. To quantify the effect of the clear layer MC results have been fitted with the solution of the diffusion equation for the homogeneous medium in order to reconstruct the optical properties of the diffusive medium and to investigate their deviations from the actual values. We point out that both the radiosity-diffusion [3] theory and the expression for the boundary conditions proposed by Ripoll et al. [4] are based on the assumption that the light incident on the nonscattering region is already diffuse. Since to reach the diffusive regime of propagation light must travel at least a few transport mean free paths [7] these approaches are not applicable to the clear layer at the surface of the diffusive medium. In fact, in this case the nonscattering region is illuminated directly by the source and by many photons that have undergone only few scattering events near to the entrance point of the light beam.

2. Description of the MC code and of the fitting procedure

For numerical simulations we used an elementary MC code developed to investigate light propagation through layered media [8]. The light source was modelled as a pencil beam normally impinging the layered medium. Reflection and refraction due to the refractive index mismatch at the boundary of each layer have been taken into account by using the Snell law and the Fresnel reflection coefficient for non-polarized light. We denote with n_d, n_cl, and n_e the refractive index of the diffusive medium, of the clear layer and of the external medium respectively. Numerical simulations provide the temporal point spread function, i.e., the probability to detect the emitted photon per unit time and per unit area of the receiver. The statistical error has been obtained from the number of photons received within each temporal interval. To reduce the computation time we used a scattering function with the asymmetry factor g=0. Since in practical applications measurements of transmittance are usually carried out for optical mammography, and measurements of reflectance are carried out on muscular tissues, we used different optical properties for simulations of transmittance and reflectance. For simulations of transmittance we used µ′_s=1 mm^-1 for the reduced scattering coefficient and µ_a=0.003 mm^-1 for the absorption coefficient to mimic the optical properties of the breast; for reflectance we used µ′_s=1 and 0.5 mm^-1 and µ_a=0.01 mm^-1, values representative of many biological tissues.

To quantify the effect of the clear layer on measurements of time resolved reflectance or transmittance, the MC results have been fitted with the solution of the DE for the homogeneous medium obtained with the extrapolated boundary condition [2]. The extrapolation length z_e, which also accounts for the effect of reflections at the interface between the diffusive medium and the exterior, has been evaluated as [9,2]

where R(µ) is the reflection coefficient and cos^-1(µ) is the incidence angle. When a clear layer is present at the surface of the diffusive medium reflections occur both at the inner and outer surface of the clear layer and the total boundary reflectivity for a photon striking the inner boundary at an angle cos^-1(µ) is given by [10]

where R_ij is the polarization-averaged Fresnel reflectivity of the ij interface. Durian [10] showed that the solution of the DE obtained with the extrapolation length given by Eqs. (1) and (2) provides an excellent description for the total transmittance and reflectance. However, since the DE cannot include the displacement of photons due to multiple reflections inside the clear layer and the corresponding time of flight, we expect a poor accuracy of the DE in describing the spatial and temporal distribution of emerging photons.

3. Numerical results

3.1 Diffuse transmittance

The dependence of the transmittance through a diffusive slab on the thickness, s_cl, of the transparent walls that enclose the medium, has been investigated with reference to a diffusive slab of thickness s_d=40 mm between two layers of Plexiglas in air (n_cl=1.49, n_e=1). Examples of numerical results are reported in Fig. 1 for a diffusive medium having µ′_s=1 mm^-1, µ_a=0.003 mm^-1, and n_d=1.33. These values are representative for a compressed breast at near infrared wavelengths. The results have been reported for different values of the thickness of the clear layers in the range between s_cl=0 (no layers) and s_cl=10 mm. The figure reports the time resolved transmittance, T(t), for two values of the distance ρ of the receiver from the pencil light beam: ρ=0 (coaxial receiver) and ρ=20 mm. The coaxial receiver had a radius of 3 mm, that at ρ=20 mm was a ring 2 mm thick. The acceptance angle was 90 degrees. These results show that the two layers slightly affect measurements of transmittance. In particular, the temporal point spread functions for s_cl≤1 mm are almost indistinguishable, indicating that multiple reflections inside the thin clear layers have a small effect on the total time of flight and that the clear layers do not significantly affect the effective reflection coefficient of the boundary. This result is in agreement with predictions of Eqs. (1) and (2) that give very similar results for the extrapolation length in the two cases: z_e=1.677/µ′_s for water-air, and z_e=1.753/µ′_s for water-Plexiglas-air. When s_cl increases we observe an appreciable decrease of T(t) for short times and a shift of the maximum toward longer times. The perturbation decreases as the distance from the light beam increases. For the cw transmittance variations remain within 45% for ρ=0 and within 30% for ρ=20 mm.

 

Fig. 1. Dependence of the time resolved transmittance on the thickness of the transparent walls of the cell containing the diffusive medium. The medium has s_d=40 mm, µ′_s=1 mm-1, µ_a=0.003 mm^-1, n_d=1.33, n_cl=1.49, and n_e=1.

Download Full Size | PDF

 

Fig. 2. Absorption and reduced scattering coefficients retrieved from data of Fig. 1 by the fit with the solution of the DE for the homogeneous medium as a function of the thickness of the transparent walls.

Download Full Size | PDF

To evaluate the effect of the clear layers when measurements of time resolved transmittance are carried out on a diffusive medium enclosed between transparent walls, a fit with the solution of the DE for a homogeneous slab 40 mm thick has been carried out on MC results. We used for the extrapolation length the value z_e=1.753/µ′_s obtained with the reflectivity of the water-Plexiglas-air boundary. Four parameters have been fitted: µ_afit, µ′_sfit, an amplitude factor F, and a temporal shift t₀. For the fit we used only the response within the range delimited by data corresponding to 1% of the maximum value. Furthermore, since the DE is not very accurate for early-received photons [2] we disregarded data for t<2.5t_b where t_b is the time of flight for ballistic photons. The temporal shift would account for the time spent by received photons inside the walls if this parameter were independent of the overall time of flight. The results obtained for µa_fit and µ′_sfit are reported in Fig. 2 for two receivers at ρ=0 and 20 mm. For a better readability of the results we chose a non-linear scale for the x-axis. The reduced χ ² was already smaller than about 1 indicating an excellent agreement between the curves retrieved by the fit and the MC results. These results show that the distortion on the time resolved transmittance due to the presence of the Plexiglas causes a small error on the optical properties of the diffusive medium. The shift of the maximum of T(t) toward longer times observed when s_cl increases, mainly causes an increase of the reduced scattering coefficient. The effect is larger for the coaxial receiver. The variation remains within 15% even for s_cl=10 mm.

The thickness and the optical properties of the slab considered in MC simulations are representative of a breast compressed for optical mammography. The results we obtained suggest that for accurate measurements of optical properties of the breast it is preferable to avoid the use of thick clear layers. For the compression, necessary to obtain a regular geometry, it is therefore preferable to use highly absorbing layers (e.g. black polyvinyl chloride) with small transparent windows for the entrance of the light beam and the exit of the diffuse light, that ensure boundary conditions well modelled by the DE. However, for measurements of optical mammography, whose purpose is the detection and the location of the diseased tissue from the analysis of the relative perturbation that it provokes on measurements of transmittance, we expect a negligible effect of the compression plates, since while they appreciably perturb the background response, their effect on the relative perturbation is expected to be negligible.

3.2 Diffuse reflectance

Reflectance is much more influenced than transmittance by a clear layer at the surface of the diffusive medium. Results of simulations are reported in Figs. 3 and 4. Figure 3 shows the time resolved reflectance R(t) for a semi-infinite medium having µ′_s=1 mm^-1, µ_a=0.01 mm^-1, n_d=1.33, n_cl=1.49, and n_e=1. These values of µ′_s and µ_a are typical of biological tissue at near infrared wavelengths. The thickness of the clear layer ranges between s_cl=0 (no layer) and s_cl=10 mm. The figure reports the results for: ρ=10, 20, 30, and 50mm. The receivers were rings with thicknesses 0.2, 0.6, 1.6, and 4 mm respectively, and acceptance angle of 90 degrees. The results show that a thin clear layer at the surface of the diffusive medium does not appreciably perturb light propagation: even at ρ=50 mm the responses for s_cl=0 (no layer) 0.01, and 0.1 mm are indistinguishable. On the contrary, a thick clear layer provokes a strong distortion: The reflectance at short times strongly increases and the maximum of the curves, especially for large distances, strongly shifts toward shorter times. Nevertheless, the slope at long times remains almost unchanged. Since in an inversion procedure based on the solution of the DE for the homogeneous medium the retrieved reduced scattering coefficient mainly depends on the position of the maximum, and the absorption coefficient on the slope at long times, we expect large errors especially on µ′_s. This is shown in Fig. 4 where the results of the fit have been reported as a function of the source-receiver distance for different values of s_cl. Also for this figure, for a better readability of the results we chose a non-linear scale for the x-axis. As for transmittance, four parameters have been fitted and the value z_e=1.753/µ′_s has been used. For s_cl≤0.1 mm the results of the fit are in excellent agreement with the actual values, the standard error is small (smaller than the marks) and the values of the reduced χ ² smaller than about 1. As s_cl increases discrepancies rapidly increase. Also the values of the reduced χ ² rapidly increase, indicating that the solution of the DE is not suitable to fit the time resolved reflectance when a thick clear layer is present. As expected there are larger discrepancies for the reduced scattering coefficient, that is strongly underestimated, while the values of the absorption coefficient remain reasonably close to the actual values: apart from the datum for ρ=10 mm and s_cl=10 mm discrepancies are within 50%.

Figures 3 and 4 indicate that measurements of time resolved diffuse reflectance carried out on a diffusive medium bounded by a clear layer are no longer suitable to determine the optical properties of the medium (especially the reduced scattering coefficient) when the thickness of the layer is larger than about 0.1 mm. Also measurements of cw reflectance are strongly affected by the clear layer: MC results showed that for the medium considered in Fig. 3 the ratio between the cw reflectance measured with and without the clear layer is R(ρ,s_cl)/R(ρ,s_cl=0)=1.03, 1.5, 2.3, 1.6, and 2.0 for ρ=10 mm and s_cl=0.1, 1, 3, 5, and 10 mm respectively, and becomes 1.05, 2.8, 59.2, 251, and 960 for ρ=50 mm.

 

Fig. 3. Dependence of the time resolved reflectance on the thickness of a clear layer at the surface of a semi-infinite diffusive medium having: µ′_s=1 mm^-1, µ_a=0.01 mm^-1, n_d=1.33, n_cl=1.49, and n_e=1. The results have been reported for four values of the distance of the receiver from the light beam.

Download Full Size | PDF

 

Fig. 4. Absorption and reduced scattering coefficients retrieved from data of Fig. 3 by the fit with the solution of the DE for the homogeneous medium as a function of the thickness of the clear layer.

Download Full Size | PDF

4. Discussion and conclusions

With MC simulations we investigated the effect of a clear layer at the surface of a diffusive medium on measurements of diffuse transmittance and reflectance. To quantify the perturbation due to the clear layer MC results have been fitted with the solution of the DE for the homogeneous medium. Significantly different results have been obtained for transmittance and for reflectance. Measurements of transmittance are slightly perturbed by the clear layer: In a geometry typical for measurements on a compressed breast the reduced scattering coefficient retrieved by the fit differs no more than 15% with respect to the actual value even when the clear layer is 10 mm thick. Even smaller differences have been obtained on the absorption coefficient. We therefore expect that measurements of optical mammography, in which the detection of the diseased tissue is based on measurements of relative perturbation, are slightly affected by the compression plates. We point out that the results we have presented refer to an infinitely extended slab. If a clear layer limits the diffusive medium also laterally (this is the case for a transparent scattering cell for measurements on liquid diffusive media) we expect significantly larger perturbations due to the light piping through the lateral walls.

Measurements of reflectance are strongly perturbed by the clear layer: when s_cl>1 mm the information on µ′_s is completely lost and an error of about 50% can be made on µ_a in the experimental situation typical for measurements on biological tissue. We point out that for MC simulations the clear layer was assumed perfectly transparent (µ′s_cl=µ_acl=0) with perfectly smooth plane surfaces. Small differences with respect to this idealized situation can greatly perturb propagation through the clear layer and significantly different results can be obtained. This is shown in the example of Fig. 5 in which the comparison among the time resolved reflectance simulated for three clear layers having slightly different optical properties has been reported. We also point out that, since n_cl>n_d>n_e, there are the conditions for a guided propagation through the clear layer that, in the idealized situation of perfectly smooth plane surfaces, cannot be established since light cannot penetrate into the clear layer with angles greater than the limit angle. However, recent experiments carried out to develop a phantom for studying light propagation through layered media [11] showed that, due to the roughness of the surfaces or to the scattering of sedimented particles, the clear layer acts as a lossy guide and even a very thin membrane (few tens of microns) immersed into the diffusive medium can significantly perturb measurements of reflectance.

 

Fig. 5. Dependence of the time resolved reflectance on the optical properties of a clear layer 10 mm thick at the surface of a semi-infinite diffusive medium having: µ′_s=1 mm-1, µ_a=0.01 mm^-1, n_d=1.33, n_cl=1.49, and n_e=1. The results have been reported for three combinations of µ′_scl and µ_acl, and for two values of the distance of the receiver from the light beam.

Download Full Size | PDF

Numerical simulations have been also carried out to investigate how the results are affected by the characteristics of the light source and of the receiver, and by the scattering properties of the medium. To investigate the effect of the light source we considered a source emitting a conical beam with a semiaperture of 20° (to mimic a medium illuminated with an optical fiber). The results we obtained were almost indistinguishable with respect to those for the pencil beam. Similar results have been obtained from simulations carried out for a diffusive medium having a scattering function with an asymmetry factor g=0.92, similar to that expected for biological tissues, indicating that the single scattering properties of the diffusive medium have a small effect on light propagation. As for the receiver we investigated the influence of the acceptance angle α : We repeated simulations for α=10, 30, and 90 degrees. Simulations showed that the shape of the temporal response is slightly affected by the acceptance angle. Consequently, the values of µ_afit and µ′_sfit retrieved by the fit have a weak dependence on α. As an example, from the time resolved reflectance referring to a clear layer with s_cl=5 mm and to a receiver with α=10 degrees (collimated detection) we obtained: µ_afit=0.0064, 0.0083, 0.0088, and 0.0103 mm^-1 and µ_s ′_fit=0.35, 0.13, 0.09, and 0.06 mm^-1 for ρ=10, 20, 30, and 50 mm respectively. The corresponding results for α=90 degrees (open detector) were: µ_afit=0.0057, 0.0136, 0.0098, and 0.0104 mm^-1 and µ_s′_fit=0.25, 0.04, 0.09, and 0.05 mm^-1.

To investigate the effect of the scattering properties of the medium we repeated MC simulations for a diffusive medium with µ′_s=0.5 mm^-1 and µ_a=0.01 mm^-1. The results of the fit on reflectance data are summarized in Fig. 6. Also these results show the strong effect of the clear layer on the retrieved optical properties. Although we observe an overall behaviour similar to the one of Fig. 4 referring to µ′_s=1 mm^-1, it is not possible to describe the effect of the clear layer on the fitting results as a systematic error. Therefore, even if a reference measurement on a reference medium could be available, a correction procedure of the fitting results seems unrealistic.

 

Fig. 6. Absorption and reduced scattering coefficients retrieved from simulations for a diffusive medium having µ′_s=0.5 mm^-1 and µ_a=0.01 mm^-1 by the fit with the solution of the DE for the homogeneous medium as a function of the thickness of the clear layer.

Download Full Size | PDF

In conclusion, numerical results reported in Sect. 3 and experimental results presented in [11] indicate that measurements of time resolved reflectance carried out on a diffusive medium bounded by a clear layer are no longer suitable for an accurate reconstruction of the optical properties of the medium even when the thickness of the layer is very small. These conclusions are based on results obtained using for the inversion procedure the solution of the DE for the homogeneous medium, but in our opinion they remain valid even if a more realistic model would be available for the inversion (for instance based on the repetition of MC simulations). In fact, as shown by examples of Fig. 5, large variations of reflectance are caused by even small variations of the optical properties of the clear layer (scattering and absorption, refractive index, roughness). Thus, the boundary effects of the clear layer can be unlikely predicted and accounted for in an inversion procedure based on measurements of reflectance.