1.

Introduction

There is a great deal of interest in the development of noninvasive optical techniques for tissue diagnosis.¹ The main drawback is that light is heavily scattered within tissue and this leads to uncertainty about the volume from which information is retrieved. Our recent research^{2 3} has focused on the use of polarized light to characterize layered media for such applications as measurement of burn and melanoma thicknesses. In burn treatment, thickness is the most important parameter for clinicians diagnosing the need to perform a skin graft;⁴ e.g., a deep second-degree burn penetrates into the dermis to a depth on the order of 1 to 2 mm.⁵ Thickness is important in melanoma prognosis because the cancer may spread if the epidermis–dermis boundary is broken.⁶

Polarized light techniques^{7 8 9 10 11 12} utilize the property that light depolarizes as it propagates and the initial polarization of incident light is lost within relatively few scattering events. This can be used to localize volumes close to the surface. In addition, it has been observed that there are varying rates of depolarization of different initial polarization states with scattering¹³ and it has been demonstrated^{2 3} that this has the potential to characterize layered scattering media and make coarse optical sectioning possible. Previously³ we showed that the spatial distribution of polarized light backscattered from a layered medium is sensitive to variations in optical absorption and thickness of the layer. This represents only part of the problem, and in this work we investigated whether such measurements can be used to determine the absorption coefficients and thickness of a two-layer scattering medium.

Other research groups have investigated methods, to determine the optical properties of layered media. Most notably, Pham et al.;¹⁴ fitted a layered diffusion model to obtain the optical properties from frequency-domain measurements. They varied one parameter at a time to determine the accuracy of their approach and then increased the number of variables. This allows the effectiveness of the fitting algorithm at different stages of complexity of the medium to be established. The results are considered quantitatively accurate if the errors are less than 10 and qualitatively useful if they are between 10 and 20. When three or more parameters are varied, the results become inaccurate. Alexandrakis et al.;¹⁵ investigated a similar problem but used a frequency-domain hybrid Monte Carlo diffusion model to obtain the scattering and absorption of both layers and the thickness of the layers simultaneously. It was found that the hybrid model is more accurate than a diffusion model in recovering optical properties of the upper layer and the thickness, but the errors are still fairly large (>30). However, diffusion theory cannot be used to model polarized light and so we considered using neural networks trained on data obtained from Monte Carlo simulations to recover the optical properties of layered media. Encouraging results have been obtained when neural networks are used to extract the optical coefficients of semi-infinite media,^{16 17} but in this work the more difficult problem of layered media was considered.

The following section presents the analysis methods, including details of the Monte Carlo simulation; the samples used; and the neural network applied. Section 3 presents the results of an investigation to determine whether a neural network trained on polarized light measurements can be used to characterize layered scattering media. Discussions and conclusions follow in Secs. 4 and 5, respectively.

2.

Theory

2.2.

Samples and Sample Geometries

Figure 1 shows the general form of the sample under investigation. The model is capable of simulating multiple layers, each infinite in the x–y plane with a semi-infinite lower layer. However, for simplicity here we consider only a two-layer medium, with an upper planar layer of thickness d and a semi-infinite lower layer. There is a mismatch of 1.4 in the refractive index at the air–tissue interface; at the tissue–tissue interface the layers are index matched. The medium is composed of a monodispersion of Mie scatterers with the mean cosine of the scattering angle, g=0.92 and a size parameter ka=13.9, which are consistent with typical tissue scatterers.^{1 20} Each layer has different scattering (μ_s) and absorption (μ_a) coefficients and absorption is added postsimulation using Lambert-Beer’s law, depending on the propagation distance within each layer. The absorption and scattering properties of the media analyzed are stated in mean-free paths (mfp) where 1 mfp=1/μ_s so that the results can be easily scaled for a wide range of media.

Figure 1

General geometry of the sample.

3.

Results

We investigated the ability of a neural network to determine μ_a1, μ_a2, and d of a two-layer scattering medium from polarized light measurements. In all cases we assumed that the values of g and the scattering coefficients of both layers are known from in vitro studies. We used an approach similar to that of Pham et al.;¹⁴ and initially varied one parameter at a time and then extended the number of variables. This allows better characterization of the performance of the neural networks at different stages in the complexity of the inversion. Sections 3.1 and 3.2 describe measurements varying μ_a2 only and μ_a1 only, respectively, keeping one layer at a constant absorption. In Sec. 3.3, both μ_a1 and μ_a2 are varied. In Sec. 3.4, the measurement of the layer is thickness alone is considered. Recovery of both μ_a1 and d is discussed in Sec. 3.5. Finally, the effects of restricting the range of variables are considered in Sec. 3.6. For all the tables, data are presented to two significant digits.

3.1.

Varying Absorption (μ_a2) of the Lower Layer

This section describes the ability of a neural network to recover μ_a2 over the range 7.55e-3 to 0.04005 mfp⁻¹ in steps of 2.5e-4 mfp⁻¹. The remaining optical coefficients are fixed at μ_a1=0.001 mfp ⁻¹, μ_s1=1 mfp ⁻¹, and μ_s2=0.5 mfp ⁻¹. Three thicknesses are considered; d=5, 10, and 30 mfp. A typical plot (d=5 mfp, network trained on the first-order moments of channels 1 and 2) of actual versus recovered μ_a2 is shown in Fig. 2 to demonstrate the effectiveness of the method. A straight line representing the ideal case is also shown to aid visualization. As observed by Pham,¹⁴ at low absorption values the percentage of errors is high even though the absolute errors are small.

Figure 2

The μ_a2 values obtained with a neural network trained on first-order moments of channels 1 and 2 (d=5 mfp). A straight line is shown to aid visualization.

The percentage error for networks trained with different combinations of the first- and second-order normalized moments for different thicknesses of a layer is shown in Table 2. In principle, the network should be able to take all input parameters and weight each input accordingly to provide the optimum performance. However, to gain a better physical insight into the problem, the network is also trained using either the first- or second- or both first- and second-order moments of the linear channels (1 and 2) or the circular channels (3 and 4). This will reveal whether all the required information is contained in a particular combination of measurements. To determine whether polarization measurements are essential, the moments of the total intensity (unpolarized) measurements are also included. The mean values of rows and columns are shown to emphasize trends in the results.

Table 2

Percentage error in μa2 when it is varied over the range 7.55e-3 to 0.04005 mfp−1 and μa1 is constant (0.001 mfp−1).
d (mfp) Training data	5 ()	10 ()	30 ()	Mean ()
Channels 1 and 2, first-order moment	7.1	17	35	20
Channels 3 and 4, first-order moment	4.8	5.1	34	15
Total intensity, first-order moment	20	2.4	27	17
Channels 1 and 2, second-order moment	18	49	59	42
Channels 3 and 4, second-order moment	14	14	49	26
Total intensity, second-order moment	12	9.7	17	13
Channels 1, 2, 3, and 4, first- and second-order moments	7.7	23	54	28
Total intensity, first- and second-order moments	12	9.8	39	20
Mean ()	12	16	39	22

For the d=5 mfp and d=10 mfp layers, the majority of the results are within the acceptable margins of error (<20). In general, the polarized light measurements offer no significant improvement over total intensity measurements. For the thinnest layer (d=5 mfp), the error in μ_a2 is lowest because the upper layer has relatively little effect on the measurements.

3.4.

Varying Layer Thickness (d)

Previously³ we demonstrated the sensitivity of the different polarization channels to thickness where thickness was modeled using a single-layer medium. This is equivalent to an upper layer with a varying d above a second layer that is totally absorbing. The advantage of this approach is that a single Monte Carlo simulation can be used to model different d is by recording the maximum visitation depth of a photon and discarding those absorbed in the second medium. This improves the efficiency of the simulations and allows the moments of the distributions over a range of layer thicknesses to be calculated (d=2 to 20 mfp in steps of 2 mfp and d=20 to 60 mfp in steps of 5 mfp). However, it should be noted that this represents the extreme contrast for a two-layer medium. For typical tissue-scattering mean-free paths (0.1 mm), this corresponds to a thickness of 0.2 to 6 mm, which covers the range of burn thicknesses and the thickness of many skin lesions. For studies of superficial skin lesions, more investigation is required although we anticipate the trends observed in this case to be valid. Figure 3 contains the values of d for networks trained on the first- and second-order moments of the polarization channels 1, 2, 3, and 4 [Figs. 3(a) and 3(b)]. A straight line representing the ideal case is also shown to aid visualization. When both scattering and absorption are fixed, all measurements accurately extract the layer’s thickness (first-order moment error=2.5, second-order moment error=1.2).

Figure 3

Values of d obtained from networks trained on (a) first-order moments of channels 1, 2, 3, and 4; (b) second-order moments of channels 1, 2, 3, and 4. A straight line is shown to aid visualization.

4.

Discussion

We have investigated whether polarized light measurements can be used to determine the absorption coefficient and thickness of a two-layer scattering medium. Inversion of the measured data cannot be achieved by analytical solutions such as diffusion theory, owing to the presence of weakly scattered polarization-maintaining light. A neural network therefore provides a useful tool for analyzing such data and determining the measurable parameters that are most sensitive to absorption or thickness of the layer.

Our results show that polarized light measurements are more sensitive to the properties of superficial tissue than unpolarized measurements. This is demonstrated by the lower errors obtained with polarized light measurements when the upper layer’s absorption is extracted as described in Secs. 3.3, 3.5, and 3.6, and d is obtained as described in Sec. 3.5 and 3.6. This is because the destruction of polarization information by scattering causes sensitivity to superficial tissue.^{2 3 9 10} However, for the same reason, polarized light measurements offer no significant improvement over unpolarized measurements when they are used to obtain μ_a2.

In the majority of cases, training the network on all the available polarized data provides the best performance. The advantage of using combined first- and second-order moments rather than only the first- or second-order moments was discussed in a previous paper³ using a method in which contours were plotted for measured values of first- and second-order moments for different values of μ_a1 and μ_a2. The steep gradients and orthogonal contours that were observed indicated that the data are robust and suitable for inversion. Therefore although networks trained using data from only the first-order moments generally outperform those trained using only normalized second-order moment data, there is still more information to be obtained by combining the two sets. This indicates that this application benefits from the ability of neural networks to extract subtleties from the data.

To completely characterize a two-layer scattering medium, seven parameters need to be determined: d, μ_a1, μ_a2, μ_s1, μ_s2, and the values of g in both layers. This is an ill-conditioned problem and so it is necessary to assume prior knowledge of some of the parameters. In this paper we considered the recovery of absorption and thickness while leaving g and the scattering coefficients constant. In the majority of cases considered, polarized light measurements are capable of determining the absorption coefficient relatively well. Clearly, when only the absorption of a single layer is varied (Tables 2 and 3), it is a comparatively simple problem to determine the absorption coefficient, and the only significant increase in error occurs when the top layer is thin (5 mfp) and light does not adequately sample this region. In the more difficult cases of varying the absorption and thickness of the layer, it is still possible to determine the absorption coefficient to an acceptable accuracy if the most appropriate measurables are selected. Tables 4 and 5 show that if neural networks are trained using both the first- and second-order moments of the polarized light measurements, then acceptable accuracy can be achieved. The only exception is obtaining μ_a1 when both μ_a1 and μ_a2 are varied (Table 3) and in this case neural networks trained using both the first- and second-order moments of the polarized light measurements still provide the lowest errors.

Determination of the thickness of a layer is more difficult than determining absorption because when both absorption and thickness are varied, d can only be determined when all the available polarized light data are used (Table 5). When the range of absorption is restricted (Table 6), the accuracy of unpolarized light measurements is improved significantly, but the improvement in polarized light measurements is insignificant. When the range of thickness is restricted, significant improvements in the performance of both polarized and unpolarized measurements are observed. It is not unreasonable to make assumptions about the optical properties of the tissue under investigation since these are well documented.¹ For certain applications, it is useful to restrict the range of absorption values to known in vitro measurements to provide accurate inversion, e.g., necrotic tissue overlying healthy tissue in characterization of burns or a lesion containing melanin overlying melanin-free tissue. Further improvements could possibly be achieved by using a larger training set since at present relatively few independent Monte Carlo simulations are used in the training process.

We have presented preliminary results that demonstrate the potential of using polarized light and neural networks for characterizing layered media. However, several factors need to be considered for practical implementation. As previously discussed, some assumption of the optical properties or the range of optical properties from known in vitro values is necessary. In addition, we have assumed an ideal geometry with layers that are uniform in thickness and infinite in the lateral dimension. In this case photons that are multiply scattered and sample a large volume provide better data for training the neural networks. In practice, the layer’s thickness will vary and the spatial localization obtained by the subtraction of different polarization channels may be beneficial. As an indication of the potential of this approach, we have obtained preliminary results for networks trained on data from subtracted polarization channels for the case considered in Table 5. The error in μ_a1 was 5, but the error in d was relatively high (35). Future research will investigate this approach further.

One potential approach is to obtain the properties of the upper layer using polarized light measurements and then use this knowledge to obtain the properties of the lower layer from unpolarized light measurements. To ensure accurate determination of the moments, a high dynamic range detector is required, owing to the range of intensities observed in backscattered distributions. First-order moments are more easily measured because the normalized second-order moments are more heavily weighted by photons emerging further from the source and are susceptible to noise. An alternative method we are currently investigating is whether the neural network approach can be applied to single-point polarized light measurements at a range of wavelengths. This would be advantageous because it would allow a full-field measurement to be obtained using a simple CCD camera configuration.¹⁰ In the specific case of diagnosis of melanoma, other parameters,^{6 23} such as melanoma shape or size, could also be useful additions to the neural network training data. Other issues that need to be resolved before implementation of such techniques becomes practical include the effects of the birefringence of collagen,²⁴ layer nonuniformity, and a mismatch of the refractive index between the tissue layers.

5.

Conclusions

The potential use of neural networks for determining the properties of a two-layer scattering medium has been demonstrated. Polarized light measurements provide better performance than unpolarized measurements in obtaining the absorption coefficient of the upper layer and the layer’s thickness because of the sensitivity of polarized light to superficial tissue. However, polarized light measurements offer no significant benefit in obtaining the absorption of the lower layer. Improvements in performance can be achieved by restricting the range of optical coefficients to those corresponding to the documented range of tissue values.