2.1.

Spatial Frequency Domain Reflectance

Spatial frequency domain spectroscopy involves illumination of media with a spatial pattern of the form,

The depth-sensitivity of spatial frequency domain imaging has been established by several publications.^{5, 8, 9, 10} Cuccia ⁵ demonstrated that spatially modulated illumination facilitates quantitative wide-field optical property mapping and sensitivity to buried heterogeneities in turbid media. They performed spatial frequency domain imaging of tissue simulating phantoms with embedded heterogeneities using 42 spatial frequencies f _x between 0 and 0.6 mm⁻¹. They showed that changes in demodulated reflectance at low versus high spatial frequencies are sensitive to the lower versus upper embedded heterogeneities. This work showed that spatial frequency domain spectroscopy can detect contrast between background and heterogeneity, but did not provide a quantitative technique for determining the optical properties of the heterogeneity itself. Konecky ⁸ used spatial frequency domain imaging to detect tube heterogeneities buried in homogeneous tissue simulating phantoms. They measured spatial frequency dependent reflectance from the heterogeneous phantoms at 11 spatial frequencies. They then used an inverse method based on the diffusion approximation to reconstruct tomographic contrast images of the buried tubes. This model, however, may not be applicable if the highly absorbing heterogeneity is close to the surface (i.e., the epidermis of the skin).

2.2.

Artificial Neural Networks

The present work takes a semi-empirical approach toward quantifying heterogeneity in two layered media. Instead of estimating the spatial frequency dependent reflectance with an analytical or approximate model, we performed multiple Monte Carlo simulations and then fit a machine learning algorithm—an artificial neural network—to the output data. An artificial neural network is a data structure that can accurately approximate a nonlinear relationship between a set of input and output parameters from multiple samples of input–output pairs.²³ Unlike approximate models such as the diffusion approximation, a neural network can be trained to predict tissue reflectance for strongly and weakly absorbing media. Furthermore, a neural network can be trained to directly estimate an inverse relationship between measured tissue reflectance and the tissue's optical properties, thus avoiding iterative techniques such as nonlinear least-squares fitting.

Neural network-based approaches to determining optical scattering and absorption coefficients of biological tissue from tissue and phantom reflectance measurements have been proposed by other investigators. ^{24, 25, 26, 27, 28, 29, 30} For example, Farrell, trained a neural network to determine the absorption and reduced scattering coefficient of homogeneous biological tissue from spatially resolved diffuse reflectance measurements at eight source-detector separations.^{27, 28} The authors solved for the spatially resolved diffuse reflectance function for multiple input optical properties using the spatially resolved diffusion approximation. The artificial neural network was then trained to map the spatially resolved reflectance to the set of optical properties. This resulted in a functional inverse relationship between a measurement of reflectance and tissue optical properties without the need to perform iterative least-squares fitting. In fact, the model proposed by Farrell ^{27, 28} was later used by Bruulsema ³¹ to measure changes in the skin's scattering coefficient as a function of changes in blood glucose concentration. Since the underlying function used to generate the training set was the diffusion approximation in a semi-infinite, homogeneous medium, the analysis presented by Farrell is limited to wavelengths in the NIR and for weakly absorbing media that are approximately homogeneous.^{27, 28} On the other hand, Pfefer developed an artificial neural network-based technique for extracting the absorption and reduced scattering coefficients from spatially resolved reflectance measurements of highly absorbing media.²⁹ Their approach was similar to that of Farrell,^{27, 28} however, Monte Carlo simulations were used as the underlying photonics model in strongly absorbing media. Additionally, Wang developed a neural network to detect the optical properties of two layer media using spatially resolved reflectance measurements.³² They produced a training set from Monte Carlo simulations and then validated the trained neural network on two layer tissue simulating phantoms. They then used their inverse model to detect the absorption and reduced scattering coefficients of two layer media in the ultraviolet and visible ranges, for which absorption coefficient is typically greater than reduced scattering coefficient. However, they reported large prediction errors in all their parameters (absorption and scattering coefficients of the top and bottom layer) that ranged between 0.15 and 1.21 mm⁻¹ (20% to 120%).

This study that we have carried out presents a forward and inverse model designed for spatial frequency domain measurements of two layer media. First, the practical limitations of spatial frequency domain imaging are discussed in terms of coupling between top layer absorption coefficient and thickness, and insensitivity of measured reflectance to the top layer's reduced scattering coefficient. Then, a practical description of a two layer tissue model is developed and tested in simulated reflectance spectra from human skin.

3.1.

Two Layer Tissue Model

Figure 1 shows the two layer geometry, optical properties, and illumination considered in this study. The superficial layer was illuminated by a collimated light source. The spatial frequency f _x of the illumination pattern was considered to range between 0 and 0.25 mm⁻¹. The index of refraction, absorption coefficient, reduced scattering coefficients, and thickness of layer 1 are denoted by n ₁, μ_{a, 1}, [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ , and d ₁, respectively. The index of refraction n ₁ was assumed to be 1.40 to represent biological tissue.³³ The absorption coefficient μ_{a, 1} was assumed to range between 0.10 and 2.00 mm⁻¹ (Refs. 33, 34, 35, 36, 37, 38, 39). The reduced scattering coefficient [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ was assumed to range between 0.50 and 2.00 mm⁻¹. Finally, the thickness d ₁ was assumed to range between 15 and 150 μm which is typical of many regions of the human body.^{40, 41, 42}

Fig. 1

Illustration of the two layer geometry, optical properties, and illumination considered.

The index of refraction, absorption coefficient, and reduced scattering coefficients of layer 2 are denoted by n ₂, μ_{a, 2}, and [TeX:] ${\rm \mu}^ \prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ , respectively. For simplicity, the index of refraction of layer 1 was assumed to be equal to that of layer 2 but not equal to that of air (i.e., n ₁ = n ₂ = 1.40). The absorption coefficient of layer 2, μ_{a, 2}, was assumed to range between 0.01 and 0.20 mm⁻¹, while reduced scattering coefficient [TeX:] ${\rm \mu}^ \prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ was assumed to range between 0.50 and 2.00 mm⁻¹ (Refs. 33, 34, 35, 36, 37, 38, 39).

The spatial frequency dependent reflectance [TeX:] $R({n}_1,{d}_1,{\rm \mu }_{{\rm a},1} {\rm,\mu}^ \prime_{{\rm s},1} {\rm,\mu }_{{\rm a},2} {\rm,\mu}^ \prime_{{\rm s},2},{f}_{x})$ $R(n_1,d_1,{\mu }_{\mathrm{a},1}{,\mu }_{\mathrm{s},1}^{\prime }{,\mu }_{\mathrm{a},2}{,\mu }_{\mathrm{s},2}^{\prime },f_x)$ was determined with Monte Carlo simulations. Monte Carlo simulation software developed by Wang and Jacques⁴³ was used to calculate the radial diffuse reflectance function [TeX:] $R({n}_1,{d}_1,{\rm \mu }_{{\rm a},1} {\rm,\mu}^ \prime_{{\rm s},1} {\rm,\mu }_{{\rm a},2} {\rm,\mu}^ \prime_{{\rm s},2},{\rm \rho })$ $R(n_1,d_1,{\mu }_{\mathrm{a},1}{,\mu }_{\mathrm{s},1}^{\prime }{,\mu }_{\mathrm{a},2}{,\mu }_{\mathrm{s},2}^{\prime },\rho )$ , where ρ was the radial distance from the simulation's origin. Then, the spatial frequency domain diffuse reflectance function was calculated with the Hankel transform using the method suggested by Cuccia,⁶ namely,

Figure 2 shows an example of this procedure and illustrates the minimal effect of large changes in [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ on R(f _x). It shows estimates of the spatial frequency diffuse reflectance R(f _x) as a function of spatial frequency f _x for n ₁ = 1.40, d ₁ between 15 and 150 μm, μ_{a, 1} equaling 1 mm⁻¹, [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ between 0.50 and 2.50 mm⁻¹, μ_{a, 2} equaling 0.01 mm⁻¹, [TeX:] ${\rm \mu}^ \prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 2.00 mm⁻¹, and f _x between 0 and 0.25 mm⁻¹ predicted by Monte Carlo simulations and Eq. 2. Each bundle of curves was generated by varying [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ between 0.50 and 2.50 mm⁻¹ while keeping all other parameters constant. Figure 2 illustrates the weak dependence of R(f _x) on [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ for the range of d ₁ considered. Large changes in [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ resulted in minimal changes in the reflectance of the two layer medium. Thus, a limitation of the present model is that it is insensitive to [TeX:] ${\rm \mu}^ \prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ for the range of d ₁ considered. Therefore, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ was assumed to be equal to 1 mm⁻¹ in developing the forward and inverse models presented here.

Fig. 2

Estimates of the reflectance as a function of spatial frequency f _x for n ₁ = 1.4, d ₁ between 15 and 150 μm, μ_{a, 1} equaling 1 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ between 0.50 and 2.50 mm⁻¹, μ_{a, 2} equaling 0.01 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 2.00 mm⁻¹, and f _x between 0 and 0.25 mm⁻¹ predicted by Monte Carlo simulations. Each bundle of curves was generated by sweeping [TeX:] ${\mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ between 0.50 and 2.50 mm⁻¹ while keeping all other parameters constant.

Additionally, Fig. 3 shows estimates of the spatial frequency dependent reflectance as a function of spatial frequency f _x between 0 and 0.25 mm⁻¹ predicted by Monte Carlo simulations. It illustrates the strong coupling between μ_{a, 1} and d ₁. For example, the solid curve indicated by Fig. 3a was generated with μ_{a, 1} equaling 0.1 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, d ₁ equaling 100 μm, μ_{a, 2} equaling 0.01 mm⁻¹ and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 1.7 mm⁻¹. On the other hand, the broken curve that overlays the solid curve was generated with μ_{a, 1} equaling 0.21 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 0.5 mm⁻¹, d ₁ equaling 50 μm, and the same values of μ_{a, 2} and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ . It is apparent that increasing μ_{a, 1} and decreasing d ₁ can produce nearly identical spatial frequency dependent reflectance function. Figure 3 also shows a solid curve indicated by (b) that was generated with μ_{a, 1} equaling 0.2 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 2 mm⁻¹, d ₁ equaling 20 μm, and μ_{a, 2} equaling 0.01 mm⁻¹ and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 1.0 mm⁻¹. Again, a similar spatial frequency dependent reflectance function was generated with μ_{a, 1} equaling 0.1 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, and d ₁ equaling 50 μm for the same values of μ_{a, 2} and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ . Increasing the top layer's thickness has approximately the same effect as increasing its absorption coefficient. This coupling phenomenon was also observed for all [TeX:] $\tau^\prime_1 > 0$ ${\tau }_1^{\prime }>0$ . In fact, for [TeX:] $\tau^\prime_1$ ${\tau }_1^{\prime }$ greater than 4, the top layer was almost completely occluded the bottom layer and the two layer medium effectively became a single layer medium with the optical properties of the top layer. Thus, a limitation of the present model is that the individual values of μ_{a, 1} and d ₁ were not detectable from spatial frequency dependent reflectance. For this reason, this study was limited to determination of the reduced optical thickness [TeX:] $\tau^\prime_1$ ${\tau }_1^{\prime }$ defined as

Fig. 3

Estimates of the diffuse reflectance as a function of spatial frequency f _x between 0 and 0.25 mm⁻¹ predicted by Monte Carlo simulations. The solid curve indicated by (a) was generated with μ_{a, 1} equaling 0.1 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, d ₁ equaling 100 μm, μ_{a, 2} equaling 0.01 mm⁻¹, and [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 1.7 mm⁻¹, while the broken curve was generated with μ_{a, 1} equaling 0.21 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 0.5 mm⁻¹, d ₁ equaling 50 μm and the same values of μ_{a, 2} and [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ . The solid curve indicated by (b) was generated with μ_{a, 1} equaling 0.2 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 2 mm⁻¹, d ₁ equaling 20 μm, μ_{a, 2} equaling 0.01 mm⁻¹, and [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ equaling 1.0 mm⁻¹ while the broken curve was generated with μ_{a, 1} equaling 0.1 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, d ₁ equaling 50 μm and the same values of μ_{a, 2} and [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ .

3.2.

Generating Training Set

The model parameters d ₁, μ_{a, 1}, [TeX:] ${\rm \mu }_{{\rm a},2},\;{\rm and}\;{\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{a},2},\mathrm{and}{\mu }_{\mathrm{s},2}^{\prime }$ were sampled according to a uniform distribution,

3.3.

Generating a Validation Set

A validation set was generated to test the performance of the neural network. Ten values for each model parameter d ₁, μ_{a, 1}, [TeX:] ${\rm \mu }_{{\rm a},2},\;{\rm and}\;{\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{a},2},\mathrm{and}{\mu }_{\mathrm{s},2}^{\prime }$ were selected along a uniform four dimension grid for a total of 10,000 samples. The number of test samples was chosen such that the entries of the covariance between the model parameters d ₁, μ_{a, 1}, μ_{a, 2}, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ and R(f _x = 0) for both the training and validation sets were within 1% of each other. This ensured that the test set was statistically close to the training set. As with the training set, the scattering coefficient [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ was equal to 1 mm⁻¹. Then, Monte Carlo simulations were performed to determine the spatial frequency domain diffuse reflectance for each test sample. Training was performed on the training set and assessment of model accuracy was performed with the validation set.

3.4.

Training

A neural network was trained to map a set of input model parameters to a frequency domain diffuse reflectance,

A second neural network was trained to map the spatial frequency domain reflectance R(f _x) to a set of model parameters [TeX:] $\tau _1,\mu _{\rm a,2},\mu ^\prime_{\rm s,2}$ ${\tau }_1,{\mu }_{\mathrm{a},2},{\mu }_{\mathrm{s},2}^{\prime }$ , namely,

Prior to training, the model parameters and the spatial frequency dependent reflectance were normalized to range between −1 and 1. Training was performed using the MATLAB software package (The MathWorks, Incorporated, Natick, Massachusetts) with Neural Networks and Chemometrics toolbox routines. Each neural network converged in less than 1000 iterations.

4.1.

Forward Problem: NN _f

Figures 4a, 4b compare estimates for 10,000 validation samples of the diffuse reflectance by the neural network [TeX:] $NN_f ({{d}_1,{\rm \mu }_{{\rm a},1},{\rm \mu }_{{\rm a},2},{\rm \mu} ^\prime_{{\rm s},2} })$ $N{N}_f\left(d_1,{\mu }_{\mathrm{a},1},{\mu }_{\mathrm{a},2},{\mu }_{\mathrm{s},2}^{\prime }\right)$ and Monte Carlo simulation for f _x equaling 0 and 0.21 mm⁻¹, respectively, for the validation set with d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹. The average relative difference between estimates of R(f _x = 0) by Monte Carlo simulation and the neural network was 0.25%, while the maximum absolute difference was 3.33%. Similarly, the average relative difference between estimates of R(f _x = 0.21) by Monte Carlo simulation and the neural network was 0.38%, while the maximum absolute difference was 4.34%. Figure 4 illustrates that the neural network generalized the relationship expressed in Eq. 5 from the 50,000 training examples. The goodness of fit between estimates of the reflectance by Monte Carlo simulations and the neural network to a linear model (y = x) was calculated for 21 values of f _x between 0 and 0.25 mm⁻¹ and found to be greater than 0.9998 for all cases considered. An r-squared value of unity suggests a perfect linear relationship. It is apparent that the present neural network model exhibits a nearly perfect correlation to Monte Carlo simulations for all f _x considered.

Fig. 4

Estimates of the reflectance as for n ₁ = 1.4, d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${ \mu} ^\prime_{{\rm s,1}}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${ \mu} ^\prime_{{\rm s,2}}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹, and f _x equaling (a) 0.00 mm⁻¹ and (b) 0.21 mm⁻¹ predicted by Monte Carlo simulations and the trained neural network for the validation set.

4.2.

Inverse Problem with the Least-Squares Fitting

The forward model [TeX:] $NN_f ({{d}_1,{\rm \mu }_{{\rm a},1},{\rm \mu }_{{\rm a},2},{\rm \mu} ^\prime_{{\rm s},2} })$ $N{N}_f\left(d_1,{\mu }_{\mathrm{a},1},{\mu }_{\mathrm{a},2},{\mu }_{\mathrm{s},2}^{\prime }\right)$ defined in Eq. 5 was used along with an iterative inverse method to estimate parameters [TeX:] ${\rm \hat d}_1,{\rm \hat \mu }_{{\rm a},1},{\rm \hat \mu }_{{\rm a},2},{\rm and},{\rm \hat \mu ^\prime }_{{\rm s},2}$ ${\widehat{\mathrm{d}}}_1,{\widehat{\mu }}_{\mathrm{a},1},{\widehat{\mu }}_{\mathrm{a},2},\mathrm{and},{{\widehat{\mu }}^{\prime }}_{\mathrm{s},2}$ from a spatial frequency dependent reflectance R _ref(f _x). This is done by choosing [TeX:] ${\rm \hat d}_1,{\rm \hat \mu }_{{\rm a},1},{\rm \hat \mu }_{{\rm a},2},\;{\rm and}\;{\rm \hat \mu ^\prime }_{{\rm s},2}$ ${\widehat{\mathrm{d}}}_1,{\widehat{\mu }}_{\mathrm{a},1},{\widehat{\mu }}_{\mathrm{a},2},\mathrm{and}{{\widehat{\mu }}^{\prime }}_{\mathrm{s},2}$ to minimize a quadratic cost function,

The values of d ₁ and μ_{a, 1} could not be found with any certainty with this method. In fact, the converged values [TeX:] $\hat d_1$ ${\widehat{d}}_1$ and [TeX:] $\hat \mu _{\rm a,1}$ ${\widehat{\mu }}_{\mathrm{a},1}$ were strongly dependent on the initial conditions chosen for minimization while [TeX:] $\hat \mu _{\rm a,2}$ ${\widehat{\mu }}_{\mathrm{a},2}$ and [TeX:] $\hat \mu ^\prime_{\rm s,2}$ ${\widehat{\mu }}_{\mathrm{s},2}^{\prime }$ where not. However, the product [TeX:] $\hat \tau ^\prime = \hat d_1 (\hat \mu _{\rm a,1} + \hat \mu ^\prime_{\rm s,1})$ ${\widehat{\tau }}^{\prime }={\widehat{d}}_1({\widehat{\mu }}_{\mathrm{a},1}+{\widehat{\mu }}_{\mathrm{s},1}^{\prime })$ was stable with respect to initial conditions. Figure 5a compares the error between the true value of [TeX:] $\hat \tau ^\prime $ ${\widehat{\tau }}^{\prime }$ and the value estimated by minimizing L in Eq. 7 for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹. Figure 5b also shows a histogram of the relative percent error in estimating [TeX:] $\hat \tau ^\prime $ ${\widehat{\tau }}^{\prime }$ . In this range, the average absolute percent relative error between the input and estimated parameters was 57%. To identify the reason for this high average relative error, we defined a sensitivity of R _ref(f _x) to τ′ as,

Fig. 5

(a) Estimates of [TeX:] $\hat \tau ^\prime $ ${\widehat{\tau }}^{\prime }$ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by minimizing L in Eq. 7. (b) Histogram of the relative percent estimation error.

Fig. 6

Sensitivity S[R(f _x), τ′] as a function of spatial frequency.

Figure 7a shows the input and estimated values of μ_{a, 2} for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by minimizing L in Eq. 7. Figure 8b shows a histogram of the relative percent estimation error in estimating μ_{a, 2}. In this range, the average absolute percent relative error between the input and estimated parameters was 14.7%. Similarly, Fig. 8a shows the input and estimated values of [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by minimizing L in Eq. 7. Figure 8b shows a histogram of the relative percent estimation error in estimating [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ . In this range, the average absolute percent relative error between the input and estimated parameters was 4.3%.

Fig. 7

(a) Estimates of μ_{a, 2} for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by minimizing L in Eq. 7. (b) Histogram of the relative percent estimation error.

Fig. 8

(a) Estimates of [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by minimizing L in Eq. 7. (b) Histogram of the relative percent estimation error.

4.3.

Direct Approach with Neural Network: NN _i

Iterative least-squares fitting techniques are effective for determining optical properties from diffuse reflectance measurements. However, the accuracy and computational efficiency of least-squares fitting may be susceptible to initial conditions. A direct mapping between a spatial frequency dependent reflectance and optical properties is desired and was denoted in this study by NN _i. A neural network was trained on the same set of data presented in Sec. 4.2. However, six spatial frequencies between 0 and 0.25 mm⁻¹ were used as inputs and the parameters [TeX:] $\tau ^\prime_1,\mu _{\rm a,2}$ ${\tau }_1^{\prime },{\mu }_{\mathrm{a},2}$ and [TeX:] $\mu ^\prime_{\rm s,2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ were used as outputs in an effort to create a direct relationship between measured tissue reflectance and its optical properties

Figure 9a compares the error between the true value of [TeX:] $\hat \tau ^\prime $ ${\widehat{\tau }}^{\prime }$ and the value estimated with Eq. 6 for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹. Figure 9b also shows a histogram of the relative percent error in estimating [TeX:] $\hat \tau ^\prime $ ${\widehat{\tau }}^{\prime }$ . In this range, the average absolute percent relative error between the input and estimated parameters was 43%. In fact, the direct neural network approach applied to 6 spatial frequencies performed on average 11% better than the least-squares approach applied to 21 spatial frequencies. Additionally, if d ₁ was restricted to be greater than 50 μm and while μ_{a, 2} was kept smaller than 0.09 mm⁻¹, the average absolute percent relative error between the input and estimated parameters fell to only 15% for reasons already discussed.

Fig. 9

(a) Estimates of τ′ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by the direct neural network approach. (b) Histogram of the relative percent estimation error.

Figure 10a shows the input and estimated values of μ_{a, 2} for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by the direct neural network approach. Figure 10b shows a histogram of the relative percent estimation error in estimating μ_{a, 2}. In this range, the average absolute percent relative error between the input and estimated parameters was 8.0%. Indeed, the mean absolute percent estimation relative error for the direct neural network model is almost half of the error associated with the least-squares fitting method presented in Sec. 4.2.

Fig. 10

(a) Estimates of μ_{a, 2} for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by the direct neural network approach. (b) Histogram of the relative percent estimation error.

Figure 11a shows the input and estimated values of [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by the direct neural network approach. Figure 11b shows a histogram of the relative percent estimation error in estimating μ_{a, 2}. In this range, the average absolute percent relative error between the input and estimated parameters was 3.8%. The neural network performed slightly better in determining [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ than the least-squares fitting method.

Fig. 11

(a) Estimates of [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ for d ₁ between 15 and 150 μm, μ_{a, 1} between 0.10 and 2.00 mm⁻¹, [TeX:] ${\rm \mu} ^\prime_{{\rm s},1}$ ${\mu }_{\mathrm{s},1}^{\prime }$ equaling 1 mm⁻¹, μ_{a, 2} between 0.01 and 0.20 mm⁻¹, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ between 0.50 and 2.00 mm⁻¹ determined by the direct neural network approach. (b) Histogram of the relative percent estimation error.

Table 1 summarizes the performance of the iterative and direct inverse methods. It shows the mean, mean absolute, and standard deviation of the relative percent difference between input values of [TeX:] $\tau^\prime_1$ ${\tau }_1^{\prime }$ , μ_{a, 2}, and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ and their estimates. The mean error is an indicator of the average bias in prediction of a parameter. For example, the iterative inverse method underpredicts the optical thickness [TeX:] $\tau^\prime_1$ ${\tau }_1^{\prime }$ by 11% while the direct method overpredicts it by 23%. The mean absolute error is an estimate of model accuracy without regard for sign. For example, the direct method exhibits a mean absolute error of 8% in prediction of μ_{a, 2} while the iterative inverse method exhibits a larger error of 14.7%. Finally, the standard deviations reported in Table 1 represent the width of the error distribution. It is apparent that prediction of [TeX:] $\tau^\prime_1$ ${\tau }_1^{\prime }$ exhibits a larger variance than the other parameters and may thus be considered less reliable. Table 1 indicates that the computationally efficient direct method performs as well as the iterative inverse method in predicting μ_{a, 2} and [TeX:] ${\rm \mu} ^\prime_{{\rm s},2}$ ${\mu }_{\mathrm{s},2}^{\prime }$ , but exhibits a larger bias in predicting [TeX:] ${\rm \tau }_1^{\rm ^\prime }$ ${\tau }_1^{\prime }$

Table 1

Mean and standard deviation of the relative percent error between estimates of the model parameters $\tau ^\prime_1,\mu _{\rm a,2}$ τ1′,μa,2 , and $\mu ^\prime_{\rm s,2}$ μs,2′ predicted by minimizing L in Eq. 7 and by the direct inverse method given by Eq. 6 with respect to Monte Carlo simulations. The mean of the absolute relative error is also shown.