2.1.

Mie Theory Interpretation

Mie theory provides an exact solution for the scattering and the anisotropy coefficients of perfect dielectric spheres of arbitrary size in a uniform background medium.^{24, 25} Using this theoretical framework, the reduced scattering spectra of bulk homogeneous samples can be approximately expressed by the following equation:

Since the reduced scattering spectra ${\mu }_s^{\prime }\left(\lambda \right)$ depends on the histogram of particle number density per unit particle size $f\left(a_i\right)$ , it is important to carefully analyze how these particular functions might impact the results of the study. Three histogram shapes have been examined, including (1) a step function, (2) a normalized Gaussian function, and (3) an exponentially decaying function. Each is used with the same average particle size and the same total number density to determine differences in ${\mu }_s^{\prime }$ caused by different histogram shape assumptions.

In Eq. 2, the particle number density only changes the magnitude of the scattering coefficients, but not the shape of the scattering spectra. Thus we are only required to consider the three parameters: (1) histogram shape, (2) average particle size, and (3) refractive index ratio $m$ when evaluating the shape of the scattering spectra. By setting two of the parameters to be the same, we can observe the influence of the third parameter on the scattering spectra.

2.2.

Scattering Spectra Power Law Fit

Rather than applying Mie scattering theory directly, a more empirical approach was first proposed by van Staveren, ²⁴ who fit the scattering spectrum of Intralipid. A number of groups have adopted the approach to characterize the spectrum of the reduced scattering coefficient observed in tissues.^{36, 39, 42, 43} In these studies, the scattering spectra are considered to satisfy a power law relationship. Empirically, when there is a broad range of scattering particle sizes, this spectrum is described by a power law curve of the type:

2.3.

Scattering from Tissue Spectra

Reduced scattering coefficients at 6 wavelengths have been obtained with an experimental clinical imaging system (described in detail elsewhere.⁴⁵) The limited number of wavelengths is not sufficient for identifying oscillations in the scattering spectra of the type that might be observed from larger Mie scattering particles, as observed in Fig. 1(d) in Sec. 3 for the $10\mu \mathrm{m}$ sized particles. Nonetheless, if it is assumed that the dominant particle sizes are smaller and thus the large particle oscillations are small, the data may be used to compare with Mie theory calculations in the wavelength range $660\text{to}850\mathrm{nm}$ . To estimate particle size and number density, the six experimental reduced scattering coefficients are fit to predictions of the power law expression in Eq. 3, first to get the scattering amplitude and scattering power, and then to generate the reduced scattering coefficients at 15 wavelengths (650, 675, 700, 725, 750, 775, 800, 825, 850, 875, 900, 925, 950, 975, and $1000\mathrm{nm}$ ). For extracting the average particle size, we normalize these 15 reduced scattering coefficients at one wavelength (we selected $800\mathrm{nm}$ ). By normalizing the scattering coefficients, we eliminate the influence of number density, so only one parameter (average particle size) is required in the first fitting. Comparing these 15 data with Mie theory results from Eq. 2 (also normalized at $800\mathrm{nm}$ ) using a least-squares minimization, the average particle size can be effectively estimated. With the average particle size established, the number density can then be estimated by comparing the original 15 data (no normalization) with Mie theory [Eq. 2] using a second least-squares minimization method. In the least-squares minimization method, the deviation of the experimental data from the Mie prediction is expressed as the error function:

2.4.

Simulation Study

Intralipid is a fat emulsion that is used clinically as an intravenously administered nutrient and provides a convenient scattering component in a tissue phantom to investigate propagation of light in tissue.²⁴ For Intralipid, there is considerable evidence to suggest its particle distribution is exponential, especially considering the dominant fraction of Rayleigh scattered light that occurs in the blue region of the spectrum.⁴⁶ In the van Stanveren paper,²⁴ the size distribution of the scattering particles in 10% Intralipid was determined by transmission electron microscopy to be exponential and the average particle size was reported to be $97\pm 3\mathrm{nm}$ . If the functional form of $f\left(a_i\right)$ is a normalized exponential function with respect to $a_i$ ,

To validate the method, simulated data were used to test the fitting process. The mean particle size in the simulation study was selected in the smaller size range $(<1000\mathrm{nm})$ , which is the size range of collagen fiber networks in the extracellular matrix, mitochondria, and other intracellular vesicles,⁴⁷ but do not include the larger size, such as that of cell nuclei, typically $5\text{to}15\mu \mathrm{m}$ in diameter. The first five groups of simulated data included the following mean size and mean number densities: (1) $⟨a⟩=50\mathrm{nm}$ , $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ ; (2) $⟨a⟩=100\mathrm{nm}$ , $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ ; (3) $⟨a⟩=200\mathrm{nm}$ , $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ ; (4) $⟨a⟩=500\mathrm{nm}$ , $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ ; and, (5) $⟨a⟩=1000\mathrm{nm}$ , $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ . These data were generated computationally using the expression in Eq. 2 for the six wavelengths that are available in our tomography system. These data were used as synthetic scattering spectra to determine how accurately the fitting process can be completed.

2.5.

Phantom Studies

The tomography system was used to validate the particle size fitting approach with data from tissue-simulating phantoms. Liquid phantoms were used, composed of Intralipid at 11 different concentrations, to obtain a reduced scattering coefficient spectrum at six wavelengths. The data at each concentration was analyzed with the method already described for estimating the average particle size and number density. By varying the concentration, each spectrum should result in the same particle size ( $97\pm 3\mathrm{nm}$ , according to the van Staveren paper²⁴) as only the number density in the liquid is altered when the concentration varies. Thus, the ratio of number density over concentration should be constant for these test solutions.

2.6.

Clinical Studies

In a final step, the fitting method was applied to normal breast tissue data. The study was approved by the institutional committee for the protection of human subjects. NIR imaging studies were performed on asymptomatic women recruited into the study following a negative mammogram. Informed consent was provided prior to the NIR imaging exam. Normal subjects were stratified by age (i.e., $10\text{‐}\mathrm{y}$ intervals) and by one of four radiographic density categories (fatty, scattered, HD, and ED). Data from 31 normal subjects were used; right and left breasts were analyzed separately. Scattering coefficient spectra at the six wavelengths were estimated for the entire slice tissue, and these were fit with the algorithm. Average particle size and number density were determined from the spectra, and the resulting data were grouped according to the radiographic density of the subjects.

3.1.

Influence of Particle Size Distribution

The influence of parameter selections on the resulting reduced scattering spectra is shown for a specific case in Fig. 1, where the effects of (1) size distribution function, (2) average particle size, and (3) index of refraction change are illustrated. Figure 1(a) indicates the relative shape of the three distribution functions (i.e., histogram shapes) considered, a Gaussian function, $f\left(a\right)=\exp [-{(a-1000)}^2∕2\times {10}^2]∕25$ ; a step function [if $0<a<2000$ , $f\left(a\right)=1∕2000$ ; otherwise, $f\left(a\right)=0$ ]; and an exponential decreasing function $[f\left(a\right)=9.5\times {10}^{-4}\exp (-9.5\times {10}^{-4}a)]$ . They are normalized and have the same average particle size $(⟨a⟩=1000\mathrm{nm})$ . For the Gaussian distribution with a small variance, most particles are clustered around $1000\mathrm{nm}$ . For the step function, the particles are distributed evenly between zero and $2000\mathrm{nm}$ , for the exponential function, more particles exist in the smaller size range. A comparison of scattering spectra among these three different histogram shapes with the same average particle size and same refractive index $n_1=1.311$ , $n_2=1.451$ , is shown in Fig. 1(b). In this graph, the exponential distribution has the largest scattering coefficient, while the Gaussian distribution has the smallest one. Based on this result and the distribution characteristics of these three functions, it is evident that particles with smaller sizes have more impact on the magnitude of the scattering spectrum in the wavelength range $650\text{to}1000\mathrm{nm}$ . This figure also shows the characteristic oscillatory components in the scattering spectrum associated with the Gaussian distribution that were observed in other epithelial studies of elastic scattering.

Fig. 1

(a) Comparison of Gaussian, uniform, and exponential particle size distributions with the same average particle size $⟨a⟩=1000\mathrm{nm}$ ; (b) calculated reduced scattering spectra for these three particle size distributions with the same refractive index change ( $n_1=1.311$ and $n_2=1.451$ ); (c) comparison of reduced scattering spectra for three different refractive indexes for the same size distribution with average particle size $⟨a⟩=1000\mathrm{nm}$ , where the inset plot is the comparison of these three spectra after being normalized at $800\mathrm{nm}$ ; and (d) comparison of reduced scattering spectra for four different average particle sizes with the same exponential size distribution and refractive index ( $n_1=1.36$ , $n_2=1.4$ ).

The comparison of reduced scattering coefficients among three different sets of refractive indices is shown in Fig. 1(c). These results were calculated with the same exponential size distribution, and average particle size, but with refractive index values of $n_1=1.33$ , $n_2=1.45(m=1.09)$ ; $n_1=1.33$ , $n_2=1.40(m=1.05)$ ; and the smallest set $n_1=1.33$ , $n_2=1.35(m=1.02)$ . The figure indicates that as the difference between two refractive indices increases, the scattering intensity increases. The scattering spectra were also normalized at $800\mathrm{nm}$ [inset graph in Fig. 1(c)] to observe whether differences in spectral shape are caused by changes in relative refractive index $m=n_2∕n_1$ . It is clear that in the spectral shape changes are nominal; hence, the assumption in Eq. 2 that refractive indices are the same for all particle sizes appears reasonable.

The reduced scattering coefficients obtained from four different average particle sizes, $⟨a⟩=50$ , 100, 1000, and $\mathrm{10,000}\mathrm{nm}$ for the exponential particle size distribution and same refractive indices ( $n_1=1.36$ , $n_2=1.40$ ) are presented in Fig. 1(d). The scattering spectra were normalized at $800\mathrm{nm}$ to investigate potential shape changes caused by the average particle size. Here, increasing the average particle size in the range of $50\text{to}1000\mathrm{nm}$ generates a reduced scattering coefficient spectrum that has a smaller overall slope but more oscillatory components. This result is consistent with the prior findings of Mourant ⁴³ By normalizing the scattering spectra at one wavelength and then comparing it to Mie theory, we are able to extract the average particle size by least-squares minimization.⁴⁸

3.2.

Simulation Results of the Optimal Estimated Values for Particle Size and Number Density

The least-squares minimization method described in Eq. 5 was used to estimate the optimal values for particle size and number density. For simulated test data with a true average particle size of $⟨a⟩=100\mathrm{nm}$ and $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ , the estimated particle size and number density were $110\mathrm{nm}$ and $0.75\times {10}^{19}{\mathrm{m}}^{-3}$ . These results are shown in Fig. 2 , which indicates that a unique fitted solution is available through the optimization process. The estimates of average particle size for simulated data with true values of average particle sizes of $⟨a⟩=50$ , 100, 200, 500, and $1000\mathrm{nm}$ , and number densities of $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ , were 80, 110, 150, 550, and $1100\mathrm{nm}$ , respectively, for average size and $2\times {10}^{18}$ , $7.5\times {10}^{18}$ , $2\times {10}^{19}$ , $8\times {10}^{18}$ , and $8\times {10}^{18}{\mathrm{m}}^{-3}$ , respectively, for number density. The results are also shown in graphical form in Fig. 2 and indicate that the extraction method is reasonably robust and accurate when fitting simulated data.

Fig. 2

(a) Error function for estimating average particle size using simulated test data with a true average particle size of $⟨a⟩=100\mathrm{nm}$ . The minimum of the function presents the optimal estimation of particle size. (b) The same for number density assuming an average particle size $⟨a⟩=100\mathrm{nm}$ and number density $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ . (c) Estimates of average particle size based on test data with true values of $⟨a⟩=50$ , 100, 200, 500, and $1000\mathrm{nm}$ , with $N=1\times {10}^{19}{\mathrm{m}}^{-3}$ and (d) corresponding estimates of number density, indicating that number density does not change significantly with changes in particle size.

3.3.

Estimated Values of Particle Size and Number Density of Intralipid Phantoms

Estimates of average particle size and number density were obtained for 11 different concentrations of Intralipid, and are presented in Fig. 3 . The average particle size results ranged from $75\text{to}130\mathrm{nm}$ . Over the 11 concentrations used here, the mean average particle size was $93\mathrm{nm}$ , with a standard deviation of $19\mathrm{nm}$ . This value is close to the Intralipid particle size estimated from electron microscopy reported $(97\pm 3\mathrm{nm})$ in the paper by van Staveren ²⁴ The estimated number density ranged from $1.1\times {10}^{19}\text{to}3.3\times {10}^{19}{\mathrm{m}}^{-3}$ , which when divided by Intralipid concentration spanned $2.19\times {10}^{19}\text{to}2.35\times {10}^{19}{\mathrm{m}}^{-3}$ . Overall these 11 concentrations, the mean value of number density divided by concentration was $2.26\times {10}^{19}{\mathrm{m}}^{-3}$ , with standard deviation $0.06\times {10}^{19}{\mathrm{m}}^{-3}$ . The ratio of the number density over concentration is essentially constant for the 11 concentrations as expected. In addition, it appears that Fig. 3(a) includes a systematic underestimation of the size below 1% Intralipid, and an overestimation above.

Fig. 3

Estimates of average particle size for Intralipid phantoms with varying (a) concentration and (b) number Density.

3.4.

Estimated Values of Particle Size and Number Density of Normal Subjects with Different Radiographic Density Categories

Normal breast tissue types of normal breasts were grouped into one of four radiographic density categories (fatty, scattered, HD, ED). A clear difference in the mean average particle size and number density for these four compositions of breast tissue was found and shown in Fig. 4 . For the ED breast type, the average particle size ranged from $20\text{to}65\mathrm{nm}$ , with a mean of $36\pm 16\mathrm{nm}$ . For the HD breast type, the average particle sizes spanned $25\text{to}200\mathrm{nm}$ , with a mean of $68\pm 40\mathrm{nm}$ . For the scattered type breast, the average particle sizes covered $140\text{to}1200\mathrm{nm}$ , with a mean of $318\pm 330\mathrm{nm}$ . Finally, for the fatty type breast, average particle sizes from $150\text{to}1400\mathrm{nm}$ were found, with a mean of $485\pm 461\mathrm{nm}$ . The composite data are summarized in Table 1 . Over the sequence of increasing mean particle size with breast tissue type (ED, HD, scattered, and fatty), there is a concordant decrease of the number density.

Fig. 4

Average particle sizes from individual normal subject exams grouped by (a) radiographic density and (b) number density.

Table 1

Average particle size and number density estimated from mean scattering spectrum data acquired in vivo from normal breasts, where the scattering coefficients were taken as the whole breast average obtained for each subject.