2.1.

Basic Principles for the Paired-Wavelength Approach

We have previously introduced our method to measure the oxygen saturation of hemoglobin in breast tumors (or the relative concentration of two chromophores in localized inclusions) by appropriately pairing wavelengths.⁷ Briefly, we hypothesize that an inclusion characterized by an absorption contrast $\mathrm{\Delta }{\mu }_a$ and a zero scattering contrast $(\mathrm{\Delta }{\mu }_s^{\prime }=0)$ determines a maximal relative intensity change [ $\mathrm{\Delta }I∕I_0{\mid }_{max}\left(\lambda \right)$ , where $I_0$ is the background intensity] that depends on the product ${\mu }_{s0}^{\prime }\left(\lambda \right)\mathrm{\Delta }{\mu }_a\left(\lambda \right)$ and on wavelength-independent parameters associated with geometrical factors and boundary conditions. Consequently, if two wavelengths ${\lambda }_1$ and ${\lambda }_2$ are such that $\mathrm{\Delta }I∕I_0{\mid }_{max}^{\left({\lambda }_1\right)}=\mathrm{\Delta }I∕I_0{\mid }_{max}^{\left({\lambda }_2\right)}$ , then we can conclude that ${\mu }_{s0}^{\prime }\left({\lambda }_1\right)\mathrm{\Delta }{\mu }_a\left({\lambda }_1\right)={\mu }_{s0}^{\prime }\left({\lambda }_2\right)\mathrm{\Delta }{\mu }_a\left({\lambda }_2\right)$ . In other words, the ratio of the absorption perturbations at these two wavelengths is given by the inverse of the ratio of the background reduced scattering coefficients at the same two wavelengths. Because the oxygen saturation of hemoglobin is only a function of the ratio of the optical absorption at two wavelengths, one can use this relationship and translate a measurement of the background scattering ratio into a measurement of the absorption perturbation ratio associated with the embedded inclusion. In our experiment, we use a high-contrast defect, where the absorption perturbation $\left(\mathrm{\Delta }{\mu }_a\right)$ is much greater than the background absorption $\left({\mu }_{a0}\right)$ , so that $\mathrm{\Delta }{\mu }_a$ is representative of the defect absorption. If instead of using molar concentrations (as is typically done in the case of ${\mathrm{HbO}}_2$ and Hb), one uses volume fractions (as we have done for the two chromophore solutions in our experimental test), the molar extinction coefficients of the two chromophores are replaced by the absorption coefficients of the two chromophore solutions. As a result, the expression for the relative concentration of ink $\left(R{C}_{ink}\right)$ , based on the expression for oxygen saturation reported previously,⁷ is the following:

2.2.

Experimental Procedures

The experiments were performed in a highly scattering medium (2% milk-water mixture) that simulates the optical properties of breast tissue. The optical properties of the background medium were measured at two wavelengths (690 and $830\mathrm{nm}$ ) using a multidistance frequency-domain method⁸ with a commercial frequency-domain tissue spectrometer (OxyplexTS, ISS, Inc., Champaign, Illinois). The spectral extrapolation of the reduced scattering coefficient was performed by fitting the frequency-domain data at 690 and $830\mathrm{nm}$ with the power law ${\mu }_{s0}^{\prime }=a{\lambda }^{-b}$ .⁹ The spectral measurements for our proposed method were performed with the source and detector fibers deeply embedded into the turbid medium and collinear with each other at a distance of $6\mathrm{cm}$ (simulating a typical breast thickness under mild compression). The optical inclusion is a transparent plastic tube ( $0.78\mathrm{cm}$ in diameter, $12\mathrm{cm}$ in length) containing a mixture of two test chromophores and background medium. The two test chromophores were black India ink and a blue food coloring dye. The absorption spectra of these chromophores ( ${\mu }_{aink}$ and ${\mu }_{adye}$ ) were quantitatively measured in a nonscattering regime using a standard spectrophotometer (Lambda 35, PerkinElmer Instruments, Shelton, Connecticut).

A bandpass filtered (400 to $1000\mathrm{nm}$ ) Xenon arc lamp (Model 66984, Oriel Instrument, Stratford, Connecticut) was coupled to the source optical fiber (internal diameter: $3\mathrm{mm}$ ). The light collected with the detector optical fiber (internal diameter: $4\mathrm{mm}$ ) was sent to a spectrograph (SpectraPro-150, Acton Research Corporation, Acton, Massachusetts) equipped with a 300-G/mm grating to achieve a spectral dispersion of $20\mathrm{nm}∕\mathrm{mm}$ . A charge-coupled device (CCD) camera detector (DU420-BR-DD, Andor Tech., South Windsor, Connecticut) was placed at the exit port of the spectrograph for the simultaneous data collection of the intensity spectrum, with a spectral resolution of $0.5\mathrm{nm}$ . The experimental setup is shown in Fig. 1 .

Fig. 1

Experimental setup showing the container of the tissuelike liquid phantom and the cylindrical inclusion containing the two test chromophores. CCD: charge-coupled device.

Eleven 0.78-cm diameter test tubes were prepared at the beginning of experiment, each containing various relative concentrations (by volume) of the two chromophores solutions, ranging from 0 to 100% by steps of 10% and a milk-water mixture with the same scattering coefficient as the background medium. Measurements were taken under three different conditions: (1) background medium with the tube containing the same medium as background to get the background intensity $I_0$ ; (2) tube center positioned on the midplane, [i.e., $3\mathrm{cm}$ away from both source and detector (centered case)]; (3) tube center positioned $1.5\mathrm{cm}$ away from the source fiber (off-centered case). Before taking the data for cases 2 and 3, we performed a tandem scan of source and detector along the transverse coordinate ( $x$ in Fig. 1), to determine the position of maximum intensity change. Then the intensity data was recorded at this position with a CCD exposure time of $30\mathrm{s}$ .

2.3.

Data Analysis

We measured the spectral dependence of the maximum relative intensity change with respect to the background value, $\mathrm{\Delta }I∕I_0\left(\lambda \right)$ . More precisely, $\mathrm{\Delta }I\left(\lambda \right)=I\left(\lambda \right)-I_0\left(\lambda \right)$ , where $I_0\left(\lambda \right)$ is the background intensity and $I\left(\lambda \right)$ the intensity measured when source and detector are collinear with the defect. These spectra of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ were then used in our proposed method that first identifies paired wavelengths ${\lambda }_1$ and ${\lambda }_2$ such that $\mathrm{\Delta }I∕I_0{\mid }_{max}\left({\lambda }_1\right)=\mathrm{\Delta }I∕I_0{\mid }_{max}\left({\lambda }_2\right)$ , then calculates the relative concentration of the chromophores $\left(R{C}_{ink}\right)$ according to Eq. 1. In the case of spectral regions where the $\mathrm{\Delta }I∕I_0$ spectrum is relatively flat, the two wavelengths ${\lambda }_1$ and ${\lambda }_2$ may be arbitrarily close to each other, in which case the numerator and denominator of Eq. 1 would both tend to 0. To prevent such situation, we adopted the criterion that the difference between the two paired wavelengths be greater than $15\mathrm{nm}$ $(\mid {\lambda }_1-{\lambda }_2\mid >15\mathrm{nm})$ .

Starting with the lowest wavelength ( $520\mathrm{nm}$ in this work) for ${\lambda }_1$ , we look for its paired wavelength ${\lambda }_2>{\lambda }_1$ according to the criteria already mentioned, namely $\mathrm{\Delta }I∕I_0{\mid }_{max}\left({\lambda }_1\right)=\mathrm{\Delta }I∕I_0{\mid }_{max}\left({\lambda }_2\right)$ and $({\lambda }_2-{\lambda }_1)>15\mathrm{nm}$ . For this wavelength pair, we calculate the relative concentration of ink according to Eq. 1, and we estimate its experimental error. The error in the relative concentration is induced by the experimental error on $\mathrm{\Delta }I∕I_0{\mid }_{max}$ (which in turn translates into errors in ${\lambda }_1$ and ${\lambda }_2$ , which depend on the spectral shape of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ ) and by the measurement error on the scattering spectral ratio ${\mu }_{s0}^{\prime }$ $\left({\lambda }_1\right)∕{\mu }_{s0}^{\prime }\left({\lambda }_2\right)$ .We repeat the procedure for all measured wavelengths to find a set of wavelength pairs in this two-dimensional spectral space. We then set a 5% error threshold for $R{C}_{ink}$ , so that any wavelength pair yielding an error greater than 5% in $R{C}_{ink}$ was discarded. The final measured value of $R{C}_{ink}$ for a specific ink and dye mixture is then given by the average of the $R{C}_{ink}$ values associated with all of the wavelength pairs that were not discarded, and the final error on $R{C}_{ink}$ was estimated by combining the average error in $R{C}_{ink}$ for the individual pairs and the standard error for the set of $R{C}_{ink}$ values.

We compared the results of our paired-wavelength method with a multiwavelength method that assumes that $\mathrm{\Delta }I∕I_0{\mid }_{max}$ is proportional to ${\mu }_{s0}^{\prime }\mathrm{\Delta }{\mu }_a$ (which is correct for diffusion theory within first-order perturbation when there is no scattering contrast). Both our paired-wavelength method and first-order perturbation for a pointlike defect do not require prior knowledge (and neither provide information) about the size, shape, and location of the defect. In our case, $\mathrm{\Delta }{\mu }_a$ is given by the weighted sum of the absorption coefficients of the two chromophore solutions (where the weights are given by the corresponding relative volume fractions). This linear relationship makes it easy to determine the relative concentration of the chromophores by using the multiwavelength method¹⁰ according to the following equation:

3.1.

Experimental Tests

Figure 2 shows the absorption spectra of the two test chromophores (black India ink and blue food dye) in water solution (no scattering) as measured by standard spectrophotometry. The absorption peaks of these dyes are at shorter wavelengths than those employed in near-infrared optical mammography, but our proof of principle does not necessarily need to be performed at the same wavelengths planned for the application to tumor oximetry for optical mammography. The absorption spectra of the two test dyes qualitatively mimic the absorption spectra of ${\mathrm{HbO}}_2$ and Hb in the wavelength region of interest for optical mammography (680 to $880\mathrm{nm}$ ). Namely, in this near-infrared spectral range, ${\mathrm{HbO}}_2$ shows a relatively featureless absorption spectrum (similar to black India ink), and Hb shows local absorption maximum and stronger wavelength dependence (similar to the blue food coloring dye in the 520- to $800\text{-}\mathrm{nm}$ region).

Fig. 2

Absorption spectra of the aqueous solutions of ink and blue dye as measured by standard spectrophotometry in a nonscattering regime.

Table 1 shows the optical properties of the background medium at two wavelengths (690 and $830\mathrm{nm}$ ) as measured with frequency-domain spectrometry. The absorption coefficient of the background medium is one order of magnitude smaller than the absorption coefficient of the chromophore solutions. The reduced scattering spectra is obtained by fitting the reduced scattering coefficients at 690 and $830\mathrm{nm}$ with the power function as mentioned in Sec. 2.2. We have found values of the fitting parameters $a=6.25\times {10}^4$ , $b=1.344$ for ${{\mu }_{s0}}^{\prime }$ $\left({\mathrm{cm}}^{-1}\right)$ and $\lambda$ (nm). Because we have used the same scattering medium in the cylindrical tube as in the background, we assume no scattering mismatch between the defect and background medium.

The 11 curves in Figs. 3a and 3b correspond to 11 various relative concentrations of ink $[V_{ink}∕(V_{ink}+V_{dye}\left)\right]$ , ranging from 0 to 100% by steps of 10%, for the centered cylinder case [Fig. 3a] and the off-centered cylinder case [Fig. 3b] as we have described in Sec. 2.2. Each curve has been shifted by an arbitrary offset for clarity. The top curve corresponds to the lowest ink concentration (0%), and the bottom curve corresponds to the highest ink concentration (100%). Our paired wavelength method requires that the relative intensity change caused by the defect show well-defined local maxima and/or local minima. This is not the case for the bottom three spectra in Figs. 3a and 3b, which correspond to relative ink concentrations of 80, 90, and 100%. To quantify this situation, we define the useful contrast in each spectrum as the difference between the maximum and minimum values of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ over the wavelength range where it is possible to find paired wavelengths that satisfy our criteria. The contrast-to-noise ratios (CNRs) for the highest three ink concentrations (80, 90, 100%) are 8.8, 9.2, and 8, respectively, but CNRs for all other spectra are higher than 12. As a result, we have applied our method only to the spectra corresponding to a relative ink concentration within the range 0 to 70%.

Fig. 3

Intensity perturbation spectra induced by the cylindrical ink-dye inclusion for (a) the centered case (cylinder equidistant from source and detector), and (b) the off-center case (cylinder closer to the source).

We have discussed (see Sec. 2.3) that our critical hypothesis is that the product ${\mu }_{s{0}^{\prime }}\mathrm{\Delta }{\mu }_a$ is the same at the two wavelengths ${\lambda }_1$ and ${\lambda }_2$ of each pair. Figure 4 shows the absolute values of ${\mu }_{s{0}^{\prime }}\mathrm{\Delta }{\mu }_a$ at the two wavelengths, their relative difference, the measured values of $R{C}_{ink}$ , and ${\lambda }_2$ as a function of ${\lambda }_1$ . Three representative cases are presented for actual $R{C}_{ink}$ values of 0% [panel (a)], 30% [panel (b)], and 70% [panel (c)]. One important point to observe in Fig. 4 is that the wavelength pairs for which the products ${\mu }_{s0}^{\prime }\mathrm{\Delta }{\mu }_a$ are equal, yield a measured value of $R{C}_{ink}$ that coincides with the corresponding actual value (shown as a horizontal dashed line in the $R{C}_{ink}$ graph). We observe that the collection of $R{C}_{ink}$ values associated with the various wavelength pairs may be off by up to about 15%, but because measured values both overestimate and underestimate $R{C}_{ink}$ , the average across all wavelength pairs tends to partially correct for these inaccuracies. Therefore, although our hypothesis on the functional dependence of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ on ${\mu }_{s0}^{\prime }\mathrm{\Delta }{\mu }_a$ is not accurate over the whole spectrum, the deviation does not translate into dramatic inaccuracies on the measured relative concentration, and such inaccuracies tend to cancel out in the averaging process across wavelength pairs. We also notice highly accurate (to within 1 to 2%) measurements over extended spectral regions of ${\lambda }_1$ [e.g., 525 to $545\mathrm{nm}$ in Fig. 4b and 550 to $600\mathrm{nm}$ in Fig. 4c].

Fig. 4

Absolute values of ${{\mu }_{s0}}^{\prime }\mathrm{\Delta }{\mu }_a$ as function of the paired wavelengths ${\lambda }_1$ and ${\lambda }_2$ , their relative difference, and measured values of $R{C}_{ink}({\lambda }_1,{\lambda }_2)$ as a function of ${\lambda }_1$ and ${\lambda }_2$ for relative ink concentrations of (a) 0%, (b) 30%, and (c) 70%.

With regard to the sensitivity to the scattering spectral ratio, we point out that the spectrum of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ for 70% $R{C}_{ink}$ (see Fig. 3) is relatively flat over the 550-to-600-nm wavelength region. This implies that ${\lambda }_1$ wavelength within this range (550 to $600\mathrm{nm}$ ) corresponds to approximately the same ${\lambda }_2$ $\left(638\mathrm{nm}\right)$ . The ratio ${{\mu }_{s0}}^{\prime }\left({\lambda }_1\right)∕{{\mu }_{s0}}^{\prime }\left({\lambda }_2\right)$ in this two-dimensional wavelength range is about 7%, but it translates into a variability for $R{C}_{ink}$ of only 1.3%. This result is due to the fact that the variation in the reduced scattering ratio is compensated by the different set of absorption coefficients for the ink and blue dye chromophores.

Figure 5 shows the measured relative concentration of ink $\left[R{C}_{ink}^{\left(measured\right)}\right]$ versus its actual value $\left[R{C}_{ink}^{\left(actual\right)}\right]$ for the centered [panel (a)] and off-centered [panel (b)] cases. The open circles are calculated by using the full spectrum and the assumption of linearity between $\mathrm{\Delta }I∕I_0{\mid }_{max}$ and ${{\mu }_{s0}}^{\prime }\mathrm{\Delta }{\mu }_a$ according to first-order perturbation theory, as discussed previously. The diamonds are calculated by using the new wavelength-paired method, which holds under any monotonic functional dependence of $\mathrm{\Delta }I∕I_0{\mid }_{max}$ on ${{\mu }_{s0}}^{\prime }\mathrm{\Delta }{\mu }_a$ . The line indicates the ideal result for the measured values of relative concentration. These results show that the relative concentrations of ink measured with our wavelength-paired method deviate by no more than 6% from the actual values over the 0 to 70% range regardless of the location of the defect. By contrast, the $R{C}_{ink}$ values measured using the full spectrum and the first-order perturbation assumption of proportionality between $\mathrm{\Delta }I∕I_0{\mid }_{max}$ and ${{\mu }_{s0}}^{\prime }\mathrm{\Delta }{\mu }_a$ have a much worse accuracy, with results that are off by 15 to 30%. We assign the better performance of our method with respect to first-order perturbation theory to two reasons. First, our experimental conditions are beyond the limits of validity of first-order perturbation theory in quantifying the spectral dependence of the absorption contrast. Our paired-wavelength method, instead, is used to calculate ratios of absorption contrast, which is a more robust measurement. Second, the spectral information is used in a unique way by our method, where wavelength pairs are selected according to well-defined criteria. Therefore, the key point of the method is that it provides a correct estimate of the ratios of absorption contrast by choosing particular paired wavelengths. Also, we notice that because the paired wavelengths cannot be defined in advance, any multiwavelength approach that uses a discrete number of wavelengths instead of a broadband illumination is more limited in the application of our method.

Fig. 5

Measured versus actual values of relative concentrations of ink for the cases of (a) centered cylinder and (b) off-centered cylinder. Closed diamonds are obtained with our paired-wavelength approach; open circles are obtained from the fit of the full spectrum with first-order perturbation for the case of a pointlike object.

Table 1

Absorption coefficient (μa0) and reduced scattering coefficient (μs0′) of the background medium (diluted milk) at the two wavelengths measured with multidistance, frequency-domain spectroscopy.

Figure 5 allows us to quantify the sensitivity of our method (and the full spectral method based on first-order perturbation) to the location of the inclusion. We have found that for our paired-wavelength method, the relative concentration of ink measured in the centered case is about 4 to 5% lower than that measured in the off-centered case. In the case of full spectral analysis based on first-order perturbation, the relative concentration of ink measured in the centered case is about 9 to 13% lower than that measured in the off-centered case.