## Abstract

We have devised and experimentally validated, on tissue-simulating phantoms and *in vivo*, a time-resolved spectral fitting analysis for direct assessment of chromophore concentrations and scattering parameters. Experimental data have been acquired with a time-resolved broadband system based on supercontinuum light generated in a photonic crystal fiber and a 32 channel Time Correlated Single Photon Counting system. The novel method is more robust than conventional techniques, especially at low signal-to-noise ratio.

©2006 Optical Society of America

## 1. Introduction

In recent years, diffuse spectroscopy has opened the way to noninvasive optical characterization of biological tissue and has fostered the development of several pre-clinical and clinical applications like optical mammography, human brain mapping, and tissue oximetry among others [1].

In the red and near-infrared spectral region the absorption of main tissue components (oxy-hemoglobin, HbO_{2}; deoxy-hemoglobin, Hb; water and lipid) is much lower than the scattering of light, allowing photons to deeply penetrate into tissue. At the same time, because of the many scattering events (multiple scattering regime, or photon migration) photons travel very long paths within tissue before being eventually remitted at the boundaries. Typical path length is of the order of hundreds of centimeters, therefore even a faint absorption (approximately 1/100 of the scattering) can produce significant effect on the probing light.

Estimate of tissue components concentration can be obtained by the well-known Lambert Beer law from the knowledge of the extinction coefficient (i.e. the absorption of the pure substance, typically reported in the literature) if measurements of the absorption coefficient are performed at several discrete wavelengths. Under the assumption that the absorption coefficient is a linear combination of the extinction coefficients of components weighted by their concentration, and that water, lipid, Hb and HbO_{2} are the only chromophores contributing to absorption, the minimum set of wavelengths to assess tissue concentration is four. However, in the 600–1000 nm spectral range no sharp peaks are present in the absorption spectra of Hb, HbO_{2}, water and lipids. The spectral features of these components originate in fact from overtones of vibrational transition in the infrared (e.g. OH and CH group for water and lipid respectively), and from electronic transition of pyrroles, heme (e.g. Hb and HbO_{2}) [2]. Measured absorption spectra are therefore smooth and a proper choice of the operational wavelengths is needed to avoid an indeterminate solution for the chromophore concentration.

In parallel, diffuse spectroscopy can yield information on tissue structure from the interpretation of the reduced scattering spectra on the basis of an empirical approximation to Mie theory [3, 4]. Typically the scattering amplitude and the scattering power are obtained by fitting the scattering spectrum. These parameters can be also directly related to scatter density and size respectively [5].

To enhance robustness of the fitting procedures for the determination of both tissue constituents and tissue structure parameters, recently some research groups have introduced the concept of spectrally constrained analysis [6, 7, 8, 9, 10, 11]. The spectral approach naturally reduces the number of fitting parameters, for example from 8 parameters (4 absorption coefficients and 4 scattering coefficients at 4 independent wavelengths) to 6 (4 tissue constituents and 2 scatter parameters), and it directly reconstructs tissue chromophore concentration and scattering parameters. The main advantages are a better robustness and an increased stability in the presence of higher level of noise.

Up to now, the spectrally constrained approach has been presented only for the continuous wave (CW) (both in spatially resolved set-up and in a tomographic approach) and the frequency domain (FD) case. In this paper, for the first time to our knowledge, we report on the extension of the method to the time-resolved (TR) case. Since system performances of CW, FD and TR in terms of instrumentation, algorithm and models can be noticeably different the extension to the TR case is not straightforward and has to be checked.

## 2. Materials and methods

#### 2.1 Time-resolved spectral constraint analysis

A conventional method for estimation of chromophore concentration is based on a two step procedure. In the first step optical parameters (*μ*_{a}
and *μ́*_{s}) are calculated by fitting time-resolved curves to an analytical or numerical solution of transport equation. In our case we will consider an on-axis transmittance geometry, hence time-resolved curves will be fitted to the analytical solution of the diffusion approximation to the transport equation for an infinite homogeneous slab [12] with extrapolated boundary conditions, given by:

$$\phantom{\rule{13.2em}{0ex}}\times \sum _{n=-\infty}^{+\infty}\left[{z}_{n}^{+}\mathrm{exp}\left(-\frac{3{\mu}_{s}^{\prime}{\left({z}_{n}^{+}\right)}^{2}}{4\mathit{vt}}\right)-{z}_{n}^{-}\mathrm{exp}\left(-\frac{3{\mu}_{s}^{\prime}{\left({z}_{n}^{-}\right)}^{2}}{4\mathit{vt}}\right)\right],$$

where ${z}_{n}^{+}$ = {1 - 2*n*)*d* - 4*nz*_{e}
- *z*
_{0}, ${z}_{n}^{-}$ = (1 - 2*n*)*d* - (4*n* - 2)*z*_{e}
+ *z*
_{0}, *z*
_{0} = (9*μ́*_{s},)^{-1}, *z*_{e}
= (2*A*/3*μ́*_{s}), *v* is the speed of light in the medium, *d* is thickness of the sample and A takes into account the reflections at the slab surface.

The theoretical curve is convolved with the instrumental response function (IRF) and normalized to the area of the experimental curve. By varying *μ*_{a}
and *μ́*_{s} the difference between the convolved theoretical curve and experimental data is minimized by using a standard Levenberg-Marquardt algorithm [13]. This procedure is repeated for all the signals at each wavelength and it permits calculation of absorption and scattering spectra.

In the second step the absorption spectrum is exploited to calculate the concentration of the different sample constituents. The total absorption coefficient *μ*_{a}*(λ*), which depends on the wavelength *λ*, can be written as a linear combination of constituent spectra weighted by their concentrations:

where *c*_{i}
, is the concentration and *ε*_{i}*(λ)* is the extinction coefficient of the *i*-th constituent.

Similarly, the scattering spectrum is analyzed by using an approximation of Mie scattering theory. Under the hypothesis that the medium is composed of scattering centers which are independent homogeneous spheres, the wavelength dependence of the reduced scattering coefficient *μ́*_{s} is approximately given by [3, 4]:

where *λ*_{0}
is a fixed reference wavelength (in our case 600 nm) and *a* and *b* are the scatter amplitude and the scatter power, respectively. It is worth reminding that both parameters are correlated with structural composition such as the density and sizes of scattering centers.

As previously stated it is advantageous to analyze the time-resolved curves at different wavelengths in a single step. In this work we present a novel procedure adopted to directly evaluate both chromophore concentration and structural parameters by analyzing spectral and time-resolved curves (hereafter called TR spectral fitting) as a whole. To this aim the data set of all the time-resolved curves has been considered.

Equation (1) can be rewritten as the product of two terms, one depending on the absorption and the other on the scattering properties of the medium, as follows:

By substituting Eq. (2) and Eq. (3) into the solution of transmittance time-resolved diffusion approximation to transport equation for an infinite homogeneous slab, given by Eq. (4), we have obtained a theoretical expression of transmittance signal, as a function of wavelength, which directly depends on tissue constituents concentrations *c*_{i}
and structural parameters *a* and *b*:

Then, for each wavelength, this expression is convolved with the corresponding IRF and normalized to the area of the experimental curve. This procedure is repeated for all the available wavelengths covered by detector.

In order to calculate the unknowns (*c*_{i}
, *a*, *b*), the fitting procedure is carried out by minimizing the difference between the experimental and theoretical vectors of all the time-resolved curves, by using a standard Levenberg-Marquardt algorithm [13].

#### 2.2 Broadband time-resolved set-up

A broadband time-resolved instrumentation is the best suited to experimentally validate the time-resolved spectral constraint analysis proposed in the previous section. The set-up used in this work is based on the time-resolved white light spectrophotometer for diffusive media recently developed by our group [14]. In the present work significant upgrades of system performance have been realized. The Argon laser, used to pump the Ti:Sapphire oscillator, has been replace with a solid state laser, improving the compactness of the whole set-up and increasing the stability of the pump intensity and therefore of supercontinuum light pulses. A new 32 channels PMT has been adopted, leading to a double advantage: first, the higher spectral sensitivity of this detector allows one to measure in the spectral range 600–900 nm, second, the increased number of channels leads to a better spectral resolution. Finally, a new home made 32 channels router has been designed and realized.

Hereafter a description of the set-up, shown in Fig. 1, is presented. The laser source is based on a passively mode-locked Ti: Sapphire oscillator (Tissa 50, CDP Systems, Russian Federation) pumped by a continuous-wave solid state laser emitting at 532 nm (Millennia V, Spectra Physics, USA). It emits light pulses at about 810 nm at a repetition rate of 85 MHz, with a pulse width shorter than 100 fs and an average power of 350 mW.

The laser pulse passes through a broadband Faraday isolator to reduce back reflection into the laser cavity and it is focused into an 60-cm-long photonic crystal fiber (PCF) (NL-2.4-800, Blaze Photonics, UK) by an aspheric singlet lens. The supercontinuum (SC) light pulses exiting the PCF covers the spectral range 550–1000 nm with a variation of less than 10 dB in the spectral range (600–900 nm). Through a 50-μm core graded-index fiber the SC light is directed to the measurement area, then attenuated to suitable intensity by a variable filter (NDC-100C-4, Thorlabs, USA), and then splitted into three beams. The main part (less than 20 mW) of the beam is injected onto the sample, the second one (a few hundred μW) is used as a reference beam to monitor both the shape and amplitude of the injected beam. The third beam is probed with a spectrometer (USB 2000, Ocean Optics, USA) to monitor the spectral shape and stability of the SC.

The laser pulses are injected on one side of the sample and diffuse light exiting on the other side is collected (transmittance geometry), by using a 3 mm diameter bundle connected to an imaging spectrometer (SP-2150, Acton Research, USA). The bundle consists of seven plastic fibers (Eska CK-40-Mitsubishi Rayon Co., 1 mm, NA 0.55), which are circularly arranged on one side (close to the sample) and linearly assembled on the other side. This configuration allows high photon collection efficiency from the sample, while the linear arrangement, on the other side, fits to the entrance slit of the spectrometer. One more fiber is placed at the entrance slit of the spectrometer in order to acquire the reference signal. The length of this fiber is chosen in order to temporally separate the reference and the diffused light exiting the sample.

The imaging spectrometer is coupled to a multi-anode 32-channel linear array PMT (R5900U-L32- Hamamatsu, Japan) which enables simultaneous detection in 32 wavelength bands. In all the measurements, described in this work, a 600 lines/mm grating has been used. Grating dispersion and the dimension of a single photocathode stripe of the PMT (1 mm) lead to a spectral resolution of 9 nm/channel and a spectral range of 288 nm.

Time-resolved curves were acquired with the TCSPC technique. In order to simultaneously record the signal from all channels we used a custom-built 32-channel router and an SPC-630 PC board (Becker & Hickl GmbH, Germany). The cross talk between adjacent channels has been measured to be less than 4%.

#### 2.3 Tissue phantoms

The sample is a liquid phantom, contained in a rectangular plastic tank (9×9×3 cm), having the typical optical parameters of biological tissue. It was made of a distilled water solution of Intralipid^{®} (IL) (Pharmacia, Italy) and ink (Rotring) in order to simulate scattering and absorption, respectively. In particular, in order to change the spectral dependence of the absorption we used a mixture of three inks (green, blue and black) with different absorption spectra. In Table 1, the ink concentrations of all the samples measured in this work are reported.

An initial measurement (#1 in Table 1) was carried out in order to calculate the absorption spectra of the IL solution (a sample with water and IL). In all the following measurements the IL solution is 200 ml, with an IL content of 7 ml which corresponds, at the wavelength of 633 nm, to a reduced scattering coefficient of about 10 cm^{-1}. Afterward the absorption spectra of each ink was evaluated by measuring three different samples containing one ink only (0.33 ml) (#2-4 in Table 1). The absorption spectra of each ink and water (without IL) were also measured with a spectrophotometer (V-570 Jasco).

Samples with nine different concentrations of two inks were prepared (#5–13 in Table 1). As reported in Table 1, all mixtures originate from the different combinations of the following three inks concentrations: 25%, 50%, and 75% of reference quantity. Finally, one mixture of three inks, all with the same concentration (0.11 ml), was prepared (#14 in Table 1). In all the measurements, the total amount of ink was chosen in order to have, at the wavelength of 670 nm, an absorption between 0.05–0.1 cm^{-1}.

The acquisition time for each measurement is 10 s. Each sample was measured 5 times in order to allow a subsequent statistical analysis.

A final set of measurements was performed on a tissue phantom (50% black and 50% blue) for a gradually reduced number of total photon counts. This was achieved by reducing the acquisition time, while keeping the same level of injection power. The aim was to test the robustness of the fitting methods at different signal to noise ratio.

## 3. Results and discussion

#### 3.1 Ink characterization

Preliminary measurements were devoted to the choice of inks. In order to better simulate *in vivo* measurements we looked for inks with a smooth spectral shapes in the range 600–1000 nm. Figure 2 shows the absorption spectra of the three selected inks measured with the spectrophotometer. Different reasons motivated the choice of these three inks. First, the three inks must have smooth spectra, typical of tissue constituents spectra. Second, green and blue inks are more similar and clearly differentiates from black because of sharper spectra. In particular the similar spectra of green and blue inks allows a better test of the accuracy of the fitting method to evaluate the constituent concentrations. In Fig. 2, we also reported the measured water absorption spectrum.

As for the absolute values, the measurement of ink absorption coefficient given by spectrophotometer is not reliable, since it does not discriminate absorption from scattering. Hence, absorption spectra of inks and water were also measured with TRS-PCF. Exploiting the fitting procedure, described in par. 2.1, we calculated the absorption spectra of four samples (#1–4 in Table 1): IL solution with no inks and with one ink only. In the case of IL solution we directly obtain water absorption spectrum (IL concentration is low and hence the absorption contribution of lipid is negligible), while for the three other samples we obtain the absorption spectra of water and inks. The water background was subtracted from the single-ink measurements to obtain the pure ink spectra. Figure 2 shows the absorption spectra of three inks and water measured with TRS-PCF. In order to compare the data obtained with the spectrophotometer and TRS-PCF, the absorption coefficient was normalized to the maximum in the spectral range 600–822 nm. All the analysis described in this paper will use data acquired over that spectral range.

We obtain a good agreement in spectral shape between the absorption spectra measured by TRS-PCF and the spectrophotometer. As expected, the absolute values of inks’ absorption coefficients, given by spectrophotometer, shows an overestimation of about 30 %. Hence, in the following of this work, we will adopt as reference absorption spectra of water and inks those obtained with the TRS-PCF instrumentation.

#### 3.2 Phantoms with a mixture of two and three inks

In this section we deal with the measurements on phantoms containing two (#5–13 in Table 1) and three (#14 in Table 1) inks, described in par. 2.3. All the mixtures were analyzed by using both fitting procedures previously described in par. 2.1. Table 2 reports the relative concentrations of inks, scattering parameters and volume concentration of water, obtained with the two methods. For all the samples, we report the mean values of concentrations and scattering parameters calculated over five repetitions.

Both fitting procedures provide a good agreement with nominal values. In order to quantify the comparison between the two methods we calculated, for each sample, the root mean square (RMS) between the nominal concentration and the calculated value, with the two methods. The sum of RMS values for the inks diluted in each specific sample, are lower with TR spectral fit than the two step approach in eight cases. Average RMS, obtained by TR spectral fit, results to be better than 0.9% for inks and 2.1% for water whereas for two-step approach is lower than 1.9% for inks and 5.2% for water.

As for the volume concentration of water, it is expected to be close to 100%, because the volumes of ink added in the solutions are negligible. The discrepancy we observe, with both fitting methods, with respect to the nominal values in the water volume estimation is mainly due to the small values of the water absorption coefficient in the spectral range considered. Also in this case both fitting procedures give similar results, in particular the RMS values, obtained by TR spectral fit, is better than two step approach in six cases.

Concerning scattering parameters, nominal values are not available due to the high variability of data reported in literature, hence the comparison is done between the values calculated with the two methods. The two fitting procedures lead to comparable results, in fact the difference in the estimation of *a* and *b* is always less than 1% and 4%, respectively. Moreover the values of *a* and *b* are in agreement with literature data [15].

Beyond the correct estimation of concentrations and scattering parameters it is important to determine the data dispersion. Hence, we calculated the coefficient of variation (CV) of ink concentrations, water volume concentration and scattering parameters obtained from five repeated measurements carried out for each sample. CV values are reported in Table 3. For all the samples the CV of ink and water volume concentrations is lower with TR spectral fitting than with the standard two-step approach. In particular with TR spectral approach CV results to be lower than 1% for inks and 2.6% for water, whereas for two-step approach is lower than 2.7% for inks and 6.2% for water.

In contrast to the case of absorption, CVs of scattering parameters obtained with the two methods are comparable. This is probably because, with the two-step approach, the absorption spectrum is noisier than scattering spectra. Moreover, the scattering parameters are obtained by fitting with two variables while concentrations are obtained by fitting the absorption spectra with four fitting variables. In the latter case the higher stability of one step fitting procedure results more evident.

All these considerations demonstrate that TR spectral analysis leads to results comparable to the conventional two-step approach for the estimation of concentrations, while data dispersion is lower. Concerning scattering parameters similar results are evident between the two methods in both the absolute value and data dispersion.

#### 3.3 Robustness of fitting algorithms

Another fundamental issue to assess is the robustness with the two methods at lower signal to noise ratio. We repeated the measurement on the phantom (black and blue with 50-50 % relative concentration, #7 in Table 1) with lower numbers of total photon counts (sum of all counts in all wavelength channels during one acquisition). Figure 3 shows the plot of ink and water volume calculated concentrations for different total photon counts, obtained with the two methods.

We observe a similar result for 10^{5} counts or more. Reducing count number, experimental data analyzed with the two-step procedure gives wrong estimation of ink and water volume concentration. On the contrary, TR spectral analysis gives reliable values down to 2.5 10^{4} counts. At this point, the RMS error respect to the nominal values is 13% in the estimation of the three ink concentrations. It is worth notice that the fitting procedure considers also the presence of green ink. Hence, the RMS value takes into consideration the calculated presence of green ink. By calculating the RMS error only on blue and black inks it lowers down to 5%. As for the volume concentration of water, the RMS error is about 16%.

In addition to the accuracy in the evaluation of the absolute value at lower signal to noise ratio, we tested its reproducibility. Figure 4 shows the CV of the concentrations, calculated with the two methods, for each set of five repeated measurements.

For all number of counts, the TR spectral method gives results with a CV lower than with the two-step analysis. In particular, in order to get a CV lower than 10%, for the ink concentrations, the TR spectral analysis needs a total number of counts of at least 6 10^{4} whereas the corresponding value for the two-step method is about 7 10^{5}.

This demonstrates that the TR spectral method is more robust against noise compared with the conventional two-step approach. This feature is quite important, because it allows measurements with lower total photon counts and therefore a shorter acquisition time. This advantage is relevant for those measurements where fast changes are expected and for imaging measurements. The reduction of the acquisition time allows faster image measurement time, which is compatible with clinical requirements.

The possibility to operate with a lower count rate is particularly interesting with the TRS-PCF system realized in this work. In fact, the real-time capability is obtained by means of a high parallelism with the drawback of low photon counts for each wavelength channel. The broadband diffused light exiting the sample is spread over many detectors by the spectrometer, and hence the counts number per wavelength channel is low.

#### 3.4 In vivo measurements

In this section we present preliminary results concerning the application of the TR spectral fitting on *in vivo* measurements. We carried out breast measurements in an on-axis transmittance geometry with one volunteer. We exploited the set-up, described in par. 2.2, used to perform the phantoms measurement. In particular, in order to position the volunteer the source fiber and detection bundle were coaxially mounted on the mammography system [16] available in the laboratory. Following the results obtained with the phantoms measurements, the acquisition time was set to 0.32 s which corresponds to about 2.3 10^{5} number of total photon counts. By applying the TR spectral fitting, we obtained the following constituents concentrations and scattering parameters: lipid 65.8 %, water 13.2 %, haemoglobin 4.3 μM and deoxyhaemoglobin 7.7 μM, *a* 13.4 and *b* 0.37. These values lead to an oxygen saturation (SO_{2}) of 61.3 % and a total haemoglobin content (tHB) of 12.0 μM. These values are similar to those obtained from previous optical mammography measurements [16].

Once the constituent concentrations and scattering parameters were calculated, by applying Eq. (2) and Eq. (3), we calculated the absorption and scattering spectra (solid lines in Fig. 5). In the same figure the absorption and reduced scattering spectra obtained with the standard method (markers), are sketched.

It is worth pointing out that the solid lines, shown in Fig. 5, does not represent the interpolation of marker points. The two plots represent the absorption/scattering spectra obtained with the two fitting methods. Both absorption and scattering spectra, obtained with the TR spectral analysis, are similar to the spectra obtained by directly fitting the experimental data on *μ*_{a}
and *μ́*_{s}.

This example of *in vivo* measurement demonstrates that TR spectral fitting method is suitable also for analysis of clinical data. This preliminary result let us expect that the higher robustness, observed with the phantoms, of the TR spectral fitting compared with the standard method can be achieved with data sets acquired in a clinical environment.

## 4. Conclusion

In this work we implemented and experimentally validated, on tissue-simulating phantoms, a time-resolved spectral fitting analysis for direct estimation of chromophore concentrations and scattering parameters. Moreover, we demonstrated the improvements of this method compared with the conventional two-step approach, in particular at lower signal to noise ratio. In particular we demonstrated that the higher robustness of this fitting method allows measurements with a lower number of photon counts and therefore shorter acquisition times. Finally, the time-resolved spectral fitting method was applied to the analysis of breast measurements in transmittance geometry as an example of an *in vivo* application.

Moreover, we believe that the time-resolved spectral analysis proposed in this work can improve the quantification of tissue constituents and scattering parameters, not only with a large spectral data set, but also with signals at a reduced number of wavelengths. Further studies regarding this aspect will be carried out.

It is worth mentioning that these measurements have been realized with a time-resolved spectrophotometer based on a photonic crystal fiber which allows real-time measurements over spectral range of 600–870 nm. We believe that with this system a time-resolved spectral fitting procedure is a superior analysis method to exploit its available informative content. Future perspectives will regard the use of both the TRS-PCF system and the TR spectral fitting algorithm to perform tissue constituent imaging on breast.

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