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Optimization of the frequency-domain instrument for the near-infrared spectro-imaging of the human brain

Although many results on the near infrared spectroscopy and imaging of the human brain have been already published, the signal-to-noise ratio (SNR) of the instrumentation used in these studies has never been systematically optimized for the anatomy of the human head. Assuming a typical anatomical structure of the adult human head, we have studied the SNR of the frequency-domain instrument as a function of the modulation frequency and the source-detector distance. The study was performed both experimentally (using a pulsed laser system) and numerically (using the diffusion approximation and Monte Carlo simulations). For the specified geometry we have found that changes caused by non-homogeneities in the phase and in the ratio of the modulation amplitude to the mean intensity significantly increase with the modulation frequency and exhibit peaks between 400 and 1100 MHz, depending on the source-detector distance. Assuming the shot noise, we have found that the corresponding SNR peaks between 400 and 600 MHz. Increase in the source-detector distance results in the SNR increase only for distances below 30 mm. At larger source-detector distances the higher noise decreases the SNR. Our results show that increasing modulation frequency only to 300 MHz offers a very significant improvement in the functional near-infrared spectro-imaging of the adult human brain compared to the 100 MHz instrument.


INTRODUCTION
Most of the current frequency-domain instruments operate at relatively low modulation frequencies around 100 MHz. Such a choice was determined not only by the higher cost of higher frequency electronics, but also by the lack of evidence that higher modulation frequencies result in significant signal-to-noise ratio (SNR) improvements. However, in the case of the non-homogeneous semi-infinite medium the CW method may be unacceptably inaccurate, and only the frequency-domain method provides reliable results. The goal of this paper is to examine the frequency dependence of the SNR of the frequency-domain method in a geometry topologically close to the geometry of the human head. For a geometry that is specific for a particular activated area in the human brain we study the dependence on the modulation frequency and source-detector distance of the changes caused by the absorbing inhomogeneity in two parameters: the phase and the modulation depth (MD), i.e., the ratio AC/DC, of the photon density wave. We present the results of two studies: one performed experimentally using a sub-nanosecond pulsed light source and a spherical absorbing inhomogeneity immersed in a highly scattering solution, and the other performed numerically using Monte Carlo simulations of light transport in the MRI-based digital phantom of the adult human head which includes not only highly scattering tissues, but also the low-scattering cerebro-spinal fluid.
We show that changes caused by the absorbing inhomogeneity in both phase and MD increase with frequency and reach their maximum values between 400 and 1400 MHz depending on the source-detector distance. Under the condition that the major source of the phase and modulation noise is the quantum shot noise, the standard deviations of both phase and modulation noise can be analytically related to the modulation frequency through their dependence on the mean modulation amplitude (AC). Using these relationships and experimental data we show that for the human head geometry the SNR, i.e. the ratio of the parameter change caused by the inhomogeneity to the standard deviation of the noise in this parameter, of the frequency-domain measurement can be significantly improved by increasing the modulation frequency.

THE NOISE MODEL
As discussed in Refs. 1, 2 the sources of noise in biomedical near-infrared spectro-imaging are physiological fluctuations unrelated to the changes being studied and instrumental noise. The principal instrumental noise components in the detected signal are the quantum fluctuations of the number of electrons in the detector current (caused by the combination of the quantum noise in the input light and internal fluctuations in the number of generated photoelectrons), and specific noise sources in the electronic circuits, such as the thermal noise, 1/f noise, etc. In the case where the illumination is sufficiently high, the electronic noise can be neglected compared to the quantum noise.
Our experience in using frequency domain instruments for biomedical spectroscopy and imaging shows that the electronic noise usually is not significant. Therefore, we limited our consideration of the instrument noise to the quantum shot noise. One can show that in this case the standard deviations of fluctuations in the modulation amplitude and in the phase are proportional to HDC and HDC/AC, respectively. Since the value of AC decreases with frequency, the phase noise increases. It can be also shown that the MD standard deviation is proportional to AC/DC This means that the MD noise decreases with the frequency, so that the SNR may not decrease with frequency as rapidly as the phase SNR.
Apart from the quantum noise, in-vivo measurements are affected by physiological fluctuations, in particular by hemodynamic fluctuations, respiration and vasomotion. To investigate whether the quantum noise is the dominant source of noise affecting measurements using the frequency-domain instrument, we used frequency-domain data (modulation amplitudes and phases) acquired simultaneously at a number of source-detector distances ranging from 0.5 to 4 cm using an ISS tissue Oximeter (ISS, Champaign, Il) and a probe placed on the human forehead during one of our previous studies 3 . We have found that that, in spite of the other noise sources, quantum noise is the dominant source of noise in these in vivo frequency-domain measurements.

EXPERIMENTAL SETUP
The experimental set-up consisted of a Tsunami sapphire laser system (Spectra Physics) in a femtosecond configuration, a tank containing a Liposyn (Abbott Laboratories)/black India ink solution, a photomultiplier tube (PMT), a motorized positioning system, a single photon-counting board (Beckel & Hickel model SPC-830), and a photodiode which provides the synchronization signal to the SPC-830 board. The laser system provided a 780 nm pulsed light source at a 80 MHz repetition rate. Using a 400 µm multimode silica optical fiber the laser pulse was sent into the Liposyn India ink solution which simulated human brain in terms of the optical properties ( ≈ ′ s µ 11 cm -1 , ≈ ′ a µ 0.1 cm -1 ). The halfmaximum width of the pulse at the fiber output was roughly 0.5 ns. The scattered light was collected at the surface of the medium at a variable distance from the source by means of a 3 mm fiber bundle and then sent to the PMT detector (Hammamatsu R5600P). The acquisition system was based on the SPC-830 single photon counting board which measured the detected photon arrival times and produced a time histogram representing the averaged received pulse. In order to model the brain activity, a 2 cm diameter spherical object with the same scattering but different absorbing properties ( ≈ ′ s µ 11 cm -1 , ≈ a µ 0.2 cm -1 ) was placed at 15 mm depth between the source and detector.
The distance between the source and the detector fibers was set to 15, 20, 25, or 30 mm. In order to perform measurements with and without the object, the object was moved in and out of position using a computer-controlled step-moving system which allowed movement of the object along three orthogonal axes with an accuracy of up to 5 microns/step.
In order the to avoid pulse distortions due to the pile-up effect it was important to maintain a sufficiently low light intensity at the detector input so that the photon counting rate did not exceed 100,000 photons per second. At sourcedetector distances of 25 and 30 mm the signal attenuation due to the medium was sufficient to fulfill this requirement, while at 15 and 20 mm distances additional attenuation by a neutral density filter was necessary. For this reason we were unable to maintain the same source power at all distances. Each photon-counting measurement continued for 600 s. This time duration was selected to provide a sufficient ratio of the number of detected photons to the photon noise on the one hand, and to avoid significant non-stationary instabilities of the laser output occurring at large time intervals. In order to assess the signal variability each measurement was repeated four times under identical measuring conditions. The mean time shift between the averaged pulses corresponding to different measurements was about 25 ps, which corresponds to the Fourier transform phase slope of about 1.5∞ /GHz. Fig. 1 shows changes in phase caused by the absorbing inhomogeneity, and Fig. 2 shows changes in the modulation depth (AC/DC ratio). Different lines correspond to different source-detector distances. Changes in these parameters increase significantly with frequency and show maxima at frequencies from 600 to 1200 MHz (depending on the source-detector distance) in case of the phase change, and from 500 to 700 MHz in case of the ratio AC/DC. The error bars in Figs. 1 and 2 show average errors corresponding to measurements repeated four times. In Fig. 1 one can see that the phase error increases with frequency according to the model based on quantum noise. In case of the MD signal (Fig.  2) the error does not increase with frequency, which also agrees with the quantum noise model. The fact that the MD error in Fig. 2 does not generally decrease with frequency is due to the small long-term instabilities in the laser pulse, which slightly modify its shape during different photon-counting measurements. These instabilities are specific for the pulsed laser used in this study, but not for a well designed frequency-domain instruments utilizing intensity-modulated light sources.  Since the photon counting method required using additional light attenuation at smaller distances, we could not normalize the SNR obtained at different distances. Therefore, in both Figs. 3 and 4 all curves are normalized to the corresponding SNR values at 100 MHz. Thus, Figs. 3 and 4 do not compare the SNRs for signals at different distances, but rather show the SNR changes from their respective values at 100 MHz. Fig. 3 shows that, depending on the distance, the phase SNR reaches its maximum in the range of 300-500 MHz. In this frequency range the improvement compared to 100 MHz is quite significant (2.5-3 times). According to Fig. 3 the maximum improvement is reached at source-detector distances of about 20-25 mm. At smaller distances the phase signal is too small, while at a larger distance the phase noise becomes too large.

EXPERIMENTAL RESULTS
The main difference between the phase and MD SNRs is that the phase SNR peaks between 300 and 500 MHz, while the AC/DC SNR grows monotonically up to approximately 1200 MHz. This happens because, in the case of the MD SNR, the values corresponding to the signal curves in Fig. 4 are divided by the AC values, which decline monotonically with frequency. It is interesting to note that in the range of 300-500 MHz where the phase SNR reaches maximum, the MD SNR at different distances increases about 5-10 times compared to its value at 100 MHz.

MONTE CARLO SIMULATIONS
We used the Monte Carlo code described in Ref. 4 . which models propagation of light beams (ì photonsî ) in a volume specified as a collection of voxels with specific optical properties. We derived our digital phantom of the adult human head (see Fig. 5) from the high-resolution, MRI-based, segmented, computerized head phantom 5 .  The size of each voxel was 1x1x1 mm. Although the original phantom included more than 100 tissues, we simplified it by unifying various tissues into four layers: the external layer (scalp and skull), the cerebro-spinal fluid (CSF), the nonactivated brain, and the activated brain (see Fig. 5). The absorption coefficients of these layers were 0.1 cm -1 , 0.01 cm -1 , 0.1 cm -1 , and 0.13-0.26 cm -1 (since no absolute values for the optical properties of the activated brain tissue have yet been published, in our simulations we used various values of µ a in the range 0.13-0.26 cm -1 .), respectively. The reduced scattering coefficients were 10 cm -1 , 1.0 cm -1 , 10 cm -1 , and 10 cm -1 , respectively. The values of absorption and reduced scattering coefficients for the outer layer and the brain correspond approximately to those we obtained in our experimental study 6 . The values of the optical properties for the low-scattering and low-absorbing CSF layer (µ a =0.01 cm -1 and µ' s =1 cm -1 ) were taken from the literature 4 . The activated area of diameter about 1.5 cm was selected in the primary motor cortex (see Fig. 5).
In Figs. 6 and 7 the SNR curves for simulated phase and MD changes are normalized to the values corresponding to 100 MHz and a source-detector distance of 15 mm. Thus, these figures compare the SNR behavior at different frequencies and at different distances with that at 100 MHz and 15 mm. In Fig. 6 one can see that the phase SNR significantly decreases at distances larger than 25 mm. Fig. 7 shows that, although at larger distances and higher frequencies the MD SNR very significantly exceeds its value at 15 mm and 100 MHz, the curves for 30 and 35 mm are very similar. These features indicate that while an increase in the modulation frequency leads to a significant SNR improvement, the optimal source-detector distance does not exceed 30 mm.

DISCUSSION
For the geometry topologically similar to the one of the human head we have found, both experimentally and using the Monte-Carlo simulation, that the changes caused by an absorbing inhomogeneity to the modulation depth and phase of the photon density wave exhibit a non-monotonic dependence on the modulation frequency. The change in phase reaches maximum between 400 and 1400 MHz, and the MD change achieves its maximum value in the 500-600 MHz frequency range depending on the source-detector distance. Such a resonance-like dependence of signals caused by an inhomogeneity is specific for the situation when a relatively small object is immersed in a semi-infinite medium at a depth of about 15 mm, and was not found for the slab geometry 1 . The most important result of our study is that, for the specified geometry, the SNR of the frequency-domain instrument can be significantly improved by increasing the modulation frequency. The phase SNR can be increased 2-3 times compared to the value at 100 MHz, which is comparable with the improvement demonstrated for the slab geometry in Ref. 1 . The MD SNR can be theoretically increased more than 10 times, provided that the modulation frequency is sufficiently high. Current electronics allows modulation of light sources at frequencies up to 400-500 MHz with a modulation depth of more than 30 %. Therefore, the discussed significant SNR improvement is now achievable in practice.
We have also found that, for the specified geometry, both phase and MD SNR are optimized at source-detector distances of about 20-30 mm. Increasing source-detector distances to greater than 30 mm causes the phase SNR to decrease due to the higher noise, and the lower signal.
The results of both studies are close not only qualitatively, but also quantitatively. This indicates that the demonstrated influence of the modulation frequency and source-detector distance on the phase and MD SNR is related to the common features of both models, such as the geometry and the values of the optical properties of layers, which play the most important role in the light transport. Thus we should conclude that including the CSF layer intohe digital model does not cause significant effect on the phase and modulation depth.

CONCLUSION
In phantom experiments with sub-nanosecond light pulses, and using numerical Monte-Carlo simulations we have found that changes in the phase and modulation depth of a photon-density wave caused by local functional activation in human brain exhibit non-monotonic dependence on frequency, with a maximum value corresponding to frequencies between 400 and 1400 MHz depending on the source-detector distance. We have experimentally demonstrated that for a typical frequency-domain instrument the noise in the AC and phase data acquired in vivo corresponds to the quantum shot noise model. We have shown that the SNR for these changes can be very significantly improved compared to its value at 100 MHz by increasing the modulation frequency to 400-500 MHz. The source-detector distance which optimizes the SNR for both phase and modulation depth signals is about 20-25 mm.

ACKNOWLEDGEMENT
This work was supported by the NIH SBIR grant no. NS 38378-03, and by the NIH grant 1R01MH065429-01A2.