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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12261–12274
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Signal detectability in diffusive media using phased arrays in conjunction with detector arrays

Dongyel Kang and Matthew A. Kupinski  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12261-12274 (2011)
http://dx.doi.org/10.1364/OE.19.012261


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Abstract

We investigate Hotelling observer performance (i.e., signal detectability) of a phased array system for tasks of detecting small inhomogeneities and distinguishing adjacent abnormalities in uniform diffusive media. Unlike conventional phased array systems where a single detector is located on the interface between two sources, we consider a detector array, such as a CCD, on a phantom exit surface for calculating the Hotelling observer detectability. The signal detectability for adjacent small abnormalities (2mm displacement) for the CCD-based phased array is related to the resolution of reconstructed images. Simulations show that acquiring high-dimensional data from a detector array in a phased array system dramatically improves the detectability for both tasks when compared to conventional single detector measurements, especially at low modulation frequencies. It is also observed in all studied cases that there exists the modulation frequency optimizing CCD-based phased array systems, where detectability for both tasks is consistently high. These results imply that the CCD-based phased array has the potential to achieve high resolution and signal detectability in tomographic diffusive imaging while operating at a very low modulation frequency. The effect of other configuration parameters, such as a detector pixel size, on the observer performance is also discussed.

© 2011 OSA

1. Introduction

Diffuse optical tomography (DOT) is a non-invasive imaging tool that probes the functional structure of biological tissues and has been used in imaging of the brain and breast. The harmless near-infrared light used in DOT discriminates oxyhemoglobin, de-oxyhemoglobin, and water content due to different absorption coefficients in biological tissues [1

1. C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D Appl. Phys. 36(14), R207–R227 (2003). [CrossRef]

,2

2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

]. Transmitted and/or reflected diffusive photons from a biological tissue are used tomographically to reconstruct maps of scattering and absorption coefficients of the tissue. Therefore, DOT has enormous potential to diagnose tissue abnormalities including cancerous tumors.

Data measured using a continuous wave (DC) source cannot, in general, be reconstructed to a unique solution due to the ill-posed nature of the forward scattering problem. One effective data-acquisition method to diminish this problem is to modulate the intensity of the incident beam source. In this frequency-domain measurement, modulation amplitude attenuation and phase delay of output beams are employed to reconstruct the absorption and scattering maps [3

3. S. R. Arridge and W. R. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23(11), 882–884 (1998). [CrossRef]

]. It could be mistakenly considered that higher modulation frequency fm always leads to better signal-to-noise (SNR) of measurements due to larger amplitude and phase variations as measured at individual detectors. In reality, however, the performance is not always increasing as fm increases due to noise; hence there exists an optimal fm [4

4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

]. Generally, SNR of the modulation amplitude and/or phase under appropriate noise models is considered to estimate the system performance [4

4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

7

7. U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]

]. V. Toronov et al. [6

6. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express 11(21), 2717–2729 (2003). [CrossRef] [PubMed]

] regarded a quantum shot noise as a main noise source in DOT and derived analytic expressions for amplitude and phase standard deviations as
σA=βaDC  and  σφ=σA/aAC,
(1)
respectively, where aDCand aAC are mean DC and modulation amplitude. β is a term affected by the measurement system which is also proportional to the amount of quantum shot noise. It is experimentally verified that the standard deviations in Eq. (1) are reasonable for the frequency-domain DOT [6

6. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express 11(21), 2717–2729 (2003). [CrossRef] [PubMed]

,7

7. U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]

]. It has recently been reported that the optimal fm for a single source system considering the noise model of Eq. (1) is between 400MHz and 600MHz for small tissue volumes [4

4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

]. If the distance between source and detector is closer and/or a tissue has a lower scattering coefficient, the optimal fm increases up to 800MHz [8

8. U. J. Netz, A. H. Hielscher, A. K. Scheel, and J. Beuthan, “Signal-to-noise analysis for propagation of laser radiation through a tissue-like medium by diffuse photon-density waves,” Laser Phys. 17(4), 453–460 (2007). [CrossRef]

].

Since fm in DOT is usually hundreds of MHz or higher, current CCD or CMOS detectors cannot measure the modulation amplitude and phase of output beams. This is overcome by a heterodyne method, where the modulated gain of an image intensifier makes the amplified output modulate at the beat frequency that is the difference between fm and the gain modulation frequency [9

9. K. Lee, S. D. Konecky, R. Choe, H. Y. Ban, A. Corlu, T. Durduran, and A. G. Yodh, “Transmission RF diffuse optical tomography instrument for human breast imaging,” Proc. SPIE 6629, 66291R, 66291R-6 (2007). [CrossRef]

]. Because the beat frequency is typically much lower than the bandwidth of CCD or CMOS detectors, the heterodyne output can be measured precisely, thus obtaining modulation amplitude and phase images across the entire detector array. If the gain is modulated at the exact same frequency as fm, but has a different phase from the source beam, the amplified output beam varies along the heterodyne output trajectory by changing the relative phase; this is known as a homodyne method [10

10. A. B. Thompson and E. M. Sevick-Muraca, “Near-infrared fluorescence contrast-enhanced imaging with intensified charge-coupled device homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8(1), 111–120 (2003). [CrossRef] [PubMed]

]. With the homodyne method, noise can be easily reduced by increasing the detector exposure time or acquiring multiple images and averaging; an advantage not shared by the heterodyne method. Although it is technically possible to realize gain modulation frequencies up to 1GHz or more [7

7. U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]

], the instrumentation needed to achieve this fast gain modulation is very expensive. Furthermore, the lifetime of the image intensifier’s cathode, where a modulated voltage is applied to achieve gain modulation, might be dramatically reduced under the circumstance of high frequency modulation. Therefore, it is beneficial to develop frequency-domain diffusive imaging methods that have high performance while operating at low modulation frequencies.

The use of out-of-phase modulation inputs, called a phased array, has been introduced as the effective method to increase the phase variation of modulated output beams in diffusive media [11

11. J. M. Schmitt, A. Knüttel, and J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9(10), 1832–1843 (1992). [CrossRef] [PubMed]

,12

12. D. G. Papaioannou, G. W. ‘t Hooft, S. B. Colak, and J. T. Oostveen, “Detection limit in localizing objects hidden in a turbid medium using an optically scanned phased array,” J. Biomed. Opt. 1(3), 305 (1996). [CrossRef]

]. A typical phased array system has two modulated inputs with π intensity phase difference and a single detector on the interface between two sources, where a null plane of abrupt amplitude and phase transitions is generated in homogeneous media. The amplitude and phase on a single detector are very sensitive to the variation of photon number difference from sources, which are strongly affected by inhomogeneity of the investigated phantom. Therefore, traditional phased array systems optically scan the phantom to detect a signal that perturbs the modulation amplitude and/or phase null plane. It has been reported that a phased array system is more robust to biological random noise than a single source system, and that the amplitude-phase crosstalk is almost canceled when both input beams are generated by one source [13

13. S. P. Morgan and K. Y. Yong, “Amplitude-phase crosstalk cancelation in frequency domain instrumentation,” Proc. SPIE 4250, 269–275 (2001). [CrossRef]

,14

14. X. Intes, V. Ntziachristos, and B. Chance, “Analytical model for dual-interfering sources diffuse optical tomography,” Opt. Express 10(1), 2–14 (2002). [PubMed]

]. Not confined to just scanning tissues, the idea of phased array is used in fluorescence imaging tomographic reconstructions under the assumption of diffuse approximation [14

14. X. Intes, V. Ntziachristos, and B. Chance, “Analytical model for dual-interfering sources diffuse optical tomography,” Opt. Express 10(1), 2–14 (2002). [PubMed]

16

16. B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52(5), 1409–1429 (2007). [CrossRef] [PubMed]

]. Recently, simulations have shown that the reconstructed images from a phased array show better resolution than a single source system, but it is hard to implement the system for reliable measurements due to the scanning [17

17. S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105(2), 024702 (2009). [CrossRef]

].

2. Model

The abnormality or signal to be detected is 2 × 2 × 2mm3 and is placed at various locations within the uniform phantom (see Fig. 1). The possible abnormality locations (LSig) are indicated as dark gray squares in Fig. 1, which are numbered for convenience. The separation between each LSig is 4mm. We will simulate two types of signals to be detected: absorbing and scattering signals. For absorbing signals, μs of the signal is the same as the background but μa of the signal is set to 0.01mm−1. For scattering signals, μa is the same as the background and μs is set to 5.005mm−1. These signals mimic abnormal tissues generally showing higher μa and μs than values in normal breast tissues due to highly concentrated blood vessels. 2mm-diameter sources of a phased array system are simulated with 108 photons each, which are indicated as two red arrows and dots in Figs. 1(a) and 1(b), respectively. Output photons from each source are recorded on the detector array located on the center of the phantom exit face, as shown in Fig. 1(a). 2 × 2mm (D1) and 40 × 40mm (D2) detector arrays are considered to investigate the effect of a large detector array. A relative time difference τ is introduced between two sources to model the relative phase difference for a fixed fm. The modulation amplitude and phase for the phased array system are calculated by Fourier transforming the combination of two output photons from each source, where one is shifted from the other by the amount of τ.

The observer we will focus on is an ideal linear observer called Hotelling observer, which is considered a reliable assessment tool in statistical decision problems [21

21. H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12(5), 834–852 (1995). [CrossRef]

,22

22. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

,28

28. A. R. Pineda, M. Schweiger, S. R. Arridge, and H. H. Barrett, “Information content of data types in time-domain optical tomography,” J. Opt. Soc. Am. A 23(12), 2989–2996 (2006). [CrossRef]

]. The Hotelling observer SNR for the current type of study, where both signals and background are known exactly, is defined as
SNRH2=Δg¯tΚ1Δg¯
(2)
where Δg¯=g¯1g¯0 with g¯1 being the mean data vector when the signal is present and g¯0being the mean data vector when the signal is absent. The term K=1/2(K0+K1) is the average covariance matrix of the data without and with the signal - K0and K1 covariance matrices, respectively. The dimension of K is N × N, where N is the number of detector pixels. As shown in Eq. (2), the Hotelling observer SNR can be calculated from only first and second moments of data.

It is known that Bayesian ideal observer calculates the most reliable performance but the computation of this observer is often impractical [22

22. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

]. The efficiency of other observers, such as Hotelling observer, can be defined from the ideal observer’s performance, where higher observer efficiency indicates the performance from the observer is more reliable. Hotelling observer is equivalent to Bayesian ideal observer when data distributions are Gaussian [22

22. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

]. Although statistics of modulation amplitude and phase are not Gaussian, the skewness of their probability density functions don’t deviate much from that of Gaussian distribution [32

32. D. Kang and M.A. Kupinski, “Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging,” J. Biomed. Opt. (to be submitted). [PubMed]

]. Therefore, it can be stated that Hotelling observer SNRs of Eqs. (4) and (5) are highly efficient, which validates the use of Hotelling observer assessment for this study.

3. Simulation result

Usually, the measurement of modulation amplitudes is less reliable than phase due to surface roughness of tissues and/or stray light [33

33. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005). [CrossRef] [PubMed]

]. Therefore, we mainly investigate SNRφ2 for the Hotelling observer performance of the phased array systems. The detector pixel size lD is 1 × 1mm, so N in Eq. (5) is 4 and 1600 for D1 and D2, respectively. The term β in Eqs. (4) and (5) is not physically meaningful, because this quantity just scales the system’s Hotelling observer performance. Therefore, β is arbitrarily chosen, and held constant for all simulations. The characteristics ofSNRA2 and the effect of different lD on SNRφ2 are discussed later. Figures 2(a)
Fig. 2 Hotelling observer SNRs for the task of detecting small absorbing signals atLSig = 1 and 3 in a phased array system with the detector D1 ((a) and (b)) and D2 ((c) and (d)) are shown. These SNRs are calculated with modulation phases. SNRs of all examined signal locations at fixed modulation frequencies with D1 and D2 are shown in (e) and (f), respectively.
2(d) show SNRφ2 from two different detectors D1 and D2 for absorbing signals located on LSig = 1 and 3. For each result, SNRφ2 with τ = 0, 2ns, and 4ns are shown as the function of fm, where π of the relative phase differences between two input sources happen at fm = 125MHz and 250HMz for τ = 2ns and 4ns, respectively. The system configuration for Figs. 2(a) and 2(b) is similar to that of a conventional scanning phased array system because of the point-like detector D1, where SNRφ2 values near π of the relative phase are relatively high, but show minimum at the exact π. S. Morgan et al. [19

19. S. P. Morgan and K. Y. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7(13), 540–546 (2000). [CrossRef] [PubMed]

] pointed out this feature of a phased array system with the explanation that the steepest phase gradient is generated on the phase transition plane with the source phase difference of near π, which is not possible for the exact π. The detecting mechanism of the conventional phased array system can be understood from Figs. 2(a) and 2(b), where the movement of the absorbing abnormality around the phase transition plane caused by phased array scanning dramatically changes detectability, when the phase difference between two sources are near π. Although not shown in Fig. 2, SNRφ2 forLSig = 2 shows a similar behavior with Fig. 2(b), which indicates the abrupt SNR improvement happens when the inhomogeneity is scanned around the phase transition plane.

SNRφ2 calculated from modulation phases detected from D2 is dramatically increased, as shown in Fig. 2(c) and 2(d). In addition to the dramatic improvement, it is observed that SNRφ2 from D2 still shows a minimum value at π phase difference, but the width of this dip is much wider than that of the D1 detector. Resulting from this broadening, peak SNR values for each τ are shifted to lower fm than Figs. 2(a) and 2(b). For the example of τ = 4ns and LSig = 1, peaks of SNR are observed around 80MHz for D2, and around 120MHz for D1. Another important characteristic of SNRφ2 from D2 is that the performance for all studied absorbing signal locations is relatively high at fixed fm ; this fm is called the optimal fm. This can be partially observed in Figs. 2(c) and 2(d) that bothSNRφ2s of τ = 4ns show peaks around fm = 80MHz, which is not the case for Figs. 2(a) and 2(b). The existence of the optimal fm is obvious in Figs. 2(e) and 2(f), whereSNRφ2 of all LSigat selected modulation frequencies are shown for D1 and D2, respectively. Compared to Fig. 2(e), where the variation of SNRφ2 is largely sensitive toLSig, the overall performance in Fig. 2(f) is consistently high for the entireLSig.

Hotelling observer SNRs for the task of distinguishing between adjacent signals are shown in Fig. 3
Fig. 3 Hotelling observer SNRs for two adjacent signals at for LSig = 1 and 3 in a phased array system with the detector D1 ((a) and (b)) and D2 ((c) and (d)) are shown. These SNRs are calculated with modulation phases. SNRs of all examined signal locations at fixed modulation frequencies with D1 and D2are shown in (e) and (f), respectively.
, of which notation is indicated asSNRΔφ2. For SNRΔφ2, two different system states in Eq. (2) are two small absorbing abnormalities, where one is located on LSig and the other is shifted 2mm in depth from the LSig. Different from SNRφ2 that indicates signal detectability for the absorbing signal located on LSig,SNRΔφ2 measures the resolution of reconstructed images for these two adjacent signals. It is known that the depth resolution of the reconstructed image in DOT is usually worse than the lateral resolution [34

34. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

]. Therefore, onlySNRΔφ2 for signals shifted in depth is investigated in this paper. Figure 3 shows that characteristics of the SNRΔφ2variation from D1 to D2 are similar with the case ofSNRφ2. These SNRΔφ2 values are dramatically increased by the larger detector array, and an optimal fm, where SNRΔφ2of all signal locations are relatively high and stable, exists. It is estimated from Figs. 3 and 4
Fig. 4 (a) Mean SNRφ2 calculated from allLSig are shown with different τ and μs, where the optimal fm are almost invariant to μs. (b) the variation of SNRφ2toLSig is shown for different detector sizes at τ = 6ns and fm = 55MHz, where the increasing rate ofSNRφ2 is saturated.
that considering a detector array in a phased array system can stabilize both SNRφ2 andSNRΔφ2 simultaneously at the same fixed modulation frequencies for absorbing signals. Especially, for the phantom configuration of Fig. 1, the fm is ~80MHz for τ = 4ns, which is less than 100MHz, so the instrument cost could be dramatically reduced.

Figure 4(a) shows averaged SNRφ2 from all LSig measured from D2 for differentμsand τ, where the optimal fm are at around 80MHz, 55MHz, and 40MHz for τ = 4ns, 6ns and 8ns, respectively. Although not shown here, it is observed that the behavior of averaged SNRΔφ2 is similar and peaks of these values are observed at the almost same fm as SNRφ2. As shown in Fig. 4(a), the optimal fm is decreased and the absolute peak value is slight increased, as τ increases. Therefore, it can be concluded that the higher τ is the better the Hotelling observer performance is in the CCD-based phased array. Notice that the optimal fm is ~40MHz and 55MHz for τ = 6ns and 8ns, respectively, which are much smaller than the previously reported optimal fm in a single source system. More importantly, these optimal fm for each τ are almost invariant to the background μsas determined from Fig. 4(a). This characteristic of the CCD-based phased array is highly desirable to DOT because it implies that the system can be optimized for all types of tissue properties.

Figure 4(b) shows SNRφ2 forμs = 5mm−1 at fm = 55MHz with τ = 6ns for various sizes of the detector array for the fixed detector pixel size of 1 × 1mm. It is reasonable to consider that Hotelling observer performance in Eq. (2) is increased more for the higher number of detector pixels, because of more terms added. It is observed in Fig. 4(b), however, that the increasing rate of SNRφ2 is saturated as the detector size increases. We confirmed thatSNRΔφ2 shows the similar trend, even though the results are not shown here. There are two reasons for this phenomenon. First, photon numbers detected by the pixels far from the center are much smaller than the central area, so aAC2/aDCin Eq. (5) is dramatically decreased. Second, the abrupt phase variation of a phased array is mainly concentrated on the central area, so Δφ¯2in Eq. (5) measured from outer area is also decreased. Therefore, the overall contribution from outer detector pixels toSNRφ2 becomes very small. Additional study shows that this SNRφ2 saturation by the increased detector array size happens for different source separations of 8mm and 24mm. Furthermore, amazingly, the optimal fm observed from the saturated SNRφ2 for these different source separations are almost the same. Overall, Fig. 4 indicates that applying a CCD-based phased array system generates a sensitivity area inside diffusive media. Both SNRφ2 and SNRΔφ2 for absorption inhomogeneities within the entire sensitivity area simultaneously show peak values at the same fm, where the fm is mainly determined by τ and almost invariant to other properties, such as μs and a source separation.

The effect of changing lD in a CCD-based detector array on the Hotelling observer SNR can be understood by further investigating Eq. (5), where it is easily found out that the ith SNRH2component is proportional to the number of photons measured on ith detector pixel. Therefore, if lD is decreased, the SNR value on each detector pixel is also decreased. However, the wholeSNRH2 is calculated by summing all pixel values, and the number of pixels is increased as lD is decreased (assuming the size of the detector array is the same). Resulting from this, SNRH2 remains at least the same as long as the amounts of aDC and aACon a single detector pixel are proportionally changed to lD. This implies that there should be other factors to further increase ΔaAC2 and/or Δφ¯2of SNRΔφ2 and SNRφ2 in Figs. 5(a) and 5(b), respectively. Figures 5(c) and 5(d) show one-dimensional profiles of modulation phase difference Δφ¯ from absorbing and scattering signals, respectively. Δφ¯ for absorbing signals are mainly non-negative or non-positive, which is already observed elsewhere [19

19. S. P. Morgan and K. Y. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7(13), 540–546 (2000). [CrossRef] [PubMed]

]. For this situation, when lDis reasonably small compared with the variation of Δφ¯, the summed value of Δφ¯2from all detector pixels is almost invariant to lD. Therefore, lDdoesn’t affect SNRφ2 for small absorbing abnormalities. The situation is, however, different for Δφ¯ from scattering signals, as shown in Fig. 5(d), where each Δφ¯2 value could be dramatically increased as lD decreases by reducing the abrupt oscillating part of Δφ¯ in each detector pixel area. Therefore, smaller lD achieves higher SNRφ2 as shown in Fig. 5(b). The increase of SNRΔφ2 in Fig. 5(a) can be similarly understood. Generally, most biological tissues are inhomogeneous in terms of μa and μs. Therefore, it can be concluded that imaging smaller areas on the exit surface of phantoms to each detector pixel always increases both SNRφ2 and SNRΔφ2. There is, however, the limitation to increaseSNRφ2 and SNRΔφ2by reducing lD, because the noise model of Eq. (1) is not valid when the number of photons on a single detector pixel is below a certain level [32

32. D. Kang and M.A. Kupinski, “Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging,” J. Biomed. Opt. (to be submitted). [PubMed]

]. Increasing the measurement time (detector exposure time) could solve this problem, but this is prohibited or limited in some applications of diffusive imaging.

4. Discussion and conclusion

We observed that using a detector array substantially improves detectability in phased array systems where the detectability is the Hotelling observer performance calculated from modulation phases. It is interesting to investigate the effect of a detector array onSNRA2 in Eq. (4) in a phased array system. Figures 6(a)
Fig. 6 Hotelling observer SNRs for the task of detecting small absorbing abnormalities at = 1 and 3 in phased array systems with detectors (a) D1 and (b) D2 are shown. Hotelling observer SNRs for the task of detecting two adjacent abnormalities are shown in (c) and (d) for D1 and D2, respectively. These quantities of Hotelling observer SNR are calculated with modulation amplitudes.
and 6(b) showSNRA2 from absorbing signals at LSig = 1 and 3 with D1 and D2, respectively, where a detector array of D2 increases overall SNRA2 dramatically likeSNRφ2. For both detector types, it is observed that SNRA2 of τ = 0ns is monotonically decreasing as fm increases, which is similar to the result in a single source system [4

4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

,6

6. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express 11(21), 2717–2729 (2003). [CrossRef] [PubMed]

]. For LSig = 1 with D1, introducing nonzero τ between two modulation sources makes SNRA2 oscillate to fm having minimums at π phase differences. The trend of SNRA2 variation forLSig = 3 does not differ much from the case of LSig = 1 except there are small peaks at π phase differences. Figure 6(b) shows that the detector array of D2 increases SNRA2 at LSig = 3 more than the case of LSig = 1 and additionally, SNRA2of τ = 2ns and 4ns approaches to values of τ = 0ns, which are the clear improvement caused by D2. Figures 6(c) and 6(d) show SNRΔA2 calculated from modulation amplitudes for two adjacent absorbing signals, where basic characteristics are analogous SNRA2. The noticeable difference from SNRA2 is that peaks at π phase differences between two sources are more remarkable.

Previously, conventional SNRs of modulation amplitudes and phases have been separately considered to estimate the performance of frequency-domain diffusive imaging [4

4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

,20

20. S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt. 43(10), 2071–2078 (2004). [CrossRef] [PubMed]

]. It is reasonable that observer performance is larger (smaller) when both quantities are simultaneously increased (decreased). For the most cases, though, the variations of these two quantities are not correspondent, so it is complicated to determine the system’s optimization condition. For example of the phased array with D2 for τ = 4ns, SNRA2 shows relatively high values at fm = 250MHz, but this is not the case forSNRφ2, as shown in Figs. 6(b) and 2(d), respectively. One ad hoc method for compromising this issue is to simply consider the modulation amplitude and phase together in Eq. (2). For this case, dimensions of the data vectors and covariance matrix are 2N where N is the number of detector pixels. Since the modulation amplitude and phase are uncorrelated in frequency-domain measurement [32

32. D. Kang and M.A. Kupinski, “Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging,” J. Biomed. Opt. (to be submitted). [PubMed]

], the covariance matrix doesn’t have any non-zero off-diagonal terms. Therefore, the Hotelling observer SNR for this approach is just the summation of Eqs. (4) and (5). However, the absolute value ofSNRA2 is usually much larger than SNRφ2, as shown in Figs. 2 and 6, which means that it is not reasonable to just simply consider the modulation amplitude and phase together for Hotelling observer performance. Due to less reliability of measured modulation amplitudes, the current study mainly focuses on SNRφ2 rather than SNRA2, but it is worthwhile investigating the method of effectively combining modulation amplitude and phase in frequency-domain diffusive imaging for more appropriate assessment.

The sensitivity area is observed in a CCD-based phased array, where both SNRφ2 and SNRΔφ2 are simultaneously high at a fixed fm that is called as an optimal fm in this paper. Additional simulations showed that performances for signals located outside boundaries determined by two sources are very poor, which means the sensitivity area width in the direction of connecting two sources is usually limited by the source separation, unless the distance between two sources is very small. Increasing the separation of two sources could extend the sensitivity area, but the overall performance is decreased. It is also observed for the case of small absorbing abnormalities that there is no clear boundary of the sensitivity area in the direction vertical to the line connecting sources. The sensitivity area’s vertical boundary is not strongly varied to the source separation as long as phantom properties are remained as the same. B. Chance et al. [35

35. B. Chance, E. Anday, S. Nioka, S. Zhou, L. Hong, K. Worden, C. Li, T. Murray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2(10), 411–423 (1998). [CrossRef] [PubMed]

] contrived multiple sources in a phased array system, where the relative phase difference between sources is π and point detectors are located on multiple phase transition planes. We expect that the system of multiple modulation sources with a detector array could generate much large sensitivity area functioning at some fixed fm that could be much less than 1/(2ω) by introducing some phase difference between sources, where ω is the modulation frequency of sources.

In conclusion, we investigated the effect of CCD or CMOS detectors on Hotelling observer detectability in phased array systems for the tasks of detecting small abnormalities (or signals) and for distinguishing between two adjacent signals in homogeneous diffusive media. Simulations show that the CCD-based phased array system could dramatically improve signal detectability and resolution of reconstructed images. Furthermore, we demonstrated that the modulation frequency optimizing the phased array system is generated by the detector array, where both signal detection and reconstruction resolution for small inhomogeneity located within some phantom area (i.e., sensitivity area) are simultaneously high. The value of the optimal fm can be much smaller than 100MHz by increasing the time difference between two modulated input sources, which drives down the equipment cost for these systems. These results verify that a CCD-based phased array system is a high performance and cost-effective tomographic tool for diffusive imaging of tissue, which could be developed into new imaging systems for diffusive imaging applications.

Acknowledgments

Authors acknowledge Stefano Young for his discussion about the tMCimg simulator.

References and links

1.

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D Appl. Phys. 36(14), R207–R227 (2003). [CrossRef]

2.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]

3.

S. R. Arridge and W. R. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23(11), 882–884 (1998). [CrossRef]

4.

H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]

5.

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express 9(4), 212–224 (2001). [CrossRef] [PubMed]

6.

V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express 11(21), 2717–2729 (2003). [CrossRef] [PubMed]

7.

U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]

8.

U. J. Netz, A. H. Hielscher, A. K. Scheel, and J. Beuthan, “Signal-to-noise analysis for propagation of laser radiation through a tissue-like medium by diffuse photon-density waves,” Laser Phys. 17(4), 453–460 (2007). [CrossRef]

9.

K. Lee, S. D. Konecky, R. Choe, H. Y. Ban, A. Corlu, T. Durduran, and A. G. Yodh, “Transmission RF diffuse optical tomography instrument for human breast imaging,” Proc. SPIE 6629, 66291R, 66291R-6 (2007). [CrossRef]

10.

A. B. Thompson and E. M. Sevick-Muraca, “Near-infrared fluorescence contrast-enhanced imaging with intensified charge-coupled device homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8(1), 111–120 (2003). [CrossRef] [PubMed]

11.

J. M. Schmitt, A. Knüttel, and J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9(10), 1832–1843 (1992). [CrossRef] [PubMed]

12.

D. G. Papaioannou, G. W. ‘t Hooft, S. B. Colak, and J. T. Oostveen, “Detection limit in localizing objects hidden in a turbid medium using an optically scanned phased array,” J. Biomed. Opt. 1(3), 305 (1996). [CrossRef]

13.

S. P. Morgan and K. Y. Yong, “Amplitude-phase crosstalk cancelation in frequency domain instrumentation,” Proc. SPIE 4250, 269–275 (2001). [CrossRef]

14.

X. Intes, V. Ntziachristos, and B. Chance, “Analytical model for dual-interfering sources diffuse optical tomography,” Opt. Express 10(1), 2–14 (2002). [PubMed]

15.

Y. Chen, G. Zheng, Z. H. Zhang, D. Blessington, M. Zhang, H. Li, Q. Liu, L. Zhou, X. Intes, S. Achilefu, and B. Chance, “Metabolism-enhanced tumor localization by fluorescence imaging: in vivo animal studies,” Opt. Lett. 28(21), 2070–2072 (2003). [CrossRef] [PubMed]

16.

B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52(5), 1409–1429 (2007). [CrossRef] [PubMed]

17.

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105(2), 024702 (2009). [CrossRef]

18.

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express 9(4), 212–224 (2001). [CrossRef] [PubMed]

19.

S. P. Morgan and K. Y. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7(13), 540–546 (2000). [CrossRef] [PubMed]

20.

S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt. 43(10), 2071–2078 (2004). [CrossRef] [PubMed]

21.

H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12(5), 834–852 (1995). [CrossRef]

22.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

23.

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13(5), 669–680 (1997). [CrossRef] [PubMed]

24.

Z. M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets,” Opt. Lett. 30(24), 3338–3340 (2005). [CrossRef]

25.

R. B. Schulz, J. Peter, W. Semmler, C. D’Andrea, G. Valentini, and R. Cubeddu, “Comparison of noncontact and fiber-based fluorescence-mediated tomography,” Opt. Lett. 31(6), 769–771 (2006). [CrossRef] [PubMed]

26.

G. Y. Panasyuk, Z. M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33(15), 1744–1746 (2008). [CrossRef] [PubMed]

27.

H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7(7), 1266–1278 (1990). [CrossRef] [PubMed]

28.

A. R. Pineda, M. Schweiger, S. R. Arridge, and H. H. Barrett, “Information content of data types in time-domain optical tomography,” J. Opt. Soc. Am. A 23(12), 2989–2996 (2006). [CrossRef]

29.

D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10(3), 159–170 (2002). [PubMed]

30.

N. Shah, A. Cerussi, C. Eker, J. Espinoza, J. Butler, J. Fishkin, R. Hornung, and B. Tromberg, “Noninvasive functional optical spectroscopy of human breast tissue,” Proc. Natl. Acad. Sci. U.S.A. 98(8), 4420–4425 (2001). [CrossRef] [PubMed]

31.

M. S. Nair, N. Ghosh, N. S. Raju, and A. Pradhan, “Determination of optical parameters of human breast tissue from spatially resolved fluorescence: a diffusion theory model,” Appl. Opt. 41(19), 4024–4035 (2002). [CrossRef] [PubMed]

32.

D. Kang and M.A. Kupinski, “Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging,” J. Biomed. Opt. (to be submitted). [PubMed]

33.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005). [CrossRef] [PubMed]

34.

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]

35.

B. Chance, E. Anday, S. Nioka, S. Zhou, L. Hong, K. Worden, C. Li, T. Murray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2(10), 411–423 (1998). [CrossRef] [PubMed]

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(030.6600) Coherence and statistical optics : Statistical optics
(290.7050) Scattering : Turbid media

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: March 10, 2011
Revised Manuscript: May 26, 2011
Manuscript Accepted: June 1, 2011
Published: June 9, 2011

Virtual Issues
Vol. 6, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Dongyel Kang and Matthew A. Kupinski, "Signal detectability in diffusive media using phased arrays in conjunction with detector arrays," Opt. Express 19, 12261-12274 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12261


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References

  1. C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D Appl. Phys. 36(14), R207–R227 (2003). [CrossRef]
  2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999). [CrossRef]
  3. S. R. Arridge and W. R. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23(11), 882–884 (1998). [CrossRef]
  4. H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16(22), 18082–18101 (2008). [CrossRef] [PubMed]
  5. Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express 9(4), 212–224 (2001). [CrossRef] [PubMed]
  6. V. Toronov, E. D’Amico, D. Hueber, E. Gratton, B. Barbieri, and A. Webb, “Optimization of the signal-to-noise ratio of frequency-domain instrumentation for near-infrared spectro-imaging of the human brain,” Opt. Express 11(21), 2717–2729 (2003). [CrossRef] [PubMed]
  7. U. J. Netz, J. Beuthan, and A. H. Hielscher, “Multipixel system for gigahertz frequency-domain optical imaging of finger joints,” Rev. Sci. Instrum. 79(3), 034301 (2008). [CrossRef] [PubMed]
  8. U. J. Netz, A. H. Hielscher, A. K. Scheel, and J. Beuthan, “Signal-to-noise analysis for propagation of laser radiation through a tissue-like medium by diffuse photon-density waves,” Laser Phys. 17(4), 453–460 (2007). [CrossRef]
  9. K. Lee, S. D. Konecky, R. Choe, H. Y. Ban, A. Corlu, T. Durduran, and A. G. Yodh, “Transmission RF diffuse optical tomography instrument for human breast imaging,” Proc. SPIE 6629, 66291R, 66291R-6 (2007). [CrossRef]
  10. A. B. Thompson and E. M. Sevick-Muraca, “Near-infrared fluorescence contrast-enhanced imaging with intensified charge-coupled device homodyne detection: measurement precision and accuracy,” J. Biomed. Opt. 8(1), 111–120 (2003). [CrossRef] [PubMed]
  11. J. M. Schmitt, A. Knüttel, and J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9(10), 1832–1843 (1992). [CrossRef] [PubMed]
  12. D. G. Papaioannou, G. W. ‘t Hooft, S. B. Colak, and J. T. Oostveen, “Detection limit in localizing objects hidden in a turbid medium using an optically scanned phased array,” J. Biomed. Opt. 1(3), 305 (1996). [CrossRef]
  13. S. P. Morgan and K. Y. Yong, “Amplitude-phase crosstalk cancelation in frequency domain instrumentation,” Proc. SPIE 4250, 269–275 (2001). [CrossRef]
  14. X. Intes, V. Ntziachristos, and B. Chance, “Analytical model for dual-interfering sources diffuse optical tomography,” Opt. Express 10(1), 2–14 (2002). [PubMed]
  15. Y. Chen, G. Zheng, Z. H. Zhang, D. Blessington, M. Zhang, H. Li, Q. Liu, L. Zhou, X. Intes, S. Achilefu, and B. Chance, “Metabolism-enhanced tumor localization by fluorescence imaging: in vivo animal studies,” Opt. Lett. 28(21), 2070–2072 (2003). [CrossRef] [PubMed]
  16. B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52(5), 1409–1429 (2007). [CrossRef] [PubMed]
  17. S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105(2), 024702 (2009). [CrossRef]
  18. Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express 9(4), 212–224 (2001). [CrossRef] [PubMed]
  19. S. P. Morgan and K. Y. Yong, “Controlling the phase response of a diffusive wave phased array system,” Opt. Express 7(13), 540–546 (2000). [CrossRef] [PubMed]
  20. S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt. 43(10), 2071–2078 (2004). [CrossRef] [PubMed]
  21. H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12(5), 834–852 (1995). [CrossRef]
  22. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  23. J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13(5), 669–680 (1997). [CrossRef] [PubMed]
  24. Z. M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets,” Opt. Lett. 30(24), 3338–3340 (2005). [CrossRef]
  25. R. B. Schulz, J. Peter, W. Semmler, C. D’Andrea, G. Valentini, and R. Cubeddu, “Comparison of noncontact and fiber-based fluorescence-mediated tomography,” Opt. Lett. 31(6), 769–771 (2006). [CrossRef] [PubMed]
  26. G. Y. Panasyuk, Z. M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33(15), 1744–1746 (2008). [CrossRef] [PubMed]
  27. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7(7), 1266–1278 (1990). [CrossRef] [PubMed]
  28. A. R. Pineda, M. Schweiger, S. R. Arridge, and H. H. Barrett, “Information content of data types in time-domain optical tomography,” J. Opt. Soc. Am. A 23(12), 2989–2996 (2006). [CrossRef]
  29. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10(3), 159–170 (2002). [PubMed]
  30. N. Shah, A. Cerussi, C. Eker, J. Espinoza, J. Butler, J. Fishkin, R. Hornung, and B. Tromberg, “Noninvasive functional optical spectroscopy of human breast tissue,” Proc. Natl. Acad. Sci. U.S.A. 98(8), 4420–4425 (2001). [CrossRef] [PubMed]
  31. M. S. Nair, N. Ghosh, N. S. Raju, and A. Pradhan, “Determination of optical parameters of human breast tissue from spatially resolved fluorescence: a diffusion theory model,” Appl. Opt. 41(19), 4024–4035 (2002). [CrossRef] [PubMed]
  32. D. Kang and M.A. Kupinski, “Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging,” J. Biomed. Opt. (to be submitted). [PubMed]
  33. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50(4), R1–R43 (2005). [CrossRef] [PubMed]
  34. D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” Neuroimage 23(Suppl 1), S275–S288 (2004). [CrossRef] [PubMed]
  35. B. Chance, E. Anday, S. Nioka, S. Zhou, L. Hong, K. Worden, C. Li, T. Murray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2(10), 411–423 (1998). [CrossRef] [PubMed]

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