Fast blood flow monitoring in deep tissues with real-time software correlators

We introduce, validate and demonstrate a new software correlator for high-speed measurement of blood flow in deep tissues based on diffuse correlation spectroscopy (DCS). The software correlator scheme employs standard PC-based data acquisition boards to measure temporal intensity autocorrelation functions continuously at 50− 100 Hz, the fastest blood flow measurements reported with DCS to date. The data streams, obtained in vivo for typical source-detector separations of 2.5 cm, easily resolve pulsatile heart-beat fluctuations in blood flow which were previously considered to be noise. We employ the device to separate tissue blood flow from tissue absorption/scattering dynamics and thereby show that the origin of the pulsatile DCS signal is primarily flow, and we monitor cerebral autoregulation dynamics in healthy volunteers more accurately than with traditional instrumentation as a result of increased data acquisition rates. Finally, we characterize measurement signal-to-noise ratio and identify count rate and averaging parameters needed for optimal performance. © 2016 Optical Society of America OCIS codes: (030.0030) Speckle; (030.5290) Photon Statistics; (030.5260) Photon Counting; (170.3880) Medical and Biological Imaging; (170.1610) Clinical Applications; (110.4153) Motion estimation and optical flow; (170.2655) Functional monitoring and imaging; (170.1470) Blood or tissue constituent monitoring; (290.4210) Multiple scattering; (170.5270) Photon density waves; (170.6510) Spectroscopy, tissue diagnostics; (170.6480) Spectroscopy, speckle. References and links 1. D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phy. Rev. Lett. 75, 1855–1858 (1995). 2. D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997). 3. T. Durduran and A. G. Yodh, “Diffuse correlation spectroscopy for non-invasive, micro-vascular cerebral blood flow measurement,” NeuroImage 85 Pt 1, 51–63 (2014). 4. G. Yu, “Diffuse Correlation Spectroscopy (DCS): A Diagnostic Tool for Assessing Tissue Blood Flow in Vascular-Related Diseases and Therapies,” Curr. Med. Im. Rev. 8, 194–210 (2012). #255500 Received 14 Dec 2015; revised 20 Jan 2016; accepted 21 Jan 2016; published 3 Feb 2016 (C) 2016 OSA 1 Mar 2016 | Vol. 7, No. 3 | DOI:10.1364/BOE.7.000776 | BIOMEDICAL OPTICS EXPRESS 776 5. E. M. Buckley, A. B. Parthasarathy, P. E. Grant, A. G. Yodh, and M. A. Franceschini, “Diffuse correlation spectroscopy for measurement of cerebral blood flow: future prospects,” Neurophotonics 1, 011009 (2014). 6. C. G. Favilla, R. C. Mesquita, M. Mullen, T. Durduran, X. Lu, M. N. Kim, D. L. Minkoff, S. E. Kasner, J. H. Greenberg, A. G. Yodh, and J. A. Detre, “Optical bedside monitoring of cerebral blood flow in acute ischemic stroke patients during head-of-bed manipulation,” Stroke 45, 1269–1274 (2014). 7. R. C. Mesquita, T. Durduran, G. Yu, E. M. Buckley, M. N. Kim, C. Zhou, R. Choe, U. Sunar, and A. G. Yodh, “Direct measurement of tissue blood flow and metabolism with diffuse optics,” Philos. Trans. Ser. A: Math. Phys. Eng. Sci. 369, 4390–4406 (2011). 8. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phy. 73, 076701 (2010). 9. E. M. Buckley, D. Hance, T. Pawlowski, J. M. Lynch, F. B. Wilson, R. C. Mesquita, T. Durduran, L. K. Diaz, M. E. Putt, D. J. Licht, M. A. Fogel, and A. G. Yodh, “Validation of diffuse correlation spectroscopic measurement of cerebral blood flow using phase-encoded velocity mapping magnetic resonance imaging,” J. Biomed. Opt. 17, 037007 (2012). 10. E. M. Buckley, N. M. Cook, T. Durduran, M. N. Kim, C. Zhou, R. Choe, G. Yu, S. Schultz, C. M. Sehgal, D. J. Licht, P. H. Arger, M. E. Putt, H. H. Hurt, and A. G. Yodh, “Cerebral hemodynamics in preterm infants during positional intervention measured with diffuse correlation spectroscopy and transcranial Doppler ultrasound.” Opt. Express 17, 12571–12581 (2009). 11. V. Jain, E. M. Buckley, D. J. Licht, J. M. Lynch, P. J. Schwab, M. Y. Naim, N. A. Lavin, S. C. Nicolson, L. M. Montenegro, A. G. Yodh, and F. W. Wehrli, “Cerebral oxygen metabolism in neonates with congenital heart disease quantified by MRI and optics,” J. Cereb. Blood Flow Metab. 34, 380–388 (2013). 12. T. Durduran, C. Zhou, E. M. Buckley, M. N. Kim, G. Yu, R. Choe, J. W. Gaynor, T. L. Spray, S. M. Durning, S. E. Mason, L. M. Montenegro, S. C. Nicolson, R. A. Zimmerman, M. E. Putt, J. Wang, J. H. Greenberg, J. A. Detre, A. G. Yodh, and D. J. Licht, “Optical measurement of cerebral hemodynamics and oxygen metabolism in neonates with congenital heart defects.” J. Biomed. Opt. 15, 037004 (2010). 13. T. Durduran, C. Zhou, B. L. Edlow, G. Yu, R. Choe, M. N. Kim, B. L. Cucchiara, M. E. Putt, Q. Shah, S. E. Kasner, J. H. Greenberg, A. G. Yodh, and J. A. Detre, “Transcranial optical monitoring of cerebrovascular hemodynamics in acute stroke patients.” Opt. Express 17, 3884–3902 (2009). 14. M. N. Kim, B. L. Edlow, T. Durduran, S. Frangos, R. C. Mesquita, J. M. Levine, J. H. Greenberg, A. G. Yodh, and J. A. Detre, “Continuous optical monitoring of cerebral hemodynamics during head-of-bed manipulation in brain-injured adults,” Neurocrit. Care 20, 443–445 (2014). 15. M. N. Kim, T. Durduran, S. Frangos, B. L. Edlow, E. M. Buckley, H. E. Moss, C. Zhou, G. Yu, R. Choe, E. Maloney-Wilensky, R. L. Wolf, M. S. Grady, J. H. Greenberg, J. M. Levine, A. G. Yodh, J. A. Detre, and W. A. Kofke, “Noninvasive measurement of cerebral blood flow and blood oxygenation using near-infrared and diffuse correlation spectroscopies in critically brain-injured adults,” Neurocrit. Care 12, 173–180 (2010). 16. R. C. Mesquita, M. Putt, M. Chandra, G. Yu, X. Xing, S. W. Han, G. Lech, Y. Shang, T. Durduran, C. Zhou, A. G. Yodh, and E. R. Mohler, “Diffuse optical characterization of an exercising patient group with peripheral artery disease,” J. Biomed. Opt. 18, 057007 (2013). 17. Y. Shang, K. Gurley, and G. Yu, “Diffuse Correlation Spectroscopy (DCS) for Assessment of Tissue Blood Flow in Skeletal Muscle: Recent Progress,” Anat. Physiol. 03, 128 (2013). 18. R. Choe, M. E. Putt, P. M. Carlile, T. Durduran, J. M. Giammarco, D. R. Busch, K. W. Jung, B. J. Czerniecki, J. C. Tchou, M. D. Feldman, C. Mies, M. A. Rosen, M. D. Schnall, A. DeMichele, and A. G. Yodh, “Optically Measured Microvascular Blood Flow Contrast of Malignant Breast Tumors,” PLoS ONE 9, e99683 (2014). 19. T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tchou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30, 2915 (2005). 20. U. Sunar, H. Quon, T. Durduran, J. Zhang, J. Du, C. Zhou, G. Yu, R. Choe, A. Kilger, R. Lustig, L. Loevner, S. Nioka, B. Chance, and A. G. Yodh, “Noninvasive diffuse optical measurement of blood flow and blood oxygenation for monitoring radiation therapy in patients with head and neck tumors: a pilot study,” J. Biomed. Opt. 11, 064021 (2006). 21. G. Yu, “Near-infrared diffuse correlation spectroscopy in cancer diagnosis and therapy monitoring,” J. Biomed. Opt. 17, 010901 (2012). 22. W. B. Baker, A. B. Parthasarathy, T. S. Ko, D. R. Busch, K. Abramson, S.-Y. Tzeng, R. C. Mesquita, T. Durduran, J. H. Greenberg, D. K. Kung, and A. G. Yodh, “Pressure modulation algorithm to separate cerebral hemodynamic signals from extracerebral artifacts,” Neurophotonics 2, 035004 (2015). 23. T. Durduran, G. Yu, M. G. Burnett, J. A. Detre, J. H. Greenberg, J. Wang, C. Zhou, and A. G. Yodh, “Diffuse optical measurement of blood flow, blood oxygenation, and metabolism in a human brain during sensorimotor cortex activation.” Opt. Lett. 29, 1766–1768 (2004). 24. F. Jaillon, J. Li, G. Dietsche, T. Elbert, and T. Gisler, “Activity of the human visual cortex measured noninvasively by diffusing-wave spectroscopy.” Opt. Express 15, 6643–6650 (2007). 25. J. Li, M. Ninck, L. Koban, T. Elbert, J. Kissler, and T. Gisler, “Transient functional blood flow change in the human brain measured noninvasively by diffusing-wave spectroscopy,” Opt. Lett. 33, 2233–2235 (2008). 26. N. Roche-Labarbe, A. Fenoglio, H. Radhakrishnan, M. Kocienski-Filip, S. A. Carp, J. Dubb, D. A. Boas, P. E. #255500 Received 14 Dec 2015; revised 20 Jan 2016; accepted 21 Jan 2016; published 3 Feb 2016 (C) 2016 OSA 1 Mar 2016 | Vol. 7, No. 3 | DOI:10.1364/BOE.7.000776 | BIOMEDICAL OPTICS EXPRESS 777 Grant, and M. A. Franceschini, “Somatosensory evoked changes in cerebral oxygen consumption measured noninvasively in premature neonates,” NeuroImage 85, 279–286 (2014). 27. T. Binzoni and F. Martelli, “Assessing the reliability of diffuse correlation spectroscopy models on noise-free analytical Monte Carlo data,” Appl. Opt. 54, 5320–5326 (2015). 28. W. B. Baker, A. B. Parthasarathy, D. R. Busch, R. C. Mesquita, J. H. Greenberg, and A. G. Yodh, “Modified Beer-Lambert law for blood flow,” Biomed. Opt. Express 5, 4053–4075 (2014). 29. R. Sriram Chandran, G. Devaraj, R. Kanhirodan, D. Roy, and R. M. Vasu, “Detection and estimation of liquid flow through a pipe in a tissue-like object with ultrasound-assisted diffuse correlation spectroscopy,” J. Opt. Soc. Am. A 32, 1888–1897 (2015). 30. A. Tsalach, Z. Schiffer, E. Ratner, I. Breskin, R. Zeitak, R. Shechter, and M. Balberg, “Depth selective acoustooptic flow measurement,” Biomed. Opt. Express 6, 4871–4886 (2015). 31. L. Gagnon, M. Desjardins, J. Jehanne-Lacasse, L. Bherer, and F. Lesage, “Investigation of diffuse correlation spectroscopy in multi-layered media including the human head.” Opt. Express 16, 15514–15530 (2008). 32. Y. Shang and G. Yu, “A Nth-order linear algorithm for extracting diffuse correlation spectroscopy blood flow indices in heterogeneous tissues,” Appl. Phy. Lett. 105, 133702 (2014). 33. K. Gurley, Y. Shang, and G. Yu, “Noninvasive optical quantification of absolute blood flow, blood oxygenation


Abstract:
We introduce, validate and demonstrate a new software correlator for high-speed measurement of blood fow in deep tissues based on diffuse correlation spectroscopy (DCS). The software correlator scheme employs standard PC-based data acquisition boards to measure temporal intensity autocorrelation functions continuously at 50 − 100 Hz, the fastest blood fow measurements reported with DCS to date. The data streams, obtained in vivo for typical source-detector separations of 2.5 cm, easily resolve pulsatile heart-beat fuctuations in blood fow which were previously considered to be noise. We employ the device to separate tissue blood fow from tissue absorption/scattering dynamics and thereby show that the origin of the pulsatile DCS signal is primarily fow, and we monitor cerebral autoregulation dynamics in healthy volunteers more accurately than with traditional instrumentation as a result of increased data acquisition rates. Finally, we characterize measurement signal-to-noise ratio and identify count rate and averaging parameters needed for optimal performance.
A third limitation and opportunity for improvement concerns data throughput, e.g., measurement time resolution and acquisition rate. Most DCS measurements of blood fow are slow, with measurement sampling rates ranging from 0.3 to 1 Hz. Thus DCS has only been used to measure fow variation over slow time scales, i.e., measurements every minute/hour [10, 13, 37, 38], or measurements every day [39]. Indeed, fast cerebral blood fow (CBF) measurements (25−50 Hz) can enable new applications for DCS, such as monitoring cerebrovascular autoregulation dynamics [40,41] wherein beat-to-beat variability of both blood pressure and blood fow are used to characterize autoregulation. High temporal resolution measurements will also improve identifcation of motion artifacts and thereby create potential for measuring tissue blood fow during exercise [42]. Finally, fast sampling increases measurement throughput and will Highly coherent single mode laser light is used to illuminate the sample via optical fbers. Red blood cell motion (e.g., red disks to light red disks in time τ; blood fow F) causes fuctuations in the intensity of backscattered light that is collected a distance ρ away from the source, and is directed to single photon counting avalanche photo diodes (APDs). A correlator counts the arrival of digital TTL pulses generated by the APDs to compute the DCS autocorrelation functions, (B) Sample intensity autocorrelation functions (g 2 (τ)) highlighting different fow rates. enable high spatial resolution imaging with fewer detectors. For example, photons collected from many (32 − 48) detector positions can be routed to a few (4 − 8) photon detectors via an optical switch. In these cases, besides the obvious cost advantages, fast sampling can reduce the imaging frame rate to seconds or less (rather than minutes), thus enabling dynamic imaging.
In this contribution we report on the development of a novel software correlator optimized for continuous, high-speed monitoring of deep tissue blood fow based on diffuse correlation spectroscopy (DCS). This device uses the 'shift-and-add' method [43,44] to directly compute the correlation function at a few (40), highly relevant delay times (1µs ≤ τ ≤ 250µs). Leveraging this data compression and other technological improvements, we demonstrate sustained blood fow measurement speeds up to 100 Hz with 8 simultaneous detection channels (and up to 1 kHz with 2 detection channels). To our knowledge, these experiments represent the fastest measurements of blood fow with DCS. The fast data streams easily resolve pulsatile heart-beat fuctuations in blood fow which were previously treated as noise, and they enable us to monitor cerebral autoregulation dynamics more accurately than with traditional instrumentation.
The remainder of this paper is organized as follows. We frst provide context for our work with respect to traditional fow measurement devices and other software correlators. Next, we describe the software correlator instrumentation and validate its measurements in vivo with a hardware correlator. We then highlight the utility of the fast correlator for two in vivo applications. First, we measure the pulsatile arterial blood fow with DCS; in the process we separate the tissue blood fow components in the dynamical signal from tissue absorption/scattering components. Second, we measure cerebral autoregulation dynamics in healthy adults. Finally, we characterize the effect of averaging and photon count rate on measurement signal-to-noise ratio.

Correlation methods: background and new features
The fast data throughput improvements are best appreciated by comparison to traditional methods utilized for DCS measurements [2][3][4][5]. Briefy, DCS employs coherent near-infrared light to characterize moving particles (red blood cells) in tissue via temporal light intensity fuctuations. Figure 1 shows a schematic of the typical DCS instrument. Laser light illuminates the tissue. An optical fber, placed on the surface ∼ 2 − 3 cms away from the source, collects light that has diffused through the tissue and directs it to a photon counting detector that generates an electrical Transistor-Transistor Logic (TTL) digital pulse for every detected photon. A cor-

Time ~13
Sounce p(cm)~""'°' 9 2(T) ._, -u.,1:r relator records the arrival of TTL pulses and uses the distribution of arrival times to quantify the temporal fuctuations of detected light intensity. Formally, the correlator calculates the normalized intensity autocorrelation function (g 2 (τ)) from measurements of the photon intensity, g 2 (τ) ≡ hI(t)I(t + τ)i/hI(t)i 2 , where, I(t) is the detected intensity at time t, τ is the correlation delay time, and hi represent time-averages. Blood fow is estimated by ftting the measured intensity autocorrelation function to mathematical models appropriate to the measurement geometry.
The rapid adoption of DCS for clinical fow monitoring was aided by the availability of userfriendly and convenient hardware correlators (e.g., Correlator.com, Bridgewater, New Jersey; ALB, Hessen, Germany), but this convenience also limited measurement speed. Traditionally, correlators employ embedded programming and a multi-tau algorithm [45-47] to compute the autocorrelation functions over a large range of delay times (from ∼ 1µs up to as much as 1 ∼ 2s); this design follows from early dynamic light scattering (DLS) and diffusing wave spectroscopy (DWS) experiments in mostly non-biological samples [43][44][45][46][47]. DCS intensity autocorrelation functions from deep tissue blood fow, however, typically decay at a much faster rates than in DLS experiments [3,8], and exhibit other slow dynamics at heart rate frequencies (and even faster). Deep tissue g 2 (τ) most often decays to 1 at delay times of τ ∼ 250µs or less (see Fig. 1(B)). Thus, correlation data at delay times greater than ∼ 250µs does not offer signifcant information about tissue dynamics. Furthermore, the autocorrelation function at short delay times is more sensitive to photons that travel deep into tissue, i.e., the photons we care about [28,48,49], and blood fow changes can be estimated from DCS autocorrelation functions at a single delay-time [28]. Thus, many reasons exist for data-set reduction/compression, and by reducing the number of delay-times in the correlation function calculation, our data acquisition can be made more effcient and faster.
Unfortunately, reconfguration of commercial correlators that are pre-optimized for general applications is nontrivial; reconfguration involves reprogramming of the embedded correlator circuits with specialized equipment. Nevertheless, some precedence for fast blood fow measurements exists with hardware correlators, albeit with limitations. Of note is the work of Dietsche et. al.
[50] who used a hardware correlator in the so-called 'burst' mode to measure blood fow at ∼ 40 Hz; the measurement was fast enough to detect pulsatile blood fow in the human arm and forehead at source-detector separations of 1.4 ∼ 1.9 cm. However, the measurements required either averaging of correlation functions from many (16 − 32) detectors at a single point in space [50], or the use of a dual-mode correlator and 2 detectors gated at the pulse rate [51]. Importantly, burst mode correlators store rapidly acquired normalized intensity autocorrelation functions on an internal memory buffer which only has space for approximately 1000 correlation functions [50]. Thus, they are designed for sustained high speed acquisition over short time intervals (∼ 20 s [50]) and cannot be easily used for continuous long-term monitoring. Moreover, data transfer interfaces and program drivers employed to transfer the correlation functions to a computer are not optimized for speed. Therefore, even though actual computation of correlation functions on the hardware correlator can be fast, programming overhead issues result in blood fow measurements at speeds only as fast as 1 ∼ 2 Hz (for standard correlators).
Computation of correlation functions in software rather than hardware, i.e., the software correlator, offers a fexible alternative approach that greatly facilitates optimization of the correlation measurement for deep tissue blood fow with DCS. The software correlators also utilize digital counters to record TTL pulses from the photon counting detectors [43,44,52,53]. Instead of computing the correlation function with embedded programming, however, high-level programs (e.g. LabVIEW, C++) control the counter readout and estimate the correlation function in the computer's random access memory (RAM). Proof-of-principle measurements with diffuse correlation spectroscopy via the Wiener-Khinchin theorem, i.e., the convolution of the measured temporal intensity and its time-reversed duplicate have been made with software correlators [52,53]. These studies did not optimize the correlation function computations for measurement speed, and this approach requires a large buffer to store the stream of detected photon pulses each second, e.g., data sampling rates smaller than 2 µs would fll a typical computer's buffer in less than a second [52]. Thus, continuous blood fow monitoring over long time intervals with this approach is not practical (as was the case with the burst mode correlator).
Our approach utilizes the 'shift-and-add' method [43,44], to directly compute the correlation function; this scheme is technically similar to that of hardware correlators. Importantly, we leverage several technical advances. First, we employ a data reduction/compression strategy, i.e., the correlation function is measured at only a few biologically relevant delay times (40 delays; 1µs ≤ τ ≤ 250µs). Second, we employ an improved instrument design, i.e., the photon counters are directly connected to the computer's PCI-bus, eliminating the need for USBsoftware drivers. Finally, our software design substantially reduces data transfer overheads. The present contribution thus describes an approach to simplify and optimize computation of correlation functions for deep tissue blood fow monitoring with DCS. Importantly, this fexible solution can be implemented/adapted to any existing DCS system with modest instrumentation upgrades. The design of this improved software correlator is described in detail below.

Real-time software correlator: design and instrumentation
Our real-time software correlator is implemented on a personal computer (Dell Inspiron, Intel core i5 − 4200M, Dual Core, 8GB RAM) using a dedicated 8 channel PCIe/PXIe6612 counter/timer data acquisition board (National Instruments, Austin, TX) and a custom software program (LabVIEW, National Instruments, Austin, TX). As shown schematically in Fig. 1(A), a stream of digital TTL pulses generated by the photon counting APD is directed to an edgedetecting photon counter on the PCIe6612 data acquisition board. The operation is diagrammed in Fig. 2(B). Briefy, the counter's operations are synchronized by an internal timebase (set to 80 MHz by default). At every clock-tick of the timebase, the counter seeks a TTL signal at its input terminal, and if the TTL pulse is present, then the counter increments and updates its internal count by 1. The photon counts are then transferred to an internal buffer at a user-defned sampling clock frequency ( f s = 1/Δt). By generating the sampling clock based on the data acquisition board's built-in frequency generator, we ensure that counter timing/sampling is unaffected by computer processing operations. The counter buffer (N(i)) is allowed to accumulate counts over a user-defned integration time (t int ), and then n int = f s × t int points of the buffer are transferred to the computer for calculation of the correlation function. Thus, N(i) represents the number of counts that have been accumulated through the i th sample interval in the counter buffer.
Since the counter continuously accumulates photon counts, the quasi-instantaneous photon count during the small time interval at index i (i.e., n(i)) is calculated as The normalized intensity autocorrelation at delay time τ = Δn/ f s is then estimated from: where, hi represent time averages over n avg = n int − Δn points. Notice, the smallest delay time in the computation of the autocorrelation function is (1/ f s ) s, and the correlation function detection frame rate is (1/t int ) Hz. In our implementation of the software correlator, f s = 1 MHz, while t int can vary from 1 ms to 1 s. For a typical in vivo experiment, t int ∼ 40 ms. The custom software correlator computes the normalized autocorrelation function using Eq. (1) over 40 delay times (1 µs to ∼ 125 µs) from 8 channels simultaneously. Real-time computation of the correlation function is ensured via buffered producer-consumer loops [54].
The high acquisition rate is in part due to more time-effcient software architecture. Moreover, computing the autocorrelation function over a relatively small range of delay times greatly simplifes the software operations; a smaller range of delay times permits use of the simpler single-tau correlator design wherein photons do not have to be temporally binned as in multitau correlators [44]. More signifcantly, the smaller delay-time range permits smaller integration times. For example, from Eq. (1) one can infer that the minimum number of points, n int , required to estimate the correlation function is Δn + 1. Computation of the correlation function at large delay times, i.e., larger Δn, will necessitate larger n int (and consequently larger integration times, t int ) to ensure that n Avg 1. As we have previously described, DCS intensity correlation functions decay to their minimum value at delay times < 200 µs, and measurements beyond these time scales are superfuous.
We close this section with a note about timing and sampling considerations. In general, the temporal response time of the APD will limit the maximum number of photons that can be detected in one second, while the counter timebase (80 MHz) places a limit on the frequency of TTL pulses (i.e., photons), that can be counted. The most popular/common single photon counting APD used for DCS (SPCM-AQ4C, Excelitas, Quebec, Canada) has a temporal response time of 25 ns, and a response 'dead time' of 50 ns. Thus the maximum photon count rate that the APD can detect is ∼ 13.3 MHz, which is suffciently small compared to the default counter timebase of 80 MHz. Nevertheless, the relative differences between the APD response time and the counter timebase are important design considerations for the software correlator.

Experiments and results
All blood fow measurements were carried out using a custom DCS instrument ( Fig. 2(A)) [22, 28]. Briefy, a continuous wave, long coherence length (> 5 m) fber coupled laser (785 nm, 80 mW, DL785-100-3O, CrystaLaser Inc., Reno, NV) was used to illuminate the sample via a multimode fber (200 µm diameter, OZ Optics, Ottawa, Canada). Remitted light that travelled through the sample is detected by a bundle of single mode fbers (5 µm diameter, OZ Optics, Ottawa, Canada) located 2.5 cms away from the source. Each detector fber directs light to a single photon counting APD (SPCM-AQ4C, Excelitas, Quebec, Canada). For compar- ison studies, the outputs of the detectors were split between a commercial hardware correlator (Correlator.com, Bridgewater, NJ) and our custom software correlators. Correlation functions derived from the same source-detector separation were averaged. All in vivo experiments were approved by the Institutional Review Board of the University of Pennsylvania, and a total of eight subjects were recruited for this study.

Software correlator provides accurate estimates of fow
We frst demonstrate that the real-time software correlator accurately estimates blood fow in humans under baseline conditions and during an arm cuff ischemia (see Fig. 3(A)). For these validation experiments, a blood pressure cuff was placed around the subject's bicep (on the same arm as the probe). With the subject lying supine on a comfortable bed, an optical probe with embedded sources and detector fbers (2.5 cm separation) was secured on the subject's forearm. A commercial pulse oximeter (Rad-9, Masimo, Irvine, CA) monitored the subject's heart rate, and the heart rate data was recorded on the computer. The integration times (alternatively, the correlation function frame rate) of both hardware and software correlators were fxed at 1 s. After 10 minutes of baseline condition measurements, the blood fow in the arm was reduced for 3 minutes, by infating the blood pressure cuff to 180 mmHg using a Tourniquet system (Zimmer Inc., Warsaw, IN). The experiment concluded with 5 minutes of post-occlusion data, and 10 minutes of baseline data with the software correlator set to record at a frame rate of 10 Hz (i.e., increased from 1 Hz).
The accuracy of the software correlator for estimation of the intensity autocorrelation function is evident from representative intensity correlation curves in Fig. 3(B). The correlation function measured with the software correlator overlaps the 'gold-standard' hardware correlator values. Note, both correlators average over the 2 detector channels, with an average photon count rate of ∼ 200 KHz. Also note, the correlation functions decay to a minimum value of 1 at delay times of ∼ 200 µs, which is close to the maximum delay time used by the software correlator. Correlation measurements beyond these time scales are not useful. Thus, the software correlator estimates fow from the intensity correlation functions at the delay times most sensitive to blood fow changes. We refer the interested reader to Fig. 10 in Appendix 3 for comprehensive comparisons of blood fow measured with the hardware and software correlators from 8 subjects; differences in the blood fow indices measured by the two devices were not statistically signifcant (p = 0.13).
Figure 3(C) shows the blood fow dynamics estimated from hardware and software correlation data. The ∼ 100% reduction in blood fow, due to cuff-ischemia, is clearly monitored by the software correlator. Tissue blood fow indices were estimated by ftting the intensity correlation functions to a semi-infnite geometry solution of the correlation diffusion equation (Eq. (2), Appendix 1). For this representative subject, the baseline tissue optical properties were measured to be µ a = 0.16 cm −1 and µ 0 = 4.28 cm −1 , using a frequency domain diffuse optis cal spectroscopy instrument (Imagent, ISS Inc., IL, USA). These measurements also clearly demonstrate that 40 delay times between 1 µs and 125 µs are suffcient (more than suffcient) to accurately ft for a tissue blood fow index.

High speed measurements of baseline blood fow reveals pulsatile fow dynamics
We next demonstrate the ability of the real-time software correlator to measure high speed blood fow dynamics in human subjects under baseline conditions. Figure 4 displays the results of high-speed blood fow monitoring of baseline fow in the arm (i.e., using the last 10 minutes of data from the previous experiment, Fig. 3(A)). Figure 4(A), shows representative intensity autocorrelation functions measured with a total integration time of 1 s. The solid red line is the correlation function measured by the hardware correlator (1 curve obtained at an integration time of 1 s). The solid blue circles, represent the correlation function values obtained at discrete delay times (i.e., 40 delay times between 1 µs to ∼ 250 µs) derived over the same time period with the high speed software correlator (data associated with 10 curves obtained at an integration time of 0.1 s each). In effect, the hardware correlator smears out the rapid fuctuations of the intensity correlation function. Figure 4(B) displays the blood fow index estimated using both the hardware (solid red line) and the high-speed software correlator (solid blue line), during the baseline period of 10 minutes; fow indices were determined by ftting the measured intensity autocorrelation functions to a semi-infnite geometry solution of the correlation diffusion equation (Eq. (2), Appendix 1).
At frst glance, the blood fow index estimated using the high temporal resolution software correlator data appears to be very noisy. However, a more careful observation of the data, such as shown in the 15 s extracted time-window in Fig. 4(C), reveals signifcant temporal structure that corresponds to the pulsatile nature of tissue blood fow. The hemodynamics of the entire cardiac cycle is captured, including the 'dicrotic notch' which is the result of a brief increase in pressure (and thus fow) following closure of the aortic valve. The resolution of the dicrotic notch is particularly exciting, since it is rarely observed, i.e., it is seen only when using high quality high speed instrumentation such as arterial line tracings. Further, as is evident from the solid red line, the blood fow index estimated using the hardware correlator averages out these fuctuations. The cardiac pulsatility is further confrmed by frequency spectrum of the data shown in Fig. 4(D). We refer the interested reader to Fig. 11 in Appendix 3 for comparisons of (2), Appendix 1) (C) ∼ 15 s extract of baseline blood fow fuctuations, clearly demonstrating that the fuctuations in the blood fow index are a result of the pulsatile nature of blood fow. Notice, the entire cardiac cycle is clearly resolved, including the 'dicrotic notch', i.e., the second fow peak of smaller magnitude within the cycle, corresponding to aortic valve closure. (D) The frequency spectrum of the baseline blood fow indices measured with the software correlator, highlighting the heart rate as ∼ 0.9 Hz, with corresponding harmonics at 1.8, 2.7 and 3.6 Hz . the heart rates estimated with DCS, and a pulse oximeter from 8 healthy volunteers; statistically signifcant differences in the heart rate frequencies measured by the two devices were not found (p = 0.76).
The clear resolution of pulsatile fow dynamics, and the entire cardiac cycle, is the frst signifcant physiological result of our paper. These fuctuations are often misconstrued as noise, and indeed, researchers have traditionally used integration times of 1 to ∼ 2.5 seconds, in order to average out these fuctuations. Insuffcient measurement speed, averages out useful information including beat-to-beat variations in blood fow. Moreover, measurements at intermediate  sampling rates (e.g. 1 − 2 Hz) do not fully resolve pulsatile dynamics and can lead to aliasing artifacts. For clarity, we have shown fuctuations in the correlation functions, sampled at 10 Hz in this characteristic example. In practice, our in vivo data is sampled at 20 − 50 Hz; the measured photon count rates have an impact on the signal-to-noise of the measurements. We discuss these signal to noise considerations in a separate section.

Fluctuations in DCS blood fow index are primarily due to changes in blood fow
To elucidate the nature and origin of the fast blood fow index fuctuations more precisely, we carried out clarifying DOS and DCS experiments. In general, the DCS blood fow index depends parametrically on tissue blood fow and tissue absorption (µ a ) and scattering (µ 0 ) co-  Accordingly, the baseline fuctuations in tissue optical properties (µ a and µ 0 ), were monitored s (at ∼ 10 Hz) on the forearm and forehead of 3 volunteers, using a commercial frequency domain Diffuse Optical Spectroscopy instrument (Imagent, ISS Inc., IL), operating at 788 nm. Then, the high-speed software correlator was used to record the baseline fuctuations in the intensity autocorrelation functions from the same measurement spots.
The intensity correlation data gives the variation in DCS optical density. The average deviation from baseline of tissue optical properties were calculated from 10 minutes of DOS data. Table 1 summarizes these tissue absorption (µ 0 ± Δµ a ), and scattering (µ 00 ± Δµ 0 ) varia s s ations. The absorption and scattering contributions to the DCS optical density, respectively, are d a (τ, ρ)Δµ a and d s (τ, ρ)Δµ 0 ; they are readily estimated (see Appendix 2 for details) using s the measured fuctuations in optical properties (Table 1). Finally, the fractional absorption and scattering contributions to the DCS fuctuations were estimated from Eq. (3) in Appendix 2 as ΔOD DCS (τ, ρ)/d a (τ, ρ)Δµ a , and ΔOD DCS (τ, ρ)/d s (τ, ρ)Δµ 0 respectively. Figure 5 displays s the results of these comparisons for the forearm and the brain. From this analysis, it is apparent that more than 90% of DCS fuctuations are driven by blood fow changes. This physiological fnding, is the second important result of this paper. Pulsatile variation in the DCS signal refect variations in blood fow.

Real-time software correlator can estimate cerebral autoregulation dynamics
We next explore the utility of the real-time software correlator in the context of a critical clinical application wherein rapid acquisition of blood fow information is needed: measurement of cerebral autoregulation dynamics [41]. Briefy, cerebral autoregulation (CVAR) refers to the mechanism by which normal (i.e. healthy) brain maintains relatively constant cerebral blood fow (CBF) despite fuctuations in mean arterial blood pressure (MAP) [55]. Importantly, CVAR is often impaired after brain injury; in this scenario, CBF can vary in response to MAP variation [56,57]. Moreover, the degree of CVAR impairment correlates with the initial severity of brain injury and is an independent predictor of outcome [41, 56,58]. CVAR is typically measured using static or dynamic techniques [40] that rely on the detection of cerebral blood fow velocity, e.g., derived by trans-cranial Doppler ultrasound. Here, we showcase the potential for monitoring dynamic autoregulation with DCS using the high speed real-time software correlator. We employ a standard approach for measuring CVAR dynamics. In particular, we measure the time-dependent changes in CBF resulting from transient increases in cardiac output, i.e., following defation of blood pressure cuffs applied to the thigh [41]. Figure 6(A) details the protocol used to measure dynamic cerebral autoregulation [41] from one healthy volunteer. With the subject lying supine, the optical probe was placed on the subject's forehead, over the frontal cortex. Two blood pressure (BP) cuffs were wrapped around the subject's thighs, about 10 cm above the knee. The subject's blood pressure was continuously monitored using a non-invasive fnger pressure monitor (Finometer Pro, Finapress Medical Systems, Netherlands), and cerebral blood fow was measured with the real-time software correlator at a data acquisition rate of 20 Hz. After a 5 minute baseline measurement, the thigh cuffs were both infated and held at 30 to ∼ 40 mmHg above the subject's baseline systolic blood pressure (here 170 mmHg) for a period of 4 minutes. At the end of the infation period, both cuffs were rapidly defated by disconnecting the pressure pump. Two blood pressure manipulation trials were carried out, i.e., the thigh cuffs were infated and defated twice. The experiment concluded with 4 minutes of baseline measurements. Typically, cerebral hemodynamics (i.e., CBF and BP) from multiple thigh cuff 'trials' on the same subject are averaged. Here, we showcase the speed and measurement fdelity of the real-time software correlator, by estimating cerebral autoregulation dynamics using measurements from a single trial.
Cerebral blood fow (CBF) and blood pressure (BP) were continuously monitored through- out the experiment at a data acquisition rate of 20 Hz. The relative unfltered change in CBF (rCBF = CBF(t)/CBF 0 ) and BP (rBP = BP(t)/BP 0 ) from a single trial of blood pressure manipulation is displayed in Fig. 6(B). Here, time t = 0 (vertical dashed line) denotes the start of bilateral cuff defation, and the solid red and blue lines denote the pulsatile dynamics of CBF and blood pressure, respectively. Both CBF and BP are normalized to their values during the 10 s pre-defation baseline period. The sudden cuff defation, causes a rapid increase in venous return and cardiac output; these effects produce a transient decrease in BP and CBF. This ∼ 20% and ∼ 40% decrease in average BP and CBF is clearly evident, even in the unfltered measurements. Note that the high temporal resolution of the software correlation technique reveals a phase difference between CBF and blood pressure (see inset in Fig. 6(B)). Ultimately, it may be possible to use the dynamics of this phase shift as a biomarker of cerebral autoregulation [59][60][61]. We note however, that differences in pulse transit times to cerebral/peripheral vasculature may lead to 'offsets' in the phase-shift between the CBF and BP. Pulse transit times can potentially be measured by simultaneous blood fow and blood pressure recordings in the brain and arm. A more complete investigation of these issues is planned; for example, correction/calibration factors may need to be developed before the absolute value of the phase-shift can be utilized as a biomarker for cerebral autoregulation. Reduced data rates were achieved by averaging/integrating highfrequency software correlator intensity temporal autocorrelation functions (20 Hz, gray) to 0.5 Hz (blue, a common hardware correlator data rate), 1 Hz (red) and 2 Hz (green). Measurements at 0.5 and 1 Hz are highly averaged and capture the ∼ 20% baseline fuctuation in CBF. The 1 Hz data rate identifes, but only poorly resolves the heart rate fuctuations. None of lower frequency data can accurately capture the instantaneous decrease in CBF due to cuff defation. Note that this form of averaging is an accurate representation of data integration in the hardware correlators. (B) Reduced data rates achieved by down-sampling the high-frequency software correlator intensity autocorrelation functions. Down-sampled CBF data is quite noisy.
In order to quantify dynamic cerebral autoregulation, the measured CBF and BP dynamics were frst fltered to remove heart rate effects using a low pass flter with a cutoff frequency set to 75% of the heart rate. The fltered autoregulation measurements from a single trial are displayed in Fig. 6(C), i.e., rCBF by solid red line and rBP by solid blue lines, respectively. Importantly, Fig. 6(C) also shows the change in cerebrovascular resistance (solid magenta line): rCV R = rBP/rCBF. The vertical dashed black line indicates the start of cuff defation.
In combination, the changes in CBF, BP and CVR, describe the autoregulation process. The sudden decrease in BP and CBF is characterized by an almost instantaneous increase in vascular resistance, followed by a gradual return to baseline due to the autoregulation process. The rate of change of rCV R, drCV R/dt, can be estimated from the linear decrease in rCV R between the two vertical dashed green lines (see inset in Fig 6(B)). Finally, a rate of regulation can be calculated, ROR = (drCV R/dt)/ΔBP, wherein ΔBP is the maximum decrease in rBP from baseline. For this representative subject, the rate of regulation is 0.66 sec; i.e., a 66% change in resistance is required per second in order to autoregulate a 1% change in blood pressure. We emphasize that this entire analysis was carried out from a single trial without averaging.
The utility of the real-time software correlator for measurements of cerebral autoregulation dynamics is evident from comparisons with lower speed CBF measurements of traditional hardware correlators. We frst averaged (Fig. 7(A)) the intensity correlation functions acquired at 20 Hz (gray lines) to CBF measurements at 0.5 Hz (blue lines), 1 Hz (red lines) and 2 Hz (green lines). For example, for every CBF measurement at 1 Hz, 20 intensity autocorrelation functions originally measured at 20 Hz were binned and averaged. This manner of averaging simulates increased integration time in hardware correlators. A second approach, down-samples (Fig. 7(B)) the 20 Hz data to a lower data rate.
Once averaged (or down-sampled), a blood fow index was estimated by ftting the averaged (or down-sampled) intensity autocorrelation function with the solution to the correlation diffu-  Figure 7(A) shows the effects of averaging on measurement of cerebral autoregulation dynamics from a single trial. Unsurprisingly, the integrated CBF measurements at 0.5 Hz and 1 Hz completely average the pulsatile blood fow fuctuations. At 2 Hz, the pulsatile fow is identifed, but is aliased and is therefore poorly resolved. When compared to the fltered autoregulation curves in Fig. 6(C), these averaged measurements exhibit greater baseline noise, appear to exhibit timing inaccuracies with respect to cuff defation (i.e. vertical dashed line), and do not show an instantaneous decrease in CBF due to cuff defation. These detrimental effects are clearly evident in the down-sampled data (Fig. 7(B)) wherein the fuctuations/noise in CBF measurements are sometimes indistinguishable from CBF changes due to the thigh cuff defation. In both cases, additional fltering (or averaging) can reduce the noise, but such averaging/fltering also temporally broadens the autoregulation 'signal', in large part due to reduced temporal resolution. Note, an experiment with the hardware correlator will be affected by averaging and (to a lesser extent) down-sampling, due to data transfer lags and software overheads. The best quality data is obtained by measuring CBF dynamics at the highest data rates possible, i.e., with the real-time software correlator. This observation highlights the value of the real-time correlator for continuous monitoring of autoregulation. The demonstration of instrumentation to monitor cerebral autoregulation dynamics in this manner is arguably the most important result of this paper.

Signal-to-noise ratio considerations for fast blood fow measurements with DCS
We conclude this paper with a discussion about signal-to-noise ratio (SNR) considerations for fast fow measurements with DCS. The ability of the real-time software correlator to detect high frequency fow dynamics (Fig. 4(A)) has been demonstrated. Ultimately however, the ability to discern meaningful fow information, i.e., blood fow index variation, will depend on the fdelity of measured autocorrelation functions, which, in turn, depends on the number of detected photons, i.e., the detected light intensity, and the amount of averaging i.e., measurement integration time. The precise dependence of the measurement SNR on light intensity and integration time can be ascertained using a photon correlation noise model adapted for diffuse light (i.e., DCS noise model) [62].
To confrm the accuracy of the DCS noise model at the short integration times permitted by our new software correlator, we systematically characterized correlation noise in a liquid tissue phantom. The liquid tissue phantom consisted of 21.7 ml/l of 30% Intralipid (Fresenius Kabi, Uppsala, Sweden) with 1.88 ml/l of India ink (Higgins, Black India 44201, MA) resulting in optical properties of µ a = 0.16 cm −1 and µ s 0 = 4.32 cm −1 . An optical probe with 1.5 cm sourcedetector separation was placed on the liquid surface to simulate a semi-infnite geometry. The integration time was systematically varied as depicted in Fig. 8(A). Further, the photon count rate was also systematically varied using an attenuator on the source arm.
Following the conventions of a previous DCS noise model [62], we defne 'noise' to be the standard deviation of the measured intensity autocorrelation function, σ (τ) (i.e., σ (τ) is the standard deviation of g 2 (τ) measured over the duration of the experiment). We then defne the SNR to be ζ (τ) = (g 2 (τ) − 1)/σ (τ). Here, we make the inherent assumption that fuctuations in the measured autocorrelation function are due to random noise (appropriate for these tissue phantom experiments). Figure 8 displays the result of the SNR characterization from the liquid phantom. In panels (B) and (D), we have plotted the measurement noise as a function of the integration time of the software correlator at delay times of 20 µs and 80 µs respectively, for three different photon detection signal levels -20 kHz (blue circles), 50 kHz (red squares), and 94 kHz (black diamonds). In each case, the measurements are ft to a DCS noise model [62]. In a similar vein, panels (C) and (E) show the corresponding measurements of signal-to-noise ratio. For each condition examined in Fig. 8, the measured noise (dots) agrees well with the DCS noise model (solid lines). Thus, we confrm that the DCS noise model provides a theoretical framework that would allow an experimenter to pick the right photon count rates, integration times, and delay times to achieve a desired SNR. Figures 8(B) and (D) clearly show that the measurement noise decreases with increased averaging (increased integration times) and improved signal (increased detected photon count rates). Correspondingly, as is evident from Figs. 8(C) and (E), the signal-to-noise ratio increases with integration time and photon count rates. From these measurements, one can observe that a SNR of 1 at acquisition rates of 25 Hz (40 ms integration time), requires a photon count rate of ∼ 20 kHz. More realistic signal levels of ∼ 50 kHz will permit acquisition rates of ∼ 50 Hz at SNR of 1.
More practically, this experiment and analysis enables us to estimate optimum operating parameters for fast in vivo measurements of blood fow. For in vivo experiments, we are interested in the ability of the correlator to resolve dynamics at particular frequencies.  Fig. 9. Optimization of experimental parameters to isolate heart rate with fast blood fow measurements. (A) Natural fuctuations in blood fow index acquired on the arm are plotted as a function of time for 19 kHz (solid blue lines) and 182 kHz (solid red lines) photon count rates and 5 ms integration times. (B) Corresponding frequency spectra of blood fow index dynamics show a clear peak at the heart rate frequency for the higher photon count rate data. (C) Scatter plot showing which photon count rates and integration times permit the identifcation of the heart rate in in vivo data. Red crosses indicate parameters where heart rate could not be identifed, while solid blue circles indicate parameters where heart rate was successfully identifed.
quency of interest is that of the heart rate, especially because physiological perturbations in fow are oftentimes slower than the heart rate. With a setup similar to Fig. 2, we measured baseline blood fow dynamics, at integration times ranging from 1 ms to 100 ms, with photon count rates manipulated via laser attenuation. Data was collected for a total of 30 ∼ 40 s at each setting. Figure 9(A) shows the representative fuctuations in blood fow index over a period of 30 seconds measured with a 5 ms integration time, at photon count rates of 19 kHz in blue and 182 kHz in red. Figure 9(B) shows the corresponding frequency spectra; a clear peak at ∼ 1 Hz is visible in the measurements at 182 kHz indicating that the heart rate is well resolved at the high photon count rate, but not at the smaller signal levels. Figure 9(C) shows the result of the experiment wherein multiple integration times and photon count rates were tested on similar data. The solid blue circles indicate the measurement parameters where the heart rate was clearly resolved, i.e., the maximum frequency component in the spectrum was between 0.8 Hz and 1.2 Hz. Red crosses, indicate parameters were the heart rate was not resolved. From these measurements, we conclude that photon count rates of ∼ 50 − 100 kHz are ideal for fast fow measurements over a large range of acquisition rates. Importantly, this experiment provides a  practical framework to select an appropriate data acquisition rate for accurate estimation of blood fow.

Discussion
Traditionally, blood fow has been measured with DCS at relatively slow data rates (0.5 −1 Hz). Therefore fuctuations due, for example, to heart beats have often been considered as noise that should be averaged in the measured autocorrelation function (for example, Fig. 4(A)); this noise assignment, in turn, prompted the need for increased temporal averaging. Interestingly, although heartbeat oscillations are well recognized in the NIRS community [63][64][65], the DCS community has been comparatively slow to appreciate the full potential of fast blood fow measurements.
The primary goal of the present contribution is to report development and testing of new, optimized instrumentation and software that fulflls an unmet clinical need for continuous fast measurements of blood fow with DCS. Aside from validating the accuracy of the software correlator, the work provides a rigorous overview of timing and sampling considerations. Further, for the frst time, we have characterized the signal-to-noise characteristics of DCS at low integration times, and in the process we have validated the prevailing DCS noise models for fast measurements of blood fow. Critically, these characterization experiments provide a framework for the identifcation of optimum data acquisition parameters.
We demonstrated sustained/continuous blood fow measurements at speeds up to 100 Hz in vivo at typical source detector separations of 2.5 cm. Measurement speeds are limited only by the available photon count rates. To our knowledge, the results represent the fastest reported blood fow measurements achieved with DCS, using either hardware or software correlators. We have leveraged this high temporal resolution data to learn about new fow physiology, and to highlight new opportunities for monitoring of cerebral health.
1. We showed that blood fow accounts for over 90% of the pulsatile DCS signal, thereby identifying the origin of the fuctuations in the pulsatile DCS signal in vivo. This discovery was facilitated by the improved speed afforded by the software correlator.
2. Our work shows that cerebral autoregulation dynamics can be studied using DCS. The high data rates afforded by the software correlator were critical for this application (see for example Fig. 7). This demonstration is important because cerebral autoregulation holds tremendous potential as a biomarker of brain injury, and the use of DCS for this application will permit non-invasive and real-time monitoring of brain injury.
3. Our measurements of blood fow pulsatility clearly resolve features of the entire cardiac cycle including the dicrotic notch, demonstrating data quality akin to high speed arterial line tracings. Such high temporal resolution holds potential for measurements of compliance in cerebrovascular circulation; for example, fow pulsatility measured on arm and the brain could reveal differences in arterial elasticity. Measurements of cerebrovascular compliance can potentially identify arterial stiffening or dissection, i.e., information indicative of diseases such as atherosclerosis, amyloidosis and stroke.
The software correlator we have demonstrated is relatively easy to implement on a standard personal computer. When compared to the fastest DCS hardware correlators [50], our measurements are faster, can be sustained for longer durations, and do not require averaging over many detectors. With the new software correlator design, we adopt a slightly different approach compared to previous studies, i.e., we use direct computation of the autocorrelation with the shift-and-add method [44], instead of highly effcient fast Fourier transforms (FFT) [52,53]. Fig. 11. Comparison of heart rates estimated using the software correlator (e.g. from frequency spectrum in Fig. 4(C)) and a commercial pulse oximeter, under baseline conditions from 8 subjects. (A) Scatter plot of heart rate estimate estimated using the software correlator (HR DCS , x-axis) and commercial pulse oximeter (HR Oximeter , y-axis). Solid blue circles represent each measurement, the dashed green line is a 1 : 1 line, and the solid red line is a line of linear regression line. The slope of the regression line is 1 denoting excellent agreement between the two techniques. (B) Bland-Altman plot that represents the average (x-axis) and difference (y-axis) of the estimated heart rates. All measurements are within the 95% confdence lines (dashed horizontal black lines) indicating good agreement between the techniques.
show excellent agreement between the two measurements. This agreement is confrmed in a Bland-Altman analysis, Fig. 11(B), which clearly shows no signifcant difference between the two techniques (p = 0.76).