Abstract

Quantification of chromophore concentrations in reflectance mode remains a major challenge for biomedical optics. Spectroscopic Optical Coherence Tomography (SOCT) provides depth-resolved spectroscopic information necessary for quantitative analysis of chromophores, like hemoglobin, but conventional SOCT analysis methods are applicable only to well-defined specular reflections, which may be absent in highly scattering biological tissue. Here, by fitting of the dynamic scattering signal spectrum in the OCT angiogram using a forward model of light propagation, we quantitatively determine hemoglobin concentrations directly. Importantly, this methodology enables mapping of both oxygen saturation and total hemoglobin concentration, or alternatively, oxyhemoglobin and deoxyhemoglobin concentration, simultaneously. Quantification was verified by ex vivo blood measurements at various pO2 and hematocrit levels. Imaging results from the rodent brain and retina are presented. Confounds including noise and scattering, as well as potential clinical applications, are discussed.

© 2015 Optical Society of America

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2014 (8)

N. Bosschaart, G. J. Edelman, M. C. Aalders, T. G. van Leeuwen, and D. J. Faber, “A literature review and novel theoretical approach on the optical properties of whole blood,” Lasers Med. Sci. 29(2), 453–479 (2014).
[Crossref] [PubMed]

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

V. J. Srinivasan and H. Radhakrishnan, “Optical Coherence Tomography angiography reveals laminar microvascular hemodynamics in the rat somatosensory cortex during activation,” Neuroimage 102(Pt 2), 393–406 (2014).
[Crossref] [PubMed]

C. Du, N. D. Volkow, A. P. Koretsky, and Y. Pan, “Low-frequency calcium oscillations accompany deoxyhemoglobin oscillations in rat somatosensory cortex,” Proc. Natl. Acad. Sci. U.S.A. 111(43), E4677–E4686 (2014).
[Crossref] [PubMed]

P. Y. Teng, J. Wanek, N. P. Blair, and M. Shahidi, “Response of inner retinal oxygen extraction fraction to light flicker under normoxia and hypoxia in rat,” Invest. Ophthalmol. Vis. Sci. 55(9), 6055–6058 (2014).
[Crossref] [PubMed]

M. Kraszewski, M. Trojanowski, and M. R. Strąkowski, “Comment on Quantitative comparison of analysis methods for spectroscopic optical coherence tomography,” Biomed. Opt. Express 5(9), 3023–3033 (2014).
[Crossref] [PubMed]

N. Bosschaart, T. G. van Leeuwen, M. C. Aalders, and D. J. Faber, “Quantitative comparison of analysis methods for spectroscopic optical coherence tomography: reply to comment,” Biomed. Opt. Express 5(9), 3034–3035 (2014).
[Crossref] [PubMed]

J. Yi, S. Chen, V. Backman, and H. F. Zhang, “In vivo functional microangiography by visible-light optical coherence tomography,” Biomed. Opt. Express 5(10), 3603–3612 (2014).
[PubMed]

2013 (4)

2012 (1)

2011 (4)

R. V. Kuranov, J. Qiu, A. B. McElroy, A. Estrada, A. Salvaggio, J. Kiel, A. K. Dunn, T. Q. Duong, and T. E. Milner, “Depth-resolved blood oxygen saturation measurement by dual-wavelength photothermal (DWP) optical coherence tomography,” Biomed. Opt. Express 2(3), 491–504 (2011).
[PubMed]

X. Liu, K. Zhang, Y. Huang, and J. U. Kang, “Spectroscopic-speckle variance OCT for microvasculature detection and analysis,” Biomed. Opt. Express 2(11), 2995–3009 (2011).
[Crossref] [PubMed]

F. E. Robles, C. Wilson, G. Grant, and A. Wax, “Molecular imaging true-colour spectroscopic optical coherence tomography,” Nat. Photonics 5(12), 744–747 (2011).
[Crossref] [PubMed]

P. Cimalla, J. Walther, M. Mittasch, and E. Koch, “Shear flow-induced optical inhomogeneity of blood assessed in vivo and in vitro by spectral domain optical coherence tomography in the 1.3 μm wavelength range,” J. Biomed. Opt. 16(11), 116020 (2011).
[Crossref] [PubMed]

2010 (6)

P. J. Drew, A. Y. Shih, J. D. Driscoll, P. M. Knutsen, P. Blinder, D. Davalos, K. Akassoglou, P. S. Tsai, and D. Kleinfeld, “Chronic optical access through a polished and reinforced thinned skull,” Nat. Methods 7(12), 981–984 (2010).
[Crossref] [PubMed]

A. L. Vazquez, K. Masamoto, M. Fukuda, and S. G. Kim, “Cerebral oxygen delivery and consumption during evoked neural activity,” Front Neuroenergetics 2, 11 (2010).
[PubMed]

A. L. Vazquez, M. Fukuda, M. L. Tasker, K. Masamoto, and S. G. Kim, “Changes in cerebral arterial, tissue and venous oxygenation with evoked neural stimulation: implications for hemoglobin-based functional neuroimaging,” J. Cereb. Blood Flow Metab. 30(2), 428–439 (2010).
[Crossref] [PubMed]

J. Yi and X. Li, “Estimation of oxygen saturation from erythrocytes by high-resolution spectroscopic optical coherence tomography,” Opt. Lett. 35(12), 2094–2096 (2010).
[Crossref] [PubMed]

F. E. Robles, S. Chowdhury, and A. Wax, “Assessing hemoglobin concentration using spectroscopic optical coherence tomography for feasibility of tissue diagnostics,” Biomed. Opt. Express 1(1), 310–317 (2010).
[Crossref] [PubMed]

Y. Wang and R. Wang, “Autocorrelation optical coherence tomography for mapping transverse particle-flow velocity,” Opt. Lett. 35(21), 3538–3540 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (3)

2007 (4)

E. M. Hillman, “Optical brain imaging in vivo: techniques and applications from animal to man,” J. Biomed. Opt. 12(5), 051402 (2007).
[Crossref] [PubMed]

L. Kagemann, G. Wollstein, M. Wojtkowski, H. Ishikawa, K. A. Townsend, M. L. Gabriele, V. J. Srinivasan, J. G. Fujimoto, and J. S. Schuman, “Spectral oximetry assessed with high-speed ultra-high-resolution optical coherence tomography,” J. Biomed. Opt. 12(4), 041212 (2007).
[Crossref] [PubMed]

H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. H. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90(5), 053901 (2007).
[Crossref]

R. N. Graf and A. Wax, “Temporal coherence and time-frequency distributions in spectroscopic optical coherence tomography,” J. Opt. Soc. Am. A 24(8), 2186–2195 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (2)

2004 (2)

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
[Crossref] [PubMed]

D. J. Faber, M. C. Aalders, E. G. Mik, B. A. Hooper, M. J. van Gemert, and T. G. van Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. 93(2), 028102 (2004).
[Crossref] [PubMed]

2003 (1)

2000 (3)

1993 (1)

S. Ogawa, R. S. Menon, D. W. Tank, S. G. Kim, H. Merkle, J. M. Ellermann, and K. Ugurbil, “Functional brain mapping by blood oxygenation level-dependent contrast magnetic resonance imaging. A comparison of signal characteristics with a biophysical model,” Biophys. J. 64(3), 803–812 (1993).
[Crossref] [PubMed]

1988 (1)

1986 (1)

R. N. Pittman, “In vivo photometric analysis of hemoglobin,” Ann. Biomed. Eng. 14(2), 119–137 (1986).
[Crossref] [PubMed]

1974 (1)

1964 (1)

L. H. Gray and J. M. Steadman, “Determination of the Oxyhaemoglobin Dissociation Curves for Mouse and Rat Blood,” J. Physiol. 175(2), 161–171 (1964).
[Crossref] [PubMed]

1910 (1)

A. V. Hill, “The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves,” J. Physiol. 40, 7 (1910).

Aalders, M. C.

Akassoglou, K.

P. J. Drew, A. Y. Shih, J. D. Driscoll, P. M. Knutsen, P. Blinder, D. Davalos, K. Akassoglou, P. S. Tsai, and D. Kleinfeld, “Chronic optical access through a polished and reinforced thinned skull,” Nat. Methods 7(12), 981–984 (2010).
[Crossref] [PubMed]

Andreasson, K. I.

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

Antaris, A. L.

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

Atochin, D. N.

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

Backman, V.

Barry, S.

Bigelow, C. E.

Bjornerud, A.

K. Briely-Sabo and A. Bjornerud, “Accurate de-oxygenation of ex vivo whole blood using sodium Dithionite,” Proc. Intl. Sot. Mag. Reson. Med. 8, 2025 (2000).

Blair, N. P.

P. Y. Teng, J. Wanek, N. P. Blair, and M. Shahidi, “Response of inner retinal oxygen extraction fraction to light flicker under normoxia and hypoxia in rat,” Invest. Ophthalmol. Vis. Sci. 55(9), 6055–6058 (2014).
[Crossref] [PubMed]

Blinder, P.

P. J. Drew, A. Y. Shih, J. D. Driscoll, P. M. Knutsen, P. Blinder, D. Davalos, K. Akassoglou, P. S. Tsai, and D. Kleinfeld, “Chronic optical access through a polished and reinforced thinned skull,” Nat. Methods 7(12), 981–984 (2010).
[Crossref] [PubMed]

Boppart, S. A.

Bosschaart, N.

Briely-Sabo, K.

K. Briely-Sabo and A. Bjornerud, “Accurate de-oxygenation of ex vivo whole blood using sodium Dithionite,” Proc. Intl. Sot. Mag. Reson. Med. 8, 2025 (2000).

Cable, A. E.

Campagnola, P.

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rCSLM),” J. Biomed. Opt. 13(4), 041309 (2008).
[Crossref] [PubMed]

Chang, J. L.

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

Chen, C. X.

G. S. Hong, S. Diao, J. L. Chang, A. L. Antaris, C. X. Chen, B. Zhang, S. Zhao, D. N. Atochin, P. L. Huang, K. I. Andreasson, C. J. Kuo, and H. J. Dai, “Through-skull fluorescence imaging of the brain in a new near-infrared window,” Nat. Photonics 8(9), 723–730 (2014).
[Crossref]

Chen, S.

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C. Du, N. D. Volkow, A. P. Koretsky, and Y. Pan, “Low-frequency calcium oscillations accompany deoxyhemoglobin oscillations in rat somatosensory cortex,” Proc. Natl. Acad. Sci. U.S.A. 111(43), E4677–E4686 (2014).
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A. L. Vazquez, M. Fukuda, M. L. Tasker, K. Masamoto, and S. G. Kim, “Changes in cerebral arterial, tissue and venous oxygenation with evoked neural stimulation: implications for hemoglobin-based functional neuroimaging,” J. Cereb. Blood Flow Metab. 30(2), 428–439 (2010).
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H. F. Zhang, K. Maslov, M. Sivaramakrishnan, G. Stoica, and L. H. V. Wang, “Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy,” Appl. Phys. Lett. 90(5), 053901 (2007).
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P. Y. Teng, J. Wanek, N. P. Blair, and M. Shahidi, “Response of inner retinal oxygen extraction fraction to light flicker under normoxia and hypoxia in rat,” Invest. Ophthalmol. Vis. Sci. 55(9), 6055–6058 (2014).
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Figures (14)

Fig. 1
Fig. 1

Wavelength-dependence of oxyhemoglobin and deoxyhemoglobin absorption (red and blue, respectively), whole blood scattering (black), brain tissue scattering (pink), and water absorption (green). Hemoglobin absorption spectra and blood scattering spectra (compiled averages from literature) were obtained from [23] and [18] respectively. The water absorption spectrum was obtained from [24] while the reduced scattering coefficient of brain tissue was obtained from [25], and the scattering coefficient was derived assuming a constant anisotropy of g=0.9. As shown in the plot, hemoglobin absorption is more significant relative to scattering in the visible wavelength range, as compared to the near-infrared wavelength range.

Fig. 2
Fig. 2

A) Visible light SOCT setup. The supercontinuum source (SC) delivered the light via a photonic crystal (PC) fiber. FC: fiber collimator; F: spectral filters; BS: beam splitter; M: mirror; NDF: neutral density filter; DG: diffraction grating; L1-3: 30mm achromatic doublets; L4: 75 mm achromatic doublet pairs. The figure is not drawn to scale. B) Spectrum measured by spectrometer after filtering of the SC light source.

Fig. 3
Fig. 3

Spectrometer calibration requires correct accounting for nonlinearities in wavelength sampling. A) The locations of two calibration wavelengths were determined using auxiliary laser diodes with central wavelengths at 532 nm and 635 nm respectively (blue circles show calibration wavelengths). The linear wavelength fit is shown as a blue dashed line. Another linear wavelength curve, empirically chosen to optimize sensitivity roll-off, is plotted (green dashed line). Finally, the calibrated curve, derived from both the measured phase and calibration wavelengths, is shown (red solid line; see description in article for details). B) Wavenumbers corresponding to wavelengths (A) are shown. C) The axial profiles of two spectral fringe patterns after resampling based on the linear wavelength fit, followed by Fourier transformation, are shown (blue dashed line). Poor sensitivity roll-off and axial profile broadening were observed at larger depths, indicating incorrect resampling. By contrast, by empirically stretching the linear wavelength curve, improved roll-off could be achieved, at the expense of inaccurate calibration (green dashed line). D) Using our proposed calibration procedure, axial profiles show good sensitivity roll-off and negligible broadening (red solid line). Furthermore, unlike empirical linear wavelength calibration (green dashed line in A), calibrated wavelengths matched the two calibration wavelengths (red solid line in A).

Fig. 4
Fig. 4

A) Fresh whole blood was harvested from Long Evans rats by cardiac puncture. B) Blood samples were placed in a closed chamber with a gas mixture supply (i.e. N2, O2, & CO2) to maintain physiological pCO2 levels and to manipulate pO2 levels. The blood sample was stirred by a magnetic stirrer to ensure homogenous mixing of the blood and equilibration with the surrounding gas mixture. C) Each blood sample was pumped through FEP tubing (inner diameter of 200 μm) and imaged under the SOCT system. D) Blood gas and pH readings were taken before and after imaging sessions to confirm physiological pCO2 (35-45 mmHg) and pH (7.35-7.45). Low pO2 values were achieved by adding sufficient sodium dithionite to fully deoxygenate the hemoglobin [30]. E) The hematocrit of the blood sample was measured using a micro-centrifuge.

Fig. 5
Fig. 5

Validation of ex vivo rat blood oxygen saturation measured using visible light SOCT. A) Estimated SO2 (%) using SOCT was plotted versus the corresponding pO2 measurements from a blood gas analyzer. The oxygen-hemoglobin dissociation curve (OHDC) was fit using the Hill equation [41] for our data set as well as Gray & Steadman’s data [28]. B) There is a strong linear correlation between the SO2 values estimated by SOCT and those determined from pO2 and the OHDC fit of Gray & Steadman’s data [28]. The 95% confidence intervals are plotted as dashed blue lines.

Fig. 6
Fig. 6

Ex vivo calibration with rat blood shows a linear relationship between packed cell volume and attenuation slope (A, C), maximum OCT signal (A, D), and hemoglobin concentration obtained by fitting of the modified Beer-Lambert Law (B, E). The y-intercept of the fit of OCT signal slope vs. hematocrit (C) does not pass through the origin, indicating that the two quantities are not strictly proportional. Notably, the maximum OCT signal (D) and modified Beer-Lambert Law fitting (E) are proportional to hematocrit. However, in flowing blood, red blood cell orientation effects make signal slope and maximum signal (related to attenuation and backscattering, respectively) poor correlates of hemoglobin content [42]. The predicted line in E (red dashed line) is based on typical rat blood hematocrit of 45% and hemoglobin concentration of 138 g/L.

Fig. 7
Fig. 7

Quantification of total hemoglobin concentration (CHbT) in the rat inner retina using axial profile of parameters obtained from spectroscopic fitting. A) OCT angiogram with a red box containing the axial scans used to determine axial profiles in B-C. B) The signal from the angiogram shows orientation effects with higher backscattering near the vessel walls due to orientation of RBCs in shear flow. C) Fitting to determine the path length times the hemoglobin concentration (LCHbT) yields a profile that is flat above the vessel, linearly increases inside the vessel, then flattens out below the vessel. The slope of LCHbT within the vessel is quantitatively related to the local hematocrit, and is relatively insensitive to orientation effects.

Fig. 8
Fig. 8

CHbT mapping of the rat inner retina. A) R2 from the fit of the absorbance spectrum of inner retinal vessels. Note that the largest R2 values are observed near the bottom of large vessels. B) Map of LCHbT showing the largest values near the bottom of large vessels where the cumulative path length is largest. C) Comparison of absorption MIP (maximum intensity projection of LCHbT over the axial direction) and scattering MIP (maximum intensity projection of standard angiogram, obtained by high-pass filtering along the slow axis). The absorption MIP shows more self-consistent values along large vessels. D) Comparison of histograms for absorption and scattering angiograms along a vessel (blue and red arrows in C) shows a threefold lower coefficient of variation using the absorption-based method. This is likely because the standard angiogram is sensitive to RBC orientation and vignetting effects, whereas the absorption-based angiogram is not.

Fig. 9
Fig. 9

Quantification of chromophores in the mouse neocortex in cross-section. A) R2 values from the fit (Eq. (19)) show the highest values near the distal side of vessels, with a decrease in R2 in the multiple scattering tails. B) The parameter Φ accounts for RBC scattering effects. C) Saturation map, showing clear distinctions between arteries and veins. D-E) Maps of the product of oxygenated or deoxygenated hemoglobin concentrations and distance exhibit a characteristic downward “crescent” shape, due to larger cumulative distances at the distal end of the vessel. F) Similarly the product of total hemoglobin concentration and distance shows an increase with depth until the distal end of the vessel, and remains constant in the multiple scattering tail. All maps were displayed with transparency based on the local R2 value after averaging over six adjacent transverse (y) locations. An artery (a) and vein (v) are labelled.

Fig. 10
Fig. 10

Quantification of chromophores in the mouse brain in an en face view. A) Maximum intensity projection of R2 values from the fit (Eq. (19)) shows the highest values near the centers of vessels, with a decrease at the edges. B) The parameter Φ accounts for RBC scattering effects. C) Saturation map, showing clear distinctions between arteries and veins. D) Map of the maximum of the product of oxygenated hemoglobin concentration and distance shows that veins and arteries contain oxyhemoglobin. E) By comparison, under the given experimental conditions, most of the deoxyhemoglobin is contained in veins. F) The map of the maximum of the product of total hemoglobin concentration and distance shows larger values in larger vessels, with localized increases at vessel crossings. It should be noted that quantitative measurements of chromophores can be achieved by integrating the maps (D-F) in the transverse plane (x and y dimensions). All maps were displayed with transparency based on the local R2 values at each transverse location, averaged over depth. An artery (a) and vein (v) are labelled.

Fig. 11
Fig. 11

Quantification of chromophores in the rat retina in cross-section. A) Saturation map, showing clear distinctions between arteries and veins. B) The product of total hemoglobin concentration and distance shows an increase with depth until the distal end of the vessel, and remains constant in the multiple scattering tail. C-D) Maps of the product of oxygenated or deoxygenated hemoglobin concentrations and distance increase towards the distal end of the vessel. While all vessels contain oxyhemoglobin (C), the three locations in D) with significant deoxyhemoglobin absorption at the distal end of the vessel correspond to veins. All maps were displayed using an alpha map based on the local R2 value after averaging over six adjacent transverse (y) locations. An artery (a) and vein (v) are labelled.

Fig. 12
Fig. 12

Quantification of chromophores in the rat retina in an en face view. A) Saturation map, showing clear distinctions between arteries and veins. B) The map of the maximum of the product of total hemoglobin concentration and distance shows larger values in larger vessels. C) Map of the maximum of the product of oxygenated hemoglobin concentration and distance shows that veins and arteries contain oxyhemoglobin. D) By comparison, under the given experimental conditions, most of the deoxyhemoglobin is contained in veins. It should be noted that quantitative measurements of chromophores can be achieved by integrating the maps (B-D) in the transverse plane (x and y dimensions). All maps were displayed using an en face alpha map based on the local R2 values at each transverse location, averaged over depth.

Fig. 13
Fig. 13

Delivery of oxygen from an artery to surrounding tissue is demonstrated by high-resolution mapping. The oxygen saturation is parametrically determined as a function of distance along the flow direction shown as a dotted line on the angiogram (A) and saturation map (B). C) A statistically significant linear decrease in saturation along the direction of flow is demonstrated. Saturations from fits with an R2 value of >0.6 were included for this figure.

Fig. 14
Fig. 14

Changes in intravascular oxygenation and chromophore concentrations after cardiac arrest. A-D) Maps of saturation, deoxyhemoglobin, oxyhemoglobin, and total hemoglobin at baseline. E-H) Maps of the same parameters after cardiac arrest. In particular, the saturation is reduced to zero (A,E) and deoxyhemoglobin content increases (B,F), while oxyhemoglobin is eliminated from the brain vasculature (C,G).

Equations (23)

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S[ k( x i ) ]cos{ 2k( x i )z+ φ disp [ k( x i ) ] },
Δφ( x i )=2k( x i )Δz+ φ offset ,
k( x i )= Δφ( x i ) φ offset 2Δz .
k c ( x n ) 2π λ n n=1,,N,
φ ^ offset = argmin φ offset n=1 N [ k( x n ) k c ( x n ) ] 2 .
k ^ ( x i )= Δφ( x i ) φ ^ offset 2Δz , λ ^ ( x i )= 2π k ^ ( x i ) .
a'( x,z,t )=a[ x,zδz(x,t),t ] e jδθ( x,t ) .
a'(x,z,t,k)=a ' s (x,z,k)+a ' d (x,z,t,k)+a ' n (x,z,t,k).
I(x,z,k)= I s (x,z,k)+ I d (x,z,k)+ I n (z,k).
F(z,k)=STFT{ f(z) }= z' f(z')w(z'z) e j2kz' .
A ^ ( x,z,k )= 1 2 ln[ I ^ vessel ( x,z,k ) I ^ norm ( z,k ) ],
I ^ vessel ( x,z,k )= I ^ d ( x,z,k )= t | STFT{ h hpf ( t )a'( x,z,t ) } | 2 I ^ d,bkgnd ( z,k ),
I ^ norm ( z,k )= I ^ s ( z,k )= 1 N x x [ t | STFT{ h lpf ( t )a'( x,z,t ) } | 2 I ^ s,bkgnd ( z,k ) ] .
I ^ d,bkgnd ( z,k )= t | STFT{ h hpf ( t ) a bkgnd ( z,t ) } | 2 ,
I ^ s,bkgnd ( z,k )= t | STFT{ h lpf ( t ) a bkgnd ( z,t ) } | 2 .
A( x,z,k )= C HbT ( x,z' ){ S( x,z' ) ε Hb O 2 (k)+[ 1S(x,z') ] ε Hb (k) } dz'+Φ( x,z ).
A( x,z,k )= C HbT ( x )L(x,z){ S( x ) ε Hb O 2 (k)+[ 1S(x) ] ε Hb (k) }+Φ( x,z ),
A( x,z,k )=L(x,z)[ C Hb O 2 ( x ) ε Hb O 2 (k)+ C Hb ( x ) ε Hb (k) ]+Φ( x,z ),
A ^ ( x,z,k )= a 1 (x,z) ε Hb O 2 ( k )+ a 2 (x,z) ε Hb ( k )+ a 3 (x,z)+n(x,z,k).
a 1 L C HbT S=L C Hb O 2 ,
a 2 L C HbT ( 1S )=L C Hb .
a 1 + a 2 L C HbT .
z 0 (x)= argmax z { R 2 [ A ^ ( x,z,k ) ] },

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