Abstract

We propose a simple and affordable imaging technique to evaluate transcutaneously multiple physiological parameters by using a digital red-green-blue camera. In this method, the RGB-values were converted into tristimulus values in the CIE (Commission Internationale de l’Eclairage) XYZ color space, which is compatible with the common color spaces. Monte Carlo simulation for light transport in biological tissue was then performed to specify the relationship among the XYZ-values and the concentrations of oxygenated hemoglobin, deoxygenated hemoglobin, bilirubin, and melanin. The concentration of total hemoglobin and tissue oxygen saturation were also calculated from the estimated concentrations of oxygenated and deoxygenated hemoglobin. In vivo experiments with bile duct ligation in rats demonstrated that the estimated bilirubin concentration increased after ligation of the bile duct and reached around 22 mg/dl at 116 h after the onset of ligation, which corresponds to the ground truth value of bilirubin measured by a commercially available transcutaneous bilirubinometer. Experiments with rats while varying the fraction of inspired oxygen demonstrated that oxygenated hemoglobin and deoxygenated hemoglobin decreased and increased, respectively, as the fraction of inspired oxygen decreased. Consequently, tissue oxygen saturation dramatically decreased. We further extended the method to a non-contact imaging photo-plethysmograph and estimation of the percutaneous oxygen saturation. An empirical formula to estimate percutaneous oxygen saturation was derived from the pulse wave amplitudes of oxygenated and deoxygenated hemoglobin. The estimated percutaneous oxygen saturation dropped remarkably when a faction of inspired oxygen was below 19%, indicating the onset of hypoxemia due to hypoxia, whereas the tissue oxygen saturation decreased gradually according to the reduction of the faction of inspired oxygen. The results in this study indicate the potential of this method for imaging of multiple physiological parameters in skin tissue and evaluating an optical biomedical imaging technique that enables cost-effective, easy-to-use, portable, remotely administered, and/or point-of-care solutions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

A. Nkengne, J. Robic, P. Seroul, S. Gueheunneux, M. Jomier, and K. Vie, “SpectraCam®: a new polarized hyperspectral imaging system for repeatable and reproducible in vivo skin quantification of melanin, total hemoglobin, and oxygen saturation,” Skin Res. Technol. 24(1), 99–107 (2018).
[Crossref]

2015 (1)

F. Vasefi, N. MacKinnon, R. B. Saager, A. J. Durkin, R. Chave, E. H. Lindsley, and D. L. Farkas, “Polarization-sensitive hyperspectral imaging in vivo: a multimode dermoscope for skin analysis,” Sci. Rep. 4(1), 4924 (2015).
[Crossref]

2014 (2)

2011 (4)

D. A. Boas and M. A. Franceschini, “Haemoglobin oxygen saturation as a biomarker: the problem and a solution,” Philos. Trans. R. Soc., A 369(1955), 4407–4424 (2011).
[Crossref]

I. Nishidate, N. Tanaka, T. Kawase, T. Maeda, T. Yuasa, Y. Aizu, T. Yuasa, and K. Niizeki, “Noninvasive imaging of human skin hemodynamics using a digital red-green-blue camera,” J. Biomed. Opt. 16(8), 086012 (2011).
[Crossref]

I. Nishidate, A. Wiswadarma, Y. Hase, N. Tanaka, T. Maeda, K. Niizeki, and Y. Aizu, “Non-invasive spectral imaging of skin chromophores based on multiple regression analysis aided by Monte Carlo simulation,” Opt. Lett. 36(16), 3239–3241 (2011).
[Crossref]

I. Kuzmina, I. Diebele, D. Jakovels, J. Spigulis, L. Valeine, J. Kapostinsh, and A. Berzina, “Towards noncontact skin melanoma selection by multispectral imaging analysis,” J. Biomed. Opt. 16(6), 060502 (2011).
[Crossref]

2009 (2)

S.-H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in vivo,” Opt. Express 17(17), 14599–14617 (2009).
[Crossref]

J. O’Doherty, P. McNamara, N. T. Clancy, J. G. Enfield, and M. J. Leahy, “Comparison of instruments for investigation ofmicrocirculatory blood flow and red blood cell concentration,” J. Biomed. Opt. 14(3), 034025 (2009).
[Crossref]

2007 (1)

J. Allen, “Photoplethysmography and its application in clinical physiological measurement,” Physiol. Meas. 28(3), R1–R39 (2007).
[Crossref]

2005 (2)

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[Crossref]

B. S. Sorg, B. J. Moeller, O. Donovan, Y. Cao, and M. W. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt. 10(4), 044004 (2005).
[Crossref]

2004 (4)

G. N. Stamatas and N. Kollias, “Blood stasis contributions to the perception of skin pigmentation,” J. Biomed. Opt. 9(2), 315–322 (2004).
[Crossref]

I. Nishidate, Y. Aizu, and H. Mishina, “Estimation of melanin and hemoglobin in skin tissue using multiple regression analysis aided by Monte Carlo simulation,” J. Biomed. Opt. 9(4), 700–710 (2004).
[Crossref]

F. Groenendaal, J. van der Grond, and L. S. de Vries, “Cerebral metabolism in severe neonatal hyperbilirubinemia,” Pediatrics 114(1), 291–294 (2004).
[Crossref]

M. J. Maisels, E. M. Ostrea Jr, S. Touch, S. E. Clune, E. Cepeda, E. Kring, K. Gracey, C. Jackson, D. Talbot, and R. Huang, “Evaluation of a new transcutaneous bilirubinometer,” Pediatrics 113(6), 1628–1635 (2004).
[Crossref]

2003 (1)

2001 (4)

G. Zonios, J. Bykowski, and N. Kollias, “Skin melanin, hemoglobin, and light scattering properties can be quantitatively assessed in vivo using diffuse reflectance spectroscopy,” J. Invest. Dermatol. 117(6), 1452–1457 (2001).
[Crossref]

F. F. Rubaltelli, G. R. Gourley, N. Loskamp, N. Modi, M. Roth-Kleiner, A. Sender, and P. Vert, “Transcutaneous bilirubin measurement: a multicenter evaluation of a new device,” Pediatrics 107(6), 1264–1271 (2001).
[Crossref]

A. A. Stratonnikov and V. B. Loschenov, “Evaluation of blood oxygen saturation in vivo from diffuse reflectance spectra,” J. Biomed. Opt. 6(4), 457–467 (2001).
[Crossref]

G. Zonios, J. Bykowski, and N. Kollias, “Skin melanin, hemoglobin, and light scattering properties can be quantitatively assessed in vivo using diffuse reflectance spectroscopy,” J. Invest. Dermatol. 117(6), 1452–1457 (2001).
[Crossref]

1999 (2)

M. G. Sowa, J. R. Payette, M. D. Hewko, and H. H. Mantsch, “Visible-near infrared multispectral imaging of the rat dorsal skin flap,” J. Biomed. Opt. 4(4), 474–481 (1999).
[Crossref]

N. Tsumura, H. Haneishi, and Y. Miyake, “Independent-component analysis of skin color image,” J. Opt. Soc. Am. A 16(9), 2169–2176 (1999).
[Crossref]

1997 (1)

S. L. Jacques, I. S. Saidi, A. Ladner, and D. Oelberg, “Developing an optical fiber reflectance spectrometer to monitor bilirubinemia in neonates,” Proc. SPIE 2975, 115–124 (1997).
[Crossref]

1996 (1)

S. L. Jacques, R. D. Glickman, and J. A. Schwartz, “Internal absorption coefficient and threshold for pulsed laser disruption of melanosomes isolated from retinal pigment epithelium,” Proc. SPIE 2681, 468–477 (1996).
[Crossref]

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML: Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref]

1994 (1)

D. J. Newton, D. K. Harrison, C. J. Delaney, J. S. Beck, and P. T. McCollum, “Comparison of macro- and maicro-lightguide spectrophotometric measurements of microvascular haemoglobin oxygenation in the tuberculin reaction in normal human skin,” Physiol. Meas. 15(2), 115–128 (1994).
[Crossref]

1992 (1)

D. K. Harrison, S. D. Evans, N. C. Abbot, J. S. Beck, and P. T. McCollum, “Spectrophotometric measurements of haemoglobin saturation and concentration in skin during the tuberculin reaction in normal human subjects,” Clin. Phys. Physiol. Meas. 13(4), 349–363 (1992).
[Crossref]

1990 (1)

G. Agati and F. Fusi, “New trends in photobiology (invited review). Recent advances in bilirubin photophysics,” J. Photochem. Photobiol., B 7(1), 1–14 (1990).
[Crossref]

1989 (2)

J. W. Feather, M. Hajizadeh-Saffar, G. Leslie, and J. B. Dawson, “A portable scanning reflectance pectrophotometer using visible wavelengths for the rapid measurement of skin pigments,” Phys. Med. Biol. 34(7), 807–820 (1989).
[Crossref]

A. A. Kamal, J. B. Harness, G. Irving, and A. J. Mearns, “Skin photoplethysmography–a review,” Comput. Meth. Programs Biomed. 28(4), 257–269 (1989).
[Crossref]

1987 (1)

A. U. Ferrari, A. Daffonchio, F. Albergati, and G. Mancia, “Inverse relationship between heart rate and blood pressure variabilities in rats,” Hypertension 10(5), 533–537 (1987).
[Crossref]

1984 (1)

T. Sarna and R. C. Sealy, “Photoinduced oxygen consumption in melanin systems. Action spectra and quantum yields for eumelanin and synthetic melanin,” Photochem. Photobiol. 39(1), 69–74 (1984).
[Crossref]

Abbot, N. C.

D. K. Harrison, S. D. Evans, N. C. Abbot, J. S. Beck, and P. T. McCollum, “Spectrophotometric measurements of haemoglobin saturation and concentration in skin during the tuberculin reaction in normal human subjects,” Clin. Phys. Physiol. Meas. 13(4), 349–363 (1992).
[Crossref]

Agati, G.

G. Agati and F. Fusi, “New trends in photobiology (invited review). Recent advances in bilirubin photophysics,” J. Photochem. Photobiol., B 7(1), 1–14 (1990).
[Crossref]

Aizu, Y.

I. Nishidate, A. Wiswadarma, Y. Hase, N. Tanaka, T. Maeda, K. Niizeki, and Y. Aizu, “Non-invasive spectral imaging of skin chromophores based on multiple regression analysis aided by Monte Carlo simulation,” Opt. Lett. 36(16), 3239–3241 (2011).
[Crossref]

I. Nishidate, N. Tanaka, T. Kawase, T. Maeda, T. Yuasa, Y. Aizu, T. Yuasa, and K. Niizeki, “Noninvasive imaging of human skin hemodynamics using a digital red-green-blue camera,” J. Biomed. Opt. 16(8), 086012 (2011).
[Crossref]

I. Nishidate, Y. Aizu, and H. Mishina, “Estimation of melanin and hemoglobin in skin tissue using multiple regression analysis aided by Monte Carlo simulation,” J. Biomed. Opt. 9(4), 700–710 (2004).
[Crossref]

Albergati, F.

A. U. Ferrari, A. Daffonchio, F. Albergati, and G. Mancia, “Inverse relationship between heart rate and blood pressure variabilities in rats,” Hypertension 10(5), 533–537 (1987).
[Crossref]

Allen, J.

J. Allen, “Photoplethysmography and its application in clinical physiological measurement,” Physiol. Meas. 28(3), R1–R39 (2007).
[Crossref]

Andermann, M. L.

Bargo, P.

Bargo, P. R.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[Crossref]

Beck, J. S.

D. J. Newton, D. K. Harrison, C. J. Delaney, J. S. Beck, and P. T. McCollum, “Comparison of macro- and maicro-lightguide spectrophotometric measurements of microvascular haemoglobin oxygenation in the tuberculin reaction in normal human skin,” Physiol. Meas. 15(2), 115–128 (1994).
[Crossref]

D. K. Harrison, S. D. Evans, N. C. Abbot, J. S. Beck, and P. T. McCollum, “Spectrophotometric measurements of haemoglobin saturation and concentration in skin during the tuberculin reaction in normal human subjects,” Clin. Phys. Physiol. Meas. 13(4), 349–363 (1992).
[Crossref]

Berzina, A.

I. Kuzmina, I. Diebele, D. Jakovels, J. Spigulis, L. Valeine, J. Kapostinsh, and A. Berzina, “Towards noncontact skin melanoma selection by multispectral imaging analysis,” J. Biomed. Opt. 16(6), 060502 (2011).
[Crossref]

Bish, S. F.

Blair, G.

P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and empirical light transport model during endoscopy,” J. Biomed. Opt. 10(3), 034018 (2005).
[Crossref]

Boas, D. A.

Bolay, H.

Bykowski, J.

G. Zonios, J. Bykowski, and N. Kollias, “Skin melanin, hemoglobin, and light scattering properties can be quantitatively assessed in vivo using diffuse reflectance spectroscopy,” J. Invest. Dermatol. 117(6), 1452–1457 (2001).
[Crossref]

G. Zonios, J. Bykowski, and N. Kollias, “Skin melanin, hemoglobin, and light scattering properties can be quantitatively assessed in vivo using diffuse reflectance spectroscopy,” J. Invest. Dermatol. 117(6), 1452–1457 (2001).
[Crossref]

Cao, Y.

B. S. Sorg, B. J. Moeller, O. Donovan, Y. Cao, and M. W. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt. 10(4), 044004 (2005).
[Crossref]

Cepeda, E.

M. J. Maisels, E. M. Ostrea Jr, S. Touch, S. E. Clune, E. Cepeda, E. Kring, K. Gracey, C. Jackson, D. Talbot, and R. Huang, “Evaluation of a new transcutaneous bilirubinometer,” Pediatrics 113(6), 1628–1635 (2004).
[Crossref]

Chave, R.

F. Vasefi, N. MacKinnon, R. B. Saager, A. J. Durkin, R. Chave, E. H. Lindsley, and D. L. Farkas, “Polarization-sensitive hyperspectral imaging in vivo: a multimode dermoscope for skin analysis,” Sci. Rep. 4(1), 4924 (2015).
[Crossref]

Clancy, N. T.

J. O’Doherty, P. McNamara, N. T. Clancy, J. G. Enfield, and M. J. Leahy, “Comparison of instruments for investigation ofmicrocirculatory blood flow and red blood cell concentration,” J. Biomed. Opt. 14(3), 034025 (2009).
[Crossref]

Clune, S. E.

M. J. Maisels, E. M. Ostrea Jr, S. Touch, S. E. Clune, E. Cepeda, E. Kring, K. Gracey, C. Jackson, D. Talbot, and R. Huang, “Evaluation of a new transcutaneous bilirubinometer,” Pediatrics 113(6), 1628–1635 (2004).
[Crossref]

Daffonchio, A.

A. U. Ferrari, A. Daffonchio, F. Albergati, and G. Mancia, “Inverse relationship between heart rate and blood pressure variabilities in rats,” Hypertension 10(5), 533–537 (1987).
[Crossref]

Dale, A. M.

Dawson, J. B.

J. W. Feather, M. Hajizadeh-Saffar, G. Leslie, and J. B. Dawson, “A portable scanning reflectance pectrophotometer using visible wavelengths for the rapid measurement of skin pigments,” Phys. Med. Biol. 34(7), 807–820 (1989).
[Crossref]

de Vries, L. S.

F. Groenendaal, J. van der Grond, and L. S. de Vries, “Cerebral metabolism in severe neonatal hyperbilirubinemia,” Pediatrics 114(1), 291–294 (2004).
[Crossref]

Delaney, C. J.

D. J. Newton, D. K. Harrison, C. J. Delaney, J. S. Beck, and P. T. McCollum, “Comparison of macro- and maicro-lightguide spectrophotometric measurements of microvascular haemoglobin oxygenation in the tuberculin reaction in normal human skin,” Physiol. Meas. 15(2), 115–128 (1994).
[Crossref]

Delgado Atencio, J. A.

J. A. Delgado Atencio, S. L. Jacques, and S. Vázquezy Montiel, “Monte Carlo Modeling of Light Propagation in Neonatal Skin,”in Applications of Monte Carlo Methods in Biology, Medicine and Other Fields of Science, C. J. Mode, ed. (InTechOpen, 2011).

Devor, A.

Dewhirst, M. W.

B. S. Sorg, B. J. Moeller, O. Donovan, Y. Cao, and M. W. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt. 10(4), 044004 (2005).
[Crossref]

Diebele, I.

I. Kuzmina, I. Diebele, D. Jakovels, J. Spigulis, L. Valeine, J. Kapostinsh, and A. Berzina, “Towards noncontact skin melanoma selection by multispectral imaging analysis,” J. Biomed. Opt. 16(6), 060502 (2011).
[Crossref]

Donovan, O.

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B. S. Sorg, B. J. Moeller, O. Donovan, Y. Cao, and M. W. Dewhirst, “Hyperspectral imaging of hemoglobin saturation in tumor microvasculature and tumor hypoxia development,” J. Biomed. Opt. 10(4), 044004 (2005).
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M. G. Sowa, J. R. Payette, M. D. Hewko, and H. H. Mantsch, “Visible-near infrared multispectral imaging of the rat dorsal skin flap,” J. Biomed. Opt. 4(4), 474–481 (1999).
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I. Kuzmina, I. Diebele, D. Jakovels, J. Spigulis, L. Valeine, J. Kapostinsh, and A. Berzina, “Towards noncontact skin melanoma selection by multispectral imaging analysis,” J. Biomed. Opt. 16(6), 060502 (2011).
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G. N. Stamatas and N. Kollias, “Blood stasis contributions to the perception of skin pigmentation,” J. Biomed. Opt. 9(2), 315–322 (2004).
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A. A. Stratonnikov and V. B. Loschenov, “Evaluation of blood oxygen saturation in vivo from diffuse reflectance spectra,” J. Biomed. Opt. 6(4), 457–467 (2001).
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M. J. Maisels, E. M. Ostrea Jr, S. Touch, S. E. Clune, E. Cepeda, E. Kring, K. Gracey, C. Jackson, D. Talbot, and R. Huang, “Evaluation of a new transcutaneous bilirubinometer,” Pediatrics 113(6), 1628–1635 (2004).
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I. Nishidate, A. Wiswadarma, Y. Hase, N. Tanaka, T. Maeda, K. Niizeki, and Y. Aizu, “Non-invasive spectral imaging of skin chromophores based on multiple regression analysis aided by Monte Carlo simulation,” Opt. Lett. 36(16), 3239–3241 (2011).
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Figures (15)

Fig. 1.
Fig. 1. Flowchart of the process for estimating the concentration of melanin Cm, bilirubin Cbil, oxygenated hemoglobin CHbO, deoxygenated hemoglobin CHbR, total hemoglobin CHbT, and tissue oxygen saturation, StO2 from the measured RGB-values.
Fig. 2.
Fig. 2. Schematic illustration of setup for RGB imaging of skin tissue. A white LED is used to illuminate the skin surface, which is imaged using an RGB CCD camera. The computer acquires raw RGB images.
Fig. 3.
Fig. 3. (a) Diffuse reflectance spectra of a rat dorsal skin after bile duct occlusion and (b) simulated diffuse reflectance spectra by the Monte Carlo model under the several bilirubin concentrations.
Fig. 4.
Fig. 4. Typical sequential images obtained from rat dorsal skin including RGB color images, Cbil, CHbO, CHbR, CHbT, StO2, and Cm after bile duct ligation. The scale bar in each color image is 10 mm.
Fig. 5.
Fig. 5. Typical time courses of transcutaneous bilirubin concentration Cbil estimated by the proposed method and those measured by a commercially available bilirubinometer (JM-105) after bile duct ligation.
Fig. 6.
Fig. 6. Comparison between the transcutaneous bilirubin concentration Cbil estimated by the proposed method and those measured by a commercially available bilirubinometer (JM-105) after bile duct ligation (n = 6).
Fig. 7.
Fig. 7. Histograms of estimated values for (a) the concentration of melanin Cm, (b) the concentration of total hemoglobin CHbT, and (c) the tissue oxygen saturation StO2, obtained from six samples during bile duct ligation (n = 6).
Fig. 8.
Fig. 8. Typical sequential images obtained from rat dorsal skin including RGB color images, CHbO, CHbR, CHbT, StO2, Cm, and Cbil during changes in FiO2. The scale bar in each color image is 5 mm.
Fig. 9.
Fig. 9. Time courses of average chromophore concentration and tissue oxygen saturation averaged over the ROI for (a) oxygenated hemoglobin CHbO, (b) deoxygenated hemoglobin CHbR, (c) total hemoglobin CHbT, (d) tissue oxygen saturation StO2, (e) melanin Cm, and (f) bilirubin Cbil. Data are expressed as mean ± SD (n = 11).
Fig. 10.
Fig. 10. Typical time courses of CHbT and the extracted PPG signal averaged over the region of interest on in vivo rat scalp under the normoxic condition (FiO2=21%).
Fig. 11.
Fig. 11. Comparison between the estimated HR obtained by the proposed method and the ground truth HR measured by the commercially available pulse oximeter. (n = 4)
Fig. 12.
Fig. 12. Typical PPG signals of CHbO and CHbR obtained from the scalp of rat under the condition of (a) FiO2=95%, (b) FiO2=21%, (c) FiO2=14%, and (d) FiO2=6%.
Fig. 13.
Fig. 13. Dependence of SpO2 measured by the pulse oximeter on the value of ϕ obtained from the proposed method.
Fig. 14.
Fig. 14. Typical resultant images for (a) raw RGB color, (b) SpO2, (c) StO2, (d) Cm, and (e) Cbil obtained from the scalp of rat while changing FiO2. The scale bar in each color image is 3 mm.
Fig. 15.
Fig. 15. Comparison between the estimated melanin concentration obtained by the proposed method and the ground truth melanin concentration.

Tables (1)

Tables Icon

Table 1. Summary of expected impacts of changes in light scattering properties and epidermal thickness on the estimated chromophore]e concentration.

Equations (15)

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[ R G B ] = L 1 [ X Y Z ] ,
X = κ E ( λ ) x ¯ ( λ ) Θ ( λ ) ,
Y = κ E ( λ ) y ¯ ( λ ) Θ ( λ ) ,
Z = κ E ( λ ) z ¯ ( λ ) Θ ( λ ) ,
κ = 100 E ( λ ) y ¯ ( λ ) .
Θ = I I 0 = [ 0 P e ( μ s , e , l e ) exp ( μ a , m l e ) d l e ] × [ 0 P d ( μ s . d , l d ) exp [ ( μ a , HbO + μ a , HbR + μ a , b i l ) l d ] d l d ] ,
μ a = C × ε .
[ X Y Z ] = N 1 [ R G B ] .
C m = α 0 + α 1 X + α 2 Y + α 3 Z ,
C HbO = β 0 + β 1 X + β 2 Y + β 3 Z ,
C HbO = γ 0 + γ 1 X + γ 2 Y + γ 3 Z ,
C bil = ω 0 + ω 1 X + ω 2 Y + ω 3 Z ,
N 2 = [ α 0 α 1 α 2 α 3 β 0 β 1 β 2 β 3 γ 0 γ 1 γ 2 γ 3 ω 0 ω 1 ω 2 ω 3 ] .
[ C m C HbO C HbR C bil ] = N 2 [ 1 X Y Z ] .
S p O 2 ( ϕ ) = 100 A 1 + exp ( ϕ B C ) .

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