Abstract

Functional near-infrared spectroscopy (fNIRS) can non-invasively measure hemodynamic responses in the cerebral cortex with a portable apparatus. However, the observation signal in fNIRS measurements is contaminated by the artifact signal from the hemodynamic response in the scalp. In this paper, we propose a method to separate the signals from the cortex and the scalp by estimating both hemodynamic changes by diffuse optical tomography (DOT). In the inverse problem of DOT, we introduce smooth regularization to the hemodynamic change in the scalp and sparse regularization to that in the cortex based on the nature of the hemodynamic responses. These appropriate regularization models, with the spatial information of optical paths of many measurement channels, allow three-dimensional reconstruction of both hemodynamic changes. We validate our proposed method through two-layer phantom experiments and MRI-based head-model simulations. In both experiments, the proposed method simultaneously estimates the superficial smooth activity in the scalp area and the deep localized activity in the cortical area.

© 2013 Optical Society of America

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2012 (6)

M. Ferrari, V. Quaresima, “A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application,” NeuroImage 63, 921–935 (2012).
[CrossRef] [PubMed]

E. Kirilina, A. Jelzow, A Heine, M. Niessing, H. Wabnitz, R. Brühl, B. Ittermann, A. M. Jacobs, I. Tachtsidis, “The physiological origin of task-evoked systemic artefacts in functional near infrared spectroscopy,” NeuroImage 61, 70–81 (2012).
[CrossRef] [PubMed]

L. Gagnon, R. J. Cooper, M. A. Yücel, K. L. Perdue, D. N. Greve, D. A. Boas, “Short separation channel location impacts the performance of short channel regression in NIRS,” NeuroImage 59, 2518–2528 (2012).
[CrossRef]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61, 1120–1128 (2012).
[CrossRef] [PubMed]

V. C. Kavuri, Z. J. Lin, F. Tian, H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3, 943–957 (2012).
[CrossRef] [PubMed]

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, M. Sato, “Hierarchical Bayesian estimation improves depth accuracy and spatial resolution of diffuse optical tomography,” Opt. Express 20, 20427–20446 (2012).
[CrossRef] [PubMed]

2011 (3)

C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage 59, 3201–3211 (2011).
[CrossRef] [PubMed]

R. B. Saager, N. L. Telleri, A. J. Berger, “Two-detector Corrected Near Infrared Spectroscopy (C-NIRS) detects hemodynamic activation responses more robustly than single-detector NIRS,” NeuroImage 55, 1679–1685 (2011).
[CrossRef] [PubMed]

T. Takahashi, Y. Takikawa, R. Kawagoe, S. Shibuya, T. Iwano, S. Kitazawa, “Influence of skin blood flow on near-infrared spectroscopy signals measured on the forehead during a verbal fluency task,” NeuroImage 57, 991–1002 (2011).
[CrossRef] [PubMed]

2010 (2)

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenergetics 2, 1–8 (2010).

Q. Fang, “Mesh-based Monte Carlo method using fast ray-tracing in Plücker coordinates,” Biomed. Opt. Express 1, 165–175 (2010).
[CrossRef] [PubMed]

2009 (3)

Q. Zhang, G. E. Strangman, G. Ganis, “Adaptive filtering to reduce global interference in non-invasive NIRS measures of brain activation: how well and when does it work?” NeuroImage 45, 788–794 (2009).
[CrossRef] [PubMed]

T. Yamada, S. Umeyama, K. Matsuda, “Multidistance probe arrangement to eliminate artifacts in functional near-infrared spectroscopy,” J. Biomed. Opt. 14, 064034 (2009).
[CrossRef]

Q. Fang, D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[CrossRef] [PubMed]

2007 (3)

D. Tsuzuki, V. Jurcak, A. K. Singh, M. Okamoto, E. Watanabe, I. Dan, “Virtual spatial registration of stand-alone fNIRS data to MNI space,” NeuroImage 34, 1506–1518 (2007).
[CrossRef] [PubMed]

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12, 062111 (2007).
[CrossRef]

Q. Zhang, E. N. Brown, G. E. Strangman, “Adaptive filtering for global interference cancellation and real-time recovery of evoked brain activity: a Monte Carlo simulation study,” J. Biomed. Opt. 12, 044014 (2007).
[CrossRef] [PubMed]

2005 (3)

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

Y. Zhang, D. H. Brooks, M. A. Franceschini, D. A. Boas, “Eigenvector-based spatial filtering for reduction of physiological interference in diffuse optical imaging,” J. Biomed. Opt. 10, 011014 (2005).
[CrossRef]

D. A. Boas, A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. 44, 1957–1968 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage 23, 806–826 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

K. J. Worsley, C. H. Liao, J. Aston, V. Petre, G. H. Duncan, F. Morales, A. C. Evans, “A general statistical analysis for fMRI data,” NeuroImage 15, 1–15 (2002).
[CrossRef] [PubMed]

2001 (1)

M. Sato, “Online model selection based on the variational Bayes,” Neural Comput. 13, 1649–1681 (2001).
[CrossRef]

1999 (2)

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

C. M. Bishop, “Variational principal components,” In Proc. of ICANN 1, 509–514 (1999).

1998 (1)

1997 (1)

N. Lange, S. L. Zeger, “Non-linear Fourier time series analysis for human brain mapping by functional magnetic resonance imaging,” Appl. Statist. 46, 1–29 (1997).
[CrossRef]

1995 (1)

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995).
[CrossRef] [PubMed]

1993 (4)

Y. Hoshi, M. Tamura, “Detection of dynamic changes in cerebral oxygenation coupled to neuronal function during mental work in man,” Neurosci. Lett. 150, 5–8 (1993).
[CrossRef] [PubMed]

T. Kato, A. Kamei, S. Takashima, T. Ozaki, “Human visual cortical function during photic stimulation monitoring by means of near-infrared spectroscopy,” J. Cereb. Blood Flow Metab. 13, 516–520 (1993).
[CrossRef] [PubMed]

A. Villringer, J. Planck, C. Hock, L. Schleinkofer, U. Dirnagl, “Near infrared spectroscopy (NIRS): a new tool to study hemodynamic changes during activation of brain function in human adults,” Neurosci. Lett. 154, 101–104 (1993).
[CrossRef] [PubMed]

B. Chance, Z. Zhuang, C. UnAh, C. Alter, L. Lipton, “Cognition-activated low-frequency modulation of light absorption in human brain,” Proc. Natl. Acad. Sci. U.S.A. 90, 3770–3774 (1993).
[CrossRef] [PubMed]

1977 (1)

F. F. Jöbsis, “Noninvasive, infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters,” Science 198, 1264–1267 (1977).
[CrossRef] [PubMed]

Akaike, H.

H. Akaike, “Likelihood and the Bayes procedure,” in Bayesian Statistics,J. M. Bernardo, M. H. De Groot, D. V. Lindley, A. F. M. Smith, eds. (Univ. Press, Valencia, 1980), 143–166.

Alter, C.

B. Chance, Z. Zhuang, C. UnAh, C. Alter, L. Lipton, “Cognition-activated low-frequency modulation of light absorption in human brain,” Proc. Natl. Acad. Sci. U.S.A. 90, 3770–3774 (1993).
[CrossRef] [PubMed]

Amita, T.

T. Shimokawa, T. Kosaka, O. Yamashita, N. Hiroe, T. Amita, Y. Inoue, M. Sato, “Hierarchical Bayesian estimation improves depth accuracy and spatial resolution of diffuse optical tomography,” Opt. Express 20, 20427–20446 (2012).
[CrossRef] [PubMed]

S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12, 062111 (2007).
[CrossRef]

Arridge, S. R.

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

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

Aston, J.

K. J. Worsley, C. H. Liao, J. Aston, V. Petre, G. H. Duncan, F. Morales, A. C. Evans, “A general statistical analysis for fMRI data,” NeuroImage 15, 1–15 (2002).
[CrossRef] [PubMed]

Atsumori, H.

T. Funane, H. Atsumori, T. Katura, A. N. Obata, H. Sato, Y. Tanikawa, E. Okada, M. Kiguchi, “Quantitative evaluation of deep and shallow tissue layers’ contribution to fNIRS signal using multi-distance optodes and independent component analysis,” NeuroImage (in press).

Attias, H.

H. Attias, “Inferring parameters and structure of latent variable models by variational Bayes,” Proc. 15th Conf. on Uncertainty in Artificial Intelligence, Morgan Kaufmann, 21–30 (1999).

Bays, R.

Berger, A. J.

R. B. Saager, N. L. Telleri, A. J. Berger, “Two-detector Corrected Near Infrared Spectroscopy (C-NIRS) detects hemodynamic activation responses more robustly than single-detector NIRS,” NeuroImage 55, 1679–1685 (2011).
[CrossRef] [PubMed]

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenergetics 2, 1–8 (2010).

Bishop, C. M.

C. M. Bishop, “Variational principal components,” In Proc. of ICANN 1, 509–514 (1999).

Boas, D. A.

L. Gagnon, R. J. Cooper, M. A. Yücel, K. L. Perdue, D. N. Greve, D. A. Boas, “Short separation channel location impacts the performance of short channel regression in NIRS,” NeuroImage 59, 2518–2528 (2012).
[CrossRef]

Q. Fang, D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[CrossRef] [PubMed]

D. A. Boas, A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt. 44, 1957–1968 (2005).
[CrossRef] [PubMed]

Y. Zhang, D. H. Brooks, M. A. Franceschini, D. A. Boas, “Eigenvector-based spatial filtering for reduction of physiological interference in diffuse optical imaging,” J. Biomed. Opt. 10, 011014 (2005).
[CrossRef]

L. Gagnon, M. A. Yücel, D. A. Boas, R. J. Cooper, “Further improvement in reducing superficial contamination in NIRS using double short separation measurements,” NeuroImage (in press).

Brooks, D. H.

Y. Zhang, D. H. Brooks, M. A. Franceschini, D. A. Boas, “Eigenvector-based spatial filtering for reduction of physiological interference in diffuse optical imaging,” J. Biomed. Opt. 10, 011014 (2005).
[CrossRef]

Brown, E. N.

Q. Zhang, E. N. Brown, G. E. Strangman, “Adaptive filtering for global interference cancellation and real-time recovery of evoked brain activity: a Monte Carlo simulation study,” J. Biomed. Opt. 12, 044014 (2007).
[CrossRef] [PubMed]

Brühl, R.

E. Kirilina, A. Jelzow, A Heine, M. Niessing, H. Wabnitz, R. Brühl, B. Ittermann, A. M. Jacobs, I. Tachtsidis, “The physiological origin of task-evoked systemic artefacts in functional near infrared spectroscopy,” NeuroImage 61, 70–81 (2012).
[CrossRef] [PubMed]

Chance, B.

B. Chance, Z. Zhuang, C. UnAh, C. Alter, L. Lipton, “Cognition-activated low-frequency modulation of light absorption in human brain,” Proc. Natl. Acad. Sci. U.S.A. 90, 3770–3774 (1993).
[CrossRef] [PubMed]

Chen, C.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61, 1120–1128 (2012).
[CrossRef] [PubMed]

Cooper, C. E.

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995).
[CrossRef] [PubMed]

Cooper, R. J.

L. Gagnon, R. J. Cooper, M. A. Yücel, K. L. Perdue, D. N. Greve, D. A. Boas, “Short separation channel location impacts the performance of short channel regression in NIRS,” NeuroImage 59, 2518–2528 (2012).
[CrossRef]

L. Gagnon, M. A. Yücel, D. A. Boas, R. J. Cooper, “Further improvement in reducing superficial contamination in NIRS using double short separation measurements,” NeuroImage (in press).

Cope, M.

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995).
[CrossRef] [PubMed]

Culver, J. P.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61, 1120–1128 (2012).
[CrossRef] [PubMed]

N. M. Gregg, B. R. White, B. W. Zeff, A. J. Berger, J. P. Culver, “Brain specificity of diffuse optical imaging: improvements from superficial signal regression and tomography,” Front. Neuroenergetics 2, 1–8 (2010).

Dale, A. M.

Dan, I.

D. Tsuzuki, V. Jurcak, A. K. Singh, M. Okamoto, E. Watanabe, I. Dan, “Virtual spatial registration of stand-alone fNIRS data to MNI space,” NeuroImage 34, 1506–1518 (2007).
[CrossRef] [PubMed]

Dehghani, H.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61, 1120–1128 (2012).
[CrossRef] [PubMed]

Delpy, D. T.

E. Okada, D. T. Delpy, “Near-infrared light propagation in an adult head model. I. Modeling of low-level scattering in the cerebrospinal fluid layer,” Appl. Opt. 42, 2906–2914 (2003).
[CrossRef] [PubMed]

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995).
[CrossRef] [PubMed]

Dirnagl, U.

A. Villringer, J. Planck, C. Hock, L. Schleinkofer, U. Dirnagl, “Near infrared spectroscopy (NIRS): a new tool to study hemodynamic changes during activation of brain function in human adults,” Neurosci. Lett. 154, 101–104 (1993).
[CrossRef] [PubMed]

Dögnitz, N.

Doya, K.

M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage 23, 806–826 (2004).
[CrossRef] [PubMed]

Duncan, G. H.

K. J. Worsley, C. H. Liao, J. Aston, V. Petre, G. H. Duncan, F. Morales, A. C. Evans, “A general statistical analysis for fMRI data,” NeuroImage 15, 1–15 (2002).
[CrossRef] [PubMed]

Eggebrecht, A. T.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. Chen, Y. Zhan, A. Z. Snyder, H. Dehghani, J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fMRI cortical mapping,” NeuroImage 61, 1120–1128 (2012).
[CrossRef] [PubMed]

Elwell, C. E.

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope, D. T. Delpy, “Performance comparison of several published tissue near-infrared spectroscopy algorithms,” Anal. Biochem. 227, 54–68 (1995).
[CrossRef] [PubMed]

Evans, A. C.

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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage 59, 3201–3211 (2011).
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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage 59, 3201–3211 (2011).
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S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12, 062111 (2007).
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T. Takahashi, Y. Takikawa, R. Kawagoe, S. Shibuya, T. Iwano, S. Kitazawa, “Influence of skin blood flow on near-infrared spectroscopy signals measured on the forehead during a verbal fluency task,” NeuroImage 57, 991–1002 (2011).
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S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12, 062111 (2007).
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D. Tsuzuki, V. Jurcak, A. K. Singh, M. Okamoto, E. Watanabe, I. Dan, “Virtual spatial registration of stand-alone fNIRS data to MNI space,” NeuroImage 34, 1506–1518 (2007).
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A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage 59, 3201–3211 (2011).
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Q. Zhang, G. E. Strangman, G. Ganis, “Adaptive filtering to reduce global interference in non-invasive NIRS measures of brain activation: how well and when does it work?” NeuroImage 45, 788–794 (2009).
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Y. Hoshi, M. Tamura, “Detection of dynamic changes in cerebral oxygenation coupled to neuronal function during mental work in man,” Neurosci. Lett. 150, 5–8 (1993).
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M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage 23, 806–826 (2004).
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S. Kohno, I. Miyai, A. Seiyama, I. Oda, A. Ishikawa, S. Tsuneishi, T. Amita, K. Shimizu, “Removal of the skin blood flow artifact in functional near-infrared spectroscopic imaging data through independent component analysis,” J. Biomed. Opt. 12, 062111 (2007).
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T. Yamada, S. Umeyama, K. Matsuda, “Multidistance probe arrangement to eliminate artifacts in functional near-infrared spectroscopy,” J. Biomed. Opt. 14, 064034 (2009).
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D. Tsuzuki, V. Jurcak, A. K. Singh, M. Okamoto, E. Watanabe, I. Dan, “Virtual spatial registration of stand-alone fNIRS data to MNI space,” NeuroImage 34, 1506–1518 (2007).
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K. J. Worsley, C. H. Liao, J. Aston, V. Petre, G. H. Duncan, F. Morales, A. C. Evans, “A general statistical analysis for fMRI data,” NeuroImage 15, 1–15 (2002).
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T. Yamada, S. Umeyama, K. Matsuda, “Multidistance probe arrangement to eliminate artifacts in functional near-infrared spectroscopy,” J. Biomed. Opt. 14, 064034 (2009).
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M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage 23, 806–826 (2004).
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L. Gagnon, R. J. Cooper, M. A. Yücel, K. L. Perdue, D. N. Greve, D. A. Boas, “Short separation channel location impacts the performance of short channel regression in NIRS,” NeuroImage 59, 2518–2528 (2012).
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Figures (11)

Fig. 1
Fig. 1

Setup of two-layer phantom experiment. (a) Schematic of phantom. The liquid filling the tank represents the cortical area. Silicone plate on the tank’s bottom represents the scalp area. Measurements were done in rest and task conditions. In the task condition, spherical absorbers were inserted in the liquid. The absorption coefficient of the silicone plate in the task condition was slightly higher than that in the rest condition. (b) Arrangement of source and detector probes on bottom of tank. Red circles represent the source, and blue squares represent the detector. x-y positions are superposed where the spherical absorbers were settled in measurements. One absorber was settled at (i) (0, 0, d) in the one-absorber experiment. Two absorbers were settled at (ii) ( l 2 , 0, d) and ( l 2 + h , 0, d) in the two-absorber experiment. Here, d denotes the depth or the vertical distance between the observation surface and the center of the absorber, h denotes the horizontal distance between the centers of the two absorbers, and l denotes the probe interval.

Fig. 2
Fig. 2

Example of 3D reconstruction image. Top 5-mm-thick layer is the silicone layer representing the scalp, and the remaining region deeper than 5 mm is the liquid layer representing the cortex. Three-dimensional image of the absorption change (tomography) is represented in gray scale, and the x-y maximum intensity projection (topography) of the absorption change in the liquid layer is represented in green scale on the z = 30 surface. (a) True distribution of absorption change. Positions of centers of two absorbers are (−9.2, 0, 15) [mm] and (5.8, 0, 15) [mm]. Red cylinders represent the source, and blue square prisms represent the detector. (b) Solution of sensitivity-normalized Tikhonov regularization. (c) Solution of hierarchical Bayesian estimation method when only using the sparse regularization model. (d) Solution of hierarchical Bayesian estimation method using both the sparse and smooth regularization models.

Fig. 3
Fig. 3

Estimation results of one-absorber experiment. (a) Estimation results under various depths and probe-interval conditions. From left, results under depth conditions of the spherical absorption change: d = 10 mm, 12.5 mm, 15 mm, 17.5 mm, and 20 mm. From top, results under two probe-interval conditions: l = 18.4 mm, 13 mm. White circles represent the true position of the 5-mm-diameter spherical absorption change in the liquid layer. Estimation result is surrounded by a bold frame if the positional error of the spherical absorption change was within one voxel (2.5 mm) in each x, y, z direction and the maximum estimation value exceeded 0.025 mm−1. (b) Projection region is the central 7.5-mm-thick layer containing the true distribution of the spherical absorption change. Estimation images of (a) were obtained by projecting the maximum estimation value within the projection region in the y direction onto the x-z space. (c) Estimated depth of spherical absorption changes as a function of the true depth.

Fig. 4
Fig. 4

Estimation results for 18.4-mm probe-interval condition. From left, results under four conditions of horizontal distance: h = 17.5, 15, 12.5, and 10 mm. From top, results under three depth conditions: d = 12.5, 15, and 17.5 mm. White circles represent the true positions of the spherical absorption change. Estimation image is surrounded by a bold frame if the positional errors of both spherical absorbers were within one voxel (2.5 mm) in each x, y, z direction and both maximum estimation values exceeded 0.025 mm−1.

Fig. 5
Fig. 5

Arrangement of source and detector probes. Red points (e.g., labeled ‘T1’) represent the source probe, and the blue points (e.g., labeled ‘R1’) represent the detector probe. Region considered for DOT reconstruction image is illustrated as a cuboid.

Fig. 6
Fig. 6

Example of estimation. (a) (top) Spatial distribution of the true oxygenated hemoglobin concentration change. Center position of the local hemoglobin concentration change in the cortex was (x, y, z) = (21, 41, 19) [mm]. Cross-section view of y = 41 plane is shown. (middle) Red solid and blue dashed lines represent time courses of true ΔHbO2 and ΔHbR, respectively, of the peak value’s voxel in the cortex. (bottom) Time course of true value of ΔHbO2 and ΔHbR in the peak value’s voxel in the scalp. (b) (top) Observation value of all channels. Observation value is the linear summation of the cortical signal, scalp signal, and observation noise. (middle) Cortical signal resulting from the hemoglobin concentration changes in cortical area. (bottom) Scalp signal resulting from hemoglobin concentration changes in scalp area. (c) (top) Spatial distribution of estimated oxygenated hemoglobin concentration change. (middle) Red solid and blue dashed lines represent time courses of estimated ΔHbO2 and ΔHbR, respectively, of the peak value’s voxel in cortex. (bottom) Time course of estimated value of ΔHbO2 and ΔHbR in the peak value’s voxel in scalp.

Fig. 7
Fig. 7

Localization error map. A point’s color in the gray matter area represents the value of the localization error. The point’s position represents the center position of the true hemoglobin concentration change. Red and blue points represent the positions of the source and detector probes, respectively.

Fig. 8
Fig. 8

Estimation errors as functions of true depth of cortical hemoglobin concentration change. Red x and blue + marks represent the error values of ΔHbO2 and ΔHbR, respectively, of the 84 positions shown in Fig. 7. Red solid and blue dashed lines represent average values of x and + marks, respectively. These average values were calculated for every 1-mm range of true depth. (a) Localization error of cortical hemoglobin concentration change. (b) Depth of peak voxel of estimated cortical hemoglobin concentration change. (c) Ratio of estimated peak value to true peak value of cortical hemoglobin concentration change. (d) Root mean squared error of hemoglobin concentration change in the scalp.

Fig. 9
Fig. 9

Estimation errors with three conditions of signal-to-noise ratio. Dotted, solid, and dashed lines represent estimation errors of ΔHbO2 with observation noise levels of half, standard, and double strength, respectively, compared to data shown in Fig. 8. (a) Localization error of cortical hemoglobin concentration change. (b) Ratio of estimated peak value to true peak value of cortical hemoglobin concentration change.

Fig. 10
Fig. 10

Comparison of regression and proposed methods for scalp artifact removal. Each line represents localization error of ΔHbO2 in the cortex. Dotted line represents the estimation error without scalp artifact removal, where only the cortical hemodynamic change was estimated by the hierarchical Bayesian method. Dashed line represents the estimation error with scalp artifact removal by the regression method, where the cortical hemodynamic change was estimated after removing scalp signals by the regression method. Solid line represents the estimation error obtained by the proposed method, where both the cortical and scalp hemodynamic changes were estimated for scalp artifact removal.

Fig. 11
Fig. 11

Estimation errors when there was a discrepancy between the data generation model and the estimation model. Dotted and dashed lines represent the cases where there was discrepancy in the optical parameter and the anatomical structure, respectively. The solid lines represent the case where there was no discrepancy. (a) Localization error of ΔHbO2 in the cortex. (b) Estimated depth of ΔHbO2 in the cortex as a function of the true depth.

Equations (29)

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y = A c x c + A s x s + ξ ,
P ( Y | X c , X s , σ ) = t = 1 T k = 1 K N ( y k , t ; A k c x k , t c + A k s x k , t s , σ 1 Σ y , k ) ,
P ( σ ) σ 1 .
P ( Z c | λ ) = t = 1 T k = 1 K N ( z k , t c ; 0 , Λ 1 ) ,
P ( X c | λ ) = t = 1 T k = 1 K N ( x k , t c ; 0 , W Λ 1 W T ) .
P ( λ ) = i = 1 N c Γ ( λ i ; λ ¯ 0 i , γ 0 i ) ,
P ( Z s | η ) = t = 1 T k = 1 K N ( z k , t s ; 0 , ( η I N s ) 1 ) ,
P ( X s | η ) = t = 1 T k = 1 K N ( x k , t s ; 0 , L 1 η 1 L 1 T ) .
P ( η ) = Γ ( η ; η ¯ 0 , γ η 0 ) ,
P ( X c , X s , λ , η , σ | Y ) = P ( Y , X c , X s , λ , η , σ ) P ( Y ) ,
F ( Q ) = d X c d X s d λ d η d σ Q ( X c , X s , λ , η , σ ) log P ( Y , X c , X s , λ , η , σ ) Q ( X c , X s , λ , η , σ ) , = log P ( Y ) KL [ Q ( X c , X s , λ , η , σ ) | | P ( X c , X s , λ , η , σ | Y ) ] ,
( Δ μ a , 1 Δ μ a , K ) = ( ε 1 HbO 2 ε 1 HbR ε K HbO 2 ε K HbR ) ( Δ HbO 2 Δ HbR ) E ( Δ HbO 2 Δ HbR ) ,
( Δ HbO 2 Δ HbR ) = ( E T E ) 1 E T ( Δ μ a , 1 Δ μ a , K ) .
Q ( X , λ , η , σ ) = Q X ( X ) Q λ ( λ , η , σ ) ,
log Q X ( X ) = log P ( Y , X , λ , η , σ ) Q λ ( λ , η , σ ) = Σ t = 1 T Σ k = 1 K log N ( x k , t ; x ¯ k , t , Σ x , k ) ,
log Q λ ( λ , η , σ ) = log P ( Y , X , λ , η , σ ) Q X ( X ) = Σ i = 1 N c log Γ ( λ i ; λ ¯ i , γ i ) + log Γ ( η ; η ¯ , γ η ) + log Γ ( σ ; σ ¯ , γ σ ) ,
F ( Q ) = 1 2 Σ t = 1 T Σ k = 1 K [ σ ¯ Σ y , k 1 2 ( y k , t A k x ¯ k , t ) 2 2 + Λ ¯ 1 2 W 1 x ¯ k , t c 2 2 + η ¯ L x ¯ k , t s 2 2 ] + Σ i = 1 N c γ 0 i [ log ( λ ¯ i / λ ¯ 0 i ) ( λ ¯ i / λ ¯ 0 i ) + 1 ] + γ η 0 [ log ( η ¯ / η ¯ 0 ) ( η ¯ / η ¯ 0 ) + 1 ] + Σ 2 T Σ k = 1 K [ log | Σ x , k | + log | Λ ¯ | + N s log η ¯ + log | σ ¯ Σ y , k 1 | ] T 2 Σ k = 1 K tr [ Σ x , k ( σ ¯ A k T Σ y , k 1 A k + ( W 1 T Λ ¯ W 1 0 0 L T η ¯ L ) ) ] + const . ,
C D ( X ; λ ) = Σ t = 1 T Σ k = 1 K [ Σ y , k 1 2 ( y k , t A k x k , t ) 2 2 + λ D k 1 2 x k , t 2 2 ] ,
D k = diag ( ρ k + β 1 N ) ,
ρ k , i = ( A k T Σ y , k 1 A k ) i i .
Σ k : = σ ¯ 1 Σ y , k + A k c W Λ ¯ 1 W T A k c T + A k s L 1 η ¯ 1 L 1 T A k s T ,
h c : = Σ k = 1 K diag [ Λ ¯ 1 W T A k c T Σ k 1 Y k Y k T Σ k 1 A k c W + T ( 1 N c Λ ¯ 1 W T A k c T Σ k 1 A k c W ) ] ,
h s : = Σ k = 1 K tr [ η ¯ 1 L 1 T A k s T Σ k 1 Y k Y k T Σ k 1 A k s L 1 + T ( I N s η ¯ 1 L 1 T A k s T Σ k 1 A k s L 1 ) ] ,
S : = Σ k = 1 K tr [ σ ¯ 1 Σ y , k Σ k 1 Y k Y k T Σ k 1 + T ( Λ ¯ 1 W T A k c T Σ k 1 A k c W + η ¯ 1 L 1 T A k s T Σ k 1 A k s L 1 ) ] .
γ i : = γ 0 i + K 2 T , γ i λ ¯ i 1 : = γ 0 i λ ¯ 0 i 1 + 1 2 h i c λ ¯ i 1 ,
γ η : = γ η 0 + K 2 N s T , γ η η ¯ 1 : = γ η 0 η ¯ 0 1 + 1 2 h s η ¯ 1 ,
γ σ : = 1 2 M T , γ σ σ ¯ 1 : = 1 2 S σ ¯ 1 .
X ^ k c = W Λ ¯ 1 W T A k c T Σ k 1 Y k ,
X ^ k s = L 1 η 1 L 1 T A k s T Σ k 1 Y k .

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