Abstract

3D-polarized light imaging (3D-PLI) reconstructs nerve fibers in histological brain sections by measuring their birefringence. This study investigates another effect caused by the optical anisotropy of brain tissue – diattenuation. Based on numerical and experimental studies and a complete analytical description of the optical system, the diattenuation was determined to be below 4 % in rat brain tissue. It was demonstrated that the diattenuation effect has negligible impact on the fiber orientations derived by 3D-PLI. The diattenuation signal, however, was found to highlight different anatomical structures that cannot be distinguished with current imaging techniques, which makes Diattenuation Imaging a promising extension to 3D-PLI.

© 2017 Optical Society of America

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2016 (1)

2015 (4)

M. Menzel, K. Michielsen, H. De Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc. Interface 12, 20150734 (2015).
[Crossref] [PubMed]

M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourg, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2015).
[Crossref]

J. Reckfort, H. Wiese, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, “A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3D-PLI,” Front. Neuroanat. 9(118), 1–11 (2015).
[Crossref]

M. Dohmen, M. Menzel, H. Wiese, J. Reckfort, F. Hanke, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, ‘Understanding fiber mixture by simulation in 3D Polarized Light Imaging,” NeuroImage 111, 464–475 (2015).
[Crossref] [PubMed]

2014 (2)

H. Wiese, D. Grässel, U. Pietrzyk, K. Amunts, and M. Axer, “Polarized Light Imaging of the human brain – a new approach to the data analysis of tilted sections,” Proc. SPIE 9099, 90990 (2014).
[Crossref]

W. Wang, L.G. Lim, S. Srivastava, J. S. Yan, A. Shabbir, and Q. Liu, “Roles of linear and circular polarization properties and effect of wavelength choice on differentiation between ex vivo normal and cancerous gastric samples,” J. Biomed. Opt. 19(4) 046020 (2014).
[Crossref] [PubMed]

2013 (4)

2012 (1)

J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
[Crossref]

2011 (5)

H. Axer, S. Beck, M. Axer, F. Schuchardt, J. Heepe, A. Flücken, M. Axer, A. Prescher, and O. W. Witte, “Microstructural analysis of human white matter architecture using polarized light imaging: views from neuroanatomy,” Front. Neuroinform. 5(28), 1–12 (2011).
[Crossref]

N. Ghosh and I.A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
[Crossref]

M. Axer, D. Grässel, M. Kleiner, J. Dammers, T. Dickscheid, J. Reckfort, T. Hütz, B. Eiben, U. Pietrzyk, K. Zilles, and K. Amunts, “High-resolution fiber tract reconstruction in the human brain by means of three-dimensional polarized light imaging,” Front. Neuroinform. 5(34), 1–13 (2011).
[Crossref]

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. 30(6), 431–451 (2011).
[Crossref] [PubMed]

2010 (3)

N. Ghosh, J. Soni, M. F. G. Wood, M. A. Wallenberg, and I. A. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana – J. Phys. 75(6), 1071–1086 (2010).
[Crossref]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18(2), 854–876 (2010).
[Crossref] [PubMed]

S. Klein, M. Staring, K. Murphy, M. A. Viergever, and J. P. W. Pluim, “elastix: A toolbox for intensity-based medical image registration,” IEEE Trans. Med. Imaging 29(1), 196–205 (2010).
[Crossref]

2009 (1)

2008 (1)

N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

2007 (1)

L. Larsen, L. D. Griffin, D. Grässel, O. W. Witte, and H. Axer, “Polarized light imaging of white matter architecture,” Microsc. Res. Tech. 70(10), 851–863 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (1)

2004 (2)

2003 (3)

2002 (1)

S. Jiao and L. V. Wang, “Quantification of polarization in biological tissue by optical coherence tomography,” Proc. SPIE 4617, 200–203 (2002).
[Crossref]

2001 (1)

M. H. Smith, “Interpreting Mueller matrix images of tissues,” Proc. SPIE 4257, 82–89 (2001).
[Crossref]

1997 (1)

R. Storn and K. Price, “Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[Crossref]

1996 (2)

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, and G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41(3), 369–382 (1996).
[Crossref] [PubMed]

S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996).
[Crossref]

1993 (1)

1992 (1)

1989 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Engin. 28(2), 280290 (1989).

1980 (1)

B. de Campos Vidal, M. L. Silveira Mello, A. C. Caiseiro-Filho, and C. Godo, “Anisotropic properties of the myelin sheath,” Acta Histochem. 66, 32–39 (1980).
[Crossref] [PubMed]

1939 (1)

F. O. Schmitt and R. S. Bear, “The ultrastructure of the nerve axon sheath,” Biol. Rev. Camb. Philos. Soc. 14, 27–50 (1939).
[Crossref]

1913 (1)

G. F. Göthlin, “Die doppelbrechenden Eigenschaften des Nervengewebes – ihre Ursachen und ihre biologischen Konsequenzen,” Kungl. Svenska Vet. Akad. Handlingar. 51(1), 1–91 (1913).

Allé, P.

Amunts, K.

M. Menzel, K. Michielsen, H. De Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc. Interface 12, 20150734 (2015).
[Crossref] [PubMed]

J. Reckfort, H. Wiese, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, “A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3D-PLI,” Front. Neuroanat. 9(118), 1–11 (2015).
[Crossref]

M. Dohmen, M. Menzel, H. Wiese, J. Reckfort, F. Hanke, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, ‘Understanding fiber mixture by simulation in 3D Polarized Light Imaging,” NeuroImage 111, 464–475 (2015).
[Crossref] [PubMed]

H. Wiese, D. Grässel, U. Pietrzyk, K. Amunts, and M. Axer, “Polarized Light Imaging of the human brain – a new approach to the data analysis of tilted sections,” Proc. SPIE 9099, 90990 (2014).
[Crossref]

J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
[Crossref]

M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
[Crossref]

M. Axer, D. Grässel, M. Kleiner, J. Dammers, T. Dickscheid, J. Reckfort, T. Hütz, B. Eiben, U. Pietrzyk, K. Zilles, and K. Amunts, “High-resolution fiber tract reconstruction in the human brain by means of three-dimensional polarized light imaging,” Front. Neuroinform. 5(34), 1–13 (2011).
[Crossref]

Axer, H.

M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
[Crossref]

H. Axer, S. Beck, M. Axer, F. Schuchardt, J. Heepe, A. Flücken, M. Axer, A. Prescher, and O. W. Witte, “Microstructural analysis of human white matter architecture using polarized light imaging: views from neuroanatomy,” Front. Neuroinform. 5(28), 1–12 (2011).
[Crossref]

L. Larsen, L. D. Griffin, D. Grässel, O. W. Witte, and H. Axer, “Polarized light imaging of white matter architecture,” Microsc. Res. Tech. 70(10), 851–863 (2007).
[Crossref] [PubMed]

Axer, M.

J. Reckfort, H. Wiese, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, “A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3D-PLI,” Front. Neuroanat. 9(118), 1–11 (2015).
[Crossref]

M. Dohmen, M. Menzel, H. Wiese, J. Reckfort, F. Hanke, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, ‘Understanding fiber mixture by simulation in 3D Polarized Light Imaging,” NeuroImage 111, 464–475 (2015).
[Crossref] [PubMed]

M. Menzel, K. Michielsen, H. De Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc. Interface 12, 20150734 (2015).
[Crossref] [PubMed]

H. Wiese, D. Grässel, U. Pietrzyk, K. Amunts, and M. Axer, “Polarized Light Imaging of the human brain – a new approach to the data analysis of tilted sections,” Proc. SPIE 9099, 90990 (2014).
[Crossref]

J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
[Crossref]

H. Axer, S. Beck, M. Axer, F. Schuchardt, J. Heepe, A. Flücken, M. Axer, A. Prescher, and O. W. Witte, “Microstructural analysis of human white matter architecture using polarized light imaging: views from neuroanatomy,” Front. Neuroinform. 5(28), 1–12 (2011).
[Crossref]

H. Axer, S. Beck, M. Axer, F. Schuchardt, J. Heepe, A. Flücken, M. Axer, A. Prescher, and O. W. Witte, “Microstructural analysis of human white matter architecture using polarized light imaging: views from neuroanatomy,” Front. Neuroinform. 5(28), 1–12 (2011).
[Crossref]

M. Axer, D. Grässel, M. Kleiner, J. Dammers, T. Dickscheid, J. Reckfort, T. Hütz, B. Eiben, U. Pietrzyk, K. Zilles, and K. Amunts, “High-resolution fiber tract reconstruction in the human brain by means of three-dimensional polarized light imaging,” Front. Neuroinform. 5(34), 1–13 (2011).
[Crossref]

M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
[Crossref]

M. Menzel, M. Axer, H. De Raedt, and K. Michielsen, “Finite-Difference Time-Domain Simulation for Three-dimensional Polarized Light Imaging”, in Brain-Inspired Computing – Second International Workshop, BrainComp 2015, Cetraro, Italy, July 6–10, 2015, Revised Selected Papers, K. Amunts, L. Grandinetti, T. Lippert, and N. Petkov, eds. (Springer International Publishing, 2016), pp. 73–85.

Banerjee, C.

Bear, R. S.

F. O. Schmitt and R. S. Bear, “The ultrastructure of the nerve axon sheath,” Biol. Rev. Camb. Philos. Soc. 14, 27–50 (1939).
[Crossref]

Beck, S.

H. Axer, S. Beck, M. Axer, F. Schuchardt, J. Heepe, A. Flücken, M. Axer, A. Prescher, and O. W. Witte, “Microstructural analysis of human white matter architecture using polarized light imaging: views from neuroanatomy,” Front. Neuroinform. 5(28), 1–12 (2011).
[Crossref]

Benoit, A.-M.

Beuthan, J.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, and G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41(3), 369–382 (1996).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th Ed. (Cambridge University Press, 2011).

Breuer, L.

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M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourg, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2015).
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M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
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J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
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M. Axer, K. Amunts, D. Grässel, C. Palm, J. Dammers, H. Axer, U. Pietrzyk, and K. Zilles, “A novel approach to the human connectome: Ultra-high resolution mapping of fiber tracts in the brain,” NeuroImage 54(2), 1091–1101 (2011).
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M. Axer, D. Grässel, M. Kleiner, J. Dammers, T. Dickscheid, J. Reckfort, T. Hütz, B. Eiben, U. Pietrzyk, K. Zilles, and K. Amunts, “High-resolution fiber tract reconstruction in the human brain by means of three-dimensional polarized light imaging,” Front. Neuroinform. 5(34), 1–13 (2011).
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[Crossref]

M. Menzel, K. Michielsen, H. De Raedt, J. Reckfort, K. Amunts, and M. Axer, “A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue,” J. R. Soc. Interface 12, 20150734 (2015).
[Crossref] [PubMed]

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W. Wang, L.G. Lim, S. Srivastava, J. S. Yan, A. Shabbir, and Q. Liu, “Roles of linear and circular polarization properties and effect of wavelength choice on differentiation between ex vivo normal and cancerous gastric samples,” J. Biomed. Opt. 19(4) 046020 (2014).
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J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
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S. B. Mehta, M. Shribak, and R. Oldenbourg, “Polarized light imaging of birefringence and diattenuation at high resolution and high sensitivity,” J. Opt. 15, 1–13 (2013).
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B. de Campos Vidal, M. L. Silveira Mello, A. C. Caiseiro-Filho, and C. Godo, “Anisotropic properties of the myelin sheath,” Acta Histochem. 66, 32–39 (1980).
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W. Wang, L.G. Lim, S. Srivastava, J. S. Yan, A. Shabbir, and Q. Liu, “Roles of linear and circular polarization properties and effect of wavelength choice on differentiation between ex vivo normal and cancerous gastric samples,” J. Biomed. Opt. 19(4) 046020 (2014).
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M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourg, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2015).
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M. Koike-Tani, T. Tani, S. B. Mehta, A. Verma, and R. Oldenbourg, “Polarized light microscopy in reproductive and developmental biology,” Mol. Reprod. Dev. 82(7–8), 548–562 (2015).
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[Crossref]

Vitkin, I. A.

N. Ghosh, J. Soni, M. F. G. Wood, M. A. Wallenberg, and I. A. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana – J. Phys. 75(6), 1071–1086 (2010).
[Crossref]

N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
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Vitkin, I.A.

N. Ghosh and I.A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
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Wallenberg, M. A.

N. Ghosh, J. Soni, M. F. G. Wood, M. A. Wallenberg, and I. A. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana – J. Phys. 75(6), 1071–1086 (2010).
[Crossref]

Wang, L. V.

Wang, W.

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M. Dohmen, M. Menzel, H. Wiese, J. Reckfort, F. Hanke, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, ‘Understanding fiber mixture by simulation in 3D Polarized Light Imaging,” NeuroImage 111, 464–475 (2015).
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[Crossref]

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N. Ghosh, M. F. Wood, and I. A. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13(4), 044036 (2008).
[Crossref] [PubMed]

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N. Ghosh, J. Soni, M. F. G. Wood, M. A. Wallenberg, and I. A. Vitkin, “Mueller matrix polarimetry for the characterization of complex random medium like biological tissues,” Pramana – J. Phys. 75(6), 1071–1086 (2010).
[Crossref]

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W. Wang, L.G. Lim, S. Srivastava, J. S. Yan, A. Shabbir, and Q. Liu, “Roles of linear and circular polarization properties and effect of wavelength choice on differentiation between ex vivo normal and cancerous gastric samples,” J. Biomed. Opt. 19(4) 046020 (2014).
[Crossref] [PubMed]

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Yao, G.

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[Crossref]

M. Dohmen, M. Menzel, H. Wiese, J. Reckfort, F. Hanke, U. Pietrzyk, K. Zilles, K. Amunts, and M. Axer, ‘Understanding fiber mixture by simulation in 3D Polarized Light Imaging,” NeuroImage 111, 464–475 (2015).
[Crossref] [PubMed]

J. Dammers, L. Breuer, M. Axer, M. Kleiner, B. Eiben, D. Grässel, T. Dickscheid, K. Zilles, K. Amunts, N. J. Shah, and U. Pietrzyk, “Automatic identification of gray and white matter components in polarized light imaging,” NeuroImage 59(2), 1338–1347 (2012).
[Crossref]

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[Crossref]

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[Crossref]

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M. Axer, D. Grässel, M. Kleiner, J. Dammers, T. Dickscheid, J. Reckfort, T. Hütz, B. Eiben, U. Pietrzyk, K. Zilles, and K. Amunts, “High-resolution fiber tract reconstruction in the human brain by means of three-dimensional polarized light imaging,” Front. Neuroinform. 5(34), 1–13 (2011).
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W. Wang, L.G. Lim, S. Srivastava, J. S. Yan, A. Shabbir, and Q. Liu, “Roles of linear and circular polarization properties and effect of wavelength choice on differentiation between ex vivo normal and cancerous gastric samples,” J. Biomed. Opt. 19(4) 046020 (2014).
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L. Larsen, L. D. Griffin, D. Grässel, O. W. Witte, and H. Axer, “Polarized light imaging of white matter architecture,” Microsc. Res. Tech. 70(10), 851–863 (2007).
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[Crossref]

J. Reckfort, New Approaches to the Interpretation of 3D-Polarized Light Imaging Signals for an Advanced Extraction of Fiber Orientation, PhD Thesis (University of Wuppertal, 2015).

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Figures (15)

Fig. 1
Fig. 1

Design of the study.

Fig. 2
Fig. 2

(a–c) Schematic of the setups for the 3D-PLI, XP, and DI measurements: For the 3D-PLI measurement (a), the brain section is placed between a pair of crossed linear polarizers (polarizer/analyzer) and a quarter-wave retarder. For the XP measurement (b), only the crossed linear polarizers are used while for the DI measurement (c) only the polarizer is used. For all measurement setups, the employed filters are rotated simultaneously by discrete rotation angles ρ around the stationary specimen. (d) The transmitted light intensity is calculated using the Müller-Stokes calculus, in which each optical element is represented by a Müller matrix (ξ, δ, D, τ) as defined in Eqs. (5) to (7). (e–g) Analytically computed normalized light intensity profiles for the different measurement setups, assuming a retardance of δ = arcsin(0.8), a fiber direction of φ = 80°, and ideal filter properties (Dx = Dy = 1, τx = τy = 1/2; γ = π/2, τΛ = 1): For the 3D-PLI and XP measurements, the diattenuation and absorption of the brain tissue were neglected (D = 0, τ = 1). For the DI measurement, the tissue diattenuation was assumed to be D = 4 %. The phase φ of the intensity profiles (in red) is a measure for the in-plane fiber direction, while the amplitude (in blue) is correlated to the out-of-plane fiber inclination. (h) The three-dimensional fiber orientation is defined in spherical coordinates by the in-plane direction angle φ and the out-of-plane inclination angle α.

Fig. 3
Fig. 3

Predicted impact of the non-ideal system properties (specified in Eq. (42)) on the fiber orientation (φP, αP) derived from the simulation of the 3D-PLI measurement (see Sec. 3.1). The plots show the difference between (φP, αP) and the actual fiber orientation (φ, α) for different α and tissue diattenuations D. The fiber direction is assumed to be φ = 0°: (a) Difference between αP and α. (b) Difference between α̃P and α. (c) Difference between φP and φ. (d) Enlarged view of (c) for D ≤ 4 %.

Fig. 4
Fig. 4

Predicted impact of the non-ideal system properties (specified in Eq. (42)) on the direction angle φX derived from the simulation of the XP measurement (see Sec. 3.2). The plots show the difference between φX and the actual fiber direction φ plotted against the tissue diattenuation D for different fiber inclination angles α. The actual fiber direction was set to φ = 45° such that |φXφ| becomes maximal. (a) Difference between φX and φ. (b) Enlarged view of (a) for D ≤ 4 %.

Fig. 5
Fig. 5

Predicted impact of the non-ideal system properties (specified in Eq. (42)) on the diattenuation �� and the direction angle φD derived from the simulation of the DI measurement (see Sec. 3.3): (a) Difference between �� and the actual tissue diattenuation D plotted against the fiber direction φ for different D. The solid lines correspond to horizontal fibers with inclination α = 0°, the dashed lines to vertical fibers with α = 90°. The curves for fibers with intermediate inclination angles lie in between. (b) Maximum difference (pos./neg.) between �� and D plotted against D for different α. Note that the values min(��D) correspond to the bottom curve for all α. (c) Difference between φD and φ plotted against φ for different D. The solid lines correspond to horizontal fibers with inclination α = 0°, the dashed lines (all lying along the zero line) to vertical fibers with α = 90° and D > 0. The curves for fibers with intermediate inclination angles lie in between. (d) Maximum difference (pos./neg.) between φD and φ plotted against D for different α.

Fig. 6
Fig. 6

Transmittance images of the five investigated rat brain sections. For reference, some anatomical structures are labeled exemplary in section s0177.

Fig. 7
Fig. 7

Correlation between measured retardation and diattenuation: (a) Retardation rP and diattenuation �� shown exemplary for one rat brain section (s0175). The yellow arrows indicate the regions with maximum retardation and diattenuation, respectively. (b) 2D histogram showing rP plotted against �� evaluated for all five brain sections. The number of bins is 100 for both axes. The dashed vertical line marks the region (�� > 1 %) for which the diattenuation signal is expected to be mainly caused by the brain tissue and not by non-ideal system components (cf. Sec. 4.2.3).

Fig. 8
Fig. 8

Histogram showing the difference between the direction angle φD determined from the DI measurement and the direction angle φP determined from the 3D-PLI measurement (bin width = 0.5°). Due to the 180°-periodicity, the data range has been reduced to [−45°, 135°). To ensure that the diattenuation signal is mainly caused by the brain tissue and not by non-ideal system components (cf. Sec. 4.2.3), only regions with �� > 1 % were used for evaluation. The highlighted regions show the 2σ-environments of the fitted Gaussian distributions: green (53.97 % of the selected pixels: μ = 2.1°, σ = 11.0°), red (19.06 % of the selected pixels: μ = 91.1°, σ = 5.8°).

Fig. 9
Fig. 9

Diattenuation images of the five investigated rat brain sections: Regions that show different types of diattenuation are highlighted in green (D+) and red (D), corresponding to the angle ranges defined in Fig. 8 (for �� > 1 %).

Fig. 10
Fig. 10

(a) 2D histogram showing the difference between the direction angle φD derived from the DI measurement and the direction angle φX derived from the XP measurement plotted against the measured diattenuation ��. The dashed cyan lines correspond to the maximum difference (pos./neg.) as predicted by the numerical study (see Fig. 5(d), for α = 0°). (b) 2D histogram showing the difference between the direction angle φP derived from the 3D-PLI measurement and φX plotted against ��. To ensure a sufficient signal-to-noise ratio for φX (cf. Fig. 15(b)), only regions with retardations rP > 0.1 were selected for evaluation. The number of bins in the 2D histograms is 100 for both axes, respectively. The dashed vertical lines mark the region (�� > 1 %) for which the diattenuation signal is expected to be mainly caused by the brain tissue and not by non-ideal system components (cf. Sec. 4.2.3).

Fig. 11
Fig. 11

Histogram showing the difference between the direction angle φP derived from the 3D-PLI measurement and the direction angle φX derived from the XP measurement computed for regions with diattenuation of type D+ (green) and D (red) according to Fig. 9 (bin width = 0.1°). The green and red dashed lines indicate the respective mean values: 0.19° for D+ and −1.07° for D. The highlighted areas contain 75 % of the respective data: [−2.45°, 2.84°] for D+ and [−3.27°, 1.13°] for D. To ensure a sufficient signal-to-noise ratio for φX (cf. Fig. 15(b)), only regions with retardations rP > 0.1 were selected for evaluation.

Fig. 12
Fig. 12

Images and histograms of light source, polarizer, retarder, and analyzer. Note that the contrast of the images is different (the maximum measurable intensity values are depicted in white and the minimum values in black, respectively).

Fig. 13
Fig. 13

Normalized light intensity profiles: The solid curves show the profiles obtained from the filter measurements described in Tab. 1. The dashed curves show the modeled profiles computed from the polarization parameters defined in Tab. 2, first row.

Fig. 14
Fig. 14

Parameter maps of one rat brain section (s0175) obtained from the 3D-PLI (red), XP (green), and DI (yellow) measurements

Fig. 15
Fig. 15

(a) 2D histogram showing the difference between the direction angle φD derived from the DI measurement and the direction angle φX derived from the XP measurement plotted against the measured retardation rP. (b) 2D histogram showing the difference between the direction angle φP derived from the 3D-PLI measurement and φX plotted against rP. The vertical dashed line marks the region (rP < 0.1) for which the signal-to-noise ratio of φX is low. The number of bins in the 2D histograms is 100 for both axes, respectively.

Tables (2)

Tables Icon

Table 1 Configuration of the filter measurements to determine the polarization properties of the optical components: The angle position ξ of the filters (polarizer/retarder/analyzer) is defined in terms of the rotation angle ρ as described in Sec. 3.

Tables Icon

Table 2 Fitted polarization properties of filters, light source, and camera: The parameters in the first row minimize the sum of squared differences (Δmin = 2138.52). The other parameters show the average, standard deviation, and relative standard deviation (divided by the average) for the best 20 fits with Δ = 2138.52, . . . , 2154.68.

Equations (50)

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δ 2 π λ d Δ n cos 2 α ,
D = I max I min I max + I min , 0 D 1 .
τ = I max + I min 2 I 0 , 0 τ 1 ,
S = ( I I p cos ( 2 ψ ) cos ( 2 χ ) I p sin ( 2 ψ ) cos ( 2 χ ) I p sin ( 2 χ ) ) , p = S 1 2 + S 2 2 + S 3 2 S 0 ,
( δ , D , τ ) = τ ( 1 D 0 0 D 1 0 0 0 0 1 D 2 cos δ 1 D 2 sin δ 0 0 1 D 2 sin δ 1 D 2 cos δ ) ,
R ( ξ ) = ( 1 0 0 0 0 cos ( 2 ξ ) sin ( 2 ξ ) 0 0 sin ( 2 ξ ) cos ( 2 ξ ) 0 0 0 0 1 ) .
( ξ , δ , D , τ ) = R ( ξ ) ( δ , D , τ ) R ( ξ ) .
S = S ,
Polarizer : P x ( ρ , D x , τ x ) ( ρ , 0 , D x , τ x ) ,
Retarder : Λ ( ρ , γ , τ Λ ) ( ρ 45 ° , γ , 0 , τ Λ ) ,
Brain Tissue : M ( φ , δ , D , τ ) ( φ , δ , D , τ ) ,
Analyzer : P y ( ρ , D y , τ y ) ( ρ + 90 ° , 0 , D y , τ y ) .
S P ( ρ ) = P y ( ρ , D y , τ y ) M ( φ , δ , D , τ ) Λ ( ρ , γ , τ Λ ) P x ( ρ , D x , τ x ) S unpol
I P ( ρ ) = τ τ x τ y τ Λ I 0 [ 1 + D x D y sin γ sin δ 1 D 2 sin ( 2 ( ρ φ ) ) D x D y cos γ ( cos 2 ( 2 ( ρ φ ) ) + 1 D 2 cos δ sin 2 ( 2 ( ρ φ ) ) ) + D ( D x cos γ D y ) cos ( 2 ( ρ φ ) ) ] ,
I P ( ρ ) = a 0 P + a 2 P cos ( 2 ρ ) + b 2 P sin ( 2 ρ ) + a 4 P cos ( 4 ρ ) + b 4 P sin ( 4 ρ ) ,
a 0 P = τ τ x τ y τ Λ I 0 ( 1 1 2 D x D y cos γ ( 1 + 1 D 2 cos δ ) ) ,
a 2 P = τ τ x τ y τ Λ I 0 ( D ( D x cos γ D y ) cos ( 2 φ ) 1 D 2 D x D y sin γ sin δ sin ( 2 φ ) ) ,
b 2 P = τ τ x τ y τ Λ I 0 ( D ( D x cos γ D y ) sin ( 2 φ ) + 1 D 2 D x D y sin γ sin δ cos ( 2 φ ) ) ,
a 4 P = 1 2 τ τ x τ y τ Λ I 0 D x D y cos γ ( 1 1 D 2 cos δ ) cos ( 4 φ ) ,
b 4 P = 1 2 τ τ x τ y τ Λ I 0 D x D y cos γ ( 1 1 D 2 cos δ ) sin ( 4 φ ) .
I T , P = 2 a 0 P .
φ P = atan 2 ( a 2 P , b 2 P ) 2 [ 0 , π ) .
r P | sin δ P | = a 2 P 2 + b 2 P 2 a 0 P ,
α ˜ P = arccos ( arcsin ( r P ) arcsin ( r max ) ) .
S X ( ρ ) = P y ( ρ , D y , τ y ) M ( φ , δ , D , τ ) P x ( ρ , D x , τ x ) S unpol
I X ( ρ ) = τ τ x τ y I 0 [ 1 D x D y + ( 1 1 D 2 cos δ ) sin 2 ( 2 ( ρ φ ) ) + D ( D x D y ) cos ( 2 ( ρ φ ) ) ]
I X ( ρ ) = a 0 X + a 2 X cos ( 2 ρ ) + b 2 X sin ( 2 ρ ) + a 4 X cos ( 4 ρ ) + b 4 X sin ( 4 ρ ) ,
a 0 X = τ τ x τ y I 0 ( 1 1 2 D x D y ( 1 + 1 D 2 cos δ ) ) ,
a 2 X = τ τ x τ y I 0 D ( D x D y ) cos ( 2 φ ) ,
b 2 X = τ τ x τ y I 0 D ( D x D y ) sin ( 2 φ ) ,
a 4 X = 1 2 τ τ x τ y I 0 D x D y ( 1 1 D 2 cos δ ) cos ( 4 φ ) ,
b 4 X = 1 2 τ τ x τ y I 0 D x D y ( 1 1 D 2 cos δ ) sin ( 4 φ ) .
φ X = atan 2 ( b 4 X , a 4 X ) 4 [ 0 , π / 2 ) .
S D ( ρ ) = M ( φ , δ , D , τ ) P x ( ρ , D x , τ x ) S unpol
I D ( ρ ) = τ τ x I 0 ( 1 + D D x cos ( 2 ( ρ φ ) ) )
I D ( ρ ) = a 0 D + a 2 D cos ( 2 ρ ) + b 2 D sin ( 2 ρ ) ,
a 0 D = τ τ x I 0 ,
a 2 D = τ τ x I 0 D D x cos ( 2 φ ) ,
b 2 D = τ τ x I 0 D D x sin ( 2 φ ) .
φ D = atan 2 ( b 2 D , a 2 D ) 2 [ 0 , π ) .
D D = a 2 D 2 + b 2 D 2 D x a 0 D .
D x 0.98 , D y 0.97 , γ 0.49 π , S L = ( 1 5 × 10 3 8 × 10 4 5 × 10 7 ) , S c ( 1 8 × 10 3 1 × 10 3 5 × 10 4 ) .
S L = ( 1 p L cos ( 2 ψ L ) cos ( 2 χ L ) p L sin ( 2 ψ L ) cos ( 2 χ L ) p L sin ( 2 χ L ) ) , S c = ( 1 p c cos ( 2 ψ c ) cos ( 2 χ c ) p c sin ( 2 ψ c ) cos ( 2 χ c ) p c sin ( 2 χ c ) ) .
Δ = j , k [ ( I meas ( ρ ) I model ( ρ ) ) 2 σ 2 ( ρ ) ( I meas , max I meas , min ) 2 ] k .
S L ( 1 5 × 10 3 8 × 10 4 5 × 10 7 ) , S c ( 1 8 × 10 3 1 × 10 3 5 × 10 4 ) .
δ = arccos ( E ± E 2 + 4 ( F G ) ( H G ) 2 ( G H ) ) ,
E a 2 P 2 + b 2 P 2 a 0 P 2 cos γ D x D y 1 D 2 ( 1 1 2 cos γ D x D y ) ,
G sin 2 γ D x 2 D y 2 ( 1 D 2 ) ,
F a 2 P 2 + b 2 P 2 a 0 P 2 ( 1 1 2 cos γ D x D y ) 2 D 2 ( cos γ D x D y ) 2 ,
H a 2 P 2 + b 2 P 2 4 a 0 P 2 cos 2 γ D x 2 D y 2 ( 1 D 2 ) .

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