Abstract

Tissue polarimetry has demonstrated its great potential in biomedical field presently. In this study, the polarization characteristics of red blood cell (RBC) suspensions in a back-detection geometry have been investigated with experimental measurements and Monte Carlo (MC) simulation based on Mueller matrix decomposition. It is revealed that the simulated dependence of degree of polarization (DOP) and diattenuation on the distance away from incident point is qualitatively consistent with experimental result. DOP and diattenuation decay with increasing radial distance except in the region adjacent to the incident point. Further analysis shows that the number of scattering events and the scattering angle simultaneously influence the trends of DOP and diattenuation curves in the central region.

© 2012 OSA

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2011 (2)

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

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

2010 (2)

2009 (4)

X. Li and G. Yao, “Mueller matrix decomposition of diffuse reflectance imaging in skeletal muscle,” Appl. Opt. 48(14), 2625–2631 (2009).
[CrossRef] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[CrossRef]

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

2007 (2)

2006 (1)

2005 (5)

2004 (1)

2003 (2)

P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42(21), 4389–4395 (2003).
[CrossRef] [PubMed]

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[CrossRef] [PubMed]

2001 (1)

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

1999 (2)

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

M. J. Raković, G. W. Kattawar, M. B. Mehrübeoğlu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiment,” Appl. Opt. 38(15), 3399–3408 (1999).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34(6), 1558–1568 (1995).
[CrossRef]

Antonelli, M. R.

Baum, B. A.

Benali, A.

Bigio, I. J.

Boulvert, F.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Buddhiwant, P.

Cameron, B. D.

Cariou, J.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Chen, Z.

Chipman, R. A.

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]

J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34(6), 1558–1568 (1995).
[CrossRef]

Chung, J.

Coté, G. L.

Côté, D.

De Martino, A.

Dörschel, K.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

Eick, A. A.

Freyer, J. P.

Friebel, M.

M. Friebel, J. Helfmann, and M. C. Meinke, “Influence of osmolarity on the optical properties of human erythrocytes,” J. Biomed. Opt. 15(5), 055005 (2010).
[CrossRef] [PubMed]

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

Gayet, B.

Ghosh, N.

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

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[CrossRef]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006).
[CrossRef] [PubMed]

Goudail, F.

Guo, X.

Gupta, P. K.

Hahn, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

Hammer-Wilson, M. J.

Helfmann, J.

M. Friebel, J. Helfmann, and M. C. Meinke, “Influence of osmolarity on the optical properties of human erythrocytes,” J. Biomed. Opt. 15(5), 055005 (2010).
[CrossRef] [PubMed]

Hielscher, A. H.

Hostetler, C. A.

Hu, X. H.

J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

Hu, Y. X.

Itoh, M.

M. Itoh, M. Yamanari, Y. Yasuno, and T. Yatagai, “Polarization characteristics of multiple backscattering in human blood cell suspensions,” Opt. Quantum Electron. 37(13-15), 1277–1285 (2005).
[CrossRef]

Jacques, S. L.

Jung, W.

Kattawar, G. W.

Lai, J.

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

Le Brun, G.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Le Jeune, B.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Li, R. K.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

Li, S. H.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

Li, X.

Li, Z.

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

Lu, J. Q.

J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

Lu, S.-Y.

Manhas, S.

Martin, L.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Mehrübeoglu, M.

Mehrübeoglu, M. B.

Meinke, M. C.

M. Friebel, J. Helfmann, and M. C. Meinke, “Influence of osmolarity on the optical properties of human erythrocytes,” J. Biomed. Opt. 15(5), 055005 (2010).
[CrossRef] [PubMed]

Morio, J.

Mourant, J. R.

Müller, G.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

Novikova, T.

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34(6), 1558–1568 (1995).
[CrossRef]

Pierangelo, A.

Prahl, S. A.

Rakovic, M. J.

Ramella-Roman, J. C.

Rastegar, S.

Roggan, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

Shen, D.

Singh, J.

Smith, M. H.

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

Sun, C. W.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[CrossRef] [PubMed]

Swami, M. K.

Validire, P.

Vitkin, A.

Vitkin, I. A.

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

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[CrossRef]

D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express 13(1), 148–163 (2005).
[CrossRef] [PubMed]

Wang, L. V.

Wang, X.

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[CrossRef] [PubMed]

Wei, H.

Weisel, R. D.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

Wilder-Smith, P.

Wilson, B. C.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

Winker, D. M.

Wood, M. F. G.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[CrossRef]

X. Guo, M. F. G. Wood, and A. Vitkin, “Monte Carlo study of pathlength distribution of polarized light in turbid media,” Opt. Express 15(3), 1348–1360 (2007).
[CrossRef] [PubMed]

Yamanari, M.

M. Itoh, M. Yamanari, Y. Yasuno, and T. Yatagai, “Polarization characteristics of multiple backscattering in human blood cell suspensions,” Opt. Quantum Electron. 37(13-15), 1277–1285 (2005).
[CrossRef]

Yang, C. C.

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[CrossRef] [PubMed]

Yang, L.

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

Yang, P.

J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

P. Yang, H. Wei, G. W. Kattawar, Y. X. Hu, D. M. Winker, C. A. Hostetler, and B. A. Baum, “Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase,” Appl. Opt. 42(21), 4389–4395 (2003).
[CrossRef] [PubMed]

Yao, G.

Yasuno, Y.

M. Itoh, M. Yamanari, Y. Yasuno, and T. Yatagai, “Polarization characteristics of multiple backscattering in human blood cell suspensions,” Opt. Quantum Electron. 37(13-15), 1277–1285 (2005).
[CrossRef]

Yatagai, T.

M. Itoh, M. Yamanari, Y. Yasuno, and T. Yatagai, “Polarization characteristics of multiple backscattering in human blood cell suspensions,” Opt. Quantum Electron. 37(13-15), 1277–1285 (2005).
[CrossRef]

Appl. Opt. (4)

J Biophotonics (1)

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics 2(3), 145–156 (2009).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[CrossRef]

J. Biomed. Opt. (5)

J. Q. Lu, P. Yang, and X. H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

X. Wang, L. V. Wang, C. W. Sun, and C. C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8(4), 608–617 (2003).
[CrossRef] [PubMed]

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

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, and G. Müller, “Optical properties of circulating human blood in the wavelength range 400-2500 nm,” J. Biomed. Opt. 4(1), 36–46 (1999).
[CrossRef]

M. Friebel, J. Helfmann, and M. C. Meinke, “Influence of osmolarity on the optical properties of human erythrocytes,” J. Biomed. Opt. 15(5), 055005 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm of an experimental Mueller matrix,” Opt. Commun. 282(5), 692–704 (2009).
[CrossRef]

Opt. Eng. (1)

J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34(6), 1558–1568 (1995).
[CrossRef]

Opt. Express (7)

M. R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De Martino, “Mueller matrix imaging of human colon tissue for cancer diagnostics: how Monte Carlo modeling can help in the interpretation of experimental data,” Opt. Express 18(10), 10200–10208 (2010).
[CrossRef] [PubMed]

D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express 13(1), 148–163 (2005).
[CrossRef] [PubMed]

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13(12), 4420–4438 (2005).
[CrossRef] [PubMed]

J. C. Ramella-Roman, S. A. Prahl, and S. L. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part II,” Opt. Express 13(25), 10392–10405 (2005).
[CrossRef] [PubMed]

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14(1), 190–202 (2006).
[CrossRef] [PubMed]

X. Guo, M. F. G. Wood, and A. Vitkin, “Monte Carlo study of pathlength distribution of polarized light in turbid media,” Opt. Express 15(3), 1348–1360 (2007).
[CrossRef] [PubMed]

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. Shen, J. P. Freyer, and I. J. Bigio, “Diffuse backscattering Mueller matricesof highly scattering media,” Opt. Express 1(13), 441–453 (1997).
[CrossRef] [PubMed]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Itoh, M. Yamanari, Y. Yasuno, and T. Yatagai, “Polarization characteristics of multiple backscattering in human blood cell suspensions,” Opt. Quantum Electron. 37(13-15), 1277–1285 (2005).
[CrossRef]

Proc. SPIE (2)

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

X. Wang, L. Yang, J. Lai, and Z. Li, “Polar decomposition applied to light back-scattering by erythrocyte suspensions,” Proc. SPIE 8192, 81924T, 81924T-6 (2011).
[CrossRef]

Other (1)

http://omlc.ogi.edu/software/polarization/

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup. H W 1 is a quartz half-wavelength plate, Q W 1 is a quartz quarter-wavelength plate, Q W 2 is a mica zero-order quarter-wavelength plate, A is a linear analyzer, L is a imaging lens and CCD is a imaging camera

Fig. 2
Fig. 2

The geometry of multiple scattering events in a slab of turbid medium

Fig. 3
Fig. 3

(a) DOP and (b) diattenuation patterns in the backscattering plane for RBC suspensions with 4% concentration

Fig. 4
Fig. 4

Experimental results: (a) DOP and (b) diattenuation curves on central horizontal axis in the backscattering plane for RBC suspensions with different concentrations (4%, 6%, 8%, 10%)

Fig. 5
Fig. 5

Numerical results for infinitely narrow beam: (a) DOP and (b) Diattenuation curves on central horizontal axis in the backscattering plane for RBC suspensions with different concentrations (4%, 6%, 8%, 10%)

Fig. 6
Fig. 6

Numerical results for Gaussian beam: (a) DOP and (b) Diattenuation curves on central horizontal axis in the backscattering plane for RBC suspensions with different concentrations (4%, 6%, 8%, 10%)

Fig. 7
Fig. 7

Indexed DOP dependence on radial distance in the backscattering plane. N denotes the number of scattering events. The symbols are MC calculation results and the lines are a guide for eyes.

Fig. 8
Fig. 8

Indexed diattenuation dependence on radial distance in the backscattering plane. N denotes the number of scattering events. The symbols are MC calculation results and the lines are a guide for the eye.

Equations (9)

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M=[ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] = 1 4 [ HH+HV+VH+VV HH+HVVHVV PH+PVQHQV RH+RVLHLV HHHV+VHVV HHHVVH+VV PH+QVQHPV RH+LVLHRV HPHQ+VPVQ HP+VQVPHQ PP+QQQPPQ RP+LQLPRQ HRHL+VRVL VL+HRHLVR QL+PRPLQR LL+RRRLLR ]
C(x,y)= G(x x ' ,y y ' )S( x ' , y ' )d x ' d y '
S( x ' , y ' )= S 0 exp[2( x ' 2 + y ' 2 )/ R 2 ]
S 0 =2P/(π R 2 )
M= M Δ M R M D
DOP= M Δ (2,2)+ M Δ (3,3)+ M Δ (4,4) 3
δ= cos 1 { [ ( M R (2,2)+ M R (3,3)) 2 + ( M R (3,2) M R (2,3)) 2 ] 1/2 1}
ψ= 1 2 tan 1 [ M R (3,2) M R (2,3) M R (2,2)+ M R (3,3) ]
D= [ { M D (1,2)} 2 + { M D (1,3)} 2 + { M D (1,4)} 2 ] 1/2 M D (1,1)

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