Abstract

We present a theoretical framework for strain estimation in optical coherence elastography (OCE), based on a statistical analysis of displacement measurements obtained from a mechanically loaded sample. We define strain sensitivity, signal-to-noise ratio and dynamic range, and derive estimates of strain using three methods: finite difference, ordinary least squares and weighted least squares, the latter implemented for the first time in OCE. We compare theoretical predictions with experimental results and demonstrate a ~12 dB improvement in strain sensitivity using weighted least squares compared to finite difference strain estimation and a ~4 dB improvement over ordinary least squares strain estimation. We present strain images (i.e., elastograms) of tissue-mimicking phantoms and excised porcine airway, demonstrating in each case clear contrast based on the sample’s elasticity.

© 2012 OSA

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  1. J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
    [Crossref] [PubMed]
  2. J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
    [Crossref] [PubMed]
  3. J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
    [Crossref] [PubMed]
  4. R. C. Chan, A. H. Chau, W. C. Karl, S. Nadkarni, A. S. Khalil, N. Iftimia, M. Shishkov, G. J. Tearney, M. R. Kaazempur-Mofrad, and B. E. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
    [Crossref] [PubMed]
  5. J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
    [Crossref] [PubMed]
  6. H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
    [Crossref] [PubMed]
  7. W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer-Verlag, Berlin, 2008).
  8. B. F. Kennedy, T. R. Hillman, A. Curatolo, and D. D. Sampson, “Speckle reduction in optical coherence tomography by strain compounding,” Opt. Lett. 35(14), 2445–2447 (2010).
    [Crossref] [PubMed]
  9. B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
    [Crossref] [PubMed]
  10. T. R. Hillman, A. Curatolo, B. F. Kennedy, and D. D. Sampson, “Detection of multiple scattering in optical coherence tomography by speckle correlation of angle-dependent B-scans,” Opt. Lett. 35(12), 1998–2000 (2010).
    [Crossref] [PubMed]
  11. R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
    [Crossref]
  12. R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
    [Crossref]
  13. X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
    [Crossref] [PubMed]
  14. S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18(25), 25519–25534 (2010).
    [Crossref] [PubMed]
  15. X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
    [Crossref] [PubMed]
  16. B. F. Kennedy, X. Liang, S. G. Adie, D. K. Gerstmann, B. C. Quirk, S. A. Boppart, and D. D. Sampson, “In vivo three-dimensional optical coherence elastography,” Opt. Express 19(7), 6623–6634 (2011).
    [Crossref] [PubMed]
  17. B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express 13(11), 3931–3944 (2005).
    [Crossref] [PubMed]
  18. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  19. F. Kallel and J. Ophir, “A least-squares strain estimator for elastography,” Ultrason. Imaging 19(3), 195–208 (1997).
    [PubMed]
  20. B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
    [Crossref] [PubMed]
  21. A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
    [Crossref] [PubMed]
  22. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  23. J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
    [Crossref] [PubMed]
  24. I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
    [Crossref]
  25. J. Fox, Applied Regression Analysis, Linear Models and Related Methods (Sage, Thousand Oaks, Calif., 1997).
  26. T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: the strain filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(1), 164–172 (1997).
    [Crossref] [PubMed]
  27. J. I. Jackson and L. J. Thomas, “Ultrasound-based strain rate estimation of moving, fully developed speckle,” in 2001 IEEE Ultrasonic Symposium (IEEE, 2001), Vol. 2, pp. 1593–1596.
  28. M. Pircher, B. Baumann, E. Götzinger, H. Sattmann, and C. K. Hitzenberger, “Phase contrast coherence microscopy based on transverse scanning,” Opt. Lett. 34(12), 1750–1752 (2009).
    [Crossref] [PubMed]
  29. A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography—limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
    [Crossref] [PubMed]
  30. S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
    [Crossref] [PubMed]
  31. G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
    [Crossref] [PubMed]
  32. S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
    [Crossref] [PubMed]
  33. J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
    [PubMed]
  34. C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Sixth International Conference on Computer Vision, 1998 (IEEE, 1998), 839–846.

2011 (2)

B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
[Crossref] [PubMed]

B. F. Kennedy, X. Liang, S. G. Adie, D. K. Gerstmann, B. C. Quirk, S. A. Boppart, and D. D. Sampson, “In vivo three-dimensional optical coherence elastography,” Opt. Express 19(7), 6623–6634 (2011).
[Crossref] [PubMed]

2010 (5)

2009 (3)

2008 (2)

2007 (1)

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[Crossref]

2006 (2)

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[Crossref] [PubMed]

2005 (2)

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express 13(11), 3931–3944 (2005).
[Crossref] [PubMed]

2004 (2)

R. C. Chan, A. H. Chau, W. C. Karl, S. Nadkarni, A. S. Khalil, N. Iftimia, M. Shishkov, G. J. Tearney, M. R. Kaazempur-Mofrad, and B. E. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
[Crossref] [PubMed]

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

2003 (1)

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

1999 (1)

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

1998 (1)

1997 (3)

F. Kallel and J. Ophir, “A least-squares strain estimator for elastography,” Ultrason. Imaging 19(3), 195–208 (1997).
[PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: the strain filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(1), 164–172 (1997).
[Crossref] [PubMed]

1995 (1)

I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
[Crossref]

1992 (1)

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

1991 (1)

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Adie, S. G.

Alam, S. K.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

Alexandrov, S. A.

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[Crossref] [PubMed]

Armstrong, J. J.

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[Crossref] [PubMed]

Bamber, J.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Baumann, B.

Bert, J. L.

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

Boppart, S. A.

Bouma, B.

Bouma, B. E.

R. C. Chan, A. H. Chau, W. C. Karl, S. Nadkarni, A. S. Khalil, N. Iftimia, M. Shishkov, G. J. Tearney, M. R. Kaazempur-Mofrad, and B. E. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Brenner, M.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Brezinski, M. E.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Cense, B.

Cespedes, I.

I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
[Crossref]

Céspedes, I.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Chan, R. C.

Chaney, E. J.

Chau, A. H.

Chen, Z.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Colt, H.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Crecea, V.

Curatolo, A.

de Boer, J.

El-Abbadi, N. H.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Fatemi, M.

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

Fujimoto, J. G.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Garcia, L.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Garra, B.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

Gerstmann, D. K.

Götzinger, E.

Greenleaf, J. F.

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

Grimwood, A.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Han, S.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Hanna, N.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Hillman, T. R.

Hinds, M.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[Crossref]

Hitzenberger, C. K.

Holmes, J.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Iftimia, N.

Insana, M.

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
[Crossref]

John, R.

Jung, W. G.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Kaazempur-Mofrad, M. R.

Kallel, F.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

F. Kallel and J. Ophir, “A least-squares strain estimator for elastography,” Ultrason. Imaging 19(3), 195–208 (1997).
[PubMed]

Karl, W. C.

Kennedy, B. F.

B. F. Kennedy, X. Liang, S. G. Adie, D. K. Gerstmann, B. C. Quirk, S. A. Boppart, and D. D. Sampson, “In vivo three-dimensional optical coherence elastography,” Opt. Express 19(7), 6623–6634 (2011).
[Crossref] [PubMed]

B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
[Crossref] [PubMed]

T. R. Hillman, A. Curatolo, B. F. Kennedy, and D. D. Sampson, “Detection of multiple scattering in optical coherence tomography by speckle correlation of angle-dependent B-scans,” Opt. Lett. 35(12), 1998–2000 (2010).
[Crossref] [PubMed]

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18(25), 25519–25534 (2010).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, A. Curatolo, and D. D. Sampson, “Speckle reduction in optical coherence tomography by strain compounding,” Opt. Lett. 35(14), 2445–2447 (2010).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[Crossref] [PubMed]

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[Crossref] [PubMed]

Khalil, A. S.

Kirkpatrick, S.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[Crossref]

Kirkpatrick, S. J.

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

Ko, H. J.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[Crossref] [PubMed]

Konofagou, E.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

Kowalczyk, A.

Krouskop, T.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

Li, X.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Liang, X.

Ma, Z. H.

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

Mahmood, U.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

McLaughlin, R. A.

Mina-Araghi, R.

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Mujat, M.

Nadkarni, S.

Oldenburg, A. L.

Ophir, J.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

F. Kallel and J. Ophir, “A least-squares strain estimator for elastography,” Ultrason. Imaging 19(3), 195–208 (1997).
[PubMed]

T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: the strain filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(1), 164–172 (1997).
[Crossref] [PubMed]

I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
[Crossref]

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Pankhurst, Q. A.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Paré, P. D.

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

Park, B.

Patel, N. A.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

Pierce, M. C.

Pircher, M.

Pitris, C.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Ponnekanti, H.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Quirk, B. C.

Rains, J. K.

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

Roberts, C. R.

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

Rogowska, J.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

Sampson, D. D.

B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
[Crossref] [PubMed]

B. F. Kennedy, X. Liang, S. G. Adie, D. K. Gerstmann, B. C. Quirk, S. A. Boppart, and D. D. Sampson, “In vivo three-dimensional optical coherence elastography,” Opt. Express 19(7), 6623–6634 (2011).
[Crossref] [PubMed]

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18(25), 25519–25534 (2010).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, A. Curatolo, and D. D. Sampson, “Speckle reduction in optical coherence tomography by strain compounding,” Opt. Lett. 35(14), 2445–2447 (2010).
[Crossref] [PubMed]

T. R. Hillman, A. Curatolo, B. F. Kennedy, and D. D. Sampson, “Detection of multiple scattering in optical coherence tomography by speckle correlation of angle-dependent B-scans,” Opt. Lett. 35(12), 1998–2000 (2010).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[Crossref] [PubMed]

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[Crossref] [PubMed]

Sattmann, H.

Saunders, C. M.

B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
[Crossref] [PubMed]

Schmitt, J. M.

Shishkov, M.

Southern, J. F.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Stack, R.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[Crossref] [PubMed]

Szkulmowska, A.

Szkulmowski, M.

Tan, W.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[Crossref] [PubMed]

Tearney, G.

Tearney, G. J.

R. C. Chan, A. H. Chau, W. C. Karl, S. Nadkarni, A. S. Khalil, N. Iftimia, M. Shishkov, G. J. Tearney, M. R. Kaazempur-Mofrad, and B. E. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
[Crossref] [PubMed]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Tomlins, P.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Varghese, T.

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: the strain filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(1), 164–172 (1997).
[Crossref] [PubMed]

Wang, R. K.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[Crossref]

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

Wojtkowski, M.

Woolliams, P.

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Yazdi, Y.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

Yun, S. H.

Annu. Rev. Biomed. Eng. (1)

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[Crossref]

Heart (1)

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

I. Cespedes, M. Insana, and J. Ophir, “Theoretical bounds on strain estimation in elastography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(5), 969–972 (1995).
[Crossref]

T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: the strain filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44(1), 164–172 (1997).
[Crossref] [PubMed]

J. Appl. Physiol. (1)

J. K. Rains, J. L. Bert, C. R. Roberts, and P. D. Paré, “Mechanical properties of human tracheal cartilage,” J. Appl. Physiol. 72(1), 219–225 (1992).
[PubMed]

J. Biomed. Opt. (1)

B. F. Kennedy, A. Curatolo, T. R. Hillman, C. M. Saunders, and D. D. Sampson, “Speckle reduction in optical coherence tomography images using tissue viscoelasticity,” J. Biomed. Opt. 16(2), 020506 (2011).
[Crossref] [PubMed]

Opt. Express (8)

J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
[Crossref] [PubMed]

R. C. Chan, A. H. Chau, W. C. Karl, S. Nadkarni, A. S. Khalil, N. Iftimia, M. Shishkov, G. J. Tearney, M. R. Kaazempur-Mofrad, and B. E. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
[Crossref] [PubMed]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[Crossref] [PubMed]

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18(25), 25519–25534 (2010).
[Crossref] [PubMed]

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[Crossref] [PubMed]

B. F. Kennedy, X. Liang, S. G. Adie, D. K. Gerstmann, B. C. Quirk, S. A. Boppart, and D. D. Sampson, “In vivo three-dimensional optical coherence elastography,” Opt. Express 19(7), 6623–6634 (2011).
[Crossref] [PubMed]

B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express 13(11), 3931–3944 (2005).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Med. Biol. (2)

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[Crossref] [PubMed]

A. Grimwood, L. Garcia, J. Bamber, J. Holmes, P. Woolliams, P. Tomlins, and Q. A. Pankhurst, “Elastographic contrast generation in optical coherence tomography from a localized shear stress,” Phys. Med. Biol. 55(18), 5515–5528 (2010).
[Crossref] [PubMed]

Proc. Inst. Mech. Eng. H (1)

J. Ophir, S. K. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, and T. Varghese, “Elastography: ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. H 213(3), 203–233 (1999).
[Crossref] [PubMed]

Respiration (1)

S. Han, N. H. El-Abbadi, N. Hanna, U. Mahmood, R. Mina-Araghi, W. G. Jung, Z. Chen, H. Colt, and M. Brenner, “Evaluation of tracheal imaging by optical coherence tomography,” Respiration 72(5), 537–541 (2005).
[Crossref] [PubMed]

Science (1)

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276(5321), 2037–2039 (1997).
[Crossref] [PubMed]

Tissue Eng. (1)

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[Crossref] [PubMed]

Ultrason. Imaging (2)

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

F. Kallel and J. Ophir, “A least-squares strain estimator for elastography,” Ultrason. Imaging 19(3), 195–208 (1997).
[PubMed]

Other (6)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. I. Jackson and L. J. Thomas, “Ultrasound-based strain rate estimation of moving, fully developed speckle,” in 2001 IEEE Ultrasonic Symposium (IEEE, 2001), Vol. 2, pp. 1593–1596.

J. Fox, Applied Regression Analysis, Linear Models and Related Methods (Sage, Thousand Oaks, Calif., 1997).

C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Sixth International Conference on Computer Vision, 1998 (IEEE, 1998), 839–846.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer-Verlag, Berlin, 2008).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

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Figures (5)

Fig. 1
Fig. 1

Theoretical predictions of strain estimation using FD (dashed red) and OLS (solid blue) methods: (a) sensitivity; and (b) SNR versus strain axial resolution. (c) SNR versus change in displacement, Δd, in one resolution element.

Fig. 2
Fig. 2

(a) Schematic diagram of the OCE system; RB: rigid boundary; RA: ring actuator; SM: scanning mirror; BS: beam splitter; RM: reference mirror; SLD: superluminescent diode; (b) Saw-tooth pattern of lateral scanning beam synchronized with motion of ring actuator. (c) The measured phase difference with no lateral scanning and (d) with lateral scanning.

Fig. 3
Fig. 3

(a) Strain sensitivity; (b) mean strain; and (c) strain SNR of Phantom 1. Experimental results are presented for FD (red dots), OLS (blue dots), WLS (green dots), and GS-WLS (black dots) strain estimation. Theoretical results are presented for FD (red line) and OLS (blue line) strain estimation.

Fig. 4
Fig. 4

Phantom 2: (a) OCT structural image; and (b) FD; (c) OLS; (d) WLS; and (e) GS-WLS elastograms. In (f)-(j), corresponding lateral traces are shown for the depth indicated by the blue arrow in (a).

Fig. 5
Fig. 5

(a) OCT structural image; and (b) GS-WLS elastogram of Phantom 3. (c) OCT structural image; and (d) GS-WLS elastogram of a thin section of excised porcine airway with layers as labeled in (c).

Equations (33)

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Y= σ ε b = F A Δl l 0 ,
ε l = Δd Δz ,
I (j) = { ( z i , a i (j) exp(ι φ i (j) ))| z i =(i1)δz+ z 1 R, a i (j) R + , φ i (j) [π,π) } i=1 N ,
D= { ( z i , d i )| d i = λ( ϕ i (2) ϕ i (1) ) 4πn = λΔ ϕ i 4πn } i=1 N ,
σ Δ φ i = 1 SN R OC T i .
ε i f = d i+m1 d i z i+m1 z i = Δd Δz .
σ Ε i f 2 = σ D i 2 + σ D i+m1 2 2cov( D i , D i+m1 ) Δ z 2 ,
σ Ε i f 2 = 2 σ D 2 Δ z 2 .
μ Ε i f =ε.
d=ε(z z i1 )+c.
R= j=i i+m1 [ d j ε( z j z i1 )c ] 2 .
ε i o = ( j=i i+m1 1 )( j=i i+m1 ( z j z i1 ) d j )( j=i i+m1 ( z j z i1 ) )( j=i i+m1 d j ) ( j=i i+m1 1 )( j=i i+m1 ( z j z i1 ) 2 ) ( j=i i+m1 ( z j z i1 ) ) 2 ,
ε i o = j=i i+m1 ( κ 0 ( z j z i1 ) κ 1 κ 0 κ 2 κ 1 2 ) d j ,
κ x = j=i i+m1 ( z j z i1 ) x ,  x=0,1,2.
ε i o = j=i i+m1 ( κ 0 (ji+1)δz κ 1 κ 0 κ 2 κ 1 2 ) d j .
κ 0 = j=i i+m1 ( (ji+1)δz ) 0 =m,
κ 1 = j=i i+m1 ( (ji+1)δz ) 1 = m(m+1) 2 δz,
κ 2 = j=i i+m1 ( (ji+1)δz ) 2 = m(m+1)(2m+1) 6 δ z 2 .
ε i o = j=i i+m1 ( 6(2(ji)m+1) δz( m 3 m) ) d j .
μ Ε i o = j=i i+m1 ( 6(2(ji)m+1) δz( m 3 m) ) μ D j .
μ Ε i o = j=i i+m1 ( 6(2(ji)m+1) δz( m 3 m) ) ( ε(ji+1)δz+c )=ε.
σ Ε i o 2 = j=i i+m1 ( 6(2(ji)m+1) δz( m 3 m) ) 2 σ D 2 = 12 σ D 2 δ z 2 ( m 3 m) 12 σ D 2 Δ z 2 m .
S Ε i f/o = σ Ε i f/o .
SN R Ε i f/o = μ Ε i f/o σ Ε i f/o .
D R Ε i f/o = ( Δ d max Δz ) σ Ε i f/o = λ 4n σ Ε i f/o Δz .
S Ε i f = 2 σ D Δz ;
SN R Ε i f = μ Ε i f Δz 2 σ D ; and
D R Ε i f = λ 32 σ D n .
S Ε i o = 12 σ D Δz m ;
SN R Ε i o = μ Ε i o Δz m 12 σ D ; and
D R Ε i o = λ m 192 σ D n .
ε i w = ( j=i i+m1 w j )( j=i i+m1 w j ( z j z i1 ) d j )( j=i i+m1 w j ( z j z i1 ) )( j=i i+m1 w j d j ) ( j=i i+m1 w j )( j=i i+m1 w j ( z j z i1 ) 2 ) ( j=i i+m1 w j ( z j z i1 ) ) 2 ,
w j = 1 σ D j 2 .

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