Abstract

Perturbation Monte Carlo (pMC) has been previously proposed to rapidly recompute optical measurements when small perturbations of optical properties are considered, but it was largely restricted to changes associated with prior tissue segments or regions-of-interest. In this work, we expand pMC to compute spatially and temporally resolved sensitivity profiles, i.e. the Jacobians, for diffuse optical tomography (DOT) applications. By recording the pseudo random number generator (PRNG) seeds of each detected photon, we are able to “replay” all detected photons to directly create the 3D sensitivity profiles for both absorption and scattering coefficients. We validate the replay-based Jacobians against the traditional adjoint Monte Carlo (aMC) method, and demonstrate the feasibility of using this approach for efficient 3D image reconstructions using in vitro hyperspectral wide-field DOT measurements. The strengths and limitations of the replay approach regarding its computational efficiency and accuracy are discussed, in comparison with aMC, for point-detector systems as well as wide-field pattern-based and hyperspectral imaging systems. The replay approach has been implemented in both of our open-source MC simulators - MCX and MMC (http://mcx.space)

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates

Jin Chen, Vivek Venugopal, and Xavier Intes
Biomed. Opt. Express 2(4) 871-886 (2011)

Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations

Frederic Leblond, Hamid Dehghani, Dax Kepshire, and Brian W. Pogue
J. Opt. Soc. Am. A 26(6) 1444-1457 (2009)

Time-resolved diffuse optical tomography with patterned-light illumination and detection

Jin Chen, Vivek Venugopal, Frederic Lesage, and Xavier Intes
Opt. Lett. 35(13) 2121-2123 (2010)

References

  • View by:
  • |
  • |
  • |

  1. A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
    [Crossref] [PubMed]
  2. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [Crossref]
  3. D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
    [Crossref]
  4. D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
    [Crossref]
  5. D. Contini, L. Zucchelli, L. Spinelli, M. Caffini, R. Re, A. Pifferi, R. Cubeddu, and A. Torricelli, “Brain and muscle near infrared spectroscopy/imaging techniques,” J. Near Infrared Spectrosc. 20, 15–27 (2012).
    [Crossref]
  6. A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
    [Crossref] [PubMed]
  7. M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
    [Crossref]
  8. A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
    [Crossref] [PubMed]
  9. A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
    [Crossref]
  10. M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).
  11. M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
    [Crossref]
  12. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [Crossref]
  13. M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
    [Crossref]
  14. M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
    [Crossref] [PubMed]
  15. V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
    [Crossref] [PubMed]
  16. N. Ducros, C. D’andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett. 35, 3676–3678 (2010).
    [Crossref] [PubMed]
  17. L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [Crossref] [PubMed]
  18. C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
    [Crossref]
  19. Q. Fang, “Mesh-based Monte Carlo method using fast ray-tracing in Plücker coordinates,” Biomed. Opt. Express 1, 165–175 (2010).
    [Crossref] [PubMed]
  20. H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. & Biol. 55, 947 (2010).
    [Crossref]
  21. Q. Fang and 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]
  22. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
    [Crossref]
  23. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
    [Crossref]
  24. J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
    [Crossref] [PubMed]
  25. J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
    [Crossref] [PubMed]
  26. A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
    [Crossref]
  27. Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
    [Crossref] [PubMed]
  28. J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).
  29. Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
    [Crossref] [PubMed]
  30. Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
    [Crossref] [PubMed]
  31. A. Sassaroli, “Fast perturbation Monte Carlo method for photon migration in heterogeneous turbid media,” Opt. Lett. 36, 2095–2097 (2011).
    [Crossref] [PubMed]
  32. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
    [Crossref]
  33. A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29, 2110–2117 (2012).
    [Crossref]
  34. J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
    [Crossref] [PubMed]
  35. M. A. O’Leary, “Imaging with diffuse photon density waves,” Ph.D. thesis, University of Pennsylvania, Philadelphia (1996).
  36. L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).
  37. D. A. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002).
    [Crossref] [PubMed]
  38. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
    [Crossref]
  39. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
    [Crossref]
  40. P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
    [Crossref] [PubMed]
  41. M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.
  42. R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography with compressive sensing based preconditioning,” Biomed. Opt. Express 6, 4887–4898 (2015).
    [Crossref] [PubMed]
  43. V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
    [Crossref]
  44. L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
    [Crossref]
  45. R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
    [Crossref] [PubMed]
  46. M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
    [Crossref]
  47. V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.
  48. Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.
  49. Q. Fang and D. A. Boas, “Tetrahedral mesh generation from volumetric binary and grayscale images,” in “IEEE Int. Symp. on Biomed. Imaging: From Nano to Macro, 2009. ISBI’09,” (IEEE, 2009), pp. 1142–1145.
  50. D. A. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001).
    [Crossref] [PubMed]
  51. P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 10, 36–52 (1989).
    [Crossref]
  52. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
    [Crossref] [PubMed]

2018 (1)

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

2017 (1)

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

2016 (3)

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (2)

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

2013 (3)

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

2012 (3)

2011 (5)

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

A. Sassaroli, “Fast perturbation Monte Carlo method for photon migration in heterogeneous turbid media,” Opt. Lett. 36, 2095–2097 (2011).
[Crossref] [PubMed]

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

2010 (5)

2009 (3)

2008 (3)

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (2)

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

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
[Crossref]

2004 (1)

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

2002 (1)

2001 (2)

2000 (1)

1999 (1)

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
[Crossref]

1998 (1)

1995 (1)

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

1994 (1)

1989 (1)

P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 10, 36–52 (1989).
[Crossref]

Aikawa, E.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Alerstam, E.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

Alianelli, L.

Andersson-Engels, S.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

Angelo, J. P.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Arridge, S.

Arridge, S. R.

Bartels, K. E.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Bassi, A.

Beuthan, J.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Bevilacqua, F.

Blaschke, S.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Blumetti, C.

Boas, D. A.

Bunting, C. F.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Caffini, M.

Chen, C.-W.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Chen, J.

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
[Crossref] [PubMed]

J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
[Crossref] [PubMed]

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Chen, S.-J.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Chen, Y.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Choe, R.

Contini, D.

Corlu, A.

Cubeddu, R.

Culver, J.

Culver, J. P.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

D’andrea, C.

Dai, G.

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

Dayal, R.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

de Kleine, R. H.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Dehghani, H.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[Crossref]

Ducros, N.

Dunn, A.

Dunn, A. K.

Durduran, T.

Eggebrecht, A. T.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Fang, Q.

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

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

Q. Fang and 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]

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Q. Fang and D. A. Boas, “Tetrahedral mesh generation from volumetric binary and grayscale images,” in “IEEE Int. Symp. on Biomed. Imaging: From Nano to Macro, 2009. ISBI’09,” (IEEE, 2009), pp. 1142–1145.

Feng, T.-C.

Ferradal, S. L.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Fisher, J. P.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Foi, A.

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

Gardner, A. R.

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

Gaudette, T.

Geimer, S. D.

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Gibson, A.

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

Gioux, S.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Grosenick, D.

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

Haskell, R. C.

Hassanpour, M. S.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Hayakawa, C. K.

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[Crossref]

Hebden, J.

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

Hershey, T.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Hielscher, A.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Hielscher, A. H.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
[Crossref]

Hoi, J.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Holyoak, G. R.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Illing, G.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Intes, X.

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography with compressive sensing based preconditioning,” Biomed. Opt. Express 6, 4887–4898 (2015).
[Crossref] [PubMed]

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
[Crossref] [PubMed]

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
[Crossref] [PubMed]

J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
[Crossref] [PubMed]

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Ismaelli, A.

Jacques, S. L.

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

Ji, R.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Jiang, Z.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Kaeli, D.

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

Khalil, M.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Kim, H.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Kim, H. K.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Kim, I.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Kirsch, D. G.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Lee, V. K.

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

Lesage, F.

Liu, Q.

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

Lynch, D. R.

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Makitalo, M.

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

Martelli, F.

McAdams, M. S.

Meaney, P. M.

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Montejo, L. D.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Muller, G. A.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Netz, U. J.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Nguyen, B.-N. B.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Niedre, M. J.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

Nina-Paravecino, F.

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

Ntziachristos, V.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

O’Leary, M. A.

M. A. O’Leary, “Imaging with diffuse photon density waves,” Ph.D. thesis, University of Pennsylvania, Philadelphia (1996).

Ochoa, M.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Ochoa, M. I.

M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

Ozturk, M. S.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

Paulsen, K. D.

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Pian, Q.

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
[Crossref] [PubMed]

R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography with compressive sensing based preconditioning,” Biomed. Opt. Express 6, 4887–4898 (2015).
[Crossref] [PubMed]

M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.

Piao, D.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Pifferi, A.

Pimpalkhare, M.

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

Pogue, B. W.

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Re, R.

Rinneberg, H.

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

Ripoll, J.

Ritchey, J. W.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Robichaux-Viehoever, A.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Rosen, M. A.

Rudge, T.

Sassaroli, A.

Schnall, M. D.

Schotland, J. C.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

Schweiger, M.

Shen, H.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. & Biol. 55, 947 (2010).
[Crossref]

Shrikhande, G.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Sinsuebphon, N.

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

Slobodov, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Snyder, A. Z.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Sonneveld, P.

P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 10, 36–52 (1989).
[Crossref]

Spanier, J.

Spinelli, L.

Stott, J.

Streltsov, A. V.

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Sunar, U.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Svaasand, L. O.

Svensson, T.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

Taroni, P.

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

Torricelli, A.

Tromberg, B. J.

Tsay, T.-T.

Valentini, G.

Venugopal, V.

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
[Crossref] [PubMed]

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Venugopalan, V.

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[Crossref]

Wang, G.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. & Biol. 55, 947 (2010).
[Crossref]

Wang, L.

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

Wang, L. V.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).

Weissleder, R.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Wu, H.-I.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).

Xu, G.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Yalavarthy, P. K.

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Yamada, Y.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
[Crossref]

Yao, R.

Yodh, A. G.

You, J. S.

Yu, L.

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

Zaccanti, G.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
[Crossref]

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[Crossref]

Zhao, L.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
[Crossref] [PubMed]

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

Zheng, L.

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

Zhu, C.

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

Zucchelli, L.

Zwaka, P. A.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Ann. Biomed. Eng. (1)

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (4)

Comput. Methods Programs Biomed. (1)

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

Current Opinion in Biotech. (1)

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
[Crossref]

Eur. J. Vasc. Endovasc. Surg. (1)

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

IEEE Trans. Med. Imaging (1)

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

IEEE Trans. on Image Processing (1)

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

IEEE Trans. on Medical Imaging (1)

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Inverse Probl. (1)

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

J. Near Infrared Spectrosc. (1)

J. of Biomed. Imaging (1)

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. of Biomed. Optics (7)

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

J. Opt. Soc. Am. A (3)

Med. Phys. (2)

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Nat. Photonics (2)

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (5)

Phys. Med. & Biol. (2)

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. & Biol. 55, 947 (2010).
[Crossref]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
[Crossref]

Phys. Med. Biol. (1)

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

Proc Natl Acad Sci U S A (1)

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

SIAM J. Sci. Statist. Comput. (1)

P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 10, 36–52 (1989).
[Crossref]

Other (7)

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.

Q. Fang and D. A. Boas, “Tetrahedral mesh generation from volumetric binary and grayscale images,” in “IEEE Int. Symp. on Biomed. Imaging: From Nano to Macro, 2009. ISBI’09,” (IEEE, 2009), pp. 1142–1145.

M. A. O’Leary, “Imaging with diffuse photon density waves,” Ph.D. thesis, University of Pennsylvania, Philadelphia (1996).

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).

M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Supplementary Material (1)

NameDescription
» Visualization 1       The complete profile time-gated Jacobians for all 50 time gates, associated with Figure 3.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic of the baseline MC (left) and replay MC (right) in a photon replay algorithm
Fig. 2
Fig. 2 Comparison between the adjoint and replay Jacobians: (a–c) absorption Jacobian, positive and negative components of scattering Jacobian from the adjoint approach; (d–f) corresponding Jacobians from the replay approach; (d) is also the weighted average pathlength; (g) weighted average scattering; (h–i) comparisons between the adjoint and replay Jacobians.
Fig. 3
Fig. 3 Validation of time-resolved replay Jacobian from (a) an early (0.25-0.35 ns) and (b) a late (1.15–1.25 ns) gate with the same accumulated photon weights, and (c) comparison of the change of TPSFs between pMC and replay with an inclusion of absorption perturbation. An animation of Jacobians from all time gates can be found in Visualization 1.
Fig. 4
Fig. 4 Reconstructions with Born (top row) and Rytov approximations (bottom): the recovered (a, d) absorption and (b, e) scattering profiles are shown on the z = 10 mm and y = 20 mm planes, and (c, f) compare reconstructed and ground truth values along y-axis on the z = 10 mm plane. The locations of 64 sources (green triangles) and 36 detectors (white crosses) are overlaid in (a).
Fig. 5
Fig. 5 The (a) top and (b) front view of reconstruction results with 5 wavelengths, compared to the ground truth (black dashed lines); (c) wavelength dependent absorption coefficients for background medium, 0.008‰ Epolight 2735 and 0.024‰ India Ink; (d) reconstructed crosstalk and concentration ratios of two absorbers with different wavelength channels.
Fig. 6
Fig. 6 Contour plots of the replay run-times divided by the adjoint run-times at various source-detector numbers: (a) a point-source/detector system, (b) a single-pixel camera hyperspectral system, and (c) a multispectral point-detection system with iterative reconstructions.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

w ^ = w ( μ ^ s μ s ) p exp ( ( μ ^ s μ s ) L ) exp ( ( μ ^ a μ a ) L ) ,
w ^ = w j = 1 M ( μ ^ s ( Ω j ) μ s ( Ω j ) ) p ( Ω j ) exp ( δ μ s ( Ω j ) L ( Ω j ) ) exp ( δ μ a ( Ω j ) L ( Ω j ) ) ,
W ^ ( T ) = k = 1 N T w k j = 1 M ( μ ^ s ( Ω j ) μ s ( Ω j ) ) p k ( Ω j , T ) exp ( δ μ s ( Ω j ) L k ( Ω j , T ) ) exp ( δ μ a ( Ω j ) L k ( Ω j , T ) ) ,
δ W μ ^ a ( Ω j ) = W ^ μ ^ a ( Ω j ) W = k = 1 N T w k [ exp ( δ μ a ( Ω j ) L k ( Ω j , T ) ) 1 ] ,
δ W μ ^ s ( Ω j ) = W ^ μ ^ s ( Ω j ) W = k = 1 N T w k ( μ s ( Ω j ) + δ μ s ( Ω j ) μ s ( Ω j ) ) p k ( Ω j , T ) exp ( δ μ s ( Ω j ) L k ( Ω j , T ) ) ,
J μ a ( Ω j , T ) = lim δ μ a ( Ω j ) 0 δ W μ ^ a ( Ω j ) δ μ a ( Ω j ) = k = 1 N T w k L k ( Ω j , T ) ,
J μ s ( Ω j , T ) = lim δ μ s ( Ω j ) 0 δ W μ ^ s ( Ω j ) δ μ s ( Ω j ) = k = 1 N T w k ( p k ( Ω j , T ) μ s ( Ω j ) L k ( Ω j , T ) ) .
J μ a ( Ω j , T ) = L ¯ ( Ω j , T ) ,
J μ s ( Ω j , T ) = p ¯ ( Ω j , T ) μ s ( Ω j ) L ¯ ( Ω j , T ) ,
J μ a ( Ω j ) = Ω j G ( r j , r s ) Φ ( r d , r j ) Φ ( r d , r s ) d V ,
J μ s ( Ω j ) = Ω j G ( r j , r s ) Φ ( r d , r j ) 3 μ s 2 ( 1 g ) Φ ( r d , r s ) d V ,
Φ ( r d , r ) = R ( r d , r ) = detected w A d all w 0 ,
Φ ( r d , r ) = J n ( r d , r ) = J ( r d , r ) n = detected w ( v n ) A d all w 0 ,
J μ a ( Ω j ) = α Ω j G ( r j , r s ) G ( r j , r d ) Φ ( r d , r s ) d V ,
J μ s ( Ω j ) = α Ω j G ( r j , r s ) G ( r j , r d ) 3 μ s 2 ( 1 g ) Φ ( r d , r s ) d V .
Δ μ = J + Δ Φ ,
Δ μ = ( J T J + λ L T L ) 1 J T Δ Φ ,
Δ μ = ( L T L ) 1 J T [ J ( L T L ) 1 J T + λ I ] 1 Δ Φ ,

Metrics