Abstract

Monte Carlo methods commonly used in tissue optics are limited to a layered tissue geometry and thus provide only a very rough approximation for many complex media such as biological structures. To overcome these limitations, a Meshed Monte Carlo method with flexible phase function choice (fpf-MC) has been developed to function in a mesh. This algorithm can model the light propagation in any complexly shaped structure, by attributing optical properties to the different mesh elements. Furthermore, this code allows the use of different discretized phase functions for each tissue type, which can be simulated from the microstructural properties of the tissue, in combination with a tool for simulating the bulk optical properties of polydisperse suspensions. As a result, the scattering properties of tissues can be estimated from information on the microstructural properties of the tissue. This is important for the estimation of the bulk optical properties that can be used for the light propagation model, since many types of tissue have never been characterized in literature. The combination of these contributions, made it possible to use the MMC-fpf for modeling the light porapagation in plant tissue. The developed Meshed Monte Carlo code with flexible phase function choice (MMC-fpf) was successfully validated in simulation through comparison with the Monte Carlo code in Multi-Layered tissues (R2 > 0.9999) and experimentally by comparing the measured and simulated reflectance (RMSE = 0.015%) and transmittance (RMSE = 0.0815%) values for tomato leaves.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]

2014 (2)

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

2013 (1)

2012 (1)

R. Watté, B. Aernouts, and W. Saeys, “A multilayer Monte Carlo method with free phase function choice,” Proc. SPIE 8429, 84290S (2012).

2011 (2)

J. R. Austin and L. A. Staehelin, “Three-dimensional architecture of grana and stroma thylakoids of higher plants as determined by electron tomography,” Plant Physiol. 155(4), 1601–1611 (2011).
[Crossref] [PubMed]

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

2010 (1)

2009 (2)

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

2008 (4)

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

M. Steele, A. A. Gitelson, and D. Rundquist, “Nondestructive estimation of leaf chlorophyll content in grapes,” Am. J. Enol. Viticult 59, 299–305 (2008).

G. Raszewski, B. A. Diner, E. Schlodder, and T. Renger, “Spectroscopic properties of reaction center pigments in photosystem II core complexes: revision of the multimer model,” Biophys. J. 95(1), 105–119 (2008).
[Crossref] [PubMed]

2007 (4)

T. A. Dittmer, N. J. Stacey, K. Sugimoto-Shirasu, and E. J. Richards, “LITTLE NUCLEI genes affecting nuclear morphology in Arabidopsis thaliana,” Plant Cell 19(9), 2793–2803 (2007).
[Crossref] [PubMed]

G. P. Henderson, L. Gan, and G. J. Jensen, “3-D ultrastructure of O. tauri: electron cryotomography of an entire eukaryotic cell,” PLoS ONE 2(8), e749 (2007).
[Crossref] [PubMed]

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

S. K. Sharma, S. Banerjee, and M. K. Yadav, “Light propagation in a fractal tissue model: a critical study of the phase function,” J. Opt. A, Pure Appl. Opt. 9(1), 49–55 (2007).
[Crossref]

2006 (1)

2005 (4)

D. Passos, J. C. Hebden, P. N. Pinto, and R. Guerra, “Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution,” J. Biomed. Opt. 10(6), 064036 (2005).
[Crossref] [PubMed]

S. K. Sharma and S. Banerjee, “Volume concentration and size dependence of diffuse reflectance in a fractal soft tissue model,” Med. Phys. 32(6), 1767–1774 (2005).
[Crossref] [PubMed]

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D Appl. Phys. 38(15), 2543–2555 (2005).
[Crossref]

W. Hong, “SNAREs and traffic,” Biochim. Biophys. Acta 1744(2), 120–144 (2005).
[Crossref] [PubMed]

2001 (1)

2000 (2)

S. Jacquemoud, “Comparison of four radiative transfer models to simulate plant canopies reflectance direct and inverse mode,” Remote Sens. Environ. 74(3), 471–481 (2000).
[Crossref]

C. Daughtry, “Estimating corn leaf chlorophyll concentration from leaf and canopy reflectance,” Remote Sens. Environ. 74(2), 229–239 (2000).
[Crossref]

1999 (1)

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” JOSA A 16(12), 2935–2945 (1999).
[Crossref]

1998 (1)

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

1997 (1)

1996 (1)

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41(10), 2221–2227 (1996).
[Crossref] [PubMed]

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(2), 131–146 (1995).
[Crossref] [PubMed]

1992 (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Abera, M.

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Aernouts, B.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref] [PubMed]

R. Watté, B. Aernouts, and W. Saeys, “A multilayer Monte Carlo method with free phase function choice,” Proc. SPIE 8429, 84290S (2012).

Asner, G. P.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Austin, J. R.

J. R. Austin and L. A. Staehelin, “Three-dimensional architecture of grana and stroma thylakoids of higher plants as determined by electron tomography,” Plant Physiol. 155(4), 1601–1611 (2011).
[Crossref] [PubMed]

Avrillier, S.

G. Bernard, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms : a fractal approach Phase function simulation in tissue phantoms : a fractal approach,” Pure Appl. Opt. 377 (1996).

Badal, A.

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

Badano, A.

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

Banerjee, S.

S. K. Sharma, S. Banerjee, and M. K. Yadav, “Light propagation in a fractal tissue model: a critical study of the phase function,” J. Opt. A, Pure Appl. Opt. 9(1), 49–55 (2007).
[Crossref]

S. K. Sharma and S. Banerjee, “Volume concentration and size dependence of diffuse reflectance in a fractal soft tissue model,” Med. Phys. 32(6), 1767–1774 (2005).
[Crossref] [PubMed]

Banh, D. P.

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

Barry, K. M.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

Bashkatov, A. N.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D Appl. Phys. 38(15), 2543–2555 (2005).
[Crossref]

Baumbach, T.

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Bernard, G.

G. Bernard, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms : a fractal approach Phase function simulation in tissue phantoms : a fractal approach,” Pure Appl. Opt. 377 (1996).

Beullens, K.

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

Bevilacqua, F.

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(17), 1335–1337 (2001).
[Crossref] [PubMed]

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” JOSA A 16(12), 2935–2945 (1999).
[Crossref]

Bidel, L. P. R.

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Bigio, I. J.

Bobelyn, E.

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

Bongaers, E.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Boyer, J.

Daughtry, C.

C. Daughtry, “Estimating corn leaf chlorophyll concentration from leaf and canopy reflectance,” Remote Sens. Environ. 74(2), 229–239 (2000).
[Crossref]

De Block, J.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

Defraeye, T.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Delport, F.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

Depeursinge, C.

F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” JOSA A 16(12), 2935–2945 (1999).
[Crossref]

Diner, B. A.

G. Raszewski, B. A. Diner, E. Schlodder, and T. Renger, “Spectroscopic properties of reaction center pigments in photosystem II core complexes: revision of the multimer model,” Biophys. J. 95(1), 105–119 (2008).
[Crossref] [PubMed]

Dittmer, T. A.

T. A. Dittmer, N. J. Stacey, K. Sugimoto-Shirasu, and E. J. Richards, “LITTLE NUCLEI genes affecting nuclear morphology in Arabidopsis thaliana,” Plant Cell 19(9), 2793–2803 (2007).
[Crossref] [PubMed]

Dunn, A. K.

Fang, Q.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Feret, J. B.

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Féret, J. B.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

François, C.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Frank, J.

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

Fuselier, T.

Gan, L.

G. P. Henderson, L. Gan, and G. J. Jensen, “3-D ultrastructure of O. tauri: electron cryotomography of an entire eukaryotic cell,” PLoS ONE 2(8), e749 (2007).
[Crossref] [PubMed]

Genina, E. A.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D Appl. Phys. 38(15), 2543–2555 (2005).
[Crossref]

Gitelson, A.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

Gitelson, A. A.

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

M. Steele, A. A. Gitelson, and D. Rundquist, “Nondestructive estimation of leaf chlorophyll content in grapes,” Am. J. Enol. Viticult 59, 299–305 (2008).

Grassucci, R. A.

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

Guerra, R.

D. Passos, J. C. Hebden, P. N. Pinto, and R. Guerra, “Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution,” J. Biomed. Opt. 10(6), 064036 (2005).
[Crossref] [PubMed]

Hayakawa, C. K.

Hebden, J. C.

D. Passos, J. C. Hebden, P. N. Pinto, and R. Guerra, “Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution,” J. Biomed. Opt. 10(6), 064036 (2005).
[Crossref] [PubMed]

Helfen, L.

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Henderson, G. P.

G. P. Henderson, L. Gan, and G. J. Jensen, “3-D ultrastructure of O. tauri: electron cryotomography of an entire eukaryotic cell,” PLoS ONE 2(8), e749 (2007).
[Crossref] [PubMed]

Herremans, E.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Hertog, M.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Ho, Q.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Ho, Q. T.

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Hong, W.

W. Hong, “SNAREs and traffic,” Biochim. Biophys. Acta 1744(2), 120–144 (2005).
[Crossref] [PubMed]

Jacquemoud, S.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

S. Jacquemoud, “Comparison of four radiative transfer models to simulate plant canopies reflectance direct and inverse mode,” Remote Sens. Environ. 74(3), 471–481 (2000).
[Crossref]

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(2), 131–146 (1995).
[Crossref] [PubMed]

Jensen, G. J.

G. P. Henderson, L. Gan, and G. J. Jensen, “3-D ultrastructure of O. tauri: electron cryotomography of an entire eukaryotic cell,” PLoS ONE 2(8), e749 (2007).
[Crossref] [PubMed]

Johnson, T. M.

Kienle, A.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41(10), 2221–2227 (1996).
[Crossref] [PubMed]

Kochubey, V. I.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D Appl. Phys. 38(15), 2543–2555 (2005).
[Crossref]

Kyprianou, I.

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

Lammertyn, J.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref] [PubMed]

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

le Maire, G.

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Martin, R. E.

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

Merchiers, M.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

Mourant, J. R.

Nicolai, B.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Nicolaï, B.

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

Nicolaï, B. M.

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Palmer, G. M.

Panigada, C.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

Passos, D.

D. Passos, J. C. Hebden, P. N. Pinto, and R. Guerra, “Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution,” J. Biomed. Opt. 10(6), 064036 (2005).
[Crossref] [PubMed]

Patterson, M. S.

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41(10), 2221–2227 (1996).
[Crossref] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Peirs, A.

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

Pinto, P. N.

D. Passos, J. C. Hebden, P. N. Pinto, and R. Guerra, “Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution,” J. Biomed. Opt. 10(6), 064036 (2005).
[Crossref] [PubMed]

Ramanujam, N.

Raszewski, G.

G. Raszewski, B. A. Diner, E. Schlodder, and T. Renger, “Spectroscopic properties of reaction center pigments in photosystem II core complexes: revision of the multimer model,” Biophys. J. 95(1), 105–119 (2008).
[Crossref] [PubMed]

Renger, T.

G. Raszewski, B. A. Diner, E. Schlodder, and T. Renger, “Spectroscopic properties of reaction center pigments in photosystem II core complexes: revision of the multimer model,” Biophys. J. 95(1), 105–119 (2008).
[Crossref] [PubMed]

Richards, E. J.

T. A. Dittmer, N. J. Stacey, K. Sugimoto-Shirasu, and E. J. Richards, “LITTLE NUCLEI genes affecting nuclear morphology in Arabidopsis thaliana,” Plant Cell 19(9), 2793–2803 (2007).
[Crossref] [PubMed]

Richardson, A. D.

J. B. Féret, C. François, A. Gitelson, G. P. Asner, K. M. Barry, C. Panigada, A. D. Richardson, and S. Jacquemoud, “Optimizing spectral indices and chemometric analysis of leaf chemical properties using radiative transfer modeling,” Remote Sens. Environ. 115(10), 2742–2750 (2011).
[Crossref]

Rogge, S.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Rundquist, D.

M. Steele, A. A. Gitelson, and D. Rundquist, “Nondestructive estimation of leaf chlorophyll content in grapes,” Am. J. Enol. Viticult 59, 299–305 (2008).

Saeys, W.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref] [PubMed]

R. Watté, B. Aernouts, and W. Saeys, “A multilayer Monte Carlo method with free phase function choice,” Proc. SPIE 8429, 84290S (2012).

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
[Crossref]

Schlodder, E.

G. Raszewski, B. A. Diner, E. Schlodder, and T. Renger, “Spectroscopic properties of reaction center pigments in photosystem II core complexes: revision of the multimer model,” Biophys. J. 95(1), 105–119 (2008).
[Crossref] [PubMed]

Sempau, J.

A. Badal, I. Kyprianou, D. P. Banh, A. Badano, and J. Sempau, “penMesh--Monte Carlo radiation transport simulation in a triangle mesh geometry,” IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009).
[Crossref] [PubMed]

Sharma, S. K.

S. K. Sharma, S. Banerjee, and M. K. Yadav, “Light propagation in a fractal tissue model: a critical study of the phase function,” J. Opt. A, Pure Appl. Opt. 9(1), 49–55 (2007).
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S. K. Sharma and S. Banerjee, “Volume concentration and size dependence of diffuse reflectance in a fractal soft tissue model,” Med. Phys. 32(6), 1767–1774 (2005).
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Spanier, J.

Srivastava, S.

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

Stacey, N. J.

T. A. Dittmer, N. J. Stacey, K. Sugimoto-Shirasu, and E. J. Richards, “LITTLE NUCLEI genes affecting nuclear morphology in Arabidopsis thaliana,” Plant Cell 19(9), 2793–2803 (2007).
[Crossref] [PubMed]

Staehelin, L. A.

J. R. Austin and L. A. Staehelin, “Three-dimensional architecture of grana and stroma thylakoids of higher plants as determined by electron tomography,” Plant Physiol. 155(4), 1601–1611 (2011).
[Crossref] [PubMed]

Steele, M.

M. Steele, A. A. Gitelson, and D. Rundquist, “Nondestructive estimation of leaf chlorophyll content in grapes,” Am. J. Enol. Viticult 59, 299–305 (2008).

Sugimoto-Shirasu, K.

T. A. Dittmer, N. J. Stacey, K. Sugimoto-Shirasu, and E. J. Richards, “LITTLE NUCLEI genes affecting nuclear morphology in Arabidopsis thaliana,” Plant Cell 19(9), 2793–2803 (2007).
[Crossref] [PubMed]

Theron, K.

B. Nicolaï, K. Beullens, E. Bobelyn, A. Peirs, W. Saeys, K. Theron, and J. Lammertyn, “Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: A review,” Postharvest Biol. Technol. 46(2), 99–118 (2007).
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Tinet, E.

G. Bernard, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms : a fractal approach Phase function simulation in tissue phantoms : a fractal approach,” Pure Appl. Opt. 377 (1996).

Tromberg, B. J.

Tsuta, M.

Tualle, J. M.

G. Bernard, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms : a fractal approach Phase function simulation in tissue phantoms : a fractal approach,” Pure Appl. Opt. 377 (1996).

Tuchin, V. V.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm,” J. Phys. D Appl. Phys. 38(15), 2543–2555 (2005).
[Crossref]

Ustin, S. L.

J. B. Féret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

J. B. Feret, C. François, G. P. Asner, A. A. Gitelson, R. E. Martin, L. P. R. Bidel, S. L. Ustin, G. le Maire, and S. Jacquemoud, “PROSPECT-4 and 5: Advances in the leaf optical properties model separating photosynthetic pigments,” Remote Sens. Environ. 112(6), 3030–3043 (2008).
[Crossref]

G. M. Palmer, N. Ramanujam, M. M. Verstraete, and S. L. Ustin, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45(5), 1062–1071 (2006).
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Van Beers, R.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
[Crossref]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref] [PubMed]

Venugopalan, V.

Verboven, P.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

Verlinden, B.

E. Herremans, P. Verboven, T. Defraeye, S. Rogge, Q. Ho, M. Hertog, B. Verlinden, E. Bongaers, M. Wevers, and B. Nicolai, “X-ray CT for quantitative food microstructure engineering: The apple case,” Nucl. Instr. & Meth. Phys. B 324, 88–94 (2014).
[Crossref]

Verschoor, A.

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

Verstraete, M. M.

Wang, L.

Warner, J. R.

A. Verschoor, J. R. Warner, S. Srivastava, R. A. Grassucci, and J. Frank, “Three-dimensional structure of the yeast ribosome,” Nucleic Acids Res. 26(2), 655–661 (1998).
[Crossref] [PubMed]

Watté, R.

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
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S. K. Sharma, S. Banerjee, and M. K. Yadav, “Light propagation in a fractal tissue model: a critical study of the phase function,” J. Opt. A, Pure Appl. Opt. 9(1), 49–55 (2007).
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F. Bevilacqua and C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” JOSA A 16(12), 2935–2945 (1999).
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S. K. Sharma and S. Banerjee, “Volume concentration and size dependence of diffuse reflectance in a fractal soft tissue model,” Med. Phys. 32(6), 1767–1774 (2005).
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B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid® phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref] [PubMed]

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium: experimental validation,” Opt. Express 22, 20112–20238 (2014).
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R. Watté, B. Aernouts, and W. Saeys, “A multilayer Monte Carlo method with free phase function choice,” Proc. SPIE 8429, 84290S (2012).

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P. Verboven, E. Herremans, L. Helfen, Q. T. Ho, M. Abera, T. Baumbach, M. Wevers, and B. M. Nicolaï, “Synchrotron X-ray computed laminography of the 3-D anatomy of tomato leaves,” Plant J. (to be published).

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Figures (15)

Fig. 1
Fig. 1 Flowchart of the MMC-fpf algorithm.
Fig. 2
Fig. 2 Visualisation of the MMC-fpf algorithm: the figure illustrates how an absorption profile is generated and how the photon packets move from one element to another. The scattering process is defined by a predefined phase function, which can be computed from Mie theory (instead of an Henyey-Greenstein approximation). Even though the resulting anisotropy factor is the same, the phase functions are different.
Fig. 3
Fig. 3 Barycentric coordinate system.
Fig. 4
Fig. 4 Four-layered homogeneous slab used for in silico validation.
Fig. 5
Fig. 5 (top) Particle size distribution of scatterers in the tissue (bottom left); Phase function computed through Mie theory, starting from the PSD (bottom right); Henyey-Greenstein approximation with the same value of the anisotropy factor.
Fig. 6
Fig. 6 (A) Synchrotron computed laminography image of a brick-like section of a tomato leaf; (B) corresponding mesh for MMC-fpf. The different colors are the different types of tissues in the tomato leaf. Different optical properties were attributed to different types of tissue.
Fig. 7
Fig. 7 Resulting phase function at 680 nm, expressed in a logarithmic scale. (A), (B), (C), (D), (E) and (F) correspond to mitochondria, peroxisomes, nuclei, golgi stacks, ribosomes in the epidermis domain, and grana in the chloroplast domain, respectively.
Fig. 8
Fig. 8 Estimated bulk scattering coefficient spectra for the different cell organelles in the epidermis as a function of the wavelength.
Fig. 9
Fig. 9 Scatter plot of reflectance, fraction of absorbed energy and transmittance values simulated with MMC-fpf and MCML for 4 different homogeneous slabs.
Fig. 10
Fig. 10 Absorption profiles MCML vs. MMC-fpf in log10-scale (fraction absorbed in each pixel, log10 scale).
Fig. 11
Fig. 11 Ratio of absorption profiles generated with MCML vs. MMC-fpf.
Fig. 12
Fig. 12 Comparison of measured (solid lines, with confidence intervals) and simulated (stars) transmittance and reflectance spectra for the 5 tomato leaves (Growdena).
Fig. 13
Fig. 13 (A) Cross section of a synchrotron computed laminography image (dark blue: air; medium blue: cytosol; light blue: epidermis; red: chloroplast; brown: vacuole); (B) Percentage of the photon energy stored in each pixel, expressed on a log10-scale.
Fig. 14
Fig. 14 (A) Cross section of a synchrotron computed laminography image (dark blue: air; medium blue: cytosol; red: chloroplast; brown: vacuole); (B) Percentage of photon energy stored in each pixel, expressed on a log10-scale.
Fig. 15
Fig. 15 Simulated 3D absorbance profile (percentage of energy absorbed) for the cytosol layers where absorbance is expressed on a log10 scale.

Tables (2)

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Table 1 Organelles and their sizes used for computing the optical properties of the different subcellular structures

Tables Icon

Table 2 Overview of the optical configurations used for the in silico validation. All the simulations use the same meshed multi-layered grid. Simulations for 1-layered tissues were accomplished by using the same optical properties for all 4 layers of the mesh.

Equations (8)

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

x= λ 1 x 1 + λ 2 x 2 + λ 3 x 3 y= λ 1 y 1 + λ 2 y 2 + λ 3 y 3
λ 1 ( x 1 x 3 )+ λ 2 ( x 2 x 3 )+ λ 3 ( x 3 x )=0 λ 1 ( y 1 y 3 )+ λ 2 ( y 2 y 3 )+ λ 3 ( y 3 y )=0
Tλ=r r 3 λ= T 1 (r r 3 )
0< λ i <1,i
ΔW=W μ a μ a + μ s
W= W 0 exp( μ a L)
PS D V ( r )= o e ( r r ¯ ) 2 2 σ r 2 σ r 2π dr
μ s,bulk = μ s,i p( θ )=( μ s,i p i ( θ ) )/ μ s,bulk

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