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October  2015, 8(5): 953-967. doi: 10.3934/dcdss.2015.8.953

Numerical algorithm for tracking cell dynamics in 4D biomedical images

Karol Mikula 1, , Róbert Špir 1, and  Nadine Peyriéras 2,

1. 

Department of Mathematics, Slovak University of Technology, Radlinskeho 11, 813 68 Bratislava, Slovak Republic
2. 

Institut de Neurobiologie Alfred Fessard, CNRS UPR 3294, Av. de la Terrasse, 91198 Gif-sur-Yvette, France

Received  January 2014 Revised  July 2014 Published  July 2015

The paper presents new numerical algorithm for an automated cell tracking from large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. The cell trajectories are extracted as centered paths inside segmented spatio-temporal tree structures representing cell movements and divisions. Such paths are found by using a suitably designed and computed constrained distance functions and by a backtracking in steepest descent direction of a potential field based on these distance functions combination. The naturally parallelizable discretization of the eikonal equation which is used for computing distance functions is given and results of the tracking method for real 4D image data are presented and discussed.
Keywords: 4D images, image processing., cell dynamics, distance function, Cell tracking.
Mathematics Subject Classification: 35Q9.
Citation: Karol Mikula, Róbert Špir, Nadine Peyriéras. Numerical algorithm for tracking cell dynamics in 4D biomedical images. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 953-967. doi: 10.3934/dcdss.2015.8.953
References:
[1]

Y. Bellaīche, F. Bosveld, F. Graner, K. Mikula, M. Remešíková and M. Smíšek, New Robust Algorithm for Tracking Cells in Videos of Drosophila Morphogenesis Based on Finding an Ideal Path in Segmented Spatio-Temporal Cellular Structures, Proceeding of the 33rd Annual International IEEE EMBS Conference, Boston Marriott Copley Place, Boston, MA, USA, August 30 - September 3, 2011, IEEE Press, 2011. Google Scholar

[2]

P. Bourgine, R. Čunderlík, O. Drblíková, K. Mikula, N. Peyriéras, M. Remešíková, B. Rizzi and A. Sarti, 4D embryogenesis image analysis using PDE methods of image processing, Kybernetika, 46 (2010), 226-259.  Google Scholar

[3]

P. Bourgine, P. Frolkovič, K. Mikula, N. Peyriéras and M. Remešíková, Extraction of the intercellular skeleton from 2D microscope images of early embryogenesis, Lecture Notes in Computer Science 5567 (Proceeding of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision, Voss, Norway, June 1-5,2009), Springer, (2009), 38-49. Google Scholar

[4]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 694-699. doi: 10.1109/ICCV.1995.466871.  Google Scholar

[5]

Y. Chen, B. C. Vemuri and L. Wang, Image denoising and segmentation via nonlinear diffusion, Comput. Math. Appl., 39 (2000), 131-149. doi: 10.1016/S0898-1221(00)00050-X.  Google Scholar

[6]

P. Frolkovič, K. Mikula, N. Peyriéras and A. Sarti, A counting number of cells and cell segmentation using advection-diffusion equations, Kybernetika, 43 (2007), 817-829.  Google Scholar

[7]

S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301. doi: 10.1007/BF00379537.  Google Scholar

[8]

Z. Krivá, K. Mikula, N. Peyriéras, B. Rizzi, A. Sarti and O. Stašová, 3D early embryogenesis image filtering by nonlinear partial differential equations, Medical Image Analysis, 14 (2010), 510-526. Google Scholar

[9]

C. Melani, Algoritmos de Procesamiento de Imagenes Para la Reconstruccion Del Desarrollo Embrionario Del Pez Cebra, Computer Science PhD Thesis, UBA, FCEN. Buenos Aires, Argentina, 2013. Google Scholar

[10]

K. Mikula, N. Peyriéras, M. Remešíková and M. Smíšek, 4D numerical schemes for cell image segmentation and tracking, Finite Volumes in Complex Applications VI, Problems & Perspectives, Eds. J. Fořt et al. (Proceedings of the Sixth International Conference on Finite Volumes in Complex Applications, Prague, June 6-10, 2011), Springer Verlag, 4 (2011), 693-701. doi: 10.1007/978-3-642-20671-9_73.  Google Scholar

[11]

K. Mikula, N. Peyriéras, M. Remešíková and O. Stašová, Segmentation of 3D cell membrane images by PDE methods and its applications, Computers in Biology and Medicine, 41 (2011), 326-339. doi: 10.1016/j.compbiomed.2011.03.010.  Google Scholar

[12]

K. Mikula and J. Urbán, 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, Proceedings of the Third International Conference on Scale Space Methods and Variational Methods in ComputerVision, Ein Gedi, Israel, 2011, Lecture Notes in Computer Science, 6667 (2012), 640-652. doi: 10.1007/978-3-642-24785-9_54.  Google Scholar

[13]

R. Mikut, T. Dickmeis, W. Driever, P. Geurts, F. A. Hamprecht, B. X. Kausler, M. J. Ledesma-Carbayo, R. Marée, K. Mikula, P. Pantazis, O. Ronneberger, A. Santos, R. Stotzka, U. Strähle and N. Peyriéras, Automated processing of zebrafish imaging data: A survey, Zebrafish, 10 (2013), 401-421. doi: 10.1089/zeb.2013.0886.  Google Scholar

[14]

E. Rouy and A. Tourin, Viscosity solutions approach to shape-from-shading, SIAM Journal on Numerical Analysis, 29 (1992), 867-884. doi: 10.1137/0729053.  Google Scholar

[15]

A. Sarti, R. Malladi and J. A. Sethian, Subjective surfaces: A method for completing missing boundaries, Proceedings of the National Academy of Sciences of the UnitedStates of America, 97 (2000), 6258-6263. doi: 10.1073/pnas.110135797.  Google Scholar

[16]

C. Zanella, M. Campana, B. Rizzi, C. Melani, G. Sanguinetti, P. Bourgine, K. Mikula, N. Peyrieras and A. Sarti, Cells segmentation from 3-D confocal images of early zebrafish embryogenesis, IEEE Transactions on Image Processing, 19 (2010), 770-781. doi: 10.1109/TIP.2009.2033629.  Google Scholar

show all references

References:
[1]

Y. Bellaīche, F. Bosveld, F. Graner, K. Mikula, M. Remešíková and M. Smíšek, New Robust Algorithm for Tracking Cells in Videos of Drosophila Morphogenesis Based on Finding an Ideal Path in Segmented Spatio-Temporal Cellular Structures, Proceeding of the 33rd Annual International IEEE EMBS Conference, Boston Marriott Copley Place, Boston, MA, USA, August 30 - September 3, 2011, IEEE Press, 2011. Google Scholar

[2]

P. Bourgine, R. Čunderlík, O. Drblíková, K. Mikula, N. Peyriéras, M. Remešíková, B. Rizzi and A. Sarti, 4D embryogenesis image analysis using PDE methods of image processing, Kybernetika, 46 (2010), 226-259.  Google Scholar

[3]

P. Bourgine, P. Frolkovič, K. Mikula, N. Peyriéras and M. Remešíková, Extraction of the intercellular skeleton from 2D microscope images of early embryogenesis, Lecture Notes in Computer Science 5567 (Proceeding of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision, Voss, Norway, June 1-5,2009), Springer, (2009), 38-49. Google Scholar

[4]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 694-699. doi: 10.1109/ICCV.1995.466871.  Google Scholar

[5]

Y. Chen, B. C. Vemuri and L. Wang, Image denoising and segmentation via nonlinear diffusion, Comput. Math. Appl., 39 (2000), 131-149. doi: 10.1016/S0898-1221(00)00050-X.  Google Scholar

[6]

P. Frolkovič, K. Mikula, N. Peyriéras and A. Sarti, A counting number of cells and cell segmentation using advection-diffusion equations, Kybernetika, 43 (2007), 817-829.  Google Scholar

[7]

S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301. doi: 10.1007/BF00379537.  Google Scholar

[8]

Z. Krivá, K. Mikula, N. Peyriéras, B. Rizzi, A. Sarti and O. Stašová, 3D early embryogenesis image filtering by nonlinear partial differential equations, Medical Image Analysis, 14 (2010), 510-526. Google Scholar

[9]

C. Melani, Algoritmos de Procesamiento de Imagenes Para la Reconstruccion Del Desarrollo Embrionario Del Pez Cebra, Computer Science PhD Thesis, UBA, FCEN. Buenos Aires, Argentina, 2013. Google Scholar

[10]

K. Mikula, N. Peyriéras, M. Remešíková and M. Smíšek, 4D numerical schemes for cell image segmentation and tracking, Finite Volumes in Complex Applications VI, Problems & Perspectives, Eds. J. Fořt et al. (Proceedings of the Sixth International Conference on Finite Volumes in Complex Applications, Prague, June 6-10, 2011), Springer Verlag, 4 (2011), 693-701. doi: 10.1007/978-3-642-20671-9_73.  Google Scholar

[11]

K. Mikula, N. Peyriéras, M. Remešíková and O. Stašová, Segmentation of 3D cell membrane images by PDE methods and its applications, Computers in Biology and Medicine, 41 (2011), 326-339. doi: 10.1016/j.compbiomed.2011.03.010.  Google Scholar

[12]

K. Mikula and J. Urbán, 3D curve evolution algorithm with tangential redistribution for a fully automatic finding of an ideal camera path in virtual colonoscopy, Proceedings of the Third International Conference on Scale Space Methods and Variational Methods in ComputerVision, Ein Gedi, Israel, 2011, Lecture Notes in Computer Science, 6667 (2012), 640-652. doi: 10.1007/978-3-642-24785-9_54.  Google Scholar

[13]

R. Mikut, T. Dickmeis, W. Driever, P. Geurts, F. A. Hamprecht, B. X. Kausler, M. J. Ledesma-Carbayo, R. Marée, K. Mikula, P. Pantazis, O. Ronneberger, A. Santos, R. Stotzka, U. Strähle and N. Peyriéras, Automated processing of zebrafish imaging data: A survey, Zebrafish, 10 (2013), 401-421. doi: 10.1089/zeb.2013.0886.  Google Scholar

[14]

E. Rouy and A. Tourin, Viscosity solutions approach to shape-from-shading, SIAM Journal on Numerical Analysis, 29 (1992), 867-884. doi: 10.1137/0729053.  Google Scholar

[15]

A. Sarti, R. Malladi and J. A. Sethian, Subjective surfaces: A method for completing missing boundaries, Proceedings of the National Academy of Sciences of the UnitedStates of America, 97 (2000), 6258-6263. doi: 10.1073/pnas.110135797.  Google Scholar

[16]

C. Zanella, M. Campana, B. Rizzi, C. Melani, G. Sanguinetti, P. Bourgine, K. Mikula, N. Peyrieras and A. Sarti, Cells segmentation from 3-D confocal images of early zebrafish embryogenesis, IEEE Transactions on Image Processing, 19 (2010), 770-781. doi: 10.1109/TIP.2009.2033629.  Google Scholar

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