Topological Flow Structures in a Mathematical Model for Rotation-Mediated Cell Aggregation
In this paper we applyvector field topology methods to amathematical model for the fluid dynamics of reaggregation processes in tissue engineering. The experimental background are dispersed embryonic retinal cells, which reaggregate in a rotation culture on a gyratory shaker, according to defined rotation and culture conditions. Under optimal conditions, finally complex 3D spheres result. In order to optimize high throughput drug testing systems of biological cell and tissue models, a major aim is to understand the role which the fluid dynamics plays in this process. To allow for a mathematical analysis, an experimental model system was set up, using micro-beads instead of spheres within the culture dish. The qualitative behavior of this artificial model was monitored in time by using a camera. For this experimental setup amathematical model for the bead-fluid dynamics was derived, analyzed and simulated. The simulations showed that the beads form distinctive clusters in a layer close to the bottom of the petri dish. To analyze these patterns further, we perform a topological analysis of thevelocity field within this layer of the fluid. We find that traditional two-dimensional visualization techniques like path lines, streak lines and currenttime-dependent topology approaches are not able to answer the posed questions, for example they do not allow to find the location of clusters. We discuss the problems of these techniques and develop a new approach that measures thedensity of advected particles in the flow to find the moving point of particleaggregation. Using thedensity field the path of the movingaggregation point is extracted.
KeywordsVisualization Technique Density Grid Path Line Lagrangian Coherent Structure Gyratory Shaker
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We like to thank C. Garth for supplying his DOPRI5 code and W. Hackbusch, S. Luckhaus, and J.J.L. Velázquez for fruitful discussions on themathematical model and the numerical code. During the course of this work A. Wiebel was supported by DFG grant SCHE 663/3-8 and hired by the BSV group at the University of Leipzig. R. Chan and C. Wolf were supported by the DFG Graduate College InterNeuro (GRK 1097) as well as A. Robitzki and A. Stevens as members and supervisors of this College. Further, A. Robitzki was supported by the DFG via SFB 610 - Z5. A. Stevens’ work for this project was done, while she was hired by the Max-Planck-Institute for Mathematics in the Sciences, Leipzig.
- 1.G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, 1967. Cambridge Mathematical Library Edition.Google Scholar
- 2.B. Cabral and L. C. Leedom. Imaging Vector Fields Using Line Integral Convolution. In SIGGRAPH ’93, pages 263–270. ACM Press, 1993.Google Scholar
- 3.Raymond Chan. A Biofluid Dynamic Model for Centrifugal Accelerated Cell Culture Systems. PhD thesis, Fakultät für Mathematik und Informatik, Universität Leizig, DFG-Graduate College InterNeuro (GRK 1097), 2008. Preprint.Google Scholar
- 5.C. Garth, X. Tricoche, and G. Scheuermann. Tracking of Vector Field Singularities in Unstructured 3D Time-Dependent Datasets. In IEEE Visualization 2004, pages 329 – 336. IEEE Computer Society, October 2004.Google Scholar
- 7.J. L. Helman and L. Hesselink. Surface Representations of Two- and Three-Dimensional Fluid Flow Topology. In IEEE Visualization ’90, pages 6–13. IEEE Computer Society Press, 1990.Google Scholar
- 11.F. Sadlo and R. Peikert. Efficient visualization of lagrangian coherent structures by filtered AMR ridge extraction. IEEE TVCG, 13(6):1456–1463, 2007.Google Scholar
- 12.Ravi Samtaney, Deborah Silver, Norman Zabusky, and Jim Cao. Visualizing Features and Tracking Their Evolution. IEEE Computer, 27(7):20 – 27, 1994.Google Scholar
- 13.G. J. Sheard, T. Leweke, M. C. Thompson, and K. Hourigan. Flow around an impulsively arrested circular cylinder. Physics of Fluids, 19(8), 2007.Google Scholar
- 14.Kuangyu Shi, Holger Theisel, Tino Weinkauf, Helwig Hauser, Hans-Christian Hege, and Hans-Peter Seidel. Path line oriented topology for periodic 2D time-dependent vector fields. In Proc. EuroVis 2006, pages 139–146, Lisbon, Portugal, May 2006.Google Scholar
- 15.H. Theisel, J. Sahner, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Extraction of Parallel Vector Surfaces in 3D Time-Dependent Fields and Application to Vortex Core Line Tracking. In IEEE Visualization 2005, pages 631–638, October 2005.Google Scholar
- 16.H. Theisel and H.-P. Seidel. Feature Flow Fields. In VISSYM ’03: Proc. of the Sym. on Data Visualisation 2003, pages 141–148. Eurographics Association, 2003.Google Scholar
- 18.Xavier Tricoche, Thomas Wischgoll, Gerik Scheuermann, and Hans Hagen. Topology Tracking for the Visualization of Time-Dependent Two-Dimensional Flows. Computers & Graphics, 26(2):249 – 257, 2002.Google Scholar
- 19.A. Wiebel, X. Tricoche, D. Schneider, H. Jänicke, and G. Scheuermann. Generalized streak lines: Analysis and visualization of boundary induced vortices. IEEE TVCG, 13(6):1735–1742, 2007.Google Scholar