BIOSYSTMe Bio S www biw kuleuven be Ellipsoid
BIOSYST-Me. Bio. S www. biw. kuleuven. be Ellipsoid tessellation algorithm for modelling fruit microstructure H K Mebatsion
Why modelling fruit microstructure? BIOSYST-Me. Bio. S “We have to investigate the smallest details in order to appreciate the wider spectrum. ” “ Multiscale Modelling” Fruit microstructures are a web of interconnected cells Microstructures are the building block of the macrostructure Yet, microstructures are not homogeneous Elegant modelling procedure not affected by heterogeneity is a challenge
Microstructural heterogeneity What you see is what you can see BIOSYST-Me. Bio. S c a a) cortex b b) vascular bundle c) v-c transition
2 D microstructural modelling Voronoi tessellation (Mebatsion et al. , 2006 a) BIOSYST-Me. Bio. S Porosity=15. 95% CVD Porosity=15. 51% Apple parenchyma PVD Porosity=16. 17%
BIOSYST-Me. Bio. S Finite element simulation (Greenstar) Cell Pore O 2 concentration profile Tri Ho (unpublished)
Deeper investigation of microstructure Middle lamella Individual cells BIOSYST-Me. Bio. S a ESEM image of Granny Smith, Nieto et al. , 2004 c Cell wall b Synchrotron image (C. Pear) d Cells TEM images of Jonagold (c) and conference pear (d)
BIOSYST-Me. Bio. S Ellipse tessellation (Mebatsion et al. , 2006 b) Porosity= 16% Porosity=17% Cell Intercellular space Cell wall
BIOSYST-Me. Bio. S Investigation of O 2 profile at the microscale Conference pear Tri Ho (unpublished)
BIOSYST-Me. Bio. S 3 D microstructural modelling Cells in 2 D are elliptical. Should they be ellipsoidal in 3 D? How to determine the geometrical properties of ellipsoids (size, position…)? Is there something like ellipsoid fitting algorithm in 3 D? Minimum Volume Circumscribing Ellipsoid (MVCE)
MVCE BIOSYST-Me. Bio. S Given points in 3 D, there a unique ellipsoid with a minimum volume that encloses these sets of point An ellipsoid can be given by the following standard equation n (1)
BIOSYST-Me. Bio. S Then the equation of ellipsoid in the centre form reduces to (1. 1)
We want points to be inside the ellipsoid hence, they must satisfy (1. 2) BIOSYST-Me. Bio. S And the volume of the ellipsoid, is given by (2) the volume of unit sphere
BIOSYST-Me. Bio. S Hence, the optimization problem is all about To guarantee the positive volume (avoid trivial solution) lift the set of points Each point is lifted to a hyperplane
BIOSYST-Me. Bio. S The MVCE is determined as The solution is obtained using Khachiyan's algorithm The optimization problem is converted into concave optimization problem Lagrangian dual approach
MVCE algorithm ( Conditional Gradient Ascent Method) initialization BIOSYST-Me. Bio. S N Y
BIOSYST-Me. Bio. S Previous Minimum Volume Circumscribing Ellipsoid (MVCE) algorithms took into account all sets of points In our approach, we determine the MVCE of a polytope determined by the Convhull of the set of points Our approach is many times faster!!!
Comparison of previous and our approaches 101 points BIOSYST-Me. Bio. S 45 points 2. 2 times as fast
Evaluation of MVCE code (2 D) BIOSYST-Me. Bio. S MVCE Centroid =[101. 3835 153. 8091] l 1=167. 8051 l 2= 271. 0583 Ellipse fitting Centroid =[101. 4004 152. 66] l 1=161. 8438 l 2= 252. 3563
Evaluation of MVCE code (3 D) MVCE Centroid =[0 0 0] BIOSYST-Me. Bio. S R 1=R 2=R 3=1 Centroid =[0. 3887, -0. 2099, -0. 0954] * 1. 0 e-016 R 1=R 2=R 3=1
Implementation of MVCE BIOSYST-Me. Bio. S Image acquisition Synchrotron experiment
BIOSYST-Me. Bio. S Implementation… 120 phase contrast images were used Digitization of cells in different slices is impractical Digitization at every 10 th slice (compromised) Getting 3 D coordinates of individual cells MVCE calculation of individual cells Visualization tool
BIOSYST-Me. Bio. S Results MVCE
BIOSYST-Me. Bio. S Results
BIOSYST-Me. Bio. S Results MVCE
BIOSYST-Me. Bio. S Comparison of our approach and Amira Three. Dview version 1. 1 Conference pear Amira version 4. 2
Ellipsoid Tessellation algorithm Start MVCE (i) MVCE (j) BIOSYST-Me. Bio. S Overlap ? Y Non-overlap region (i) N MVCE (i) (NOR) + (MVCE (i)) Y N Y Intersection of regions N End
Ellipsoid tessellation A brand new approach BIOSYST-Me. Bio. S Volume distribution Pear parenchyma tissue
Ellipsoid tessellation BIOSYST-Me. Bio. S A brand new approach Volume distribution Porosity=9. 3% Pear dermal tissue
BIOSYST-Me. Bio. S Conclusion and future plan A very fast MVCE for the 3 D visualization of cells was developed Due to the nature of the synchrotron image automatic contour detection was not possible A brand new ellipsoid tessellation algorithms was developed These geometries will be exported to finite element/ volume codes (Ansys)
BIOSYST-Me. Bio. S Thank you
Nothing amuses more harmlessly than computation… BIOSYST-Me. Bio. S Mike L Klien
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