BIOSYST-MeBioSwww.biw.kuleuven.be Ellipsoid tessellation algorithm for modelling fruit microstructure H K Mebatsion

BIOSYST-MeBioS Why modelling fruit microstructure? “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

BIOSYST-MeBioS Microstructural heterogeneity a b c a) cortex b) vascular bundle c) v-c transition What you see is what you can see

BIOSYST-MeBioS Voronoi tessellation (Mebatsion et al.,2006a) CVD PVD Porosity=15.95% Porosity=15.51%Porosity=16.17% Apple parenchyma 2D microstructural modelling

BIOSYST-MeBioS Finite element simulation (Greenstar) Pore Cell O 2 concentration profile Tri Ho (unpublished)

BIOSYST-MeBioS Middle lamella Individual cells Cells ESEM image of Granny Smith, Nieto et al., 2004 Cell wall TEM images of Jonagold (c) and conference pear (d) Cells Synchrotron image (C. Pear) Deeper investigation of microstructure a b c d

BIOSYST-MeBioS Ellipse tessellation (Mebatsion et al.,2006b) Cell Cell wall Intercellular space Porosity= 16% Porosity=17%

BIOSYST-MeBioS Investigation of O 2 profile at the microscale Conference pear Tri Ho (unpublished)

BIOSYST-MeBioS Cells in 2D are elliptical. Should they be ellipsoidal in 3D? How to determine the geometrical properties of ellipsoids (size, position…)? Is there something like ellipsoid fitting algorithm in 3D? Minimum Volume Circumscribing Ellipsoid (MVCE) 3D microstructural modelling

BIOSYST-MeBioS MVCE Given points in 3D, 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-MeBioS Then the equation of ellipsoid in the centre form reduces to (1.1)

BIOSYST-MeBioS We want points to be inside the ellipsoid hence, they must satisfy And the volume of the ellipsoid, is given by the volume of unit sphere (2) (1.2)

BIOSYST-MeBioS 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-MeBioS 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

BIOSYST-MeBioS MVCE algorithm ( Conditional Gradient Ascent Method) N Y initialization

BIOSYST-MeBioS 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!!!

BIOSYST-MeBioS Comparison of previous and our approaches 101 points 45 points 2.2 times as fast

BIOSYST-MeBioS Evaluation of MVCE code (2D) MVCE Centroid =[ ] l1= l2= Ellipse fitting Centroid =[ ] l1= l2=

BIOSYST-MeBioS Evaluation of MVCE code (3D) Centroid =[0 0 0] R1=R2=R3=1 Centroid =[0.3887, , ] * 1.0e-016 R1=R2=R3=1 MVCE

BIOSYST-MeBioS Implementation of MVCE Image acquisition Synchrotron experiment

BIOSYST-MeBioS Implementation… 120 phase contrast images were used Digitization of cells in different slices is impractical Digitization at every 10th slice (compromised) Getting 3D coordinates of individual cells MVCE calculation of individual cells Visualization Visualization tool

BIOSYST-MeBioS Results MVCE

BIOSYST-MeBioS Results

BIOSYST-MeBioS Results MVCE

BIOSYST-MeBioS Comparison of our approach and Amira Conference pear ThreeDview version 1.1 Amira version 4.2

BIOSYST-MeBioS Ellipsoid Tessellation algorithm MVCE (i) MVCE (j) Overlap ? Non-overlap region (i) MVCE (i) (NOR) + (MVCE (i)) Y N Y N Intersection of regions End Y N Start

BIOSYST-MeBioS Ellipsoid tessellation A brand new approach Volume distribution Pear parenchyma tissue

BIOSYST-MeBioS Ellipsoid tessellation A brand new approach Volume distribution Pear dermal tissue Porosity=9.3%

BIOSYST-MeBioS Conclusion and future plan A very fast MVCE for the 3D 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-MeBioS Thank you

BIOSYST-MeBioS Nothing amuses more harmlessly than computation… Mike L Klien