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Abstract:
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A new developmental model for growth of the echinoid test is based on a review of the growth patterns in regular echinoids . Echinoids are structurally composed of a tessellation of hundreds to thousands of individual plates . The two major aspects of echinoid growth are treated separately . (1 ) New plates are added in accordance with the Ocular Plate Rule and plate addition is hypothesized to be constitutively active but inhibited by a morphogen originating in coronal plates . Morphogen production is modeled as an inverse function of plate size and the concentration of inhibiting morphogen at a plate nucleation point is inversely proportional to the distance from surrounding plate centers . Plate addition is triggered whenever the inhibiting morphogen concentration falls below a threshold value . (2 ) The growth of individual plates is described using the Bertalanffy growth equation to model change in plate perimeter . The geometric model is based on a spherical frame of reference , and all calculations of position and growth are modeled over the surface of the sphere (i .e . , along geodesics ) . The data structure defined to maintain the geometric parameters is based on a spherical Delaunay triangulation of plate centers , and the edge geometry approximated by the dual Voronoi polygonalization . Echinoid plates are thus modeled as Voronoi polygons covering the sphere . Growth is modeled by the increasing radius of the sphere and the changing topology of the plates as new plates are added and existing plates grow . Final form of the complete test is generated by an affine deformation of the sphere . The growth model is implemented as the program EFORECHINOID , coded in the object -oriented programming language C++ with significant usage of the Standard Template Library (STL ) for efficient coding and memory management . Most parameters are available to a user via a Graphical User Interface (GUI ) , and output of 3 -dimensional simulations is via standard 3 -D AutoCAD[trademark] Drawing Exchange Format (DXF ) files . Program efficiency is O (nlogn ) and reasonably parameterized growth simulations with several hundred time steps can be performed in a matter of minutes per run . |