## Ellipsoid Factor The Structure Model Index has some well-known flaws, including not accounting for concave surfaces, which make it unsuitable for use on real bone geometries. Ellipsoid Factor is a new replacement for the structure model index and other methods for measuring rod/plate geometry. It uses the axis lengths from prolate, oblate and intermediate elipsoids to determine how prolate or oblate the trabecular space is at a particular point. Highly prolate (javelin-shaped, rod-like) ellipsoids have a single long axis (c) and two short axes (a, b) such that a < b << c , whereas highly oblate (discus-shaped, plate-like) ellipsoids have two longer axes (b, c) and one much shorter axis (a) so that a << b < c. Calculating EF as the difference in ratios, EF = a/b - b/c leads to a useful scale ranging from -1 (oblate, a/b ~ 0; b/c ~ 1) to +1 (prolate, a/b ~ 1; b/c ~ 0). EF of 0 indicates an intermediate ellipsoid where a/b = b/c, which is the case for spheres (a = b = c) and other ellipsoids with axis ratios a:qa:q²a. Ellipsoid Factor runs Skeletonise 3D, to get the medial axis of the trabeculae, which is used as the seed for sampling. Ellipsoids are seeded from each voxel on the medial axis. A combination of dilation, contraction, rotation and a small amount of translation is run iteratively until the ellipsoid increases no further in volume.

The EF at a point in the structure is determined as the EF of the most voluminous ellipsoid which contains that point.

Doube M (2015). The Ellipsoid Factor for quantification of rods, plates and intermediate forms in 3D geometries. Frontiers in Endocrinology 6:15. doi: 10.3389/fendo.2015.00015

Salmon PL, Ohlsson C, Shefelbine SJ, Doube M (2015) Structure model index does not measure rods and plates in trabecular bone. Frontiers in Endocrinology 6:162. doi:10.3389/fendo.2015.00162

• Input
• Binary stack
• Options
• Sampling options
• Sampling increment: distance between sample points on each vector; should be less than the pixel spacing.
• Vectors: number of vectors to sample at each seed point.
• Skeleton points per ellipsoid: allows dense or sparse sampling, a value of 1 means that an ellipsoid is sampled at every seed point
• Contact sensitivity: how many vectors must touch the background before dilation stops.
• Maximum iterations: how hard to try to find larger ellipsoids - fitting will stop if no improvement has been made after this number of iterations.
• Maximum drift: how far the centroid may be displaced from its seed point.
• Output options
• EF image: stack containing EF values for each point contained by at least one ellipsoid and NaN elsewhere
• Ellipsoid ID image: stack containing the ID of the biggest ellipsoid at each point, ranked in descending order (0 is the largest ellipsoid)
• Volume image: image showing the volume of the largest ellipsoid containing that point
• Axis ratio images: images showing a/b and b/c ratios foreach point containing at least one ellipsoid and NaN elsewhere
• Flinn peak plot: plot of a/b vs b/c weighted by volume, so bright pixels indicate relatively more of the structure has that axis ratio
• Gaussian sigma: amount to blur the Flinn peak plot - set to 0 for a precise but less 'beautiful' result
• Flinn plot: unweighted Flinn plot - every ellipsoid is represented by the same sized point regardless of ellipsoid size
• Output
• EF image: stack containing EF values. NaN (not a number) values are used in the background. Summary statistics can be obtained by running Analyze > Histogram
• Short-Mid image: stack containing the a/b ratios
• Mid-Long image: stack contining the b/c ratios
• Volume image: stack containing ellipsoid volumes
• Max id image: stack containing the ID of the largest ellipsoid at each point; IDs relate to the sort order based on volume, so ID = 0 is the largest ellipsoid. -1 is foreground and background is labelled with a large negative number.
• Flinn diagram: plot of a/b versus b/c values present in the volume
• Weighted Flinn plot: Flinn diagram with peaks of intensity proportional to volume occupied by each (a/b, b/c) ratio
• Test data from the manuscript  This file last modified 1345hrs 29 May 2020 © Michael Doube 2004-2020 :: Designed to be interoperable and standards-compliant