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Structure model index (SMI) is widely used to measure rods and plates in trabecular bone. It exploits the change in surface curvature that occurs as a structure varies from spherical (SMI = 4), to cylindrical (SMI = 3) to planar (SMI = 0). The most important assumption underlying SMI is that the entire bone surface is convex and that the curvature differential is positive at all points on the surface. The intricate connections within the trabecular continuum suggest that a high proportion of the surface could be concave, violating the assumption of convexity and producing regions of negative differential. We implemented SMI in the BoneJ plugin and included the ability to measure the amounts of surface that increased or decreased in area after surface mesh dilation, and the ability to visualize concave and convex regions. We measured SMI and its positive (SMI(+)) and negative (SMI(-)) components, bone volume fraction (BV/TV), the fraction of the surface that is concave (CF), and mean ellipsoid factor (EF) in trabecular bone using 38 X-ray microtomography (XMT) images from a rat ovariectomy model of sex steroid rescue of bone loss, and 169 XMT images from a broad selection of 87 species' femora (mammals, birds, and a crocodile). We simulated bone resorption by eroding an image of elephant trabecule and recording SMI and BV/TV at each erosion step. Up to 70%, and rarely <20%, of the trabecular surface is concave (CF 0.155-0.700). SMI is unavoidably influenced by aberrations induced by SMI(-), which is strongly correlated with BV/TV and CF. The plate-to-rod transition in bone loss is an erroneous observation resulting from the close and artifactual relationship between SMI and BV/TV. SMI cannot discern between the distinctive trabecular geometries typical of mammalian and avian bone, whereas EF clearly detects birds' more plate-like trabecule. EF is free from confounding relationships with BV/TV and CF. SMI results reported in the literature should be treated with suspicion. We propose that EF should be used instead of SMI for measurements of rods and plates in trabecular bone.