Cross‐ and Triple‐Ratios of Human Body Parts During Development

Abstract

Recently developed landmark-based geometric morphometry has been used to depict the morphological development of organisms. In geometry, four landmarks can be mapped to any other four by Möbius transformations, if the cross-ratio of the landmarks is invariant and vice versa. To geometrically analyze the morphological development of the human body, we examined the cross-ratio of three consecutive body parts that are segmented by four landmarks in their configuration. Moreover, we introduced the triple-ratio of five landmarks that segments four consecutive parts (e.g., the shoulder, upper arm, forearm, and hand) and examined their growth patterns. The cross-and triple-ratios of the upper limb and shoulder girdle in fetuses were constant when biomechanical landmarks were used, although the cross-ratio of the upper limb varied when anatomical landmarks were used. The cross-ratios of the lower limbs, trunk, and pelvic girdles in fetuses differed from their corresponding cross-ratios in adults. These results suggest Möbius growth in the fetal upper limb and shoulder girdle but not in the other body parts examined. However, the growth balance of the three contiguous body parts was represented by the developmental change in the cross-ratio. Therefore, the cross-and triple-ratios may be applicable for simple but significant assessments of growth balance or proportion of the body parts. Anat Rec, 294:1360Rec, 294: -1369Rec, 294: , 2011. V V C 2011 Wiley-Liss, Inc.

GFP-Forked, a genetic reporter for studyingDrosophilaoocyte polarity

“…Previous work from our lab and others revealed similarities between oocyte and bristle cytoskeleton organization (Abdu et al, 2006;Amsalem et al, 2013;Bitan et al, 2010;Dubin-Bar et al, 2011;Dubin-Bar et al, 2008;Lundh et al, 2011;Otani et al, 2015;Shapiro and Anderson, 2006). It was shown that the complex of proteins containing Spn-F, Ik2 and Jvl plays a similar role in both oocyte and bristle actin network organization.…”

Correlation between musculoskeletal structure of the hand and primate locomotion: Morphometric and mechanical analysis in prehension using the cross- and triple-ratios

“…Hence, the cross-ratio can represent the positional relationship among the CMC, MCP, PIP, and distal interphalangeal (DIP) upon finger flexion and extension. The cross-ratio of a digit was calculated as follows [7,15]:…”

“…To address these questions, we introduced the cross-and triple-ratio [7], which is calculated from the lengths of the metacarpal bone, and the proximal, middle, and distal phalanges, as parameters associated with the intrinsic proportions of rays. For the first question, we examined whether or not the primate classification using these ratios was consistent with the classification based on their behavior, i.e., arboreality, semi-arboreality, or terrestriality, which is involved in prehensile function.…”

Mathematical Analysis of Mandibular Morphogenesis by Micro‐CT‐Based Mouse and Alizarin Red S‐Stained‐Based Human Studies During Development

“…The shape of a biological structure such as the mandible is a function of the relative position of the landmarks (Rohlf, 2002). Triple-ratio can be used to examine the ratio among the contiguous four lengths between the objects in the sense of conformality (Lundh et al, 2011); however, triple-ratio is difficult to be used to compare the multidimensional shape, which is represented by the coordinates of multiple landmarks, between the objects. By the mathematical analysis used in the present study, we can compare the shape of the mandible between the individuals, but not the size.…”