Just Around the Bend: A New Open Source Tool for Measuring the 2nd Moment of Area of Complex Shapes

What does a construction engineer and vertebrate morphologist* have in common?

*Someone who studies the morphology, or shape and structure, of vertebrates.

“*Lunch atop a Skyscraper*, 1932″ by Charles C. Ebbets show construction workers eating lunch on an I-beam.

Well, more than you might think! Whether it’s beams or bones, it’s important to understand the strength and stability of the material that is supporting your home…and your legs!

One metric for understanding the strength of beams and bones is called the “Second Moment of Area.” This metric is all about the shape of the structure (let’s say a beam for now). The way a beam is shaped, i.e. the way the material is arranged along a certain axis and with a certain direction of force, can tell us a lot about how it might bend under stress.

Let’s take the simple I-beam for example. An I-beam is designed to resist bending, by orienting most of it’s material in line with the incoming force, and away from the midline (where the “neutral axis” is). However, if you turn an I-beam on it’s side, this orientation is much more prone to bending. Generally, the more material that is oriented further away from this neutral axis, the less it will bend.

A simple diagram showing the cross-section of different shapes relative to their 2nd moment of area. The I-beam in it’s intended orientation is on the far left, demonstrating maximum resistance to bending. The I-beam on the far right, flipped 90 degrees, has a low 2nd moment of area and will bend easier with force.

Simple, right? Well, not quite. This is easy to understand with human-engineered shapes, but what happens when the shapes get more complicated? Let’s go back to bones. The cross sectional shape of a bone is variable, and varies greatly from bone to bone, animal to animal, and even along the length of the bone itself.

To understand how bones are shaped to resist stress, vertebrate morphologists have to look along the entire length of the bone, representing hundreds of little cross-sectional slices, and then look at the 2nd moment of area for each of those slices. This presents two challenges: 1) how do we look “inside” bones to see all those cross-sectional slices? and 2) how can we quantify these irregular shapes and then sum all their tiny contributions to understand it’s resistance to bending?

Figure taken from the presently discussed paper, Huie et al. “A) A bone in a cantilevered loading regime, with a cross-section taken at mid-shaft to show the orientation of the loading and neutral axes. (B) Examples of measurements taken from the cross-sectional geometries include the CSA – cross-sectional area of the shaded region, C – centroid, R – distance to the furthest point from the minor axis, D – maximum Feret diameter. (C) Different cross-sectional shapes ranked from lowest to highest second moment of area. (D) Different cross-sectional shapes ranked from lowest to highest second moment of area when force is applied along the dorsoventral axis. (E) Example cross-sections found in nature (not to scale) – 1. Frog femur, 2. Gharial lower jaw, 3. Catfish pectoral spine, 4. Sea otter radius, 5. Chameleon humerus, 6. Bird femur, 7. Horn shark lower jaw, 8. Bat humerus, 9. Chipmunk humerus, 10. Gar fish body cavity, 11. Salamander humerus.”

In a new paper by Huie, Summers, and Kawano they tackle the challenges of measuring the 2nd moment of area of complex shapes and provide a general guideline for the utility, assumptions, and limitations of this metric. Huie et al. recognize that with the advent of CT-scans, vertebrate morphologists can now easily look “inside” bones, slice by slice, and ask new questions about the evolution and mechanical properties of different bone shapes. BUT, with all this possible new data comes the need for a powerful tool to analyze it. Huie et al. aimed to fill this gap by developing a new, open-source software, termed “SegmentGeometry.”

“SegmentGeometry” can process up to thousands of these individual slices in a short time frame. The authors also demonstrated the usefulness of this new tool through two case-studies, looking at the 2nd moment of area of shark jaws and salamander legs. And not only that, but this software is not limited to calcified structures only. Plant material, invertebrates, and even data outside of CT-scans can be analyzed by this software: all you need is series of cross-sectional images!

Figure taken from the presently discussed paper, Huie et al. “Variation in the second moment of area across the humerus of an Aneides lugubris salamander. (A) The orientation of the humerus while the animal is mid-stance and how the bone was oriented in 3D Slicer. (B) Second moment of area varies along the length of the bone. The arrow indicates a point on the bone where there is high di- rectionality; specifically, bending resistance around the dorsoventral axis is nearly 4 times higher than the bending resistance around the craniocaudal axis (recall that the direction of force is perpendicular to the axis around which bending occurs). (C) Three example cross- sections. The first two rows of numbers report the correspondence in bending resistance of each cross-section about the craniocaudal ( I CC ) and dorsoventral ( I DV ) axes relative to the highest bending resistance about the minor principal axis ( I minor ). The third row reports material normalized I CC values.”

This new software, implemented in 3D-Slicer (an open-access image analysis software), is an exciting new tool for not only vertebrate morphologists, but any organismal biologist that wants to understand the relationship between structure and bending resistance through the 2nd moment of area.

I foresee some exciting new studies using this tool, just around the bend!😉

This post was written by Kelsi Rutledge, a 4th year Ph.D. Candidate at UCLA studying the functional morphology and fluid dynamics of olfaction in batoid fishes. Check out her website here or follow her on Twitter here.