Measuring the elongation of vertebrae
September 20, 2013
Let’s take another look at that Giraffatitan cervical. MB.R.2180:C5, from a few days ago:
That’s a pretty elongate vertebra, right? But how elongate, exactly? How can we quantify whether it’s more or less elongate than some other vertebra?
The traditional answer is that we quantify elongation using the elongation index, or EI. This was originally defined by Upchurch (1998:47) as “the length of a vertebral centrum divided by the width across its caudal face”. Measuring from the full-resolution version of the image above, I make that 1779/529 pixels, or 3.36.
But then those doofuses Wedel et al. (2000:346) came along and said:
When discussing vertebral proportions Upchurch (1998) used the term elongation index (EI), defined as the length of the centrum divided by the width of the cotyle. Although they did not suggest a term for the proportion, Wilson & Sereno (1998) used centrum length divided by the height of the cotyle as a character in their analysis. We prefer the latter definition of this proportion, as the height of the cotyle is directly related to the range of motion of the intervertebral joint in the dorsoventral plane. For the purposes of the following discussion, we therefore redefine the EI of Upchurch (1998) as the anteroposterior length of the centrum divided by the midline height of the cotyle.
Since then, the term EI has mostly been used in this redefined sense — but I think we all agree now that it would have been better for Wedel et al to have given a new name to Wilson and Sereno’s ratio rather than apply Upchurch’s name to it.
Aaaanyway, measuring from the image again, I give that vertebra an EI (sensu Wedel et al. 2000) of 1779/334 = 5.33. Which is 58% more elongate than when using the Upchurch definition! This of course follows directly from the cotyle being 58% wider than tall (529/334 pixels).
So one of principal factors determining how elongate a vertebra seems to be is the shape of its cotyle. And that’s troublesome, because the cotyle is particularly subject to crushing — and it’s not unusual for even consecutive vertebrae from the same column to be crushed in opposite directions, giving them (apparently) wildly different EIs.
Here’s an example (though not at all an extreme one): cervicals 4 and 6 of the same specimen, MB.R.2180 (formerly HM SI), as the multi-view photo above:
Measuring from the photos as before, I make the width:height ratio of C4 683/722 pixels = 0.95, and that of C6 1190/820 pixels = 1.45. So these two vertebrae — from the same neck, and with only one other vertebrae coming in between them — differ in preserved cotyle shape by a factor of 1.53.
And by the way, this is one of the best preserved of all sauropod neck series.
Let’s take a look at the canonical well-preserved sauropod neck: the Carnegie Diplodocus, CM 84. Here are the adjacent cervicals 13 and 14, in posterior view, from Hatcher (1901: plate VI):
For C14 (on the left), I get a width:height ratio of 342/245 pixels = 1.40. For C13 (on the right), I get 264/256 pixels = 1.03. So C14 is apparently 35% broader than its immediate predecessor. I absolutely don’t buy that this represents how the vertebrae were in life.
FOR EXTRA CREDIT: what does this tell us about the reliability of computer models that purport to tell us about neck posture and flexibility, based on the preserved shapes of their constituent vertebrae?
So what’s to be done?
The first thing, as always in science, is to be explicit about what statements we’re making. Whenever we report an elongation index, we need to clearly state whether it’s EI sensu Upchurch 1998 or EI sensu Wedel et al. 2000. Since that’s so cumbersome, I’m going propose that we introduce two new abbreviations: EIH (Elongation Index Horizonal), which is Upchurch’s original measure (length over horizontal width of cotyle) and EIV (Elongation Index Vertical), which is Wilson and Sereno’s measure (length over vertical height of cotyle). If we’re careful to report EIH and EIV (or better still both) rather than an unspecified EI, then at least we can avoid comparing apples with oranges.
But I think we can do better, by combining the horizontal and vertical cotyle measurements in some way, and dividing the length by the that composite. This would give us an EIA (Elongation Index Average), which we could reasonably expect to preserve the original cotyle size, and so to give a more reliable indication of “true” elongation.
The question is, how to combine the cotyle width and height? There are two obvious candidates: either take the arithmetic mean (half the sum) or the geometric mean (the square root of the product). Note that for round cotyles, both these methods will give the same result as each other and as EIH and EIV — which is what we want.
Which mean should we use for EIA? to my mind, it depends which is best preserved when a vertebra is crushed. If a 20 cm circular cotyle is crushed vertically to 10cm, does it tend to smoosh outwards to 30 cm (so that 10+30 = the original 20+20) or to 40 cm (so that 10 x 40 = the original 20 x 20)? If the former, then we should use arithmetic mean; if the latter, then geometric mean.
Does anyone know how crushing works in practice? Which of these models most closely approximates reality? Or can we do better than either?
Update (8:48am): thanks for Emanuel Tschopp for pointing out (below) what I should have remembered: that Chure et al.’s (2010) description of Abydosaurus introduces “aEI”, which is the same as one of my proposed definitons of EIA. So we should ignore the last four paragraphs of this post and just use aEI. (Their abbreviation is better, too.)
- Hatcher, Jonathan B. 1901. Diplodocus (Marsh): its osteology, taxonomy and probable habits, with a restoration of the skeleton. Memoirs of the Carnegie Museum 1:1-63 and plates I-XIII.
- Upchurch, Paul. 1998. The phylogenetic relationships of sauropod dinosaurs. Zoological Journal of the Linnean Society 124:43-103.
- Wedel, Mathew J., Richard L. Cifelli and R. Kent Sanders. 2000b. Osteology, paleobiology, and relationships of the sauropod dinosaur Sauroposeidon. Acta Palaeontologica Polonica 45(4):343-388.
- Wilson, J. A. and Paul C. Sereno. 1998. Early evolution and higher-level phylogeny of sauropod dinosaurs. Society of Vertebrate Paleontology, Memoir 5:1-68.