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I was at the Natural History Museum of Los Angeles County yesterday to do some research in the ornithology collection. After lunch I was working on this pelican skeleton and I thought, “Geez, there is just no way to do this thing justice with still photos. I should make a video.” Here it is. You’ll want to see it full-screen–this being my first time out making a video, I didn’t realize that I was holding the phone the wrong way for efficient viewing on other devices.

The specimen is LACM Ornithology 86262. I’m posting this video with the knowledge and kind permission of the ornithology collection staff.

For previous things in this vein, please see:

If you like it that stuff like this exists, please support your local natural history museum, especially the LACM, which has some really fantastic education and outreach programs.

This came out two months ago, and I should have blogged about it then, but as usual I am behind. I’m blogging about it now because it deals with a question that has been on my mind for about 10 years now. If you want to skip my blatherations and get on to the good stuff, here’s the paper (Martin and Palmer 2014).

An Unsolved Problem

Back in 2004 I realized that if one had CTs or other cross-sections of a pneumatic bone, it was possible to quantify how much of the cross-sectional space was bone, and how much was air, a ratio I called the Air Space Proportion (ASP). That was the subject of my 2004 SVP talk, and a big part–arguably the most important part–of my chapter in The Sauropods in 2005. Of course the same calculation works for marrow-filled bones as well, where you would refer to it as an MSP rather than an ASP. If you can quantify the areas of bone, air, and marrow, you can figure out how dense the element was. One-stop shopping for all the relevant (simple) math is in this post.

(From Wedel 2005)

(From Wedel 2005)

Sometimes in science you end up with data that you don’t know what to do with, and that was my situation in 2004. Since I had CTs and other cross-sectional images of sauropod vertebrae, I could calculate ASPs for them, but I didn’t know what those results meant, because I didn’t have anything to compare them to. But I knew where to get I could get comparative data: from limb bone cross-sections. John Currey and R. McNeill Alexander had published a paper in 1985 titled, “The thickness of the walls of tubular bones”. I knew about that paper because I’d become something of an R. McNeill Alexander junkie after reading his book, Dynamics of Dinosaurs and Other Extinct Giants (Alexander 1989). And I knew that it had data on the cross-sectional properties of the limb bones in a host of animals, including crocs, birds, mammals, and–prophetically–pterosaurs.

If you know the inner and outer radii of a tubular bone, it is trivial to convert that to an ASP. So I could take the data from Currey and Alexander (1985) and calculate ASPs for the pneumatic bird and pterosaur bones in their study. Cubo and Casinos (2000) had a much larger sample of bird limb bones, and those got fed into my 2005 paper as well.

I was alert to the possibility that a mid-shaft cross-section might not be representative of the whole bone, and I hedged a bit in describing the bird ASPs (Wedel 2005: p. 212):

For the avian long bones described above, data were only presented for a single cross sec- tion located at midshaft. Therefore, the ASP values I am about to discuss may not be representative of the entire bones, but they probably approximate the volumes (total and air) of the diaphyses. For tubular bones, ASP may be determined by squaring K (if r is the inner diameter and R the outer, then K is r/R, ASP is πr^2/πR^2 or simply r^2/R^2, and ASP = K^2). For the K of pneumatic bones, Currey and Alexander (1985) report lower and upper bounds of 0.69 and 0.86, and I calculate a mean of 0.80 from the data presented in their table 1. Using a larger sample size, Cubo and Casinos (2000) found a slightly lower mean K of 0.77. The equivalent values of ASP are 0.48 and 0.74, with a mean of 0.64, or 0.59 for the mean of Cubo and Casinos (2000). This means that, on average, the diaphysis of a pneumatic avian long bone is 59%–64% air, by volume.

Now, even though I hedged and talked about diaphyses (shafts of long bones) rather than whole bones, I honestly expected that the ASP of any given slice would not change much along the length of a bone. Long bones tend to be tubular near the middle, with a thick bony cortex surrounding the marrow or air space, and honeycombed near the ends, with much thinner cortices and lots of bony septa or trabeculae (for marrow-filled bones, this is called spongy or trabecular bone, and for air-filled bones it is best referred to as camellate pneumatic bone). I figured that the decrease in cortical bone thickness near the ends of the bone would be offset by the increase in internal bony septa, and that the bone-to-air ratio through the whole element would be under some kind of holistic control that would keep it about even between the middle of the bone and the ends.

It is fair to ask why I didn’t just go check. The answer is that research is to some extent a zero-sum game, in that every project you take on means another that gets left waiting in the wings or abandoned completely. I was mainly interested in what ASP had to say about sauropods, not birds, and I had other fish to fry.

So that’s me from 2004-2012: aware that mid-shaft cross-sections of bird and pterosaur long bones might not be representative of whole elements, but not sufficiently motivated to go check. Then at SVPCA in Oxford that fall, Liz Martin rocked my world.

journal.pone.0097159.g001

Figure 1. CT scan images from two different regions of pterosaur first wing phalanx. A and B show the unmodified CT scans from A) the distal end of UP WP1 and B) the mid-shaft of UP WP1, while C and D show the modified and corrected images used in the calculation. Air space proportion (ASP) is calculated by determining the cross-sectional area of the internal, air filled cavity (the black centre of D) and dividing that by the total cross-sectional area, including the white cortical tissue and the black cavity. In areas with trabeculae, like C, the calculation of the air space includes the air found in individual trabeculae around the edges. Scale = 10 mm. doi:10.1371/journal.pone.0097159.g001 (From Martin and Palmer 2014)

A Paper in the Can

At SVPCA 2012, Liz Martin gave a talk titled, “A novel approach to estimating pterosaur bone mass using CT scans”, the result of her MS research with Colin Palmer at the University of Bristol. In that talk–the paper for which has been submitted to JVP–Liz and Colin were interested in using CT scans of pterosaur bones to quantify the volume of bone, in order to refine pterosaur mass estimates. I was fully on board, since estimating the masses of extinct animals is a minor obsession of mine. But what really caught my attention is that Liz and Colin had full stacks of slices spanning the length of each element–and therefore everything they needed to see how or if ASPs of pterosaur wing bones changed along their lengths.

At the next available break I dashed up to Liz, opened up my notebook, and started scribbling and gesticulating and in general carrying on like a crazy person. It’s a wonder she didn’t flee in terror. The substance of my raving was that (1) there was this outstanding problem in the nascent field of ASP research, and (2) she had everything she needed to address it, all that was required was a little math using the data she already had (I say this as if running the analyses and writing the paper were trivial tasks–they weren’t). Fortunately Liz and Colin were sufficiently interested to pursue it. Their paper on ASPs of pterosaur wing bones was submitted to PLOS ONE this February, and published on May 9 (while their earlier paper continues to grind its way through JVP).

And I’m blogging about it because the results were not what I expected.

Pterosaur wing bone ASPs - Martin and Palmer 2014

Figure 2. Plot of air space proportion over the length in six pterosaur wing bones. These plots show a polynomial line fit for each bone to show the general shape distribution. Exact measurements can be seen in Table S1. (From Martin and Palmer 2014).

Here’s the graph that tells the tale. Each line traces the ASP per slice along the length of a single pterosaur wing bone. A few things jump out:

  • Almost all of the lines drop near the left end. This is expected–if you’re cutting slices of a bone and measuring the not-bone space inside, then as you approach the end of the bone, you’re cutting through progressively more bone and less space. A few of the lines also drop near the right. I’m puzzled by that–if my explanation is correct, the ASP should plunge about equally at both ends. And the humerus USNM 11925 doesn’t follow the same pattern as the rest. As Martin and Palmer write, “It is unknown if this is a general feature of humeri, or this single taxon and more investigation is needed.”
  • Almost all of the bones have MUCH lower ASPs at mid-shaft than near the ends, on the order of 10% or more. So mid-shaft cross-sections of pterosaur wing bones tend to significantly underestimate how pneumatic they were. It would be interesting to know if the same holds true for bird long bones, or for the vertebrae of pterosaurs, birds, and sauropods. As Martin and Palmer point out, more work is needed.
  • The variation in ASP along the length of a single bone is in some cases greater than the variation between elements and individuals. That’s pretty cool. On the happy side, it means that getting into the nitty-gritty of ASP is not just stamp-collecting; you really need to know what is going on along the length of a bone before you can say anything intelligent about ASP or the density of the element. On the less happy side, that’s going to be a righteous pain in the butt for sauropod workers, because vertebrae are tough to get good scans of, assuming they will fit through a CT scanner at all (most don’t).
  • Finally, pterosaurs turn out to be even more pneumatic than you would think from looking at the already-freakishly-thin-walled shafts of their long bones. That’s pretty awesome, and it dovetails nicely with the emerging picture that pneumaticity in ornithodirans was more prevalent and more interesting than even I had suspected–it’s in prosauropods (Yates et al. 2012) and brachiosaur tails (Wedel and Taylor 2013) and rebbachisaur hips (Fanti et al. 2013) and saltasaur shoulders (Cerda et al. 2012) and, er, a couple of places that I can’t mention just yet. So life is good.

A few last odds and ends:

You can read more of this story at Liz Martin’s blog, scattered over several recent posts.

If you have CTs of bones and you want to follow in the footsteps of Martin and Palmer, you can do a lot of the work, and maybe all of it, in BoneJ, a free plug-in for ImageJ, which is also free.

A final note: this is Liz Martin’s first published paper, so congratulations are in order. Well done, Liz!

Almost Immediate Update: As soon as I posted this, I sent the link to Liz to see if I’d missed anything important. She writes, “It may be worth mentioning that it’s a question that I am actively following up on in my PhD, and looking into it with birds too hopefully. And it is indeed all possible using ImageJ, as that’s how I did the whole thing!”

References

OLYMPUS DIGITAL CAMERA

Now considered a junior synonym of Supersaurus, on very solid grounds.

Incidentally, unlike the neural spines of most non-titanosaurian sauropods, the neural spine of this vertebra is not simply a set of intersecting plates of bone. It is hollow and has a central chamber, presumably pneumatic. Evidence:

OLYMPUS DIGITAL CAMERA

You know the drill: lotsa pretty pix, not much yap.

IMG_5024

Our first stop of the day was the Fruita Paleontological Area, which has a fanstastic diversity of Morrison animals, including the mammal Fruitafossor and the tiny ornithopod Fruitadens.

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Plus it’s a pretty epic landscape, especially with the clouds and broken light we had this morning.

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I found a bone! Several bits, actually, a few meters away from the Fruitadens type quarry. I’d like to think that this proximal femur might be Fruitadens, but I don’t know the diagnostic characters and haven’t had time to look them up. Anyone know how diagnostic this honorary shard of excellence might be?

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After lunch, John Foster took us on a short hike to the quarry where Elmer Riggs got the back half of the Field Museum Apatosaurus. The front half came from a site in southern Utah, several decades later.

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The locals brought Riggs out in the 1930s for the dedication of two monuments–this one at the Apatosaurus quarry, and another like it at the Brachiosaurus quarry some miles away. Tragically, both monuments have the names of the dinosaurs misspelled!

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In the afternoon we visited the Mygatt-Moore Quarry and the Camarasaurus site in Rabbit Valley. Can you see the articulated Camarasaurus neck in this photo?

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Here’s a hint: the neural arches of two posterior cervical vertebrae in transverse horizontal cross-section.

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This Camarasaurus is apparently a permanent feature. If you’re wondering why no-one has excavated it, it’s because it’s buried in sandstone that is stupid-dense. The expenditure of time and resources just isn’t worth it, when right down the hill dinosaurs are pouring out of the much softer sediments of the Mygatt-Moore Quarry like water from a hydrant. This is the lesson I am learning about the Morrison: finding dinosaurs is easy. Finding dinosaurs you can get out of the ground and prepare–that’s something else.

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Our last stop of the day was Gaston Design, where Rob Gaston showed us how he molds, casts, and mounts everything from tiny teeth to good-sized skeletons.

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Like this Deinosuchus that is about to chomp on Jim Kirkland. Jim doesn’t look too worried.

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Here’s a nice cast of a busted sauropod dorsal, probably from Apatosaurus or Diplodocus, showing the pneumatic internal structure. Compare to similar views of dorsals in this post and this one. This is actually one half of a matched set that includes both halves of the centrum. I left with one of those sets of my own, a few dollars poorer and a whole lot happier.

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The end–for now.

Illustration talk slide 47

Illustration talk slide 48

Illustration talk slide 49

Illustration talk slide 50

That last one really hurts. Here’s the original image, which should have gone in the paper with the interpretive trace next to it rather than on top of it:

Sauroposeidon C6-C7 scout

The rest of the series.

Papers referenced in these slides:

“Look at all the things you’ve done for me
Opened up my eyes,
Taught me how to see,
Notice every tree.”

So sings Dot in Move On, the climactic number of Stephen Sondheim’s Pulitzer Prize-winning music Sunday in the Park with George, which on the surface is about the post-impressionist painter Georges Seurat, but turns out to be a study of obsession and creativity.

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Un dimanche après-midi à l’Île de la Grande Jatte – 1884 [A Sunday Afternoon on the Island of La Grande Jatte – 1884]

“Taught me how to see”? What kind of talk is that? One the surface, it seems silly — we all know how to see. We do it constantly, without thinking. Yet it’s something that artists talk about all the time. And anyone who’s sat down and seriously tried to paint or draw something will have some understanding of what the phrase means. We have such strong implicit ideas of what things look like that we tend to reproduce what we “know” is there rather than what’s actually there. Like I said, we see without thinking.

In fact, the psychology of perception is complicated and sophisticated, and the brain does an extraordinary amount of filtering of the visual signals we get, to save us the bother of having to consciously process way too much data. This is a whole scientific field of its own, and I’m going to avoid saying very much about it for fear of making a fool of myself — as scientists so often do when wandering outside their own field. But I think it’s fair to say that we all have a tendency to see what we expect to see.

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Phylogeny of Sauropoda, strict consensus of most parsimonious trees according to Wilson (2002:fig. 13a)

In the case of sauropods, this tendency has meant that we’ve all been startlingly bad at seeing pneumaticity in the caudal vertebrae of sauropods. Because the literature has trained us to assume it’s not there. For example, in the two competing sauropod phylogenies that dominated the 2000s, both Wilson (2002) and Upchurch et al. (2004) scored caudal pneumaticity as very rare: Wilson’s character 119, “Anterior caudal centra, pneumatopores (pleurocoels)”, was scored 1 only for Diplodocus and Barosaurus; and  Upchurch et al. (2004:286) wrote that “A few taxa (Barosaurus, Diplodocus, and Neuquensaurus) have pleurocoel-like openings in the lateral surfaces of the cranial [caudal] centra that lead into complex internal chambers”. That’s all.

And that’s part of the reason that every year since World War II, a million people have walked right past the awesome mounted brachiosaur in the Museum Für Naturkunde Berlin without noticing that it has pneumatic caudals. After all, we all knew that brachiosaur caudals were apneumatic.

But in my 2005 Progressive Palaeontology talk about upper limits on the mass of land animals estimated through the articular area of limb-bone cartilage, I included this slide that shows how much bigger the acetabulum of Giraffatitan is than the femoral head that it houses:

Screenshot from 2014-01-24 17:30:30

And looking at that picture made me wonder: those dark areas on the sides of the first few caudals (other than the first, which is a very obvious plaster model) certainly look pneumatic.

Then a few years later, I was invited to give a talk at the Museum Für Naturkunde Berlin itself, on the subject “Brachiosaurus brancai is not Brachiosaurus“. (This of course was drawn from the work that became my subsequent paper on that subject, Taylor 2009) And as I was going through my photos to prepare the slides of that talk, I thought to myself: darn it, yes, it does have pneumatic caudals!

So I threw this slide into the talk, just in passing:

Screenshot from 2014-01-24 17:32:06

Those photos were pretty persuasive; and a closer examination of the specimen on that same trip was to prove conclusive.

Meanwhile …

Earlier in 2009, I’d been in Providence, Rhode Island, with my Index Data colleagues. I’d managed to carve a day out of the schedule to hope along the coast to the Yale Peabody Museum in New Haven, Connecticut. My main goal was to examine the cervicals of the mounted Apatosaurus (= “Brontosaurus“) excelsus holotype (although it was also on that same trip that I first saw the Barosaurus holotype material that we’ve subsequently published a preprint on).

The Brontosaurus cervicals turned out to be useless, being completely encased in plaster “improvements” so that you can’t tell what’s real and what’s not. hopefully one day they’ll get the funding they want to take that baby down off its scaffold and re-prep the material.

But since I had the privilege of spending quality time with such an iconic specimen, it would have been churlish not to look at the rest of it. And lo and behold, what did I see when I looked at the tail but more pneumaticity that we thought we knew wasn’t there!

Wedel and Taylor (2013b: Figure 10).

An isolated pneumatic fossa is present on the right side of caudal vertebra 13 in Apatosaurus excelsus holotype YPM 1980. The front of the vertebra and the fossa are reconstructed, but enough of the original fossil is visible to show that the feature is genuine. (Wedel and Taylor 2013b: Figure 10).

What does this mean? Do other Giraffatitan and Apatosaurus specimens have pneumatic tails? How pervasive is the pneumaticity? What are the palaeobiological implications?

Stay tuned! All will be revealed in Matt’s next post (or, if you can’t wait, in our recent PLOS ONE paper, Wedel and Taylor 2013b)!

References

A few bits and pieces about the PLOS Collection on sauropod gigantism that launched yesterday.

2013-10-29-SauropodEbook1-thumb

First, there’s a nice write-up of one of our papers (Wedel and Taylor 2013b on pneumaticity in sauropod tails) in the Huffington Post today. It’s the work of PLOS blogger Brad Balukjian, a former student of Matt’s from Berkeley days. The introduction added by the PLOS blogs manager is one of those where you keep wanting to interrupt, “Well, actually it’s not quite like that …” but the post itself, once it kicks in, is good. Go read it.

Brad also has a guest-post on Discover magazine’s Crux blog: How Brachiosaurus (and Brethren) Became So Gigantic. He gives an overview of the sauropod gigantism collection as a whole. Well worth a read to get your bearings on the issue of sauropod gigantism in general, and the new collection in particular.

PLOS’s own community blog EveryONE also has its own brief introduction to the collection.

And PLOS and PeerJ editor Andy Farke, recently in these pages because of his sensational juvenile Parasaurolophus paper, contributes his own overview of the collection, How Big? How Tall? And…How Did It Happen?

Finally, if you’re at SVP, go and pick up your free copy of the collection. Matt was somehow under the impression that the PLOS USB drives with the sauropod gigantism collection would be distributed with the conference packet when people registered. In fact, people have to go by the PLOS table in the exhibitor area (booth 4 in the San Diego ballroom) to pick them up. There are plenty of them, but apparently a lot of people don’t know that they can get them.

References

This is an exciting day: the new PLOS Collection on sauropod gigantism is published to coincide with the start of this year’s SVP meeting! Like all PLOS papers, the contents are free to the world: free to read and to re-use.  (What is a Collection? It’s like an edited volume, but free online instead of printed on paper.)

There are fourteen papers in the new Collection, encompassing neck posture (yay!), nutrition (finally putting to bed the Nourishing Vomit Of Eucamerotus hypothesis), locomotion, physiology and evolutionary ecology. Lots every sauropod-lover to enjoy.

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Taylor and Wedel (2013c: Figure 12). CT slices from fifth cervical vertebrae of Sauroposeidon. X-ray scout image and three posterior-view CT slices through the C5/C6 intervertebral joint in Sauroposeidon OMNH 53062. In the bottom half of figure, structures from C6 are traced in red and those from C5 are traced in blue. Note that the condyle of C6 is centered in the cotyle of C5 and that the right zygapophyses are in articulation.

Matt and I are particularly excited that we have two papers in this collection: Taylor and Wedel (2013c) on intervertebral cartilage in necks, and Wedel and Taylor (2013b) on pneumaticity in the tails of (particularly) Giraffatitan and Apatosaurus. So we have both ends of the animal covered. It also represents a long-overdue notch on our bed-post: for all our pro-PLOS rhetoric, this is the first time either of has had a paper published in a PLOS journal.

Wedel and Taylor (2013b: Figure 4). Giraffatitan brancai tail MB.R.5000 (‘Fund no’) in right lateral view. Dark blue vertebrae have pneumatic fossae on both sides, light blue vertebrae have pneumatic fossae only on the right side, and white vertebrae have no pneumatic fossae on either side. The first caudal vertebra (hatched) was not recovered and is reconstructed in plaster.

It’s a bit of a statistical anomaly that after a decade of collaboration in which there was never a Taylor & Wedel or Wedel & Taylor paper, suddenly we have five of them out in a single year (including the Barosaurus preprint, which we expect to eventually wind up as Taylor and Wedel 2014). Sorry about the alphabet soup.

Since Matt is away at SVP this week, I’ll be blogging mostly about the Taylor and Wedel paper this week. When Matt returns to civilian life, the stage should be clear for him to blog about pneumatic caudals.

Happy days!

References

Here is Tataouinea, named by Fanti et al. (2013) last week — the first sauropod to be named after a locality from Star Wars (though, sadly, that is accidental — the etymology refers to the Tataouine Governatorate of Tunisia).

FantiEtAl2013-tataouinea-fig3

Fanti et al. (2013: figure 3) T. hannibalis selected elements and reconstruction. (a) Sacral neural arches 1-3, right lateral view; (b) sacral neural spine 4, right lateral view; (c) sacral neural spine 5, right lateral view; (d) caudal vertebra 2 and fragment of caudal 1 postzygapophyses, left lateral view; (e) caudal vertebra 1, left lateral view; (f) sacral centrum 1, ventral view; (g) sacral centra 2-5, ventral view; (h-j) caudal vertebra 3, anterior (h), left lateral (i), posterior (j) views; (k) left ilium, lateral view; (l) right ischium, medial view; and (m) skeletal reconstruction of T. hannibalis. Missing elements based on other nigersaurines. Scale bar: 10 cm (a-l), 1 m (m). a, acetabulum; f, fossa; hr, hyposphenal ridge; ip, ischial peduncle; ll, lateral lamina; pf, pneumatic foramen; pl, pleurocoel; poz, postzygapophysis; pp, pubic peduncle; psdf, prezygospinodiapophyseal foramen; sdl, spinodiapophyseal lamina; spol, spinopostzygapophyseal lamina; spzl, spinoprezygapophyseal lamina; sr, sacral rib; tp, transverse process. The asterisk indicates the fossa bounded by the spzl and the sdl.

No doubt Matt willl have much more to say about this animal, and especially its pneumatic features. I just thought it was time for a picture-of-the-week post.

UPDATE: Matt here, just a few quick thoughts (I’m in the middle of my summer anatomy lectures so they will be less extensive than this animal deserves). First, it’s awesome to see so much pneumaticity, and in elements that have not previously been reported as pneumatic in sauropods. The authors make a good case that we’re looking at actual pneumaticity here, for example in the pelvic elements, and not something else. So that’s cool.

What’s even cooler is that we’re seeing this in a diplodocoid:  Tataouinea is a rebbachisaurid. We’ve seen extreme pneumaticity in saltasaurines, and now we’ve got a parallel evolution of this character complex in diplodocoids. That’s cool by itself, and it’s further evidence that the underlying generating mechanism–the air sacs and their diverticula–were all in place long before they started leaving traces on the skeleton. The case for a birdlike lung-air sac system in sauropods, in saurischians, and in ornithodirans generally only keeps getting stronger. That is, we’re seeing more evidence not just that air sacs were there, but that they were bird-like in their layout, e.g., pneumatization of the pectoral girdle by clavicular air sacs, in both saltasaurines and theropods (avian and otherwise), and now extensive pelvic pneumatization (i.e., going beyond what we’ve seen previously in saltasaurines) by abdominal air sacs in rebbachisaurids and theropods (and pterosaurs, can’t forget about them). Happy times.

Reference

Fanti, Federico, Andrea Cau, Mohsen Hassine and Michela Contessi. 9 July 2013. A new sauropod dinosaur from the Early Cretaceous of Tunisia with extreme avian-like pneumatization. Nature Communications 4:2080. doi:10.1038/ncomms3080

Currey Alexander 1985 fig 1

Figure 1 from Currey and Alexander (1985)

This post pulls together information on basic parameters of tubular bones from Currey & Alexander (1985), on ASP from Wedel (2005), and on calculating the densities of bones from Wedel (2009: Appendix). It’s all stuff we’ve covered at one point or another, I just wanted to have it all in one convenient place.

Definitions:

  • R = outer radius = r + t
  • r = inner radius = R – t
  • t = bone wall thickness = R – r

Cross-sectional properties of tubular bones are commonly expressed in R/t or K (so that r = KR). K is defined as the inner radius divided by the outer radius (r/R). For bones with elliptical or irregular cross-sections, it’s best to measure two radii at right angles to each other, or use a different measure of cross-sectional geometry (like second moment of area, which I’m not getting into here).

R/t and K can be converted like so:

  • R/t = 1/(1-K)
  • K = 1 – (1/(R/t))

ASP (air space proportion) and MSP (marrow space proportion) measure the cross-sectional area of an element not taken up by bone tissue. ASP and MSP are the same measurement–the amount of non-bone space in a bony element divided by the total–we just use ASP for air-filled bones and MSP for marrow-filled bones. See Tutorial 6 and these posts: one, two, three.

For tubular bones, ASP (or MSP) can be calculated from K:

  • ASP = πr^2/πR^2 = r^2/R^2 = (r/R)^2 = K^2

Obviously R/t and K don’t work for bones like vertebrae that depart significantly from a tubular shape. But if you had a vertebra or other irregular bone with a given ASP and you wanted to see what the equivalent tubular bone would look like, you could take the square root of ASP to get K and then use that to draw out the cross-section of that hypothetical tubular bone.

To estimate the density of an element (at least near the point of a given cross-section), multiply the proportional areas of bone and air, or bone and marrow, by the specific gravities of those materials. According to Currey and Alexader (1985: 455), the specific gravities of fatty marrow and bone tissue are 0.93 and 2.1, respectively.

For a marrow-filled bone, the density of the element (or at least of the part of the shaft the section goes through) is:

  • 0.93MSP + 2.1(1-MSP)

Air is matter and therefore has mass and density, but it is so light (0.0012-0.0013 g/mL) that we can effectively ignore it in these calculations. So the density of a pneumatic element is: 2.1(1-ASP) For the three examples in the figure at the top of the post, the ASP/MSP values and densities are:

  • (b) alligator femur (marrow-filled), K = 0.35, MSP = K^2 = 0.12, density = (0.93 x 0.12) + (2.1 x 0.88) = 1.96 g/mL
  • (c) camel tibia (marrow-filled), K = 0.57, MSP = K^2 = 0.32, density = (0.93 x 0.32) + (2.1 x 0.68) = 1.73 g/mL
  • (d) Pteranodon first phalanx (air-filled), K = 0.91, ASP = K^2 = 0.83, density = (2.1 x 0.17) = 0.36 g/mL

What if we switched things up, and imagined that the alligator and camel bones were pneumatic and the Pteranodon phalanx was marrow-filled? The results would then be:

  • (b) alligator femur (hypothetical air-filled), K = 0.35, ASP = K^2 = 0.12, density = (2.1 x 0.88) = 1.85 g/mL
  • (c) camel tibia (hypothetical air-filled), K = 0.57, ASP = K^2 = 0.32, density = (2.1 x 0.68) = 1.43 g/mL
  • (d) Pteranodon first phalanx (hypothetical marrow-filled), K = 0.91, MSP = K^2 = 0.83, density = (0.93 x 0.83) + (2.1 x 0.17) = 1.13 g/mL

In the alligator femur, the amount of non-bone space is so small that it does much matter whether that space is filled by air or marrow–replacing the marrow with air only lowers the density of the element by 5-6%. The Pteranodon phalanx is a lot less dense than the alligator femur for two reasons. First, there is much less bony tissue–the hypothetical marrow-filled phalanx is 42% less dense as the alligator femur. Second, the marrow is replaced by air, which reduces the density by an additional 40% relative to the alligator.

Next time: how to write punchier endings for tutorial posts.

References

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