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

I recently reread Dubach (1981), “Quantitative analysis of the respiratory system of the house sparrow, budgerigar and violet-eared hummingbird”, and realized that she reported both body masses and volumes in her Table 1. For each of the three species, here are the sample sizes, mean total body masses, and mean total body volumes, along with mean densities I calculated from those values.

  • House sparrow, Passer domesticus, n = 16, mass = 23.56 g, volume = 34.05 mL, density = 0.692 g/mL
  • Budgerigar, Melopsittacus undulatus, n = 19, mass = 38.16 g, volume = 46.08 mL, density = 0.828 g/mL
  • Sparkling violetear,* Colibri coruscans, n = 12, mass = 7.28 g, volume = 9.29 mL, density = 0.784 g/mL

* This is the species examined by Dubach (1981), although not specified in her title; there are four currently-recognized species of violetears. And apparently ‘violetear’ has overtaken ‘violet-eared hummingbird’ as the preferred common name. And as long as we’re technically on a digression,  I’m almost certain those volumes do not include feathers. Every volumetric thing I’ve seen on bird masses assumes plucked birds (read on).

This is pretty darned interesting to me, partly because I’m always interested in how dense animals are, and partly because of how the results compare to other published data on whole-body densities for birds. The other results I am most familiar with are those of Hazlehurst and Rayner (1992) who had this to say:

There are relatively few values for bird density. Welty (1962) cited 0.9 g/mL for a duck, and Alexander (1983) 0.937 g/mL for a domestic goose, but those values may not take account of the air sacs. Paul (1988) noted 0.8 g/mL for unspecified bird(s). To provide more reliable estimates, the density of 25 birds of 12 species was measured by using the volume displacement method. In a dead, plucked bird the air-sac system was reinflated (Saunder and Manton 1979). The average density was 0.73 g/mL, suggesting that the lungs and air sacs occupy some quarter of the body.

That result has cast a long shadow over discussions of sauropod masses, as in this paper and these posts, so it’s nice to see similar results from an independent analysis.  If you’re curious, the weighted mean of the densities calculated from Duchard’s Dubach’s (1981) data is 0.77. I’d love to see the raw data from Hazlehurst and Rayner (1992) to see how much spread they got in their density measurements.  Unfortunately, they did not say which birds they used or give the raw data in the paper (MYDD!), and I have not asked them for it because doing so only just occurred to me as I was writing this post.

There will be more news about hummingbirds here in the hopefully not-too-distant future. Here’s a teaser:

SkeletonFULL

Yes, those are its hyoids wrapped around the back of its head–they go all the way around to just in front of the eyes, as in woodpeckers and other birds that need hyper-long tongue muscles. There are LOADS of other interesting things to talk about here, but it will be faster and more productive if I just go write the paper like I’m supposed to be doing.

Oh, all right, I’ll say a little more. This is a  young adult female Anna’s hummingbird, Calypte anna, who was found by then-fellow-grad-student Chris Clark at a residential address in Berkeley in 2005. She was unable to fly and died of unknown causes just a few minutes after being found. She is now specimen 182041 in the ornithology collection at the Museum of Vertebrate Zoology at Berkeley. Chris Clark and I had her microCTed back in 2005, and that data will finally see the light of day thanks to my current grad student, Chris Michaels, who generated the above model.

This bird’s skull is a hair over an inch long, and she had a body mass of 3.85 grams at the time of her death. For comparison, those little ketchup packets you get at fast-food burger joints each contain 8-9 grams of ketchup, more than twice the mass of this entire bird when it was alive!

References

  • Dubach, M. 1981. Quantitative analysis of the respiratory system of the house sparrow, budgerigar and violet-eared hummingbird. Respiration Physiology 46(1): 43-60.
  • Hazlehurst, G.A., and Rayner, J.M. 1992. Flight characteristics of Triassic and Jurassic Pterosauria: an appraisal based on wing shape. Paleobiology 18(4): 447-463.
Brachiosaurus sp. BYU 12866 c5? in left lateral view with CT slices, some corrected for distortion.

Brachiosaurus sp. BYU 12866 c5? in left lateral view with CT slices, some corrected for distortion.

Last Tuesday Mike popped up in Gchat to ask me about sauropod neck masses.  We started throwing around some numbers, derived from volumetric estimates and some off-the-cuff guessing. Rather than tell you more about it, I should just paste our conversation, minimally edited for clarity and with a few hopefully helpful links thrown in.

Mike: Dud. Neck masses.
Matt: What about ’em?
Mike: Taylor (2009:803) measured the neck of Giraffatitan by GDI as 4117 liters.
Matt: k
Mike: I didn’t convert that to a mass, but I guess density of 0.5 is as good as any, which gives us (say) 2 tonnes.
Matt: That works for me.
Mike: That’s for an 8.5 m neck. So Supersaurus at 15 …
Matt: Yep. Almost twice as long, and not much more slender, and from what I’ve seen, ASP about the same.
Mike: Is 1.76 times as long. If it was isometric with the G. neck, it would be 5.5 times as heavy, which is 11 tonnes.
Matt: Oh.
Mike: So first: yeesh. Like, that is the mass of a whole freaking Diplo. Now we surely have to say isometry is unlikely.
Matt: Prolly.
Mike: But just multiplying out by length is unrealistic too. So maybe I should guess at mass =~ l^2? If I went with that, I’d get 6410 kg, which is elephant mass.
Matt: Something just occurred to me. Like, just now. For my 2006 poster, I calculated the mass of the cervical series in Giraffatitan, by summing over the CT slices from Brachiosaurus sp. BYU 12866 and multiplying by appropriate scale factors for the rest of the verts. We could “skin” that in muscle, and actually figure this out, for various muscle thicknesses, for one sauropod.
Mike: We should totally do that … if we had some idea how heavily muscled it was.
Matt: Well, obviously the thing to do is what Hutch et al. did for the tyrannosaurs, and put on several soft tissue envelopes. Crazy skinny, our best guess, markedly unfit, OMG, etc. It’s not that much more work. In fact, that could be my SVPCA talk this year.
Mike: Sure, but that’s just how to mitigate our ignorance. All we’d be doing at this point is taking n guesses instead of one. But, yeah, we should do it. Or you should if you prefer.
Matt: Let’s make it a Wedel and Taylor. I’ll crunch the numbers, but I want your input.
Mike: Works for me!
Matt: Good. Now let’s file it until April at least.
BYU 12613, a posterior cervical probably referable to Diplodocus, in dorsal (top), left lateral (left), and posterior (right) views. It compares most favourably with C14 of D. carnegii CM 84/94 (Hatcher 1901: plate 3) despite being less than half as large, with a centrum length of 270 mm compared to 642 mm for C14 of D. carnegii. From Wedel and Taylor (in press).

BYU 12613, a posterior cervical probably referable to Diplodocus, in dorsal (top), left lateral (left), and posterior (right) views. It most closely resembles C14 of D. carnegii CM 84/94 (Hatcher 1901: plate 3) despite being less than half as large, with a centrum length of 270 mm compared to 642 mm for C14 of D. carnegii. From Wedel and Taylor (in press).

Matt: Oh!
Matt: Also.
Matt: You know that little Diplo cervical from BYU that we figure in our in-press paper?
Mike: I think I know the one, yeah.
Matt: I am SUCH a moron. I have CT scans of the whole thing.
Mike: Good.
Matt: I forgot that Kent and I scanned it back in 2008. Even blogged about it, fer cryin’ out loud.  So I can do the sum-over-slices, scale-for-other-verts thing for Diplodocus, too. Which is at least closer to Supes than JANGO is.
Mike: Remind me, is it from a juvenile?
Matt: Maybe, maybe not. It IS tiny, but the neural spine is fused, the internal structure is crazy complex, and it doesn’t have any obvious juvenile characters other than just being small. The ASP is about as high as it gets in diplodocids. Which, as you may remember, is not nearly as high as it gets in titanosauriforms–that’s another paper that needs writing. Damn it. To know all this stuff and not have told it yet is killing me.
Mike: PeerJ!
Matt: I know!
Mike: Bottom line, it’s nuts that no-one has ever even tried to weigh a sauropod neck.* We should definitely do it, even if we do a really crappy job, if only so that others feel obliged to rebut.
Matt: Quite. Let’s do it. For reals.
Mike: In April. Done.

* R. McNeill Alexander (1985, 1989) did estimate the mass of the neck of Diplodocus, based on the old Invicta model and assuming a specific gravity of 1.0. Which was a start, and waaay better than no estimate at all. Still, let’s pretend that Mike meant “tried based on the actual fossils and what we know now about pneumaticity”.

The stuff about putting everything off until April is in there because we have a March 31 deadline to get a couple of major manuscripts submitted for an edited thingy. And we’ve made a pact to put off all other sciencing until we get those babies in. But I want to blog about this now, so I am.

Another thing Mike and I have been talking a lot about lately is the relation between blogging and paper-writing. The mode we’ve seen most often is to blog about something and then repurpose or rewrite the blog posts as a paper. Darren paved the way on this (at least in our scientific circle–people we don’t know probably did it earlier), with “Why azhdarchids were giant storks“, which became Witton and Naish (2008). Then last year our string of posts (starting here) on neural spine bifurcation in Morrison sauropods became the guts–and most of the muscles and skin, too–of our in-press paper on the same topic.

But there’s another way, which is to blog parts of the science as you’re doing them, which is what Mike was doing with Tutorial 20–that’s a piece of one of our papers due on March 31.

Along the way, we’ve talked about John Hawks’ model of using his blog as a place to keep his notes. We could, and should, do more of that, instead of mostly keeping our science out of the public eye until it’s ready to deploy (which I will always favor for certain projects, such as anything containing formal taxonomic acts).

And I’ve been thinking that maybe it’s time for me–for us–to take a step that others have already taken, and do the obvious thing. Which is not to write a series of blog posts and then decide later to turn it into a paper (I wasn’t certain that I’d be writing a paper on neural spine bifurcation until I had written the second post in that series), but to write the paper as a series of blog posts, deliberately and from the outset, and get community feedback along the way. And I think that the sauropod neck mass project is perfect for that.

Don’t expect this to become the most common topic of our posts, or even a frequent one. We still have to get those manuscripts done by the end of March, and we have no shortage of other projects waiting in the wings. And we’ll still post on goofy stuff, and on open access, and on sauropod stuff that has nothing to do with this–probably on that stuff a lot more often than on this. But every now and then there will be a post in this series, possibly written in my discretionary blogging time, that will hopefully move the paper along incrementally.

References

I don’t have time to write about this properly, but a few people have asked me about the new Sellers et al. (2012) paper on measuring the masses of extinct animals — in particular, the Berlin Giraffatitan — by having a CAD program generate minimal complex hulls around various body regions. Rather than write something new about it, I’m going to publish the comments that I sent Ed Yong for his Discover piece on the new technique:

Hi, Ed, good to hear from you. Yes, it’s a good paper: a useful new technique that has some useful properties, most importantly that it requires no irreproducible judgements on the part of the person using it, and that it’s ground-truthed on solid data from extant animals.

It’s a reassuring sanity-check to find that my (2009) mass estimate falls well within their method’s 95% confidence interval, and is in fact within 0.6% of their best estimate.

There are a couple of problems with this study, which I hope will be addressed in followups. The authors are honest enough to touch on all of these problems themselves, though! They are:

1. All the extant animals used to determine the fudge factor are mammals, which means they are not necessarily completely relevant to dinosaurs. In particular I would very much like to have seen regression lines and correlation coefficients for this method for birds and crocodilians, both of which are much more closely related to Giraffatitan.

2. Much depends on the reconstruction of the torso, particular the position of the ribs, which is very difficult to do well and confidently with dinosaurs. In my volumetric analysis (Taylor 2009:803) I found that the torso accounts for 70% of total body volume in Giraffatitan, so rib orientation will make a big difference to overall mass. Sauropod ribs that are well preserved and undistorted along their whole length are extremely rare.

3. Use of a single density value for the whole animal, while appropriate for mammals, really isn’t for brachiosaurs, in which the very long neck likely had a density no more than half that of the legs. I’m not sure what can be done about this, though, since any attempt to correct for density variation involves subjective guesswork. Then again, so do all guesses at overall body density in dinosaurs.

Issue 1 bothers me most, because the convex hulls of limb segments in mammals will be proportionally much larger than in sauropods, due to the complex shapes of mammalian long-bone ends. I worry that using mammals as a baseline will underestimate sauropod leg mass.

Still, even with these caveats, it’s a good exposition of an important new method which I expect to see widely adopted.

Hope that’s helpful.

In short: good work, widely applicable, and probably the best mass-estimation technique we now have available for complete and near-complete skeletons. It would be good to see it applied to (say) the Yale, AMNH and CM apatosaurs.

Composite illustration from Sellers et al.’s press release. Top left: bear skeleton from the Oxford University Natural History Museum, presumably Ursus maritimus: original skeleton, derived point cloud and convex hulls (also used as Sellers et al. 2012:fig. 1). Top right: shedloads of awesome. Bottom: complex hulls around body segments of Giraffatitan.

References

Sellers, W. I., J. Hepworth-Bell, P. L. Falkingham, K. T. Bates, C. A. Brassey, V. M. Egerton and P. L. Manning. 2012. Minimum convex hull mass estimations of complete mounted skeletons. Biology Letters, online ahead of print. doi:10.1098/rsbl.2012.0263

Taylor, Michael P. 2009a. A re-evaluation of Brachiosaurus altithorax Riggs 1903 (Dinosauria, Sauropoda) and its generic separation from Giraffatitan brancai (Janensch 1914). Journal of Vertebrate Paleontology 29(3):787-806.

Why we do mass estimates

Mass estimates are a big deal in paleobiology. If you want to know how much an animal needed in terms of food, water, and oxygen, or how fast it could move, or how many offspring it could produce in a season, or something about its heat balance, or its population density, or the size of its brain relative to its body, then at some point you are going to need a mass estimate.

All that is true, but it’s also a bit bogus. The fact is, people like to know how big things are, and paleontologists are not immune to this desire. We have loads of ways to rationalize our basic curiosity about the bigness of extinct critters. And the figuring out part is both very cool and strangely satisfying. So let’s get on with it.

Two roads diverged

There are two basic modes for determining the mass of an extinct animal: allometric, and volumetric. Allometric methods rely on predictable mathematical relationships between body measurements and body mass. You measure a bunch of living critters, plot the results, find your regression line, and use that to estimate the masses of extinct things based on their measurements. Allometric methods have a couple of problems. One is that they are absolutely horrible for extrapolating to animals outside the size range of the modern sample, which ain’t so great for us sauropod workers. The other is that they’re pretty imprecise even within the size range of the modern sample, because real data are messy and there is often substantial scatter around the regression line, which if faithfully carried through the calculations produces large uncertainties in the output. The obvious conclusion is that anyone calculating extinct-animal masses by extrapolating an allometric regression ought to calculate the 95% confidence intervals (e.g. “Argentinosaurus massed 70000 kg, with a 95% confidence interval of 25000-140000 kg), but, oddly, no-one seems to do this.

Volumetric methods rely on creating a physical, digital, or mathematical model of an extinct animal, determining the volume of the model, multiplying by a scale factor to get the volume of the animal in life, and multiplying that by the presumed density of the living animal to get its mass. Volumetric methods have three problems: (1) many extinct vertebrates are known from insufficient material to make a good 3D model of the skeleton; (2) even if you have a complete skeleton, the method is very sensitive to how you articulate the bones–especially the ribcage–and the amount of flesh you decide to pack on, and there are few good guidelines for doing this correctly; and (3) relatively small changes in the scale factor of the model can produce big changes in the output, because mass goes with the cube of the linear measurement. If your scale factor is off by 10%, you mass will be off by 33% (1.1^3=1.33).

On the plus side, volumetric mass estimates are cheap and easy. You don’t need hundreds or thousands of measurements and body masses taken from living animals; you can do the whole thing in your kitchen or on your laptop in the space of an afternoon, or even less. In the old days you’d build a physical model, or buy a toy dinosaur, and use a sandbox or a dunk tank to measure the volume of sand or water that the model displaced, and go from there. Then in the 90s people started building digital 3D models of extinct animals and measuring the volumes of those.

But you don’t need a physical model or a dunk tank or even a laptop to do volumetric modeling. Thanks to a method called graphic double integration or GDI, which is explained in detail in the next section, you can go through the whole process with nothing more than pen and paper, although a computer helps.

Volumetric methods in general, and GDI in particular, have one more huge advantage over allometric methods: they’re more precise and more accurate. In the only published study that compares the accuracy of various methods on extant animals of known mass, Hurlburt (1999) found that GDI estimates were sometimes off by as much as 20%, but that allometric estimates were much worse, with several off by 90-100% and one off by more than 800%. GDI estimates were not only closer to the right answers, they also varied much less than allometric methods. On one hand, this is good news for GDI afficionados, since it is the cheapest and easiest of all the mass estimation methods out there. On the other hand, it should give us pause that on samples of known mass, the best available method can still be off by as much as a fifth even when working with complete bodies, including the flesh. We should account for every source of error that we can, and still treat our results with appropriate skepticism.

Graphic Double Integration

GDI was invented by Jerison (1973) to estimate the volumes of cranial endocasts. Hurlburt (1999) was the first to apply it to whole animals, and since then it has been used by Murray and Vickers-Rich (2004) for mihirungs and other extinct flightless birds, yours truly for small basal saurischians (Wedel 2007), Mike for Brachiosaurus and Giraffatitan (Taylor 2009), and probably many others that I’ve missed.

GDI is conceptually simple, and easy to do. Using orthogonal views of a life restoration of an extinct animal, you divide the body into slices, treat each slice as an ellipse whose dimensions are determined from two perspectives, compute the average cross-sectional area of each body part, multiply that by the length of the body part in question, and add up the results. Here’s a figure from Murray and Vickers-Rich (2004) that should clarify things:

One of the cool things about GDI is that it is not just easy to separate out the relative contributions of each body region (i.e., head, neck, torso, limbs) to the total body volume, it’s usually unavoidable. This not only lets you compare body volume distributions among animals, it also lets you tinker with assigning different densities to different body parts.

An Example: Plateosaurus

Naturally I’m not going to introduce GDI without taking it for a test drive, and given my proclivities, that test drive is naturally going to be on a sauropodomorph. All we need is an accurate reconstruction of the test subject from at least two directions, and preferably three. You could get these images in several ways. You could take photographs of physical models (or toy dinosaurs) from the front, side, and top–that could be a cool science fair project for the dino-obsessed youngster in your life. You could use the white-bones-on-black-silhouette skeletal reconstructions that have become the unofficial industry standard. You could also use orthogonal photographs of mounted skeletons, although you’d have to make sure that they were taken from far enough away to avoid introducing perspective effects.

For this example, I’m going to use the digital skeletal reconstruction of the GPIT1 individual of Plateosaurus published by virtual dino-wrangler and frequent SV-POW! commenter Heinrich Mallison (Mallison et al 2009, fig. 14). I’m using this skeleton for several reasons: it’s almost complete, very little distorted, and I trust that Heinrich has all the bits in the right places. I don’t know if the ribcage articulation is perfect but it looks reasonable, and as we saw last time that is a major consideration. Since Heinrich built the digital skeleton in digital space, he knows precisely how big each piece actually is, so for once we have scale bars we can trust. Finally, this skeleton is well known and has been used in other mass estimate studies, so when I’m done we’ll have some other values to compare with and some grist for discussion. (To avoid accidental bias, I’m not looking at those other estimates until I’ve done mine.)

Of course, this is just a skeleton, and for GDI I need the body outline with the flesh on. So I opened the image in GIMP (still free, still awesome) and drew on some flesh. Here we necessarily enter the realm of speculation and opinion. I stuck pretty close to the skeletal outline, with the only major departures being for the soft tissues ventral to the vertebrae in the neck and for the bulk of the hip muscles. As movie Boromir said, there are other paths we might take, and we’ll get to a couple of alternatives at the end of the post.

This third image is the one I used for actually taking measurements. You need to lop off the arms and legs and tote them up separately from the body axis. I also filled in the body outlines and got rid of the background so I wouldn’t have any distracting visual clutter when I was taking measurements. I took the measurements using the measuring tool in GIMP (compass icon in the toolbar), in orthogonal directions (i.e., straight up/down and left/right), at regular intervals–every 20 pixels in this case.

One thing you’ll have to decide is how many slices to make. Ideally you’d do one slice per pixel, and then your mathematical model would be fairly smooth. There are programs out there that will do this for you; if you have a 3D digital model you can just measure the voxels (= pixels cubed) directly, and even if all you have is 2D images there are programs that will crank the GDI math for you and measure every pixel-width slice (Motani 2001). But if you’re just rolling with GIMP and OpenOffice Calc (or Photoshop and Excel, or calipers and a calculator), you need to have enough slices to capture most of the information in the model without becoming unwieldy to measure and calculate. I usually go with 40-50 slices through the body axis and 9 or 10 per limb.

The area of a circle is pi*r^2, and the area of an ellipse is pi*r*R, where r and R are the radii of the minor and major axes. So enter the widths and heights of the body segments in pixels in two columns (we’ll call them A and B) in your spreadsheet, and create a third column with the function 3.14*A1*B1/4. Divide by four because the pixel counts you measured on the image are diameters and the formula requires radii. If you forget to do that, you are going to get some wacky numbers.

One obvious departure from reality is that the method assumes that all of the body segments of an animal have elliptical cross-sections, when that is often not exactly true. But it’s usually close enough for the coarse level of detail that any mass estimation method is going to provide, and if it’s really eating you, there are ways to deal with it without assuming elliptical cross-sections (Motani 2001).

For each body region, average the resulting areas of the individual slices and multiply the resulting average areas by the lengths of the body regions to get volumes. Remember to measure the lengths at right angles to your diameter measurements, even when the body part in question is curved, as is the tail of Heinrich’s Plateosaurus.

For sauropods you can usually treat the limbs as cylinders and just enter the lateral view diameter twice, unless you are fortunate enough to have fore and aft views. It’s not a perfect solution but it’s probably better than agonizing over the exact cross sectional shape of each limb segment, since that will be highly dependent on how much flesh you (or some other artist) put on the model, and the limbs contribute so little to the final result. For Plateosaurus I made the arm circular, the forearm and hand half as wide as tall, the thigh twice as long as wide, and the leg and foot round. Don’t forget to double the volumes of the limbs since they’re paired!

We’re not done, because so far all our measurements are in pixels (and pixels cubed). But already we know something cool, which is what proportion each part of the body contributes to the total volume. In my model based on Heinrich’s digital skeleton, segmented as shown above, the relative contributions are as follows:

  • Head: 1%
  • Neck: 3%
  • Trunk: 70%
  • Tail: 11%
  • Forelimbs (pair): 3%
  • Hindlimbs (pair): 12%

Already one of the great truths of volumetric mass estimates is revealed: we tend to notice the extremities first, but really it is the dimensions of the trunk that drive everything. You could double the size of any given extremity and the impact on the result would be noticeable, but small. Consequently, modeling the torso accurately is crucial, which is why we get worried about the preservation of ribs and the slop inherent in complex joints.

Scale factor

The 170 cm scale bar in Heinrich’s figure measures 292 pixels, or 0.582 cm per pixel. The volume of each body segment must be multiplied by 0.582 cubed to convert to cubic cm, and then divided by 1000 to convert to liters, which are the lingua franca of volumetric measurement. If you’re a math n00b, your function should look like this: volume in liters = volume in pixels*SF*SF*SF/1000, where SF is the scale factor in units of cm/pixel. Don’t screw up and use pixels/cm, or if you do, remember to divide by the scale factor instead of multiplying. Just keep track of your units and everything will come out right.

If you’re not working from an example as perfect as Heinrich’s digital (and digitally measured) skeleton, you’ll have to find something else to use for a scale bar. Something big and reasonably impervious to error is good. I like the femur, if nothing else is available. Any sort of multi-segment dimension like shoulder height or trunk length is going to be very sensitive to how much gloop someone thought should go between the bones. Total length is especially bad because it depends not only on the intervertebral spacing but also on the number of vertebrae, and even most well-known dinos do not have complete vertebral series.

Density

Finally, multiply the volume in liters by the assumed density to get the mass of each body segment. Lots of people just go with the density of water, 1.0 kg/L, which is the same as saying a specific gravity (SG) of 1. Depending on what kind of animal you’re talking about, that may be a little bit off or it may be fairly calamitous. Colbert (1962) found SGs of 0.81 and 0.89 for an extant lizard and croc, which means an SG of 1.0 is off by between 11% and 19%. Nineteen percent–almost a fifth! For birds, it’s even worse; Hazlehurst and Rayner (1992) found an SG of 0.73.

Now, scroll back up to the diagram of the giant moa, which had a mass of 257.5 kg “assuming a specific gravity of 1”. If the moa was as light as an extant bird–and its skeleton is highly pneumatic–then it might have had a mass of only 188 kg (257.5*0.73). Or perhaps its density was higher, like that of a lizard or a croc. Without a living moa to play with, we may never know. Two points here: first, the common assumption of whole-body densities of 1.0 is demonstrably incorrect* for many animals, and second, since it’s hard to be certain about the densities of extinct animals, maybe the best thing is to try the calculation with several densities and see what results we get. (My thoughts on the plausible densities of sauropods are here.)

* Does anyone know of actual published data indicating a density of 1.0 for a terrestrial vertebrate? Or is the oft-quoted “bodies have the same density as water” basically bunk? (Note: I’m not disputing that flesh has a density close to that of water, but bones are denser and lungs and air spaces are lighter, and I want to know the mean density of the whole organism.)

Back to Plateosaurus. Using the measurements and calculations presented above, the total volume of the restored animal is 636 liters. Here are the whole body masses (in kg) we get using several different densities:

  • SG=1.0 (water), 636 kg
  • SG=0.89 (reptile high), 566 kg
  • SG=0.81 (reptile low), 515 kg
  • SG=0.73 (bird), 464 kg

I got numbers. Now what?

I’m going to describe three possible things you could do with the results once you have them. In my opinion, two of them are the wrong the thing to do and one is the right thing to do.

DON’T mistake the result of your calculation for The Right Answer. You haven’t stumbled on any universal truth. Assuming you measured enough slices and didn’t screw up the math, you know the volume of a mathematical model of an organism. If you crank all the way through the method you will always get a result, but that result is only an estimate of the volume of the real animal the model was based on. There are numerous sources of error that could plague your results, including: incomplete skeletal material, poorly articulated bones, wrong scale factor, wrong density, wrong amount of soft tissue on the skeleton. I saved density and gloop for last because you can’t do much about them; here the strength of your estimate relies on educated guesses that could themselves be wrong. In short, you don’t even know how wrong your estimate might be.

Pretty dismal, eh?

DON’T assume that the results are meaningless because you don’t know the actual fatness or the density of the animal, or because your results don’t match what you expected or what someone else got. I see this a LOT in people that have just run their first phylogenetic analysis. “Why, I could get any result I wanted just by tinkering with the input!” Well, duh! Like I said, the method will always give you an answer, and it won’t tell you whether the answer is right or not. The greatest advantage of explicit methods like cladistics and GDI is that you know what the input is, and so does everyone else if you are honest about reporting it. So if someone disagrees with your character coding or with how much the belly sags on your model sauropod, you can have a constructive discussion and hopefully science as a whole gets closer to the right answer (even if we have no way of knowing if or when we arrive, and even if your pet hypothesis gets trampled along the way).

DO be appropriately skeptical of your own results without either accepting them as gospel or throwing them out as worthless. The fact that the answer changes as you vary the parameters is a feature, not a bug. Investigate a range of possibilities, report all of those results, and feel free to argue why you think some of the results are better than others. Give people enough information to replicate your results, and compare your results to those of other workers. Figure out where yours differ and why.

Try to think of more interesting things you could do with your results. Don Henderson went from digitally slicing critters (Henderson 1999) to investigating floating sauropods (Henderson 2004) to literally putting sauropods through their paces (Henderson 2006)–not to mention working on pterosaur flight and swimming giraffes and other cool stuff. I’m not saying you should run out and do those exact things, but rather that you’re more likely to come up with something interesting if you think about what you could do with your GDI results instead of treating them as an end in themselves.

How massive was GPIT1, really?

Beats me. I’m not the only one who has done a mass estimate based on that skeleton. Gunga et al. (2007) did not one but two volumetric mass estimates based on GPIT1, and Mallison (2010) did a whole series, and they published their models so we can see how they got there. (In fact, many of you have probably been reading this post in slack-jawed horror, wondering why I was ignoring those papers and redoing the mass estimate the hard way. Now you know!) I’m going to discuss the results of Gunga et al. (2007) first, and come back to Mallison (2010) at the end.

Here’s the “slender” model of Gunga et al. 2007 (their fig. 3):

and here’s their “robust” model (Gunga et al. 2007:fig. 4):

(These look a bit…inelegant, let’s say…because they are based on the way the physical skeleton is currently mounted; Heinrich’s model looks much nicer because of his virtual remount.)

For both mass estimates they used a density of 0.8, which I think is probably on the low end of the range for prosauropods but not beyond the bounds of possibility. They got a mass of 630 kg for the slender model and 912 kg for the robust one.

Their 630-kg estimate for the slender model is deceptively close to the upper end of my range; deceptive because their 630-kg estimate assumes a density of 0.8 and my 636-kg one assumes a density of 1.0. The volumes are more directly comparable: 636 L for mine, 790 L for their slender one, and 1140 L for their robust one. I think that’s pretty good correspondence, and the differences are easily explained. My version is even more skinnier than their slender version; I made it about as svelte as it could possibly have been. I did that deliberately, because it’s always possible to pack on more soft tissue but at some point the dimensions of the skeleton establish a lower bound for how voluminous a healthy (i.e., non-starving) animal could have been. The slender model of Gunga et al. (2007) looks healthier than mine, whereas their robust version looks, to my eye, downright corpulent. But not unrealistically so; fat animals are less common than skinny ones but they are out there to be found, at least in some times and places. It pays to remember that the mass of a single individual can fluctuate wildly depending on seasonal food availability and exercise level.

For GPIT1, I think something like 500 kg is probably a realistic lower bound and 900 kg is a realistic upper bound, and the actual mass of an average individual Plateosaurus of that size was somewhere in the middle. That’s a big range–900 kg is almost twice 500 kg. It’s hard to narrow down because I really don’t know how fleshy Plateosaurus was or what it’s density might have been, and I feel less comfortable making guesses because I’ve spent much less time working on prosauropods than on sauropods. If someone put a gun to my head, I’d say that in my opinion, a bulk somewhere between that of my model and the slender model of Gunga et al. is most believable, and a density of perhaps 0.85, for a result in the neighborhood of 600 kg. But those are opinions, not hypotheses, certainly not facts.

I’m happy to see that my results are pretty close to those of Mallison (2010), who got 740 L, which is also not far off from the slender model of Gunga et al. (2007). So we’ve had at least three independent attempts at this and gotten comparable results, which hopefully means we’re at least in the right ballpark (and pessimistically means we’re all making mistakes of equal magnitude!). Heinrich’s paper is a goldmine, with loads of interesting stuff on how the skeleton articulates, what poses the animal might have been capable of, and how varying the density of different body segments affects the estimated mass and center of mass. It’s a model study and I’d happily tell you all about it but you should really read it for yourself. Since it’s freely available (yay open access!), there’s no barrier to you doing so.

Conclusion

So: use GDI with caution, but do use it. It’s easy, it’s cool, it’s explicit, it will give you lots to think about and give us lots to talk about. Stay tuned for related posts in the not-too-distant future.

References

How fat was Camarasaurus?

January 16, 2011

For reasons that will soon become apparent (yes, that’s a teaser), Matt and I wanted to figure out how heavy Camarasaurus was.  This is the story of how I almost completely badgered up part of that problem.  I am publishing it as a cautionary tale because I am very secure and don’t mind everyone knowing that I’m an idiot.

Those who paid close attention to my recent paper on Brachiosaurus and Giraffatitan will remember that when I estimated their mass using Graphic Double Integration (Taylor 2009: 802-804) I listed separately the volumes of the head, neck, forelimbs, hindlimbs, torso and tail of each taxon.  In Giraffatitan, the torso accounted for 71% of the total volume (20588 of 29171 litres), and in Brachiosaurus, 74% (26469 of 35860 litres), so it’s apparent that torso volume hugely dominates that of the whole animal.  In the giant balloon-model Giraffatitan of Gunga et al.’s (1995, 1999) estimates, the torso accounted for 74% of volume (55120 of 74420 litres) so even though their fleshing out of the skeleton was morbidly obese, the relative importance of the torso came out roughly the same.  Finally, Gunga et al’.s (2008) revised, less bloated, model of the same Giraffatitan had the torso contributing 68% of volume (32400 of 47600 litres).  So far as I know, these are all of the published accounts that give the volumes of separate parts of a sauropod body, but if there are any more, please tell me in the comments!   (Odd that they should all be for brachiosaurids.)

3D “slim” version of reconstruction of the “Brachiosaurus” brancai mounted and exhibited at the Museum of Natural History in Berlin (Germany).  A. Side view, upper panel; B. top view, lower panel.  The cross in the figure of upper panel indicates the calculated center of gravity.  (Gunga et al. 2008: figure 2)

So it’s evident that, in brachiosaurs at least, the torso accounts for about 70% total body volume, and therefore for about that much of the total mass.  (The distribution of penumaticity means that it’s denser than the neck and less dense than the limbs, so that its density is probably reasonably close to the average of the whole animal.)

Now here’s the problem.  How fat is the sauropod?  Look at the top-view of Giraffatitan in the Gunga et al. figure above: it’s easy to imagine that the torso could be say 20% narrower from side to side, or 20% broader.  Those changes to breadth would affect volume in direct proportion, which would mean (if the torso is 70% of the whole animal) a change in total body volume of 14% either way.  Significant stuff.

So what do we know about the torso breadth in sauropods?  It obviously dependant primarily on the orientation of the ribs and their articulation to the dorsal vertebrae.  And what do we know about that?

Nothing.

Well, OK, I am over-simplifying a little.  It’s been mentioned in passing in a few papers, but it’s never been discussed in any detail in a published paper that I know of.  (There’s a Masters thesis out there that starts to grapple with the subject, but I don’t know whether I should talk about that while it’s still being prepared for publication, so I won’t say anything more.)  The most important published contribution is more than a century old — Holland’s (1910) smackdown of Tornier’s and Hay’s comical Diplodocus postures, which included the following cross-sections of the torsos of several animals at the seventh dorsal vertebra:

(This figure previously appeared on SV-POW! in Matt’s post, Sauropods were tacos, not corn dogs, which as far as I am aware is the only existing non-technical treatment of sauropod torso-shape.)

Holland unfortunately did not discuss the torso shape that he illustrated, merely asserting it.  Presumably it is based on the mounted skeleton of the Diplodocus carnegii holotype CM 84, which is at the Carnegie Museum in Pittsburgh, where Holland was based.  I have no reason to doubt it; just noting that it wasn’t discussed.

All right then — what about Camarasaurus?  I think it’s fair to say that it’s generally considered to be fairly rotund among sauropods, as for example this skeletal reconstruction by Greg Paul shows:

Camarasaurus lentus skeletal reconstruction, in dorsal and right lateral views. (Paul 2010:197)

Measuring off the height and width of the torso at the seventh dorsal vertebra, using GIMP, I find that they are 341 and 292 pixels respectively, so that the eccentricity is 341/292 = 1.17.  This compares with 1760/916 = 1.92 for Holland’s Diplodocus above, so if both figures are accurate, then Camarasaurus is much fatter than Diplodocus.

But is Paul’s Camarasaurus ribcage right?  To answer that, I went back to my all-time favourite sauropod paper, Osborn and Mook’s (1921) epic descriptive monograph of Camarasaurus (and Cope’s other sauropods).  I knew that this awesomely comprehensive piece of work would include plates illustrating the ribs; and in fact there are four plates that each illustrate a complete set of dorsal ribs (although the associations are doubtful).  Here they all are:

Left dorsal ribs of Camarasaurus (Osborn and Mook 1921:pl. LXXVIII)

Left dorsal ribs of Camarasaurus (Osborn and Mook 1921:pl. LXXIX)

Left dorsal ribs of Camarasaurus (Osborn and Mook 1921:pl. LXXX)

Left dorsal ribs of Camarasaurus (Osborn and Mook 1921:pl. LXXXI)

But hang on a minute — what do you get if you articulate these ribs with the dorsal vertebrae?  Osborn and Mook also provided four plates of sequences of dorsal vertebrae, and the best D7 of the four they illustrate is probably the one from plate  LXX.  And of the four 7th ribs illustrated above, the best preserved is from plate LXXIX.  So I GIMPed them together, rotated the ribs to fit as best I could and …

What on earth?!

I spent a bit of time last night feeling everything from revulsion to excitement about this bizarre vertebra-and-rib combination.  Until I happened to look again Osborn and Mook — earlier on, in the body of the paper, in the section about the ribs.  And here’s what I saw:

(Note that this is the vertebra and ribs at D4, not D7; but that’s close enough that there’s no way there could be a transition across three vertebrae like the change between this and the horrible sight that I presented above.)

What’s going on here?  In the plates above, the ribs do not curve inwards as in this cross-section: they are mostly straight, and in many case seem to curve negatively — away from the torso.  So why do O&M draw the ribs in this position that looks perfectly reasonable?

And figure 70, a few pages earlier, makes things even weirder: it clearly shows a pair of ribs curving medially, as you’d expect them to:

So why do these ribs look so totally different from those in the plates above?

I’ll give you a moment to think about that before I tell you the answer.

Seriously, think about it for yourself.  While you’re turning it over in your mind, here is a picture of the beautiful Lego kit #10198, the Blockade Runner from the original Star Wars movie.  (I deeply admire the photography here: clear as a bell.)

OK, welcome back.

Got it?  I bet most of you have.

The answer was right there in figure 71:

Osborn and Mook 1921:fig. 71. Left rib of Camarasaurus supremus Cope. Rib 4 (Amer. Mus. Cope Coll. No. 5761/R-A-24). (A) direct external view when placed as in position in the body; (B) direct anterior when placed as in position in the body. Capit. capitulum; Sh. shaft; Tub. tuberculum. Reconstructed view, portion in outline.

Osborn and Mook 1921:fig. 71. Left rib of Camarasaurus supremus Cope. Rib 4 (Amer. Mus. Cope Coll. No. 5761/R-A-24). (A) direct external view when placed as in position in the body; (B) direct anterior when placed as in position in the body. Capit. capitulum; Sh. shaft; Tub. tuberculum. Reconstructed view, portion in outline.

And, my word, isn’t it embarrassingly obvious once you see it?  I’d been blithely assuming that the ribs in O&M’s plates were illustrated in anterior view, with the capitula (which articulate with the parapophyses) located more medially, as well as more ventrally, than the tubercula (which articulate with the diapophyses).  But no: as in fact the captions of the plates state perfectly clearly — if I’d only had the wits to read them — the ribs are shown in “external” (i.e. lateral) view.  Although it’s true that the capitula in life would indeed have been more medially positioned than the tubercula, it’s also true that they were more anteriorly positioned, and that’s what the plates show at the rib heads.  And the curvature that I’d been stupidly interpreting as outward, away from the midline, is in fact posteriorly directed: the ribs are “swept back”.  The ventral portions of the ribs also curve medially, away from the viewer and into the page … but of course you can’t see that in the plates.

The important truth — and if you take away nothing else from this post, take this — is that I am dumb bones are complex three-dimensional objects, and it’s impossible to fully understand their shape from single-view illustrations.  It’s for this reason that I make an effort, when I can, to illustrate complex bones from all cardinal directions — in particular, with the Archbishop bones, as for example “Cervical S” in the Brachiosaurus coracoid post.

Because ribs, in particular, are such complex shapes — because their curvature is so unpredictable, and because their articulation with the dorsal vertebrae is via two points which are located differently on successive vertebrae, and because this articulation still allows a degree of freedom of movement — orthogonal views, even from all cardinal directions, are of limited value.  Compositing figures will give misleading results … as demonstrated above.  PhotoShop is no more use here.  Fly, you fools!

Paradoxically, our best source of information on the shapes of saurpod torsos is: mounted skeletons.  I say “paradoxically” because we’ve all grown used to the idea that mounts are not much use to us as scientists, and are really there only as objects of awe.  As Brian Curtice once said, “A mounted skeleton is not science.  It’s art.  Its purpose is to entertain the public, not to be a scientifically accurate specimen”.  In many respects, that’s true — especially in skeletons like that of the “Brontosaurus” holotype, YPM 1980, where the bones are restored with, and in some cases encased in, plaster so you can’t tell what’s what.  But until digital scanning and modelling make some big steps forward, actual mounted skeletons are the best reference we have for the complex articulations of ribs.

Giraffatitan brancai paralectotype HMN SII, composite mounted skeleton, torso in left posteroventrolateral view (photograph by Mike Taylor)

And I finish this very long (sorry!) post with yet another note of caution.  Ribs are long and thin and very prone to damage and distortion.  It’s rare to find complete sauropod ribs (look closely at the O&M plates above for evidence), but even when we do, we shouldn’t be quick to assume that the shape in which they are preserved is necessarily the same as the shape they had in life.  (If you doubt this, take another look at rib #6 in the third of the four O&M plates above.)  And as if that weren’t enough to discourage us, we should also remember that the vertebra-rib joints would have involved a lot of cartilage, and we don’t know its extent or shape.

So bearing in mind the complicated 3D shape of ribs and of dorsal vertebrae, the tendency for both to distort during and after fossilisation, and the complex and imperfectly known nature of the joints between them, I think that maybe I wasn’t too far wrong earlier when I said that what we know about sauropod torso shape is: nothing.

It’s a sobering thought.

References

Dorsal vertebrae from Argentinosaurus (center) and Supersaurus (either side). The vert on the left is the holotype of Ultrasauros, and the one on the right is the holotype of Dystylosaurus, but both of those taxa have been sunk into Supersaurus. Found on teh intert00bz.

As often happens here, a comment thread got to be more interesting than the original post and ended up deserving a post of its own. In this case, I’m talking about the thread following the recent Mamenchisaurus tail club post, which got into some interesting territory regarding mass estimates for the largest sauropods. This post was inspired by a couple of comments in particular.

Zach Armstrong wrote:

I don’t trust Mazzetta et al.’s (2004) estimate, because it is based off of logarithmic-based regression analyses of certain bone lengths, which a recent paper by Packard et al. (2009) have shown to overestimate the mass by as much as 100 percent! This would mean the estimate of 73 tonnes given my Mazzetta would be reduced to 36 tonnes.

To which Mike replied:

Zach, Mazzetta et al. used a variety of different techniques in arriving at their Argentinosaurus mass estimate, cross-checked them against each other and tested their lines for quality of fit. I am not saying their work is perfect (whose is?) but I would certainly not write it off as readily as you seem to have.

Weeeeell…Mazzetta et al. did use a variety of measurements to make their mass estimates, but they did it in a way that hardly puts them above criticism. First, their estimates are based on limb-bone allometry, which is known to have fairly low accuracy and precision (like, often off by a factor of 2, as Zach noted in his comment). Second, the “raw data” for their allometric equation consists of volumetric mass estimates. So their primary estimation method was calibrated against…more estimates. Maybe I’m just lazy, but I would have skipped the second step and just used volumetric methods throughout. Still, I can see the logic in it for critters like Argentinosaurus where we have limb bones but no real idea of the overall form or proportions of the entire animal.

Anyway, the accuracy of their allometric estimates is intertwingled with their volumetric results, so if their volumetric estimates are off…. The volumetric estimates used a specific gravity of 0.95, which to me is unrealistically high. Taking into account the skeletal pneumaticity alone would lower that to 0.85 or 0.8, and if the critter had air sacs comparable to those of birds, 0.75 or even 0.7 is not beyond the bounds of possibility (as discussed here and also covered by Zach in his comment).

Now, Mazzetta et al. (2004) were not ignorant of the potential effects of pneumaticity. Here’s  what they wrote about density (p. 5):

The values from Christiansen (1997) were recalculated using a slightly higher overall density (950 kg/m^3), as the 900 kg/m^3 used in that paper may be slightly too low. Most neosauropods have extensively pneumatised vertebrae, particularly the cervicals, which would tend to lower overall density. However, these animals are also very large, implying a proportionally greater amount of skeletal tissue (Christiansen, 2002), particularly appendicular skeletal tissue, and consequently, they should have had a higher overall density.

This is pretty interesting: they are arguing that the positive allometry of skeletal mass as a fraction of body mass–which is well documented in extant critters–would offset the mass reduction from pneumaticity in animals as big as sauropods. I haven’t given that enough thought, and I definitely need to. My guess–and it is a guess–is that the effects of skeletal allometry were not enough to undo the lightening imposed by both PSP (~10%) and pulmonary air sacs (another ~10%, separate from the lungs), but I haven’t done any math on this yet. Fodder for another post, I reckon.

Getting back to Mazzetta et al., some of the volumes themselves strike me as too high, like ~41,500 liters for HM SII. That’s a LOT more voluminous than Greg Paul, Don Henderson, or Mike found for the same critter. The 16 metric ton Diplodocus and 20.6 metric ton Apatosaurus used by Mazzetta et al. are also outside the bounds of other recent and careful estimates. Not necessarily wrong, but definitely at the upper end of the current spectrum.

Mazzetta et al. got a mass estimate of 73,000 kg for Argentinosaurus, but (1) they used a density that I think is probably too high even if skeletal allometry is considered, (2) at least some of the volumetric mass estimates that form the “data” for the limb-bone regressions are probably too high, and (3) even if those problems were dealt with, there is still the general untrustworthiness of limb-bone regression as a mass estimation technique. 1 and 2, if fixed to my satisfaction, would tend to push the estimated mass of Argentinosaurus down, perhaps significantly (the effect of 3 is, if not unknowable, at least unknown to me). Given that, Zach’s ~52 metric ton estimate for Argentinosaurus is very defensible. (Probably worth remembering that I am a sparse-wing fanatic, though.)

None of this means that Mazzetta et al. (2004) were sloppy or that their estimate is wrong. Indeed, one of the reasons that we can have such a deep discussion of these points is that every link in their chain is so well documented. And there is room for honest disagreement in areas where the fossils don’t constrain things as much as we’d like. You cannot simply take a skeleton, even a complete one, and get a single whole-body volume. The body masses of wild animals often fluctuate by a third over the course of a single year, which pretty well buries any hope of getting precise estimates based on skeletons alone. And no one knows how dense–or sparse–sauropods were. I haven’t actually done any math to gauge the competing effects of skeletal allometry on one hand and PSP and air sacs on the other–and, AFAIK, no one else has either (Mazzetta et al. were guessing about pneumaticity as much as I’m guessing about skeletal allometry). Finally, Argentinosaurus is known from a handful of vertebrae and a handful of limb bones and that’s all, at least for now. If we can’t get a single body volume even when we have a complete skeleton, we have to get real about how precise we can be in cases where we have far less material.

The upshot is not that Argentinosaurus massed 73 metric tons or 52 or any other specific number. As usual, the two-part take home message is that (1) mass estimates of sauropods are inherently imprecise, so all we can do is make our assumptions as clear as possible, and (2) even the biggest sauropods might have been smaller than you think. ;-)

Reference

Mazzetta, G.V., Christiansen, P., and Farina, R.A. 2004. Giants and bizarres: body size of some southern South American Cretaceous dinosaurs. Historical Biology 2004:1-13.

Lovers of fine sauropods will be well aware that, along with the inadequately described Indian titanosaur Bruhathkayosarus, the other of the truly super-giant sauropods is Amphicoelias fragillimus.  Known only from a single neural arch of a dorsal vertebra, which was figured and briefly described by Cope (1878) and almost immediately either lost or destroyed, it’s the classic “one that got away”, the animal that sauropod aficionados cry into their beer about late at night.

Amphicoelias fragillimus, holotype dorsal vertebral neural arch in posterior view. From Osborn and Mook (1921:fig. 21), which in turn was gently tweaked from Cope (1878:unnumbered and only figure).

I’m not going to write about A. fragillimus in detail here, because Darren’s so recently covered it in detail over at Tetrapod Zoology — read Part 1 and Part 2 right now if you’ve not already done so.  The bottom line is that it was a diplodocoid roughly twice as big as Diplodocus in linear dimension (so about eight times as heavy).  That makes it very very roughly 50 m long and 100 tonnes in mass.

But Mike!, you say, Isn’t it terribly naive to go calculating masses and all from a single figure of part of a single bone?

Why, yes!  Yes, it is!  And that is what this post is about.

As I write, the go-to paper on A. fragillimus is Ken Carpenter’s (2006) re-evaluation, which carefully and tentatively estimated a length of 58 m, and a mass of around 122,400 kg.

As it happens, Matt and a colleague submitted a conference abstract a few days ago, and he ran it past me for comments before finalising.  In passing, he’d written “there is no evidence for sauropods larger than 150 metric tons and it is possible that the largest sauropods did not exceed 100 tons”.  I replied:

I think that is VERY unlikely. […] the evidence for Amphicoelias fragillimus looks very convincing, Carpenter’s (2006) mass estimate of 122.4 tonnes is conservative, being extrapolated from Greg Paul’s ultra-light 11.5 tonne Diplodocus.

Carpenter’s estimate is based on a reconstruction of the illustrated vertebra, which when complete he calculated would have been 2.7 m tall.  That is 2.2 times the height of the corresponding vertebra in Diplodocus, and the whole animal was considered as it might be if it were like Diplo scaled up by that factor.  Here is his reconstruction of the vertebra, based on Cope’s figure of the smaller but better represented species Amphicoelias altus:

One possible reconstruction of the Alphicoelias fragillimus vertebra, from Carpenter (2006:fig. 1).  Part A is Cope’s original figure annotated with lamina designations; part C is Cope’s illustration of an Amphocoelias altus dorsal; part B is Carpenter’s reconstruction of the former after the latter.

Matt’s answer to me was:

First, Paul’s ultra-light 11.5 tonne Dippy is not far off from my 12 tonne version that you frequently cite, and mine should be lighter because it doesn’t include large air sacs (density of 0.8 instead of a more likely 0.7). If my Dippy had an SG of 0.7, it would have massed only 10.25 tonnes. Second, Carpenter skewed […] in the direction of large size for Amphicoelias. I don’t necessarily think he’s wrong, but his favoured estimate is at the extreme of what the data will support. Let’s say that Amphicoelias was evenly twice as large as Dippy in linear terms; that could still give it a mass as low as 90 tonnes. And that’s not including the near-certainty that Amphicoelias had a much higher ASP than Diplodocus. If Amphicoelias was to Diplodocus as Sauroposeidon was to Brachiosaurus—pneumatic bones about half as dense—then 1/10 of its volume weighed ½ as much as it would if it were vanilla scaled up Dippy, and we might be able to knock off another 5 tonnes.

There’s lots of good stuff here, and there was more back and forth following, which I won’t trouble you with.  But what I came away with was the idea that maybe the scale factor was wrong.  And the thing to do, I thought, was to make my own sealed-room reconstruction and see how it compared.

So I extracted the A.f. figure from Osborn and Mook, and deleted their dotted reconstruction lines.  Then I went and did something else for a while, so that any memory of where those lines might have been had a chance to fade.  I was careful not look at Carpenter’s reconstruction, so I could be confident mine would be indepedent.  Then I photoshopped the cleaned A. fragillimus figure into a copy the A. altus figure, scaled it to fit the best as I saw it, and measured the results.  Here it is:

My scaling of a complete Amphicoelias fragillimus vertebra: on the left, Cope’s figure of the only known vertebra; on the right, Cope’s figure of an A. altus dorsal vertebra, scaled to match the preserved parts of the former.  Height of the latter scaled according to the measured height of the former.

As you can see, when I measured my scaled-to-the-size-of-A.f. Amphicoelias vertebra, it was “only” 2293 mm tall, compared with 2700 mm in Ken’s reconstruction.  In other words, mine is only 85% as tall, which translates to 0.85^3 = 61% as massive.  So if this reconstruction is right, the big boy is “only” 1.87 times as long as Diplodocus in linear dimension — maybe 49 meters long — and would likely come in well below the 100-tonne threshhold.  Using Matt’s (2005) 12-tonne estimate for Diplodocus, we’d get a mere 78.5 tonnes for Amphicoelias fragillimus.  So maybe Matt called that right.

Amphicoelias altus dorsal vertebra, almost certainly the holotype, in left lateral view, lying on its back.  Photograph by Matt Wedel, from the collections of the AMNH.  I can’t believe — can’t BELIEVE — that I didn’t take ten minutes to look at this vertebra when I was in that basement last February.  What a doofus.

The Punchline

Folks — please remember, the punchline is not “Amphicoelias fragillimus only weighed 78.5 tonnes rather than 122.4 tonnes”.  The punchline is “when you extrapolate the mass of an extinct animal of uncertain affinities from a 132-year-old figure of a partial bone which has not been seen in more than a century, you need to recognise that the error-bars are massive and anything resembling certainty is way misplaced.”

Caveat estimator!

References

  • Carpenter, Kenneth.  2006.  Biggest of the big: A critical re-evalustion of the mega-sauropod Amphicoelias fragillimus Cope, 1878.  pp. 131-137 in J. Foster and S. G. Lucas (eds.), Paleontology and Geology of the Upper Jurassic Morrison Formation.  New Mexico Museum of Natural History and Science Bulletin 36.
  • Cope, Edward Drinker.  1878.  Geology and Palaeontology: a new species of Amphicoelias.  The American Naturalist 12 (8): 563-566.
  • Osborn, Henry Fairfield, and Charles C. Mook.  1921.  Camarasaurus, Amphicoelias and other sauropods of Cope.  Memoirs of the American Museum of Natural History, n.s. 3:247-387, and plates LX-LXXXV.

Bifid Brachiosaurs, Batman!

September 6, 2009

These are the days of miracle and wonder, especially for all you right-minded people out there who are lovers of fine brachiosaurs.  I heard yesterday evening about a new paper in Proceedings of the Royal Society B: You and Li’s (2009, duh) description of a new brachiosaur, the first one known from the Cretaceous of Asia: Qiaowanlong kangxii. Best of all, it’s based primarily on vertebral material:

You and Li (2009:fig. 2)  Cervical vertebrae of Qiaowanlong kangxii holotype FRDC GJ 07-14.

You and Li (2009:fig. 2) Cervical vertebrae of Qiaowanlong kangxii holotype FRDC GJ 07-14. (a) Photograph and (b) interpretative line drawing of C4-C7 in left lateral view; (c) a distal portion of a cervical rib; C9 in (d) cranial, (e) left lateral, (f) caudal, (g) right lateral, (h) dorsal and (i) ventral views. di, diapophysis; f1-f5, fossa 1-fossa 5; pa, parapophysis; poz, postzygapophysis; prz, prezygapophysis; sp, neural spine. Scale bars, 10 cm.

Brachiosaur aficionados will be gazing slack-jawed at parts d, f and h of this figure (the anterior, posterior and dorsal views of C9), which clearly show that the neural spines of the new taxon are bifid (i.e. have two peaks side by side and a trough between them) — just like the cervical neural spines of flagellicaudatans (diplodocids and dicraeosaurs) and camarasaurs.  And mamenchisaurs.  And some titanosaurs.  And Erketu.  Finding this feature yet again — apparently independently evolved in brachiosaurs — makes it about the most plastic character in the matrix.  Very exciting.

That is, it’s exciting if this really is a brachiosaurid.  Now as it happens, Matt was one of the reviewers for this paper (and by the way did an amazingly professional job of not telling me about it until it came out, the git).  He’s told me in email that he’s satisfied that Qiaowanlong really is a brachiosaur, and I hesitate to question that identification given that (A) unlike the authors I’ve never seen the material, and (B) unlike Matt, I’ve spent most of my brachiosaur-presacral quality time with dorsals rather than cervicals.  But, with that caveat, I’m not sure that a compelling case has yet been made for a brachiosaurian identity.

The authors cite three characters in support of a brachiosaurid identity:

  • The most persuasive is the deeply excavated cervical neural spines.
  • Next is a transition in neural spine height: this is quite abrupt in “Brachiosaurusbrancai between cervicals 6 and 7, and also in Sauroposeidon — presumably also between C6 and C7, but that can’t be known for sure, since it’s only the assumption that this is the case that led to the identification of the four preserved Sauroposeidon cervicals as C5-C8 in the first place.  In Qiaowanlong, this transition is “much less pronounced”, with spines increasing in height by only 25% rather then 100% in the other taxa — and occurs between C8 and C9.  All in all, not really very similar to the condition in “B.” brancai.
  • The final character supporting the brachiosaurid identity of Qiaowanlong is the absence of an anterior centrodiapophyseal lamina.  As the authors point out, though, this lamina does exist in “B.” brancai and is absent only in Sauroposeidon; so if this is evidence of anything, it’s a synapomorphy of a clade uniting Qiaowanlong and Sauroposeidon to the absence of other brachiosaurs — something that seems very unlikely given the proportions of the vertebrae.

Putting it all together, there seems to be only one convincing brachiosaur character cited; and that stands against several non-brachiosaur characters, most obviously the bifurcation of the neural spine and the low Elongation Index (centrum length divided by cotyle height) but also by a few other characters that are not discussed in the paper.  For example, Matt has previously noted that in brachiosaur cervicals, the diapophyses are more anteriorly positioned than the parapophyses whereas in diplodocids the opposite is the case: as shown in fig 2(b) above, C6 at least of Qiaowanlong resembles diplodocids in this respect.

To try to get more of a handle on this, I put together a comparative figure of the 8th and 9th cervicals of various sauropods:

8th/9th cervicals vertebrae of various sauropods, scaled to the same centrum length.  "Brachiosaurus" brancai, Sauroposeidon; Qiaowanlong, Diplodocus; Haplocanthosaurus, Camarasaurus

8th/9th cervicals vertebrae of various sauropods, scaled to the same centrum length. From top to bottom and left to right: "Brachiosaurus" brancai, Sauroposeidon; Qiaowanlong, Diplodocus; Haplocanthosaurus, Camarasaurus. Six sauropod vertebrae for the price of one!

Based on overall proportions, I don’t find it intuitively obvious that the Qiaowanlong (middle row, left) more closely resembles the brachiosaurs (top row) than it does the other three.

What does all this mean?  Probably nothing: most likely there are further reasons for the brachiosaurid identification of the new taxon, and lack of space prevented their explanation and illustration.  We can hope that the authors, having placed an initial brief description in Proc. B, will follow it up with a more comprehensive description and analysis in a journal that does not impose such tight length restrictions.  But for now at least, my feeling is that the case for a bifid brachiosaur has yet to be made.

Moving on … Qiaowanlong is also represented by some nice appendicular material: the entire right side of the pelvis (ilium, ischium and pubis).  The ilium certainly looks brachiosaury, so that is another bit of support for the systematic hypothesis, but the proportions of the pelvic bones are very odd:

Right pelvis of "Brachiosaurus" brancai (left), based on composite of Janensch's (1961) figures, and Qiaowanlong (from You and Li 2009: fig. 3a).  Scaled to same ilium length.

Right pelvis of "Brachiosaurus" brancai (left), based on composite of Janensch's (1961) figures, and Qiaowanlong (from You and Li 2009: fig. 3a). Scaled to same ilium length.

You and Li (2009) describe their pelvis as having a “much reduced ischium”, but as is apparent by comparison with the pelvis of “Brachiosaurusbrancai, the ischium is in reasonable proportion to the ilium, and the oddity is more that the pubis is enormous.  So much so that it makes me feel a little ill looking at it, and it makes me wonder how certain it is that all three of these bones are from the same individual — sadly, the paper doesn’t discuss the association of the material.

[Not to flog a dead horse, but this kind of omission shows once more the perils of publishing new taxa in general-interest journals such as Proc. B that impose draconian length limits.  This paper just creeps onto page 7, and I simply don’t believe that it’s possible to do anything like justice to the description of a new taxon in that little space, especially when there is also geography, geology, phylogeny and discussion to be got through.  I don’t want to go all This Is How To Do It, but I can’t help remembering that Darren and I took 18 pages, nearly three times as long, to describe the single partial vertebra that is Xenoposeidon (Taylor and Naish 2007), and it’s not as though that paper wastes a lot of words.  To give You and Li credit, they did squeeze in photos of a representative vertebra from all six cardinal directions, which is great; but only at the cost of the photos being too tiny to be much use.  Please, folks: send your new taxon descriptions to a proper descriptive journal, not to a tabloid!  </hobbyhorse>]

Back on the Dinosaur Mailing List, B tH asked how big Qiaowanlong was.  According to the BBC, the authors say that “the dinosaur would have been a relatively small sauropod about 12m long, 3m high, and weighing perhaps 10 tonnes”.  Can we confirm that?  Well, the excellently comprehensive table of measurements in the paper gives centrum lengths, not counting the condyle, totalling 267 cm for the seven vertebrae C5-C11.  Janensch (1950a:44) gave measurements for the corresponding vertebrae of “Brachiosaurusbrancai HMN SII totalling 577 cm, which is more than twice as long.  If Qiaowanlong was 267/577 = 0.46 times as long as HMN SII, which Janensch (1950b:102) gave as 22.46 m, then it would have been 10.4 m long; it’s not obvious how the authors got the larger figure of 12 m unless they had reason to think the neck was proportionally shorter than in HMN SII.  If Qiaowanlong was isometrically similar to HMN SII, then it was 0.46^3 = 0.99 0.099 times as heavy.  Using my own in-press mass of 23337 kg for HMN SII, this would make Qiaowanlong only 2312 kg in mass — pretty pathetic for a sauropod.

That’s it for now.  I’d be the first to admit that there’s an awful lot of speculation in this post based on relatively little published information.  Hopefully You Hai-Lu will drop by and comment — I’ll be letting him know that I’ve posted this.

References

  • Janensch, Werner.  1950.  Die Wirbelsaule von Brachiosaurus brancai. Palaeontographica (Suppl. 7) 3: 27-93.
    Janensch, Werner.  1950.  Die Skelettrekonstruktion von Brachiosaurus brancai.  Palaeontographica (Suppl. 7) 3: 97-103.
    Janensch, Werner.  1961.  Die Gliedmaszen und Gliedmaszengurtel der Sauropoden der Tendaguru-Schichten.  Palaeontographica, suppl. 7 (1), teil 3, lief. 4: 177-235.
    Taylor, Michael P. and Darren Naish.  2007.  An unusual new neosauropod dinosaur from the Lower Cretaceous Hastings Beds Group of East Sussex, England.  Palaeontology 50 (6): 1547-1564.  doi: 10.1111/j.1475-4983.2007.00728.x
    You, Hai-Lu, and Li, Da-Qing.  2009.  The first well-preserved Early Cretaceous brachiosaurid dinosaur in Asia.  Proceedings of the Royal Society B: Biological Sciences.  doi: 10.1098/rspb.2009.1278.
  • Janensch, Werner.  1950.  Die Wirbelsaule von Brachiosaurus brancai. Palaeontographica (Suppl. 7) 3: 27-93.
  • Janensch, Werner.  1950.  Die Skelettrekonstruktion von Brachiosaurus brancai.  Palaeontographica (Suppl. 7) 3: 97-103.
  • Janensch, Werner.  1961.  Die Gliedmaszen und Gliedmaszengurtel der Sauropoden der Tendaguru-Schichten.  Palaeontographica, suppl. 7 (1), teil 3, lief. 4: 177-235.
  • Taylor, Michael P. and Darren Naish.  2007.  An unusual new neosauropod dinosaur from the Lower Cretaceous Hastings Beds Group of East Sussex, England.  Palaeontology 50 (6): 1547-1564.  doi: 10.1111/j.1475-4983.2007.00728.x
  • You, Hai-Lu, and Li, Da-Qing.  2009.  The first well-preserved Early Cretaceous brachiosaurid dinosaur in Asia.  Proceedings of the Royal Society B: Biological Sciences.  doi: 10.1098/rspb.2009.1278.

And finally … two announcements!

Traumador the Tyrannosaur has asked us to point out that over on ART Evolved (the palaeo-art blog), the next big art gallery is to be sauropod themed.  Details are on the site, so get over there and submit your sauropod art!

And Matt and I will shortly be teaming up with Andy Farke, the open-source paleontologist, on a new project where we plan to actually do some of this Shiny Digital Future that we keep on talking about.  Andy will be announcing the details on Tuesday 8th September.  Mark the date well!  For now, I shall say no more …

How big was Alamosaurus?

September 2, 2009

Alamosaurus skeleton reference 480

Here’s a skeletal reconstruction of Alamosaurus modified from Lehman and Coulson (2002:fig. 11). I cloned the neck and rotated it a few degrees to see where it would put the head.

The skeleton in the figure is scaled to the size of the individuals in the Smithsonian and at UT Austin. The scale bar is 1 meter, which by my calculations gives that individual the following dimensions:

  • Total length: 15.8 meters (52 feet)
  • Neck length: 5.2 meters (17 feet)
  • Shoulder height: 4 meters (13 feet)
  • Head height (with neck raised): 8.4 meters (27.5 feet)

Big Bend Alamosaurus dig

Here are a couple of articles on a giant sauropod found in Big Bend in 1999. This critter is generally assumed to be Alamosaurus but it could be something new (I have no evidence either way); the material is currently under study at the Dallas Museum of Nature and Science.
http://www.nps.gov/bibe/naturescience/alamosaurus.htm
http://www.geocities.com/stegob/texasdino.html

According to the articles, 10 cervical vertebrae were found in a string 23 feet long. From the pictures, those ten vertebrae look like the ten largest, which should account for almost all of the neck except for the first few cervicals behind the head. Let’s assume that this big individual therefore had a neck just a little longer than 23 feet, and we find that it is almost exactly 1.5 times bigger than the one listed above. If its proportions follow those of the Lehman and Coulson recon, its measurements would be:

  • Total length: 24 meters (79 feet)
  • Neck length: 7.8 meters (25.5 feet)
  • Shoulder height: 6 meters (19.5 feet)
  • Head height: 12.6 meters (41 feet)

In the second article Homer Montgomery speculates that the complete neck would have been more than 30 feet long. That’s certainly not impossible, since 30-foot-plus necks are known for the largest individuals in several clades (e.g., Mamenchisaurus, Supersaurus, Sauroposeidon, probably Puertasaurus, possibly Futalognkosaurus, but probably not Aegyptosaurus) If so, then you could just about double all of the proportions from the first individual described above, which would give a truly prodigious animal. The 52-foot animal probably had a mass around 15 tons, so the 79-footer would have been about 50 tons (1.5^3 = 3.375), and the hypothetical 100-footer would have been 120 tons, which is up in Amphicoelias/Bruhathkayosaurus territory. For what it’s worth, I think the numbers for the 79-foot animal are more plausible, but who knows. Anytime you’ve got a partial neck that is longer than the complete neck of Diplodocus, you’re dealing with a wacky big animal.

Reference

Lehman, T.M. & Coulson, A.B. 2002. A juvenile specimen of the sauropod Alamosaurus sanjuanensis from the Upper Cretaceous of Big Bend National Park, Texas. Journal of Paleontology 76(1): 156-172.