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
Crosssectional 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 crosssections, it’s best to measure two radii at right angles to each other, or use a different measure of crosssectional 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/(1K)
 K = 1 – (1/(R/t))
ASP (air space proportion) and MSP (marrow space proportion) measure the crosssectional area of an element not taken up by bone tissue. ASP and MSP are the same measurement–the amount of nonbone space in a bony element divided by the total–we just use ASP for airfilled bones and MSP for marrowfilled 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 crosssection of that hypothetical tubular bone.
To estimate the density of an element (at least near the point of a given crosssection), 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 marrowfilled bone, the density of the element (or at least of the part of the shaft the section goes through) is:
 0.93MSP + 2.1(1MSP)
Air is matter and therefore has mass and density, but it is so light (0.00120.0013 g/mL) that we can effectively ignore it in these calculations. So the density of a pneumatic element is: 2.1(1ASP) For the three examples in the figure at the top of the post, the ASP/MSP values and densities are:
 (b) alligator femur (marrowfilled), 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 (marrowfilled), 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 (airfilled), 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 marrowfilled? The results would then be:
 (b) alligator femur (hypothetical airfilled), K = 0.35, ASP = K^2 = 0.12, density = (2.1 x 0.88) = 1.85 g/mL
 (c) camel tibia (hypothetical airfilled), K = 0.57, ASP = K^2 = 0.32, density = (2.1 x 0.68) = 1.43 g/mL
 (d) Pteranodon first phalanx (hypothetical marrowfilled), 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 nonbone 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 56%. The Pteranodon phalanx is a lot less dense than the alligator femur for two reasons. First, there is much less bony tissue–the hypothetical marrowfilled 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
 Currey, J. D., and Alexander, R. McN. 1985. The thickness of the walls of tubular bones. Journal of Zoology 206:453–468.
 Wedel, M.J. 2005. Postcranial skeletal pneumaticity in sauropods and its implications for mass estimates; pp. 201228 in Wilson, J.A., and CurryRogers, K. (eds.), The Sauropods: Evolution and Paleobiology. University of California Press, Berkeley.
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 offthecuff 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.
* 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 paperwriting. 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 inpress 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.

Alexander, R.M. 1985. Mechanics of posture and gait of some large dinosaurs. Zoological Journal of the Linnean Society, 83(1): 125.

Alexander, R.M. 1989. Dynamics of Dinosaurs and Other Extinct Giants. Columbia University Press.
 Hutchinson, J.R., Bates, K.T., Molnar, J., Allen, V., and Makovicky, P.J. 2011. A computational analysis of limb and body dimensions in Tyrannosaurus rex with implications for locomotion, ontogeny, and growth. PLoS ONE 6(10): e26037. doi:10.1371/journal.pone.0026037
 Taylor, M.P. 2009. A reevaluation of Brachiosaurus altithorax Riggs 1903 (Dinosauria, Sauropoda) and its generic separation from Giraffatitan brancai (Janensch 1914). Journal of Vertebrate Paleontology 29(3):787806.
 Wedel, M.J., and Taylor, M.P. In press. Neural spine bifurcation in sauropod dinosaurs of the Morrison Formation: ontogenetic and phylogenetic implications. PalArch’s Journal of Vertebrate Paleontology.
 Witton, M.P., and Naish, D. 2008. A reappraisal of azhdarchid pterosaur functional morphology and paleoecology. PLoS ONE 3(5): e2271. doi:10.1371/journal.pone.0002271
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 groundtruthed on solid data from extant animals.
It’s a reassuring sanitycheck 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 longbone 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 massestimation technique we now have available for complete and nearcomplete skeletons. It would be good to see it applied to (say) the Yale, AMNH and CM apatosaurs.
References
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 extinctanimal 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 25000140000 kg), but, oddly, noone 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 90100% 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 VickersRich (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 crosssectional 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 VickersRich (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 dinoobsessed youngster in your life. You could use the whitebonesonblacksilhouette 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 dinowrangler and frequent SVPOW! 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 pixelwidth 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 4050 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 crosssections, 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 crosssections (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 multisegment 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 wellknown 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 wholebody 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 oftquoted “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 slackjawed 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 630kg estimate for the slender model is deceptively close to the upper end of my range; deceptive because their 630kg estimate assumes a density of 0.8 and my 636kg 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., nonstarving) 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 nottoodistant future.
References
 Gunga, H.C., Suthau, T., Bellmann, A., Friedrich, A., Schwanebeck, T., Stoinski, S., Trippel, T., Kirsch, K., Hellwich, O. 2007. Body mass estimations for Plateosaurus engelhardti using laser scanning and 3D reconstruction methods. Naturwissenschaften 94(8):623630.
 Hazlehurst, G.A., and Rayner, J.M. 1992. Flight characteristics of Triassic and Jurassic Pterosauria: an appraisal based on wing shape. Paleobiology 18(4):447463.
 Henderson, D.M. 1999. Estimating the mass and centers of mass of extinct animals by 3D mathematical slicing. Paleobiology 25:88106.
 Henderson, D.M. 2004. Tipsy punters: sauropod dinosaur pneumaticity, buoyancy and aquatic habits. Proceedings: Biological Sciences 271 (Supplement):S180S183.
 Henderson, D.M. 2006. Burly gaits: centers of mass, stability and the trackways of sauropod dinosaurs. Journal of Vertebrate Paleontology 26:907921.
 Hurlburt, G. 1999. Comparison of body mass estimation techniques, using Recent reptiles and the pelycosaur Edaphosaurus boanerges. Journal of Vertebrate Paleontology 19:338–350.
 Jerison, H.J. 1973. Evolution of the Brain and Intelligence. Academic Press, New York, NY, 482 pp.
 Mallison, H., Hohloch, A., and Pfretzschner, H.U. 2009. Mechanical digitizing for paleontology–new and improved techniques. Palaeontologica Electronica 12(2):4T, 41 pp.
 Mallison, H. 2010. The digital Plateosaurus I: Body mass, mass distribution, and posture assessed by using CAD and CAE on a digitally mounted complete skeleton. Palaeontologica Electroncia 13(2):8A, 26 pp.
 Motani, R. 2001. Estimating body mass from silhouettes: testing the assumption of elliptical body crosssections. Paleobiology 27(4):735–750.
 Murray, P.F. and VickersRich, P. 2004. Magnificent Mihirungs. Indiana University Press, Bloomington, IN, 410 pp.
 Taylor, M.P. 2009. A reevaluation of Brachiosaurus altithorax Riggs 1903 (Dinosauria, Sauropoda) and its generic separation from Giraffatitan brancai (Janensch 1914). Journal of Vertebrate Paleontology 29(3):787806.
 Wedel, M.J. 2007. What pneumaticity tells us about ‘‘prosauropods,’’ and vice versa. Special Papers in Palaeontology 77:207–222.
Argentinosaurus: smaller than you think?
April 15, 2010
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 logarithmicbased 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, crosschecked 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 limbbone 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 limbbone regressions are probably too high, and (3) even if those problems were dealt with, there is still the general untrustworthiness of limbbone 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 sparsewing 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 wholebody 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 twopart 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 2004:113.
How big was Amphicoelias fragillimus? I mean, really?
February 19, 2010
Lovers of fine sauropods will be well aware that, along with the inadequately described Indian titanosaur Bruhathkayosarus, the other of the truly supergiant 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.
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 goto paper on A. fragillimus is Ken Carpenter’s (2006) reevaluation, 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 ultralight 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:
Matt’s answer to me was:
First, Paul’s ultralight 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 nearcertainty 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 sealedroom 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:
As you can see, when I measured my scaledtothesizeofA.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 100tonne threshhold. Using Matt’s (2005) 12tonne estimate for Diplodocus, we’d get a mere 78.5 tonnes for Amphicoelias fragillimus. So maybe Matt called that right.
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 132yearold figure of a partial bone which has not been seen in more than a century, you need to recognise that the errorbars are massive and anything resembling certainty is way misplaced.”
Caveat estimator!
References
 Carpenter, Kenneth. 2006. Biggest of the big: A critical reevalustion of the megasauropod Amphicoelias fragillimus Cope, 1878. pp. 131137 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): 563566.
 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. 3:247387, and plates LXLXXXV.
How big was Alamosaurus?
September 2, 2009
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)
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 30footplus 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 52foot animal probably had a mass around 15 tons, so the 79footer would have been about 50 tons (1.5^3 = 3.375), and the hypothetical 100footer would have been 120 tons, which is up in Amphicoelias/Bruhathkayosaurus territory. For what it’s worth, I think the numbers for the 79foot 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): 156172.