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 just got this message from Rana Ashour of Paleontology Journal, an open-access journal published by Hindawi, who are generally felt to be a perfectly legitimate publisher:

Dear Dr. Taylor,

I am writing to invite you to submit an article to Paleontology Journal which is a peer-reviewed open access journal for original research articles as well as review articles in all areas of paleontology.

Paleontology Journal is published using an open access publication model, meaning that all interested readers are able to freely access the journal online  without the need for a subscription, and authors retain the copyright of their work. All manuscripts that are submitted to the journal during June 2013 will not be subject to any page charges, color charges, or article processing charges.

[snip]

(Apart from anything else, the waiving of APCs pretty clearly indicates that this is not a scam journal.)

I replied:

Hi, Rana. Thanks for this invitation. I am supportive of Hindawi as a good-quality, low-cost open-access publisher. In particular I want Paleontology Journal to do well: it has at least one colleague of mine among its editors. I am particularly pleased to see that no APCs are payable on submissions made during June 2013.

But as a matter of principle I never respond to “academic spam”. Messages sent as bulk mailings to a broad group of potential authors are at best impolite, and at worst actively damage the reputation of the journal and its publisher — see point M on Jefffey Beall’s Criteria for Determining Predatory Open-Access Publishers.

I urge you to use what influence you have to discontinue the use of spam to advertise Paleontology Journal. If the journal is good, it can be advertised by publicising the papers that appear in it.

Thanks,

Dr. Michael P. Taylor
Department of Earth Sciences
University of Bristol
ENGLAND

Let’s hope they go with it. I’d love them to build another low-cost, high-quality, journal in the palaeontology OA space, to compete with Acta Palaeontologica Polonica, Palaeontologia Electronica, PalArch and of course PLOS ONE and PeerJ. But they won’t do it by spamming.

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.