## Tutorial 24: variables for tubular bones, ASP, MSP, and bone density

### June 5, 2013

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**

- 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. 201-228 in Wilson, J.A., and Curry-Rogers, K. (eds.),
*The Sauropods: Evolution and Paleobiology*. University of California Press, Berkeley.

June 29, 2013 at 10:13 am

It would be easy to do this in ImageJ using the BoneJ software we developed– http://bonej.org/, and do it for whole bones, using CT scans.

July 8, 2014 at 4:34 pm

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

April 11, 2018 at 11:42 pm

[…] area of the inner circle or ellipse is equal to k^2, where k = r/R. Or d/D. (See Wedel 2005 and Tutorial 24 for the derivation of that.) For the Haestasaurus radius (the bone, not the geometric dimension), […]