While the tibia and femur do exhibit allometry across species with increasing size in pterosaurs, the allometry of the hind limb elements is relatively weak (while the allometry of the forelimb elements is very strong). Allometry equations need to include an intercept if you are trying to predict the one major body dimension from another (for example, how body mass scales relative to limb bone circumference). If you want to make that sort of prediction, then you'll need a full allometry equation, generated from known specimens, that predicts the output variable (like total size) from the variable you have (such as hind limb lengths). 

However, it sounds like you are just trying to scale known elements up to larger versions of the same elements. In that case, all you really need are the exponents. So, for example, you have a femur of known length. You want to make it four times longer (I'm assuming that's what you mean by 4 times as large). The diameter in this case would increase by a bit more than four times, because there is an allometric effect. That means that the regression of femur diameter on femur length would have an exponent different from 1 (since they are both linear dimensions, the isometric condition is a regression exponent of 1). You need to check the original literature in question (such as Brower and Veinus) to see what that exponent is. In practice, allometric regressions are often log transformed and the allometric exponent recovered as a slope, rather than an exponent. This makes the algebra a bit easier, but does not fundamentally change the result.

For sake of example, imagine that the exponent for diameter as compared to length is 1.2. That would mean that when the length increased by 4 fold, the diameter increases by 4^1.2 fold, or about 5.27 fold.

It should be noted here that apparent allometry of linear dimensions can result in isometry of major mechanical variables. So, for example, to maintain the same bending strength of a bone relative to body mass (essentially a form of mechanical isometry), the diameter of the bone must show a strong allometric increase relative to bone length. In pterosaurs, the hind limb elements show positive geometric allometry in the strict sense (they get a bit more robust in big animals) but not by enough to keep up with body size, so the *mechanical* geometry relative to size in the hind limb actually shows slight negative allometry across some of the major pterosaur clades - that is, the hind limbs are actually a bit weaker relative to estimated mass for some of the larger species.

Hi Michael,

thank your for the reply. I have read the Browers and Veinus paper and found that The allometric equations for bone diameter are: (Length of leg) = 2.46 (Mass)0.372 and (Length of leg) = 0.428 (Length of wing)0.946. The exponent for the tibia is 0.3915 and a = 0.9752.

If this fit into the formula of y=a*Mb  then  

y=0.9752*40.3915 = 1.678

Does this mean that, for example, allometric growth of the tibia would be for every 4x the lengthening of the tibia, the diameter would increase 1.678x?

Thanks again!

Danny

That depends on what you mean by "the exponent for the tibia is 0.3915". Is that the allometric exponent for a regression of tibial diameter against tibial length? The other equations you listed regressed bone length against estimated mass and a bone length against another bone length. Neither of those would give you information on bone diameter. I doubt that the exponent you listed is for tibial diameter, because a 1.678 increase in diameter for a 4 fold increase in length would make the bone *much* more slender at large sizes, and therefore a great deal weaker (large pterosaur hind limb elements do tend to be a bit weaker relative to body dimensions, but not by as much as you're suggesting above).

Keep in mind that there is no universal exponent for a bone - every exponent and scalar coefficient combination will  be specific to a particular measurement comparison. In order to scale the tibia the way you're trying to, you specifically need an allometric relationship for tibial length to tibial diameter. It looks like perhaps Browers and Veinus didn't report that comparison.