Understanding the Biomechanics of Bone using Scaling Laws (adapted from Mark Peterson, Mt.
Holyoke College)
Introduction (Read before Lab)
You may have read in your textbook that larger animals (at least those who walk upon land) have thicker
bones than do much smaller animals. Let’s look at some skeletons for animals large and small.
Figure 1: Animals of different sizes have bones that must support dramatically different bodies. Images are
obviously not to scale (left): mouse skeleton (http://www.biospace.fr/media/ct_skeleton_big.gif ) and
(right) skeleton of an apatosaurus (a large sauropod dinosaur)
(http://www.copyrightexpired.com/earlyimage/bones/large/display_osborn_apatosaurus.htm )
For each animal skeleton in Figure 1, observe the large bone of the upper leg, called the femur. (This is the
analogy to your own thigh bone.) The idea is that, because of their dramatically different masses, bones
for the apatosaurus need to be proportionally thicker for their length than those for the mouse: that is,
their width-to-length ratio should be larger for the larger the animal. As early as the 1600’s, the early
physicist Galileo was interested in this question because he observed that living things do not grow
arbitrarily large. He wanted to know what physical constraints set the limits to animal size, a topic with
contemporary relevance as we face certain knowledge that planets are common in the Universe as a
whole, and wonder what life may look like there.
Galileo explained this empirical fact (i.e, that there are not arbitrarily large animals) by arguing that the
forces on bones must be offset by enough strength to prevent the bones from breaking in routine daily
existence. The proportionally thicker bones of larger animals compensate for their greater mass. Since
after a point, it becomes impractical to keep having thicker bones, this sets an effective limit to animal
size.
Galileo formulated his theories in the 1600’s in the work Dialogues Concerning Two New Sciences1.
Implicitly, he was assuming that some mechanism (modern scientists would say natural selection and
evolution) enforced this agreement between maximum force and mechanical design. Of course, people
still care about the mechanics of bones, e.g., for understanding how to design prosthetics that replace
natural bone. In another important case, the elderly often suffer from a condition called osteoporosis, in
1
Galileo Galilei, Dialogues Concerning Two New Sciences. My edition is translated by Henry Crew and Alfonso de
Salvio ( Dover, New York 1954) and this argument appears on page 131, in the section “Second Day [168]”. There
Galileo makes the argument we’ve conveyed about bone area, though not animal mass. Interestingly, the
accompanying illustration of small and large bones makes it clear that Galileo actually does not get the scaling
exponent attributed to him. For example, he refers to bending strength, not compressive strength, and does not
appear to have thought through the argument about mass scaling. In fact, Darin Hayton (History Department,
Haverford College) reports that Galileo is unlikely to have had access to actual bones from a collection, and that he
probably reasoned from books containing etchings of animal skeletons popular at the time. Thus, it’s unlikely he
was reasoning from actual quantitative data (or that he felt that was appropriate or necessary!)
which bones lose mineral content and become brittle. This often leads to debilitating fractures and a
downward spiral of serious medical complications. Osteoporosis is currently diagnosed by measuring the
density of bone minerals, even though it is known that bones take on different shapes (remodel) with age.
Physicians are studying how age-dependent changes in bone geometry interact with losses in bone
mineral density to determine bone strength. Thus, arguments about what makes bones strong enough to
withstand the rigors of daily life are very timely.
Galileo’s theory leads to a quantitative prediction about how bone width ought to vary with animal size,
and hence bone length. We will now present the way this theory is represented in many modern texts.
Let’s argue that the forces in daily life are all proportional to body mass, M. That’s because the force of
gravity Fg = Mg pushes down on the bones and the momentum is also proportional to mass (and hence,
any forces that must slow down a moving animal are also.) We call pushing forces compressive forces.
We further argue that mass, M, ought to scale with body size in a particular way. Animals have
complicated shapes, so let’s simplify things by assuming that the animal’s body is approximately a cube
of uniform density, . In general, mass = density × volume, so for a cube with edges L long, with the
same density , we get:
M =  L3
Of course, even physicists don’t think animal bodies are cubes! This argument simply builds on
the reasonable assumption that the actual body mass is proportional to body length x width x
depth, and that these dimensions each are proportional to long bone length. We now argue that
we ought to be able to generalize this rule for any shape animal for any length in the body, such
as femur length L. So, in general, we should have:
M ~ L3
where we introduce the symbol “~”, which means “is proportional to”. We write the equation
this way because for the actual shape of animals, we don’t know the multiplicative constant
before L3, but we do know how M ought to depend on L.
Now, how does bone strength depend on body geometry? By strength we mean the maximum
force a bone can withstand without breaking. We will argue that it ought to depend on the area
of the bones supporting the body, which seems plausible enough. Thicker bones do seem better
able to support more weight. By this reasoning, the maximum force, Fmax, the bone can support,
depends upon its width, D, as:
Fmax ~ Area ~ D2
cylinder with diameter D
For a square bone with sides D wide, the area varies as D2; for a cylindrical bone, its area varies
as  (D/2)2, where D is now diameter. Again, we have only included the way strength depends
upon width here, although surely Fmax also depends on the bone’s properties (how dense it is, its
detailed shape, etc.) Galileo assumed that these quantities do not vary between animals, and so
will we for now.
By Galileo’s reasoning, if we have a bone such as the femur that has to support the body mass
above it then, we should have a balance between the forces depending on gravity (Fg) and the
strength:
M ~ Fmax , and so:
L3 ~ D2
This is what physicists call a scaling law. It is a prediction of how bone width, D, is expected to depend
on bone length, L:
D ~ L3/2 = L1.5
This theory explains why bones become proportionately thicker for larger animals. For example, if we
have two animals, one with femur length 3 times greater than another (e.g., one has length 3L, the other
L), then the diameter,D, of the femur for the larger animal ought to be L1.5 = 31.5  5.2 greater than that of
the smaller animal—the effect we began with. (That is, one with have femur diameter 5.2D, the other D.)
While Galileo’s version of this argument relies on an analysis of compressive forces, in General Physics
(2nd edition, Morton Sternheim & Joseph Kane, pg. 215 & 225-226, 1992) the authors present an
alternative derivation using buckling forces on a vertical column that results in the same scaling law; they
also show that this in fact accurately describes the way the diameter and height of tree trunks scale, for
example.
Scaling laws also are an important tool in the study of animal evolution, in the analysis of networks and
other topics of current scientific research, although most systems have more complex scaling behavior
than Galileo’s model. Haverford’s own Aaron Clauset (Haverford Physics & Computer Science ’01, now
a professor at University of Colorado, Boulder) for example studies how to apply scaling laws to
empirical data and the scaling of terrorist acts, among other topics.
Figure 2 Left: A scaling law plot of the frequency of terrorist attack measures vs. the severity of the
event, from the paper (“On the frequency of severe terrorist events,” A Clauset, M Young, KS Gleditsch
Journal of Conflict Resolution, 58-87 (2007) ;
http://arxiv.org/PS_cache/physics/pdf/0606/0606007v3.pdf). Right: Researcher Aaron Clauset, now an
assistant professor of computer science at University of Colorado, Boulder.
Testing the scaling law hypothesis
In your experiment, you will investigate scaling law behavior of bone biomechanics. For a scaling law,
there are several ways to determine the quantity of interest, the exponent, d, in the scaling law:
D ~ Ld
One is to fit D to this functional form using a computer data analysis program, like LoggerPro. Another is
to fit to this functional form by hand. Either way, it is a good idea to first plot the data for D vs. L in a
special format called a log-log plot. For a scaling law, when we have:
D = A Ld (where A is some constant),
To investigate this relationship more easily, let’s review the concept of logarithms. If we have a number
X where:
Y = 10x
Then the logarithm base 10 of Y is:
x = log Y = log (10x)
Note two additional useful facts about logarithms:
1) Multiplying two numbers makes their logarithms add:
log (AY) = log A + log Y because:
10 log (AY) = 10 log A + log Y = 10 log A 10 log Y = A Y
2) Raising a number to a power multiplies its logarithm by the value of that power:
log (Yd) = d log Y because:
𝑑
10𝑑 𝑙𝑜𝑔 𝑌 = (10𝑙𝑜𝑔 𝑌 ) = 𝑌 𝑑
Returning to our earlier problem and using logarithms in the base 10 system, we get this relation for the
logarithms of D and L:
D = A Ld
log D = log A + d log L
Thus, plotting the log of D and L thus ought to result in a straight line with slope d = 1.5. If the data
follows a different scaling law, it will still follow a straight line, but with different d. Departures from a
linear plot will indicate the scaling law hypothesis is incorrect. We see this at work in Figure 2(a), where
the terrorist attack data has been plotted as the log along both the x and y axes, revealing a linear
dependence for part of the data.
Prelab problem 1: Compute the exponent for the body length vs. body mass data below. To do this, you
must compute the change in the power of 10 along each axis, for that region where the data is linear on
the log-log plot. For example, if you had data that changed along the x-axis by 100 to 101 and along the yaxis by 101 to 103, then the slope would be y/x = (3 – 1)/(1 – 0) = 2, so the scaling law exponent is d =
2. Does your exponent agree with Galileo’s theory?
Prelab problem 2: The figure below shows a diagram that accompanies Galileo’s actual scaling
argument in his book; he discusses this diagram in some detail so it’s supposed to demonstrate his point
mathematically. Get out a measuring device like a ruler and determine how D depends on L for this
figure. What exponent did Galileo actually think resulted from his argument? (Hint: it’s not d = 1.5!
Galileo is a big authority figure in science and even now I find only one reference to this error.)
Galileo’s bones (sketch & his actual finger)
Prelab problem 3: Now derive a new scaling law, using a different set of assumptions. Thin-walled
tubes will buckle (i.e., deform as shown below), when they are subjected to a maximum force that varies
as showing below
𝜋2 EI
𝐹 = (𝐾𝐿)2 ,
where L = column length, E = the bulk modulus, I = area moment of inertia, which gives the resistance to
bending, I = R4/4 for a cylinder with radius R (as derived in Sternheim and Kane, General Physics), and
K is a constant with no dimensions that depends only on how the ends are supported. Assume that the
force F is proportional to mass, and hence we again have F ~ L3, as above. Determine how the diameter,
D, varies with L for buckling in this situation: D ~ Lb; that is, find the new exponent, b = ? (Note this is
different from the result for buckling cited above!)
Figure 3 shows buckling in columns, from Wikipedia
The Experiments
Part 1: Taking a hard look at Galileo’s ideas
Galileo’s theory is often quoted as if it were established fact, as in the textbooks cited above and in the
images from a university website below. (From our “we can’t make this stuff up” collection: This is such
a popular idea that you can even buy a dog toy shaped like Galileo’s illustration and carefully scaled to
his specifications from “petite” to “souper”.) Maybe we should all be more careful, though. He got the
wrong scaling exponent, and he apparently didn’t even test it on real bones!
Figure 4 Galileo must be right, since everyone reproduces his arguments and he even has his own brand
of dog bones! http://www.arch.virginia.edu/~km6e/arch324/content/lectures/lec-25/pres.html (accessed 12-09-09)
Before we do any experiments, though, let’s ask ourselves some hard questions about Galileo’s theory.
Discuss the following points with your team members and come up with at least two answers or
possible responses to each point and arguments to support them. (These should not be drawn from
your lab manual writeup.) Write up your responses on your lab report form. We will form a group
again to share your responses after 10 minutes.
A. What are the assumptions underlying the theory derived from Galileo’s ideas? (Example:
perhaps the forces on bones aren’t proportional to mass, or aren’t mostly determined by mass.
What might determine them instead?)
B. What else might determine bone strength that his theory about compressive strength ignores?
Name all the quantities you can think of that might play a role. When and where might they
stay constant, and when might they change?
C. What other factors might determine bone dimensions apart from arguments about strength?
PART 2: Formulating your hypotheses & designing your experiment
Now that we have some provocative questions to work with, your team will select an experiment to
perform from a list we will provide. Think through what hypothesis you are testing in your experiment,
and what you need to measure to probe it thoroughly. The materials you are provided with are
described in the table attached.
You might think that once you have formulated your experiment, you are ready to take some data.
However, figuring out how to take your measurements and determining your experimental
uncertainties is one of the crucial parts of this experiment. Talk this over with your classmates and
decide how to assign uncertainties to your measurements. Is it enough to say you can measure a
distance across a femur at some location along its length to some precision? Why or why not? What
angle will your measuring device make with the femur’s long axis? Which points along the bone
determine its length? The entire length, or the length between two other points? How uniform is the
bone? What other factors matter for this experiment? You are trying to measure the diameter as a way
of estimating area, but bones are not cylinders—what is the best way to do this? Check with your
instructor that the measurement methods you have in mind makes sense before taking a lot of data.
NOTE: We see from the mass scaling prelab problem that one really needs data that varies over a
few factors of 10 to do a good job fitting data to a scaling law. Be wary of scaling arguments based
on only a factor of 2 or 3 change in someone’s data! You will want to make measurements that vary
by a factor of 10 or more in lab—don’t set yourself up for a measurement that will not cover at
least this much variation in bone length.
Write up your experimental design, including any hypotheses you are testing and how your design
tests them, in your report. Include a description of your method for taking data and determining
experimental uncertainties (error bars as well as underlying sources of accuracy and precision.)
Explain how you will measure D and L for the bones provided (this should not just be “We used a
meter stick/calipers to measure D and L”. WHERE did you measure D and L? Why that dimension?
Were there alternatives, and if so why did you choose this one? How does it best relate to your
hypothesis? You may find the figure of a human femur on your report form helpful. We will begin
taking data in about 15 minutes, so make good use of this time!
PART 3: Taking data & testing your hypothesis
You will be determining whether your data agree with scaling law arguments (or other functional forms)
using log-log plots in Logger Pro and then fitting your data by hand. First plot your data using LoggerPro
and print it out. In Logger Pro, enter your data for D and L (for example) in the spreadsheet. Make a
plot of these quantities. Then, double-click on the x and y axis labels. This should bring up a menu box
called “Graph Options”; go to the “axes options” tab. This has a section for the “y-axis” and “x-axis”
that allows you to click on a box called “Log axis”. Click on both these boxes then click OK. You should
now have a plot of the log of D vs. the log of L. Ask for help if you are having problems.
Before you do any fits, look over your plots as a team and determine which regions looks approximately
linear (if any) then fit a straight line to these. Estimate your exponent d as in the prelab exercise. By
estimating how widely your slope can vary and still plausibly fit your data, estimate the error bars in d as
well. This will involve drawing the lines with the maximum and minimum slopes possible that still agree
with your data within error bars, then measuring their slopes.
Remember that your answer may be your data DO NOT LOOK LINEAR on a log-log plot. That is, they
may not follow a scaling law at all. In that case, you have several possibilities.
1. Perhaps only some of your data agree with a scaling law. To see if that is the case, trying labeling the
datapoints to see if there are any systematic trends as to which points might agree to a scaling law
2. If 1) is not true, do they follow a different functional relationship (a constant plus a power law?
sinusoidal? exponential? polynomial?)
3. You might conclude that there is no clear trend in your data (i.e., L does not depend on D, either
because it has the same value within error bars for all D, or because it exhibits no clear trends when
plotted vs. D; while there are good statistical measures of this, we’ll stick with the eyeball test for
this for now.)
Make up a plot for your report that shows your best fit, your estimate for the exponent d and its error
bars, or your other best fit parameters and functional form. Label your plot axes and indicate units.
(The data collection and fitting part should take about 45 minutes at most. You may talk with the
instructors and other teams while you do this part.)
After you have your data set in hand, come to the front computer and enter your data in the whole
class’s spreadsheet so we can compare results as a whole.
Materials
Measuring devices: digital calipers, rulers, meter sticks, 2 meter sticks, protractors, string, and anything else you
can think of that we have.
Software: you can of course browse the web for much additional information (to look at actual skeletons? To get
other supporting information?), and you can use LoggerPro for analyzing your data.
Bones: We have a variety of bones from different types of animals of many sizes. All are plastic casts unless noted.
Analyzing images from other sources (books, the web, etc.)
Let’s assume you have information that comes from other scholarly articles, another research group’s
website, etc. You have a scale bar (a reliable measurement of some object on the image). How do you
proceed to get other measurements? I have found that students are often unsure how to proceed, so
I’m giving an explicit guide here.
First you find the scale bar (here we are told the total length of the bone is 6 foot 1 inches high (73”),
which we will assume means its maximum height top to bottom when displayed vertically as shown in
the image. In order to measure another distance on this image, we need to get an effective scale.
1)
Using a ruler and the image, measure how long the bone is on your image itself. Let’s say that
it’s L cm long = 73”, and the diameter is D cm long on the image. Then, in real units of inches in
the real world D (true inches) = D (cm on the image) x (73”/L (cm on the image)).
2) Using an image analysis program, such as Adobe Photoshop, to get the number of pixels that
correspond to the known scale. For instance, in Photoshop, there is a ruler tool (right click on
the eyedropper tool to find it; it looks like a small ruler.) that you can use to click and drag a
measuring line from one end of the femur to the other. Doing so, we learn that it’s 437 pixels
high = 73” (the length shows up as L1 = 437 pixels at top of the screen.) Then, use that same
ruler tool to measure other distances and scale accordingly. For example, the approximate
diameter = 82 pixels X (73”/ 437 pixels ) = 13.7” (where I kept an extra significant figure to avoid
rounding errors only.)
There are obviously many sources of uncertainty here. Measuring
the dimensions yourself is far superior, but sometimes you need
this information, so you should be quite clear of its limitations,
which include: 1) You must trust the scalebar (and surprisingly
often it’s unclear what they correspond to, and they are often
even missing and must be inferred!) 2) You cannot measure
diameter apart from the projection shown from one perspective.
3) You must trust that the image was not distorted in any way
side-to-side relative to up-and-down.
For this reason, any of your own research images ought to be
produced to high standards to avoid these same problems!
http://www.angelfire.com/mi/dinosaurs/dinosaurs_sauropods.ht
ml
Two side views of the femur of the largest land mammal to ever live:
the baluchitherium (a relative of the modern rhinoceros)
Baluchitherium osborni (? syn. Indricotherium turgaicum, Borrissyak)
Clive Forster Cooper
Philosophical Transactions of the Royal Society of London. Series B, Containing Papers of a Biological Character,
Vol. 212, (1924), pp. 35-66