Understanding tissue loading, tissue damage, and running injuries

In my article on biomechanical training load, I covered the basics behind how biomechanical loading is related to the development of running injuries.

The basic idea is pretty straightforward: every time you take a step, your tendons, bones, and joints experience a loading cycle: a build-up and release of mechanical force.

Each loading cycle does a tiny amount of damage, depending on the magnitude of the force and the structural integrity of the tissue. If this damage accumulates faster than your body can repair it, the result is an overuse injury.

In this article, we’re going to take a deeper dive into exactly how this process of tissue damage works. Our goal is to build up an understanding of cumulative damage: a way of quantifying “how much damage” you’ve done to a specific piece of tissue.

We’ll use the recurring example of damage done to the Achilles tendon, since it’s a common injury and a relatively straightforward tissue in terms of understanding both its biomechanical loading and its tissue properties.

So, what is the actual mechanical process behind tissue damage in running? Let’s dive in and find out.

Our roadmap for understanding this topic will be this 2018 review paper by Brent Edwards at the University of Calgary, titled “Modeling Overuse Injuries in Sport as a Mechanical Fatigue Phenomenon.”

When I talk to people who are interested in learning more about the biomechanics behind overuse injuries I often point them to this paper, but I had forgotten that (a) the paper is quite technical and (b) it’s also not open access!

So, this article is in some ways a “journal club” that uses the Edwards 2018 paper as a jumping-off point for getting a scientific perspective on how tissue damage happens—we’ll pull in additional and more recent experimental findings as needed to understand the topic. I’ll signpost any technical points I’m including for completeness that aren’t 100% necessary for an intuitive understanding

Paperclips redux: the basic intuitions behind running injuries

In my article on biomechanical training load, I introduced a straightforward toy model of overuse injuries: bending a paperclip back and forth until it breaks.

The paperclip example is not an analogy; it’s the exact same process of “mechanical fatigue” that causes bone, tendon, and cartilage to sustain damage.^[1]

Loading cycles, loading magnitude, and structural integrity

Every time you bend a paperclip, you induce one loading cycle. In running, the same thing happens every time you take one step. Each loading cycle induces some amount of damage, which depends on two factors: the loading magnitude and the structural integrity of the tissue (or paperclip) being stressed.

Tracking loading cycles is easy—you just count them. Even in running, counting loading cycles is straightforward: all GPS watches measure cadence, which you can easily use to calculate steps taken during any given time period.^[2]

In our Achilles example, if we are examining the damage done to your right Achilles tendon, we’re interested in the total number of right-side steps you take (which is just total steps divided by two).

Tracking loading magnitude requires a little more precision. The exact kind of load we’re most concerned with differs from one tissue to the next—in tendons, we care about tensile force, while in bones, we care about the combined effects of compression, torsion, and shear.

In our Achilles example, we want to know the peak mechanical stress per step: tensile force in the tendon, divided by the tendon’s cross-sectional area.^[3] Estimating both of these variables is possible, at least in a state-of-the-art biomechanics lab (we’ll go into more detail on how you’d actually go about measuring these later).

The structural integrity of a material is typically described by its stress-strain curve: a plot that shows how much strain—deformation in the material, as a percentage of its original length—you see at a given level of mechanical stress (i.e. force per unit cross-sectional area).

In our Achilles example, we would need to plot out how much the tendon stretches out as a function of the mechanical stress applied to it. One way to get this measurement would be to do materials testing on cadaver tendons (which has been done), though you can also measure tendon deformation in vivo using ultrasound.

Here’s a real stress-strain plot for an Achilles tendon:^[4]

Stress, strain, and structural integrity

The stress-strain plot tells us a lot about the structural integrity of the Achilles tendon. From an intuitive perspective, this makes sense: For a given amount of force, a stiff and strong tendon would not stretch out very much.

So, a stronger tendon would have a “steeper” stress-strain curve (using “steep” loosely here, since the curve is in fact nonlinear).^[5] Notice we’re talking about the “intrinsic” material strength of the tendon: we’ve already normalized out the size (i.e. cross-sectional area) via mechanical stress, which is force divided by cross-sectional area.

As we’ll see below, using strain (i.e. percentage deformation) instead of stress is a useful way to account for differences in material properties when we have cases where different samples of material might have different intrinsic material properties—tendons from different individuals being a perfect example.

Stress, strain, and loading cycles to failure

Now that we understand loading cycles, loading magnitude, and structural integrity, we can start building an understanding of how each factor contributes to tissue damage.

Unsurprisingly, engineers take a very practical approach: they just get some raw material and repeatedly apply loading cycles at a given magnitude until it breaks. Then rinse and repeat.

This process is called failure testing, and it’s not hard to imagine doing it with dozens or hundreds of paperclips, using a machine to bend them back and forth until failure. What you get is something called an S-N curve – that’s “S” for stress, and “N” for number of loading cycles until failure.

Amazingly, some intrepid biomechanics researchers have done the same kind of failure testing with human tendons (again, from cadavers).^[6] Here’s the S-N curve from a 2003 study that stressed 25 human Achilles tendons until failure:

A few things to notice in this plot: first, the X axis is a log scale—so, small changes in stress lead to very big changes in the number of loading cycles until failure. This is a very important takeaway! It implies that, in running, small decreases in tendon force (or bone stress, etc.) can have a strong protective effect against injury.

Second, the fit is not that good. There’s quite a bit of “scatter” in the actual data versus the fitted curve. Part of this scatter is because different tendons from different people have different material properties. We can get rid of most of this material-properties variation by instead looking at an S-N curve fitted using strain, not stress; indeed, the fit becomes much better (but is still not perfect, for reasons we’ll return to later):

Very often, S-N curves are modeled mathematically using a power law, of the form:

   

where N_f is the number of loading cycles until failure, is stress (or strain, in which case is replaced with symbolically), and and are coefficients that are fitted to your experimental data—essentially the “slope” and “intercept” of the curve, if you are thinking of it like linear regression.

For the Achilles tendon (and other tendons), the “damage” being caused here is kinks and ruptures to the collagen fibers that make up the tendon at the microscopic level. That’s (partly) why biopsies of severe cases of tendinopathy show a mess of disorganized collagen at the cellular level.^[7] Eventually the damage gets so bad that the tendon fails.

Caveat: that S-N plot is definitely wrong

I mentioned earlier that this fatigue testing was done in cadavers. There is a pretty obvious limitation here: old age and disease are going to massively deteriorate the quality of the tendon.

Other work has also suggested that some of the “failures” in this study had to do with failures in the mechanism that clamped the tendons to the testing rig. Other studies on human tendons show stronger properties (more cycles to failure), but a similar-ish slope to the S-N curve (which is what matters for us).

As we’ll see below, the S-N curve above is very unrealistic for healthy athletes: it predicts that the Achilles would rupture after just a few minutes of high-speed running! So, we will not put too much stock in the predicted number of loading cycles until failure—the relative difference across strain magnitudes, though, will be more informative.

Connecting tissue damage to failure with Miner’s rule

So far we’ve looked at loading cycles to failure: a complete rupture, in the case of tendon. But most cases of tendinopathy merely involve some amount of damage that falls short of failure. How can we connect failure testing with some continuous estimate of how much damage has been done to the tissue?

The simplest approach to this problem is via using Miner’s rule, which has the bonus perk of also enabling us to deal with cases where tissue is loaded with a series of loading cycles at different magnitudes (which happens in running all the time—for example: 2 miles warmup + 4 miles fast + 2 miles cooldown).

Here’s the idea with Miner’s rule: suppose an Achilles tendon is loaded the following way:

2000 loading cycles at a peak strain of 0.03 per load

then, afterwards:

1000 loading cycles at a peak strain of 0.04 per load

Miner’s rule says that we can compute the total amount of damage done by calculating what percentage of the “fatigue life” of the material we consumed with each series of loading. Once that percentage of fatigue life hits 100%, we expect the material to fail. So, the “damage” done is just a fraction from 0.0 to 1.0.

From the fitted S-N curve to the figure above, we can look up the expected number of loading cycles at 0.03 and 0.04 peak strain:

Peak strain	Cycles to failure
0.03	692,688
0.04	47,424

Then, plugging in our loading protocol above, we get:

Peak strain	Cycles to failure	Actual cycles	Damage done (% life)
0.03	692,688	2000	0.289%
0.04	47,424	1000	2.112%

Which comes out to a total of 2.401%, i.e. “2.4 percent of the way” to tendon failure.

There are several more sophisticated approaches to pooling together loading cycle data, but the key intuition here is that Miner’s rule, along with its more sophisticated descendants, give us a way to put a number on the amount of damage incurred across different “protocols” of loading on a piece of tissue.

Explicitly modeling damage done to a tendon

❗ This section is a little math-heavy; skip ahead if you don't like building intuitions from equations

Earlier, we saw that the expected cycles to failure for a tendon can be modeled with a curve (an S-N curve) fitted either to stress or strain data. From above, we saw that:

   

Where is the expected number of loading cycles to failure,  is the peak strain per loading cycle, and and are fitted constants to experimental data. In our Achilles data, A = 4.42e-9 and b = 9.32.

Miner’s rule, from earlier, is:

   

With failure expected when damage () = 1.0. So, we can swap in the fitted S-N curve to get the following expression: 

   

which gives us two powerful intuitions for understanding the biomechanics of running injury:

First, adding more loading cycles—i.e. taking more steps while running, by running further—increases damage in a linear fashion. From 1000 to 2000 to 3000 steps taken on a run, damage goes up in equal increments.

Second, increasing mechanical strain per step—for example, by running faster, which increases force per step, or by changing your biomechanics in a way that increases tendon force—increases damage in an exponential fashion.

By increasing Achilles tendon force from 600 pounds per step to 700 pounds per step, you would expect a much bigger increase in damage per step, compared with increasing from 500 to 600 pounds per step.  

The actual magnitude of this exponential increase depends on the parameter b from the S-N curve, so it’s an intrinsic property of the material.

For the Achilles data above, b = 9.3. That means that a 10% increase in Achilles tendon strain leads to a 2.4-fold increase in Achilles tendon damage!

These two findings are very important for developing a way to compare “equivalent damage” across different training sessions. We should expect mechanical strain per step—and therefore force per step—to have a major influence on damage done.

Worked example: Achilles tendon damage at different speeds

We now know enough to run some real biomechanical calculations to look at how damage to the Achilles tendon—and therefore injury risk—changes at different speeds.

One important point here is that we need to ask “tendon damage per what?”—per kilometer of running? Per minute of running?—to be able to make good comparisons. Tendon damage per step is only part of the story. We’ll look both at damage per kilometer and damage per minute of running.

Some aspects of our calculations will be individualized for the particular runner we’re looking at—I’ll point out which aspects those are, and how they might change from one runner to the next.

Likewise, this is an illustrative example, not a rigorous scientific paper, so we’re going to go through the simplest possible case: Miner’s rule and simply using peak tendon strain, not worrying about more advanced topics like creep damage, impulse, or probabilistic models of failure.^[8]

The runner we’re looking at is a male in his mid-20s. We’re interested in how his Achilles tendon damage changes across four speeds, from very slow jogging (13:20/mi, 8:20/km) to fast running (5:20/mi, 3:20/km).

At each speed, we need to know:

1. The number of steps he takes per kilometer (and per minute)
2. The peak strain in his (right) Achilles tendon

And then we can apply Miner’s rule to compare damage per km and damage per minute at each speed.

Getting steps from cadence (which is highly individualized)

As noted earlier, getting steps per minute is easy—that’s just cadence. And multiplying cadence by pace gives you steps per kilometer (steps/min × min/km = steps/km).

However, two aspects of cadence are highly individualized. First, cadence at a given speed differs from one runner to another—most people know this. Second, something which is less-commonly known is that how much your cadence changes as you go faster—your speed–cadence strategy—is also highly individualized.

Some people run faster mostly by increasing their stride length, so their cadence increases only gradually as they increase their speed. Others are more cadence-dominant, so their cadence increases sharply as they go faster.^[9]

And most people are somewhere in the middle, using a balance of cadence and stride length increases to achieve faster speeds. Here are a few examples of these divergent strategies, plus the speed–cadence strategy for our runner, who is pretty average in his profile:

So, we should not expect our runner’s results to apply universally: your speed–cadence strategy will have a major impact on how much tendon damage (and bone damage, joint damage, etc.) you incur at different speeds.^[10]

What’s interesting here is that, for this runner at least, steps per mile goes down with speed: so, even though there is presumably going to be more tendon force per step when running faster, you take fewer steps per mile, so it’s not obvious a priori whether running faster will do more or less damage per mile than running slower.

(That’s not the case with steps per minute; the faster you run the more steps you take per minute, so both factors are pushing in the direction of more damage)

Getting tendon strain from biomechanical data

The other thing we need to calculate tendon damage is the peak strain per step at each speed in our runner’s Achilles tendon. This sounds impossible, but turns out to be merely “difficult”—with a well-equipped biomechanics lab, getting a good estimate of tendon strain is not an insurmountable challenge.

The first thing we need is the force per step in the Achilles tendon. With motion data and force plate data from a biomechanics lab, we can calculate a pretty good estimate of the force in your calf muscles, and therefore your Achilles tendon force, using musculoskeletal simulation software like OpenSim:^[11]

Here’s what peak Achilles tendon force looks like across the range of speeds covered by our runner:

Something to note: these forces are enormous!And yes, they are within the range of what’s been measured in-vivo. You can easily generate well over 1,000 pounds (450 kg) of force in your Achilles each step, even at a slow pace.^[12]

Now that we have force, how do we go from tendon force to tendon strain? Two possible methods have been proposed in the scientific literature.

The simplest approach is to just use standard data on average tendon stiffness and cross-sectional area, and calculate an estimate of strain treating the Achilles as a simple spring with constant stiffness (e.g. as done in this study). This is the method we’ll be using for our worked example.

A more involved method involves actually measuring Achilles tendon length changes under load, using a dynamometer (basically a weight machine with a torque meter) and an ultrasound probe. Combining these data, you get what is essentially a “lookup chart” that gives you tendon stiffness as a function of tendon force (e.g. as done here).

Summing up tendon damage per step with Miner’s rule

For each speed, we can calculate the damage per kilometers (or per minute) as follows:

1. Calculate how many steps are required at that speed to cover 1 km (or 1 min)
2. Calculate the peak tendon strain at that speed
3. Calculate the tissue damage using Miner’s rule, expressing it as the fraction of total fatigue life consumed per kilometer (or per minute)

Remember from earlier that Miner’s rule just says:

   

And we get n from step (1) above, and from step (2). For A and b, the empirical coefficients, we can use the curve we fit to the cadaver tendon data (A = 4.42e-9, b = 9.32).

Here’s what the results look like:

A few things to point out here.

First, and most importantly, Achilles tendon damage per kilometer increases with speed, and quite dramatically too. The situation is even more dramatic with damage per minute.

Will this always be the case? My bet is that the answer is yes, but you would need to run this analysis on athletes with different speed–cadence profiles, different gait mechanics, and different Achilles tendon stiffnesses to know for sure.

Second, damage per kilometer is "flatter" than damage per minute. A perfect metric for biomechanical load would be flat (i.e. same damage across all speeds), and neither distance nor time come close to meeting this criteria, but damage per minute scores worse: the ramp up is even more aggressive as you get faster. That's because as you run faster, tendon force and steps per minute both go up (whereas with distance, steps per km goes down as you go faster).

Third, we can clearly see the limitations of relying on S-N curve data from cadaver tendons: the absolute damage estimates are way too large.

In truth, the Achilles tendon strength in young healthy athletes has to be greater than what we saw in the cadaver data from the Wren et al. study we saw earlier—clearly, most healthy athletes are not going to rupture their Achilles (i.e. reach 1.0 damage under Miner’s rule) with three kilometers at 3:20/km pace.^[13]

I’m more sanguine about the relative estimates of damage, though, because looking back at the S-N curve, what matters for the relative damages between speeds is the slope of the line (which is dictated by the b parameter).

It’s easier to imagine a “baseline shift” in that line, pushing up the fatigue life across all strain levels, that keeps the slope about the same. And in any case, you can still do quite a lot with relative damage estimates across different speeds.

Sketching out an equivalent-damage mileage framework

One promising direction for the kind of individualized tendon damage analysis that we just went through is the ability to compare runs and workouts across different speeds.

Suppose this athlete we’ve been analyzing is returning to training after an Achilles injury, and wants to start up with speedwork. How much is safe to do? The tendon damage framework from above gives us a framework for calculating “damage-equivalent mileage” by rescaling the damage per mile to some common reference point.

You can choose any speed as a reference; in this case, let’s say our athlete usually does easy runs at 9:00/mi. Fitting a curve and rescaling our data from above, we get this plot showing damage as a function of speed:

Or, rescaled according to pace,^[14] we get the following plot:

Taken at face value, these results are quite shocking: they suggest that—for this runner, at least—running at 5:20/mi (3:20/km) does 5.7 times as much damage per mile (or per km) as running at 9:00/mi pace! In other words, a 3000m race in 10:00 does as much Achilles tendon damage as running nearly 11 mi at 9:00/mi.

Now, we should remember that there’s a lot of uncertainty at play here. We’re relying on cadaver testing for S-N curve data, average values from the literature for tendon stiffness, and Achilles tendon forces estimated from gait lab data (which are pretty good, but not perfect).

Moreover, this result is individualized to this athlete: a runner with different gait mechanics and a different speed–cadence strategy would get a different shape to this equivalent damage curve.

For example, I re-ran this analysis on my own biomechanical data, and at 5:20/mi pace, I’m “only” sustaining about four times as much damage per mile versus 9:00/mi (though interestingly, compared with our example athlete, I do sustain more Achilles damage in an absolute sense across all speeds).

I know these results sound crazy, but let’s think through the implications. Suppose you ran 40 miles per week, all at 9:00/mi. Then suppose you decided to replace one run per week with 4 x 400m at 5:20/mi (= 5.7 equivalent miles, same as a normal run). That would be pretty tolerable—it might even leave you no more beat up than a normal easy run.

But suppose you replaced one run per week with 16 x 400m at 5:20/mi (= 22.8 equivalent miles). That would definitely leave your calves and Achilles feeling quite beat up, and might even lead to injury.

Would introducing that session be as reckless as doing a 22.8-mile run? I’m not sure—my instincts suggest that the long run is more reckless—but using my own data, where 5 mi at 5:20/mi is “equal” to 20 mi at 9:00/mi, seems a little more plausible.

If the equivalent damage model is wrong, it’s because of the b coefficient

Whether you believe in these dramatic increases in damage per mile essentially comes down to how much you believe in the b coefficient from the S-N curve—that’s what governs the relationship between strain and damage.

Biomechanical models of tendon strain are pretty good—at least, within a standard deviation or two of what’s been measured “in vivo” with strain gauges. Not off by crazy amounts. And the other parameters we’re modeling, like cadence, are very accurate.

With the b coefficient, though, things are much less precise. It comes from cadaver data, not healthy athletes, and though we estimated it as b = 9.3, the 95% confidence interval for b from the power law model gives a range of 6.0 to 12.7 (!).

The bottom line is that we need better data on the material properties of healthy tendons in well-trained athletes. Better materials testing is one avenue for solving this problem, but another promising method is to work “top-down”—follow runners and see who doesn’t get hurt, to put upper bounds on tendon damage and/or tendon repair.

Research into joint damage and arthritis has taken a similar approach. The idea, as laid out in this study,  is as follows:

• Runners as a group do not get knee arthritis at a higher rate than non-runners
• When you run the math on cartilage damage per step, the results say that basically every single runner should blow out their knees by age 55
• Since most runners do not get knee arthritis, knee cartilage must either be able to repair itself or “condition” itself to increase its structural integrity after exposure to training

I think a similar approach should work for other locations of running injury. Disentangling damage and repair will be a challenge, of course, but not an insurmountable one.

Implications and conclusion

Understanding the biomechanics of tissue damage gives us some powerful tools for designing training and rehabilitation programs.

The most important takeaway is that small increases in tendon force lead to big increases in tendon damage—even if we can’t fully trust the exact numbers we get out of testing on cadaver tendons, the broad picture is still the same.

Another important takeaway is that for many runners, tendon damage per mile (and per km) ramps up, sometimes dramatically, as a function of speed. Even though you take fewer steps per mile at higher speeds, the increase in tendon force is often enough to counteract the reduction in loading cycles per mile. 

Third, remember that the analysis above is an illustrative example for one athlete. Tendon damage as a function of speed is highly individual, and depends on (a) the stiffness of your tendons, (b) your speed–cadence relationship, and (c) your gait mechanics, which determine your tendon force as a function of speed.

While we don’t yet have a universal framework for comparing equivalent damage across different training sessions, we can see the contours of a strategy that will work. Here is how it would work for the Achilles tendon—specifics would differ slightly for different tissues, but the high-level picture would be the same:

1. Estimate the stiffness of the athlete’s Achilles tendon^[15]
2. Determine the force in the Achilles per step across different speeds^[16]
3. Determine the athlete’s speed–cadence relationship across different speeds
4. Calculate expected damage per mile across different speeds, using Miner’s rule or some more advanced theory of cumulative damage
5. Express equivalent damage relative to one mile at some reference speed (for example, the athlete’s usual easy run pace)

Extensions to this model are obvious: comparing damage across different inclines and declines, for example, or different footwear (spikes vs. trainers).

In all cases, the reference speed helps eliminate many of the uncertainties associated with estimating absolute probabilities of failure. Athletes usually already have a rough idea of how much easy running they can handle; what they need is an “exchange rate” to compare damage across different sessions—and just maybe, with a little more research, the cumulative damage framework can give us exactly that.

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Footnotes

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^[1] To avoid confusion with all the other uses of the word “fatigue” in running, I’ll be using the term “mechanical damage” instead of “mechanical fatigue” in this article.

^[2] Also, many GPS watches keep a step count field in the raw activity file, even though it never gets displayed for runs on most data dashboards.

^[3] Dividing by cross-sectional area deals with the fact that a thicker tendon with the same material properties will experience less localized stress for a given force, since there is more area to “carry” the load.

^[4] Experienced engineers might notice that, unlike steel, tendon collagen does not have a “linear elastic” region, so it’s not quite as easy as specifying a Young’s modulus for tendon tissue.

^[5] Engineering has a plotting convention that I find extremely annoying, which is that the dependent variable is often plotted on the X axis instead of the Y axis. This is the case for both stress-strain plots (we apply a stress, and measure a strain, but engineers plot strain on the X axis) and also for S-N curves (we apply a stress, and observe how many cycles until failure, but the number of cycles is plotted on the X axis). I have reversed them here, so that equations fitted to the data appropriately deal with uncertainty.

^[6] One issue with cadaver tendons: they tend to be weaker than tendons in young, healthy people—most people do die old, and often spend a lot of time in not-so-great health before checking out. Unsurprisingly, bed rest and poor health do not exactly do wonders for the material properties of your tendons.

^[7] The other reason why collagen from tendons with severe tendinopathy looks like a plate of spaghetti is that the tendon healing process gets disrupted over and over, leading to new collagen and blood vessels being laid down in a haphazard, disordered way. Often, damaged tendons get thicker because there is so much new collagen that isn’t being laid down in the normal orderly fashion.

^[8] See this paper by Firminger et al. and this paper by Van Hooren et al. for more advanced approaches to this same question.

^[9] Speed, cadence, and stride length are mathematically linked: speed = stride length × cadence. It’s worth writing out the units and proving to yourself that this is the case. A consequence of this mathematical link is that if you have two of the three, you can solve for the third. This is how devices like the Garmin HRM “measure” your stride length—they aren’t literally measuring horizontal displacement; they just divide your GPS-measured speed by your cadence.

^[10] Across individuals, there’s also a correlation between cadence at a given speed and biomechanical forces per step at that speed. Higher cadences lead to lower biomechanical force per step. This makes sense, because when you run a given speed at a higher cadence, you are essentially “spreading out” the mechanical work you need to do into a greater number of steps.

^[11] Essentially, the process for estimating Achilles tendon force looks like this: Use the dimensions of our runner’s body (height, leg length, hip width, etc.) to scale a “generic” model of the bones, muscles, and joints of the body to create a “virtual twin” of our runner. Then make this virtual twin move the same way our runner moved while running, and generate the same forces going into the ground that our runner generated. Then use an optimization algorithm to infer the most likely pattern of muscle activations that are consistent with the motion and forces we actually measured. Summing up the calf muscle forces then gives you Achilles tendon force. This sort of technique involves a lot of assumptions but tends to agree pretty well with actually-measured Achilles tendon force (e.g. from studies that use surgically implanted strain gauges).

^[12] One way to get intuitions on why Achilles tendon forces can be so big: ground reaction forces are 2-3x your body weight, and in mid/late-stance, are applied close to the ball of your foot. For me, that’s about eight inches / 20 cm in front of my ankle joint. This ground reaction force creates a torque at your ankle joint, and if you are going to propel yourself forward, your Achilles has to create an even bigger torque to oppose it. But the Achilles is much closer to the ankle joint—about two inches (5 cm) for me. So, that’s a four-fold difference in the amount of leverage. A ground reaction force of 2x your body weight could easily require an Achilles tendon force of 8x your body weight to overcome it. These rough calculations are not quite biomechanically correct but it’s close enough to get good intuitions on the magnitude of the forces involved!

^[13] It is at least somewhat plausible that an elderly bedridden person (the kind of person whose Achilles tendon would end up in a cadaver study) might suffer an Achilles rupture if they were somehow able to run 3 km at 3:20/km!

^[14] This plot shows why thinking in terms of pace does not always lead to intuitive results. Paces “compress” the high end, leading to the dramatic rise in damage at faster paces. The uptick is still present when looking at it in terms of speed, but is less dramatic.

^[15] Alternatively, you could use mean and SD values for tendon stiffness from the literature and then do uncertainty analysis, to see the extent to which tendon stiffness affects the relative damage incurred by different speeds. Stiffness will of course affect absolute damage sustained, but relative damage across speeds may be affected to a different (maybe lesser, maybe greater?) extent.

^[16] You could estimate this in a research-grade gait lab, try to estimate it from wearable sensor data (this was the topic of my PhD dissertation), or—and I’m most excited about this prospect since it is highly scalable—use pose detection + smartphone cameras in the real world, e.g. as done for the OpenCap project

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