Jensen's Inequality and training injuries
Last time, I started going down the rabbit hole of non-linear responses to training and chronic injuries. I wanted to expand on that a little. Humans love to put the world into linear boxes. It shows up all the time. For example, in the weightlifting community they often measure the total "tonnage" lifted in a day/week/cycle. That tonnage is generally reps x weight. That is 100 pounds lifted 10 times, or 200 pounds lifted 5 times, or 500 pound lifted 2 times is the same tonnage.
Now I know that no one really thinks that 10x100 lbs is the same as 2x500 lbs, but the fact that this shows up in the literature is evidence of the linear thinking that folks have.
The world is non-linear and our bodies are definitely non-linear. That is, doubling the dose doesn't necessarily double the response. It could be that doubling the dose gives you less of a response - this shows up a lot in training, you think that training once a day is great, so you go to two-a-days but don't get double the response. Or what happens when you keep the dose the same and the response levels off? You know, that plateau that happens after the beginner-gains are gone. You're putting the same effort in but you're frustrated in the lack of results. You're spending more time chasing smaller marginal returns. These are your "concave" functions. It takes more and more effort for less gain.
It could be non-linear in the other way where doubling the dose more than doubles the response. This is the typical hormetic stressor story, or "the poison is in the dose". A small amount of wine (might) be good for you, but we know that too much ruins your liver, for example. Or jumping 10 times off of a 1 meter box is not the same as jumping once off of a 10 meter box. We call this a "convex" function when the result is much more for a small input. All of the good stuff comes from learning how to take advantage of the convex functions.
You can't have 10 small glasses a wine a night and call it good. You need to have a little bit and then wait for your body to adapt. There is a time component to the human body can recovering from hormetic stressors. You can see this pattern in our programming. We rarely program any day to be longer than 45 minutes, often the day is even shorter. Your workout is a small fraction of your day, and you have the rest of the time to recover (provided that you go home and get good sleep). Even within the workout, we follow that plan: short bouts of fast ballistic or heavy work followed by rest.
The amount of weight lifted, or the speed you run is a convex function. I think we'd all agree that a person who can lift 500 pounds is more than 50 times stronger than someone who can lift 10 pounds for reps. Who also has the higher risk of injury? The person who lifts 500 pounds and goes home gets a long time to recover. The person who lifts 10 pounds over and over again all day long will probably get a sore lower back from all that bending over and picking stuff up.
It's better to train for a shorter time and rest than a longer time because the micro-injuries add up over time and don't get a chance to repair.
For any single lift, though, the person who lift 500 pounds is at a higher risk of injury than the person who lifts 10 pounds, it's just that the person who lifts 500 pounds is only exposed to that risk once where as the person lifting 10 pounds for reps has more exposure and they're not giving their body a break.
Let's multiply the risks: our variables are the time invested in the activity, and the intensity of the activity. Below the blue line shows the risk of injury increasing as the time spent in the activity increases. The red line shows the risk of injury as the intensity (or load, or speed) of the work increases.
Time and work don't live alone in a vacuum and any one of those plots won't tell you anything. Work is done over time. So we have to look at both risks at the same time. That can be a short exposure to low intensity work where the risk of injury is negligible -- but you'll get nothing out of it. It could be a long exposure to high intensity work. I think we'd all agree that that's bad.
What happens to a long exposure to very low intensity work (say below MAF level aerobic work)? The zero risk of injury on intensity dominates and it doesn't matter how long your exposed to it. Likewise, a very short exposure to high intensity work (within reason) means there's not enough time to get injured. Sounds a lot like our A+A work -- Long Easy Distance plus short bouts of work with lots of rest. That is, "The Barbell Strategy."
The big problem is spending a moderate amount of time doing moderate intensity work - you know, the glycolytic acid bath that everyone loves when they "feel the burn." The product of the exposures magnify. That's where we see the chronic injuries that take everyone out of the game.