The most basic assumption made here is that the energy used in running an 'all-out' race can be thought of as coming from two sources, one which is in more or less constant supply (the 'aerobic' component, roughly) and another which can be exhausted (the 'anaerobic' part). I also had to make an assumption about the relation of power output to running speed. It (in the form of VO2 max plus other components such as accumulation of blood lactate) is often assumed to be directly proportional to running speed, with a small additional term for air resistance. I assumed it to be proportional to the square of the speed, based on a (perhaps oversimplified) physics based model in which the energy is being used largely to accelerate and decelerate the legs and arms.

The next step was to decide how much aerobic power was available. This varies from runner to runner, but correlates fairly well with performance in distance events. HOWEVER, two runners with, say, the same mile time might have quite different aerobic power output (per unit body mass) since this race is partly anaerobic as well. Simply put, a runners 'speed' might compensate or a lack of 'endurance' or vice versa and it's impossible to tell from a single race how much of each was exploited. For this reason, I decided to solve the problem using TWO performances, since two equations are needed to solve fortwo unknowns!

My first results were somewhat realistic, but the performance curve wasn't quite the right shape, being too optimistic for distances outside the two input and slightly pessimistic in between. Any of my assumptions could have been wrong, but I decided to modify the anaerobic part by making it depend on race distance, the idea being that you simply can't get as 'exhausted' during a very short race as during a very long one (though this idea of exhaustion does NOT exactly correspond to lactic acid in the blood, which is not the only component of the phenomenon anyway). I little playing around with functions of distance quickly brought the curve BEAUTIFULLY in line with the large amount of data I had to check it against, which included my own and that of athletes, aquaintances, and friends of widely varying ability as well as that of world-class runners. The only significant departures occurred at very long distances (such as over 10 miles) and sprints (under 400 m). Another term, perhaps corresponding to the phenomenon of glycogen depletion, brought an appropriate correction to the very long distances and other terms, such as reaction and acceleration time corrections, have brought the sprints very nicely in line as well, though this demo version doesn't include races under 400 meters.

Generally, I've been able to connect these mathematical tweakings with real physiological phenomena, but whether the labeling is correct or not, the model reflects reality very well. I am still wondering if my anaerobic adjustment really reflects, at least partly, a small error in the overall power-proportional-to-speed-squared assumption. I now believe, after reading more about it and curve-fitting treadmill test data I've seen (see Coe and Martin reference below) that the function is best described by power proportional to speed to a power between 1.2 and 1.7, at least if it's fit to a power law. It can also be modeled in more complex ways, such as the sum of two or more power-law parts, but the results I've gotten seem so good that I'm reluctant to tinker with this.

In short, the program finds a curve, based on a very reasonable physical/physiological model, that fits the individual's abilities - regardless of whether these are genetic in nature or due to training specificty - and makes predictions as well as evaluations of relative strengths (i.e. one's best race distance). This is not all the program does, however, as will be discussed below.

The above refers specifically to the core performance curve algorithms. Runpaces does so much more, including the effects of hills, wind, and uneven pacing. In these cases I started with physics and physiology based models, then made adjustments to various factors to bring the results in line with actual data. Some of this data I found online, but a great deal of it I collected personally, with spells of months when the majority of my own training doubled as carefully controlled data collection, in which effort was guaged primarily by heart rate. Likewise, most of my heart rate versus pace data used myself as a subject. Where possible, I corroborated the results against similar studies I found in books or online.

Other Runpaces outputs, such as the effect of ageing or performance levels and percetiles, were not physics-based, but rather the results of exhaustive reductions of data I found, mostly online, using a mixture of various sources and empirical fits to develop the algorithms. The Workout Planner products are distilled from decades of personal experience, combined with a dozen years of coaching high school track and cross country, and reading various books and online sources. In all cases, my goal was to find a most-likely consensus. as evaluations of relative strengths (i.e. one's best race distance). This is not all the program does, however, as will be discussed below.