VO2 (mL . kg-1 . min-1) = (0.2 . S) + (0.9 . S . G) + 3.5 mL. kg-1.min-1
VO2 (mL . kg-1 . min-1) = 1.8(work rate) / (BM) + 3.5 mL. kg-1.min-1+ 3.5 mL. kg-1.min-1
To compare the two, I had a snoop around to see if there was any agreement on a standardized model for watts produced running. There isn't. The best I found was from this conversation amongst well-informed men, and ran with Chung's assumption of a 23.9% efficiency rate, to which I simply plug into a well established calorie formula:
METs = VO2/3.5
METs = VO2/3.5
METs * kg= kCal per hour
joules\cal = 4.18
EnergyOut = (kCal\hr * 4.18 * .239)
Energy out/3600 *1000 = watts
I noticed immediately when plugging my own snails pace and rhinoceros weight in that my wattage, at least by this calculation, matches my bike FTP nearly to the watt. Sure enough, when returned into the ACSM formulae, my VO2 levels were nearly identical. Obviously, the efficiency part is a big variable. If I want a 1:1 efficiency rate, then 23% works for me. I like round numbers, but it's just a plug. Anyway...
What does this mean? There's a lot of dismissive chatter on training based on VO2Max, and I get the point. I've read several studies now that show VO2Max (assuming it's measured accurately and not estimated, which is one hell of an assumption in some of these studies) doesn't change in elite athletes, and hardly changes at all in amateur athletes. This would support Noakes' work that there exists a 'central governor' wholly different from the supposed 'VO2Max' level. The works aren't as radically different as one would think though; Noakes doesn't reject VO2, just the concept of there being a 'Max', suggesting it's more likely a function of other factors, notably muscle composition and neurological factors. However it's hard to dismiss training without some respect to VO2, if for no other reason than it includes a variable for weight that is absent otherwise. I've personally trained my VO2 tolerance higher and higher, without knowing (or caring) what my 'Max' level was. If elite athletes have been measured with a VO2Max level above 70 l/min, that would at least suggest to me there's a lot of work left to be done for us dopes mucking around in the high 40s and low 50s. Lots of training anecdotes line up with this model as well; If I play with the inputs, 5 pounds equates to 10 seconds of pace in running, which we've all heard before via 1 pound = 2 seconds. If I plug in some of my teammates FTPs and paces, the model correctly guesses their weights. When I put in my peak cycling FTP, I also get the run pace I managed to hold 'magically' one particular race, even though I hadn't been running any more or less leading up to it. I'd simply been riding more.
So I'm going to proceed with modeling workouts for now not based on TSS, but based on time spent at VO2 levels. I'll make the 'Max' level nothing more than the observed best. If your best 10K run put you at a VO2 of 51.5 l/min and you weighed 170 pounds at the time, then I would expect your corresponding bike FTP to be right around 310 (the formula produces 312, but I measure my FTP in increments of 5, as most do).
There has to be some function for time decay, however, as well as age. Riegel's formula for predicting run times in on the right track for decay, and for age I've been using these for age and gender.
I have nothing for swimming yet, and from my research so far, nobody else does either. It's hard to measure much of anything while swimming (even now, optical heart rate monitors are *just* becoming available for underwater usage at a consumer level), so it seems the science hasn't caught up. I've been using your best 800 yard pace as a proxy, basing percentages off that, and it seems to be a pretty good measure, at least as a predictive tool. I've tested it out on several of my teammates over ~30 different races at different times, and the results have been promisingly close to reality.
At the end of the day, I'm digging at all of this because TSS seems so inherently flawed. If you need permission to use math, it probably isn't math, it's a marketing term. If you work out and produce a TSS of 90 or 91 - or even 85 or 95, it doesn't mean anything on its own. It has to be looked at as a function of time, and even then, it's only an indication that you might be over- or under-training. It doesn't appear to have an upper bound, because there is no system of debit and credit, you simply accumulate more and more TSS over time. I've seen suggestions that it might have an *optimal* upper bound of 250, but that number suggests it's impossible that any age grouper could finish an ultra distance race in less than 10 hours, when of course we observe otherwise.
What I'm coming around to is that we are really only training three central things - our ability to create power, our ability to tolerate the toll it takes, and how quickly we recover. That's not just about lactic acid, or VO2Max, or muscle composition, or psychology, or neurology, or heart rate...it's about all of those things. For what I'm seeing, VO2 as a scale does a very good job of linking together performance and prediction across sports. If your last race result put you at 80% of your *best* VO2 at the time, and you manage to raise your VO2 in training since then (including all of the factors that go into it, including the ones you can't directly measure), then you're by definition faster. Whether you did it by tolerating pain better, doing drugs (I'm not condoning this, just saying...), or whatever, it comes out in VO2, of which you can reverse the math to find the pace.