Looking for rationale behind O2 minimum at PSF?

[Ted Andresen]

Hello Everyone,

I am stuck on a problem and I thought that someone might have an idea or an insight that could help. If there is another forum that is more related to this type of discussion, I would appreciate a link to that group.

I’ve been using a combined spring mass and planar swing limb model in the hope of understanding the minimums in O2 consumption that occur at a runner’s Preferred Step Frequency (PSF).

Perhaps erroneously, I thought that the minimum in metabolic power might be reflected in a power minimum in a passive bipedal mechanical model. I assumed that the minimum was due to a combination of two power functions, one decreasing and one increasing.

I did some computer experiments with the SM model. Keeping the anthropometric parameters constant, the power decreased with increasing step frequency. (See attachment). If the SM is the source of the decreasing power function, then perhaps the power function in a swing limb model could be the increasing function?

Have others explored rationales behind the minimums in O2 consumption?

Ted Andresen
St. Petersburg, FL
tjacmc@aol.com

[Rodger Kram]

Re: Looking for rationale behind O2 minimum at PSF?

I like Farley & Gonzalez's explanation.

http://www.ncbi.nlm.nih.gov/pubmed/8849811

But, I am biased since Claire Farley is my wife.
Leg springs do not operate for free, muscle force must be generated to allow the tendons to stretch/recoil.

Lieberman has a new take on this in a very recent issue of J Exp Biology.
I am not as convinced by that approach.

Alberto Minetti and Giovanni Cavagna have their own versions in which leg swing power is large at fast freqs and external work is greater at slow freqs. Both M & C ignore the metabolic cost of generating force.

Rodger Kram
Univ of Colorado

[Ross Miller]

Re: Looking for rationale behind O2 minimum at PSF?

This study was for walking (http://www.ncbi.nlm.nih.gov/pubmed/17766303) and I don't know if a similar analysis has been done for running, but it suggests the positive joint power required is lowest when taking relatively long steps, and the efficiency of producing that power was highest when taking relatively short steps.

Seems reasonable that a midway "tradeoff" between those two things could result in minimum energy consumption, but I'm not 100% confident in that suggestion.

Ross

[Riley Sheehan]

Re: Looking for rationale behind O2 minimum at PSF?

Though we only investigated downhill running (http://www.ncbi.nlm.nih.gov/pubmed/23628623), our findings suggest that the minimum metabolic cost at PSF may, at least in part, be related to a tradeoff between the cost of muscle activity in the stance phase which decreases with SF, and the cost of muscle activity in the swing phase which increases with SF. This seems to go along with the idea that the PSF strikes a balance between energy costs for the stance and swing phases, similar to the proposal of Minetti and Cavagna.

- Riley

[Ted Andresen]

Re: Looking for rationale behind O2 minimum at PSF?

Thank you all for responding to my post. I wanted to make sure I carefully responded to all of your comments; it took some time to do the supplemental analysis.

First, let me emphasize that I am working on a lossless mechanical or passive dynamic model. All restoring forces are conservative. Muscle activation is important in the real world, but it is not part of this model. I want to see if a purely mechanical model could be constructed that exhibits power minimums at a physical runner’s PSF. Such a model might be used to help understand the mid-speed performance differences observed between runners in the lab.

I appreciate the link to Farley & Gonzalez’s (F&G) research. I am using a constant leg-stiffness (kLeg) in the SM model. It certainly makes sense that leg-stiffness would change because of the difference in knee flexure between Groucho and quick-step running. I did my best to extract the leg-stiffness values from F&G’s research and then calculate the power expenditure with the SM model. The comparison is shown in the figure below.

There was a difference in the spring leg power expenditure (0.5*Kleg*(DeltaLength^2)*SF), but it did not lead to a minimum in power at the PSF. Instead, it also showed a decrease in power with increasing frequency.

In the past Prof. Cavagna kindly sent me several papers. It was gratifying to observe that one of the additive functions he presented (1) showed a decrease in the internal power (W(dot)int,push) power with SF. Also, note that the power range of 4 to 9 w/kg is close to that computed from the SM model.

Kindly let me know if you disagree, but I think that the decrease in power with an increase in frequency for a passive dynamic system is unusual when compared with oscillating systems.

An example is that of a mass on a torsion spring where the power varies as kTorsion*(ThetaMax^2)*SF.

For the bipedal runner, the ThetaMax’s of the limbs appear to decrease with step frequency. But, that decrease may not be enough to offset the linear increase with step frequency. So, a body segment oscillating with roughly the same amplitude at different step frequencies, would exhibit a power function that increases with SF.

Could that power function’s increase combine with the power decrease from the SM give a minimum in combined power in the SF-range?

At one time I thought that the minimum would be where the two functions crossed. Now, I know that is incorrect. After looking at the graphs in reference (1), I now believe that the PSF’s, indicated by the dark arrows, are located where the first derivatives or slopes of the functions are equal but opposite.

Is there any research using a 3-D system that might give a clearer picture of the internal and external power functions near the PSF?

Ted

1. http://www.ncbi.nlm.nih.gov/pmc/arti...180038/?page=8

[Rodger Kram]

Re: Looking for rationale behind O2 minimum at PSF?

Hi
you continue to ignore the metabolic cost of generating force that is needed for the tendons to operate as springs.
Farley & Gonzalez used the idea that 1/contact time can be a proxy for the rate of force generation. At faster frequencies, tc decreases and thus the force must be developed faster, which requires faster, less economical muscle fibers.
You can bark up the mechanical power tree as long as you want, but ignoring muscle physiology will likely lead you away from understanding the OSF phenomenon. OSF = optimal stride frequency.
rodger

[Ted Andresen]

Re: Looking for rationale behind O2 minimum at PSF?

I appreciate your comments, but I am not sure they are relevant to a mechanical approach to this problem. It is my understanding that the h&v GRF’s generated by the SM model compare favorably with those measured in the lab. This is especially true in the case a mid or fore-foot strikers. If differences do exist, are they significant?

For the case of a subject running at their PSF, the SM model can be extended using the above mentioned technique to yield a minimum in the Total Power/BM at the runner’s PSF.

Unfortunately the depth of the Total Power/BM minimums with constant leg stiffness are shallow (about 0.4%) compared to the 3% to 4%found by Cavanugh & Williams (C&W) and Hunter & Smith (H&S). However, if the leg stiffness is allowed to vary in the manner indicated in the F&G paper, the minimums are in the area of 3% to 4%. (See attached figures below.)

There are still some disturbing issues in using this type of extended SM model:

The magnitude of the Total Power/BM (8 W/kg) at the PSF seems too large.

Other than the graphs from Cavagna’s paper, I don’t have a physics-based reason for adding the linear Reflected Power function to the SM Power/BM.

Unfortunately, I am extracting data from research that presents group data, yet this model is intended for single-subject analysis. I don’t need the VO2 data, just PSF(v) data and the individual’s height and weight. Does someone have a few samples of that type of data that I could access?

Any comments, criticisms or questions would be appreciated.

Ted
F&G Power vs SF.gif