## Saturday, February 11, 2017

### Actionable Intelligence for Running Part 3 : Leg Spring Stiffness & Speed-Time Relationships

LSS is a theoretical concept under the linear spring mass model of running, which has proven to remarkably explain several characteristics of running. Mathematical models teach first principles. If most of what you want to know can be explained by a simple model, the additional time spent on increasing the complexity of modeling is not justified.

In Part 2 of this series, I found that change in Ground Contact Time (GCT) is a better explainer of change in Leg Spring Stiffness (LSS) than is change in cadence (step frequency or SF).

In this post, I continue looking into LSS variation with running speed for both treadmill runs and outdoor runs on predominatly asphalt. Running subject is me again. The "intelligence" gained from this post is summarized in blue italics. Please visit Part 2 for abbreviations used in the writeup.

LSS and Speed Relationship for Treadmill Runs

Around 12,527 datapoints were collected over strictly controlled treadmill runs executed on a Cybex 770T from January 1 - February 9, 2017. The data was cleaned by eliminating null values of LSS; my Stryd footpod is expected to give such data from time to time, mostly during the start of runs. The LSS vs running speed plot is shown below.

Fig.1 : Scatter plot of Stryd measured LSS vs running speed for treadmill running. Sample frequency = 0.5 Hz.

The question is whether the data suggests the relationship is linear or not. The Excel based R^2 value of 0.7902 did not inform me correctly. A better method was to use a correlation test.

A Ryan Joiner test to test the null hypothesis that the data was a random sample from a normal distribution. The probability plot of LSS from this dataset is shown below. Since the probability line is not straight, the RJ value < 1 and p-value was  .001

Fig.2 : Supplementary plot showing non-normality in LSS data collected during treadmill runs.

Next, a Spearman's rank order correlation test for non-normal data was done. The correlation coefficient was 0.104, p value = 0. This told me that the data has a monotonically increasing relation and that the probability of seeing the observed relationship was 0 if the relationship was anything other. The fact that the sample is large may show that Spearman's rank order correlation of 0.104 has high power.

Fig. 1 also shows presence of outliers - datapoints of high LSS for the same magnitude of runs as the majority of parent data. What's going on here? When I peer into just these points, I find evidence that high LSS values (13kN/m or greater) can belong in either a high cadence region or a low cadence region.

The low cadence region 1 consists of high vertical oscillation (VO) and high GCT. The high cadence region 2 consists of low VO and low GCT. After inspecting these subsets of data individually, region 1 was found to correspond with low speed strides I did on the treadmill (for want of some variation in a long run). Region 2 corresponded with high speed, high cadence running. These two regions are shown below in Fig.3.

Fig.3 : Plot showing two regimes where LSS data was found to be greater than 13 kN/m. Region 1 is low SF region with high VO and GCT, while region 2 is high SF region with low VO and GCT.

The fact that leg stiffness is high at high stride frequencies is consistent with literature. I didn't find confirmatory evidence in research for high values of leg stiffness at low step frequencies and high GCT. However, it logically makes sense that one would have to neuromechanically adjust leg stiffness to cushion a higher degree of fall from greater VO. I'll keep looking in future for data within region 1.

LSS and Speed Relationship for Outdoor Runs

I also looked for the relationship of LSS with speed for outdoor runs executed on predominantly paved surfaces. 10,164 datapoints were inspected from the period January 1 - February 9, 2017 and found to show a near monotonic relationship between speed and LSS. The Ryan Joiner test on this dataset rejected the null hypothesis that the data was a random drawing from a normal distribution (.001

Fig.4 : Scatter plot of Stryd measured LSS vs running speed for outdoor runs. Sample frequency = 0.5 Hz. This dataset includes a run done at -4 deg C ambient temperature when the running surface author encountered was covered with thin ice.

A negative correlation co-efficient was somewhat puzzling to me. I have to point out that 2,347 of these datapoints in the sample corresponded to a run done in the UK when ambient temperature was minus 4 degree C. There was still icy frost on the ground. During this run, I was trying consciously to stay upright and not slip on an unlit running path besides the Thames river at 6 in the morning. Neuromechanically, this might mean I could have changed leg stiffness to maintain center of gravity. That leg stiffnesses are adjusted by human runners based on type of running surface was shown by Farley et.al.

Following that theory, I eliminated these 2,347 datapoints from the dataset and followed through with the rank ordered Spearman test on the rest of the 7,736 datapoints. The correlation coefficient was 0.022, p-value = 0.057. This told me LSS was typically monotonically increasing with running speed and within this dataset, the probability that it is a chance event is small.

The takeaway : Given a more or less fixed step rate, leg spring stiffness is correlated with speed. The data shows that leg spring stiffness monotonically increases with speed. Majority of LSS values can be in a tight window, for example for all my runs they were typically between 10-12 kN/m.

A cautionary note is that higher than typical LSS values may be exhibited during periods of slow speed, low step rate running where vertical oscillation is high (such as during strides). It may also be exhibited during periods of high speed, high step rate running where vertical oscillation is small. It is important to survey the behavior of these connected parameters when trying to assess why LSS numbers are higher than your typical values. When such data are accounted for in outdoor runs and indoor runs (running motion oddities due to the effect of "slippery" icy surfaces for example), the relationship between LSS and running speed is appreciably monotonic positive.

LSS vs Speed and Time In Endurance Runs Greater than 1 Hour

I've not done too many distance runs greater than 1 hour 20 minutes. That said, there are two runs of between 1:20:00 - 1:40:00 which I thought could be inspected to see if long duration running and it's accompanying fatiguing aspects could translate to decreases in leg spring stiffness. The two runs with speed and LSS are displayed in Figs. 5 & 6. Plot captions indicate the conditions under which they were done.

Fig.5 : 1 hz data from endurance run #1 executed in Abu Dhabi on February 3, 2017 under 17 deg C ambient temperature and 21 kph winds (around Beufort Scale 6). Number of sample points = 2,962. Plot shows LSS, GCT and speed variation with time. Mean LSS = 10.233 kN/m, mean speed = 3 m/s, mean GCT = 247.22 ms.

Fig.6 : 1 hz data from endurance run #2 executed in Staines-on-Thames on January 23, 2017 under -4 deg C ambient temperature and low winds (around Beufort Scale 6). Number of sample points = 2,448. Plot shows LSS, GCT and speed variation with time. Mean LSS = 10.54 kN/m, mean speed = 2.67 m/s, mean GCT = 262.83 ms.

In Fig.5, the Spearman rank ordered correlation between duration and LSS showed a correlation coefficient of -0.337, p value = 0. In Fig.6, the Spearman rank ordered correlation between duration and LSS showed a correlation coefficient of only -0.094, p value = 0. The relationships in both these runs between duration and LSS was mildly monotonic decreasing.

What is interesting is that the faster, warmer, longer duration run in windier conditions shows a stronger negative correlation between LSS and duration than does the somewhat shorter, colder and slower run in cold temperature.

A contour plot of LSS, speed and duration for the run from Fig.5 is shown below. It indicates that there is a cluster of low LSS values in the range 5.0-7.5 kN/m corresponding to high speed and high duration. This is interesting and I need to concentrate on these instances both within the data and in reality to understand what is going on. For example, it hasn't been uncommon for my left feet to exhibit fatigue from long duration running.

Fig.7 : Contour plot of LSS vs Speed and Time in a long run on a windy day in Abu Dhabi on January 23, 2017

However, due to the lack of further long runs and lack of high intensity data, I cannot fully elaborate on fatigue and it's effects on LSS. It will be something I will keep a watch on. Perhaps introducing power and form power into these metrics will form a clearer picture.

The takeaway : A preliminary analysis of two long duration runs suggest there is a monotonic decreasing relationship between duration of run and LSS. It perhaps is likely that changes in running form and/or fatigue can influence LSS as evidenced by the contour plot. This aspect needs a much more comprehensive treatment with other markers of performance.

In the next post Part 4, I will move to power based metrics and see how indoor and outdoor runs compare. I will also inspect my power data from the recent RAK half marathon and see how power based pacing helps in duration specific stamina and how that methodology compares to speed based pacing. Stay tuned.