Quadrant Analysis

Friday, October 10, 2008 | By Andy R. Coggan, Ph.D.
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Quadrant Analysis (QA) is a means of analyzing power meter data files to obtain additional insight into the neuromuscular demands of a race or training session. QA is available as a feature in WKO+ 3.0 as well as TrainingPeaks Premium.

The Problem

The overall power requirement of competitive cycling varies dramatically depending on the nature and the duration of the event (e.g., match sprint vs. RAAM). In addition, there is often considerable fluctuation in a rider’s power output on a moment-to-moment basis, due to changes in speed, wind, road or trail gradient, etc. The wide range and highly variable nature of cycling power has significant physiological implications, not only in terms of the acute responses to a single ride, but also in terms of the chronic adaptations to repeated training sessions. Accordingly, tools such as normalized power, intensity factor, and training stress score, which explicitly recognize the seemingly stochastic nature of cycling power output, have been developed to help coaches and athletes better understand the actual physiological demands of a given race or workout. Since metabolic fitness – commonly measured as lactate threshold – is the most important physiological determinant of performance in all endurance sports, including cycling, these tools have been primarily designed to account for the metabolic consequences of varying power output.

Support for this concept and this approach comes from the observation that normalized power during a hard, ~1 h long mass start race is usually quite close to an individual’s functional threshold power. Even so, to completely understand the physiological consequences of large variations in power requires also understanding how they impact neuromuscular function, i.e., the actual forces and velocities that the leg muscles must generate to produce a given power output. Such effects are recognized by the algorithm used to calculate normalized power, but only to the extent that they influence metabolism, e.g., via altering fiber type recruitment patterns. While, e.g., maximal force (strength) per se is rarely a limiting factor in cycling, neuromuscular factors nonetheless can still sometimes play an important role in determining performance. Thus, there is a need for a means of analyzing or presenting powermeter data that captures this important information in a form that can be readily grasped even by non-experts.

The Approach

Some information about the neuromuscular demands of a given workout or race can be obtained by examining a frequency distribution histogram of the rider's cadence. Such plots are automatically prepared by most, if not all, powermeter software programs, and thus provide a convenient means of data analysis. However, velocity of muscle contraction (as indicated by cadence) is only one of two determinants of power, with the other of course being force. Unfortunately, at present no powermeter directly measures the force(s) applied to the pedal. However, it is possible to derive the average (i.e., over 360º) effective (i.e., tangential to the crank) pedal force (both legs combined) from power and cadence data as follows:

AEPF = (P*60)/(C*2*Pi*CL)

where AEPF = average effective pedal force (in N), C = cadence (in rev/min), CL = crank length (in m), and the constants 60, 2, and Pi serve to convert cadence to angular velocity (in rad/s). Additional insight into the neuromuscular demands of race or training session can then be obtained by preparing a frequency distribution histogram for AEPF just like for cadence, as shown in Fig. 1. (Note that as with all such plots, graphs like this one do not take into consideration how long AEPF was continuously within a given "bin" or range. This is not an issue, however, because unlike e.g., heart rate, neuromuscular responses/demands are essentially instantaneous. Indeed, it is the generation of specific velocities and forces via muscle contraction that essentially drives all other physiological responses.)

While providing some additional insight, simply examining the frequency distributions of AEPF and cadence does not reveal the relationship between these two variables - this can only be done by plotting force vs. velocity. Such force-velocity diagrams have been used by muscle physiologists to describe the contractile properties or characteristics of muscle ever since the early 1920's, when A.V. Hill derived the equation describing this relationship from myothermal measurements made on isolated frog muscle. Fig. 2 therefore provides an example of a force (AEPF)-velocity (circumferential pedal velocity) scatterplot for the same training session as used to generate Fig. 1. Circumferential pedal velocity, i.e., how fast the pedal moves around the circle it makes while pedaling, is derived from cadence as follows:

CPV = C*CL*2*Pi/60

Where CPV = circumferential pedal velocity (in m/s), C = cadence (in rev/min), CL = crank length (in m), and again the constants 2, Pi, and 60 serve to convert the data to the proper units. While technically muscle shortening velocity or at least joint angular velocity should be used instead of CPV, CPV has been shown to be an excellent predictor of joint angular velocity and, by extension, muscle shortening velocity. Indeed, since crank length is generally constant, especially for a given individual, one could just as well use cadence instead of CPV. However, the latter has been used here to be consistent with scientific convention and to emphasize the relationship of such cycling-specific plots to the more general force-velocity curve of muscle.

A scatterplot of force and velocity such as that shown in Fig. 2 therefore presents information that cannot be obtained from just frequency distribution plots of AEPF and CPV. However, it can be difficult to detect subtle and sometimes even not-so-subtle differences between roughly similar rides based on such "shotgun blast" patterns, especially if the scaling of the X and Y axes is allowed to vary. Furthermore, without additional information such force-velocity scatterplots are entirely relative in nature, i.e., there are no fixed anchors points or values that can be used as a frame of reference. It is the latter issue that quadrant analysis was specifically developed to address, as discussed below.

Once again, threshold power (and the associated cadence) provides a useful basis for comparison, and in particular for separating relatively low force from relatively high force pedaling effort. (It cannot be overemphasized that the absolute forces generated while cycling are usually quite low, such that strength is rarely a limiting factor to performance. An example of how quadrant analysis can be used to demonstrate this point is provided at the end of this article.) In particular, one factor contributing to the curvilinear relationship between exercise intensity and various metabolic responses (e.g., glycogen utilization, blood lactate concentration) is the recruitment of type II, or fast twitch, muscle fibers. Specifically, when pedaling at a typical cadence and a power output well below lactate threshold, there is little engagement or utilization of fast twitch fibers, but with progressive increases in power output, a progressively greater fraction of the total motor unit pool will be recruited to generate the required force.

The Quadrants

Based on scientific studies using a wide variety of techniques (e.g., EMG spectral analysis, muscle biopsies), it appears that threshold power represents not only a threshold in terms of the power that an athlete can sustain, but also somewhat of a threshold in terms of fast-twitch fiber recruitment. To state it another way: when pedaling at a typical self-selected cadence, functional threshold power appears to occur at the power (and thus force) at which significant fast twitch fiber recruitment first begins. Thus, AEPF and CPV at an individual's threshold power can be used to divide the force-velocity scatterplot from any of their rides into four quadrants, as shown in Fig. 3. This division is somewhat arbitrary, in part because of the gradation in force and thus motor unit recruitment that occurs when cycling. Also, exercise duration plays an important role in fiber type recruitment, but this is not considered in the figure (to do so would require a three-dimensional plot of AEPF vs. CPV vs. time, which is too complex for routine use). Nevertheless, data points that fall into these four quadrants can be interpreted as follows:

Quadrant I (upper right)

High force and high velocity. At the extreme, this would represented by sprinting, but most any extended supra-threshold effort on level ground (e.g., attack or bridge attempt during a race) would entail "quadrant I pedaling". Perhaps not surprisingly, mass start racing on the track (e.g., points race) invariably entails a significant amount of such high force, high velocity pedaling, due to the typical aggressive nature of such racing and the use of a fixed gear.

Quadrant II (upper left)

High force but low velocity. Typically, "quadrant II pedaling" occurs when climbing or accelerating, especially from a low speed. Indeed, a standing start, in which the initial CPV is zero, is the one situation in cycling where strength is truly limiting, i.e., only when CPV is zero will AEFP be maximal. Racing off-road (i.e., cyclocross or mountain bike racing) also often involves a significant amount of such high force, low velocity pedaling. However, even a race held on pavement may require a large percentage of such pedaling, if the climbs are steep and/or the rider is overgeared.

Because AEPF is sufficiently high, pedaling in both quadrant I and quadrant II would be expected to entail significant recruitment of fast twitch fibers.

Quadrant III (lower left)

Low force and low velocity. Rides that entail a very large percentage of pedaling that falls into "quadrant III" would typically be those used for recovery or for social purposes (e.g., coffee shop rides), not for actual training. However, a mass start race in which power is highly variable may also involve a good deal of such low force, low velocity pedaling, e.g., when recovering from harder efforts when there is little possibility of an attack, or when soft-pedaling in a large bunch.

Quadrant IV (lower right)

Low force but high velocity. Perhaps the most obvious example of "quadrant IV pedaling" would be use of a low fixed gear or rollers in an attempt to improve pedaling smoothness. Racing, however, may also involve a significant amount of such low force, high velocity pedaling, especially during events in which there is a frequent need to accelerate rapidly (e.g., criteriums).


To further illustrate application of this method and the insights it may provide, examples from different types of workout and races are provided below. These examples were specifically chosen because, except for the 40k TT provided first as a reference, the average (not normalized) power in each case is close to 250 W. As can be seen, however, the combination and distribution of pedaling forces and velocities accounting for this power output differed significantly. In particular, note the differing patterns evident in the plots of the constant power and microinterval ergometer training sessions - the utility of preparing a force-velocity scatterplot is especially evident in this case, as it reveals an important difference between the two workouts that cannot really be discerned based on average or normalized power, average cadence, etc.

Figure 4. Quadrant analysis of flat 40k TT (average power 294 W, average cadence 80 rpm).

Figure 5. Quadrant analysis of constant power ergometer workout (average power 250 W, average cadence 88 rpm).

Figure 6. Quadrant analysis of "microinterval" (15 s on/15 s off) workout on ergometer (average power 245 W, average cadence 91 rpm).

Figure 7. Quadrant analysis of road race on flat to rolling terrain (average power 250 W, average cadence 78 rpm).

Figure 8. Quadrant analysis of flat criterium (average power 244 W, average cadence 72 rpm).

Figure 9. Quadrant analysis of all examples plotted together.

Finally, force-velocity scatterplots can also be used to illustrate how strength per se (i.e., the maximal force generating capacity of the pedaling muscles) rarely, if ever, limits power output. This is shown in Fig. 10, in which all five examples have been replotted with an expanded Y axis and a horizontal line has been added to show this individual's maximal AEPF of 918 N (determining using the inertial load testing method - thanks to Dr. Jim Martin, University of Utah, for providing this information). As can be seen in this figure, even though this rider is somewhat weaker than average, they rarely use more than 50% of their maximal AEPF when training or racing. Indeed, even when time trialing at a relatively low cadence of only 80 rpm, to generate 300 W still only requires that they utilize less than one-fourth of their maximal cycling-specific strength. As indicated previously, however, there are some situations (e.g., standing start) in which strength may truly be limiting, as shown among these other examples.

Caveats and Limitations

Several limitations should be kept in mind when interpreting force-velocity scatterplots based on quadrant analysis. As indicated previously, the distinction between relatively low force and relatively high force pedaling is actually somewhat arbitrary, since recruitment of fast twitch motor units really occurs in a graded, rather than a threshold, fashion. In addition, muscle force and velocity, while important in determining fiber type utilization, are not the only factors that determine recruitment patterns - for example, as mentioned previously the duration of exercise also plays an important role, and there are other factors as well (e.g., the threshold for fast twitch motor unit recruitment is reduced during ballistic contractions). Because of such considerations, "quadrant analysis" should be viewed as a general indicator of fiber type recruitment, not as an absolute measurement.

Another issue that must be considered is the source of the original data file used for analysis, i.e., what type of powermeter was used. Unless specified otherwise, the data in the examples provided were collected using an SRM Professional road crank set to record at 1 s intervals. This is possibly the ideal situation, in that the recording interval is short enough to adequately "capture" more extreme variations in AEPF and CPV, while still being relatively unaffected by sampling errors. However, if another brand of powermeter had been used (even simultaneously) then the precise pattern observed would be slightly different, due to e.g., aliasing effects and/or errors in determining the precise cadence. The duration over which data are averaged can also make a difference - this is perhaps best illustrated by realizing that if power and cadence are averaged over an entire ride and then used to calculate AEPF and CPV, that workout would be represented by only one symbol on the plot, thus defeating the purpose of performing the analysis. However, analysis of the same file using rolling averages of different durations (e.g., 2 s, 5 s) demonstrates that as long as the data aren't smoothed excessively, then the general overall pattern of the scatterplot will still adequately reflect the demands of the workout. Consequently, quadrant analysis can still be performed on data generated by e.g., an Ergomo or Polar S710/720 powermeters, which report the power and cadence at 5 s intervals (average over 5 s for the Ergomo, average over 1 s for the Polar). Such plots will, however, appear less densely populated that those generated from SRM or PowerTap data files, such that some caution is advised in interpreting plots prepared from different powermeter data files.

Finally, it should be emphasized again that powermeters currently on the market do not directly measure the force applied to the pedal, such that AEPF must be derived from other data (i.e., power and cadence). Because force (and thus power) varies in a sinusoidal manner from close to zero when the cranks are vertical to some maximum when the cranks are nearly horizontal, AEPF will underestimate the maximal force by a factor of approximately two. However, there is relatively little variation between or within individuals in precisely how forces are applied when pedaling - thus, quadrant analysis can still be used to provide insight into neuromuscular demands, even if the values actually plotted are averages across one (or more) complete pedal revolutions.

Figure 1. Frequency distribution of AEPF during a level 2 training ride over varied terrain.

Figure 2. Scatterplot of AEPF vs. CPV for a level 2 training ride over varied terrain.

Figure 3. Quadrant analysis of a level 2 training ride over varied terrain.

Figure 4. Quadrant analysis of a flat 40 km TT.

Figure 5. Quadrant analysis of a constant power ergometer workout.

Figure 6. Quadrant analysis of a "microinterval" (15 s on/15 s off) ergometer workout.

Figure 7. Quadrant analysis of a road race over flat to rolling terrain.

Figure 8. Quadrant analysis of a flat criterium.

Figure 9. Quadrant analysis of all examples plotted together.

Figure 10. Quadrant analysis showing role of maximal AEPF (strength).

Figure 11. Quadrant analysis of microintervals.Average power during all four training sessions = ~295 W.

Special thanks to 1) Bill Howard, for the loan of the fluid trainer, and 2) John Verheul, for his suggestion that I perform the latter trial in the first place.

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