Modeling the effects of training on performance has been of interest in the scientific literature since Calvert et al. (13) first proposed an impulse-response (IR) modeling approach more than 3 decades ago. They hypothesized that an acute training bout (impulse) elicited 2 antagonistic responses: (a) an initial negative component (attributable to “fatigue”) that detracted from performance and (b) a delayed positive component (attributable to “fitness”) that ultimately contributed to improved performance after dissipation of the negative component. A quantitative model was proposed of the form:
where pt is the predicted performance at time t; p0 is initial performance level; ka and kf are scaling factors for the positive and negative components of performance attributed to “fitness” and “fatigue,” respectively; τa and τf are decay constants for fitness and fatigue, respectively; and ws is the training stimulus input. Subsequent to its introduction, several investigators have introduced modifications intended to improve the robustness and/or accuracy of this original approach (7,20,22,28,29). The robustness of the IR modeling approach is evident in the numerous published reports in which performance modeling has been successfully applied to various diverse athletic disciplines such as swimming, running, triathlon, and hammer throwing (2,3,8,21,22). Although IR modeling has found success in the scientific literature, this success does not appear to have translated to utilization by practitioners (e.g., coaches and/or athletes) in the field. This may be in part, as Busso and Thomas (12) and Taha and Thomas (38) have argued, that model predictions are insufficiently accurate on an individual basis to be practical for training plan design. The IR model may, therefore, have limited utility in the field because of its inability to specifically predict performances on an individual basis. Despite this limitation, beneficial insights regarding the constructs of “fitness” and “fatigue” and approaches to tapering have been derived from IR modeling studies (3,8,13,20,30,40).
Aside from issues related to the accuracy of predictions, other factors may limit the original IR model or its more recent derivatives from practical application. First, a complete understanding of the IR modeling approach requires a more advanced level of mathematical knowledge and resources (e.g., Matlab) than most practitioners may possess. Therefore, it would be of value if the mathematics of IR modeling could be simplified while still retaining the character of the approach.
In essence, all derivations of the IR model can be distilled to:
and a simplified modeling method that captures this relationship is of interest.
To that end, Coggan has recently proposed such an approach that eliminates much of the complex math, and as such can be performed using simple spreadsheet calculations (15). The simplicity of Coggan's approach lies, in part, with the elimination of the scaling factors (ka and kf) and the use of exponentially weighted moving averages, as opposed to integrals for the calculation of fitness and fatigue components from Equation 1. This results in a simple relationship:
where p(t) is the performance at time t, and CTL and ATL are the chronic and acute training loads comprised of an exponentially weighted rolling average of the training impulse with long-term (e.g., 42 days) and short-term (e.g., 7 days) decay constants, respectively. It should be noted that by using this approach, the IR model output is not intended to be an exact prediction of performance but more an indicator of performance trends, whereby an increasing or positive p(t) should be indicative of proportionally good performance and a declining or negative p(t) should be indicative of the converse. Although not quantitative per se, this information may still be of practical importance to the coach/athlete.
This approach alleviates another potential limitation to practical implementation of IR modeling in highly trained athletes: valid modeling using previous methods requires frequent standardized testing to obtain accurate scaling constants and model fits (12,21,38). In the case of athletes/coaches engaged in an intense, rigorous training regimen or competitive season, this may not be practical. By removing scaling factors, model fitting becomes less important and proportional changes in performance may be assumed from the CTL/ATL relationship, thus facilitating the use of performance modeling without the need for formalized testing procedures.
Therefore, the purpose of this study was to test if the calculated response of a simplified IR model would be correlated to actual performances of an elite middle-distance runner over the course of 7 years, including Olympic 1500-m performances for 2000 (finals) and 2004. It was anticipated that, because this model is simpler than traditional IR modeling approaches and if it indeed correlated to actual performances, it could prove useful for practitioners training athletes in the field.
Experimental Approach to the Problem
To use an applied approach to relating IR modeling to actual competition performances, training logs were obtained from the subject for the years 2000 to 2006. These logs were used to calculate a training stress score (TSS) metric for each day, which was then used as the impulse component for the IR model, and the response was modeled against actual recorded performances from competition. It was assumed that because performances were obtained from competitions, they represented “best efforts” on the days recorded, under the existing conditions. As such they should be reflective of the optimal performances on the given day considering the individual's state of fatigue and fitness. To compare performances across different distances for the athlete, times for the competition performances were converted to Mercier scores (MS), which allow cross-event comparisons in athletics. Actual performance MS were compared by correlation and regression curve fitting with IR model outputs, and the individual model components, to provide additional insight into the potential value of this approach.
The subject was an elite, international-caliber middle-distance runner (2-time Olympian, 8-time national champion, national record holder, 4-time National Collegiate Athletic Association champion), who in the first year corresponding to data collected for this study (2000) was age 25 years, 181 cm tall, and weighed 68 kg. The subject gave written informed consent prior to the investigation and all procedures were approved by the local Human Subjects Review Board at Eastern Michigan University.
Training Stress Score
The IR model as originally proposed by Calvert et al. (13) uses heart rate (HR) as the training impulse (TRIMPS). Although TRIMPS has been used with success in some previous studies (2,4,29,37,40), heart rate measurement is typically not used as an indicator of training load and often not even measured by athletes/coaches in disciplines such as running and swimming. Therefore, a more relevant measure of training load might be of value for a sport such as running. Further, in a discipline such as middle-distance running, HR may not be indicative of the true intensity of the efforts that often exceed O2max or incur an oxygen deficit and therefore would not be reflected by HR (9). To address this final limitation, in the current study a pace-based metric (TSS) was used, analogous to that used by Coggan for cycling (1).
The subject maintained detailed logs of each training session throughout the course of the 7-year period under investigation, but training was not manipulated by the investigators. The logs included duration, distance, comments such as type of workout (e.g., hills, tempo runs, repeats) and perception (e.g., fatigue, injury, increased stamina) during the training session. Relevant parameters (pace and duration) were entered into an Excel (Microsoft, Redmond, Washington, USA) spreadsheet to calculate a TSS. The TSS system proposed by Coggan for cycling is represented as follows:
The FTP has been defined as the maximal power that can be sustained for 1 hour and is analogous to maximal steady-state power (J/s) (1).
Therefore, equation 3 can be transformed to:
and as a result, if running paces can be determined for NP and maximal steady-state pace (MSSp), these can be used for the calculation of IF and TSS in lieu of power. Using this formula, a maximal 1-hour effort would result in a TSS of 100 points.
For training sessions that were reportedly comprised of constant intensity, “steady-state” efforts, a single NP was assigned for the session. In the case of interval training sessions, or sessions where clearly distinct effort levels were reported, NPs were determined for each segment.
Weighted Segment Pace
Based on the exponential relationship between intensity and various physiological parameters (e.g., epinephrine, glycogen utilization, lactate appearance, etc.), several investigators (1,29,30) have proposed an exponential weighting of intensity for the purposes of training stimulus quantification. Therefore, pace for each segment was raised to the fourth power.
Individual segments were averaged and the fourth root was obtained to determine an NP for the entire workout.
For implementation of this simplified model, 2 parameters can be varied: the time constants of decay for CTL (fitness) and ATL (fatigue). In the first case, the analogous time constants in the literature have varied from 38 to 60 days, and it has been reported that the fits using traditional IR modeling approaches are relatively insensitive to manipulation of this parameter (8,10,21,38,40); therefore, a decay constant of 42 days was chosen for the CTL parameter. With regard to the ATL, the analogous decay constants have varied in the literature between 2 and 15 days (38). For the current study, a decay constant of 7 days was chosen a priori and tested vs. 3 and 14 days to determine model sensitivity to this parameter.
Maximal steady-state pace was determined pre-season and in-season by extrapolation from actual performances using Daniels' tables (17). These tables are analogous to other nomograms, most notably Mercier's tables (6,27), and are based on studies performed by Daniels (18). They are not only physiologically sound, but they are widely available to practitioners and the lay public and therefore lend to the applied aspect of this work.
In an approach similar to that used by Pyne et al. (37), who used an objective point score system utilized by FINA (Federation Internationale Natation Amateur) for swimming, MS were determined to allow for quantitative cross-event comparison. The MS system as described by Athletics Canada is used to determine selection of athletes for major competitions, including the Olympics, and is therefore an important benchmark for athletes in this federation. MS was determined using an online calculator that can be found at the following Website: http://myweb.lmu.edu/jmureika/track/Mercier/index.html.
This system allows an individual to enter the race distance and time, resulting in MS (34). MS has been determined to be the “end-result of a linear fit to the weighted average of the 5th, 10th, 20th, 50th, and 100th World-ranked performances in each event over the past four years” (34). The use of this system allowed the utilization of performances in different distances (e.g., 800 m, 1,500 m, and 1 mile) to increase the available sample size of performances.
All statistical analyses including Pearson's product correlations were performed using SPSS 14.0 (Chicago, Illinois, USA) and significance was established by an α = 0.05. Analysis of variance (ANOVA) was performed where appropriate, and in the event of statistical significance, a Tukey's post hoc analysis was performed. Model component relationships were determined by regression curve fitting, also α = 0.05.
Training Load Distribution
The typical approach to yearly training distribution of the athlete under study can be seen in Figure 1. During the 2000 season, the runner appeared to exhibit the highest training load at the beginning of the calendar year, which was reduced over the course of the competitive season until the target event, the Olympics (red arrow). After the competitive season, training was halted for a period of time until resumption, where training load was increased to near maximal levels by the end of the calendar year. This general pattern was exhibited in all years under study (data not shown).
Training load distribution over the course of the 7 seasons studied can be seen in Figure 2. Significant differences in average TSS points were observed between several of the years. Presumably, these differences can be attributed, for the most part, to an injury incurred during the 2002 season that recurred over the course of 2002 and 2003, impairing the athlete's ability to train on several occasions and negatively affecting the overall training load in those years.
Competition performance as assessed by MS was not significantly different between years over the 7-year course of study (Figure 3). Additional analysis can be seen in Figure 4, where box plots of the MS distribution by year are presented. It is apparent by inspection of both figures that the athlete's performance was maintained at a consistently high level over the 7-year period of study.
A moderate, highly significant, positive correlation was observed between MS and p(t) (p < 0.001), whereas a weak negative correlation was observed between MS and ATL (p < 0.05; Table 1). No linear correlation was observed between MS and CTL or TSS. As might be expected, because TSS is the input parameter of the IR model used for this study, p(t) was negatively correlated to TSS and ATL (p < 0.01; Table 1). It is interesting, however, that p(t) was also negatively correlated, albeit weakly, to CTL, the presumed positive “fitness” component of the model (p < 0.01; Table 1).
Model Component Curve Fits
A significant quadratic relationship (Figure 5) between MS and p(t) was observed that was slightly stronger than the correlation presented in Table 1 (r = 0.486; F = 22.5, p < 0.001). Although no correlation was observed between MS and CTL, a significant quadratic relationship was also identified between these 2 parameters (r = 0.315, F = 7.61, p < 0.01).
In the current study, several relationships were observed between a simplified IR model and actual competition performances in an elite 1,500-m runner over 7 seasons. A novel finding of this study was that, although a correlation between IR model output response (p(t)) and performance (MS) was observed (Table 1), a slightly stronger quadratic relationship was identified (Figure 5) that suggested an optimal balance between ATL (fatigue) and CTL (fitness), which resulted in best performances over the course of 7 seasons. Further, although no correlation was observed between CTL and MS, a quadratic relationship was identified that indicated an optimal “fitness” level that would result in best performances. Finally, to our knowledge, we are the first to report the use of a pace-based metric (TSS) as the training impulse in running that was derived from logs of an elite athlete under normal, uncontrolled training conditions.
The training metric TSS was significantly different between years (Figure 2) but was not correlated with performance (MS; Table 1). The model parameter ATL was negatively correlated (p = 0.01), and model response output, p(t), was positively correlated (p < 0.01) with performance (MS). Similarly, although CTL was not linearly correlated with performance (r = 0.05, p = 0.5), a significant quadratic relationship was observed (r = 0.3, p < 0.01) (Figure 6). This is an interesting observation because the CTL parameter of the model is analogous to the “fitness” component of the original Banister model (11,13). The fitness component of the Banister IR model has recently been shown to correlate with both running speed at ventilatory threshold (VTRS) and running economy (RE) in another highly trained middle-distance runner (40). Because 2 important aspects of success in distance running are the LT (as approximated by the VTRS) and RE, the fitness component of the model would appear to capture these attributes effectively (5,16,24).
The observation of a parabolic relationship between fitness and performance has been alluded to previously (7,25). It is self-evident that low fitness (low CTL) would not result in optimal performance. The question often posed by athletes and coaches alike is, how much training is enough, or how much is too much? It may seem counterintuitive that, because “fitness” is related to VTRS and RE (40), at some point high fitness may become counterproductive to performance. Busso et al. (7), however, observed an “inverted U” relationship between training dose and performance, which is reminiscent of the relationship between CTL and performance in the current study. However, the training dose in the Busso (7) study is analogous to TSS in the current study, which exhibited no relationship to performance. Therefore, it is difficult to draw a direct comparison between the inverted U relationship observed by Busso and the quadratic relationship observed in the current study.
The relationship between CTL and MS may be related to the nature of the effort of specialty for this subject in the current study (i.e., aerobic vs. anaerobic). Although the 1,500 m is considered primarily an aerobic event, it has been estimated that national-level and world record performances in the 1,500 m derive approximately 22 to 24% of energy from anaerobic sources, respectively (9). Anaerobic performance is often at odds with aerobic performance (14,19,39), and it may be that the training load required to acquire a high fitness level (CTL) is detrimental to anaerobic capability required for optimal performance in an event such as the 1,500 m. Alternatively, the quadratic relationship between CTL and MS, resulting in declining performance at high CTL values, could simply be reflective of the proposed phenomenon of “overtraining” reported in elite athletes (25,26). Regardless, we are unaware of previous reports of such a relationship between fitness and performance in the literature. This could be of valuable insight for middle-distance athletes but remains to be determined if such a relationship exists for longer distance athletes. It should be noted that the Banister IR model containing a “fitness” component has been used to model performance in anaerobic athletes, such as hammer throwers and weight lifters, and the model accounted for performance quite well (8,11). So, it is unclear if this quadratic relationship between fitness and performance would exist for these athletes.
Although the relationship between p(t) and MS was not strong as estimated by r2, that may in part depend on the training status of the athlete(s) under study. For example, as with a number of other IR modeling studies in the literature, Busso (7) used previously untrained individuals as subjects and observed substantial improvements in performance (∼30%) and, as such, very strong correlations between model outputs and actual performance (adjusted r2 = 0.93-0.96). As is the case with other studies performed using highly trained/elite athletes, performance differences for the individual in the current study were slight. Despite the fact that the athlete suffered a major injury in 2002, which hampered training and competition in both the 2002 and 2003 seasons, performances were not significantly different between any of the years (Figure 3). Of note, in 2003, although the mean performance (MS) for this athlete was the lowest of the years under study, his “poorest” performances of that year were relatively high compared to other years (Figure 4). A possible explanation for this phenomenon is that because the athlete was injured, he may have avoided competitions where he would have performed poorly so as to avoid exacerbating the injury. This approach, however, would skew the assumed relationship between model parameters because performance measures were not collected when the athlete was in a relatively low trained state. Regardless, this athlete maintained a consistently high level of performance and, as a result, is less likely to obtain high correlations between model outputs and performance, regardless of the IR modeling approach used (i.e., complex or simplified). For example, in a recent study in elite swimmers (22), despite the use of 2 more complex modeling approaches to optimize model fits to actual performances, linear correlations ranged from r2 = 0.23 to 0.49 using the Bannister IR model and r2 = 0.21 to 0.51 using a modified IR model approach in the 7 subjects over several years. The subjects in the Hellard et al. (22) study exhibited a 0.26 to 1.56% improvement in best performance over the course of 4 years. Therefore, the comparably consistent level of performance over 7 years in the current study is likely, in part, responsible for the moderate fit of p(t) with MS. Additionally, the field (as opposed to laboratory) nature of data collection compared to studies such as Busso's may also be a contributory factor to the moderate relationship between p(t) and MS (7).
The practical importance of the relationship between CTL, p(t), and performance for this particular athlete can be seen when comparing plots of model parameters for 2 Olympic seasons (2000 and 2004) (Figure 7). In 2000 the best performance was exhibited in June (MS = 966) when CTL = 50 and p(t) = +11 (red square). Later in the year at the Olympic finals (MS = 923), CTL had declined to 40, whereas p(t) was +10 (red circle). So, p(t) was comparable at both time points, but during the Olympics the positive p(t) came at the expense of fitness (CTL). In contrast, in June 2004 performance was sufficient to qualify for the Olympics (MS = 913), CTL = 53, and p(t) = +10, but later in the season at the Olympic semifinals, performance improved (MS = 934; season best), with a CTL = 47 and a p(t) = +8. It can be seen by inspection from Figure 6 the quadratic relationship between CTL and MS results in a steep drop in performance as CTL declines and approaches 40. To further investigate this relationship, when performances were analyzed on a percentile basis, the top 10th percentile of performances occurred with a CTL between 50 and 56, and the top 25th percentile occurred at a CTL between 39 and 63 (data not shown). This analysis was extended to determine, from a practical standpoint, at what model output, p(t), the best performances occurred. In this case, the top 10th percentile of performances occurred with a p(t) between +4 and +9, whereas the top 25th percentile of performances occurred with a p(t) between +2 and +12 (data not shown). Taking this into account when viewing the model plots (Figure 7), the grey shading covers the area of optimal CTL based on the top 10th percentile of performances. As can be seen in 2000 (Figure 7a), the CTL drops below the shaded area well before the Olympic performances (red circle). However, in 2004 (Figure 7B), the CTL just drops below the shaded area prior to the Olympics, indicating that the athlete, presumably, had a higher level of fitness at the 2004 Olympics. Therefore, it might be argued that the athlete “tapered” too much in the months leading up to the 2000 Olympics. Again, the positive p(t) coincided with a relatively lower CTL and compromised fitness. If a concerted effort had been made to maintain CTL above 40 in the weeks prior to the Olympics in that year, given a proper short-term taper (31-33), performance may have been sustained to the level observed earlier in the season when CTL was higher. As can be seen from Figure 7A, a number of performances prior to the Olympics in 2000 were on par with, or better than (red square), the Olympic performances in the time frame until CTL dropped below the shaded area (CTL = 50). In fact, the season personal best, and national record, occurred earlier in the season (red square), where CTL was higher than during the Olympics. During 2004, although performances overall were not as high as in 2000, the best performance of the season was observed at the Olympics, where CTL was maintained at a higher level than in 2000. Because it would be expected that the high level of competition and motivation that would come at an event of the priority of the Olympics would serve as a strong stimulus for optimal performance, it is even more striking that the best performances came earlier in the season in 2000, lending credence to the importance of a higher CTL for this athlete to perform well. A combination of higher “fitness” in conjunction with a slightly positive p(t) during priority competitions may be the best physiological preparation as other factors (e.g., tactical, emotional, psychological, seeding, caliber of competition, etc.) play a major factor in performance success also (5,23,35,36).
This work has practical application in that it presents an approach to modeling performance in middle-distance running that can be performed using readily available tools (e.g., spreadsheet and performance tables) and does not require advanced mathematical skills/knowledge or resources (e.g., Matlab computer program). Using this approach we showed that the output response of this simplified IR model correlated with actual competition performances in the absence of formalized testing. Therefore, this simplified IR model may be utilized by coaches without the need for high-level mathematical tools and/or formalized testing that may be disruptive to training and competition schedules. Further, for this specific athlete, we were able to determine the optimal range of fitness (CTL) and p(t) that would result in best performances. Although practitioners using this method may not be able to obtain definitive answers in the short term (e.g., 1 season) regarding optimal training load for best performances, the modeling approach demonstrated here may help provide context for the constructs of “fitness” and performance potential as they relate to modulation of training load. Therefore, in some cases, with accumulated data over the course of several years, more concrete guidelines (e.g., CTL and p(t) range) for best performances may be provided. Practitioners should be cognizant, however, that results of this modeling approach are not strong predictors of performance and should be used as a tool in conjunction with the coach's and/or athlete's intuition and experience regarding training approaches and good performance.
The authors wish to thank the subject of the study for diligently recording his training logs and graciously providing the information for this study. Additionally, we would like to thank Steven Lawrence for assistance in the initial development of the training quantification approach. Finally, we would like to acknowledge Erik Bollt, PhD, for providing mathematical insight and Andrew Coggan, PhD, for his insights and discussions regarding the performance modeling approach.
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Keywords:© 2009 National Strength and Conditioning Association
middle distance running; impulse response; TRIMPS