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J Strength Cond Res. 2005 February; 19(1): 67–75.

doi: 10.1519/14853.1

PMCID: PMC5352828

INSERM Subrepository

Philippe Hellard,^{1} Marta Avalos,^{2,}^{*} Grégoire Millet,^{3} Lucien Lacoste,^{1} Frédéric Barale,^{1} and Jean-Claude Chatard^{4}

The aim of this study was to model the residual effects of training on the swimming performance and to compare a model including threshold saturation (MM) to the Banister model (BM). Seven Olympic swimmers were studied over a period of 4 ± 2 years. For three training loads (low-intensity *w ^{LIT}*, high-intensity

The training-performance relationship is an important issue for elite sports coaches in search of reproducible indicators useful for organizing the athlete’s training program. Many authors have studied the relative influence of training (22, 27) and found that reactions to training depend on volume, intensity and frequency of the training sessions. Others have reported divergent results (4, 9), perhaps related to the fact that residual effects were not taken into account (3, 5, 8, 9). These residual effects are defined both in terms of the retention of physical changes following a summation of many training sessions (delayed effects) and in terms of the results of a summation of many training sessions (accumulative effects) (9).

The model described by Banister, (4, 5) and its variations (6, 7, 8) have commonly been used to describe the dynamics of training. This model is based on two antagonistic functions, both calculated from the training impulse (4, 5). Studies on cellular adaptability reactions to exercise (4) have demonstrated that the negative function can be taken to be the complete set of fatigue reactions caused by training. The positive function can be compared to a fitness gain resulting from the organism’s adaptation to training (4, 5, 8, 23). Expressed as an exponential, the functions account for the decreasing impact of the training effect. When iterative training sessions are considered, the time course of performance is described by:

$${p}_{t}={p}_{0}+{k}_{a}\sum _{s=0}^{t-1}{e}^{-(t-s)/{\tau}_{a}}{w}_{s}-{k}_{f}\sum _{s=0}^{t-1}{e}^{-(t-s)/{\tau}_{f}}{w}_{s}$$

where *p _{t}* is the known performance at week (or day)

In the initial Banister model (BM), the training load was quantified as the product of training quantity (distance or duration) x training intensity, measured by heart rate (5), oxygen consumption (6), or lactate concentration (23). However, since swimmers train with a wide range of different exercises (low intensity, high intensity, strength training), the immediate and long-term training effects cannot be grouped only in a single regimen. A new approach, taking into account the residual effects of the various types of training loads, would be preferable (4, 23).

In the classical linear periodized model, a distinction has always been made between volume training - which aims to develop aerobic capacities - and intensity training, designed to develop qualities specifically linked to performance such as anaerobic capacities related to efficient technique (2, 3, 9, 13, 14, 22). For example, it is recommended to engage in volume training in the early part of the season (2, 9, 13, 14, 27), and to increase high intensity specific training as the season progresses and during taper phases (5, 16, 23). This model was suggested in order to prevent overtraining and to peak physical performance for major competitions (2, 9, 11, 13). Such a schedule is also based on the assumption that the different physiological systems vary in the retention and rate of loss or gain of trainedness (9, 13, 14, 24). Nevertheless, the impacts of the various types of training loads on performance have an upper limit above which training does not elicit further adaptation of the subjects (12, 19, 21). Mader (19) described the balance between protein synthesis and degradation as a function of protein degradation rate by a transcription-translation activation control loop. Steady-state protein balance and active adaptation vary according to the level of functional activity induced by the training load. If the training stimulus is too intense, protein degradation exceeds synthesis, leading to catabolic processes, excessive and damaging immune system response, chronic tissue disruption, and subsequent muscular atrophy and degradation of physical capacities (5, 12, 19). Other observations have emphasized the importance of maintaining the intensity and duration of the training stimulus below a threshold limit in order to obtain an optimal development of physical capacities (8, 20, 21). It is noteworthy that Busso (8) suggested recently that the relationship between daily amounts of training and performance may be stronger if defined by a parabolic relationship. Such a relationship would mean that when the amount of training exceeds the optimal level, performance could decline because of the fatigue induced by over-solicitation (8, 19).

The two hypotheses tested in the present study were 1) volume training has a long-term, whereas intensity training has a short-term, positive effect on performance, and 2) the impact of training on performance is non-linear and has an upper limit (for BM this impact is linear: *k _{a} w_{s}* and

To investigate the hypothesis of this study, a modeling *post facto* longitudinal research design was applied. Indeed, this study was the first quantifying training loads during an Olympic cycle in finalist and medalist Olympic swimmers. These high-performance athletes require personalized intensity, frequency, and duration of taper of training adapted to their intensity responses, when preparing for events such as the Olympic Games (3, 22). Because of these different responses, the design of studies may be problematic since the variables, including amount of exercise per training period, training format, taper pattern and rest periods, have to be individually tailored. Therefore when experimental design is difficult, modeling approach provide an attractive solution (3). In the first part of the study, in order to determinate the residual effect of training, multiple regression analysis was computed between performance (output variable) and the training variables (input variables) for three training phases: short-term (*STE*), i.e. three weeks before the performance (weeks 0, −1, −2), intermediate-term (*ITE*) (weeks −3, −4 and −5 before the performance) and long-term (*LTE*) (weeks −6, −7, −8). In the second part of the study, a modified model (MM), including a saturation threshold above which training did not elicit subjects adaptations to BM was tested.

The training characteristics and performances of 4 female and 3 male elite swimmers were analyzed over a period of 4 ± 2 years (mean ± SD). Their mean age was 19.3 ± 2.3 years, mean body weight 60 ± 3 kg, and mean height 169 ± 3 cm for females, and 20.2 ± 2.9 years, 74 ± 4 kg, 185 ± 4 cm for males at the beginning of the study. The height and the weight of the swimmers remained stable throughout the entire duration of the study, signifying the absence of the pubertal maturing process. Subject #1 was an Olympic medal winner, subjects #2, 3, 6 and 7 were Olympic finalists, and subjects #4 and 5 were European Junior level swimmers. The study was reviewed and approved by the local University Committee on Human Research and written informed consent was obtained from each participant. Each swimmer trained according to the program prescribed by their coaches, and the characteristics of the training regimens or competition schedules were not modified by the present study.

Intensity levels for swim workouts were determined as proposed by Mujika *et al.* (23). An incremental test to exhaustion was performed at the beginning of each season to determine the relationship between blood lactate concentration and swimming speed. Each subject swam 6 x 200-m at progressively higher percentages of their best personal competition time over this distance, until exhaustion. Blood lactate concentration was measured in blood samples collected from the fingertip during 1-min recovery periods separating the 200-m swims. All swimming sessions were divided into five intensity levels according to the individual results obtained during this test. Intensities *I1*, *I2*, and *I3* represented swimming speeds below (≈2 mmol· l^{−1}), at (≈4 mmol· l^{−1}) and just above (≈6 mmol· l^{−1}) the onset of blood lactate accumulation, respectively. High swimming work producing blood lactate concentrations of ≈10 mmol· l^{−1} was defined as intensity *I4*, and maximal swimming work as intensity *I5* (23). Training was quantified in m covered in each intensity zone. The measurements were repeated four times per season, and training intensity was adjusted to the swimmer’s response to training (25).

The subjects participated in a supervised strength-training program, with a training frequency of 4 sessions per week during the volume phase, 3 sessions per week during the intensity phase and 2 sessions per week during the taper phase. Strength training (*I6*) included dry land workouts, which involved various strength exercises. After a standardized 20-minute warm up, each training session included 1 exercise for the leg extensor muscles (bilateral knee extension exercises), 1 exercise for the arm extensor muscles (bench press) and 5 exercises for the main muscle groups of the body (chest press, shoulder press, isokinetic swim bench, surgical tubing, for the upper body; abdominal crunch for the trunk extensors). Each exercise was performed at 50–60% of a single maximal repetition (1RM) at the stroke rate corresponding to the specific swimming stroke rate. During the volume phase the subjects performed 20–40 repetitions per set and 2–3 sets of each exercise. During the intensity phase the number of sets was reduced and the subjects were required to complete as many repetitions as possible at the stroke rate corresponding to the specific swimming stroke rate. Finally, during the taper phase, the number of repetitions was reduced to 8–16, and the subjects had to maintaining the specific swimming stroke rate but to performed each repetition as rapidly as possible.

Strength training was quantified in minutes of active exercise excluding resting periods. As each swimmer’s stroke rate remained more or less stable during the course of the study, the method used to quantify strength training in terms of time spent (in minutes) appears to gage total volume correctly.

Quantification of the training stimulus was performed as proposed by Avalos *et al.* (3). The weekly amount of training in each training zone was notated as *V _{i, t}* where

In the first part of the study (analysis of the residual effects of training), three weekly training loads were determined according to three training zones. Low intensity training load
${w}_{t}^{\mathit{LIT}}$ was the mean of the *x _{1,t}*,

For each swimmer, performances were measured during real competitions for the same event, and during the entire study period. Performance at time *t* designated *p _{t}* was expressed as a percentage of the personal record of each swimmer.

All values were reported as mean ± SD. For all variables the hypothesis of a normal distribution was tested (*P* <0.05) with the Shapiro Wilk test, for small samples (performances), and the Kolmogorov test, for large samples (training loads) (26). The Bartlett test was used to control performances unequal variances (26). If heteroscedasticity or an abnormal distribution were observed, logarithmic (natural) transformation of the data was performed. Statistical significance was accepted as less than or equal to the type I error rate of 0.05.

A linear model of periodization characterized the training cycles (2, 11, 13): each training cycle, lasting between 8 and 12 weeks, commenced with high training volume and low intensity. As training progressed, volume decreased and intensity increased. For the whole group, 3 training phases were identified during each training cycle. The last three weeks prior the competitive period (weeks 0, −1, −2) defined the taper phase. The intensity phase was defined as weeks −3, −4 and −5 prior the competitive period. Weeks −6, −7 and −8 defined the volume phase. The training effects of these three phases were defined as short-term (*STE*), intermediate-term (*ITE*) and long-term (*LTE*) effects for the taper, intensity and volume phase, respectively. Ultimately, nine distinct training variables were defined:
$\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{HIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{ST},\phantom{\rule{0.16667em}{0ex}}\mathit{ITE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{ITE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{HIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{ITE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{ST},\phantom{\rule{0.16667em}{0ex}}\mathit{LTE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{LTE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{HIT}},\phantom{\rule{0.16667em}{0ex}}\mathit{LTE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{ST}$. The content of the three phases of 137 training cycles (*w ^{LIT}*,

Respective contents of the volume, intensity and taper phases during the entire study period (137 training cycles).

To analyze the relationships between loads and performances within each training cycle, multiple regression analysis was computed between performances (output variable) and the training variables (input variables). Each training variable was transformed by a quadratic function (26) to take into account a potential parabolic relationship between the quantity of training loads and the performances.

After testing the normality and homoscedasticity of the residuals, 95% confidence intervals (CI) were calculated for regression parameters.

With BM, the training impulse effect was represented as a linear function of the amount of this impulse limit: *k _{a} w_{s}* and

$$\mathit{Hill}(w)=\kappa \frac{{w}^{\gamma}}{{\delta}^{\gamma}+{w}^{\gamma}}$$

where *κ* is the value of the saturation threshold above which training loads no longer have an effect. The parameter *γ* expresses the sensitivity to training load and controls the time to reach *κ* (the higher the value of *γ*, the shorter the delay). The parameter *δ* is the inertia of the function to the threshold value. A low value of *δ* expresses a strong effect of training load on performance. The effects of three different values of *γ* and *δ* are shown in Figure 1.

Hill function pattern for three different *γ* values when *δ* = 1 and *κ* = 10 (A); and for three different *δ* values when *κ* = 10 and *γ* = 1 (B). The saturation threshold is rapidly reached for high *γ* and **...**

The positive and negative functions for the *t*^{th} week (*ω _{p,t}* and

$${\omega}_{p,t}={\kappa}_{p}\frac{{w}_{t}^{\gamma}}{{\delta}^{\gamma}+{w}_{t}^{\gamma}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}{\omega}_{n,t}={\kappa}_{n}\frac{{w}_{t}^{\gamma}}{{\delta}^{\gamma}+{w}_{t}^{\gamma}}$$

where *κ _{n}* and

$${p}_{t}={p}_{0}+\sum _{s=0}^{t-1}{\omega}_{p,s}{e}^{-(t-s)/{\tau}_{p}}-\sum _{s=0}^{t-1}{\omega}_{n,s}{e}^{-(t-s)/{\tau}_{n}}$$

where *p _{0}* is the initial basic performance level corresponding to the genetic endowment of the subject (6) expressed in the same units as performance, as a percentage (%) of each swimmer’s personal record during the entire study period;

Model parameters were estimated for each subject using the iterative method of nonlinear least squares, by minimizing the residual sum of quadratic differences between the real and the modeled performances with a Gauss-Newton type algorithm (26). The starting values were chosen as follows: *p _{0}* = 0.95,

The determination coefficient was calculated as follows: r^{2} = 1 − (RSS / TSS), where RSS is the residual sum of squares and TSS the total sum of squares. Since increasing the number of parameters increases the determination coefficient, the adjusted determination coefficient (
${\mathrm{r}}_{\mathit{Adj}}^{2}$) was calculated as follows:
${\mathrm{r}}_{\mathit{Adj}}^{2}=1-([\text{number}\phantom{\rule{0.16667em}{0ex}}\text{of}\phantom{\rule{0.16667em}{0ex}}\text{parameters}-1]\phantom{\rule{0.16667em}{0ex}}\text{RSS}/\text{TSS})$.
${\mathrm{r}}_{\mathit{Adj}}^{2}$ takes into account the fitting gain with respect to the two parameters (*γ, δ*) added by modifying BM.

Since BM and MM were not nested, the C_{p} score was computed as a comparison criteria (10, 26, 31):

$${\mathrm{C}}_{\mathrm{p}}=[\text{RSS}/(\text{number}\phantom{\rule{0.16667em}{0ex}}\text{of}\phantom{\rule{0.16667em}{0ex}}\text{observation})+2(\text{number}\phantom{\rule{0.16667em}{0ex}}\text{of}\phantom{\rule{0.16667em}{0ex}}\text{parameters}){\widehat{\sigma}}^{2}/(\text{number}\phantom{\rule{0.16667em}{0ex}}\text{of}\phantom{\rule{0.16667em}{0ex}}\text{observations})],$$

where ^{2} is the standard unbiased estimator of the residual variance. A small value of C_{p} indicates a small prediction error (10, 26, 31).

While r^{2} is one of the most important indicators of adequacy of regression equations, a high r^{2} value is not a guarantee of accurate prediction. Several complementary measures are needed to confirm accuracy and sensitivity (31, 26). The analysis of variance applied to the residual sum of squares was not suitable to compare BM with MM since they were not nested (models are nested when the parameters of one model are a subset of the parameters of another). The calculation of C_{p} is a useful statistical method that rewards models for good fit, but imposes a penalty for unnecessary parameters (10, 26, 31). To summarize, r^{2},
${\mathrm{r}}_{\mathit{Adj}}^{2}$ measure the goodness of fit, whereas C_{p} is a measurement of prediction error and CI of the accuracy of estimation (10, 17).

The bootstrap method (10) was used to compute the limits of agreements of estimated performances. Briefly, the procedure consisted of resampling the original data set with replacement, to create a number of “bootstrap replicate” data sets of the same size as the original data set. A random number generator was used to determine which data from the original data set to include in a replicate data set. Therefore a given data could be used more than once in the replicate data set, or not at all. This was repeated 1000 times. For each performance, the estimates that fell between the 2.5^{th} and the 97.5^{th} percentiles of the 1000 estimates were used to construct a 95% CI for the performances estimated (10). The number of actual performances included into the 95% CI for the performances estimated was compared for BM and MM.

To separate the short-term negative effects of the training doses from their long-term benefit, the positive and negative effects of training on performance were estimated as previously described (8, 22). The effect on performance on week *t* attributable to the amount of training during week *s*, for both BM and MM, was quantified as: *E(s/t) = k*_{1}
*w ^{s}e*

A negative value indicated a negative, while a positive value indicated a positive effect of training on performance. Effects of training impulses at 100%, 65% and 35% of the maximal training load were compared for BM vs. MM in subjects #2 and #3.

In the whole study group, the training volume measured during a season was 1922 ± 417 km. Contents of the volume, intensity and taper phases during the entire study period (*w ^{LIT}*,

Training volume and strength training decreased between the volume and the intensity phase (*P <*0.05), whereas training volume, low intensity, high intensity and strength training decreased between the intensity and the taper phase (*P <*0.05). The percentage of high intensity training increased as the percentage of strength training decreased between the intensity and the taper phase (*P <*0.05). The total training load (*w _{t}*) was 34.0 ± 14.2% of the maximal training stimulus (range 0.12–85.3%) measured throughout the period studied.

During the entire study period, the mean number of performances recorded for each swimmer were 48.7 ± 9.1. For the whole group, the mean performance was 96.6 ± 1.9% (range 92.8%–100%). Best performances between the beginning and end of the study improved by 0.67% (range 0.27%–1.56%) (Table 2).

The best solution (r^{2} = 0.30, F = 8.73, *P* <0.01) for the multiple regression was:

$${p}_{t}=0.97+-0.46\ast {(\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}})}^{2}+0.28\ast (\mathit{LTE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}})+0.25\ast (\mathit{ITE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{HIT}}).$$

Only significant variables were included in the multiple regression (*P* <0.05). A better adjustment of the transformed variable
${(\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}})}^{2}$ indicated a parabolic relationship between short-term, low intensity amounts of training and performance (Figure 2). The 95% CI was [−0.26; −0.65] for
${(\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}})}^{2}$, [0.12; 0.43] for
$\mathit{LTE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}}$, and [0.10; 0.40] for
$\mathit{ITE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{HIT}}$.

Parabolic relationship between short-term low-intensity training load (
$\mathit{STE}\phantom{\rule{0.16667em}{0ex}}{w}_{t}^{\mathit{LIT}}$) and performances (*p*_{t}), for the whole group of subjects. Performance on the vertical axis is expressed as a percentage of the personal record of each subject. Training load **...**

The parameters of BM and MM are presented in Table 3. The relationships between training and performance were significant (*P <*0.01) for the two models in all subjects. The determination coefficients (r^{2}) were higher for MM than for BM: 0.42 ± 0.1 (0.30 ≤ r^{2} ≤ 0.53) in BM versus 0.52 ± 0.1 (0.32 ≤ r^{2} ≤ 0.58) in MM. With
${\mathrm{r}}_{\mathit{Adj}}^{2}$, the fit of BM:
$0.36\pm 0.1\phantom{\rule{0.16667em}{0ex}}(0.23\le {\mathrm{r}}_{\mathit{Adj}}^{2}\le 0.49)$ was slightly lower than that of MM:
$0.43\pm 0.1\phantom{\rule{0.16667em}{0ex}}(0.21\le {\mathrm{r}}_{\mathit{Adj}}^{2}\le 0.51)$.

When comparing MM with BM, C_{p} score was lower in five (#1, #3, #4, #5, #7) and higher in two (#2, #6) subjects. The fitting difference between BM and MM associated with CI for subject #5 is shown as an example in figure 3.

Modeled (line) and actual performances (square) for subject #5, calculated with MM and BM. Performance on the vertical axis is expressed as a percent of the personal record. Time on the horizontal axis is expressed in weeks.
${\mathrm{r}}_{\mathit{Adj}}^{2}$ and 95% CI for modeled **...**

The mean interval width of 95% CI in BM and MM (1.92 ± 0.42 vs. 1.91 ± 0.41%) were similar. The number of measured performances included into the 95% CI was higher in BM than in MM (25.7 ± 4.8 vs. 20.1 ± 3.1). The 95% CI for subject #5 in both models is shown in figure 3.

The effects of training impulses at 100, 65 and 35% of the maximal training load were different for BM and MM. Two practical examples in subjects #2 and #3 are displayed in figure 4A and 4B, respectively. In BM, the magnitude of the responses was proportionally related to the amount of training impulses in both subjects. In MM, in subject #2, the two training doses (100 and 65%) induced markedly different responses (0.009 and 0.007 a.u., respectively) while a positive effect of similar magnitude (0.003 a.u.) was observed for the 100 and 65% training doses in subject #3 (Figure 4B).

The two main observations emerging from these analyses were: 1) the relationship between training load and performance varied according to the training phases and training loads. The short-term effect of training was related to performance by a parabolic relationship for *w ^{LIT}*, the intermediate-term effect was positive for

One of the major limitations of this research concerns its non experimental schedule. The lack of random sampling, random assignment to groups make difficult any generalizations of these findings to other situations. Experiments are better than observational studies because there are fewer grounds for doubt. Experiments often settle questions faster. Despite this, experiments are not feasible in some settings. Furthermore, the quantification method remains overly restrictive and does not take into account all intensity types of training.

The most important short-term effect was derived from training performed below and just above the onset of blood lactate accumulation (*w ^{LIT}*), which usually accounts for the greatest proportion of training in swimmers (3, 22, 23, 27). Tapering enables swimmers to recover from fatigue accumulated during intermediate- and long-term training, while maintaining previously acquired physical adaptation (16, 22, 23). Nevertheless, the best regressor in the equation was (

The effect of *w ^{HIT}* on performance was positive. This range of intensity optimizes aerobic and anaerobic energy production (22), and improves swimming techniques (28). Several authors have emphasized the importance of this training period, during which the increase in training intensity delays the stimulation of biological adaptations via an overcompensatory process (3, 13, 20, 27).

Low-intensity training had a positive effect on performance over the long term. These results suggest that an important low intensity training volume probably efficiently develops the physiological mechanisms necessary for subsequent intensity training (9, 14, 27). Aerobic training, associated with a lactate production equal to or below the onset of blood lactate accumulation increases oxidative capacity, lowers lactate production at a given swimming speed, increases critical speed, and increases training capacity while lowering the fatigue threshold (30).

The 95% CI of the different parameters of the multiple regression between performance and training variables confirmed an accurate estimation. The practical implications of these results remain to be clarified since the training variables explained only 30% of the variations in performance, suggesting several explanations. First, the swimmer’s response to a given training volume may vary among consecutive seasons, reducing the statistical significance of the relationship between training and performance (3). Indirect effects of training may also interfere. For example, aerobic training may hasten the recovery from fatigue caused by anaerobic training (9). Variations in technique may also explain a large part of the variations in performance (29). Furthermore, swimmers react differently to the same training loads (3). Finally, during the study period, performance improved by less than 1%, suggesting that, after several years of high-level training, the performance of elite athletes reaches a plateau (14, 28). Therefore, as variations in training do not directly imply variations in performances, statistical relationships are lower.

With both models, the fit between training and performance was significant in all subjects. The determining coefficients were similar to those reported by Avalos *et al.* (3) who used a linear mixed model in 13 competitive swimmers over 3 seasons. They were, however, lower than reported in earlier studies in swimmers (22, 23), probably due to a larger number of performances for each swimmer, and a longer study duration. For small samples, the mean r^{2} value may be high despite the absence of relationship between predictor and response variable (1).
${\mathrm{r}}_{\mathit{Adj}}^{2}$ values for BM were smaller than those reported by Busso (8) (0.36 ± 0.11 vs. 0.88 ± 0.04). However, in that study, sedentary subjects were trained over 15 weeks and improved their performance by approximately 30% over the period studied, an improvement much greater than can be expected in elite athletes (1–4%) (15).

Moreover both BM and MM assume that the parameters remain constant over time, an assumption that is inconsistent with observed time-dependent alterations in responses to training (3, 5, 7, 8). Although performance is specifically and largely influenced by training, athletes also adapt to other factors whose influence may increase over time, including personal involvement, intensity swimming techniques, external factors, altitude training, and time-lag during travel (3, 22, 23).

On average, the
${\mathrm{r}}_{\mathit{Adj}}^{2}$ coefficients were slightly higher (*P* <0.05) in MM than in BM (0.42 ± 0.10 vs. 0.36 ± 0.13). This result is consistent with the adjustment increase reported by Busso (8) by comparing BM to a non-linear model that took into account the magnitude and duration of exercise-induced fatigue (0.88 ± 0.04 vs. 0.94 ± 0.01). Since the mean interval width of 95% CI was similar (~1.9 ± 0.4%) in both models, the highest number of measured performances included in the 95% CI for MM indicates a higher accuracy and best specification of the latter model (31). Thus, MM can be considered to be a complementary tool to BM for modeling the relationship between training and performance. Moreover, Olympic level of the subjects, the long duration of the study, and the use of CI contributed to the validation of BM. Indeed, from a practical point of view, the CI values were accurate (e.g. for subject #5, CI was 1 s for a 100-m event performed in 1 min 3 s).

On average, positive function decay rates were shorter in MM than in BM (43.7 ± 12.1 days vs. 49.6 ± 7.7 days) while negative function decay rates were similar (19.1 ± 8.3 days vs. 17.7 ± 7.8 days). The positive and negative function values in BM and MM were close to those reported previously (4, 5, 7, 22). The fitness and fatigue acquisition coefficients *k _{a,} k_{f}* in BM were lower than the positive and negative saturation thresholds

Summarizing, in this study of elite swimmers, low intensity training was related with short-term performance by a parabolic relationship, whereas its long-term effect during the volume phase was positive. These results underscore the positive effects of low-intensity training during the volume phase and suggest this type of training should be maintained around 40–50% of the individual maximal values during the taper phase. Moreover, MM may better take into account, for each swimmer, the limit above which training does not elicit subjects’ adaptations and the delay in reaching this limit. For example, in subject #2, who was a 200-m butterfly, Olympic finalist responded quite differently to the two training doses (100 and 65%) (Figure 4A), and may respond well to high training loads. Conversely, in subject #3, who was a 200-m freestyle Olympic finalist in Atlanta 1996 and Sydney 2000, training impulses of 100 and 65% determined similar response patterns (Figure 4B), suggesting a poor response to high training loads. These results are consistent with those reported by Avalos *et al.* (3) who used a linear mixed model. Subject #3 was included in the group of swimmers responding poorly to short- or mid-term high training loads. Conversely, subject #2 was included in the group of swimmers responding well to short term high training loads (3). Finally, adaptation to training is known to be a highly individual phenomenon (3, 23). Thus, these results suggest that training programs must be highly personalized, adapted to each individual swimmer’s profile.

The present MM could be further refined. First, the change in *p _{0}* could be integrated into the modeling process, for example at the beginning of each season, since

The results of this study need to be generalized. A batch of experimental studies based on longitudinal research design would make it possible to compare several training programs over an entire training cycle. Considering the popularity of periodized training, there are surprisingly few studies examining the effectiveness of periodized training loads. The first step would be to compare linear periodization with a non periodized training involving constant volume and intensity training loads. The second step would be to investigate any residual effects and the threshold effects of training by comparing progressive multiple training load programs. For instance, it would be highly instructive to compare the effects of 2, 3 and 4 high training loads sessions per week throughout the intensity phase (weeks −3, −4, −5). By the same token, it would also be appropriate to study the optimal training volume during the volume phase (weeks −6, −7, −8). Finally, studies examining periodized models other than the traditional volume/intensity periodized model are also needed.

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