Rationale and resources for teaching the mathematical modeling of athletic training and performance

Short-duration aerobic high-intensity intervals versus moderate exercise training intensity in patients with peripheral artery disease: study protocol for a randomised controlled trial (the Angiof-HIIT Study)

INTRODUCTION

Supervised exercise training is among the first-line therapies for patients with peripheral artery disease (PAD). Current recommendations for exercise include guidance focusing on claudication pain, programme and session duration, and frequency. However, no guidance is offered regarding exercise training intensity. This study aims to compare the effects of 12-week-long supervised walking exercise training (high-intensity interval training (HIIT) vs moderate-intensity exercise (MOD)) in patients with chronic symptomatic PAD.

METHODS AND ANALYSIS

This study is a monocentric, interventional, non-blinded randomised controlled trial. 60 patients (30 in each group) will be randomly allocated (by using the random permuted blocks) to 12 weeks (three times a week) of HIIT or MOD. For HIIT, exercise sessions will consist of alternating brief high-intensity (≥85% of the peak heart rate (HRpeak)) periods (≤60 s) of work with periods of passive rest. Patients will be asked to complete 1 and then 2 sets of 5-7 (progressing to 10-15×60 s) walking intervals. For the MOD group, exercise training sessions will consist of an alternation of periods of work performed at moderate intensity (≤76% HRpeak) and periods of passive rest. Interventions will be matched by training load. The primary outcome will be the maximal walking distance. Secondary outcomes will include functional performance, functional capacity, heath-related quality of life, self-perceived walking abilities, physical activity and haemodynamic parameters.

ETHICS AND DISSEMINATION

The Angiof-HIIT Study was approved by the Human Research Ethics Committee of the Canton de Vaud (study number: 2022-01752). Written consent is mandatory prior to enrolment and randomisation. The results will be disseminated via national and international scientific meetings, scientific peer-reviewed journals and social media.

TRIAL REGISTRATION NUMBER

NCT05612945.

Development of Cycling Performance Variables and Durability in Female and Male National Team Cyclists: From Junior to Senior

AIM

This study investigated the development of power profiles and performance-related measures from the junior level (<19 yr) via U23 (19-23 yr) to senior level (>23 yr) in 19 female and 100 male Norwegian national team cyclists.

METHODS

A total of 285 tests were performed in a 3-d laboratory-standardized testing regime. The tests included power profiles with shorter duration (6-60 s) and longer durations (12-30 min) together with performance-related measures: critical power (CP), work capacity above CP (W'), power output at 4 and 2 mmol·L -1 [BLa - ] (L 4 and L 2 ), maximal aerobic power (W max ), and maximal oxygen uptake (V̇O 2max ), gross efficiency (GE), and pedaling efficiency.

RESULTS

Females and males evolve similarly when maturing from junior via U23 to senior categories (all P > 0.07), except for V̇O 2max , which increased in females (but not males) from junior to senior level (534 ± 436 mL·min -1 , P = 0.013). In general, only performances of longer durations improved with age (12 and 30 min, P = 0.028 and P = 0.042, respectively). Performance-related measures like W max , V̇O 2max , CP, L 4 , L 2 , and pedaling efficiency in the fresh state improved with age (all P ≤ 0.025). Importantly, performance in the semifatigued state during a 5-min maximal test was also improved with age ( P = 0.045) despite a higher external energy expenditure before the test ( P = 0.026).

CONCLUSIONS

Junior cyclists show highly developed sprint abilities, and the primary improvements of absolute power outputs and performance-related measures are seen for durations >60 s when maturing to U23 and senior categories. However, the durability, i.e., the capacity to maintain performance in a semifatigued state, is improved with age.

Validity and Accuracy of Impulse-Response Models for Modeling and Predicting Training Effects on Performance of Swimmers

PURPOSE

The aim of this study was to compare the suitability of models for practical applications in training planning.

METHODS

We tested six impulse-response models, including Banister's model (Model Ba), a variable dose-response model (Model Bu), and indirect-response models differing in the way they account or not for the effect of previous training on the ability to respond effectively to a given session. Data from 11 swimmers were collected during 61 wk across two competitive seasons. Daily training load was calculated from the number of pool-kilometers and dry land workout equivalents, weighted according to intensity. Performance was determined from 50-m trials done during training sessions twice a week. Models were ranked on the base of Aikaike's information criterion along with measures of goodness of fit.

RESULTS

Models Ba and Bu gave the greatest Akaike weights, 0.339 ± 0.254 and 0.360 ± 0.296, respectively. Their estimates were used to determine the evolution of performance over time after a training session and the optimal characteristics of taper. The data of the first 20 wk were used to train these two models and predict performance for the after 8 wk (validation data set 1) and for the following season (validation data set 2). The mean absolute percentage error between real and predicted performance using Model Ba was 2.02% ± 0.65% and 2.69% ± 1.23% for validation data sets 1 and 2, respectively, and 2.17% ± 0.65% and 2.56% ± 0.79% with Model Bu.

CONCLUSIONS

The findings showed that although the two top-ranked models gave relevant approximations of the relationship between training and performance, their ability to predict future performance from past data was not satisfactory for individual training planning.

Case study of IMU loads and self-reported fatigue monitoring of water polo goalkeepers preparing for the Olympic games

Introduction

Measurement of training in water polo goalkeepers has focused first on psycho-physiological variables, but also on external volume estimated with wearable sensors. However, there are limited studies exploring training monitoring in water polo goalkeepers longitudinally.

Methods

Three female senior national team goalkeepers participated in this study from May to August 2021. Internal loads were defined using session rating of perceived exertion (sRPE). Tri-axial accelerations and angular velocities were measured with an inertial measurement unit (IMU) placed on the lower back to measure external loads. Relationships between self-reported and IMU-derived metrics were explored using Spearman correlations. Two-way ANOVAs were used to assess differences between session types and between athletes.

Results

In total, 247 sessions were collected (159 practices, 67 matches and 21 game warm up), with 155 sessions having complete data. IMU metrics, such as number of kicks, number of jumps or player-load showed high correlation with each other (ρ = 0.80-0.88). There was also a moderate correlation (ρ = 0.47, 95% CI = 0.33-0.58) between sRPE and player-load measured with the IMU. ANOVA tests showed that there were significant differences between athletes for sRPE (p < 0.01) but not for player load (p = 0.47). There were no interactions between athletes and training types, except for index score (p < 0.01).

Conclusions

This study shows that monitoring of training loads can be performed successfully in water polo goalkeepers using a combination of self-reported and IMU measures. Self-reported outcomes can be expected to vary significantly across athletes within the same session, while IMU metrics vary across training situations. Finally, coaches should be mindful of missing data, as they can skew the interpretation of training loads.

High-intensity interval training: optimizing oxygen consumption and time to exhaustion taking advantage of the exponential reconstitution behaviour of D'

PURPOSE

Accumulating the time near maximum aerobic power [Formula: see text] is considered to be the most effective way to improve aerobic capacity. The aims of this study were: (1) to verify whether postponing the first recovery interval improves time to exhaustion during a high-intensity interval training (HIIT) test, and (2) to verify whether a HIIT protocol with decreasing interval duration (HIDIT) is more effective in accumulating time near [Formula: see text] compared with two classical protocols with short intervals (SI_HIIT) and long intervals (LI_HIIT).

METHODS

Nine active males (35 ± 11 years, [Formula: see text] 52 ± 5 mL·min^-1·kg^-1) performed a graded exercise test on an athletic track. Critical velocity and D' were estimated from three to five high-intensity trials to exhaustion. Then, the subjects performed three trials with a single recovery interval after 30 s (Rec_30s), after 3 min (Rec_3min) and after exhaustion (Rec_Tlim) to verify whether postponing the first recovery interval enhances the time to exhaustion. Finally, the subjects performed the three HIIT protocols mentioned above.

RESULTS

The time to exhaustion was significantly greater in Rec_Tlim (464 ± 67 s) than in Rec_3min (388 ± 48 s) (p < 0.0078) and Rec_30s (308 ± 44 s) (p > 0.0001). Additionally, it was significantly greater in Rec_3min than in Rec_30s (p = 0.0247). Furthermore, the time accumulated near [Formula: see text] was significantly longer in HIDIT (998 ± 129 s) than in SI_HIIT (678 ± 116 s) (p = 0.003) and LI_HIIT (673 ± 115 s) (p < 0.031).

CONCLUSIONS

During the trials, postponing the first recovery interval was effective in improving the time to exhaustion. Moreover, HIDIT was effective in prolonging the time near [Formula: see text].

The Fitness-Fatigue Model: What's in the Numbers?

PURPOSE

The purpose of this commentary is to outline some of the pitfalls when using the fitness-fatigue model to unravel the interaction between training load and performance. By doing so, we encourage sport scientists and coaches to interpret the parameters from the model with some extra caution.

CONCLUSIONS

Caution is needed when interpreting the fitness-fatigue model since the parameter values are influenced by the starting parameter values, the modeling technique, and the input of the model. Also, the use of general constants should be avoided since they do not account for interindividual differences and differences between training-load methods. Therefore, we advise sport scientists and coaches to use the model as a way to work more data-informed rather than working data-driven.

A Fitness-Fatigue Model of Performance in Peripheral Artery Disease: Predicted and Measured Effects of a Pain-Free Exercise Program

Banister impulse-response (IR) model estimates the performance in response to the training impulses (TRIMPs). In 100 patients with peripheral artery disease (PAD), we tested by an IR model the predictability of the effects of a 6-month structured home-based exercise program. The daily TRIMPs obtained from prescribed walking speed, relative intensity and time of exercise determined the fitness-fatigue components of performance. The estimated performance values, calculated from the baseline 6-min and pain-free walking distance (6MWD and PFWD, respectively) were compared with values measured at visits through regression models. Interval pain-free walking at controlled speed prescribed during circa-monthly hospital visits (5 ± 1) was safely performed at home with good adherence (92% of scheduled sessions, 144 ± 25 km walked in 50 ± 8 training hours). The mean TRIMP rose throughout the program from 276 to 601 a.u. The measured 6MWD and PFWD values increased (+33 m and +121 m, respectively) showing a good fit with those estimated by the IR model (6MWD: R² 0.81; PFWD: R² 0.68) and very good correspondence (correlation coefficients: 0.91 to 0.95), without sex differences. The decay of performance without training was estimated at 18 ± 3 weeks. In PAD, an IR model predicted the walking performance following a pain-free exercise program. IR models may contribute to design and verify personalized training programs.

Training Progression in Recreational Cyclists: No Linear Dose-Response Relationship With Training Load

ABSTRACT

Vermeire, KM, Vandewiele, G, Caen, K, Lievens, M, Bourgois, JG, and Boone, J. Training progression in recreational cyclists: no linear dose-response relationship with training load. J Strength Cond Res 35(12): 3500-3505, 2021-The purpose of the study was to assess the relationship between training load (TL) and performance improvement in a homogeneous group of recreational cyclists, training with a self-oriented training plan. Training data from 11 recreational cyclists were collected over a 12-week period. Before and after the training period, subjects underwent a laboratory incremental exercise test with blood lactate measurements to determine the power output associated with the aerobic threshold (PAT) and the anaerobic threshold (PANT), and the maximal power output (PMAX) was also determined. Mean weekly TL (calculated using the training impulse (TRIMP) of Banister, Edwards TRIMP, Lucia TRIMP and the individualized TRIMP) were correlated to the progression in fitness parameters using Pearson Correlation. Training intensity distribution (TID) was also determined (% in zone 1 as ANT). No significant correlations between mean weekly TRIMP values and the improvement on PMAX (r = -0.22 to 0.08), PANT (r = -0.56 to -0.31) and PAT (r = -0.08 to 0.41) were found. The TID was significant in a multiple regression with PANT as dependent variable (y = 0.0088 + 0.1094 × Z1 - 0.2704 × Z2 + 1.0416 × Z3; p = 0.02; R2 = 0.62). In conclusion, this study shows that the commonly used TRIMP methods to quantify TL do not show a linear dose-response relationship with performance improvement in recreational cyclists. Furthermore, the study shows that TID might be a key factor to establish a relationship with performance improvement.

Interaction of exercise bioenergetics with pacing behavior predicts track distance running performance

The best possible finishing time for a runner competing in distance track events can be estimated from their critical speed (CS) and the finite amount of energy that can be expended above CS (D´). During tactical races with variable pacing, the runner with the "best" combination of CS and D´ and, therefore, the fastest estimated finishing time prior to the race, does not always win. We hypothesized that final race finishing positions depend on the relationships between the pacing strategies used, the athletes' initial CS, and their instantaneous D´ (i.e., D´ balance) as the race unfolds. Using publicly available data from the 2017 International Association of Athletics Federations (IAAF) World Championships men's 5,000-m and 10,000-m races, race speed, CS, and D´ balance were calculated. The correlation between D´ balance and actual finishing positions was nonsignificant using start-line values but improved to R² > 0.90 as both races progressed. The D´ balance with 400 m remaining was strongly associated with both final 400-m split time and proximity to the winner. Athletes who exhausted their D´ were unable to hold pace with the leaders, whereas a high D´ remaining enabled a fast final 400 m and a high finishing position. The D´ balance model was able to accurately predict finishing positions in both a "slow" 5,000-m and a "fast" 10,000-m race. These results indicate that although CS and D´ can characterize an athlete's performance capabilities prior to the start, the pacing strategy that optimizes D´ utilization significantly impacts the final race outcome.NEW & NOTEWORTHY We show that the interaction between exercise bioenergetics and real-time pacing strategy predicts track distance running performance. Critical speed (CS) and the finite energy expended above CS (D´) can characterize an athlete's capabilities prior to the race start, but the pacing strategy that optimizes D´ utilization ultimately impacts whether a runner is in contention to win and whether a runner will have a fast final 400 m. Accordingly, D´ balance predicts final race finishing order.

The W' Balance Model: Mathematical and Methodological Considerations

Since its publication in 2012, the W' balance model has become an important tool in the scientific armamentarium for understanding and predicting human physiology and performance during high-intensity intermittent exercise. Indeed, publications featuring the model are accumulating, and it has been adapted for popular use both in desktop computer software and on wrist-worn devices. Despite the model's intuitive appeal, it has achieved mixed results thus far, in part due to a lack of clarity in its basis and calculation. Purpose: This review examines the theoretical basis, assumptions, calculation methods, and the strengths and limitations of the integral and differential forms of the W' balance model. In particular, the authors emphasize that the formulations are based on distinct assumptions about the depletion and reconstitution of W' during intermittent exercise; understanding the distinctions between the 2 forms will enable practitioners to correctly implement the models and interpret their results. The authors then discuss foundational issues affecting the validity and utility of the model, followed by evaluating potential modifications and suggesting avenues for further research. Conclusions: The W' balance model has served as a valuable conceptual and computational tool. Improved versions may better predict performance and further advance the physiology of high-intensity intermittent exercise.

Development of a Revised Conceptual Framework of Physical Training for Use in Research and Practice

A conceptual framework has a central role in the scientific process. Its purpose is to synthesize evidence, assist in understanding phenomena, inform future research and act as a reference operational guide in practical settings. We propose an updated conceptual framework intended to facilitate the validation and interpretation of physical training measures. This revised conceptual framework was constructed through a process of qualitative analysis involving a synthesis of the literature, analysis and integration with existing frameworks (Banister and PerPot models). We identified, expanded, and integrated four constructs that are important in the conceptualization of the process and outcomes of physical training. These are: (1) formal introduction of a new measurable component 'training effects', a higher-order construct resulting from the combined effect of four possible responses (acute and chronic, positive and negative); (2) explanation, clarification and examples of training effect measures such as performance, physiological, subjective and other measures (cognitive, biomechanical, etc.); (3) integration of the sport performance outcome continuum (from performance improvements to overtraining); (4) extension and definition of the network of linkages (uni and bidirectional) between individual and contextual factors and other constructs. Additionally, we provided constitutive and operational definitions, and examples of theoretical and practical applications of the framework. These include validation and conceptualization of constructs (e.g., performance readiness), and understanding of higher-order constructs, such as training tolerance, when monitoring training to adapt it to individual responses and effects. This proposed conceptual framework provides an overarching model that may help understand and guide the development, validation, implementation and interpretation of measures used for athlete monitoring.

One-Week High-Dose β-Alanine Loading Improves World Tour Cyclists' Time-Trial Performance

Supplementation with β-alanine is becoming a common practice in high-performance athletes. The purpose of the present study was to investigate the effects of a one-week high-dose β-alanine loading phase employing a sustained-release powder on preserving the time-trial performance capacity of world tour cyclists during overreaching training. Per day, 20 g of sustained-release β-alanine was administered during one week (7 days) of intensive team training camp in a randomised balanced placebo-controlled parallel trial design, with six participants in each β-alanine (BA) or placebo (PLA) group. A 10-min time trial (10′ TT) was carried out to analyse performance and biochemical variables. Anthropometry, paresthesia, and adverse event data were also collected. Power-based relative training load was quantified. Compared to placebo, the BA improved mean power (6.21%, 37.23 W; 95% CI: 3.98–70.48 W, p = 0.046), distance travelled (2.16%, p = 0.046) and total work (4.85%, p = 0.046) without differences in cadence (p = 0.506) or RPE. Lactate (p = 0.036) and anion gap (p = 0.047) were also higher in the BA group, without differences in pH or Bicarbonate. High daily and single doses were well tolerated. One-week high-dose β-alanine loading with a sustained-release powder blend can help attenuate 10′ TT performance losses of world tour cyclists due to intensive training.

The balance of muscle oxygen supply and demand reveals critical metabolic rate and predicts time to exhaustion

We tested the hypothesis that during whole body exercise, the balance between muscle O₂ supply and metabolic demand may elucidate intensity domains, reveal a critical metabolic rate, and predict time to exhaustion. Seventeen active, healthy volunteers (12 males, 5 females; 32 ± 2 yr) participated in two distinct protocols. Study 1 (n = 7) consisted of constant work rate cycling in the moderate, heavy, and severe exercise intensity domains with concurrent measures of pulmonary V̇o₂ and local %SmO₂ [via near-infrared spectroscopy (NIRS)] on quadriceps and forearm sites. Average %SmO₂ at both sites displayed a domain-dependent response (P < 0.05). A negative %SmO₂ slope was evident during severe-domain exercise but was positive during exercise below critical power (CP) at both muscle sites. In study 2 (n = 10), quadriceps and forearm site %SmO₂ was measured during three continuous running trials to exhaustion and three intermittent intensity (ratio = 60 s severe: 30 s lower intensity) trials to exhaustion. Intensity-dependent negative %SmO₂ slopes were observed for all trials (P < 0.05) and predicted zero slope at critical velocity. %SmO₂ accurately predicted depletion and repletion of %D' balance on a second-by-second basis (R² = 0.99, P < 0.05; both sites). Time to exhaustion predictions during continuous and intermittent exercise were either not different or better with %SmO₂ [standard error of the estimate (SEE) < 20.52 s for quad, <44.03 s for forearm] versus running velocity (SEE < 65.76 s). Muscle O₂ balance provides a dynamic physiological delineation between sustainable and unsustainable exercise (consistent with a "critical metabolic rate") and predicts real-time depletion and repletion of finite work capacity and time to exhaustion.NEW & NOTEWORTHY Dynamic muscle O₂ saturation discriminates boundaries between exercise intensity domains, exposes a critical metabolic rate as the highest rate of steady state O₂ supply and demand, describes time series depletion and repletion for work above critical power, and predicts time to exhaustion during severe domain whole body exercise. These results highlight the matching of O₂ supply and demand as a primary determinant for sustainable exercise intensities from those that are unsustainable and lead to exhaustion.

Dynamic and thermodynamic models of adaptation

The concept of biological adaptation was closely connected to some mathematical, engineering and physical ideas from the very beginning. Cannon in his "The wisdom of the body" (1932) systematically used the engineering vision of regulation. In 1938, Selye enriched this approach by the notion of adaptation energy. This term causes much debate when one takes it literally, as a physical quantity, i.e. a sort of energy. Selye did not use the language of mathematics systematically, but the formalization of his phenomenological theory in the spirit of thermodynamics was simple and led to verifiable predictions. In 1980s, the dynamics of correlation and variance in systems under adaptation to a load of environmental factors were studied and the universal effect in ensembles of systems under a load of similar factors was discovered: in a crisis, as a rule, even before the onset of obvious symptoms of stress, the correlation increases together with variance (and volatility). During 30 years, this effect has been supported by many observations of groups of humans, mice, trees, grassy plants, and on financial time series. In the last ten years, these results were supplemented by many new experiments, from gene networks in cardiology and oncology to dynamics of depression and clinical psychotherapy. Several systems of models were developed: the thermodynamic-like theory of adaptation of ensembles and several families of models of individual adaptation. Historically, the first group of models was based on Selye's concept of adaptation energy and used fitness estimates. Two other groups of models are based on the idea of hidden attractor bifurcation and on the advection-diffusion model for distribution of population in the space of physiological attributes. We explore this world of models and experiments, starting with classic works, with particular attention to the results of the last ten years and open questions.

The Influence of Different Training Load Quantification Methods on the Fitness-Fatigue Model

PURPOSE

Numerous methods exist to quantify training load (TL). However, the relationship with performance is not fully understood. Therefore the purpose of this study was to investigate the influence of the existing TL quantification methods on performance modeling and the outcome parameters of the fitness-fatigue model.

METHODS

During a period of 8 weeks, 9 subjects performed 3 interval training sessions per week. Performance was monitored weekly by means of a 3-km time trial on a cycle ergometer. After this training period, subjects stopped training for 3 weeks but still performed a weekly time trial. For all training sessions, Banister training impulse (TRIMP), Lucia TRIMP, Edwards TRIMP, training stress score, and session rating of perceived exertion were calculated. The fitness-fatigue model was fitted for all subjects and for all TL methods.

RESULTS

The error in relating TL to performance was similar for all methods (Banister TRIMP: 618 [422], Lucia TRIMP: 625 [436], Edwards TRIMP: 643 [465], training stress score: 639 [448], session rating of perceived exertion: 558 [395], and kilojoules: 596 [505]). However, the TL methods evolved differently over time, which was reflected in the differences between the methods in the calculation of the day before performance on which training has the biggest positive influence (range of 19.6 d).

CONCLUSIONS

The authors concluded that TL methods cannot be used interchangeably because they evolve differently.

Monitoring the Heart Rate Variability Responses to Training Loads in Competitive Swimmers Using a Smartphone Application and the Banister Impulse-Response Model

PURPOSE

First, to examine whether heart rate variability (HRV) responses can be modeled effectively via the Banister impulse-response model when the session rating of perceived exertion (sRPE) alone, and in combination with subjective well-being measures, are utilized. Second, to describe seasonal HRV responses and their associations with changes in critical speed (CS) in competitive swimmers.

METHODS

A total of 10 highly trained swimmers collected daily 1-minute HRV recordings, sRPE training load, and subjective well-being scores via a novel smartphone application for 15 weeks. The impulse-response model was used to describe chronic root mean square of the successive differences (rMSSD) responses to training, with sRPE and subjective well-being measures used as systems inputs. Changes in CS were obtained from a 3-minute all-out test completed in weeks 1 and 14.

RESULTS

The level of agreement between predicted and actual HRV data was R2 = .66 (.25) when sRPE alone was used. Model fits improved in the range of 4% to 21% when different subjective well-being measures were combined with sRPE, representing trivial-to-moderate improvements. There were no significant differences in weekly group averages of log-transformed (Ln) rMSSD (P = .34) or HRV coefficient of variation of Ln rMSSD (P = .12); however, small-to-large changes (d = 0.21-1.46) were observed in these parameters throughout the season. Large correlations were observed between seasonal changes in HRV measures and CS (changes in averages of Ln rMSSD: r = .51, P = .13; changes in coefficient of variation of Ln rMSSD: r = -.68, P = .03).

CONCLUSION

The impulse-response model and data collected via a novel smartphone application can be used to model HRV responses to swimming training and nontraining-related stressors. Large relationships between seasonal changes in measured HRV parameters and CS provide further evidence for incorporating a HRV-guided training approach.

A mathematical model-based approach to optimize loading schemes of isometric resistance training sessions

Individualized resistance training is necessary to optimize training results. A model-based optimization of loading schemes could provide valuable impulses for practitioners and complement the predominant manual program design by customizing the loading schemes to the trainee and the training goals. We compile a literature overview of model-based approaches used to simulate or optimize the response to single resistance training sessions or to long-term resistance training plans in terms of strength, power, muscle mass, or local muscular endurance by varying the loading scheme. To the best of our knowledge, contributions employing a predictive model to algorithmically optimize loading schemes for different training goals are nonexistent in the literature. Thus, we propose to set up optimal control problems as follows. For the underlying dynamics, we use a phenomenological model of the time course of maximum voluntary isometric contraction force. Then, we provide mathematical formulations of key performance indicators for loading schemes identified in sport science and use those as objective functionals or constraints. We then solve those optimal control problems using previously obtained parameter estimates for the elbow flexors. We discuss our choice of training goals, analyze the structure of the computed solutions, and give evidence of their real-life feasibility. The proposed optimization methodology is independent from the underlying model and can be transferred to more elaborate physiological models once suitable ones become available.

Development and field validation of an omni-domain power-duration model

Purpose: To validate and compare a novel model based on the critical power (CP) concept that describes the entire domain of maximal mean power (MMP) data from cyclists.Methods: An omni-domain power-duration (OmPD) model was derived whereby the rate of W' expenditure is bound by maximum sprint power and the power at prolonged durations declines from CP log-linearly. The three-parameter CP (3CP) and exponential (Exp) models were likewise extended with the log-linear decay function (Om3CP and OmExp). Each model bounds W' using a different nonconstant function, W'eff (effective W'). Models were fit to MMP data from nine cyclists who also completed four time-trials (TTs).Results: The OmPD and Om3CP residuals (4 ± 1%) were smaller than the OmExp residuals (6 ± 2%; P < 0.001). W'eff predicted by the OmPD model was stable between 120-1,800 s, whereas it varied for the Om3CP and OmExp models. TT prediction errors were not different between models (7 ± 5%, 8 ± 5%, 7 ± 6%; P = 0.914).Conclusion: The OmPD offers similar or superior goodness-of-fit and better theoretical properties compared to the other models, such that it best extends the CP concept to short-sprint and prolonged-endurance performance.

An Improved Version of the Classical Banister Model to Predict Changes in Physical Condition

In this paper, we formulate and provide the solutions to two new models to predict changes in physical condition by using the information of the training load of an individual. The first model is based on a functional differential equation, and the second one on an integral differential equation. Both models are an extension to the classical Banister model and allow to overcome its main drawback: the variations in physical condition are influenced by the training loads of the previous days and not only of the same day. Finally, it is illustrated how the first model works with a real example of the training process of a cyclist.

Physiological responses in different intensities of resistance exercise - Critical load and the effects of aging process

Aim was to identify critical load (CL) in young and elderly apparently healthy male cohorts. To contrast the metabolic, cardiovascular and perceptual responses on CL according to age. We evaluated 12 young (23 ± 3 years) and 10 elderly (70 ± 2 years) apparently healthy active males, who underwent: (1) 1 repetition maximum (1RM) test on a 45° Leg Press; (2) on different days, three high-intensity resistance exercise constant load tests (60%, 75% and 90% 1RM) until fatigue (Tlim). Absolute values of both the CL asymptote and curvature constant (kg) were significantly lower in elderly subjects (P < 0.05). In contrast, elderly subjects demonstrated a significantly higher number of repetitions at CL when compared with young subjects (P < 0.05). As expected, oxygen uptake (VO₂) and heart rate (HR) during maximal aerobic exercise testing were significantly reduced in older subjects. However, percent-predicted aerobic capacity were higher in older subjects (P < 0.05). In addition, blood lactate ([La^-]) corrected to Tlim and rating of perceived exertion values were greater in younger subjects at all intensities (P < 0.05). These findings, despite reduced force production in older subjects, endurance-related parameters are well preserved according to age-adjusted percent-predicted values in apparently healthy males.

Modelling of Running Performances: Comparisons of Power-Law, Hyperbolic, Logarithmic, and Exponential Models in Elite Endurance Runners

Many empirical and descriptive models have been proposed since the beginning of the 20th century. In the present study, the power-law (Kennelly) and logarithmic (Péronnet-Thibault) models were compared with asymptotic models such as 2-parameter hyperbolic models (Hill and Scherrer), 3-parameter hyperbolic model (Morton), and exponential model (Hopkins). These empirical models were compared from the performance of 6 elite endurance runners (P. Nurmi, E. Zatopek, J. Väätäinen, L. Virén, S. Aouita, and H. Gebrselassie) who were world-record holders and/or Olympic winners and/or world or European champions. These elite runners were chosen because they participated several times in international competitions over a large range of distances (1500, 3000, 5000, and 10000 m) and three also participated in a marathon. The parameters of these models were compared and correlated. The less accurate models were the asymptotic 2-parameter hyperbolic models but the most accurate model was the asymptotic 3-parameter hyperbolic model proposed by Morton. The predictions of long-distance performances (maximal running speeds for 30 and 60 min and marathon) by extrapolation of the logarithmic and power-law models were more accurate than the predictions by extrapolation in all the asymptotic models. The overestimations of these long-distance performances by Morton's model were less important than the overestimations by the other asymptotic models.

Relationships Between Training Load Indicators and Training Outcomes in Professional Soccer

BACKGROUND

In professional senior soccer, training load monitoring is used to ensure an optimal workload to maximize physical fitness and prevent injury or illness. However, to date, different training load indicators are used without a clear link to training outcomes.

OBJECTIVE

The aim of this systematic review was to identify the state of knowledge with respect to the relationship between training load indicators and training outcomes in terms of physical fitness, injury, and illness.

METHODS

A systematic search was conducted in four electronic databases (CINAHL, PubMed, SPORTDiscus, and Web of Science). Training load was defined as the amount of stress over a minimum of two training sessions or matches, quantified in either external (e.g., duration, distance covered) or internal load (e.g., heart rate [HR]), to obtain a training outcome over time.

RESULTS

A total of 6492 records were retrieved, of which 3304 were duplicates. After screening the titles, abstracts and full texts, we identified 12 full-text articles that matched our inclusion criteria. One of these articles was identified through additional sources. All of these articles used correlations to examine the relationship between load indicators and training outcomes. For pre-season, training time spent at high intensity (i.e., >90 % of maximal HR) was linked to positive changes in aerobic fitness. Exposure time in terms of accumulated training, match or combined training, and match time showed both positive and negative relationships with changes in fitness over a season. Muscular perceived exertion may indicate negative changes in physical fitness. Additionally, it appeared that training at high intensity may involve a higher injury risk. Detailed external load indicators, using electronic performance and tracking systems, are relatively unexamined. In addition, most research focused on the relationship between training load indicators and changes in physical fitness, but less on injury and illness.

CONCLUSION

HR indicators showed relationships with positive changes in physical fitness during pre-season. In addition, exposure time appeared to be related to positive and negative changes in physical fitness. Despite the availability of more detailed training load indicators nowadays, the evidence about the usefulness in relation to training outcomes is rare. Future research should implement continuous monitoring of training load, combined with the individual characteristics, to further examine their relationship with physical fitness, injury, and illness.

The Critical Power Model as a Potential Tool for Anti-doping

Existing doping detection strategies rely on direct and indirect biochemical measurement methods focused on detecting banned substances, their metabolites, or biomarkers related to their use. However, the goal of doping is to improve performance, and yet evidence from performance data is not considered by these strategies. The emergence of portable sensors for measuring exercise intensities and of player tracking technologies may enable the widespread collection of performance data. How these data should be used for doping detection is an open question. Herein, we review the basis by which performance models could be used for doping detection, followed by critically reviewing the potential of the critical power (CP) model as a prototypical performance model that could be used in this regard. Performance models are mathematical representations of performance data specific to the athlete. Some models feature parameters with physiological interpretations, changes to which may provide clues regarding the specific doping method. The CP model is a simple model of the power-duration curve and features two physiologically interpretable parameters, CP and W′. We argue that the CP model could be useful for doping detection mainly based on the predictable sensitivities of its parameters to ergogenic aids and other performance-enhancing interventions. However, our argument is counterbalanced by the existence of important limitations and unresolved questions that need to be addressed before the model is used for doping detection. We conclude by providing a simple worked example showing how it could be used and propose recommendations for its implementation.

A nonlinear model for the characterization and optimization of athletic training and performance

Study aim: Mathematical models of the relationship between training and performance facilitate the design of training protocols to achieve performance goals. However, current linear models do not account for nonlinear physiological effects such as saturation and over-training. This severely limits their practical applicability, especially for optimizing training strategies. This study describes, analyzes, and applies a new nonlinear model to account for these physiological effects. Material and methods: This study considers the equilibria and step response of the nonlinear differential equation model to show its characteristics and trends, optimizes training protocols using genetic algorithms to maximize performance by applying the model under various realistic constraints, and presents a case study fitting the model to human performance data. Results: The nonlinear model captures the saturation and over-training effects; produces realistic training protocols with training progression, a high-intensity phase, and a taper; and closely fits the experimental performance data. Fitting the model parameters to subsets of the data identifies which parameters have the largest variability but reveals that the performance predictions are relatively consistent. Conclusions: These findings provide a new mathematical foundation for modeling and optimizing athletic training routines subject to an individual’s personal physiology, constraints, and performance goals.

Validation of a novel intermittent w' model for cycling using field data

Recently, an adaptation to the critical-power (CP) model was published, which permits the calculation of the balance of the work capacity available above the CP remaining (W'bal) at any time during intermittent exercise. As the model is now in use in both amateur and elite sport, the purpose of this investigation was to assess the validity of the W'bal model in the field. Data were collected from the bicycle power meters of 8 trained triathletes. W'bal was calculated and compared between files where subjects reported becoming prematurely exhausted during training or competition and files where the athletes successfully completed a difficult assigned task or race without becoming exhausted. Calculated W'bal was significantly different between the 2 conditions (P < .0001). The mean W'bal at exhaustion was 0.5 ± 1.3 kJ (95% CI = 0-0.9 kJ), whereas the minimum W'bal in the nonexhausted condition was 3.6 ± 2.0 kJ (95% CI = 2.1-4.0 kJ). Receiver-operator-characteristic (ROC) curve analysis indicated that the W'bal model is useful for identifying the point at which athletes are in danger of becoming exhausted (area under the ROC curve = .914, SE .05, 95% CI .82-1.0, P < .0001). The W'bal model may therefore represent a useful new development in assessing athlete fatigue state during training and racing.