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Mathematical modeling, simulation, and optimization of loading schemes for isometric resistance training
Abstract
In this thesis, we present a novel mathematical model-based approach to optimize loading schemes of isometric resistance training (RT) sessions for different training goals. To this end, we develop a nonlinear ordinary differential equation model of the time course of maximum voluntary isometric (MVIC) force under external isometric loading. To validate the model, we set up multi-experiment parameter estimation problems using a comprehensive dataset from the literature. We solve these problems numerically via direct multiple shooting and the generalized Gauss-Newton method. Moreover, we use the proposed model to examine hypotheses about fatigue and recovery of MVIC force. Then, we mathematically formulate key performance indicators and optimality criteria for loading schemes of isometric RT sessions identified in sports science and incorporate these into multi-stage optimal control problems. We solve these problems numerically via direct multiple shooting and structure-exploiting sequential quadratic programming. We discuss the results from a numerical and sports scientific point of view. Based on the proposed model, we additionally formulate the estimation of critical torque as a nonlinear program. This allows us to reduce the experimental effort compared to conventional testing when estimating these quantities. Furthermore, we formulate multi-stage optimum experimental design problems to reduce the statistical uncertainty of the parameter estimates when calibrating the model. We solve these problems numerically via direct single shooting and sequential quadratic programming. We discuss the solutions from a numerical and physiological point of view. For our approach, a small amount of data obtained in a single testing session is sufficient. Our approach can be extended to more elaborate physiological models and other forms of resistance training once suitable models become available.
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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.
PurposeCritical torque (CT) is an important fatigue threshold in exercise physiology and can be used to analyze, predict, or optimize performance. The objective of this work is to reduce the experimental effort when estimating CTs for sustained and intermittent isometric contractions using a model-based approach.Materials and methodsWe employ a phenomenological model of the time course of maximum voluntary isometric contraction (MVIC) torque and compute the highest sustainable torque output by solving an optimization problem. We then show that our results are consistent with the steady states obtained when simulating periodic maximum loading schemes. These simulations correspond to all-out tests, which are used to estimate CTs in practice. Based on these observations, the estimation of CTs can be formulated mathematically as a parameter estimation problem. To minimize the statistical uncertainty of the parameter estimates and consequently of the estimated CTs, we compute optimized testing sessions. This reduces the experimental effort even further.ResultsWe estimate CTs of the elbow flexors for sustained isometric contractions to be 28% of baseline MVIC torque and for intermittent isometric contractions consisting of a 3 s contraction followed by 2 s rest to be 41% of baseline MVIC torque. We show that a single optimized testing session is sufficient when using our approach.Conclusions
Our approach reduces the experimental effort considerably when estimating CTs for sustained and intermittent isometric contractions.
SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments. This Perspective describes the development and capabilities of SciPy 1.0, an open source scientific computing library for the Python programming language.
Human fast-twitch muscle fibers generate high power in a short amount of time but are easily fatigued, while slow-twitch fibers are more fatigue resistant. The transfer of this knowledge to coaching, is hampered by the invasive nature of the current evaluation of the muscle typology by biopsies. Therefore, a non-invasive method was developed to estimate muscle typology through proton magnetic resonance spectroscopy in the gastrocnemius. The aim of this study was to investigate if male subjects with an a priori determined fast typology (FT) are characterized with a more pronounced Wingate exercise-induced fatigue and delayed recovery compared to the ones with a slow typology (ST). Ten subjects with an estimated higher percentage of fast fibers and 10 subjects with an estimated higher percentage of slow fibers underwent the test protocol, consisting of three 30 sec all-out Wingate tests. Recovery of knee extension torque was evaluated by maximal voluntary contraction combined with electrical stimulation up to 5 h after the Wingate tests. Although both groups delivered the same mean power across all Wingates, the power drop was higher in the FT group (-61%) compared to the ST group (-41%). The torque at maximal voluntary contraction had fully recovered in the ST group after 20 min, while the FT group had not yet recovered 5 h into recovery. This non-invasive estimation of muscle typology can predict the extent of fatigue and time to recover following repeated all-out exercise and may have applications as a tool to individualize training and recovery cycles.
During high intensity exercise, metabosensitive muscle afferents are thought to inhibit the motor drive command to restrict the level of peripheral fatigue to an individual’s critical threshold. No evidence exists of an individual relationship between peripheral fatigue and the decrease in voluntary activation reached after prolonged all-out exercise. Moreover, there is no explanation for the previously reported large decrease in voluntary activation despite low metabolic stress during high force contractions. Thirteen active men completed two maximal intensity isokinetic knee extension tests (160 contractions) under conditions of low force – high velocity and high force – low velocity. Neuromuscular testing including maximal torque, evoked torque and voluntary activation, was done every 20 contractions. The exponential modeling of these variables over time allowed us to predict the stable state (asymptote) and the rate of decrease (curvature constant). For both high and low force contractions the evoked torque and voluntary activation asymptotes were negatively correlated (R2 = 0.49 and R2 = 0.46, respectively). The evoked torque asymptotes of the high and low force conditions were positively correlated (R2 = 0.49). For the high force contractions, the evoked torque and voluntary activation curvature constant were negatively correlated (R2 = 0.43). These results support the idea that a restrained central motor drive keeps peripheral fatigue under this threshold. Furthermore, an individual would show similar fatigue sensibility regardless of the force generated. These data also suggest that the decrease in voluntary activation might not have been triggered by peripheral perturbations during the first high force contractions.
Abstract The maximal lactate steady state (MLSS) and the critical power (CP) are two widely used indices of the highest oxidative metabolic rate that can be sustained during continuous exercise and are often considered to be synonymous. However, while perhaps having similarities in principle, methodological differences in the assessment of these parameters typically result in MLSS occurring at a somewhat lower power output or running speed and exercise at CP being sustainable for no more than approximately 20–30 min. This has led to the view that CP overestimates the ‘actual’ maximal metabolic steady state and that MLSS should be considered the ‘gold standard’ metric for the evaluation of endurance exercise capacity. In this article we will present evidence consistent with the contrary conclusion: i.e., that (1) as presently defined, MLSS naturally underestimates the actual maximal metabolic steady state; and (2) CP alone represents the boundary between discrete exercise intensity domains within which the dynamic cardiorespiratory and muscle metabolic responses to exercise differ profoundly. While both MLSS and CP may have relevance for athletic training and performance, we urge that the distinction between the two concepts/metrics be better appreciated and that comparisons between MLSS and CP, undertaken in the mistaken belief that they are theoretically synonymous, is discontinued. CP represents the genuine boundary separating exercise in which physiological homeostasis can be maintained from exercise in which it cannot, and should be considered the gold standard when the goal is to determine the maximal metabolic steady state.
Prescribing the proper resistance training (RT) program is critical to optimize skeletal muscle hypertrophy and strength. Periodization is a strategy that entails planned manipulations of training variables to maximize fitness adaptations while minimizing the risk of overtraining. Multiple meta-analyses have shown periodized RT to be superior to non-periodized RT for enhancing muscular strength. These findings are consistent irrespective of training status or training volume. Both the linear model and the undulating model are effective for enhancing strength, although a greater benefit might be achieved through the undulating model. Despite the suggested superiority of periodized RT for strength development, some authors suggest that this might be a consequence of the study designs employed rather than the nature of periodized training. In addition, several limitations exist in the periodization literature, making it difficult to accurately assess the efficacy of periodized RT. With regard to enhancing skeletal muscle hypertrophy, both the undulating model and the linear model appear equally effective; however, this conclusion can only be generalized to untrained populations. When comparing periodized RT to non-periodized RT programs, the research is unclear on whether periodized RT is necessary to maximize skeletal muscle hypertrophy.
Isometric training is used in the rehabilitation and physical preparation of athletes, special populations and the general public. However, little consensus exists regarding training guidelines for a variety of desired outcomes. Understanding the adaptive response to specific loading parameters would be of benefit to practitioners. The objective of this systematic review, therefore, was to detail the medium to long‐term adaptations of different types of isometric training on morphological, neurological and performance variables. Exploration of the relevant subject matter was performed through MEDLINE, PubMed, SPORTDiscus and CINAHL databases. English, full‐text, peer‐reviewed journal articles and unpublished doctoral dissertations investigating medium to long‐term (≥3 weeks) adaptations to isometric training in humans were identified. These studies were evaluated further for methodological quality. Twenty‐six research outputs were reviewed. Isometric training at longer muscle lengths (0.86‐1.69%/week, ES = 0.03‐0.09/week) produced greater muscular hypertrophy when compared to equal volumes of shorter muscle length training (0.08‐0.83%/week, ES = ‐0.003‐0.07/week). Ballistic intent resulted in greater neuromuscular activation (1.04‐10.5%/week, ES = 0.02‐0.31/week vs. 1.64‐5.53%/week, ES = 0.03‐0.20/week) and rapid force production (1.2‐13.4%/week, ES = 0.05‐0.61/week vs. 1.01‐8.13%/week, ES = 0.06‐0.22/week). Substantial improvements in muscular hypertrophy and maximal force production were reported regardless of training intensity. High‐intensity (≥ 70%) contractions are required for improving tendon structure and function. Additionally, long muscle length training results in greater transference to dynamic performance. Despite relatively few studies meeting the inclusion criteria, this review provides practitioners with insight into which isometric training variables (e.g. joint angle, intensity, intent) to manipulate to achieve desired morphological and neuromuscular adaptations.
One of the most striking adaptations to exercise is the skeletal muscle hypertrophy that occurs in response to resistance exercise. A large body of work shows that a mTORC1-mediated increase of muscle protein synthesis is the key, but not sole, mechanism by which resistance exercise causes muscle hypertrophy. Whilst much of the hypertrophy signaling cascade has been identified, the initiating, resistance exercise-induced and hypertrophy-stimulating stimuli have remained elusive. For the purpose of this review, we define an initiating, resistance exercise-induced and hypertrophy-stimulating signal as "hypertrophy stimulus", and the sensor of such a signal as "hypertrophy sensor". In this review we discuss our current knowledge of specific mechanical stimuli, damage/injury-associated and metabolic stress-associated triggers as potential hypertrophy stimuli. Mechanical signals are the prime hypertrophy stimuli candidates and a Filamin-C-BAG3-dependent regulation of mTORC1, Hippo and autophagy signalling is a plausible albeit still incompletely characterised hypertrophy sensor. Other candidate mechanosensing mechanisms are nuclear deformation-initiated signalling or several mechanisms related to costameres, which are the functional equivalents of focal adhesions in other cells. Whilst exercise-induced muscle damage is probably not essential for hypertrophy, it is still unclear whether and how such muscle damage could augment a hypertrophic response. Interventions that combine blood flow restriction and especially low load resistance exercise suggest that resistance exercise-regulated metabolites could be hypertrophy stimuli but this is based on indirect evidence and metabolite candidates are poorly characterised.
Purpose
The time course of maximal voluntary isometric contraction (MVIC) force is of particular interest whenever force capacities are a limiting factor, e.g., during heavy manual work or resistance training (RT) sessions. The objective of this work was to develop a mathematical model of this time course that is suitable for optimization of complex loading schemes.
Materials and methods
We compiled a literature overview of existing models and justified the need for a new model. We then constructed a phenomenological ordinary differential equation model to describe the time course of MVIC force during voluntary isometric contractions and at rest. We validated the model with a comprehensive set of published data from the elbow flexors. For this, we estimated parameters from a subset of the available data and used those estimates to predict the remaining data. Afterwards, we illustrated the benefits of our model using the calibrated model to (1) analyze fatigue and recovery patterns observed in the literature (2) compute a work–rest schedule that minimizes fatigue (3) determine an isometric RT session that maximizes training volume.
Results
We demonstrated that our model (1) is able to describe MVIC force under complex loading schemes (2) can be used to analyze fatigue and recovery patterns observed in the literature (3) can be used to optimize complex loading schemes.
Conclusions
We developed a mathematical model of the time course of MVIC force that can be efficiently employed to optimize complex loading schemes. This enables an optimal use of MVIC force capacities.
Background
Current recommendations on resistance training (RT) frequency for gains in muscular strength are based on extrapolations from limited evidence on the topic, and thus their practical applicability remains questionable.
Objective
To elucidate this issue, we conducted a systematic review and meta-analysis of the studies that compared muscular strength outcomes with different RT frequencies.
Methods
To carry out this review, English-language literature searches of the PubMed/MEDLINE, Scopus, and SPORTDiscus databases were conducted. The meta-analysis was performed using a random-effects model. The meta-analysis models were generated with RT frequencies classified as a categorical variable as either 1, 2, 3, or 4+ times/week, or, if there were insufficient data in subgroup analyses, the training frequencies were categorized as 1, 2, or 3 times/week. Subgroup analyses were performed for potential moderators, including (1) training volume; (2) exercise selection for the 1 repetition maximum (RM) test (for both multi-joint and single-joint exercises); (3) upper and lower body strength gains; (4) training to muscular failure (for studies involving and not involving training to muscular failure); (5) age (for both middle-aged/older adults and young adults); and (6) sex (for men and for women). The methodological quality of studies was appraised using the modified Downs and Black checklist.
Results
A total of 22 studies were found to meet the inclusion criteria. The average score on the Downs and Black checklist was 18 (range 13–22 points). Four studies were classified as being of good methodological quality, while the rest were classified as being of moderate methodological quality. Results of the meta-analysis showed a significant effect (p = 0.003) of RT frequency on muscular strength gains. Effect sizes increased in magnitude from 0.74, 0.82, 0.93, and 1.08 for training 1, 2, 3, and 4+ times per week, respectively. A subgroup analysis of volume-equated studies showed no significant effect (p = 0.421) of RT frequency on muscular strength gains. The subgroup analysis for exercise selection for the 1RM test suggested a significant effect of RT frequency on multi-joint (p < 0.001), but not single-joint, 1RM test results (p = 0.324). The subgroup analysis for upper and lower body showed a significant effect of frequency (p = 0.004) for upper body, but not lower body, strength gains (p = 0.070). In the subgroup analysis for studies in which the training was and was not carried out to muscular failure, no significant effect of RT frequency was found. The subgroup analysis for the age groups suggested a significant effect of training frequency among young adults (p = 0.024), but not among middle-aged and older adults (p = 0.093). Finally, the subgroup analysis for sex indicated a significant effect of RT frequency on strength gains in women (p = 0.030), but not men (p = 0.190).
Conclusions
The results of the present systematic review and meta-analysis suggest a significant effect of RT frequency as higher training frequencies are translated into greater muscular strength gains. However, these effects seem to be primarily driven by training volume because when the volume is equated, there was no significant effect of RT frequency on muscular strength gains. Thus, from a practical standpoint, greater training frequencies can be used for additional RT volume, which is then likely to result in greater muscular strength gains. However, it remains unclear whether RT frequency on its own has significant effects on strength gain. It seems that higher RT frequencies result in greater gains in muscular strength on multi-joint exercises in the upper body and in women, and, finally, in contrast to older adults, young individuals seem to respond more positively to greater RT frequencies. More evidence among resistance-trained individuals is needed as most of the current studies were performed in untrained participants.
Purpose:
Current diet and exercise methods used to maintain or improve body composition often have poor long-term outcomes. We hypothesize that resistance exercise (RE) should aid in the maintenance of a healthy body composition by preserving lean mass (LM) and metabolic rate.
Method:
We extended a previously developed energy balance model of human metabolism to include muscle hypertrophy in response to RE. We first fit model parameters to a hypothetical individual to simulate an RE program and then compared the effects of a hypocaloric diet only to the diet with either cardiovascular exercise (CE) or RE. We then simulated a cohort of individuals with different responses to RE by varying the parameters controlling it using Latin Hypercube Sampling (LHS). Finally, we fit the model to mean data from an elderly population on an RE program.
Conclusion:
The model is able to reproduce the time course of change in LM in response to RE and can be used to generate a simulated cohort for in silico clinical studies. Simulations suggest that the additional LM generated by RE may shift the body composition to a healthier state.
Rate of force development (RFD) refers to the ability of the neuromuscular system to increase contractile force from a low or resting level when muscle activation is performed as quickly as possible, and it is considered an important muscle strength parameter, especially for athletes in sports requiring high-speed actions. The assessment of RFD has been used for strength diagnosis, to monitor the effects of training interventions in both healthy populations and patients, discriminate high-level athletes from those of lower levels, evaluate the impairment in mechanical muscle function after acute bouts of eccentric muscle actions and to estimate the degree of fatigue and recovery after acute exhausting exercise. Notably, the evaluation of RFD in human skeletal muscle is a complex task since influenced by numerous distinct methodological factors including mode of contraction, type of instruction, method used to quantify RFD, devices used for force/torque recording and ambient temperature. Another important aspect is our limited understanding of the mechanisms underpinning rapid muscle force production. Therefore, this review is primarily focused on 1) describing the main mechanical characteristics of RFD; 2) analyzing various physiological factors that influence RFD; and 3) presenting and discussing central biomechanical and methodological factors affecting the measurement of RFD. The intention of this review is to provide more methodological and analytical coherency on the RFD concept, which may aid to clarify the thinking of coaches and sports scientists in this area.
The objective of this study was to test the hypotheses that end-test torque (ET) (expressed as % maximal voluntary contraction; MVC) is higher for plantar flexors (PF) than knee extensors (KE) muscles, whereas impulse above ET (IET) is higher for KE than PF. Thus, we expected that exercise tolerance would be longer for KE than PF only during the exercise performed above ET. After the determination of MVC, 40 men performed two 5-min all-out tests to determine ET and IET. Eleven participants performed a further 4 intermittent isometric tests, to exhaustion, at ET + 5% and ET – 5%, and 1 test for KE at the exercise intensity (%MVC) corresponding to ET + 5% of PF. The IET (7243.2 ± 1942.9 vs. 3357.4 ± 1132.3 N·m·s) and ET (84.4 ± 24.8 vs. 73.9 ± 19.5 N·m) were significantly lower in PF compared with KE. The exercise tolerance was significantly longer for PF (300.7 ± 156.7 s) than KE (156.7 ± 104.3 s) at similar %MVC (∼60%), and significantly shorter for PF (300.7 ± 156.7 s) than KE (697.0 ± 243.7 s) at ET + 5% condition. However, no significant difference was observed for ET – 5% condition (KE = 1030.2 ± 495.4 s vs. PF = 1028.3 ± 514.4 s). Thus, the limit of tolerance during submaximal isometric contractions is influenced by absolute MVC only during exercise performed above ET, which seems to be explained by differences on both ET (expressed as %MVC) and IET values.
Background
Rest interval (RI) duration is an important resistance-training variable underlying gain in muscular strength. Recommendations for optimal RI duration for gains in muscular strength are largely inferred from studies examining the acute resistance training effects, and the generalizability of such findings to chronic adaptations is uncertain.
Objective
The goals of this systematic literature review are: (i) to aggregate findings and interpret the studies that assessed chronic muscular strength adaptations to resistance training interventions involving different RI durations, and (ii) to provide evidence-based recommendations for exercise practitioners and athletes.
Methods
The review was performed according to the PRISMA guidelines with a literature search encompassing five databases. Methodological quality of the studies was evaluated using a modified version of the Downs and Black checklist.
Results
Twenty-three studies comprising a total of 491 participants (413 males and 78 females) were found to meet the inclusion criteria. All studies were classified as being of good to moderate methodological quality; none of the studies were of poor methodological quality.
Conclusion
The current literature shows that robust gains in muscular strength can be achieved even with short RIs (< 60 s). However, it seems that longer duration RIs (> 2 min) are required to maximize strength gains in resistance-trained individuals. With regard to untrained individuals, it seems that short to moderate RIs (60–120 s) are sufficient for maximizing muscular strength gains.
The purpose of this paper was to conduct a systematic review of the current body of literature and a meta-analysis to compare changes in strength and hypertrophy between low- versus high-load resistance training protocols. Searches of PubMed/MEDLINE, Cochrane Library and Scopus were conducted for studies that met the following criteria: 1) an experimental trial involving both low- (≤60% 1 RM) and high- (>60% 1 RM) load training; 2) with all sets in the training protocols being performed to momentary muscular failure; 3) at least one method of estimating changes in muscle mass and/or dynamic, isometric or isokinetic strength was used; 4) the training protocol lasted for a minimum of 6 weeks; 5) the study involved participants with no known medical conditions or injuries impairing training capacity. A total of 21 studies were ultimately included for analysis. Gains in 1RM strength were significantly greater in favor of high- versus low-load training, while no significant differences were found for isometric strength between conditions. Changes in measures of muscle hypertrophy were similar between conditions. The findings indicate that maximal strength benefits are obtained from the use of heavy loads while muscle hypertrophy can be equally achieved across a spectrum of loading ranges.
Background
Periodization is an important component of resistance training programs. It is meant to improve adherence to the training regimen, allow for constant progression, help in avoiding plateaus, and reduce occurrence and severity of injuries. Previous findings regarding the effects of different periodization models on measures of muscle hypertrophy are equivocal. To provide a more in-depth look at the topic, we undertook a systematic review of the literature and a meta-analysis of intervention trials comparing the effects of linear periodization (LP) and daily undulating periodization (DUP) resistance training programs on muscle hypertrophy.
Materials and Methods
A comprehensive literature search was conducted through PubMed/MEDLINE, Scopus, Web of Science, SPORTDiscus, Networked Digital Library of Theses and Dissertations (NDLTD) and Open Access Theses and Dissertations (OATD).
Results
The pooled standardized mean difference (Cohen’s d) from 13 eligible studies for the difference between the periodization models on muscle hypertrophy was −0.02 (95% confidence interval [−0.25, 0.21], p = 0.848).
Conclusions
The meta-analysis comparing LP and DUP indicated that the effects of the two periodization models on muscle hypertrophy are likely to be similar. However, more research is needed in this area, particularly among trained individuals and clinical populations. Future studies may benefit from using instruments that are more sensitive for detecting changes in muscle mass, such as ultrasound or magnetic resonance imaging.
Detailed description of the time course of muscular adaptation is rarely found in literature. Thus, models of muscular adaptation are difficult to validate since no detailed data of adaptation are available. In this article, as an initial step toward a detailed description and analysis of muscular adaptation, we provide a case report of 8 weeks of intense strength training with two active, male participants. Muscular adaptations were analyzed on a morphological level with MRI scans of the right quadriceps muscle and the calculation of muscle volume, on a voluntary strength level by isometric voluntary contractions with doublet stimulation (interpolated twitch technique) and on a non-voluntary level by resting twitch torques. Further, training volume and isokinetic power were closely monitored during the training phase. Data were analyzed weekly for 1 week prior to training, pre-training, 8 weeks of training and 2 weeks of detraining (no strength training). Results show a very individual adaptation to the intense strength training protocol. While training volume and isokinetic power increased linearly during the training phase, resting twitch parameters decreased for both participants after the first week of training and stayed below baseline until de-training. Voluntary activation level showed an increase in the first 4 weeks of training, while maximum voluntary contraction showed only little increase compared to baseline. Muscle volume increased for both subjects. Especially training status seemed to influence the acute reaction to intense strength training. Fatigue had a major influence on performance and could only be overcome by one participant. The results give a first detailed insight into muscular adaptation to intense strength training on various levels, providing a basis of data for a validation of muscle fatigue and adaptation models.
Muscle fatigue is a temporary decline in the force and power capacity of skeletal muscle resulting from muscle activity. Because control of muscle is realized at the level of the motor unit (MU), it seems important to consider the physiological properties of motor units when attempting to understand and predict muscle fatigue. Therefore, we developed a phenomenological model of motor unit fatigue as a tractable means to predict muscle fatigue for a variety of tasks and to illustrate the individual contractile responses of MUs whose collective action determines the trajectory of changes in muscle force capacity during prolonged activity. An existing MU population model was used to simulate MU firing rates and isometric muscle forces and, to that model, we added fatigue-related changes in MU force, contraction time, and firing rate associated with sustained voluntary contractions. The model accurately estimated endurance times for sustained isometric contractions across a wide range of target levels. In addition, simulations were run for situations that have little experimental precedent to demonstrate the potential utility of the model to predict motor unit fatigue for more complicated, real-world applications. Moreover, the model provided insight into the complex orchestration of MU force contributions during fatigue, that would be unattainable with current experimental approaches.
Background
Periodization is a logical method of organizing training into sequential phases and cyclical time periods in order to increase the potential for achieving specific performance goals while minimizing the potential for overtraining. Periodized resistance training plans are proposed to be superior to non-periodized training plans for enhancing maximal strength. Objective
The primary aim of this study was to examine the previous literature comparing periodized resistance training plans to non-periodized resistance training plans and determine a quantitative estimate of effect on maximal strength. Methods
All studies included in the meta-analysis met the following inclusion criteria: (1) peer-reviewed publication; (2) published in English; (3) comparison of a periodized resistance training group to a non-periodized resistance training group; (4) maximal strength measured by 1-repetition maximum (1RM) squat, bench press, or leg press. Data were extracted and independently coded by two authors. Random-effects models were used to aggregate a mean effect size (ES), 95% confidence intervals (CIs) and potential moderators. ResultsThe cumulative results of 81 effects gathered from 18 studies published between 1988 and 2015 indicated that the magnitude of improvement in 1RM following periodized resistance training was greater than non-periodized resistance training (ES = 0.43, 95% CI 0.27–0.58; P < 0.001). Periodization model (β = 0.51; P = 0.0010), training status (β = −0.59; P = 0.0305), study length (β = 0.03; P = 0.0067), and training frequency (β = 0.46; P = 0.0123) were associated with a change in 1RM. These results indicate that undulating programs were more favorable for strength gains. Improvements in 1RM were greater among untrained participants. Additionally, higher training frequency and longer study length were associated with larger improvements in 1RM. Conclusion
These results suggest that periodized resistance training plans have a moderate effect on 1RM compared to non-periodized training plans. Variation in training stimuli appears to be vital for increasing maximal strength, and longer periods of higher training frequency may be preferred.
The curvilinear relationship between power output and the time for which it can be sustained is a fundamental and well-known feature of high-intensity exercise performance. This relationship ‘levels off’ at a ‘critical power’ (CP) that separates power outputs that can be sustained with stable values of, for example, muscle phosphocreatine, blood lactate, and pulmonary oxygen uptake (), from power outputs where these variables change continuously with time until their respective minimum and maximum values are reached and exercise intolerance occurs. The amount of work that can be done during exercise above CP (the so-called W′) is constant but may be utilized at different rates depending on the proximity of the exercise power output to CP. Traditionally, this two-parameter CP model has been employed to provide insights into physiological responses, fatigue mechanisms, and performance capacity during continuous constant power output exercise in discrete exercise intensity domains. However, many team sports (e.g., basketball, football, hockey, rugby) involve frequent changes in exercise intensity and, even in endurance sports (e.g., cycling, running), intensity may vary considerably with environmental/course conditions and pacing strategy. In recent years, the appeal of the CP concept has been broadened through its application to intermittent high-intensity exercise. With the assumptions that W′ is utilized during work intervals above CP and reconstituted during recovery intervals below CP, it can be shown that performance during intermittent exercise is related to four factors: the intensity and duration of the work intervals and the intensity and duration of the recovery intervals. However, while the utilization of W′ may be assumed to be linear, studies indicate that the reconstitution of W′ may be curvilinear with kinetics that are highly variable between individuals. This has led to the development of a new CP model for intermittent exercise in which the balance of W′ remaining () may be calculated with greater accuracy. Field trials of athletes performing stochastic exercise indicate that this model can accurately predict the time at which W′ tends to zero and exhaustion is imminent. The model potentially has important applications in the real-time monitoring of athlete fatigue progression in endurance and team sports, which may inform tactics and influence pacing strategy.
The mechanisms underlying task failure from fatiguing physical efforts have been the focus of many studies without reaching consensus. An attractive but debated model explains effort termination with a critical peripheral fatigue threshold. Upon reaching this threshold, feedback from sensory afferents would trigger task disengagement from open-ended tasks or a reduction of exercise intensity of closed-ended tasks. Alternatively, the extant literature also appears compatible with a more global critical threshold of loss of maximal voluntary contraction force. Indeed, maximal voluntary contraction force loss from fatiguing exercise realized at a given intensity appears rather consistent between different studies. However, when looking at individual data, the similar maximal force losses observed between different tasks performed at similar intensities might just be an “artifact” of data aggregation. It would then seem possible that such a difference observed between individual and aggregated data also applies to other models previously proposed to explain task failure from fatiguing physical efforts. We therefore suggest that one should be cautious when trying to infer models that try to explain individual behavior from aggregated data.
When performing a set of successive repetitions, fatigue ensues and the quality of performance during subsequent repetitions contained in the set decreases. Oftentimes, this response may be beneficial, as fatigue may stimulate the neuromuscular system to adapt, resulting in a super-compensatory response. However, there are instances in which accumulated fatigue may be detrimental to training or performance adaptations (i.e. power development). In these instances, the ability to recover and maintain repetition performance would be considered essential. By providing intermittent rest between individual repetitions or groups of repetitions within a set, an athlete is able to acutely alleviate fatigue, allowing performance to remain relatively constant throughout an exercise session. Within the scientific literature, a set that includes intermittent rest between individual repetitions or groups of repetitions within a set is defined as a cluster set. Recently, cluster sets have received more attention as researchers have begun to examine the acute and chronic responses to this relatively novel set structure. However, much of the rest-period terminology within the literature lacks uniformity and many authors attempt to compare largely different protocols with the same terminology. Additionally, the present body of scientific literature has mainly focused on the effects of cluster sets on power output, leaving the effects of cluster sets on strength and hypertrophy relatively unexplored. Therefore, the purpose of this review is to further delineate cluster set terminology, describe the acute and chronic responses of cluster sets, and explain the need for further investigation of the effects of cluster sets.
Background
A number of resistance training (RT) program variables can be manipulated to maximize muscular hypertrophy. One variable of primary interest in this regard is RT frequency. Frequency can refer to the number of resistance training sessions performed in a given period of time, as well as to the number of times a specific muscle group is trained over a given period of time.
Objective
We conducted a systematic review and meta-analysis to determine the effects of resistance training frequency on hypertrophic outcomes.
Methods
Studies were deemed eligible for inclusion if they met the following criteria: (1) were an experimental trial published in an English-language refereed journal; (2) directly compared different weekly resistance training frequencies in traditional dynamic exercise using coupled concentric and eccentric actions; (3) measured morphologic changes via biopsy, imaging, circumference, and/or densitometry; (4) had a minimum duration of 4 weeks; and (5) used human participants without chronic disease or injury. A total of ten studies were identified that investigated RT frequency in accordance with the criteria outlined.
Results
Analysis using binary frequency as a predictor variable revealed a significant impact of training frequency on hypertrophy effect size (P = 0.002), with higher frequency being associated with a greater effect size than lower frequency (0.49 ± 0.08 vs. 0.30 ± 0.07, respectively). Statistical analyses of studies investigating training session frequency when groups are matched for frequency of training per muscle group could not be carried out and reliable estimates could not be generated due to inadequate sample size.
Conclusions
When comparing studies that investigated training muscle groups between 1 to 3 days per week on a volume-equated basis, the current body of evidence indicates that frequencies of training twice a week promote superior hypertrophic outcomes to once a week. It can therefore be inferred that the major muscle groups should be trained at least twice a week to maximize muscle growth; whether training a muscle group three times per week is superior to a twice-per-week protocol remains to be determined.
During exercise, there is a progressive reduction in the ability to produce muscle forces. Processes within the nervous system, as well as within the muscles contribute to this fatigue. In addition to impaired function of the motor system, sensations associated with fatigue, and impairment of homeostasis can contribute to impairment of performance during exercise. This review discusses some of the neural changes that accompany exercise and the development of fatigue. The role of brain monoaminergic neurotransmitter systems in whole-body endurance performance is discussed, particularly with regard to exercise in hot environments. Next, fatigue-related alterations in the neuromuscular pathway are discussed in terms of changes in motor unit firing, motoneuron excitability and motor cortical excitability. These changes have mostly been investigated during single-limb isometric contractions. Finally, the small-diameter muscle afferents that increase firing with exercise and fatigue are discussed. These afferents have roles in cardiovascular and respiratory responses to exercise, and in impairment of exercise performance through interaction with the motor pathway, as well as providing sensations of muscle discomfort. Thus, changes at all levels of the nervous system including the brain, spinal cord, motor output, sensory input and autonomic function occur during exercise and fatigue. The mix of influences and the importance of their contribution varies with the type of exercise being performed.
In contrast to most scientific disciplines, sports science research has been characterized by comparatively little effort investment in the development of relevant phenomenologi-cal models. Scarcer yet is the application of said models in practice. We present a framework which allows resistance training practitioners to employ a recently proposed neu-romuscular model in actual training program design. The first novelty concerns the monitoring aspect of coaching. A method for extracting training performance characteristics from loosely constrained video sequences, effortlessly and with minimal human input, using computer vision is described. The extracted data is subsequently used to fit the underlying neuromuscular model. This is achieved by solving an inverse dynamics problem corresponding to a particular exercise. Lastly, a computer simulation of hypothetical training bouts, using athlete-specific capability parameters, is used to predict the effected adaptation and changes in performance. The software described here allows the practitioner to manipulate hypothetical training parameters and immediately see their effect on predicted adaptation for a specific athlete. Thus, this work presents a holistic view of the monitoring-assessment-adjustment loop.
In this thesis we develop algorithms for the numerical solution of problems from nonlinear
optimum experimental design (OED) for parameter estimation in differential–algebraic
equations. These OED problems can be formulated as special types of path- and control-
constrained optimal control (OC) problems. The objective is to minimize a functional on
the covariance matrix of the model parameters that is given by first-order sensitivities of the
model equations. Additionally, the objective is nonlinearly coupled in time, which make
OED problems a challenging class of OC problems. For their numerical solution, we propose
a direct multiple shooting parameterization to obtain a structured nonlinear programming
problem (NLP). An augmented system of nominal and variational states for the model
sensitivities is parameterized on multiple shooting intervals and the objective is decoupled
by means of additional variables and constraints. In the resulting NLP, we identify several
structures that allow to evaluate derivatives at greatly reduced costs compared to a standard
OC formulation.
For the solution of the block-structured NLPs, we develop a new sequential quadratic
programming (SQP) method. Therein, partitioned quasi-Newton updates are used to approximate the block-diagonal Hessian of the Lagrangian. We analyze a model problem with
indefinite, block-diagonal Hessian and prove that positive definite approximations of the
individual blocks prevent superlinear convergence. For an OED model problem, we show
that more and more negative eigenvalues appear in the Hessian as the multiple shooting grid
is refined and confirm the detrimental impact of positive definite Hessian approximations.
Hence, we propose indefinite SR1 updates to guarantee fast local convergence. We develop
a filter line search globalization strategy that accepts indefinite Hessians based on a new
criterion derived from the proof of global convergence. BFGS updates with a scaling strategy to prevent large eigenvalues are used as fallback if the SR1 update does not promote
convergence. For the solution of the arising sparse and nonconvex quadratic subproblems, a
parametric active set method with inertia control within a Schur complement approach is
developed. It employs a symmetric, indefinite LBL T -factorization for the large, sparse KKT
matrix and maintains and updates QR-factors of a small and dense Schur complement.
The new methods are complemented by two C++ implementations: muse transforms an
OED or OC problem instance to a structured NLP by means of direct multiple shooting.
A special feature is that fully independent grids for controls, states, path constraints, and
measurements are maintained. This provides higher flexibility to adapt the NLP formulation
to the characteristics of the problem at hand and facilitates comparison of different formulations in the light of the lifted Newton method. The software package blockSQP is an
implementation of the new SQP method that uses a newly developed variant of the quadratic
programming solver qpOASES. Numerical results are presented for a benchmark collection of
OED and OC problems that show how SR1 approximations improve local convergence over
BFGS. The new method is then applied to two challenging OED applications from chemical
engineering. Its performance compares favorably to an available existing implementation.
A physiology-based fatigue model was developed and tested, with the long-term objective to study optimal pacing strategies in cross-country skiing. The model considers both aerobic and anaerobic power contributions, with different demands for carbohydrate fuel. The fatigue model accumulates traces from anaerobic efforts, and dissipates fatigue exponentially. The current fatigue value affects the effective work rate output. A limited reservoir of fuel is considered. This paper discusses the numerical formulations. Examples show the relevance of the model for basic regimes of power output, and give qualitatively relevant results, but demonstrate the need for individual physiological parameters. Further examples study the model’s predictions with respect to interval training strategies, with conclusions on work rates and interval lengths.
Training models have been proposed to model the effect of physical strain on fitness. In this work we explore their use not only for analysis but also to generate training plans to achieve a given fitness goal. These plans have to include side constraints such as, e.g., maximal training loads. Therefore plan generation can be treated as a constraint satisfaction problem and thus can be solved by classical CSP solvers. We show that evolutionary algorithms such as differential evolution or CMA-ES produce comparable results while allowing for more flexibility and requiring less computational resources. Due to this flexibility, it is possible to include well known principles of training science during plan generation, resulting in reasonable training plans.
Designing Resistance Training Programs, Fourth Edition, is a guide to developing individualized training programs for both serious athletes and fitness enthusiasts. Two of the world’s leading experts on strength training explore how to design scientifically based resistance training programs, modify and adapt programs to meet the needs of special populations, and apply the elements of program design in the real world.
The fourth edition presents the most current information while retaining the studies that are the basis for concepts, guidelines, and applications in resistance training. Meticulously updated and heavily referenced, the fourth edition contains the following updates: A full-color interior provides stronger visual appeal.Sidebars focus on a specific practical question or an applied research concept, allowing readers to connect research to real-life situations.Multiple detailed tables summarize research from the text, offering an easy way to compare data and conclusions.A glossary makes it simple to find key terms in one convenient location.Newly added instructor ancillaries make the fourth edition a true learning resource for the classroom (available at www.HumanKinetics.com/DesigningResistanceTrainingPrograms).
Designing Resistance Training Programs, Fourth Edition, is an essential resource for understanding and applying the science behind resistance training for any population.
A new dynamic model for predicting road cycling individual time trials with optimal control was created. The model included both lateral and longitudinal bicycle dynamics, 3D road geometry, and anaerobic source depletion. The prediction of the individual time trial performance was formulated as an optimal control problem and solved with an indirect approach to find the pacing and cornering strategies in the respect of the physical/physiological limits of the system. The model was tested against the velocity and power output data collected by professional cyclists in two individual time trial Giro d’Italia data sets: the first data set (Rovereto, n = 15) was used to adjust the parameters of the model and the second data set (Verona, n = 13) was used to test the predictive ability of the model. The simulated velocity fell in the of the experimental data for 32 and 18% of the duration of the course for Rovereto and Verona stages, respectively. The simulated power output fell in the of the experimental data for 50 and 25% of the duration of the course for Rovereto and Verona stages respectively. This framework can be used to input rider’s physical/physiological characteristics, 3D road geometry, and conditions to generate realistic velocity and power output predictions in individual time trials. It, therefore, constitutes a tool that could be used by coaches and athletes to plan the pacing and cornering strategies before the race.
This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices.
This review used a narrative summary of findings from studies that focused on isometric strength training (IST), covering the training considerations that affect strength adaptations and its effects on sports related dynamic performances. IST has been shown to induce less fatigue and resulted in superior joint angle specific strength than dynamic strength training, and benefited sports related dynamic performances such as running, jumping and cycling. IST may be included into athletes’ training regime to avoid getting overly fatigue while still acquiring positive neuromuscular adaptations; to improve the strength at a biomechanically disadvantaged joint position of a specific movement; to improve sports specific movements that require mainly isometric contraction; and when athletes have limited mobility due to injuries. To increase muscle hypertrophy, IST should be performed at 70–75% of maximum voluntary contraction (MVC) with sustained contraction of 3–30 s per repetition, and total contraction duration of>80–150 s per session for>36 sessions. To increase maximum strength, IST should be performed at 80–100% MVC with sustained contraction of 1–5 s, and total contraction time of 30–90 s per session, while adopting multiple joint angles or targeted joint angle. Performing IST in a ballistic manner can maximize the improvement of rate of force development.
The relationship between exercise intensity and time to task failure (P-T relationship) is hyperbolic, and characterized by its asymptote (critical power[CP]) and curvature constant (W'). The determination of these parameters is of interest for researchers and practitioners, but
the testing protocol for CP and W' determination has not yet
been standardized. Conventionally, a series of constant work
rate (CWR) tests to task failure have been used to construct
the P-T relationship. However, the duration, number, and recovery
between predictive CWR and the mathematical model
(hyperbolic or derived linear models) are known to affect CP
and W'. Moreover, repeating CWR may be deemed as a cumbersome
and impractical protocol. Recently, CP and W' have
been determined in field and laboratory settings using time
trials, but the validity of these methods has raised concerns.
Alternatively, a 3-minute all-out test (3MT) has been suggested,
as it provides a simpler method for the determination of CP and
W', whereby power output at the end of the test represents
CP, and the amount of work performed above this end-test
power equates to W'. However, the 3MT still requires an initial
incremental test and may overestimate CP. The aim of this
review is, therefore, to appraise current methods to estimate
CP and W', providing guidelines and suggestions for future
research where appropriate.
System theory is classically applied to describe and to predict the effects of training load on performance. The classic models are structured by impulse-type transfer functions, nevertheless, most biological adaptations display exponential growth kinetics. The aim of this study was to propose an extension of the model structure taking into account the exponential nature of skeletal muscle adaptations by using a genetic algorithm. Thus, the conventional impulse-type model was applied in 15 resistance trained rodents and compared with exponential growth-type models. Even if we obtained a significant correlation between actual and modelled performances for all the models, our data indicated that an exponential model is associated with more suitable parameters values, especially the time constants that correspond to the positive response to training. Moreover, positive adaptations predicted with an exponential component showed a strong correlation with the main structural adaptations examined in skeletal muscles, i.e. hypertrophy (R² = 0.87, 0.96 and 0.99, for type 1, 2A and 2X cross-sectional area fibers, respectively) and changes in fiber-type composition (R² = 0.81 and 0.79, for type 1 and 2A fibers, respectively). Thus, an exponential model succeeds to describe both performance variations with relevant time constants and physiological adaptations that take place during resistance training.
Introduction:
Changes in the parameters of the power-time relationship (critical power (CP) and W') during endurance exercise would have important implications for performance. We tested the hypotheses that CP and W', estimated using the end-test power (EP) and the work done above EP (WEP), respectively, during a the 3-min all-out test (3MT), can be reliably determined, and would be lower, after completing 2-h of heavy-intensity exercise.
Methods:
In study 1, six cyclists completed a 3MT immediately following 2-h of heavy-intensity exercise on two occasions to establish the reliability of EP and WEP. In study 2, nine cyclists completed a control 3MT, and a fatigued 3MT and constant-power-output tests to 30 min or the limit of tolerance (Tlim) below and above F-EP after 2-h of heavy-intensity exercise.
Results:
In study 1, EP (273±52 vs. 276±58 W) and WEP (12.4±4.3 vs. 12.8±4.3 kJ) were not different (P>0.05) and were highly correlated (r=0.99; P<0.001). In study 2, both EP (F-EP: 282±52 vs. C-EP: 306±56 W; P<0.01) and WEP (F-WEP: 14.7±4.9 vs.
C-wep:
18.3±4.1 kJ; P<0.05) were lower following 2-h heavy-intensity exercise. However, maximum O2 uptake was not achieved during exercise >F-EP and Tlim was shorter than 30 min during exercise <F-EP (18.2±10.7 min).
Conclusion:
The EP and WEP may be reliably determined following 2-h heavy-intensity exercise. The 8% and 20% reductions in EP and WEP, respectively, have important implications for performance during endurance exercise. The physiological characterization of EP (and, by extension, CP) may differ in a fatigued compared to a rested state.
Objectives
Our goal was to systematically review the current literature and interpret the findings regarding the effects of periodized (PER) versus non-periodized (NP) resistance training programs aimed at muscular hypertrophy.
News
Controversy exists as to whether a (PER) approach to resistance training is superior to a (NP) approach for maximizing muscular hypertrophy, or vice-versa, or if no differences exist between the approaches.
Prospect and projects
Following a search of the PubMed/MEDLINE, Scopus, and Web of Science electronic databases, 12 studies comprising a total of 31 treatment groups met predetermined inclusion criteria.
Conclusion
Based on the results of our review, we conclude that similar hypertrophic effects may be achieved using either a PER or a NP approach. Importantly, the findings are specific to short-term training interventions, as the average
duration of programs across studies amounted to ∼15 weeks; and to untrained individuals, as only two studies involved resistance-trained participants. A limitation of the reviewed literature also pertains to the small number of studies (n = 3) using direct measures of hypertrophy (i.e., magnetic resonance imaging or ultrasound). Further research is needed to fill in the gaps in the current literature.
Purpose: To compare the mechanisms of fatigue and recovery between upper and lower limbs in the same subjects.
Methods: Twelve healthy young males performed a 2-min sustained maximal voluntary isometric contraction (MVC) of the knee extensors (KE) and on another day a 2-min MVC of the elbow flexors (EF). Neuromuscular function evaluations were performed with both transcranial magnetic and peripheral stimulations before (PRE), at the end of the 2-min MVCs (POSTimm), and 5 more times within 8 min of recovery.
Results: Decreases in MVC and cortical voluntary activation were ~12% (P < 0.001) and ~25% greater (P = 0.04) in KE than EF at POSTimm. Conversely, twitch response decreased ~29% more (P = 0.02) in EF than KE. Changes in motor-evoked potential with fatigue were not different between upper and lower limbs (P > 0.05) whereas the increase in silent period duration was ~30% greater in EF than KE (P < 0.05).
Conclusion: Upper and lower limbs presented different magnitudes of total, central and peripheral fatigue. Total neuromuscular fatigue and central fatigue were greater in KE than EF. Conversely, peripheral fatigue and corticospinal inhibition were greater in EF than KE.
An algorithm for the numerical solution of parameterized optimal control problems is presented, which is based on multiple shooting in connection with a recursive quadratic progrmrming technique. A condensing algorithm for the solution of the approximating linearly constrained quadratic subproblems, and high rank update procedures are introduced, which are especially suited for optimal control problems and lead to significant improvements of the convergence behaviour and reductions of computing time and storage requirements. The algorithm is completely derivative-free due to internal numerical differentiation schemes, it can be conveniently combined with indirect multiple shooting. Numerical results are given in the field of aerospace engineering.
Sustained physical exercise leads to a reduced capacity to produce voluntary force that typically outlasts the exercise bout. This "fatigue" can be due both to impaired muscle function, termed "peripheral fatigue", and a reduction in the capacity of the central nervous system to activate muscles, termed "central fatigue". In this mini-review we consider the factors that determine the recovery of voluntary force generating capacity after various types of exercise. After brief, high-intensity exercise there is typically a rapid restitution of force that is due to recovery of central fatigue (typically within 2min) and aspects of peripheral fatigue associated with excitation-contraction coupling and re-perfusion of muscles (typically within 3-5 min). Complete recovery of muscle function may be incomplete for some hours, however, due to prolonged impairment in intracellular Ca(2+) release or sensitivity. After low-intensity exercise of long duration, voluntary force typically shows rapid, partial, recovery within the first few minutes, due largely to recovery of the central, neural component. However, this ability to voluntarily activate muscles may not recover completely within 30 minutes after exercise. Recovery of peripheral fatigue contributes comparatively little to the fast initial force restitution, and is typically incomplete for at least 20-30 minutes. Work remains to identify what factors underlie the prolonged central fatigue that usually accompanies long-duration single joint and locomotor exercise, and to document how the time-course of neuromuscular recovery is affected by exercise intensity and duration in locomotor exercise. Such information could be useful to enhance rehabilitation and sports performance.
In this thesis, we advance efficient methods to solve parameter estimation problems constrained by partial differential equations (PDEs). If PDE constrained parameter estimation problems are solved by derivative based methods, here, the generalized Gauss--Newton method, and multiple shooting, the numerical effort growths drastically with the number of states. The reduced approach couples the computation of the Jacobians and the subsequent block Gaussian elimination using directional derivatives by exploiting the special structure of the constraints which arises from the shooting formulation. Thus, the computational effort is reduced to the one of single shooting. The advantages of the new method in comparison to the common approach are illustrated by means of two academic examples. Furthermore, we are the first to adapt methods of optimum experimental design for parameter estimation to processes of microbial enhanced oil recovery. We consider a nonlinear coupled PDE model which consists of two parts. The first part, the black oil model, describes two phase flow through porous media and a model of convection--diffusion--reaction type depicts the transport and growth effects of bacteria, nutrients, gas and other metabolites in the two phases. A mixed discontinuous Galerkin finite element discretization is applied in space. The discretized model is solved in time by the extended IMPES method. Under the assumption of rotational symmetry, we examine a one dimensional model formulation for parameter estimation and optimum experimental design. We follow the principles of internal numerical differentiation and algorithmic differentiation to evaluate the required derivatives, i.e., the derivatives of the model functions are computed by software tools and we solve the tangential problems with respect to the model parameters and the control variables. By optimum experimental design, a new experiment is planned to reduce the uncertainties of the estimated parameters. The designed experiment differs substantially from the experiments which are usually realized in practice. The confidence intervals for the estimated parameters are reduced by a factor of one hundred. The developed methods for parameter estimation are implemented in the software package PAREMERA which is embedded in the optimum experimental design software VPLAN. The model equations for microbial enhanced oil recovery are implemented in a simulation tool which computes not only the nominal equation but also evaluates the derivatives with respect to parameters and controls up to second order.
Introducción: el propósito de este estudio fue investigar la eficacia del entrenamiento diario de una repetición máxima (1RM) de la sentadilla en fuerza máxima. Material y método: tres levantadores de peso de competición realizaron la sentadilla durante 37 días consecutivos y se reportan como casos individuales. Participante 1 (P1) (masa corporal = 80,5 kg; edad = 28 años) y participante 3 (P3) (masa corporal = 108,8 kg; edad = 34 años) eran levantadores de fuerza; participante 2 (P2) (masa corporal = 64,1 kg; edad = 19 años) fue un levantador de pesas. Cada participante tenía por lo menos 5 años de experiencia con la posición en sentadilla de formación. Durante los días 1-35, los participantes realizaron una sentadilla de 1RM seguida por 5 conjuntos de volumen de 3 repeticiones al 85% o 2 repeticiones al 90% de la 1RM diario. En el día 36, los participantes realizan solo una serie de 1 repetición al 85% de 1RM del día 1; y el día 37 realizaron un 1RM. Resultados: cambios absolutos y porcentaje para P1 del 1 día al 37: + 5 kg/2,3% y desde el primer día al máximo (1RM era el mayor) + 12,5 kg/5,8%. P2 experimentó un aumento de 13,5 kg/10,8% en 1RM del día 1 al 37 y del día 1 al máximo. P3 demostró un aumento de 21 kg/9,5% del día 1 al 37 y del día 1 al máximo. Los tres participantes exhibieron significativa (p < 0,05) las correlaciones entre el tiempo (días) y 1RM (P1: r = 0,65, P2: r = 0,78, P3: r = 0,48). Conclusión: nuestros resultados sugieren que el entrenamiento diario de 1RM había producido efectivamente cambios significativos en la máxima fuerza en los atletas de fuerza competitiva en un periodo relativamente corto de entrenamiento.
This paper concerns some practical issues associated with the formulation of sequential quadratic programming (SQP) methods for large-scale nonlinear optimization. SQP methods find approximate solutions of a sequence of quadratic programming (QP) subproblems in which a quadratic model of the Lagrangian is minimized subject to the linearized constraints. Numerical results are given for 1153 problems from the CUTEst test collection. The results indicate that SQP methods based on maintaining a quasi-Newton approximation to the Hessian of the Lagrangian function are both reliable and efficient for general large-scale optimization problems. In particular, the results show that in some situations, quasi-Newton SQP methods are more efficient than interior methods that utilize the exact Hessian of the Lagrangian. The paper concludes with discussion of an SQP method that employs both approximate and exact Hessian information. In this approach the quadratic programming subproblem is either the conventional subproblem defined in terms of a positive-definite quasi-Newton approximate Hessian or a convexified subproblem based on the exact Hessian.
Key points:
Muscle fatigue can be defined as the transient decrease in maximal force that occurs in response to muscle use. Fatigue develops because of a complex set of changes within the neuromuscular system that are difficult to evaluate simultaneously in humans. The skeletal muscle of older adults fatigues less than that of young adults during static contractions. The potential sources of this difference are multiple and intertwined. To evaluate the individual mechanisms of fatigue, we developed an integrative computational model based on neural, biochemical, morphological and physiological properties of human skeletal muscle. Our results indicate first that the model provides accurate predictions of fatigue and second that the age-related resistance to fatigue is due largely to a lower reliance on glycolytic metabolism during contraction. This model should prove useful for generating hypotheses for future experimental studies into the mechanisms of muscle fatigue.
Abstract:
During repeated or sustained muscle activation, force-generating capacity becomes limited in a process referred to as fatigue. Multiple factors, including motor unit activation patterns, muscle fibre contractile properties and bioenergetic function, can impact force-generating capacity and thus the potential to resist fatigue. Given that neuromuscular fatigue depends on interrelated factors, quantifying their independent effects on force-generating capacity is not possible in vivo. Computational models can provide insight into complex systems in which multiple inputs determine discrete outputs. However, few computational models to date have investigated neuromuscular fatigue by incorporating the multiple levels of neuromuscular function known to impact human in vivo function. To address this limitation, we present a computational model that predicts neural activation, biomechanical forces, intracellular metabolic perturbations and, ultimately, fatigue during repeated isometric contractions. This model was compared with metabolic and contractile responses to repeated activation using values reported in the literature. Once validated in this way, the model was modified to reflect age-related changes in neuromuscular function. Comparisons between initial and age-modified simulations indicated that the age-modified model predicted less fatigue during repeated isometric contractions, consistent with reports in the literature. Together, our simulations suggest that reduced glycolytic flux is the greatest contributor to the phenomenon of age-related fatigue resistance. In contrast, oxidative resynthesis of phosphocreatine between intermittent contractions and inherent buffering capacity had minimal impact on predicted fatigue during isometric contractions. The insights gained from these simulations cannot be achieved through traditional in vivo or in vitro experimentation alone.
In dieser Arbeit werden Techniken beschrieben, die es erlauben (höhere) Ableitungen und Taylorapproximationen solcher Computerprogramme effizient zu berechnen. Auch inbesondere dann, wenn die Programme Algorithmen der numerischen linearen Algebra (NLA) enthalten. Im Gegensatz zur traditionellen algorithmischen Differentiation (AD), bei der die zugrunde liegenden Algorithmen um zusätzliche Befehlere erweitert werden, sind in dieser Arbeit die Zerlegungen durch definierende Gleichungen charakterisiert. Basierend auf den definierenden Gleichungen werden Strukturausnutzende Algorithmen hergeleitet. Genauer, neuartige Algorithmen für die Propagation von Taylorpolynomen durch die QR, Cholesky und reell-symmetrischen Eigenwertzerlegung werden präsentiert. Desweiteren werden Algorithmen für den Rückwärtsmodus der AD hergeleitet, welche im Wesentlichen nur die Faktoren der Zerlegungen benötigen. Im Vergleich zum traditionellen Ansatz, bei dem alle Zwischenergebnisse gespeichert werden, ist dies eine Reduktion von O(N^3) zu O(N^2) für Algorithmen mit O(N^3) Komplexität. N ist hier die Größe der Matrix. Zusätzlich kann bestehende, hoch-optimierte Software verwendet werden. Ein Laufzeitvergleich zeigt, dass dies im Vergleich zum traditionellen Ansatz zu einer Beschleunigung in der Größenordnung 100 führen kann. Da die NLA Funktionen als Black Box betrachtet werden, ist desweiteren auch der Berechnungsgraph um Größenordnungen kleiner. Dies bedeutet, dass Software, welche Operator Overloading benutzt, weniger Overhead hervorruft und auch weniger Speicher benötigt.