A novel equation that incorporates the linear and hyperbolic nature of the force–velocity relationship in lower and upper limb exercises

Alcázar, Julián; Pareja‐Blanco, Fernando; Gutierrez-Reguero, Hector; Sánchez-Valdepeñas, Juan; Cornejo-Daza, Pedro J.; Ara, Ignacio; Alegre, Luis M.

doi:10.1007/s00421-022-05006-1

Cited by 7 publications

(17 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, a hybrid equation 28 ( Equation ) that combines a linear and a curvilinear (Hill‐type) region was used in the current study. Briefly, this equation includes the classical linear 29 and hyperbolic 30 equations, as well as an associated coefficient to each of the two [ c 1 ( Equation ) and c 2 ( Equation ), respectively] that provides different albeit complementary weights ( c 1 + c 2 = 1) to each of them as a function of relative intensity (i.e., in this case torque relative to maximum isometric torque) 28

normalV goodbreak= c_{1} ([], \frac{T - {normalT}_{0}}{S}) goodbreak+ c_{2} ([], \frac{({normalT}_{0} - T) b}{T + a})

c_{1} goodbreak= \frac{1}{1 + e^{{((), normalk goodbreak- normalT / T_{0})}^{S}}}

c_{2} goodbreak= \frac{1}{1 + e^{{((), normalT / T_{0} goodbreak- normalk)}^{S}}}

Section: Methodsmentioning

confidence: 99%

“… 18 , 26 In both multi‐joint and single‐joint muscle actions, force decreases linearly as a function of velocity in a specific limited region (from high to moderate forces), whereas force decreases as velocity increases in a curvilinear (convex) fashion in other region (from moderate to null forces). 27 Therefore, a hybrid equation 28 ( Equation 1 ) that combines a linear and a curvilinear (Hill‐type) region was used in the current study. Briefly, this equation includes the classical linear 29 and hyperbolic 30 equations, as well as an associated coefficient to each of the two [ c 1 ( Equation 2 ) and c 2 ( Equation 3 ), respectively] that provides different albeit complementary weights ( c 1 + c 2 = 1) to each of them as a function of relative intensity (i.e., in this case torque relative to maximum isometric torque).…”

Section: Methodsmentioning

confidence: 99%

“…Briefly, this equation includes the classical linear 29 and hyperbolic 30 equations, as well as an associated coefficient to each of the two [ c 1 ( Equation 2 ) and c 2 ( Equation 3 ), respectively] that provides different albeit complementary weights ( c 1 + c 2 = 1) to each of them as a function of relative intensity (i.e., in this case torque relative to maximum isometric torque). 28

where V is velocity (angular velocity in this case), c 1 is the coefficient associated to the linear equation, c 2 is the coefficient associated to the hyperbolic equation, T is torque, T 0 is estimated maximum isometric torque, S is the slope of the linear region, a and b are Hill‐type constants, k is a constant that determines the point of transition from the linear to the hyperbolic equation (i.e., when c 1 and c 2 are both 0.5), and s is a constant that determines how smooth the model runs from the linear to the hyperbolic region. Estimated maximal unloaded velocity (V 0 ) was calculated as the intercept of the velocity axis, and S /T 0 and a /T 0 were calculated to describe the decrease in torque as a function of contraction velocity in the linear (i.e., slope) and hyperbolic (i.e., curvature) regions, respectively, of the T‐V relationship.…”

Section: Methodsmentioning

confidence: 99%

“… Analysis of the torque‐velocity (A and C) and power velocity (B and D) relationships in two participants (subject 1: A and B; subject 2: C and D). A hybrid equation 28 was applied to measured data after excluding those data that fulfilled the following criteria: Maximum isometric torque values lower than any of the registered dynamic torque values (in A); dynamic trials showing a lower torque value for a corresponding velocity compared with a faster contraction (in A); and dynamic trials showing a lower power for a corresponding velocity compared with both a slower and a faster contraction (in B and D). …”

Section: Methodsmentioning

confidence: 99%

See 3 more Smart Citations

Ten‐year longitudinal changes in muscle power, force, and velocity in young, middle‐aged, and older adults

Alcázar

Delecluse

Thomis

et al. 2023

J cachexia sarcopenia muscle

Self Cite

View full text Add to dashboard Cite

Background Maximum muscle power (P max ) is a biomarker of physical performance in all ages. No longitudinal studies have assessed the effects of aging on P max obtained from the torque-velocity (T-V) relationship, which should be considered the 'gold standard'. This study evaluated the longitudinal changes in the T-V relationship and P max of the knee-extensor muscles in young, middle-aged, and older adults after 10 years of follow-up. Methods Four hundred eighty-nine subjects (311 men and 178 women; aged 19-68 years) were tested at baseline and after a 10-year follow-up. Anthropometric data, daily protein intake, physical activity level (PAL), and knee-extension muscle function (isometric, isokinetic, and isotonic) were evaluated. A novel hybrid equation combining a linear and a hyperbolic (Hill-type) region was used to obtain the T-V relationship and P max of the participants, who were grouped by sex and age (young: 20-40 years; middle-aged: 40-60 years; and old: ≥60 years). Linear mixed-effect models were used to assess effects of time, sex, and age on T-V parameters, P max , and body mass index (BMI). Additional analyses were performed to adjust for changes in daily protein intake and PAL. Results P max decreased in young men (À0.6% per year; P < 0.001), middle-aged men and women (À1.1% to À1.4% per year; P < 0.001), and older men and women (À2.2% to À2.4% per year; P ≤ 0.053). These changes were mainly related to decrements in torque at P max at early age and to decrements in both torque and velocity at P max at older age. BMI increased among young and middle-aged adults (0.2% to 0.5% per year; P < 0.001), which led to greater declines in relative P max in those groups. S/T 0 , that is, the linear slope of the T-V relationship relative to maximal torque, exhibited a significant decline over time (À0.10%T 0 •rad•s À1 per year; P < 0.001), which was significant among middle-aged men and old men and women (all P < 0.05). Annual changes in PAL index were significantly associated to annual changes in P max (P = 0.017), so the overall decline in P max was slightly attenuated in the adjusted model (À5.26 vs. À5.05 W per year; both P < 0.001). Conclusions P max decreased in young, middle-aged, and older adults after a 10-year follow-up. The early declines in P max seemed to coincide with declines in force, whereas the progressive decline at later age was associated with declines in both force and velocity. A progressively blunted ability to produce force, especially at moderate to high movement velocities, should be considered a specific hallmark of aging.

show abstract

normalV goodbreak= c_{1} ([], \frac{T - {normalT}_{0}}{S}) goodbreak+ c_{2} ([], \frac{({normalT}_{0} - T) b}{T + a})

c_{1} goodbreak= \frac{1}{1 + e^{{((), normalk goodbreak- normalT / T_{0})}^{S}}}

c_{2} goodbreak= \frac{1}{1 + e^{{((), normalT / T_{0} goodbreak- normalk)}^{S}}}

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Ten‐year longitudinal changes in muscle power, force, and velocity in young, middle‐aged, and older adults

Alcázar

Delecluse

Thomis

et al. 2023

J cachexia sarcopenia muscle

Self Cite

View full text Add to dashboard Cite

show abstract

“…Regarding the equation chosen to model the F-V relationship, curvilinear models seem to fit registered F-V data better than linear models. However, it has been showed that F-V data deviate from the traditional rectangular hyperbola at both ends of the relationship (Alcazar et al, 2019), and so other models have been recently proposed in the literature (Alcazar et al, 2022). Even so linear models may still be valid and helpful in some cases.…”

Section: The Assessment Of the F-v Relationshipmentioning

confidence: 99%

Editorial: The force-velocity relationship: Assessment and adaptations provoked by exercise, disuse and disease

Alcázar

Csapo²,

Alegre³

2022

Front. Physiol.

View full text Add to dashboard Cite

The role of supervision and motivation during exercise on physical and mental health in older adults: a study protocol for a randomized controlled trial (PRO-Training project)

Gómez-Redondo,

Valenzuela,

Martínez-de-Quel

et al. 2024

BMC Geriatr

Self Cite

View full text Add to dashboard Cite

Background Although supervised exercise is frequently recommended for older adults, its superiority over unsupervised exercise remains uncertain. Furthermore, whether motivational techniques could help to enhance the effectiveness of the latter remains to be elucidated. The present randomized controlled trial aims to determine the role of supervision and motivational strategies on the safety, adherence, efficacy, and cost-effectiveness of different exercise programs for improving physical and mental health in older adults. Methods Participants (n = 120, aged 60–75 years) will be randomly allocated into five groups: 1-Control (CON), 2-Supervised exercise without motivational intervention (SUP), 3- Supervised exercise with motivational intervention (SUP +), 4- Unsupervised exercise without motivational intervention (UNSUP) and 5- Unsupervised exercise with motivational intervention (UNSUP +). Over 24 weeks, all exercise groups will participate in a multicomponent exercise program three times/week (performed in group classes at a center for SUP and SUP + , or home without supervision but with the help of a mobile app for UNSUP and UNSUP +), while the CON group will maintain their usual lifestyle. The motivational intervention (for SUP + and UNSUP + groups) will be based on the self-determination theory, including strategies such as phone calls, interactive workshops, motivational messages, informative infographics and videos. Primary outcomes will include safety, adherence, costs, and lower-body muscular function using a leg press machine. Secondary outcomes will include upper-body muscular function, physical and cardiorespiratory function, blood pressure and heart rate, body composition, health-related quality of life, cognitive performance, anxiety, depression, physical activity levels, sleep and sedentarism, biochemical markers, motivators and barriers to exercise. Assessments will be conducted at baseline, mid-intervention (i.e., week 13), at the end of the intervention (i.e., week 25), and 24 weeks later (i.e., week 49). Discussion The findings of this trial might provide valuable insights into the role of supervision and motivational strategies on the effectiveness of exercise programs for older adults. Additionally, the study could contribute to developing cost-effective interventions, supporting the design of future public policies for healthy aging. Trial registration NCT05619250. Registered 16 November 2022.

show abstract

A novel equation that incorporates the linear and hyperbolic nature of the force–velocity relationship in lower and upper limb exercises

Cited by 7 publications

References 27 publications

Ten‐year longitudinal changes in muscle power, force, and velocity in young, middle‐aged, and older adults

Ten‐year longitudinal changes in muscle power, force, and velocity in young, middle‐aged, and older adults

Editorial: The force-velocity relationship: Assessment and adaptations provoked by exercise, disuse and disease

The role of supervision and motivation during exercise on physical and mental health in older adults: a study protocol for a randomized controlled trial (PRO-Training project)

Contact Info

Product

Resources

About