Comparison of linear, hyperbolic and double-hyperbolic models to assess the force-velocity relationship in multi-joint exercises.

TLDR: The F–V relationship in multi-joint exercises was double-hyperbolic, which was unrelated with lower muscle excitation levels, however, linear models may be valid to assess F 0, maximal muscle power and velocity between 25% and 100% of F 0.

Abstract: This study assessed the validity of linear, hyperbolic and double-hyperbolic models to fit measured force–velocity (F–V) data in multi-joint exercises and the influence of muscle excitation on the ...

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Table 1. Parameters obtained from linear, hyperbolic and double-hyperbolic models.

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The version of record of this manuscript has been published and is available in European
Journal of Sport Science 07/April/2020
http://www.tandfonline.com/10.1080/17461391.2020.1753816
Title: Comparison of linear, hyperbolic and double-hyperbolic models to assess the force-velocity
relationship in multi-joint exercises
Authors: Julian Alcazar
1,2
, Fernando Pareja-Blanco
3
, Carlos Rodriguez-Lopez
1,2
, Roberto
Navarro-Cruz
1,2
, Pedro J. Cornejo-Daza
3
, Ignacio Ara
1,2
and Luis M. Alegre
1,2
.
Institutions:
1. GENUD Toledo Research Group, Universidad de Castilla-La Mancha. Toledo, Spain.
2. CIBER of Frailty and Healthy Aging (CIBERFES). Madrid, Spain.
3. Physical Performance & Athletic Research Center, Universidad Pablo de Olavide. Seville, Spain.
Corresponding author:
Luis M. Alegre
Email: luis.alegre@uclm.es
Institution: GENUD Toledo Research Group, Universidad de Castilla-La Mancha
Address: Avda. Carlos III, S/N, 45071, Toledo (Spain)
Phone: +34 925 268 800 (Ext: 5506)

Word count: 4214 (main text)
Abstract
This study assessed the validity of linear, hyperbolic and double-hyperbolic models to fit measured
force-velocity (F-V) data in multi-joint exercises and the influence of muscle excitation on the F-
V relationship. The force-joint angle and F-V relationships were assessed in 10 cross-training
athletes and 14 recreationally resistance-trained subjects in the unilateral leg press (LP) and
bilateral bench press (BP) exercises, respectively. A force plate and a linear encoder were installed
to register external force and velocity, respectively. Muscle excitation was assessed by surface
EMG recording of the quadriceps femoris, biceps femoris and gluteus maximus muscles during the
unilateral LP. Linear, Hill’s (hyperbolic) and Edman’s (double-hyperbolic) equations were fitted
to the measured F-V data and compared. Measured F-V data were best fitted by double-hyperbolic
models in both exercises (p<0.05). F-V data deviated from the rectangular hyperbola above a
breakpoint located at 90% of measured isometric force (F
0
) and from the linearity at ≤45% of F
0
(both p<0.05). Hyperbolic equations overestimated F
0
values by 13±11% and 6±6% in the LP and
BP, respectively (p<0.05). No differences were found between muscle excitation levels below and
above the breakpoint (p>0.05). Large associations between variables obtained from linear and
double-hyperbolic models were noted for F
0
, maximum muscle power, and velocity between 25-
100% of F
0
(r=0.70-0.99; all p<0.05). The F-V relationship in multi-joint exercises was double-
hyperbolic, which was unrelated with lower muscle excitation levels. However, linear models may
be valid to assess F
0
, maximal muscle power and velocity between 25-100% of F
0
.

Keywords: torque-velocity; muscle power; muscle mechanics; maximal unloaded velocity;
biphasic force-velocity; velocity-based training.
1. Introduction
The evaluation of the force-velocity (F-V) relationship in different populations has extended
widespread in recent years due to its functional significance (1-5). However, great discrepancies
among studies exist in terms of the equation used during F-V profiling (6). According to the F-V
relationship, the slower a muscle shortens the greater the force it can generate and vice versa (6).
The relation between force and velocity has been most frequently fitted to a rectangular
hyperbola (7). Hill’s equation has been considered a reference equation in different muscle
preparations and during in situ conditions in humans. However, several studies found deviations
from Hill’s hyperbolic function in the high-force/low-velocity region of the F-V relationship (6).
Measured isometric force (F
0
) values in muscle preparations were 20-32% lower than those
predicted by Hill’s rectangular hyperbola (8, 9). Consequently, an alternative F-V ‒ double-
hyperbolic ‒ equation that considered the biphasic nature of the F-V relation and fitted more
adequately to the high-force/low-velocity region of the F-V relationship was proposed (10).
On the other hand, recent studies investigating the F-V relationship in humans during multi-
joint exercises have reported that the F-V relationship follows a strict linear pattern (11-14). The
linearity of the F-V relationship presents some practical advantages, since it can be assessed by
collecting a few F-V data (15-17) while providing relevant information that can help optimize
physical performance (3) and fatigue monitoring (18). However, some studies have found that F-
V data collected from multi-joint exercises deviate from the linearity at very low forces (19, 20)
and the existence of a double-hyperbolic pattern has been suggested at very high forces (20). The

existence of a possible central inhibitory mechanism (lower muscle excitation levels) at specific
ranges of the F-V relationship might be behind these deviations (21), which deserves to be further
investigated. Importantly, the misconception of the shape of the F-V relationship can lead to serious
errors in the estimation of several parameters derived from linear equations, such as F
0
,
maximum
muscle power (P
max
) or maximal unloaded shortening velocity (V
0
). Of note, there might exist
differences in the shape of the F-V relationship between multi-joint lower and upper body exercises
due to the influence that muscle architecture and fiber type distribution have on the F-V relationship
and the substantial differences in these parameters that may exist between lower and upper body
skeletal muscles (22, 23).
Thus, the main goals of the present investigation were to compare the ability of linear,
hyperbolic and double-hyperbolic models to fit measured external F-V data from multi-joint lower
and upper body exercises, and to assess the influence of muscle excitation levels of quadriceps
femoris (QF), biceps femoris (BF) and gluteus maximus (GM) muscles on the F-V relation.
2. Material and methods
2.1 Subjects
Unilateral leg press (LP) assessment was conducted in 10 cross-training male athletes (age=
25.8±5.4 years; height= 1.77±0.04 m; body mass= 78.4±3.2 kg; maximum isometric force=
2831.3±299.0 N) competing at the regional and national levels with a resistance training
background of 4.3±2.6 years (8.7±2.4 h per week during the last year). On the other hand, bilateral
bench press (BP) assessment was conducted in 14 recreationally resistance-trained male subjects
(age= 24.0±4.3; height= 1.74±0.06 m; body mass= 73.7±9.3 kg; maximum isometric force=
1682.1±269.2 N) with a resistance training background of 2.8±0.8 years (2.5±0.8 h per week during

the last year). All the participants gave their written informed consent. The study was performed
in accordance with the Helsinki Declaration and approved by the local ethical committees.
2.2 Experimental setting
Unilateral (left) leg extensions were performed on a LP machine (Selection MED, Technogym,
Italy), while BP performance was measured on a Smith machine with no counterweight mechanism
(Multipower Fitness Line, Peroga, Spain). A force plate (LP: Type 9286BA, Kistler, Switzerland;
BP: T-Force System, Ergotech, Spain) and a linear encoder (LP: Linear encoder, Chronojump
Bosco System, Spain; BP: T-Force System, Ergotech, Spain) were installed on the equipment to
evaluate external force and velocity data, respectively. In the LP setting, the force plate was
mounted on the feet platform of the apparatus and the linear encoder was attached to the weights
lifted during the exercise. In the BP setting, the force plate was mounted under the bench where
the participants lay down with their feet over the bench (specifically built to be used over the force
plate) and the linear encoder was attached to the bar used during the exercise.
The position of the subjects was standardized and kept the same between repetitions in both
exercises. Both knee and elbow joint angles during the isometric contractions were measured by
video analysis (HD Pro Webcam C920 1080p, 30 Hz, Logitech, Switzerland) in the LP and BP
exercises, respectively. Superficial anatomical markers were placed on the skin of the participants
on the superior border of the greater trochanter, inferior border of the lateral condyle and inferior
border of the lateral malleolus for the LP setting, and on the lateral border of the acromion, the
lateral epicondyle and the midpoint between radial and ulnar styloids for the BP setting. The camera
was placed at a suitable distance (1.5 m), height (0.75 m) and position to capture the plane of the
movement executed in the LP and BP exercises (sagittal and transverse planes of the human body,
respectively).

Frequently asked questions

Q1. What are the contributions in this paper?

This study assessed the validity of linear, hyperbolic and double-hyperbolic models to fit measured force-velocity ( F-V ) data in multi-joint exercises and the influence of muscle excitation on the FV relationship. 

Q2. What are the future works in this paper?

In addition, further studies should evaluate the reliability of the 𝐹0 ∗/𝐹0 ratio across different days. Future studies should evaluate the validity of V0 derived from double-hyperbolic models and provide further details on the range of forces that should be measured to capture the truly characteristics of the F-V relationship in humans during multi-joint exercises. A recent review work on the shape of the F-V relationship has suggested that the apparent linear F-V relationship noted in multi-joint exercises may be in fact a misconception resulting from the relatively narrow range of forces that is usually evaluated in human studies ( 6 ). However, F0, Pmax and velocity between 25-100 % of F0 obtained from linear modelling were strongly related with those obtained from double-hyperbolic equations, which suggests that these may still be highly valuable for physical performance monitoring. 

Q3. What was used to synchronize force and EMG with velocity data?

An external custom-built trigger (USB-6501, National Instruments, USA) associated to specific software (LabView, National Instruments, USA) was used to synchronize force and EMG with velocity data. 

Q4. What should be done to improve the accuracy of the V0 relationship during multi-joint?

Double-hyperbolic models should be used to obtain more accurate information on the F-V relationship during multi-joint exercises, requiring longer testing and processing time though. 

Q5. What is the linearity of the F-V relationship in that range of forces?

The linearity of the F-V relationship in that range of forces would be facilitated by the deviations from the rectangular hyperbola noted at high forces (i.e. by the double-hyperbolic pattern). 

Q6. What should be considered when comparing their results with other studies?

F-V data were collected in this study over a partial range of movement, and thus the influence of the force-joint angle relationship and the history dependence of muscle should be considered when comparing their results with data collected at different ranges of movement. 

Q7. What was the amplitude of the lateral and vastus medialismuscles?

The EMG amplitudes from rectus femoris, vastus lateralis and vastus medialismuscles were averaged to obtain QF muscle excitation, since no differences existed between individual muscles and all of them contribute to knee extension through the patellar tendon. 

Q8. What was the simplest method used to evaluate the isometric behavior of the linear relationship?

In addition, a linear regression model was fitted to F-V data located above 50% of F0 to assess whether measured F-V data deviated from the linear equation below a certain level of force (12, 13):𝐹 = 𝑆FV 𝑉 + 𝐹0 (3)where SFV is the slope of the linear F-V relationship. 

Q9. What was the simplest method used to evaluate the isometric behavior of the F-V?

the coefficient of determination (R2) and the standard error of the estimate (SEE) were used to assess the fitting of the different models to the measured F-V data. 

Q10. What is the hypothesis behind the deviation from the rectangular hyperbola?

The latter is supported by observations made during electrically evoked muscle contractions (33) and strengthens the hypothesis that decreased force per cross-bridge may be behind this deviation (6). 

Q11. What was the level of significance of the analysis?

Statistical analyses were performed using SPSS v20 (SPSS Inc., Chicago, Illinois), and the level of significance was set at α=0.05. 

Q12. What is the significance of the linearity of the F-V relationship?

The linearity of the F-V relationship presents some practical advantages, since it can be assessed by collecting a few F-V data (15-17) while providing relevant information that can help optimize physical performance (3) and fatigue monitoring (18). 

Q13. What was the method used to evaluate the appearance of a breakpoint?

Joint angles during the isometric repetitions were assessed from the video recordings by specialized software (Tracker 4.11.0, https://physlets.org/tracker/).A semilog approach was used to evaluate the appearance of a breakpoint derived from thedouble-hyperbolic behavior of the F-V relationship (9, 10). 

Q14. What was the average change in the angle of the knee?

The average change (fatigue) observed across the different knee joint angles was ‒3.5±4.8% (‒2.1±5.1%, ‒2.5±5.8%, ‒3.8±6.1%, ‒3.9±4.3% and ‒5.1±2.9% at ~100°, ~110°, ~120°, ~130° and ~140°, respectively).