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

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 (F0) and from the linearity at ≤45% of F0 (both p<0.05). Hyperbolic equations overestimated F0 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 F0, maximum muscle power, and velocity between 25-100% of F0 (r=0.70-0.99; all p<0.05). The F-V relationship in multi-joint exercises was doublehyperbolic, which was unrelated with lower muscle excitation levels. However, linear models may be valid to assess F0, maximal muscle power and velocity between 25-100% of F0.

Exploring the velocity end of the force-velocity relationship in acyclic lower limb extensions

“…Cuk et al 2014;Jaric 2015), which validates the physiological relevance of Pmax-L. In the present study, curvilinear models showed lower value of Pmax than linear model (Table 3) in contrast to similar or higher values of Pmax previously reported (18,20). The higher Pmax values obtained from curvilinear models were certainly due to a lack of experimental force and velocity collected below 50 %F0 when fitting the linear model, which have led a lower v0-L and, in turn, a lower Pmax-L (20).…”

“…In the present study, curvilinear models showed lower value of Pmax than linear model (Table 3) in contrast to similar or higher values of Pmax previously reported (18,20). The higher Pmax values obtained from curvilinear models were certainly due to a lack of experimental force and velocity collected below 50 %F0 when fitting the linear model, which have led a lower v0-L and, in turn, a lower Pmax-L (20). Aside from this methodology, the similar values of Pmax were obtained between linear and curvilinear models due to the high number of resistive/loading conditions around the optimal velocity condition (18).…”

“…To clarify the questioned linearity of the F-v relationship in acyclic lower limb extensions, several studies have experimentally explored its ends and compared the GoF of Linear to that of the curvilinear-shaped functions. On its force end (velocity conditions < 50 %v0), the F-v relationship was shown to be linear from ~5 %v0 (18,19), beyond which some studies argue that it would exist a breakpoint (20)(21)(22)(23). From this breaking point, the force and velocity data would follow a curvilinear relationship, best modelled by Edman's exponential function (20)(21)(22).…”

“…On its force end (velocity conditions < 50 %v0), the F-v relationship was shown to be linear from ~5 %v0 (18,19), beyond which some studies argue that it would exist a breakpoint (20)(21)(22)(23). From this breaking point, the force and velocity data would follow a curvilinear relationship, best modelled by Edman's exponential function (20)(21)(22). Nevertheless, in spite of this non-continuity of the linear curve, F0 estimations from the curvilinear model were not significantly different and very close to the linear model (Alcazar et al 2020).…”

“…Nevertheless, in spite of this non-continuity of the linear curve, F0 estimations from the curvilinear model were not significantly different and very close to the linear model (Alcazar et al 2020). Consequently, the linear model could be used as the most relevant and the simplest one to accurately describe the entire F-v relationship on its force end and in turn determine F0 (20). On the velocity end of the F-v relationship (velocity conditions > 50 %v0), only the works of Yamauchi et al have included measurements of lower limb force production up to ~80-97 %v0 (e.g.…”