Abstract
Background:
Prognostic indicators of lack of strength progression or poor self-reported function may alter rehabilitation decision-making following anterior cruciate ligament reconstruction (ACLR). The torque-velocity relationship is a noninvasive measure of muscle function that is altered following ACLR, but its prognostic value has not been explored.
Purpose:
To compare the torque-velocity relationship of knee extensors (quadriceps) and flexors (hamstrings) between the ACLR and uninvolved limbs and to determine whether the torque-velocity relationship was prognostic of subsequent achievement of satisfactory strength.
Study Design:
Cohort study; Level of evidence, 3.
Methods:
Participants following ACLR with bone–patellar tendon–bone or hamstring autografts completed isokinetic knee extension and flexion at 90°/s and 180°/s bilaterally, the International Knee Documentation Committee (IKDC) form, and Knee injury and Osteoarthritis Outcome Score (KOOS) at approximately 5.5 and 8.3 months post-ACLR. The torque-velocity relationship was defined as the difference in torque production across velocities, and relationships were analyzed using 2 × 2 analyses of variance. Binomial logistic regressions were used to determine the association between the torque-velocity relationship, age, sex, and time postsurgery with satisfactory knee extension strength, IKDC, and KOOS at visit 2.
Results:
This study included 189 participants (22.4 ± 9.3 years; 55.0% female). There were significant increases in the quadriceps torque-velocity relationship from visit 1 (0.22 Nm/kg) to visit 2 (0.34 Nm/kg, P < .001) but no differences in the uninvolved limb (P = .46) or for the hamstrings (P = .20). When controlling for sex, age, and graft type, higher visit 1 quadriceps torque-velocity relationships were predictive of a higher likelihood of achieving satisfactory knee extension strength (≥1.23 Nm/kg; odds ratio [OR], 1.05; P = .04). The model was associated with an acceptable IKDC score (≥75.9, P = .01), but the only significant individual predictor was age (OR, 0.94; P < .01). The model was not associated with KOOS Sport score at visit 2 (P = .24).
Conclusion:
Quadriceps torque-velocity relationships of the ACLR limbs increased across time but remained less than uninvolved limbs. Hamstrings torque-velocity relationships remained similar between limbs and across time. These findings indicate that the torque-velocity relationship of the quadriceps changed over time and was predictive of future satisfactory strength.
Clinical Relevance:
Clinicians may use a greater quadriceps torque-velocity relationship as a positive indication of a patient's ability to achieve satisfactory strength later in rehabilitation.
Anterior cruciate ligament (ACL) injuries are increasingly common, with estimates as high as approximately 350,000 injuries each year. 34 Following anterior cruciate ligament reconstruction (ACLR), patients endure deficits in strength, perceived knee function, and quality of life that must be addressed with rehabilitation. 12 Unfortunately, complete strength and functional recovery prove challenging even with rehabilitation.39,43 Serial, objective, multidimensional assessments may be leveraged to monitor patient progress over time to characterize rehabilitation progression and identify patients with a higher likelihood of subsequent ACL injury.3,29 Identifying key prognostic indicators of progress in muscle function during postoperative rehabilitation would allow clinicians to prescribe interventions that treat specific ACLR-related deficits associated with poor outcomes before patients prepare to return to play.
Persistent quadriceps weakness following ACLR has been well documented in the months and years after surgery and has been linked to increased reinjury risk and performance deficits.15,16,30 Muscle strength is typically tested via isokinetic dynamometry at distinct contraction velocities, and single-peak torque values are used to determine muscle strength. Evaluating strength differences between contraction velocities can reveal changes in muscle function across a variety of demands throughout the strength and speed spectrum.4,22 The torque-velocity relationship is an inherent property of skeletal muscle that is characterized by the inverse relationship between muscle contraction velocity and torque production. 13 Greater differences in torque production across testing velocities, represented by increases in the slope of the torque-velocity relationship, may indicate an increase in muscle size, the percentage of type II muscle fiber distribution within muscle, and/or improved neuromuscular function in human muscle tissue.13,14,17 Decreases in the torque-velocity relationship of the quadriceps have been demonstrated following immobilization, stroke, and ACLR, but the clinical value of longitudinal monitoring of the torque-velocity relationship has yet to be determined.8,23,37
Prior work has demonstrated that the quadriceps torque-velocity relationship is altered following ACLR, but it remains to be seen if differences in the torque-velocity relationship are clinically meaningful or have prognostic value throughout the rehabilitation process.11,37 Using tools such as isokinetic dynamometry to characterize the torque-velocity relationship may have prognostic value in the postinjury assessment of patients who have had ACLR. Traditionally, following ACLR, strength is quantified using limb symmetry indices comparing the involved limb to the uninjured limb, but comparison using a single measure of peak strength may not fully highlight within-limb changes. 41 Strength testing at slower velocities can reveal changes in maximal strength, and testing at faster velocities can reveal changes in the ability to generate force quickly. 11 Testing strength at multiple velocities may allow evaluation of muscle function across a wider spectrum of demands that the muscle may undergo. While the standardized threshold for acceptable limb symmetry index is typically 90%, evidence is mixed on whether this threshold is associated with successful patient outcomes. 40 In an attempt to decrease reliance on comparison to a contralateral limb that may not represent a stable measurement from time of injury to return to sport, within-limb strength to body mass ratios have been explored, including in a study of normalized within-limb strength that determined a specific threshold of strength that patients who reported satisfactory knee strength achieved (≥1.23 Nm/kg at 90°/s). 4 Patient acceptable symptom state (PASS) thresholds have previously been identified for the International Knee Documentation Committee (IKDC) Subjective Knee Form and Knee injury and Osteoarthritis Outcome Score (KOOS) subscales in patients following ACLR and have been associated with patient strength and function.26,27 Just as specific thresholds for the knee extension strength to body mass ratio, the IKDC, and the KOOS have been developed, it is important to identify appropriate thresholds for the torque-velocity relationship that can be used as measures of patient improvement. Therefore, the aim of this study was to determine if the torque-velocity relationship in the ACLR limb was predictive of meeting satisfactory knee extension strength thresholds or self-reported function benchmarks at return-to-sport assessments. We hypothesized that the torque-velocity relationship would predict which patients who have had ACLR would demonstrate satisfactory knee extension strength and self-reported function at return-to-sport assessments.
Methods
This study was part of a longitudinal cohort study recruiting from a university health system at the point of clinical care. 3 Approval of this study was granted by the University of Virginia institutional review board (HSR #17399) for human subjects research, and all participants completed the informed consent process before beginning any study procedures.
Inclusion criteria for this study were primary ACLR with bone–patellar tendon–bone or hamstring autografts at least 4 months prior to enrollment in the study. Participants were included regardless of meniscal pathology and subsequent meniscectomy or repair. Participants were excluded if they had any previous ACLR, multiple ligament reconstruction, known muscular abnormalities, or postsurgical complications (ie, cyclops lesions, infections, arthrofibrosis). Participants completed 2 postsurgical assessments as a part of their standard of care as they attempted to work toward full clearance to return to previous levels of activity. The first visit was scheduled at least 4 months post-surgery, and the second visit was scheduled 2 to 3 months following the initial visit to measure patient progress. All patients were given the same standardized postoperative rehabilitation protocol, but adherence was not monitored for this study.
Procedures
Knee Extensor and Flexor Strength Evaluation
Following a standardized walk on a treadmill, patients were seated in a Biodex System 4 Multimode dynamometer (Biodex Medical Systems). Each participant completed 8 repetitions of isokinetic knee extension and flexion at 90°/s and 180°/s bilaterally, beginning with the uninvolved limb. Further details on the experimental setup can be found in previous publications. 37 The greatest torque produced from the 8 repetitions was selected as the maximum peak torque. The torque-velocity relationship was operationally defined as the difference in peak torque normalized to body mass (Nm/kg) produced between 90°/s and 180°/s. 37 As a result, torque-velocity was calculated as the absolute difference between the normalized torque produced at 90°/s and the normalized torque produced at 180°/s (Nm/kg).
Patient-Reported Outcome Measures
Patients also completed the IKDC Subjective Knee Form and KOOS.9,18 Both patient-reported outcome measures have been used to characterize patient function following ACLR.2,27 The KOOS Sport subscale was chosen for analysis because it was deemed the most representative of our sample of patients hoping to return to their previous level of activity and has previously been associated with knee flexion strength and performance on hopping tasks. 26
Satisfactory Thresholds
To contextualize patient-reported outcome scores, PASS thresholds were previously derived for the IKDC and KOOS subscales to determine the scores at which patients following ACLR found the current state of their knee satisfactory. 27 The thresholds for the IKDC (≥75.9) and KOOS Sport (≥75.0) had a sensitivity of 0.83 and 0.87, as well as a specificity of 0.96 and 0.88, respectively. 27 Using similar methodology, an acceptable threshold for knee extension strength at 90°/s (≥1.23 Nm/kg) was derived in patients approximately 7 months post-ACLR. 4
Statistical Analysis
The knee extensor and knee flexor torque-velocity relationships were analyzed via 2 × 2 (visit × limb) analyses of variance. Post hoc paired samples t tests were performed as indicated. Binomial logistic regressions were used to determine if the torque-velocity relationship at visit 1 was associated with achieving satisfactory knee extension strength at 90°/s, IKDC, or KOOS Sport PASS scores at visit 2 using previously established thresholds as the standard.4,27 Preliminary analyses using normalized (Nm/kg) torque values yielded very small differences that were difficult to interpret in terms of clinical magnitude. Consequently, absolute torque values (Nm) were used to compute the torque-velocity variable for all logistic regression analyses. All other descriptive summaries and analyses of variance were based on normalized values (Nm/kg) for comparability across participants. Age, sex, graft type, and time post-surgery were included in the models as covariates. Odds ratios (ORs) >1.0 indicate a greater probability of the outcome, while ratios <1.0 indicate a decreased probability. 32 The Akaike information criterion (AIC) was used to determine model fit, and a lower AIC value was considered a better model fit. 7 A priori α was set at P≤ .05. Statistical analysis was conducted in Stata 18.0. 33
Results
Torque-Velocity Relationships
A total of 189 participants were included in this study. Demographic information is in Table 1, and strength and patient-reported outcomes are in Table 2. Quadriceps torque-velocity data are visualized in Figure 1. We observed an interaction between visit and limb (P < .001, η2 = 0.03) and significant increases in the torque-velocity relationships by visit (P < .001, η2 = 0.05) and limb (P < .001, η2 = 0.52) for the quadriceps (Figure 1). The torque-velocity relationship of the ACLR limb quadriceps significantly increased from visit 1 to visit 2 in the ACLR limbs (Table 2, P < .001, d = 0.63 [0.43 to 0.84]) but not in the uninvolved limb (d = 0.05 [–0.15 to 0.25]). This relationship was different between limbs at visit 1 (P < .001, d = 1.55 [1.33-1.78]) and visit 2 (P < .001, d = 1.55 [0.98-1.41]). For the hamstrings, we observed no interaction between visit and limb (P = .20; Figure 2).
Demographic Information and Surgical Characteristics (N = 189) a
Values are presented as mean ± SD unless otherwise indicated.
Strength and Patient-Reported Outcomes a
Values are presented as mean ± SD unless otherwise indicated. ACLR, anterior cruciate ligament reconstruction; IKDC, International Knee Documentation Committee; KOOS, Knee injury and Osteoarthritis Outcome Score; PASS, patient acceptable symptom state.
Quadriceps torque velocity by limb and visit. *Significantly different from involved limb at visit 1 (P < .001). †Significantly different from involved limb at visit 2 (P < .001).
Hamstring torque velocity by limb.
Prediction of Satisfactory Strength and Function
Overall, 86.2% (163/189) of patients achieved satisfactory knee extension strength, 82.0% (155/189) achieved IKDC PASS, and 91.5% (173/189) achieved KOOS Sport PASS at visit 2. When holding sex, age, and time postsurgery constant, the torque-velocity relationship of the quadriceps was associated with increased odds of achieving satisfactory knee extension strength (≥1.23 Nm/kg) at visit 2 (OR, 1.05; P = .046; AIC, 131.66). While the overall model, including torque-velocity, sex, age, and graft type, associated with achieving an acceptable IKDC score of at least 75.9 was significant (P = .02; AIC, 176.73), age was the only significant individual predictor (OR, 0.94; P = .001). The torque-velocity relationship of the quadriceps at visit 1 was not predictive of achieving the KOOS PASS threshold at visit 2 (P = .58). The full satisfactory knee extension binomial logistic regression model results are displayed in Table 3. After accounting for all covariates, neither sex, graft type, nor time postsurgery significantly influenced the likelihood of achieving satisfactory quadriceps strength, IKDC PASS, or KOOS Sport PASS. No significant interactions were observed between torque-velocity and other covariates.
Quadriceps Torque-Velocity Logistic Regression Results a
All models included graft type, sex, age, and time postsurgery. ORs are when all other covariates (graft type, sex, age, time postsurgery) are held constant. An OR of 1.05 = 5% increase in odds of attaining satisfactory knee extension for each 1-Nm difference between a 90°/s and 180°/s increase at visit 1. IKDC, International Knee Documentation Committee; KOOS, Knee injury and Osteoarthritis Outcome Score; OR, odds ratio; PASS, patient acceptable symptom state.
Statistical significance P < .05.
Discussion
This study sought to build on previous work demonstrating decreases in the torque-velocity relationship following ACLR and to determine the predictive ability of the torque-velocity relationship of the quadriceps and hamstrings for patient progress between 5.5 and 8.3 months after ACLR. 37 These time points are representative of postoperative decision-making following ACLR. The primary finding of this study was that a 1-Nm increase in the difference in peak torque between 90°/s and 180°/s was associated with a 5% increase in the odds of achieving satisfactory knee extension and a 2% increase in the odds of achieving a satisfactory IKDC score. This is meaningful as it represents a clear increase in the odds of achieving satisfactory strength and function at later rehabilitative time points. While this indicates a benefit to increasing the quadriceps torque-velocity relationship, it also serves as a reminder that postsurgical recovery extends beyond physiological needs. Consistent with previous literature, the torque-velocity relationship of the quadriceps was lower in ACLR limbs compared to uninvolved limbs but increased over time.11,37 The torque-velocity relationship of the hamstrings did not change over time and was similar between the ACLR and uninvolved limbs, which contradicted a previous evaluation of the knee flexors following ACLR. 11 It is of note that the selected velocities for our study (90º/s and 180º/s) were closer in velocity than the velocities chosen in the aforementioned study (60º/s and 240º/s), which may explain the difference in findings. 11 Numerous authors have used varying combinations of 2-point torque-velocity calculations, and determining the most appropriate combination of velocities may affect findings and the potential additive value compared with single-velocity testing.11,20,37
While peak torque at a single velocity is a well-established indicator of strength, it only measures muscle function under 1 condition. 11 The analysis of torque production across velocities measures both maximal strength and force production across contractile demands. 11 In healthy muscle, resistance training creates the most pronounced strength gains at slower contraction velocities, but disuse may improve function at faster velocities. 1 During periods of disuse, healthy muscle undergoes fiber-type transition and atrophy of selective fibers, leading to a slow to fast fiber-type transition.24,31,35,42 In addition to the effects of disuse, following ACLR, muscle undergoes the loss of force-generating myofibril density, increased presence of factors that lead to atrophy and inflammatory cytokines, reduced satellite cell presence, and increased fat infiltration.21,24 The evaluation of torque production at a singular velocity may capture changes in maximal strength while failing to capture changes that indicate this potential fiber-type transition or allow longitudinal monitoring of prospective changes in function across velocities.
An increased torque-velocity relationship at 5.5 months, male sex, younger age, and increased time postsurgery were predictive of achieving satisfactory knee extension strength about 8 months postsurgery. Previous work has linked other measures of strength or patient function to self-reported function and psychological response.10,26 When including the torque-velocity relationship, graft type, sex, age, and time postsurgery as predictors, the entire model was associated with a higher likelihood of reporting IKDC scores above PASS thresholds, but age was the only significantly associated variable when controlling for all other covariates. The torque-velocity relationship was not predictive of passing KOOS Sport PASS thresholds. While this contradicts previous findings linking strength outcomes with KOOS Sport scores, our assessment of strength outcomes and the KOOS Sport both occurred at mid-rehabilitative time points, and strength and patient-reported outcomes may not improve uniformly. 10 Our findings indicated that increases in the quadriceps torque-velocity relationship were predictive of achieving satisfactory knee extension later in the rehabilitation process.
In the current cohort, graft type (patellar tendon vs hamstring autograft) was not a significant predictor of achieving satisfactory quadriceps strength at the follow-up visit when controlling for sex, age, and time postsurgery. The inclusion of graft type in the model did not meaningfully alter the association between the quadriceps torque-velocity relationship and postoperative strength outcomes at a subsequent time point. These findings suggest that the relationship between early quadriceps contractile performance and subsequent strength recovery is consistent across patients with bone–patellar tendon–bone or hamstring autografts. While some prior studies have reported differences in strength recovery between extensor mechanism grafts (eg, patellar and quadriceps grafts) and hamstring grafts, differences in knee extensor strength appear to dissipate over time.19,38 Graft choice is a multifactorial decision, and its potential influence on both short- and long-term outcomes must continue to be studied. 6
Clinical decision-making often relies on the synthesis of information obtained from multidimensional test batteries, including measures of patient strength and self-reported function. 39 Knee extensor strength is one of the most common prolonged deficits following ACLR, and assessing the quadriceps is paramount in any postsurgical test battery. 3 Contextualizing changes in quadriceps function and how they will affect outcomes following ACLR is a necessity, but testing at a single velocity may not give a full picture of the ability to respond to varying demands or underlying changes in physiology. 8 Isometric knee extension and isokinetic knee extension at 90°/s have been demonstrated to be indicators of satisfactory knee function at a single time point, but this study models the torque-velocity relationship across velocities to determine how it can predict function over time.4,22 Patient-reported outcome measures are a reflection of patient perception and quality of life that must be considered as they may be predictive of the ability to successfully return to previous levels of activity following ACLR. 25 The PASS thresholds for IKDC and KOOS Sport were developed to allow clinicians the ability to determine a clinically appropriate cutoff for patient-reported outcome measures. The ability to use clinically obtained data to identify factors with the highest prognostic value could allow clinicians to more readily identify intervention points in early to mid-recovery phases of rehabilitation.
The limitations of this study include the specific selected velocities and potential selection bias. While previous research has validated the use of two velocities to characterize muscle function, the velocities chosen in this study were selected based on clinical practice within the referring hospital system. 20 As a result of the referral patterns of this study, patients with multiple graft types of a wide range were included in this study. Participants in this study were tested at 2 points, reflecting only 2 time points within their rehabilitation timeline and representative of the rehab process. As this study included participants with only 2 visits within 1 year, participants may have declined repeat participation for a number of reasons that may be reflective of progress or lack thereof. Rehabilitation information was not captured for this study, but all participants had the same overall rehabilitation protocols. While these factors do add to the heterogeneity of the study, they are reflective of clinical practice and increase the external validity of the results. This study focused on strength testing following ACLR, as previous research has demonstrated a lag in the restoration of knee extension strength, even when patients are able to pass other components of return-to-sport test batteries, such as jumping.28,36 While the restoration of quadriceps strength is a critical goal during recovery, movements that use multiple joints rely on the adaptation of motor skill, along with the restoration of quadriceps strength.5,30 Although graft type was not a significant predictor in the present model, the analysis was limited to the 2 most common grafts, and future studies with larger, balanced cohorts could further clarify graft-specific recovery patterns. Future research should consider different velocities and expand longitudinal analysis to definitively determine the predictive abilities of the torque-velocity relationship.
The torque-velocity relationship following ACLR may provide a comprehensive representation of skeletal muscle function and qualities, such as fiber-type transition, that are not readily captured by a single velocity and may therefore have unique prognostic value beyond what is currently standard practice. As clinicians use longitudinal testing to monitor patient progress, determining the prognostic value of various metrics can guide clinical care. These findings indicate that the torque-velocity relationship of the quadriceps changed over time and, when captured at an earlier postsurgical time point, was predictive of the future ability to achieve satisfactory knee extensor strength.
Footnotes
Final revision submitted October 6, 2025; accepted November 9, 2025.
One or more of the authors has declared the following potential conflict of interest or source of funding: S.F.B. has received compensation for services other than consulting, including serving as faculty or speaker, as well as consulting fees, royalties, travel, and lodging from Arthrex, Exactech, and Zimmer Biomet; food and beverage from Catalyst OrthoScience, DePuy Synthes Sales, FX Shoulder USA, Fortis Surgical, Heron Therapeutics, Medacta USA, Medical Device Business Services, Smith & Nephew, and Stryker; and an editorial stipend from AOSSM Publishing. D.R.D. has received consulting fees from DePuy Synthes Products, Medical Device Business Services, OsteoCentric Technologies, Smith & Nephew, and Vericel; food and beverage from DePuy Synthes Products, DePuy Synthes Sales, Linvatec, Medical Device Business Services, OsteoCentric Technologies, Smith & Nephew, Stryker, and Zimmer Biomet; royalties or licenses from DePuy Synthes Products, OsteoCentric Technologies, and Smith & Nephew; and travel and lodging from DePuy Synthes Products, Medical Device Business Services, and Smith & Nephew. M.D.M. has received compensation for services other than consulting, including serving as faculty or speaker, from Arthrex and Pacira Pharmaceuticals; consulting fees from Arthrex and Ipsen Bioscience; food and beverage from Arthrex, Smith & Nephew, and Stryker; royalties or licenses from Arthrex; and travel and lodging from Arthrex. F.W.G. has received compensation for services other than consulting, including serving as faculty or speaker, from Arthrex; consulting fees from DePuy Synthes Products; educational support from Fortis Surgical; food and beverage from Arthrex, Exactech, Fortis Surgical, Linvatec, and Stryker; a grant from DJO; and travel and lodging from Arthrex, Fortis Surgical, and Stryker. B.C.W. has received compensation for services other than consulting, including serving as faculty or speaker, from Arthrex; consulting fees from Arthrex, KCI USA, LifeNet Health, and Pacira Pharmaceuticals; educational support from Arthrex and Fortis Surgical; food and beverage from Arthrex, Exactech, Fortis Surgical, Gentleman Orthopedic Solutions, OSSIO, Pacira Pharmaceuticals, Peerless Surgical, Piedmont Plus Innovation, Pinnacle, Pylant Medical, Saxum Surgical, Stryker, Supreme Orthopedic Systems, Titan Surgical Group, and Zimmer Biomet; honoraria from Flexion Therapeutics; and travel and lodging from Arthrex, Fortis Surgical, LifeNet Health, and Pacira Pharmaceuticals. AOSSM checks author disclosures against the Open Payments Database (OPD). AOSSM has not conducted an independent investigation on the OPD and disclaims any liability or responsibility relating thereto.
