Abstract
Recognizing the importance of both the torque-angle and torque-velocity relations, three-dimensional (3D) human strength capability, (i.e., peak torque as a function of both joint angle and movement velocity), have been increasingly reported. It is not clear, however, the degree to which these surfaces vary between joints, particularly between joints with similar biomechanical configurations. Thus, our goal was to compare 3D strength surfaces between the elbow and knee hinge joints in men and women. Peak isometric and isokinetic strength was assessed in 54 participants (30 men) using the Biodex System 3 Isokinetic Dynamometer. Normalized peak torque surfaces significantly varied between flexion and extension (within each joint) and between joints; however the normalized 3D torque surfaces did not differ between men and women. These findings suggest the underlying joint biomechanics are the primary influences on these strength surface profiles. Therefore, in applications such as digital human modeling, torque-velocity-angle relationships for each joint and torque direction must be uniquely represented to most accurately estimate human strength capability.
Keywords: muscle, isometric, isokinetic, length-tension, force-velocity, Hill’s Equation
INTRODUCTION
Biomechanical analyses of human movement have played an important role in our understanding of normal and pathokinesiology. Technological advances over the past several decades have increased our ability to make accurate and rapid estimates of dynamic joint torques for ergonomic and biomechanical task assessments. For example, inverse dynamics analyses (e.g., Brand et al., 1982; Kuo, 1998; Schache et al., 2010) and digital human modeling (e.g., Abdel-Malek et al., 2006; Delp et al., 2007) approaches provide net joint torques required to complete a dynamic task. These calculations had been previously too cumbersome to perform routinely for all but quasi-static conditions. Unfortunately, the plethora of normative strength data available is predominantly based on static torque-angle representations. To accurately estimate static and dynamic task intensity (i.e., percentage of maximum capability), normative data reflecting both known torque-angle and torque-velocity relationships are needed.
Dynamic strength assessments have largely focused on the curvilinear force-velocity relationships (Gregor et al., 1979; Thorstensson et al., 1976) first identified by Hill’s equation (1938). These nonlinear relationships have been well documented in single fiber (McDonald, 1994), isolated muscle (Brooks & Faulkner, 1991), and joint-level investigations (Gregor, et al., 1979; Thorstensson, et al., 1976). Although, often assessed at optimal muscle lengths in isolated muscle preparations, torque-velocity at the joint level is typically assessed at the angle of peak torque, potentially differing across increasing velocities (Westing et al., 1988). Few studies have compared angle-specific torque-velocity relationships across joints or torque directions.
Winter (1990) proposed a relatively straight forward three-dimensional (3D) force-velocity-length relationship at the muscle fiber level. This model included little to no interaction occurring between muscle fiber length and contraction velocity (Winter, 1990). However, joint torque clearly includes additional influential factors not present at the single fiber level, such as moment arm variations, multiple muscles acting as synergists at varying optimal muscle lengths, etc. Thus, the corresponding 3D torque-velocity-angle strength surfaces may differ substantially from those proposed for the single muscle fiber.
Recognizing the importance of both the torque-angle and torque-velocity relations, three-dimensional (3D) human strength capability, (i.e., peak torque as a function of both joint angle and movement velocity), have been increasingly reported (Anderson et al., 2007; Khalaf & Parnianpour, 2001; Khalaf et al., 2000, 2001; Khalaf et al., 1997). Moreover, these studies provide evidence that strength varies across both angle and velocity (i.e., an interaction effect), indicating the need for normative data to be reported across a range of both joint angle and velocity. It is not clear, however, the degree to which these surfaces vary between joints, particularly between joints with similar biomechanical configurations (e.g. hinge joints). Further, previous studies have utilized relatively small age- and sex-specific sample sizes (n = 7 to 10 per group) (Anderson, et al., 2007; Khalaf, et al., 2000, 2001; Khalaf, et al., 1997), thus may not be ideal for use as normative strength surfaces.
Thus, the primary purposes of this study were to 1) to evaluate the degree to which 3D strength surfaces vary between surfaces (e.g., between flexion and extension within a joint and between two different, but similar joints); 2) to compare angle-specific torque-velocity relationships within and between joints; and 3) to provide normative 3D strength data for men and women that may be useful for inverse kinematic and digital human modeling applications. While comparison of normative dynamic strength 3D surfaces across widely differing joints of the body would also be informative, our goal was limited to assessing two similar, relatively simple joints (i.e., hinge joints) in both men and women. Accordingly, we chose to evaluate the 3D peak torque surfaces for knee and elbow joints.
METHODS
Fifty-four healthy subjects participated in the study after providing written informed consent as approved by the local Institutional Review Board. All subjects denied a history of any serious medical conditions or musculoskeletal pathology and were in good general health. Subjects completed two visits; testing elbow and knee peak torque at the first and second visits, respectively.
Strength Assessment:
Joint torques were measured using an isokinetic dynamometer (Biodex System 3, New York, USA) using the manufacturer’s recommended positioning for each joint. To warm-up, subjects performed eight to ten bicep curls with a five pound weight, or 5 min of self-paced stationery cycle ergometry, prior to the respective test. Subjects were seated in the Biodex test chair and properly secured using Velcro straps to minimize extraneous and compensatory movements. The joint centers of rotation, i.e. lateral epicondyle for elbow and lateral femoral condyle for knee, were aligned with the torque dynamometer center of rotation. Range of motion (ROM) was set from full extension (0°) to 125° of elbow flexion or 110° of knee flexion. The shoulder and hip were maintained at approximately 25° and 90° of flexion, respectively, and the forearm was supinated.
Isometric torque was measured at five angles spanning the normal range of motion for each joint (15, 40, 65, 90, and 110° for elbow; 15, 35, 55, 75, and 100° for knee). The order of testing was block randomized (3 protocols) to minimize testing order effects. At each angle, subjects performed eight, three-second maximum voluntary contractions (MVCs), alternating between flexion and extension (four contractions each direction), with one-minute rest intervals.
After a five-minute rest, isokinetic testing was performed at five angular velocities (60, 120, 180, 240, and 300 °/s) using three block randomized protocols to minimize order effects. Subjects were familiarized using three submaximal repetitions at each velocity, followed by four to seven maximal repetitions, with increasing repetitions at higher velocities (similar test durations). Isokinetic testing involved continuous concentric contractions, alternating between flexion and extension throughout the full ROM. Three-minute rests were provided between each test velocity.
Analyses
The torque, position, and velocity analog outputs were digitally sampled and recorded at 1000 Hz using a custom LabVIEW 8.0 program (National Instruments, Austin, TX, USA), and further analyzed using Matlab (Math Works Inc, Natick, MA, USA). Angle-specific passive torque was subtracted from the respective peak torques (isometric and isokinetic) to account for gravitational effects. Maximum values were extracted from all trials and recorded as peak torque. Isokinetic peak torque was determined +/− 2.5° of the five target angles and within 15% of the target velocities, to minimize effects of limb acceleration and deceleration. Thus, 30 velocity-angle-specific peak torque values (5 positions, 6 velocities including zero) were extracted for the four motions tested (i.e., knee and elbow, flexion and extension) for a total of 120 data points.
If the target isokinetic velocity was not achieved within 15%, the missing data was modeled for each individual using 3D strength surfaces (TableCurve 3D, Systat Software, Inc., Chicago, IL). This estimation prevented the non-random loss of the “weakest, slowest” individuals from biasing the resulting group mean values. However, if more than 50% of subjects had missing data at a particular angle-velocity combination, that data point was not considered in the final surface analyses (e.g. see Tables 4 & 5).
Table 4.
Mean (SD) elbow flexion peak torque for males (shaded) and females (unshaded).
| Elbow Angle (°) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 40 | 65 | 90 | 110 | |||||||
| Velocity (°/sec) | 0 | 56.4 | (13.5) | 62.9 | (14.8) | 63.0 | (12.9) | 60.3 | (12.6) | 56.4 | (14.0) |
| 26.2 | (6.8) | 30.4 | (7.0) | 32.0 | (6.7) | 32.4 | (8.1) | 30.4 | (6.4) | ||
| 60 | 41.2 | (11.0) | 43.9 | (10.4) | 44.7 | (10.0) | 42.3 | (9.5) | 35.7 | (7.9) | |
| 20.4 | (3.6) | 20.8 | (4.8) | 21.4 | (4.9) | 20.8 | (5.4) | 18.3 | (5.2) | ||
| 120 | 36.1 | (9.5) | 38.0 | (8.8) | 38.7 | (8.3) | 36.8 | (7.5) | 27.9 | (6.4) | |
| 18.9 | (3.7) | 19.4 | (3.8) | 19.9 | (4.5) | 19.0 | (4.4) | 15.2 | (4.4) | ||
| 180 | 32.1 | (9.5) | 33.9 | (8.1) | 33.1 | (8.7) | 29.2 | (8.8) | 18.7 | (8.1) | |
| 15.9 | (5.0) | 18.2 | (3.9) | 17.9 | (4.7) | 15.6 | (5.2) | 11.3 | (5.9) | ||
| 240 | † | 30.0 | (6.1) | 27.9 | (6.0) | 21.5 | (6.6) | 12.0 | (7.6) | ||
| † | 14.4 | (6.2) | 15.8 | (5.0) | 13.2 | (4.9) | 8.1 | (5.4) | |||
| 300 | † | † | 25.7 | (5.7) | 18.4 | (5.0) | 8.4 | (7.7) | |||
| † | † | 14.1 | (8.0) | 13.2 | (6.8) | 8.7 | (7.5) | ||||
More than 50% of participants unable to achieve isokinetic velocities for these data.
Note: 0° elbow angle = full extension.
Table 5.
Mean (SD) elbow extension peak torque for males (shaded) and females (unshaded).
| Elbow Angle (°) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 40 | 65 | 90 | 110 | |||||||
| Velocity (°/sec) | 0 | 40.2 | (11.3) | 50.5 | (14.7) | 52.8 | (13.6) | 50.7 | (11.6) | 45.4 | (10.8) |
| 22.1 | (5.5) | 27.6 | (5.0) | 29.2 | (5.8) | 27.7 | (6.4) | 25.1 | (6.6) | ||
| 60 | 26.2 | (12.8) | 37.4 | (11.3) | 39.6 | (10.2) | 38.3 | (9.9) | 36.4 | (9.6) | |
| 13.5 | (4.7) | 19.9 | (5.8) | 22.1 | (5.8) | 21.2 | (6.6) | 20.4 | (5.7) | ||
| 120 | 21.9 | (11.1) | 33.6 | (10.6) | 35.6 | (9.3) | 35.2 | (9.2) | 33.7 | (9.1) | |
| 10.9 | (5.5) | 18.2 | (5.5) | 20.0 | (6.3) | 20.0 | (5.5) | 19.0 | 4.6) | ||
| 180 | 15.3 | (10.0) | 29.0 | (10.3) | 31.8 | (9.7) | 33.0 | (9.4) | 30.8 | (10.1) | |
| 7.6 | (5.3) | 15.3 | (5.6) | 18.1 | (5.4) | 18.3 | (4.8) | 17.4 | (4.5) | ||
| 240 | 8.7 | (10.0) | 20.5 | (10.2) | 26.1 | (7.6) | 28.5 | (7.7) | † | ||
| 6.2 | (5.2) | 12.9 | (6.0) | 15.9 | (5.6) | 15.8 | (5.4) | † | |||
| 300 | 5.5 | (7.2) | 17.1 | (9.4) | 24.6 | (7.2) | 26.7 | (7.1) | † | ||
| 3.8 | (4.2) | 10.2 | (5.4) | 13.3 | (5.6) | 12.8 | (5.4) | † | |||
More than 50% of participants unable to achieve isokinetic velocities for these data.
Note: 0° elbow angle = full extension.
Summary statistics including mean, standard deviation (SD), and coefficient of variation (CV) were calculated for male and female cohorts (SPSS 19.0, IBM). The distribution of the peak torque data were assessed using the Kolmogorov-Smirnov test statistic for normality to ensure parametric statistics were appropriate. The distribution of the CV and SD were assessed using histograms to determine the best single estimate of peak torque variance.
Relative torque was calculated as the angle- and velocity-specific strength values divided by the overall peak torque (e.g. maximum surface value equal to 1.0). The resulting 3D mean peak torque surfaces (absolute and relative torque values) were plotted (SigmaPlot, SYSTAT Software Inc, CA, USA) by sex. Repeated measures analyses of variance (ANOVA) were used to test for significant differences between and within joints, considering: joint angle, velocity, and sex for the normalized knee and elbow strength data.
Angle-specific torque-velocity relationships were also compared by dividing each set of angle-specific torques (across velocities) by the respective isometric torque (rather than to a single peak across all conditions). This produced a set of relative torque-velocity relationships extracted from each 3D surface, where each angle-specific isometric torque value was normalized to a value of 1.0. For all statistics, significance was set at alpha = 0.05.
RESULTS
Subject characteristics for the full cohort, 24 women and 30 men, are provided in Table 1. Three men and one woman did not complete the knee testing, and one woman did not complete the elbow testing. Peak torque summary statistics (mean and SD) are provided in Tables 2–5. Greater than 50% of individuals were able to achieve isokinetic target velocities for all test angles except five for the elbow, at the longest muscle lengths and fastest velocities (see Tables 4 and 5). Thus, only 115 of the original 120 data points were included in the remaining analyses. Peak torque was normally distributed for men and women for 113 of 115 joint angle and velocity combinations. Thus parametric statistics are appropriate for these strength analyses.
Table 1.
Mean (SD) Subject Demographics.
| Men | Women | p-value† | |
|---|---|---|---|
| N | 30 | 24 | |
| Age (yrs) | 24.2 (5.1) | 22.8 (4.2) | 0.27 |
| Ht (cm) | 180.1 (6.3) | 166.1 (8.0) | < 0.001* |
| Wt (kg) | 81.5 (9.2) | 67.3 (12.0) | < 0.001* |
| BMI (kg/m2) | 25.1 (2.3) | 24.5 (4.5) | 0.56 |
Independent t-test comparing men and women.
Indicates significant sex differences (p < 0.05).
Table 2.
Mean (SD) knee flexion peak torque for males (shaded) and females (unshaded).
| Knee Angle (°) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 35 | 55 | 75 | 100 | |||||||
| Velocity (°/sec) | 0 | 81.8 | (23.2) | 132.5 | (35.7) | 174.7 | (47.3) | 212.3 | (80.7) | 167.3 | (65.3) |
| 60.6 | (34.8) | 96.8 | (42.2) | 122.8 | (35.2) | 144.2 | (43.1) | 104.7 | (39. 0) | ||
| 60 | 69.9 | (17.8) | 114.1 | (30.9) | 152.4 | (41.6) | 157.5 | (45.6) | 109.6 | (45.1) | |
| 48.6 | (27.1) | 79.4 | (28.1) | 103.9 | (26.0) | 104.0 | (24.5) | 76.7 | (23.8) | ||
| 120 | 65.0 | (17.6) | 104.8 | (25.9) | 133.1 | (30.3) | 134.8 | (33.3) | 95.0 | (34.6) | |
| 46.0 | (21.8) | 74.6 | (22.5) | 94.0 | (21.4) | 92.6 | (20.6) | 66.6 | (20.0) | ||
| 180 | 58.0 | (20.1) | 91.8 | (25.8) | 116.2 | (29.1) | 115.8 | (29.5) | 85.3 | (31.4) | |
| 38.6 | (19.4) | 63.6 | (18.5) | 81.3 | (18.7) | 81.8 | (18.7) | 59.0 | (21.1) | ||
| 240 | 48.1 | (17.7) | 78.4 | (25.8) | 100.7 | (29.7) | 100.7 | (28.9) | 69.8 | (31.1) | |
| 34.9 | (17.8) | 55.0 | (17.0) | 70.4 | (16.0) | 72.1 | (15.4) | 51.5 | (18.3) | ||
| 300 | 39.5 | (17.4) | 65.8 | (24.8) | 84.4 | (28.4) | 89.3 | (29.1) | 56.6 | (29.7) | |
| 27.1 | (16.2) | 45.7 | (16.9) | 60.0 | (13.8) | 64.4 | (15.6) | 44.7 | (16.0) | ||
Note: 0° knee angle = full extension.
The 3D strength surfaces using normalized peak torques are shown in Figure 1 (males) and Figure 2 (females). The ANOVA results for within joint (i.e., flexion versus extension) and between-joint (i.e., knee versus elbow) are provided in Table 6. Normalized peak torque surfaces significantly varied as follows: between flexion and extension (within each joint), between joints (for flexion and extension), with joint angle, and with contraction velocity, but not between men and women. Further, nearly all two- and three-way interactions involving angle, velocity, direction or joint (but not sex) were significant. This indicates the torque-angle and torque-velocity relationships differed between joints and torque directions universally across both hinge joints. However, 3D strength surfaces were consistent between the sexes at each joint and joint direction. The above findings were largely consistent between the absolute and normalized peak torque data; thus only the results for the normalized data are reported here.
Figure 1.
Three-dimensional (3D) mean normalized peak torque surfaces (relative torque x velocity x angle) for the following joint and torque directions (A) knee extension; (B) elbow extension; (C) knee flexion; and (D) knee extension in males.
Figure 2.
Three-dimensional (3D) mean normalized peak torque surfaces (relative torque x velocity x angle) for the following joint and torque directions (A) knee extension; (B) elbow extension; (C) knee flexion; and (D) knee extension in females.
Table 6.
ANOVA results (p-values) for knee (flexion vs. extension), elbow (flexion vs. extension), and knee by elbow** (flexion only and extension only) analyses using normalized peak torque surface data.
| Term | Knee Flexion / Extension | Elbow Flexion / Extension | Flexion Knee / Elbow | Extension Knee / Elbow |
|---|---|---|---|---|
| Angle | <.001 | <.001 | <.001 | <.001 |
| Velocity | <.001 | <.001 | <.001 | <.001 |
| Direction | <.001 | .035 | — | — |
| Sex | .906 | .996 | .629 | .818 |
| Joint | — | — | .012 | .001 |
| Angle × Velocity | <.001 | .205 | <.001 | .002 |
| Angle × Direction | <.001 | <.001 | — | — |
| Angle × Sex | .673 | .386 | .886 | .889 |
| Velocity × Direction | .002 | <.001 | — | — |
| Velocity × Sex | .956 | .497 | .511 | .945 |
| Direction × Sex | .450 | .290 | — | — |
| Angle × Joint | — | — | <.001 | <.001 |
| Velocity × Joint | — | — | <.001 | <.001 |
| Sex × Joint | — | — | .253 | .444 |
| Angle × Velocity × Direction | <.001 | <.001 | — | — |
| Angle × Velocity × Sex | .473 | .782 | <.001 | .378 |
| Angle × Direction × Sex | .394 | .759 | — | — |
| Velocity × Direction × Sex | .211 | .101 | — | — |
| Angle × Velocity × Joint | — | — | <.001 | <.001 |
| Angle × Joint × Sex | — | — | .111 | .908 |
| Velocity × Joint × Sex | — | — | .204 | .782 |
Elbow ANOVA velocities limited to 180°/s due to missing data at 240°/s and 300°/s.
Due to missing data points, knee x elbow analyses were run with reduced number of knee data points to match elbow.
Angle-specific torque-velocity relationships, normalized to isometric strength, are shown in Figure 3. Differences between joint angles as well as between joints were observed, with less curvilinear behavior than expected. The knee torque-velocity decay was highly linear, particularly for extension; whereas the elbow torque-velocity decay dropped rapidly to 60°/s followed by a linear decay throughout the higher velocities. Further, mid-range joint angles typically exhibited the lowest relative decay in peak torque with increasing velocity.
Figure 3.
Mean (SEM) peak torque is shown as a function of velocity, normalized by the isometric peak torque at each angle (males and females combined) for: (A) knee flexion; (B) knee extension; (C) elbow flexion; and (D) elbow extension. Note the mid-range angles typically produce the least torque-velocity decay across joints and torque directions.
Men and women displayed similar torque-angle-velocity 3D surfaces when normalized to peak values; however, men were significantly stronger than women for both the knee and elbow joints (p < 0.001). This sex-difference was greater for the elbow than the knee joint. For knee flexion and extension men were 43% stronger on average than women, ranging from 25 to 94% across the angle – velocity combinations. For elbow flexion and extension men were 83% stronger on average than women, with ranges from −3 to 115% across the angle – velocity combinations.
Coefficients of variation and SDs varied with joint angle and velocity for all joint directions. Variance increased particularly at the extreme ROMs and velocities. CV did not vary between men and women (p > 0.34), but SD was significantly higher in men for all joint directions (p < 0.002). Histograms of the CVs (see Figure 4) suggest median values may provide a better point estimate of peak torque variance than mean CV. Accordingly, the median CVs by joint direction were: 0.30 (knee and elbow extension), 0.34 (knee flexion), and 0.25 (elbow flexion).
Figure 4.
Histograms of the coefficients of variation (CV) by joint direction and sex are shown for: (A) knee and (B) elbow joints. The majority of CVs centered on values of approximately 0.3; the highest CV values typically occurred at the endpoints of the range of motion and the fastest velocities.
DISCUSSION
This study provides a unique examination of 3D joint strength considering both angle and velocity for two hinge joints, noting several key findings. First, 3D joint strength surfaces do not follow simple extensions of 2D torque relationships as proposed for single muscle fibers, but demonstrate significant interactions between torque-angle and torque-velocity relationships. Thus normative data throughout the range of static and dynamic contraction conditions are needed for strength model development and validation of digital human model applications. Second, these interactions are statistically unique to each joint and torque direction, even for two similar joint configurations, but do not differ significantly between men and women. Third, torque-angle relationships are angle- and joint-specific and do not necessarily follow Hill’s nonlinear decay observed in isolated muscle preparations, but rather may produce relatively linear decays in peak torque. Lastly, CV varies across a range of angle-velocity combinations, but increases dramatically at shortened muscle lengths and/or high velocities.
The simple superimposition of the classic length-tension and force-velocity relationships at the single muscle level was proposed to create a symmetrical, three-dimensional (3D) force-length-velocity surface (Gordon et al., 1966; Hill, 1938; Winter, 1990). The in vivo equivalent of torque-velocity and torque-angle relationships at the joint level in our study did not demonstrate a similar symmetry. The nonlinear interactions observed here may be due in part to multiple possible mechanisms, including: 1) the underlying single muscle properties may in fact exhibit these nonlinearities; 2) load sharing between synergist muscles acting about a joint may induce nonlinearities even if each individual muscle does not (e.g. varying muscle moment arms and/or optimal lengths); 3) net joint torque is inherently influenced by co-contraction of antagonist muscles (Kellis, 1998) which are not considered in isolated muscle preparations; and 4) passive tension at the end-ranges of joint motion may contribute a greater proportion of total joint torque at the fastest velocities. Thus, the jump from individual muscles to strength behavior at the joint level (e.g. force-length-velocity to torque-angle-velocity) makes simple translation of well-known muscle properties to observed phenomena challenging. Accordingly, 3D strength surfaces provide additional insights on human muscle capability as a function of joint angle and contraction velocity that are not available from 2D representations of strength.
We observed significant interactions between angle and velocity, indicating the torque-velocity decay is dependent on joint angle. To our knowledge, Khalaf and colleagues are the only other investigators to have statistically examined for interaction effects between torque-velocity and torque-angle relationships in human strength assessment. They also observed significant interactions at multiple joints, but did not compare between joints and had samples of only 10 men and 10 women in each study (Khalaf & Parnianpour, 2001; Khalaf, et al., 2000, 2001; Khalaf, et al., 1997).
These interactions are further supported by our secondary analyses investigating angle-specific torque-velocity decay. Contrary to Hill, we observed linear torque decay at the knee, which are consistent with other human torque studies (Mikesky et al., 1995; Weir et al., 1996). Yet, torque-velocity decay that is in line with Hill’s curvilinear relationship has also been reported (Folland et al., 2002; Knapik & Ramos, 1980; Yoon et al., 1991). This may be explained in part by the fact we investigated angle-specific relationships, whereas most others collapsed angle-specific peak torque relationships across joint angles, ignoring the position of peak torque generation. The torque-velocity decay differed by angle more for the elbow than the knee joint. In particular, when at a shortened muscle length (i.e., 110° for flexion and 15° for extension), the 300 °/s elbow peak torque decayed to 20% or less of the peak torque observed isometrically. Whereas the knee peak torque decay reached only 35 – 40% of isometric values at 300 °/s. All joint directions showed similar levels of peak torque decay at mid-range angles.
Interestingly, the normalized 3D torque surfaces did not differ between men and women, suggesting the primary importance of the underlying joint biomechanics in determining the surface profiles, universal to both sexes. The normalized elbow and knee 3D strength surfaces qualitatively differed more between torque directions (i.e., flexion versus extension) than between joints (see Figures 1 and 2). The differences between flexion and extension may be partially due to the mechanical moment arm differences; flexor muscle moment arms vary with joint angle more than extensor muscles (Herzog & Read, 1993; Murray et al., 1995; Pigeon et al., 1996; Wretenberg et al., 1996). Further, the flexors of both joints act at the interior angles of the joints, whereas the extensors both act through pulley mechanisms. On the other hand, the knee extensors have the mechanical advantage of the sesamoid bone, the patella, which is absent for the elbow extensor musculature. Regardless, at the joint level there appears to be statistical consistency in strength 3D surfaces between the sexes and qualitative similarities in 3D strength surfaces (albeit statistically significant) between joints with similar biomechanical systems.
Study limitations that should be considered include the potential for measurement error, selection bias, and adjacent joint influences. More specifically, at the fastest velocities, there was an increased acceleration period before the participant achieved the target velocity, resulting in loss of isokinetic data for some individuals. To address this issue without having a preferential loss of the weakest or slowest individuals, we modeled any missing data prior to further analyses. However, this may result in some degree of measurement error. Second, while we provide normative 3D strength data using a larger than previously available cohort, we cannot rule out the potential for selection bias in our sample. While we were able to recruit individuals with a range of body types and strengths, we likely did not sample the weakest (e.g. may avoid a study on joint strength) or strongest (e.g. excluded elite competitive athletes) individuals possible. Lastly, these 3D strength surfaces do not take into account the neighboring joints that inherently influence muscle strength, as that was beyond the scope of this study. For example, the potential effect of shoulder position on elbow torque, or hip angle on knee torque, may alter these 3D surfaces. Future studies are needed to investigate the degree to which these 3D peak torque surfaces further change with changes to adjacent joint postures as well as with more complex joints, such as the shoulder or the trunk.
In summary, we were able to demonstrate normative 3D joint strength considering both angle and velocity for the knee and elbow. The simple, yet classic 2D torque relationships cannot simply be extended to the third dimension without error, due to significant interactions between torque-angle and torque-velocity relationships. Further, the 3D strength surfaces varied between joints and between flexion and extension within joints. Therefore, in applications such as digital human modeling, torque-velocity-angle relationships for each joint and torque direction must be uniquely represented to most accurately estimate human strength capability. Accordingly, this study provides normative strength data for the knee and elbow joints, which may be useful for estimating strength population percentiles across the 3D strength surfaces.
Table 3.
Mean (SD) knee extension peak torque for males (shaded) and females (unshaded).
| Knee Angle (°) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 35 | 55 | 75 | 100 | |||||||
| Velocity (°/sec) | 0 | 103.5 | (33.2) | 95.2 | (29.4) | 94.4 | (26.5) | 88.8 | (27.0) | 71.5 | (20.6) |
| 66.0 | (22.4) | 68.5 | (20.0) | 64.9 | (20.4) | 65.4 | (22.2) | 53.2 | (22.5) | ||
| 60 | 66.6 | (29.6) | 75.7 | (27.0) | 73.8 | (23.8) | 67.1 | (19.2) | 52.8 | (17.7) | |
| 46.5 | (13.8) | 55.2 | (13.6) | 53.7 | (15.1) | 49.6 | (16.8) | 37.7 | (19.0) | ||
| 120 | 60.5 | (21.6) | 68.5 | (22.6) | 67.0 | (20.3) | 62.1 | (17.1) | 46.6 | (17.7) | |
| 43.9 | (11.7) | 49.3 | (12.2) | 47.4 | (12.9) | 44.0 | (14.0) | 32.8 | (16.3) | ||
| 180 | 51.1 | (21.4) | 59.8 | (19.4) | 58.7 | (18.3) | 54.0 | (16.9) | 38.7 | (18.2) | |
| 41.1 | (14.0) | 43.5 | (11.0) | 41.4 | (11.3) | 37.6 | (13.3) | 25.1 | (16.2) | ||
| 240 | 47.7 | (23.8) | 56.6 | (20.1) | 53.6 | (18.7) | 46.9 | (18.5) | 31.6 | (18.8) | |
| 33.3 | (12.2) | 39.9 | (10.7) | 37.3 | (10.3) | 32.6 | (12.0) | 19.0 | (13.2) | ||
| 300 | 39.9 | (18.3) | 49.5 | (21.0) | 45.0 | (19.6) | 38.3 | (18.8) | 27.5 | (21.4) | |
| 28.3 | (14.8) | 35.4 | (11.2) | 32.8 | (9.8) | 28.2 | (12.8) | 14.2 | (10.8) | ||
Note: 0° knee angle = full extension.
ACKNOWLEDGEMENTS
This work was funded in part by the United States Council for Automotive Research, Southfield, MI; and the authors were supported in part by grants from the NIH: NRSA F31 AR056175 (KGA), K12 HD055931 (LFL), K01AR056134 (LFL); and the American Physical Therapy Association (KGA).
Footnotes
Financial Interests: Authors K. Abdel-Malek and T. Marler disclose they are part owners in Santos Human, Inc.
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