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. 2009 Jun 1;215(2):198–205. doi: 10.1111/j.1469-7580.2009.01088.x

Moment arms of the knee extensor mechanism in children and adults

Thomas D O’Brien 1, Neil D Reeves 1, Vasilios Baltzopoulos 1, David A Jones 1,2, Constantinos N Maganaris 1
PMCID: PMC2740967  PMID: 19493189

Abstract

In the present study we investigated whether there are differences in the patellar tendon moment arm (PTMA)-knee angle relationship between pre-pubertal children and adults, and whether the PTMA length scales to relevant anthropometric measurements in the two groups. Anthropometric characteristics and the PTMA length-joint angle relationships were determined in 20 adults and 20 pre-pubertal children of both genders. The anthropometric characteristics measured were height, body mass, knee circumference, medio-lateral knee breadth, anterior-posterior knee depth, leg length, femur length and tibia length. The PTMA was quantified from magnetic resonance images using the geometric centre of the femoral condyle method, at every 5° between 55° and 90° of knee flexion (0° is full extension). Adults had a significantly greater PTMA length at all joint angles (4.2 ± 0.4 vs. 3.6 ± 0.3 cm at 90°; P < 0.01), with the PTMA length decreasing from knee extension to knee flexion similarly in both adults and children. There were no significant and strong correlations between the PTMA and anthropometric measures in adults for any joint angle. In contrast, the PTMA correlated and scaled with anthropometric characteristics for the children (P < 0.05, r = 0.49–0.9) at all joint angles. The PTMA length in children was most accurately predicted at 85° of flexion from the equation PTMA = –0.25 + 0.083·tibia length + 0.02·leg length (R2 = 0.83). These findings indicate that the knee extensor mechanism in pre-pubertal children should not be considered to be a ‘scaled-down’ version of that in adults.

Keywords: growth, lever ratio, maturation, patellar tendon, puberty

Introduction

Puberty is associated with rapid growth and increased muscular strength (Parker et al. 1990). The strength of muscles is largely determined by their size but skeletal muscles, as their name implies, act through the skeletal system and across joints. As a result ‘muscle strength’ is conventionally measured as the moment acting about the joint during muscle contraction. This reflects not only the force-generating potential of the contracting muscle but also its moment arm (MA) length, which is the perpendicular distance between the muscle-tendon action line and the axis about which the moment is assumed to be generated. This is typically the joint axis of rotation although in two-dimensional models the MA is measured from the point where the axis of rotation intersects the measurement plane. As both the position of the joint centre of rotation and the length of the part of the bone proximal to the tendon's attachment point may change due to skeletal growth during adolescence, it is likely that puberty will see not only an increase in muscle size but also a change in the MA length, both of which will affect the joint moment and apparent strength.

The MA length of most muscles varies with joint angle (e.g. Rugg et al. 1990; Murray et al. 1995; Kellis & Baltzopoulos, 1999). For example, the patellar tendon moment arm (PTMA), which transmits the force from the quadriceps muscle to the tibia through the knee extensor mechanism, has been reported to decrease (e.g. Wretenberg et al. 1996; Sheehan, 2007) or follow an inverted U-shaped pattern (e.g. Kellis & Baltzopoulos, 1999; Krevolin et al. 2004) in the transition from full knee extension to full knee flexion, although there are also studies showing a constancy in PTMA length over the knee joint range of motion (Gill & O’Connor, 1996). Inter-study differences in the application of three-dimensional or two-dimensional methodologies and differences in the reference axes or points about which moments were assumed to act [e.g. the geometric centre of the femoral condyles (GCFC), trans-epicondylar line, tibio-femoral contact point or the instant centre of rotation] may partly account for the differences in the shape of the PTMA-angle relationships reported (for review see Tsaopoulos et al. 2006a). Nevertheless, for any given calculation or measurement method used, a disproportionate increase in the dimensions of knee joint structures with puberty could result in changes in the path that the knee joint centre of rotation and the patellar tendon action line follow as the knee rotates, potentially altering the shape of the PTMA length-knee angle relationship. This might explain the medio-lateral shift in the knee extension moment-angle curve in children compared with adults reported by Marginson & Eston (2001).

To date, no systematic investigation on differences in MA-angle relationships between children and adults has been carried out. Direct measurement of MA lengths requires imaging techniques, such as magnetic resonance imaging (MRI), which are expensive and time consuming (Wretenberg et al. 1996; Tsaopoulos et al. 2006b; Sheehan, 2007), or X-ray, which involves exposure to ionising radiation (e.g. Chow et al. 2006; Wang et al. 2008). Therefore, when MA lengths have been required for children [e.g. in studies investigating the effect of growth on the relationship between muscle force and size (Kanehisa et al. 2003), on tendon properties (Kubo et al. 2001), and in the application of generic computerized musculoskeletal models (as discussed by Scheys et al. 2008)] previously published MA data in adults have been scaled to the size of basic anthropometric characteristics (e.g. body height, body mass, limb length). The major underlying assumption of this approach is that the size of the basic anthropometric characteristics used for scaling are proportional to the specific joint structures that determine the MA. This assumption is appropriate for certain muscle groups, such as several of the trunk muscles, for which the MA has been shown to be proportional to measures of body height, body mass, torso depth and chest width (Reid et al. 1987; Moga et al. 1993; Jorgensen et al. 2003; Seo et al. 2003). However, the PTMA does not scale linearly as a simple ratio to, and cannot be predicted from, relevant anthropometric data in men (Tsaopoulos et al. 2006b). More importantly, the conventional scaling approach assumes that children are essentially ‘scaled-down’ adults, with the anatomical structures of interest having similar relative dimensions in the two groups. However, this assumption is based on little scientific evidence and there is a need for a simple, practical approach to estimate the PTMA length in children without the use of complex and expensive or potentially harmful equipment.

To investigate whether there are differences in the knee extensor mechanism between pre-pubertal children and adults, the current study quantified and compared the PTMA-angle relationship in these two groups, and investigated whether, in contrast to previous findings in men, the length of the PTMA can be predicted in children from relevant anthropometric measures.

Materials and methods

Participants

Twenty adults (10 men and 10 women, mean ± SD age 28.1 ± 3.7 years, height 1.75 ± 0.1 m, mass 72.8 ± 15.7 kg) and 20 children (10 boys and 10 girls, mean ± SD age 9.1 ± 0.8 years, height 1.39 ± 0.08 m, mass 37.9 ± 9.1 kg), without any history of musculoskeletal injury to the leg that was studied, volunteered to participate in this study. The limited age ranges examined (24–35 years for the adults and 8–10 years for the children) were selected to reflect two distinct points of maturation, one just prior to the onset of puberty and the other in young adulthood, with little variability within each group. Although the physical maturity of the children was not directly assessed from physical characteristics according to the scale of Tanner (1962), based on the age ranges included (8–10 years for both boys and girls) it is likely that most children in both gender groups were of pre-pubertal (stage 1) status (Tanner, 1962). Moreover, the MRI scans of the knee indicated that the growth plate of the femoral condyle was completely unfused (Grade I) in all of the children and completely fused (Grade VI) in the adults (Dvorak et al. 2007). The experimental procedures were approved by the institutional ethics committee of The Institute for Biomedical Research into Human Movement and Health.

Magnetic resonance imaging protocol

Participants lay at rest on the side of their dominant leg with the knee joint of that leg in a 0.2-T MRI scanner (E-scan, Esaote Biomedica, Genoa, Italy). MRI scans were taken in the sagittal plane using a spin-echo TI half fourier (HF) sequence with a slice thickness of 6 mm, inter-slice gap of 0.6 mm and the parameters time to repetition/echo time/number of excitations (TR/TE/NEX), 420/18/1; field of view, 160 × 160 mm2; matrix, 256 × 256 pixels. The knee joint angle was controlled by securing a large mechanical goniometer to the medial aspect of the knee in line with the femur and tibia, identified through palpation. The goniometer was locked at the appropriate angle and remained on the leg throughout the scan to prevent movement.

Moment arm length measurements

The PTMA length was measured at every 5° of knee flexion between 55 and 90° (0° is full extension) to obtain the PTMA-knee angle relationship. This knee joint angle range was chosen to include angles close to the angle of peak knee extension moment in both adults and children (Marginson & Eston, 2001).

The PTMA length was quantified with respect to the GCFC (Hollister et al. 1993; Eckhoff et al. 2001; Most et al. 2004). This method gives PTMA values in two dimensions similar to those obtained with respect to the instant centre of knee joint rotation (Tsaopoulos et al. 2009). In addition, the GCFC method accounts for the actual size of relevant anatomical structures and it was therefore considered an appropriate method for describing changes in PTMA between children and adults. Using this method the PTMA is defined as the perpendicular distance between the GCFC and the patellar tendon line of action. A MatLab-based script was used for the measurement of the PTMA (Tsaopoulos et al. 2009). Analysis was performed using three slices selected from the whole sequence: one through the medial condyle in the middle of its thickness as viewed in the frontal plane, one through the lateral condyle in the middle of its thickness as viewed in the frontal plane, and one close to the centre of the knee joint where the patellar tendon could be clearly seen (Fig. 1). The centre of the posterior portions of the medial and lateral condyles was determined from the respective slices. This was done by digitizing six points around the posterior portion of each condyle, to which a circle was fitted using a least squares method. The centre of the circle was considered as the centre of the condyle. The midpoint between the centre of the medial and lateral condyles, when superimposed onto one another, was considered as the GCFC, which was then used to quantify the PTMA length on the central image. From the field of view and image matrix in the scanning protocol used, it follows that the MRIs had a resolution of 0.06 cm2 per pixel. The PTMA length analysis procedure had an intra- and inter-observer reliability with a mean test/retest difference of < 0.03 cm and an SE of < 0.05 cm.

Fig. 1.

Fig. 1

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Example magnetic resonance images of (a) the medial condyle, (b) the lateral condyle and (c) the centre of the knee joint to quantify the patellar tendon moment arm (PTMA). The white circles in (a) and (b) were fitted to identify the position of centres, which were then used to locate the geometric centre of the femoral condyles (white cross) and quantify the length (arrow) of the PTMA in (c).

External anthropometric measurements

The external anthropometric characteristics measured were height (H), body mass (M), knee circumference (KNEEC), medio-lateral knee breadth (KNEEB), anterior-posterior knee depth (KNEED), leg length (LEGL), femur length (FEML) and tibia length (TIBL). KNEEC, KNEEB and KNEED were measured at the height of the femoral condyles and LEGL, FEML and TIBL were measured as the distances from the greater trochanter to the lateral malleolus, from the greater trocanter to the lateral femoral condyle and from the tibial plateau to the lateral malleolus, respectively. All measurements were made to the nearest millimetre using anatomical callipers for KNEEB and KNEED, and a tape measure for KNEEC, LEGL, FEML and TIBL.

Statistical analysis

Data were grouped into adult and child sets for statistical analysis. The PTMA length-knee angle relationship was analysed using a 2 × 8 mixed (age group × joint angle) analysis of variance (anova) with repeated measures on the second factor to test for any differences in PTMA length between the two age groups and across joint angles. Bonferroni corrected post-hoc tests were used to identify the joint angles at which any differences existed.

A backwards stepwise multiple regression analysis, using all anthropometric measures, was performed to identify the combination of parameters with the greatest predictive accuracy for PTMA length at each joint angle examined. Pearson's correlation coefficients were used to obtain correlations between PTMA length and each anthropometric measure at every joint angle. The level of significance was set at P ≤ 0.05. The quantities examined were considered to scale with each other as simple ratios if the 95% confidence limits for the Y-intercept in the regression equation contained the zero value.

Results

All of the anthropometric characteristics were significantly greater in adults than in children (P < 0.01 for all comparisons, Table 1).

Table 1.

Mean ± SD of anthropometric measurements for adults and children (n = 20 per group)

Adults
 Children
Mean SD Mean SD
H (cm) 174.5 9.5 139 7.5
M (kg) 72.8 15.7 37.9 9.1
KNEEC (cm) 38.3 2.9 32 3
KNEEB (cm) 11.2 1.0 9.9 1
KNEED (cm) 12.2 1 10 1
LEGL (cm) 83.3 4.1 65.2 5.2
FEML (cm) 41.4 2.8 31.3 2.9
TIBL (cm) 38.9 2.4 31.3 2.7
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Significant differences exist between adults and children for all parameters (P < 0.01).

H, height; M, body mass; KNEEC, knee circumference; KNEEB, medio-lateral knee breadth; KNEED, anterior-posterior knee depth; LEGL, leg length; FEML, femur length; TIBL, tibia length.

The adult population had significantly (P < 0.01) greater PTMA lengths than the child population at all joint angles (Fig. 2). The PTMA length decreased with increasing knee flexion in both groups (P < 0.05) and the decrease was similar in the two groups (P > 0.05).

Fig. 2.

Fig. 2

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The patellar tendon moment arm (PTMA) length-knee angle relationship in the two age groups.

In agreement with previous findings at full knee extension in men (Tsaopoulos et al. 2006b), the PTMA length in the present study could not be confidently predicted at knee angles between 55 and 90° in adults of both genders (Table 2, Equations 1–8). For the children, however, PTMA length could be predicted with R2 values between 0.62 and 0.83 at any knee joint angle tested (Table 3, Equations 9–16) and the most accurate model corresponded to 85° and included LEGL and TIBL.

Table 2.

Regression equations for the patellar tendon moment arm (PTMA) length in the adults

Angle (°) Equation R2 value
Eq. 1 55 PTMA = 3.26 + 0.019 M 0.41
Eq. 2 60 PTMA = 3.21 + 0.019 M 0.47
Eq. 3 65 PTMA = 4.661 + 0.024 M – 0.171 KNEEB 0.49
Eq. 4 70 PTMA = 3.283 + 0.016 M 0.4
Eq. 5 75 PTMA = 3.188 + 0.017 M 0.43
Eq. 6 80 PTMA = 4.411 + 0.025 M – 0.167 KNEEB 0.5
Eq. 7 85 PTMA = 3.185 + 0.016 M 0.4
Eq. 8 90 PTMA = 1.571 + 0.222 KNEED 0.27
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M, body mass; KNEEB, medio-lateral knee breadth; KNEED, anterior-posterior knee depth.

Table 3.

Regression equations for the patellar tendon moment arm (PTMA) length in the children

Angle (°) Equation R2 value
Eq. 9 55 PTMA = 0.212 + 0.028·LEGL + 0.054 KNEEC 0.62
Eq. 10 60 PTMA = 1.284 + 0.027·LEGL + 0.018 M 0.67
Eq. 11 65 PTMA = 0.414 + 0.106 TIBL 0.73
Eq. 12 70 PTMA = 0.408 + 0.105 TIBL 0.77
Eq. 13 75 PTMA = 0.214 + 0.111 TIBL 0.7
Eq. 14 80 PTMA = 0.096 + 0.114 TIBL 0.81
Eq. 15 85 PTMA = –0.25 + 0.083 TIBL + 0.019·LEGL 0.83
Eq. 16 90 PTMA = 0.204 + 0.108 TIBL 0.7
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M, body mass; KNEEC, knee circumference; LEGL, leg length; TIBL, tibia length.

In the adults the correlations between PTMA and KNEEB, KNEEC, LEGL, FEML and TIBL were either statistically non-significant (P > 0.05) or had low correlation coefficients (r < 0.5) for all knee joint angles. There were significant correlations between PTMA length and M (P < 0.01), H and KNEED (P < 0.05) at all knee joint angles, with moderate correlation coefficients (0.6–0.7 for M, 0.47–0.6 for H and KNEED). In the children, with the exception of the knee joint angle of 55° at which the P-value for the correlation with FEML was 0.056 (r = 0.43), the PTMA significantly correlated and scaled as a simple ratio with all of the anthropometric measurements for all of the joint angles (P < 0.05), with correlation coefficients between 0.49 and 0.9. These correlations were associated with statistical power values higher than 0.85, with the exception of KNEEB for which the power was 0.6. Figure 3 shows the correlations between each anthropometric characteristic and PTMA length at 85° in the two groups. At this joint angle, no significant relationships existed between PTMA and KNEEB, LEGL, FEML and TIBL in the adults. The relationships between PTMA and H, M, KNEEC and KNEED were significant but the correlation coefficients were moderate to low, with statistical power values between 0.45 and 0.63. In the children, however, all of the correlations were significant and the correlation coefficients were higher than in adults (see Fig. 3).

Fig. 3.

Fig. 3

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Scatter plots of the patellar tendon moment arm (PTMA) length at 85° of knee flexion against each anthropometric measure for each group.

Discussion

This study has shown that adults have greater PTMA lengths than children at knee flexion angles between 55 and 90° and that, in contrast to adults, the PTMA in children scales with, and can be predicted from, easily measured anthropometric characteristics.

The PTMA lengths over the knee joint angles tested were greater by ∼20% in adults compared with children. This has significant implications for the interpretation of muscle strength testing in the two populations. In several studies of children and adults, knee extensor muscle strength is measured as a joint moment (e.g. Marginson & Eston, 2001; Kanehisa et al. 2003), which is the product of the quadriceps muscle force and the PTMA length. Hence, according to the present data, the joint moment produced would be ∼20% higher in adults than in children solely due to differences in PTMA length, excluding factors affecting the force-generating potential of muscle. Therefore, interpretation of changes in knee extension joint moment with growth for the description of muscle force without accounting for the changes in PTMA length is misleading. In studies where knee extension muscle strength is quantified using a force transducer attached onto the tibia via a cable or other rigid link (e.g. Parker et al. 1990; Round et al. 1999), the external force measured is the product of the patellar tendon force and the lever ratio, i.e. the ratio of the PTMA length and the distance from the knee centre of rotation to the attachment of the force transducer (force transducer MA) (Jones & Round, 2000). Provided that the force transducer is placed at right angles to the tibia, the validity of the results obtained using this approach depends on whether the lever ratio is constant. The present strong correlation between TIBL and PTMA suggests that the lever ratio is relatively constant in children, and thus force measured on the tibia near the ankle is a good comparative indicator of the force in the patellar tendon. However, this is not the case for adults and therefore comparisons of muscle strength between children and adults measured using force transducers need to be interpreted with care.

The knee extension moment-angle relationship has been reported to be different in children than adults, with the peak joint moment being towards more extended knee angles in adults (Marginson & Eston, 2001). One possible reason for this finding is an inter-group difference in the pattern of the PTMA length-knee angle relationship. However, we showed here that, at the knee angles surrounding the angle of peak joint moment (∼80–85°, Marginson & Eston, 2001), there is no difference in the pattern of the PTMA length-angle relationship between adults and children. Therefore, other factors, such as inter-group differences in tendon stiffness, muscle architecture and agonist and antagonist muscle activation, probably play a more important role in determining the differences in the joint angle of peak moment.

In accordance with previous findings at full knee extension in 22 men (Tsaopoulos et al. 2006b), the current study showed that the PTMA length in 20 adults of both genders does not scale well with external anthropometric measurements over a number of knee joint flexion angles. This consistency gives confidence that the present result is valid but the low to moderate values of statistical power obtained indicate that a greater sample should be examined to eliminate the possibility of type II error. In contrast, the PTMA length in children correlated significantly (P < 0.01) with higher correlation coefficients than in adults and high statistical power values for all of the anthropometric measurements at all knee joint angles. These findings indicate that assuming that MAs in children scale to relevant anthropometric measures in the same way as they do in adults (Kubo et al. 2001; Kanehisa et al. 2003) is not valid for the PTMA. Thus, the practice of scaling MAs in adults to predict the corresponding values in children should be avoided.

The anthropometric parameter that had the strongest correlation with PTMA length in children was TIBL for all joint angles and the predictive equations included TIBL and/or LEGL. The relative ease with which the anthropometric characteristics examined can be measured makes the predictive Equations 9–16 practical tools for the realistic estimation of PTMA lengths in children (aged 8–10 years) of late pre-pubertal status. It may be surprising that PTMA was more closely scaled to, and more accurately predicted from, measures of ‘length’ than KNEED, KNEEB or KNEEC, which intuitively seem to have the greatest correspondence to the internal dimensions of the knee extensor mechanism that determine PTMA. Although KNEED, KNEEB and KNEEC were measured around the femoral condyles, they depend on the size of the condyles in the transverse plane, and may not be representative of the size of the condyle in the long axis down the shaft of the femur, which is the dimension of the condyle that determines the PTMA at flexed knee angles (Fig. 4). Closer to full knee extension the PTMA could correlate better with KNEED or KNEEC (Fig. 4). The inclusion of KNEEC in the predictive equation for only the most extended knee angle tested lends support to this theory.

Fig. 4.

Fig. 4

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A schematic representation of the knee at (a) full extension and (b) a flexed angle. At full extension the patellar tendon moment arm (PTMA) (Inline graphic), measured from the centre of the condyles (Inline graphic), is in the same direction as knee depth (Inline graphic), so the two parameters may be correlated. However, when the knee is flexed the PTMA is measured in a direction corresponding to the long axis of the femur, so it depends on the size of the condyle and patella in this direction (Inline graphic), which is not easily obtained from any external anthropometric measure local to the knee joint.

In applying the predictive equations presented here consideration must be given to the limitations of this study. First, the age and maturation of the children is restricted to those in the late stages of pre-puberty (aged 8–10 years) and it is not certain that the same predictive equations hold true for younger children or those in advanced pubertal stages. Second, the knee joint angles examined here were limited to those surrounding the angle of peak knee extension moment, and future studies should determine if the present findings in children are also true for more extended knee angles. Third, the conclusions drawn here are based on measures of PTMA length using the GCFC method in two dimensions (Tsaopoulos et al. 2009). These measurements do not account for the knee ‘screw-home mechanism’, which operates in three dimensions (Krevolin et al. 2004). The estimated error introduced in the PTMA length measurement by this limitation is ∼12% at full knee extension but around the knee joint angles that we examined the error is only ∼1% (Tsaopoulos et al. 2009). However, the above error estimates are based on measurements in adults, and further studies are required to confirm the applicability of these estimates in children.

In the present study, male and female data for the adults and children were grouped, which could potentially mask any difference in the way that PTMA and the anthropometric characteristics change with maturation between genders. However, the correlation between PTMA length and TIBL that was evident in children (P < 0.01, r = 0.9) remains when either the girls or the boys are removed from the analysis (P < 0.01, r = 0.87 and P < 0.01, r = 0.91, respectively), and the lack of correlation between PTMA length and TIBL in adults (P > 0.05, r = 0.36) still remains when either men or women are excluded from the analysis (P > 0.05, r = 0.0 and P > 0.05, r = 0.36, respectively). These individual correlations can be seen in Fig. 3 and indicate that the results obtained were not confounded by different growth patterns in males and females.

In conclusion, we showed that the difference in PTMA length between adults and pre-pubertal children cannot be explained by relevant anthropometric differences between the two groups. In contrast to adults, the PTMA length in late pre-pubertal children was accurately predicted from simple anthropometric measures. These findings indicate that the knee extensor mechanism in pre-pubertal children should not be considered a ‘scaled-down’ version of that in adults.

Acknowledgments

The authors thank Dimitrios E. Tsaopoulos who developed the MatLab script used for the analysis of PTMA length.

Author contributions

All authors were involved in the design of the experiment and the preparation of the submitted manuscript. Data were acquired, analysed and interpreted by T.D.O.’B., who also drafted the manuscript. Critical revisions, until the final draft was agreed, were followed by the supervisors of the project (C.N.M., V.B., N.D.R. and D.A.J.) who collectively conceived the ideas in question.

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