Normalizing Lower-Extremity Strength Data for Children Without Disability Using Allometric Scaling

Methods 

This study retrospectively examined strength results from 39 children without disability ages 4 to 17 years.19 All procedures were conducted in accordance with institutional review board standards. Demographics of the study subjects are presented in table 1. Subjects were recruited through parents within the hospital community by word of mouth.19 Subjects provided the data used to develop normalization equations for peak joint torques at the hip, knee, and ankle. Because preliminary analyses indicated no significant differences in age, mass, or peak torques between girls and boys (see table 1), the data from both sexes were pooled. All subjects underwent testing on an isokinetic dynamometer to determine the maximum torque they could produce for the ankle plantarflexors and dorsiflexors, knee flexors and extensors, and hip adductors and abductors.8, 9, 10, 20, 21 For the ankle, subjects sat on a KinCom dynamometera and had their ankle joint axis aligned with the center of rotation of the KinCom lever arm.9, 20 The pelvis and thigh were secured with self-adhesive (Velcro) straps. A custom footplate including Velcro straps was made to securely hold small feet on the plate. The hip was placed in about a 90° angle, whereas the knee was in about 25° of flexion. Wedges and other support structures were used to permit children of all sizes to achieve these positions.

Table 1. Characteristics of the Study Subjects

	Characteristics	Male (n=19)	Female (n=20)	P⁎
	Age (y)	10.3±3.8(4.3–16.5)	9.5±3.1(4.4–15.8)	.52
	Height (cm)	139.2±25.4(98.0–181.0)	136.2±17.1(109.0–163.0)	.66
	Mass (kg)	38.6±18.4(14.1–75.0)	34.3±12.8(17.3–59.5)	.41
	BMI (kg/m2)	18.6±3.3(14.7–27.2)	17.9±3.5(14.5–28.3)	.51
	Hip abduction strength (Nm)	47.5±32.1(2.4–126.8)	34.2±17.5(5.7–75.0)	.13
	Hip adduction strength (Nm)	55.3±40.9(3.0–140.3)	36.4±21.2(7.2–85.7)	.09
	Knee extension strength (Nm)	80.1±62.0(10.5–187.5)	60.2±40.0(6.4–162.0)	.25
	Knee flexion strength (Nm)	40.8±29.9(3.6–102.8)	33.5±20.2(8.1–73.8)	.38
	Ankle dorsiflexion strength (Nm)	17.6±12.6(1.6–45.8)	15.1±7.3(5.0–31.2)	.46
	Ankle plantarflexion strength (Nm)	43.0±30.5(2.4–124.2)	39.4±18.6(18.9–79.1)	.66

NOTE. Values are mean ± standard deviation (range).

⁎ Difference between males and females.

A physical therapist established the range of motion (ROM) limits for ankle dorsiflexion and plantarflexion. Each subject actively moved their ankle at 10°/s from end-range ankle dorsiflexion to end-range plantarflexion and vice versa to obtain maximum concentric contractions of the ankle plantarflexors and dorsiflexors, respectively. A movement speed of 10°/s in the passive mode was selected because previous experimentation indicated that some children could not produce enough torque to initiate movement of the KinCom’s support arm and others could not keep up with the arm at faster speeds. Isometric contractions were not used because they did not quantify torque over an entire ROM, thereby potentially missing the joint angle in which the greatest torque could be produced.8 Thus, 10°/s was a slow enough speed to be close to an isometric contraction, but strength was assessed over the entire joint ROM.

At the knee, each child was seated in a similar position as with the ankle testing, which included thigh and pelvis straps.8 The knee joint axis was aligned with the center of rotation of the KinCom lever arm. The leg of the child was attached to the KinCom support arm with Velcro straps. The range was from full knee extension to about 60° below the horizontal, and the movement speed was again 10°/s.

At the hip, each child lay supine on the KinCom dynamometer bench and had his/her hip joint abduction and adduction axis aligned with the center of rotation of the KinCom lever arm.10 The pelvis was stabilized with a belt and with the aid of a research assistant. End-range hip abduction and adduction limits were established by the physical therapist with a slight amount of hip flexion to permit the heel to clear the bench during the movement. ROM limits were set at the point in which further movement in a direction resulted in lateral pelvic tilt. Caution was taken to maintain the lower extremity in resting knee extension and neutral rotation within the child’s bony alignment limits. The movement speed was 10°/s.

Three to 5 repetitions of each movement were performed to permit the subjects to achieve their best performance; however, only the test results indicating the greatest amount of torque produced were used in the analysis. Determining the best performance was relatively simple because the dynamometer automatically overlaid the data on the monitor for consecutive tests. The maximum torque values for both dorsiflexion and plantarflexion were recorded.

The process to develop the normalization equations involved 3 steps. First, existing normalization methods were examined using linear regression. The normalization procedures evaluated included the traditional normalization method of simply dividing the strength value by subject mass (torque/mass1.0), normalization by mass by height, and normalization by BMI. If these were effective normalization schemes, then the regressions would produce slopes that were not significantly different from zero. A 2-tailed t test with α equal to .05 was used to determine if the slopes differed significantly from zero.

Next, allometric scaling equations were derived based on the previously described equation Sn=S/mb. The exponent b=bmusc was determined for each muscle group by fitting a best-fit power curve (S=Sn×mb) to the experimental data, where S is the measured torque, m is body mass, and Sn and bmusc are the empirically determined curve-fit parameters. We used 95% confidence intervals (CIs) for bmusc to determine whether the exponents differed significantly from the theoretical value of 1.0. As with the first step, each allometric normalization equation was applied to the data and tested to determine if its slope from linear regression was different from zero.

Finally, in an attempt to establish a single exponent (bavg) that could be used to normalize the strength data for the major muscles of the lower extremity (ie, hip, knee, ankle), an average value bavg was calculated from the different values obtained from each of the 6 muscle groups. The torques from each muscle group were normalized by using this average exponent, and the results were again tested to determine if the slopes from linear regression were different from zero.

Results 

The initial linear regressions with torque/mass1.0 (simple mass normalization having b=1) indicated that the slopes for the hip, knee, and ankle dorsiflexor muscles differed significantly from zero (P<.001) (table 2, fig 1). Only the slope for the ankle plantarflexors versus body mass did not differ significantly from zero (P=.28). The regressions for torque/(mass × height) had slopes that differed from zero in some cases and slopes that did not differ from zero in other cases (see table 2). All of the slopes for torque/BMI differed significantly from zero (P<.001).

Table 2. Slope of Linear Regressions Versus Body Mass and Age

	Strength Measure	Body Mass Slope ± SE (95% CI)	P	Age Slope ± SE (95% CI)	P
	Torque/mass (Nm/kg)
	Hip abduction	.012±.003(.006to.018)	<.001	6.0±0.8(4.4–7.6)	<.001
	Hip adduction	.016±.004(.008to.024)	<.001	7.9±0.9(6.2–9.8)	<.001
	Knee extension	.030±.007(.017to.043)	<.001	13.8±1.1(11.6–15.9)	<.001
	Knee flexion	.013±.003(.007to.018)	<.001	6.2±0.6(5.0–7.5)	<.001
	Ankle dorsiflexion	.004±.001(.002to.006)	<.001	2.1±0.3(1.4–2.8)	<.001
	Ankle plantarflexion	.004±.004(−.004to.013)	.28	4.3±1.0(2.4–6.3)	<.001
	Torque/mass × height (N/kg)
	Hip abduction	.002±002(−.002to.007)	.35	.016±.010(−.005to.036)	.13
	Hip adduction	.004±.003(−.002to.010)	.15	.032±.012(.008to.057)	.01
	Knee extension	.012±.004(.003to.020)	.009	.091±.015(.060to.122)	<.001
	Knee flexion	.003±.002(.000to.007)	.07	.025±.007(.010to.040)	.001
	Ankle dorsiflexion	.001±.001(−.001to.002)	.46	.001±.003(−.005to.008)	.71
	Ankle plantarflexion	−.003±.003(−.009to.003)	.36	−.011±.014(−.038to.017)	.44
	Torque/BMI (Nm3/kg)
	Hip abduction	.060±.006(.048–.072)	<.001	.27±.03(.22–.33)	<.001
	Hip adduction	.075±.009(.058–.094)	<.001	.37±.03(.30–.44)	<.001
	Knee extension	.118±.017(.084–.153)	<.001	.66±.05(.56–.76)	<.001
	Knee flexion	.058±.007(.044–.071)	<.001	.29±.02(.24–.34)	<.001
	Ankle dorsiflexion	.022±.002(.018–.027)	<.001	.10±.01(.07–.12)	<.001
	Ankle plantarflexion	.041±.009(.024–.059)	<.001	.19±.04(.10–.27)	<.001
	Torque/massbmusc
	Hip abduction	−.001±.0003(−.001to.0001)	.10	−.001±.001(−.004to.002)	.67
	Hip adduction	−.0004±.0004(−.001to.0004)	.29	.001±.002(−.003to.004)	.64
	Knee extension	−.0002±.0003(−.001to.0003)	.42	.002±.001(−.0005to.004)	.11
	Knee flexion	−.0002±.0003(−.001to.0004)	.48	.002±.001(−.001to.005)	.24
	Ankle dorsiflexion	−.0002±.0002(−.001to.0002)	.27	−.0005±.001(−.002to.001)	.57
	Ankle plantarflexion	−.001±.001(−.004to.002)	.42	−.002±.007(−.016to.011)	.73
	Torque/massbavg
	Hip abduction	−.0005±.0004(−.001to.0004)	.29	.0002±.002(−.004to.004)	.92
	Hip adduction	−.0002±.001(−.001to.001)	.66	.003±.002(−.002to.007)	.28
	Knee extension	.001±.001(−.001to.002)	.38	.011±.003(.005to.017)	<.001
	Knee flexion	−.0002±.0003(−.001to.0004)	.48	.002±.001(−.001to.005)	.24
	Ankle dorsiflexion	−.00003±.0002(−.001to.0005)	.90	.0003±.001(−.002to.003)	.81
	Ankle plantarflexion	−.001±.001(−.003to.001)	.16	−.004±.005(−.014to.006)	.41

Abbreviation: SE, standard error.

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Fig 1. Linear regression of mass-normalized torques versus body mass. Abbreviations: abd, abduction; add, adduction; DF, dorsiflexion; ext, extension; flex, flexion; max, maximum; PF, plantarflexion;

The results for determining the allometric scaling exponents indicated different values of bmusc for each muscle group (table 3). The exponents differed significantly from the theoretical value of 1.0 for all measures except ankle plantarflexion (see table 3). The slopes from the linear regressions with different values of b for each muscle group indicated no significant difference from zero (P≥.10) (fig 2). The average value of the allometric scaling exponents from the 6 muscle groups (bavg) was 1.6. The slopes from the linear regressions versus body mass using bavg for all muscle groups indicated no significant difference from zero for the hip and knee (P≥.29). However, the slope did differ significantly from zero for the ankle (P<.05). A separate common value (bankle) was, therefore, derived for the ankle by averaging the b values for ankle dorsiflexion and plantarflexion because these values did not differ significantly from each other. Using the average value of bankle equal to 1.4 produced regressions with slopes that did not differ significantly from zero.

Table 3. Allometric Scaling Parameters (bmusc) for Maximum Torque Versus Body Mass

	Motion	Scaling Parameter (95% CI)
	Hip abduction	1.66⁎(1.35–1.98)
	Hip adduction	1.68⁎(1.36–2.00)
	Knee extension	1.87⁎(1.54–2.21)
	Knee flexion	1.59⁎(1.35–1.84)
	Ankle dorsiflexion	1.47⁎(1.27–1.68)
	Ankle plantarflexion	1.32(0.98–1.67)

⁎ Significant difference from 1.0.

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Fig 2. Linear regression of torques normalized using allometric scaling with empirically determined scaling exponents (bmusc) versus body mass. Abbreviations: see fig 1.

Discussion 

The purpose of this investigation was to evaluate and develop normalization equations for lower-extremity strength in children without disabilities. The influence of growth and size on muscle strength is well recognized. However, there has been no systematic approach to adjusting for growth when assessing strength. Most studies8, 9, 10, 11, 12, 13 use either simple mass normalization or no normalization, although other approaches such as normalization by weight by height have also been applied.22

The results of this study clearly show that simple mass normalization of lower-extremity torques does not effectively adjust for size in children without disability. Joint torques continue to increase with body mass after simple mass normalization (see fig 1). Hence, this current and widely used method may not be appropriate for comparing strength longitudinally or between groups that include children of different sizes. This deviation from geometric similarity is not surprising because it is well known that growth is not proportional. The disproportionate increases in torque (b>1.0) may be caused by improved coordination, more efficient neuromuscular activity, and/or increases in muscularity as the neuromuscular and musculoskeletal systems mature. These processes may occur at different rates and to different extents across various muscle groups. It is, therefore, not clear whether different b values should be expected for different muscle groups. Our preliminary results suggest greater similarity between the hip and knee muscles, with larger differences at the ankle.

This study examined allometry as a possible alternative to simple mass normalization and other existing normalization approaches. The allometric scaling techniques used in this study provided a simple and effective means to account for growth in children without disability. To our knowledge, this study is the first to report empirically derived scaling parameters for peak muscle torques in children. More exact values with smaller CIs may be obtained in the future by applying the same techniques to a larger group of subjects. Currently, it appears that a single normalization equation could be used for the hip and knee but that a different equation is needed for the ankle. It remains to be determined whether this result holds for a larger group of subjects. It would also be desirable to test the normalization equations on a separate group of children without disability, which was not possible in this study.

Conclusions 

Because maximum torques at the hip, knee, and ankle do not scale geometrically with body mass, simple mass normalization may not be an appropriate measure for cross-sectional or longitudinal comparisons of strength in children. The allometric scaling equations derived in this study provide more effective normalization than using body mass, body mass by height, or BMI for children without disability and may also be used for comparisons between subjects without disability and other groups of children such as those with cerebral palsy.