The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

Predicting in vivo soft tissue masses of the lower extremity using segment anthropometric measures and DXA

The purpose of this study was to derive and validate regression equations for the prediction of fat mass (FM), lean mass (LM), wobbling mass (WM), and bone mineral content (BMC) of the thigh, leg, and leg + foot segments of living people from easily measured segmental anthropometric measures. The segment masses of 68 university-age participants (26 M, 42 F) were obtained from full-body dual photon x-ray absorptiometry (DXA) scans, and were used as the criterion values against which predicted masses were compared. Comprehensive anthropometric measures (6 lengths, 6 circumferences, 8 breadths, 4 skinfolds) were taken bilaterally for the thigh and leg for each person. Stepwise multiple linear regression was used to derive a prediction equation for each mass type and segment. Prediction equations exhibited high adjusted R2 values in general (0.673 to 0.925), with higher correlations evident for the LM and WM equations than for FM and BMC. Predicted (equations) and measured (DXA) segment LM and WM were also found to be highly correlated (R2 = 0.85 to 0.96), and FM and BMC to a lesser extent (R2 = 0.49 to 0.78). Relative errors between predicted and measured masses ranged between 0.7% and -11.3% for all those in the validation sample (n = 16). These results on university-age men and women are encouraging and suggest that in vivo estimates of the soft tissue masses of the lower extremity can be made fairly accurately from simple segmental anthropometric measures.     

Masses, centers-of-gravity, and moments-of-inertia of the body segments of the rhesus monkey (Macaca mulatta).

Segmental parameters (mass, center-of-gravity, and moment-of-inertia) are necessary for biomechanical analyses of a species' locomotor behavior. Seven male and eight female adult rhesus monkey cadavers were dismembered in order to determine segmental parameters. Mean values for the segment masses and moments-of-inertia are presented for males and females, separately and together. Statistical tests revealed significant differences between the sexes for these parameters. Regression equations for predicting segment masses and moments-on-inertia were developed for the sexes separately and pooled. For most segments the male and female equations did not differ significantly in slope or y-intercept. The center-of-gravity for each segment is presented as a mean percentage of the distance between the proximal and distal joint centers. The regression equations and center-of-gravity locations presented here permit biomechanical investigations of rhesus monkey locomotion without the necessity of subsequent sacrifice. The segmental parameter values determined for the rhesus monkey are compared with available data for other primate and mammalian species and the biomechanical and adaptive implications of such comparisons are discussed.     

Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines

We present a robust and computationally inexpensive method to estimate the lengths and three-dimensional moment arms for a large number of musculotendon actuators of the human lower limb. Using a musculoskeletal model of the lower extremity, a set of values was established for the length of each musculotendon actuator for different lower limb generalized coordinates (joint angles). A multidimensional spline function was then used to fit these data. Muscle moment arms were obtained by differentiating the musculotendon length spline function with respect to the generalized coordinate of interest. This new method was then compared to a previously used polynomial regression method. Compared to the polynomial regression method, the multidimensional spline method produced lower errors for estimating musculotendon lengths and moment arms throughout the whole generalized coordinate workspace. The fitting accuracy was also less affected by the number of dependent degrees of freedom and by the amount of experimental data available. The spline method only required information on musculotendon lengths to estimate both musculotendon lengths and moment arms, thus relaxing data input requirements, whereas the polynomial regression requires different equations to be used for both musculotendon lengths and moment arms. Finally, we used the spline method in conjunction with an electromyography driven musculoskeletal model to estimate muscle forces under different contractile conditions, which showed that the method is suitable for the integration into large scale neuromusculoskeletal models.     

Masses,centers-of-gravity,and moments-of-inertia of the body segments of the rhesus monkey (Macaca mulatta)

Segmental parameters (mass, center-of-gravity, and moment-of-inertia) are necessary for biomechanical analyses of a species' locomotor behavior. Seven male and eight female adult rhesus monkey cadavers were dismembered in order to determine segmental parameters. Mean values for the segment masses and moments-of-inertia are presented for males and females, separately and together. Statistical tests revealed significant differences between the sexes for these parameters. Regression equations for predicting segment masses and moments-of-inertia were developed for the sexes separately and pooled. For most segments the male and female equations did not differ significantly in slope or y-intercept. The center-of-gravity for each segment is presented as a mean percentage of the distance between the proximal and distal joint centers. The regression equations and center-of-gravity locations presented here permit biomechanical investigations of rhesus monkey locomotion without the necessity of subsequent sacrifice. The segmental parameter values determined for the rhesus monkey are compared with available data for other primate and mammalian species and the biomechanical and adaptive implications of such comparisons are discussed.     

Energetic benefits and adaptations in mammalian limbs: Scale effects and selective pressures

Differences in limb size and shape are fundamental to mammalian morphological diversity; however, their relevance to locomotor costs has long been subject to debate. In particular, it remains unknown if scale effects in whole limb morphology could partially underlie decreasing mass‐specific locomotor costs with increasing limb length. Whole fore‐ and hindlimb inertial properties reflecting limb size and shape—moment of inertia (MOI), mass, mass distribution, and natural frequency—were regressed against limb length for 44 species of quadrupedal mammals. Limb mass, MOI, and center of mass position are negatively allometric, having a strong potential for lowering mass‐specific locomotor costs in large terrestrial mammals. Negative allometry of limb MOI results in a 40% reduction in MOI relative to isometry's prediction for our largest sampled taxa. However, fitting regression residuals to adaptive diversification models reveals that codiversification of limb mass, limb length, and body mass likely results from selection for differing locomotor modes of running, climbing, digging, and swimming. The observed allometric scaling does not result from selection for energetically beneficial whole limb morphology with increasing size. Instead, our data suggest that it is a consequence of differing morphological adaptations and body size distributions among quadrupedal mammals, highlighting the role of differing limb functions in mammalian evolution.     

Mass, center of mass, and moment of inertia estimates for infant limb segments.

To quantify limb dynamics, accurate estimates are needed of anthropometric inertia parameters (mass, center-of-mass location, and moments of inertia). These estimates, however, are not available for human infants; therefore, the movement dynamics of infants have not been studied extensively. Here, regression equations for the masses, center-of-mass locations, and transverse moments of inertia of upper and lower limb segments (upper arm, forearm, and hand; thigh, leg, and foot) of 0.04 to 1.50 yr old infants are provided. A mathematical model of the human body was used to determine the anthropometric inertia parameters for upper limbs in 44 infants and for lower limbs in 70 infants. Stepwise linear regressions were used to fit the distributions of the anthropometric inertia parameters. The regression equations accounted for significant amounts of the variance (64-98%), and the R2-values compared favorably when our equations were cross-validated. Consequently, these regression equations can provide, for infants of similar ages, reasonable estimates of upper and lower limb anthropometric inertia parameters, suitable for equations of motion in the analysis of limb dynamics in human infants.     

A 3D interactive method for estimating body segmental parameters in animals: application to the turning and running performance of Tyrannosaurus rex

We developed a method based on interactive B-spline solids for estimating and visualizing biomechanically important parameters for animal body segments. Although the method is most useful for assessing the importance of unknowns in extinct animals, such as body contours, muscle bulk, or inertial parameters, it is also useful for non-invasive measurement of segmental dimensions in extant animals. Points measured directly from bodies or skeletons are digitized and visualized on a computer, and then a B-spline solid is fitted to enclose these points, allowing quantification of segment dimensions. The method is computationally fast enough so that software implementations can interactively deform the shape of body segments (by warping the solid) or adjust the shape quantitatively (e.g., expanding the solid boundary by some percentage or a specific distance beyond measured skeletal coordinates). As the shape changes, the resulting changes in segment mass, center of mass (CM), and moments of inertia can be recomputed immediately. Volumes of reduced or increased density can be embedded to represent lungs, bones, or other structures within the body. The method was validated by reconstructing an ostrich body from a fleshed and defleshed carcass and comparing the estimated dimensions to empirically measured values from the original carcass. We then used the method to calculate the segmental masses, centers of mass, and moments of inertia for an adult Tyrannosaurus rex, with measurements taken directly from a complete skeleton. We compare these results to other estimates, using the model to compute the sensitivities of unknown parameter values based upon 30 different combinations of trunk, lung and air sac, and hindlimb dimensions. The conclusion that T. rex was not an exceptionally fast runner remains strongly supported by our models-the main area of ambiguity for estimating running ability seems to be estimating fascicle lengths, not body dimensions. Additionally, the craniad position of the CM in all of our models reinforces the notion that T. rex did not stand or move with extremely columnar, elephantine limbs. It required some flexion in the limbs to stand still, but how much flexion depends directly on where its CM is assumed to lie. Finally we used our model to test an unsolved problem in dinosaur biomechanics: how fast a huge biped like T. rex could turn. Depending on the assumptions, our whole body model integrated with a musculoskeletal model estimates that turning 45 degrees on one leg could be achieved slowly, in about 1-2s.     

The application to bipeds of a geometric model of lower-limb-segment inertial properties

Drawing inferences about locomotor energetics from limb morphology, especially in regard to small differences between individuals, depends critically on valid estimates of lower-limb inertial properties. While there are numerous options for such estimations in the literature, geometric models that involve simple measures and straightforward mathematics combined with the ability to capture individual variation are rare. In this research, we apply a method, originally developed for quadrupeds, that models limb segments as elliptical columns. When the elliptical model is applied to bipeds, it provides a means of estimating limb-segment inertial properties accurately enough to test differences between individuals of similar stature and mass, but with variation in mass distribution and limb length. We test the method against commonly used equations and are able to show the validity of the method for thigh and shank segments.     

Reliability of upper and lower extremity anthropometric measurements and the effect on tissue mass predictions

Accurate modeling of soft tissue motion effects relative to bone during impact requires knowledge of the mass of soft and rigid tissues in living people. Holmes et al., [2005. Predicting in vivo soft tissue masses of the lower extremity using segment anthropometric measures and DXA. Journal of Applied Biomechanics, 21, 371–382] developed and validated regression equations to predict the individual tissue masses of lower extremity segments of young healthy adults, based on simple anthropometric measurements. However, the reliability of these measurements and the effect on predicted tissue mass estimates from the equations has yet to be determined. In the current study, two measurers were responsible for collecting two sets of unilateral measurements (25 male and 25 female subjects) for the right upper and lower extremities. These included 6 lengths, 6 circumferences, 8 breadths, and 4 skinfold thicknesses. Significant differences were found between measurers and between sexes, but these differences were relatively small in general (75–80% of between-measurer differences were <1 cm). Within-measurer measurement differences were smaller and more consistent than those between measurers in most cases. Good to excellent reliability was demonstrated for all measurement types, with intra-class correlation coefficients of 0.79, 0.86, 0.85 and 0.86 for lengths, circumferences, breadth and skinfolds, respectively. Predicted tissue mass magnitudes were moderately affected by the measurement differences. The maximum mean errors between measurers ranged from 3.2% to 24.2% for bone mineral content and fat mass, for the leg and foot, and the leg segments, respectively.     

Upper extremity soft and rigid tissue mass prediction using segment anthropometric measures and DXA

Regression equations for predicting bone mineral content (BMC), fat mass (FM), lean mass (LM), and wobbling mass (WM) of living people from simple anthropometric measures (segment lengths, circumferences, breadths, and skin folds) have been reported in the literature for the lower extremities, but are lacking for the upper extremities. Multiple linear stepwise regression was used to generate such equations for the arm, forearm, and forearm and hand segments of healthy university aged people (38 males, 38 females). Actual tissue masses were obtained from full body Dual-energy X-ray Absorptiometry (DXA) scans and were used to validate the developed equations with an independent sample of 24 participants (12 male, 12 female). Prediction equations exhibited very high adjusted R² values (range from 0.854 to 0.968), with more explained variance for LM and WM than for BMC and FM. Scatter plots of actual versus predicted tissue masses revealed a close relationship (R² range from 0.681 to 0.951). Relative errors between the predicted and actual tissue masses for the validation group ranged from ?2.2% to 15.5%, and the root-mean-squared error (RMS_error) ranged from 7.92 to 180.26 g, for BMC of the forearm and LM of the arm, respectively. These results suggest that accurate estimates of in-vivo tissue masses for the upper extremities can be predicted from simple anthropometric measurements in young adults. Access to tissue masses such as these will enable the development of more accurate models for predicting dynamic in-vivo response of the body to activities involving impact.     

Genetic variation of Polish endangered Biłgoraj horses and two common horse breeds in microsatellite loci

Genetic variation of endangered Bi?goraj horses and two common Polish horse breeds was compared with the use of 12 microsatellite loci (AHT4, AHT5, ASB2, HMS2, HMS3, HMS6, HMS7, HTG4, HTG6, HTG7, HTG10, VHL20). Lower allelic diversity was detected in all investigated populations in comparison to other studies. Large differences in the frequencies of microsatellite alleles between Bi?goraj horses and two other horse breeds were discovered. In all polymorphic loci all investigated breeds were in the Hardy-Weinberg equilibrium. Mean Fis values and the results of a test for the presence of a recent bottleneck were non-significant in all studied populations. Comparable values of observed and expected gene diversity indicate no substantial loss of genetic variation in the Bi?goraj population and two other breeds. The lowest variability observed in the investigated group of Thoroughbred horses was confirmed. About 10% of genetic variation are explained by differences between breeds. Values of pairwise Fst and two measures of genetic distance demonstrated that Bi?goraj horses are distantly related to both common horse breeds.     

Scaling of the limb long bones to body mass in terrestrial mammals

Long‐bone scaling has been analyzed in a large number of terrestrial mammals for which body masses were known. Earlier proposals that geometric or elastic similarity are suitable as explanations for long‐bone scaling across a large size range are not supported. Differential scaling is present, and large mammals on average scale with lower regression slopes than small mammals. Large mammals tend to reduce bending stress during locomotion by having shorter limb bones than predicted rather than by having very thick diaphyses, as is usually assumed. The choice of regression model used to describe data samples in analyses of scaling becomes increasingly important as correlation coefficients decrease, and theoretical models supported by one analysis may not be supported when applying another statistical model to the same data. Differences in limb posture and locomotor performance have profound influence on the amount of stress set up in the appendicular bones during rigorous physical activity and make it unlikely that scaling of long bones across a large size range of terrestrial mammals can be satisfactorily explained by any one power function. J. Morphol. 239:167–190, 1999. © 1999 Wiley‐Liss, Inc.     

Dynamic model of the equine hindlimb during the swing phase

A dynamic model is developed to describe the swing phase of the hindlimb of a normally walking horse. The limb was represented by four rigid segments constrained to move in a sagittal plane only. The mathematical equations of motion of this four-element pendulum were formulated using Lagrange's theorem. The morphometric parameters from the hindlimb segments of 3 horses were determined using high-speed film analysis. Five muscle groups were incorporated in the model. Muscle activity was derived from earlier EMG measurements. Optimization of muscle moments resulted in a simulated swing movement that approximated that in the living animal.     

Body Mass Estimates in Extinct Mammals from Limb Bone Dimensions: the Case of North American Hyaenodontids

The body mass estimation of several limb bone dimensions (shaft cross-sectional properties, articular sizes, and bone lengths) were examined using bivariate linear regression analyses. The sample included taxonomically and behaviourally diverse small to medium-sized Recent carnivorans and carnivorous marsupials. All examined limb bone dimensions indicated low errors (percentage standard error of estimate, 8–13) for the body mass estimations. Among them, humeral and femoral shaft properties correlated best with body weight, while limb bone lengths gave larger errors. Both humeral and femoral head dimensions have relatively large individual variations, and distal humeral articular dimensions seem to be influenced more by phylogenetic differences. The regressions based on each locomotor group gave slightly lower errors than those based on the total pooled sample. The results were then applied to hyaenodontid creodonts from the Eocene–Oligocene of North America. The estimated body masses (kg) are: Arfia , 5.4–9.5; Prototomus , <6.0; Pyrocyon , 2.6; Sinopa , 1.3–1.4; Tritemnodon , 7.6–13; Prolimnocyon , 1.6; Thinocyon , 0.7–2.5; Machaeroides , 12; Limnocyon , 7.8– 16; Hyaenodon , 9.1–43. The various limb bone dimensions give different body mass values, but the variation in estimates is smaller compared to those derived from dental or cranial measurements.     

Salient features in the locomotion of proboscideans revealed via the differential scaling of limb long bones

The standard differential scaling of proportions in limb long bones (length against circumference) was applied to a phylogenetically wide sample of the Proboscidea, Elephantidae and the Asian (Elephas maximus) and African (Loxodonta africana) elephants. In order to investigate allometric patterns in proboscideans and terrestrial mammals with parasagittal limb kinematics, the computed slopes between long bone lengths and circumferences (slenderness exponents) were compared with published values for mammals, and studied within a framework of the theoretical models of long bone scaling under gravity and muscle forces. Limb bone allometry in E. maximus and the Elephantidae is congruent with adaptation to bending and/or torsion induced by muscular forces during fast locomotion, as in other mammals, whereas the limb bones in L. africana appear to be adapted for coping with the compressive forces of gravity. Hindlimb bones are therefore more compliant than forelimb bones, and the resultant limb compliance gradient in extinct and extant elephants, contrasting in sign to that of other mammals, is shown to be a new important locomotory constraint preventing elephants from achieving a full‐body aerial phase during fast locomotion. Moreover, the limb bone pattern of African elephants, indicating a noncritical bone stress not increasing with increments in body weight, explains why their mean and maximal body masses are usually above those for Asian elephants. Differences in ecology may be responsible for the subtle differences observed in vivo between African and Asian elephants, but they appear to be more pronounced when revealed via mechanical patterns dictated by limb bone allometry. © 2010 The Linnean Society of London, Biological Journal of the Linnean Society, 2010, 100 , 16–29.     

Laterality in the gallop gait of horses

Bilateral asymmetry in gallop stride limb contact patterns of four Quarter Horse fillies was documented by high-speed cinematography. Horses were filmed with rider by two cameras simultaneously while galloping along a straightaway. Even though signaled for each gallop lead an equivalent number of times, horses frequently switched leads, selecting the left lead nearly twice as often as the right. Velocities and stride lengths were greater for the left lead than the right, but stride frequencies did not differ between leads. Velocity effects were partitioned out in limb contact data analysis to enable the determination of persistent gallop stride asymmetries. The contact duration for the trailing (right) fore limb on the left lead exceeded the contact duration for the trailing (left) fore limb on the right lead. Selecting the right fore limb as the trailing fore limb may have allowed horses to use it to withstand the greater stresses and caused them to preferentially gallop with the left fore limb leading. Laterality may have an important influence on equine gallop motion patterns and thereby influence athletic performance.     

Estimating femur and tibia length from fragmentary bones: an evaluation of Steele's (1970) method using a prehistoric European sample.

Steele's (1970) regression method for estimating femur and tibia length from fragmentary bones is tested on a sample of complete femora (female N = 26; male N = 33) and tibiae (female N = 16; male N = 22) from a number of European Mesolithic and Neolithic sites. Over half of the regression equations given by Steele for predicting maximum length of the bone from the length(s) of one or more of its constituent segments are shown to produce inaccurate predictions in this test sample. However, a closer evaluation of these results, including calculation of regression equations for the test sample itself, reveals that this inaccuracy does not derive from any inherent flaw in Steele's method. Rather, it is shown that differential distribution of maximum bone length among the various bone segments as defined by Steele may occur due to variation in muscular activity pattern and intensity. This argues for the retention of Steele's basic method, with care being taken to match closely the activity pattern typical of the sample from which regression equations are derived with that of the population to which the equations are to be applied. The equations calculated in this study thus are provided for use where deemed appropriate.     

Urohyal size versus body-length and weight relationships for thirteen mojarras species (Perciformes: Gerreidae) in Mexican waters

Given that it has been shown that urohyal bones can be used to identify gerreid fish to the species level, predictive regression equations between urohyal size and fish standard length and weight of thirteen species of Gerreidae living along the coasts of Mexico, were calculated. All regressions were statistically significant with coefficients of variation and coefficient of determination values around 0.90 in most of cases. These regression equations could provide a reliable tool in trophic ecology and other studies by estimating the original body size and weight of fish from measuring urohyal bone lengths. Information provide from the urohyal bone-fish length and weight relationships also facilitates the assessment of the potential role of gerreid fish found in the diet of piscivorous species from Mexican waters.     

Body mass estimates of an exceptionally complete Stegosaurus (Ornithischia: Thyreophora): comparing volumetric and linear bivariate mass estimation methods

Body mass is a key biological variable, but difficult to assess from fossils. Various techniques exist for estimating body mass from skeletal parameters, but few studies have compared outputs from different methods. Here, we apply several mass estimation methods to an exceptionally complete skeleton of the dinosaur Stegosaurus. Applying a volumetric convex-hulling technique to a digital model of Stegosaurus, we estimate a mass of 1560 kg (95% prediction interval 1082–2256 kg) for this individual. By contrast, bivariate equations based on limb dimensions predict values between 2355 and 3751 kg and require implausible amounts of soft tissue and/or high body densities. When corrected for ontogenetic scaling, however, volumetric and linear equations are brought into close agreement. Our results raise concerns regarding the application of predictive equations to extinct taxa with no living analogues in terms of overall morphology and highlight the sensitivity of bivariate predictive equations to the ontogenetic status of the specimen. We emphasize the significance of rare, complete fossil skeletons in validating widely applied mass estimation equations based on incomplete skeletal material and stress the importance of accurately determining specimen age prior to further analyses.