A general model for estimating lower extremity inertial properties of individuals with transtibial amputation

Lower extremity joint moment magnitudes during swing are dependent on the inertial properties of the prosthesis and residual limb of individuals with transtibial amputation (TTA). Often, intact limb inertial properties (INTACT) are used for prosthetic limb values in an inverse dynamics model even though these values overestimate the amputated limb’s inertial properties. The purpose of this study was to use subject-specific (SPECIFIC) measures of prosthesis inertial properties to generate a general model (GENERAL) for estimating TTA prosthesis inertial properties. Subject-specific mass, center of mass, and moment of inertia were determined for the shank and foot segments of the prosthesis (n = 11) using an oscillation technique and reaction board. The GENERAL model was derived from the means of the SPECIFIC model. Mass and segment lengths are required GENERAL model inputs. Comparisons of segment inertial properties and joint moments during walking were made using three inertial models (unique sample; n = 9): (1) SPECIFIC, (2) GENERAL, and (3) INTACT. Prosthetic shank inertial properties were significantly smaller with the SPECIFIC and GENERAL model than the INTACT model, but the SPECIFIC and GENERAL model did not statistically differ. Peak knee and hip joint moments during swing were significantly smaller for the SPECIFIC and GENERAL model compared with the INTACT model and were not significantly different between SPECIFIC and GENERAL models. When subject-specific measures are unavailable, using the GENERAL model produces a better estimate of prosthetic side inertial properties resulting in more accurate joint moment measurements for individuals with TTA than the INTACT model.     

Prehistoric mongoloid dispersals. Edited by Takeru Akazawa and Emöke J. E. Szathmáry. New York: Oxford University Press. 1996. 387 pp. ISBN 0-19-852318-1 $150.00 (cloth)

Studies of the dynamics of locomotor performances depend on knowledge of the distribution of body mass within and between limb segments. However, these data are difficult to derive. Segment mass properties have generally been estimated by modelling limbs as truncated cones, but this approach fails to take into account that some segments are of elliptical, not circular, cross section; and further, the profiles of real segments are generally curved. Thus, they are more appropriately modelled as solids of revolution, described by the rotation in space of convex or concave curves, and the possibility of an elliptical cross section needs to be taken into account. In this project we have set out to develop a general geometric model which can take these factors into account, and permit segment inertial properties to be derived from cadavers by segmentation, and from living individuals using linear external measurements. We present a model which may be described by up to four parameters, depending o the profile and serial cross section (circular or ellipsoidal) of the individual segments. The parameters are obtained from cadavers using a simplified complex-pendulum technique, and from intact specimens by calculation from measurements of segment diameters and lengths. From the parameters, the center of mass, moments of inertia, and radii of gyration may be derived, using simultaneous equations. Inertial properties of the body segments of four Pan troglodytes and a single Pongo were determined, and contrasted to comparable findings for humans. Using our approach, the mass distribution characteristics of any individual or species may be represented by a rigid-link segment model or “android.” If this is made to move according to motion functions derived from a real performance of the individual represented, we show that recordings of resulting ground reaction forces may be quite closely simulated by predictive dynamic modelling. © 1996 Wiley-Liss, Inc.     

Changes in segmental inertial properties with age

The purpose of this study was to examine how the limb segment inertial parameters vary across the decades from the 1920s to the 1970s. Sixty-six males participated in this study, ranging in age from 20 to 79 years. Pre-screening ensured that all subjects were healthy. The inertial properties of the segments were determined by modeling each segment as series of geometric solids. A multivariate analysis of variance (ANOVA) revealed statistically significant differences between decade age groups for the upper arm, forearm, shank, and thigh (p<0.01). Subsequent ANOVAs revealed statistically significant differences for all the inertial properties for the upper arm, the center of mass location for the forearm, and segment mass for the thigh. Linear regression lines were fit to the data so that each inertial parameter for each segment could be predicted by subject's age, with the slope of this regression line indicating the trend in the data. These trends were statistically significant for all forearm inertial parameters, thigh mass and longitudinal moment of inertia, and forearm center of mass location. The changes for the thigh, upper arm, and forearm were consistent with the changes, which would accompany a change in muscle mass with aging. Resultant joint moments were computed for a set of gait data using inertial properties reflective of the subjects from the age extremes in the study. The resulting differences in the knee and hip moments, young versus old, were all less than 4.5%.     

Scale Effects between Body Size and Limb Design in Quadrupedal Mammals

Recently the metabolic cost of swinging the limbs has been found to be much greater than previously thought, raising the possibility that limb rotational inertia influences the energetics of locomotion. Larger mammals have a lower mass-specific cost of transport than smaller mammals. The scaling of the mass-specific cost of transport is partly explained by decreasing stride frequency with increasing body size; however, it is unknown if limb rotational inertia also influences the mass-specific cost of transport. Limb length and inertial properties – limb mass, center of mass (COM) position, moment of inertia, radius of gyration, and natural frequency – were measured in 44 species of terrestrial mammals, spanning eight taxonomic orders. Limb length increases disproportionately with body mass via positive allometry (length ∝ body mass^0.40); the positive allometry of limb length may help explain the scaling of the metabolic cost of transport. When scaled against body mass, forelimb inertial properties, apart from mass, scale with positive allometry. Fore- and hindlimb mass scale according to geometric similarity (limb mass ∝ body mass^1.0), as do the remaining hindlimb inertial properties. The positive allometry of limb length is largely the result of absolute differences in limb inertial properties between mammalian subgroups. Though likely detrimental to locomotor costs in large mammals, scale effects in limb inertial properties appear to be concomitant with scale effects in sensorimotor control and locomotor ability in terrestrial mammals. Across mammals, the forelimb''s potential for angular acceleration scales according to geometric similarity, whereas the hindlimb''s potential for angular acceleration scales with positive allometry.     

Ontogeny of limb mass distribution in infant baboons (Papio cynocephalus)

Primates have more distally distributed limb muscle mass compared to most nonprimate mammals. The heavy distal limbs of primates are likely related to their strong manual and pedal grasping abilities, and interspecific differences in limb mass distributions among primates are correlated with the amount of time spent on arboreal supports. Within primate species, individuals at different developmental stages appear to differ in limb mass distribution patterns. For example infant macaques have more distally distributed limb mass at young ages. A shift from distal to proximal limb mass concentrations coincides with a shift from dependent travel (grasping their mother's hair) to independent locomotion. Because the functional demands placed on limbs may differ between taxa, understanding the ontogeny of limb mass distribution patterns is likely an essential element in interpreting the diversity of limb mass distribution patterns present in adult primates. This study examines changes in limb inertial properties during ontogeny in a longitudinal sample of infant baboons (Papio cynocephalus). The results of this study show that infant baboons undergo a transition from distal to proximal limb mass distribution patterns. This transition in limb mass distribution coincides with the transition from dependent to independent locomotion during infant development. Compared to more arboreal macaques, infant baboons undergo a faster transition to more proximal limb mass distribution patterns. These results suggest that functional demands placed on the limbs during ontogeny have a strong impact on the development of limb mass distribution patterns.     

Analysis of body segment parameter differences between four human populations and the estimation errors of four popular mathematical models

Calculating the kinetics of motion using inverse or forward dynamics methods requires the use of accurate body segment inertial parameters. The methods available for calculating these body segment parameters (BSPs) have several limitations and a main concern is the applicability of predictive equations to several different populations. This study examined the differences in BSPs between 4 human populations using dual energy x-ray absorptiometry (DEXA), developed linear regression equations to predict mass, center of mass location (CM) and radius of gyration (K) in the frontal plane on 5 body segments and examined the errors produced by using several BSP sources in the literature. Significant population differences were seen in all segments for all populations and all BSPs except hand mass, indicating that population specific BSP predictors are needed. The linear regression equations developed performed best overall when compared to the other sources, yet no one set of predictors performed best for all segments, populations or BSPs. Large errors were seen with all models which were attributed to large individual differences within groups. Equations which account for these differences, including measurements of limb circumferences and breadths may provide better estimations. Geometric models use these parameters, however the models examined in this study did not perform well, possibly due to the assumption of constant density or the use of an overly simple shape. Creating solids which account for density changes or which mimic the mass distribution characteristics of the segment may solve this problem. Otherwise, regression equations specific for populations according to age, gender, race, and morphology may be required to provide accurate estimations of BSPs for use in kinetic equations of motion.     

Convergence of forelimb and hindlimb Natural Pendular Period in baboons (Papio cynocephalus) and its implication for the evolution of primate quadrupedalism

The patterns of muscle mass distribution along the lengths of limbs may have important effects on the mechanics and energetics of quadrupedalism. Specifically, Myers and Steudel (J. Morphol. 234 (1997) 183) have shown that fore- and hindlimb Natural Pendular Periods (NPPs) may affect quadrupedal kinematics and must converge to reduce locomotor energetic costs. This study quantifies patterns of limb mass distribution in a live sample of Papio cynocephalus using limb inertial properties (mass, center of mass, mass moment of inertia, and radius of gyration). These inertial properties are calculated using a geometric modeling technique similar to that of Crompton et al. (Am. J. phys. Anthrop. 99 (1996) 547). The inertial properties in Papio are compared to those of Canis from Myers and Steudel (J. Morphol. 234 (1997) 183). The Papio sample has convergent fore- and hindlimb NPPs. Additionally, these limb NPPs are relatively large compared to those of Canis due to the relatively distally distributed limb mass in the Papio sample (relatively large limb masses, relatively distal centers of mass and radii of gyration, and relatively large limb mass moments of inertia). This relatively distal limb mass appears related to the grasping abilities of their hands and feet. Causal links are explored between limb shape adaptations for grasping hands and feet and the kinematics of primate quadrupedalism. In particular, if primates in general follow Papio's limb mass distribution pattern, then relatively large limb NPPs may lead to the relatively low stride frequencies already documented for primates. The kinematics of primate quadrupedalism appears to have been strongly influenced by both selection for grasping hands and feet and selection for reduced locomotor energetic costs.     

Estimation of minimum ground clearance (MGC) using body-worn inertial sensors

Objective assessment of balance and mobility in elderly populations using body-worn sensors has recently become a prevalent theme in falls-related research. Recent research by the authors identified mean absolute-valued vertical angular velocity measured using shank mounted inertial sensors during a timed-up-and-go test as having a strong association with falls history in a group of elderly adults. This study aimed to investigate the clinical relevance of this parameter by exploring the relationship between it and minimum ground clearance (MGC) measured with an optical motion capture system. MGC is an important variable when considering trip-related falls risk. This paper also presents a method of estimating properties of MGC during walking, across a range of speeds and gait patterns, using body-worn inertial sensors. We found that mean MGC and coefficient of variation (CV) MGC are correlated with mean absolute-valued vertical angular velocity and acceleration as measured by shank or foot mounted inertial sensors. Regression models generated using inertial sensor derived variables were used to robustly estimate the mean MGC and CV MGC measured by an optical marker-tracking system. Foot-mounted sensors were found to yield slightly better results than sensors on the shank. Different walking speeds and gait patterns were not found to influence the accuracy of the models. We conclude that these findings have the potential to evaluate a walking trial using body-worn inertial sensors, which could then be used to identify individuals with increased risk of unprovoked collisions with the ground during locomotion.     

Inertial characteristics of adolescent male body segments

In response to the presently limited information on body segment inertial characteristics of children and adolescents this investigation estimated the mass, centre of mass and principal moments of inertia of adolescent male body segments. Significant prediction equations based on anthropometric measurements were then sought. Thirteen subjects were measured at 6-monthly intervals for 2.5 yr to provide inertial characteristics for the leg, thigh, lower trunk and upper trunk segments. These characteristics were derived using an elliptical zone modelling technique. Following a correlation analysis, significant prediction equations of segment inertial parameters were derived from five, or fewer, anthropometric measurements. For all cases, more than 84% of the variance in the dependent variable was accounted for with a maximum R2 value of 94% being recorded for the prediction of thigh segment mass. The use of these prediction equations offered accurate and convenient estimates of body segment inertial characteristics within the limitations applicable to all modelling approaches. In contrast to recent studies, these equations accommodated the current morphological status of the subject.     

A comparison between a new model and current models for estimating trunk segment inertial parameters

Modeling of the body segments to estimate segment inertial parameters is required in the kinetic analysis of human motion. A new geometric model for the trunk has been developed that uses various cross-sectional shapes to estimate segment volume and adopts a non-uniform density function that is gender-specific. The goal of this study was to test the accuracy of the new model for estimating the trunk's inertial parameters by comparing it to the more current models used in biomechanical research. Trunk inertial parameters estimated from dual X-ray absorptiometry (DXA) were used as the standard. Twenty-five female and 24 male college-aged participants were recruited for the study. Comparisons of the new model to the accepted models were accomplished by determining the error between the models’ trunk inertial estimates and that from DXA. Results showed that the new model was more accurate across all inertial estimates than the other models. The new model had errors within 6.0% for both genders, whereas the other models had higher average errors ranging from 10% to over 50% and were much more inconsistent between the genders. In addition, there was little consistency in the level of accuracy for the other models when estimating the different inertial parameters. These results suggest that the new model provides more accurate and consistent trunk inertial estimates than the other models for both female and male college-aged individuals. However, similar studies need to be performed using other populations, such as elderly or individuals from a distinct morphology (e.g. obese). In addition, the effect of using different models on the outcome of kinetic parameters, such as joint moments and forces needs to be assessed.     

On birds: scale effects in the neognath hindlimb and differences in the gross morphology of wings and hindlimbs

Scale effects on whole limb morphology (i.e. bones together with in situ overlying muscles) are well understood for the neognath forelimb. However, scale effects on neognath gross hindlimb morphology remain largely unexplored. To broaden our understanding of avian whole limb morphology, I investigated the scaling of hindlimb inertial properties in neognath birds, testing empirical scaling relationships against the model of geometric similarity. Inertial property data – mass, moment of inertia, centre of mass distance, and radius of gyration – were collected from 22 neognath species representing a wide range of locomotor specializations. When scaled against body mass, hindlimb inertial properties scale with positive allometry. Thus, in terms of morphology, larger bodied neognaths possess hindlimbs requiring disproportionately more energy to accelerate and decelerate relative to body mass than smaller bodied birds. When scaled against limb length, hindlimb inertial properties scale according to isometry. In the subclade Land Birds (sensu Hackett et al.), hindlimb inertial properties largely scale according to positive allometry. The contrasting results of positive allometry vs. isometry in neognaths are due to how hindlimb length scales against body mass. Negative allometry of hindlimb inertial properties, which would reduce terrestrial locomotion costs, would probably make the hindlimb susceptible to mechanical failure or too diminutive for its many ecological functions. Comparing the scaling relationships of wings and hindlimbs highlights how locomotor costs influence the scaling of limb inertial properties. © 2013 The Linnean Society of London, Biological Journal of the Linnean Society, 2013, 110 , 14–31.     

Are fixed limb inertial models valid for dynamic simulations of human movement?

During human movement, muscle activation and limb movement result in subtle changes in muscle mass distribution. Muscle mass redistribution can affect limb inertial properties and limb dynamics, but it is not currently known to what extent. The objectives of this study were to investigate: (1) how physiological alterations of muscle and tendon length affect limb inertial characteristics, and (2) how such changes affect dynamic simulations of human movement. To achieve these objectives, a digital model of a human leg, custom software, and Software for interactive musculoskeletal modeling were used to simulate mass redistribution of muscle–tendon structures within a limb segment during muscle activation and joint movement. Thigh and shank center of mass and moments of inertia for different muscle activation and joint configurations were determined and compared. Limb inertial parameters representing relaxed muscles and fully active muscles were input into a simulated straight-leg movement to evaluate the effect inertial parameter variations could have on movement simulation results. Muscle activation and limb movement altered limb segment center of mass and moments of inertia by less than 0.04 cm and 1.2%, respectively. These variations in limb inertial properties resulted in less than 0.01% change in maximum angular velocity for a simulated straight-leg hip flexion task. These data demonstrate that, for the digital human leg model considered, assuming static quantities for segment center of masses and moments of inertia in movement simulations appear reasonable and induce minimal errors in simulated movement dynamics.     

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.     

Morphometry of Macaca mulatta forelimb. I. Shoulder and elbow muscles and segment inertial parameters

The present study examined the morphometric properties of the forelimb, including the inertial properties of the body segments and the morphometric parameters of 21 muscles spanning the shoulder and/or elbow joints of six Macaca mulatta and three M. fascicularis. Five muscle parameters are presented: optimal fascicle length (L(0)(M)), tendon slack length (L(S)(T)), physiological cross-sectional area (PCSA), pennation angle (alpha(0)), and muscle mass (m). Linear regressions indicate that muscle mass, and to a lesser extent PCSA, correlated with total body weight. Segment mass, center-of-mass, and the moment of inertia of the upper arm, forearm, and hand are also presented. Our data indicate that for some segments, radius of gyration (rho) predicts segment moment of inertia better than linear regressions based on total body weight. Key differences between the monkey and human forelimb are highlighted.     

儿童的惯性－惯量特征及其生长

本文应用排水法得到儿童肢体沿长轴的体积分布函数。根据关于人类肢体体积－质量的假定,肢体的质心、转动惯量及旋转半径等动力学常数可由前述的体积分布函数中求出。本文结果显示,部分力学参数与儿童的生长有着密切的联系,并受到性别的影响。因此,这些参数可以用来作为儿童的生长指标。以往数据的离散性表明,无论从系统上或技术上,这一领域都有许多工作要做。     

Steady state temperature distribution in dermal regions of an irregular tapered shaped human limb with variable eccentricity

The investigators in the past have developed some models of temperature distribution in the human limb assuming it as a regular circular or elliptical tapered cylinder. But in reality the limb is not of regular tapered cylindrical shape. The radius and eccentricity are not same throughout the limb. In view of above a model of temperature distribution in the irregular tapered elliptical shaped human limb is proposed for a three dimensional steady state case in this paper. The limb is assumed to be composed of multiple cylindrical substructures with variable radius and eccentricity. The mathematical model incorporates the effect of blood mass flow rate, metabolic activity and thermal conductivity. The outer surface is exposed to the environment and appropriate boundary conditions have been framed. The finite element method has been employed to obtain the solution. The temperature profiles have been computed in the dermal layers of a human limb and used to study the effect of shape, microstructure and biophysical parameters on temperature distribution in human limbs. The proposed model is one of the most realistic model as compared to conventional models as this can be effectively employed to every regular and nonregular structures of the body with variable radius and eccentricity to study the thermal behaviour.     

Relative variation in human proximal and distal limb segment lengths

The pattern of variation and covariation of proximal and distal limb segment lengths was examined within and between 20 geographically diverse skeletal samples of modern humans. Analyses of variance-covariance matrices (VCMs) of logarithmically transformed (ln) variates of humerus, radius, femur, and tibia length were performed to test the following hypotheses: first, within populations, the distal and proximal segments will have equal relative (i.e., size-independent) variability. However, between populations, the tibia is predicted to be more variable than the other segments. Tests of fit of computed VCMs to theoretical matrices by an iterative procedure (Anderson [1973] Ann. Stat. 1:135-141) reject the equal variance hypotheses, rather suggesting that the relative variances of the distal limb segments are greater than are those of the proximal. Males and females differ somewhat in that within females, the distal segments of both limbs have equal variance, while within males, the tibia has greater relative variance than the radius. The second hypothesis, regarding between-group variability, is somewhat supported in that between human populations, one cannot reject that the tibia has greater relative variance than the other limb segments. However, neither can one reject an alternative hypothesis that both distal limb segments (tibia and radius) are more variable than the proximal segments. Differential growth allometry is explored, and likely plays a major role in differences seen both within and between human populations.     

Body proportions in ancient Andeans from high and low altitudes

Living human populations from high altitudes in the Andes exhibit relatively short limbs compared with neighboring groups from lower elevations as adaptations to cold climates characteristic of high-altitude environments. This study compares relative limb lengths and proportions in pre-Contact human skeletons from different altitudes to test whether ecogeographic variation also existed in Andean prehistory. Maximum lengths of the humerus, radius, femur, and tibia, and femoral head breadth are measured in sex-specific groups of adult human skeletons (N = 346) from the central (n = 80) and the south-central (n = 123) Andean coasts, the Atacama Desert at 2,500 m (n = 102), and the southern Peruvian highlands at 2,000-3,800 m (n = 41). To test whether limb lengths vary with altitude, comparisons are made of intralimb proportions, limb lengths against body mass estimates derived from published equations, limb lengths against the geometric mean of all measurements, and principal component analysis. Intralimb proportions do not statistically differ between coastal groups and those from the Atacama Desert, whereas intralimb proportions are significantly shorter in the Peruvian highland sample. Overall body size and limb lengths relative to body size vary along an altitudinal gradient, with larger individuals from coastal environments and smaller individuals with relatively longer limbs for their size from higher elevations. Ecogeographic variation in relation to climate explains the variation in intralimb proportions, and dietary variation may explain the altitudinal cline in body size and limb lengths relative to body size. The potential effects of gene flow on variation in body proportions in Andean prehistory are also explored.     

Artificial selection sheds light on developmental mechanisms of limb elongation

Species diversity in limb lengths and proportions is thought to have evolved adaptively in the context of locomotor and habitat specialization, but the heritable cellular processes that drove this evolution within species are poorly understood. In this study, we take a novel “micro‐evo‐devo” approach, using artificial selection on relative limb length to amplify phenotypic variation in a population of mice, known as Longshanks, to examine the cellular mechanisms of postnatal limb development that contribute to intraspecific limb length variation. Cross‐sectional growth data indicate that differences in bone length between Longshanks and random‐bred controls are not due to prolonged growth, but to accelerated growth rates. Histomorphometric and cell proliferation assays on proximal tibial growth plates show that Longshanks’ increased limb bone length is associated with an increased number of proliferative chondrocytes. In contrast, we find no differences in other growth plate cellular features known to underlie interspecific differences in limb bone size and shape, such as the rates of chondrocyte proliferation or the size and number of hypertrophic cells in the growth plate. These data suggest that small differences among individuals in the number of proliferating chondrocytes are a potentially important determinant of selectable intraspecific variation in individual limb bone lengths, independent of body size.     

Ontogenetic and individual variation in size, shape and speed in the Australian agamid lizard Amphibolurus nuchalis

The present study investigates relationships among size, shape and speed in the Australian agamid lizard Amphibolurus nuchalis . Maximal running speed, body mass, snout-vent length, tail length, fore- and hind limb spans and thigh muscle mass were measured in 68 field-fresh individuals spanning the entire ontogenetic size range (1.3 48 g). Relative lengths of both foreand hind limbs decrease with increasing body mass (= negative allometry), whereas relative tail length and thigh muscle mass increase with body mass (= positive allometry). Repeatable and significant differences in maximal running speed exist among individuals. Maximal running speed scales as (body mass)^0.161, and 59% of the variation in maximal speed was related to body mass. Based on the results of the present and previous studies, data on scaling of body proportions alone appear inadequate to infer scaling relationships of functional characters such as top speed.
Surprisingly, individual variation in maximal speed is not related to individual variation in shape (relative limb, tail and body lengths). These components of overall shape are not independent; individuals tended to have either relatively long or relatively short limbs, tails and bodies for their body mass. Even the significant difference in multivariate shape between adult males and females has no measurable consequences for maximal speed. Speeds of field-fresh animals did not vary on a seasonal basis, and eight weeks of captivity had no effect on maximal running speeds. Gravid females and long-term (obese) captive lizards were both approximately 12% slower than field-fresh lizards.