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.     

MRI-derived body segment parameters of children differ from age-based estimates derived using photogrammetry

Body segment parameters are required when researching joint kinetics using inverse dynamics models. However, the only regression equations for estimating pediatric body segment parameters across a wide age range were developed, using photogrammetry, based on 12 boys and have not been validated to date (Jensen, R.K., 1986. Body segment mass, radius and radius of gyration proportions of children. Journal of Biomechanics 19, 359–368). To assess whether these equations could validly be applied to girls, we asked whether body segment parameters estimated by the equations differ from parameters measured using a validated magnetic resonance imaging (MRI) method. If so, do the differences cause significant differences in joint kinetics during normal gait? Body segment parameters were estimated from axial MRIs of the left thigh and shank of 10 healthy girls (9.6±0.9 years) and compared to those from Jensen's equations. Kinematics and kinetics were collected for 10 walking trials. Extrema in hip and knee moments and powers were compared between the two sets of body segment parameters. With the exception of the shank mass center and radius of gyration, body segment parameters measured using MRI were significantly different from those estimated using regression equations. These systematic differences in body segment parameters resulted in significant differences in sagittal-plane joint moments and powers during gait. Nevertheless, it is doubtful that even the greatest differences in kinetics are practically meaningful (0.3%BW×HT and 0.7%BW×HT/s for moments and power at the hip, respectively). Therefore, body segment parameters estimated using Jensen's regression equations are a suitable substitute for more detailed anatomical imaging of 8–10-year-old girls when quantifying joint kinetics during gait.     

Whole body inverse dynamics over a complete gait cycle based only on measured kinematics

This paper presents a three-dimensional (3D) whole body multi-segment model for inverse dynamics analysis over a complete gait cycle, based only on measured kinematic data. The sequence of inverse dynamics calculations differs significantly from the conventional application of inverse dynamics using force plate data. A new validated "Smooth Transition Assumption" was used to solve the indeterminacy problem in the double support phase. Kinematic data is required for all major body segments and, hence, a whole body gait measurement protocol is presented. Finally, sensitivity analyses were conducted to evaluate the effects of digital filtering and body segment parameters on the accuracy of the prediction results. The model gave reasonably good estimates of sagittal plane ground forces and moment; however, the estimates in the other planes were less good, which we believe is largely due to their small magnitudes in comparison to the sagittal forces and moment. The errors observed are most likely caused by errors in the kinematic data resulting from skin movement artefact and by errors in the estimated body segment parameters. A digital filtering cut-off frequency of 4.5Hz was found to produce the best results. It was also shown that errors in the mass properties of body segments can play a crucial role, with changes in properties sometimes having a disproportionate effect on the calculated ground reactions. The implication of these results is that, even when force plate data is available, the estimated joint forces are likely to suffer from similar errors.     

Error in the description of foot kinematics due to violation of rigid body assumptions

Kinematic data from rigid segment foot models inevitably includes errors because the bones within each segment move relative to each other. This study sought to define error in foot kinematic data due to violation of the rigid segment assumption. The research compared kinematic data from 17 different mid and forefoot rigid segment models to kinematic data of the individual bones comprising these segments. Kinematic data from a previous dynamic cadaver model study was used to derive individual bone as well as foot segment kinematics.Mean and maximum errors due to violation of the rigid body assumption varied greatly between models. The model with least error was the combination of navicular and cuboid (mean errors <=1.3°, average maximum error <=2.4°). Greatest error was seen for the model combining all the ten bones (mean errors <=4.4°, average maximum errors <=6.9°). Based on the errors reported a three segment mid and forefoot model is proposed: (1) Navicular and cuboid, (2) cuneiforms and metatarsals 1, 2 and 3, and (3) metatarsals 4 and 5. However the utility of this model will depend on the precise purpose of the in vivo foot kinematics research study being undertaken.     

Exact epidemic models on graphs using graph-automorphism driven lumping

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.     

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal''s position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat''s velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.     

Predictive regression modeling of body segment parameters using individual-based anthropometric measurements

Body segment parameters such as segment mass, center of mass, and radius of gyration are used as inputs in static and dynamic ergonomic and biomechanical models used to predict joint and muscle forces, and to assess risks of musculoskeletal injury. Previous work has predicted body segment parameters (BSPs) in the general population using age and obesity levels as statistical predictors (Merrill et al., 2017). Estimated errors in the prediction of BSPs can be as large as 40%, depending on age, and the prediction method employed (Durkin and Dowling, 2003). Thus, more accurate and representative segment parameter inputs are required for attempting to predict modeling outputs such as joint contact forces, muscle forces, and injury risk in individuals. This study aims to provide statistical models for predicting torso, thigh, shank, upper arm, and forearm segment parameters in working adults using whole body dual energy x-ray absorptiometry (DXA) scan data along with a set of anthropometric measurements. The statistical models were developed on a training data set, and independently validated on a separate test data set. The predicted BSPs in validation data were, on average, within 5% of the actual in vivo DXA-based BSPs, while previously developed predictions (de Leva, 1996) had average errors of up to 60%, indicating that the new models greatly increase the accuracy in predicting segment parameters. These final developed models can be used for calculating representative BSPs in individuals for use in modeling applications dependent on these parameters.     

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.     

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.     

Validity of the top-down approach of inverse dynamics analysis in fast and large rotational trunk movements

This study investigated the validity of the top-down approach of inverse dynamics analysis in fast and large rotational movements of the trunk about three orthogonal axes of the pelvis for nine male collegiate students. The maximum angles of the upper trunk relative to the pelvis were approximately 47°, 49°, 32°, and 55° for lateral bending, flexion, extension, and axial rotation, respectively, with maximum angular velocities of 209°/s, 201°/s, 145°/s, and 288°/s, respectively. The pelvic moments about the axes during the movements were determined using the top-down and bottom-up approaches of inverse dynamics and compared between the two approaches. Three body segment inertial parameter sets were estimated using anthropometric data sets (Ae et al., Biomechanism 11, 1992; De Leva, J Biomech, 1996; Dumas et al., J Biomech, 2007). The root-mean-square errors of the moments and the absolute errors of the peaks of the moments were generally smaller than 10 N·m. The results suggest that the pelvic moment in motions involving fast and large trunk movements can be determined with a certain level of validity using the top-down approach in which the trunk is modeled as two or three rigid-link segments.     

Comparative analysis of methods for estimating arm segment parameters and joint torques from inverse dynamics

A common problem in the analyses of upper limb unfettered reaching movements is the estimation of joint torques using inverse dynamics. The inaccuracy in the estimation of joint torques can be caused by the inaccuracy in the acquisition of kinematic variables, body segment parameters (BSPs), and approximation in the biomechanical models. The effect of uncertainty in the estimation of body segment parameters can be especially important in the analysis of movements with high acceleration. A sensitivity analysis was performed to assess the relevance of different sources of inaccuracy in inverse dynamics analysis of a planar arm movement. Eight regression models and one water immersion method for the estimation of BSPs were used to quantify the influence of inertial models on the calculation of joint torques during numerical analysis of unfettered forward arm reaching movements. Thirteen subjects performed 72 forward planar reaches between two targets located on the horizontal plane and aligned with the median plane. Using a planar, double link model for the arm with a floating shoulder, we calculated the normalized joint torque peak and a normalized root mean square (rms) of torque at the shoulder and elbow joints. Statistical analyses quantified the influence of different BSP models on the kinetic variable variance for given uncertainty on the estimation of joint kinematics and biomechanical modeling errors. Our analysis revealed that the choice of BSP estimation method had a particular influence on the normalized rms of joint torques. Moreover, the normalization of kinetic variables to BSPs for a comparison among subjects showed that the interaction between the BSP estimation method and the subject specific somatotype and movement kinematics was a significant source of variance in the kinetic variables. The normalized joint torque peak and the normalized root mean square of joint torque represented valuable parameters to compare the effect of BSP estimation methods on the variance in the population of kinetic variables calculated across a group of subjects with different body types. We found that the variance of the arm segment parameter estimation had more influence on the calculated joint torques than the variance of the kinematics variables. This is due to the low moments of inertia of the upper limb, especially when compared with the leg. Therefore, the results of the inverse dynamics of arm movements are influenced by the choice of BSP estimation method to a greater extent than the results of gait analysis.     

Has Large-Scale Named-Entity Network Analysis Been Resting on a Flawed Assumption?

The assumption that a name uniquely identifies an entity introduces two types of errors: splitting treats one entity as two or more (because of name variants); lumping treats multiple entities as if they were one (because of shared names). Here we investigate the extent to which splitting and lumping affect commonly-used measures of large-scale named-entity networks within two disambiguated bibliographic datasets: one for co-author names in biomedicine (PubMed, 2003–2007); the other for co-inventor names in U.S. patents (USPTO, 2003–2007). In both cases, we find that splitting has relatively little effect, whereas lumping has a dramatic effect on network measures. For example, in the biomedical co-authorship network, lumping (based on last name and both initials) drives several measures down: the global clustering coefficient by a factor of 4 (from 0.265 to 0.066); degree assortativity by a factor of ∼13 (from 0.763 to 0.06); and average shortest path by a factor of 1.3 (from 5.9 to 4.5). These results can be explained in part by the fact that lumping artificially creates many intransitive relationships and high-degree vertices. This effect of lumping is much less dramatic but persists with measures that give less weight to high-degree vertices, such as the mean local clustering coefficient and log-based degree assortativity. Furthermore, the log-log distribution of collaborator counts follows a much straighter line (power law) with splitting and lumping errors than without, particularly at the low and the high counts. This suggests that part of the power law often observed for collaborator counts in science and technology reflects an artifact: name ambiguity.     

Reduction of a biochemical model with preservation of its basic dynamic properties

The complexity of full-scale metabolic models is a major obstacle for their effective use in computational systems biology. The aim of model reduction is to circumvent this problem by eliminating parts of a model that are unimportant for the properties of interest. The choice of reduction method is influenced both by the type of model complexity and by the objective of the reduction; therefore, no single method is superior in all cases. In this study we present a comparative study of two different methods applied to a 20D model of yeast glycolytic oscillations. Our objective is to obtain biochemically meaningful reduced models, which reproduce the dynamic properties of the 20D model. The first method uses lumping and subsequent constrained parameter optimization. The second method is a novel approach that eliminates variables not essential for the dynamics. The applications of the two methods result in models of eight (lumping), six (elimination) and three (lumping followed by elimination) dimensions. All models have similar dynamic properties and pin-point the same interactions as being crucial for generation of the oscillations. The advantage of the novel method is that it is algorithmic, and does not require input in the form of biochemical knowledge. The lumping approach, however, is better at preserving biochemical properties, as we show through extensive analyses of the models.     

Segment‐specific muscle degeneration is triggered directly by a steroid hormone during insect metamorphosis

During metamorphosis of the hawkmoth, Manduca sexta, some larval muscles degenerate while others are respecified for new functions. In larvae, accessory planta retractor muscles (APRMs) are present in abdominal segments 1 to 6 (A1 to A6). APRMs serve as proleg retractors in A3 to A6 and body wall muscles in A1 and A2. At pupation, all APRMs degenerate except those in A2 and A3, which are respecified to circulate hemolymph in pupae. The motoneurons that innervate APRMs, the APRs, likewise undergo segment‐specific programmed cell death (PCD), as a direct, cell‐autonomous response to the prepupal peak of ecdysteroids. The segment‐specific patterns of APR and APRM death differ. The present study tested the hypothesis that APRM death is a direct, cell‐autonomous response to the prepupal peak of ecdysteroids. Prevention of the prepupal peak prevented APRM degeneration, and replacement of the peak by infusion of 20‐hydroxyecdysone restored the correct segment‐specific pattern of APRM degeneration. Surgical denervation of APRMs did not perturb their segment‐specific degeneration at pupation, indicating that signals from APRs are not required for the muscles' segment‐specific responses to ecdysteroids. The possibility that instructive signals originate from APRMs' epidermal attachment points was tested by treating the epidermis with a juvenile hormone analog to prevent pupal development. This manipulation likewise did not alter APRM fate. We conclude that both the muscles and motoneurons in this motor system respond directly and cell‐autonomously to prepupal ecdysteroids to produce a segment‐specific pattern of PCD that is matched to the functional requirements of the pupal body. © 2004 Wiley Periodicals, Inc. J Neurobiol, 2005     

A 3D Generic Inverse Dynamic Method using Wrench Notation and Quaternion Algebra

In the literature, conventional 3D inverse dynamic models are limited in three aspects related to inverse dynamic notation, body segment parameters and kinematic formalism. First, conventional notation yields separate computations of the forces and moments with successive coordinate system transformations. Secondly, the way conventional body segment parameters are defined is based on the assumption that the inertia tensor is principal and the centre of mass is located between the proximal and distal ends. Thirdly, the conventional kinematic formalism uses Euler or Cardanic angles that are sequence-dependent and suffer from singularities.

In order to overcome these limitations, this paper presents a new generic method for inverse dynamics. This generic method is based on wrench notation for inverse dynamics, a general definition of body segment parameters and quaternion algebra for the kinematic formalism.     

Effects of directional environmental change on extinction dynamics in experimental microbial communities are predicted by a simple model

Global temperatures are expected to rise between 1.1 and 6.4°C over the next 100 years, although the exact rate will depend on future greenhouse emissions, and will vary spatially. Temperature can alter an individual's metabolic rate, and consequently birth and death rates. In declining populations, these alterations may manifest as changes in the rate of that population's decline, and subsequently the timing of extinction events. Predicting such events could therefore be of considerable use. We use a small‐scale experimental system to investigate how the rate of temperature change can alter a population's time to extinction, and whether it is possible to predict this event using a simple phenomenological model that incorporates information about population dynamics at a constant temperature, published scaling of metabolic rates, and temperature. In addition, we examine 1) the relative importance of the direct effects of temperature on metabolic rate, and the indirect effects (via temperature driven changes in body size), on predictive accuracy (defined as the proximity of the predicted date of extinction to the mean observed date of extinction), 2) the combinations of model parameters that maximise accuracy of predictions, and 3) whether substituting temperature change through time with mean temperature produces accurate predictions. We find that extinction occurs earlier in environments that warm faster, and this can be accurately predicted (R² > 0.84). Increasing the number of parameters that were temperature‐dependent increased the model's accuracy, as did scaling these temperature‐dependent parameters with either the direct effects of temperature alone, or with the direct and indirect effects. Using mean temperature through time instead of actual temperature produces less accurate predictions of extinction. These results suggest that simple phenomenological models, incorporating metabolic theory, may be useful in understanding how environmental change can alter a population's rate of extinction. Synthesis Understanding how populations will respond to future climatic change is a key goal in ecology, however the exact rate of future warming will vary both spatially and temporally. Consequently, mathematical models must be used to understand the potential range of future population dynamics under various warming scenarios. We use a combination of experimentation and modelling to show that the effects of varying rates of environmental change on population dynamics can be predicted by a simple model. However, the accuracy of these predictions depends upon, amongst other things, a detailed knowledge of how temperature will change over time, rather than approximating this change to mean temperature.     

Improving joint torque calculations: optimization-based inverse dynamics to reduce the effect of motion errors

The accuracy of joint torques calculated from inverse dynamics methods is strongly dependent upon errors in body segment motion profiles, which arise from two sources of noise: the motion capture system and movement artifacts of skin-mounted markers. The current study presents a method to increase the accuracy of estimated joint torques through the optimization of the angular position data used to describe these segment motions. To compute these angular data, we formulated a constrained nonlinear optimization problem with a cost function that minimizes the difference between the known ground reaction forces (GRFs) and the GRF calculated via a top-down inverse dynamics solution. To evaluate this approach, we constructed idealized error-free reference movements (of squatting and lifting) that produced a set of known “true” motions and associated true joint torques and GRF. To simulate real-world inaccuracies in motion data, these true motions were perturbed by artificial noise. We then applied our approach to these noise-induced data to determine optimized motions and related joint torques. To evaluate the efficacy of the optimization approach compared to traditional (bottom-up or top-down) inverse dynamics approaches, we computed the root mean square error (RMSE) values of joint torques derived from each approach relative to the expected true joint torques. Compared to traditional approaches, the optimization approach reduced the RMSE by 54% to 79%. Average reduction due to our method was 65%; previous methods only achieved an overall reduction of 30%. These results suggest that significant improvement in the accuracy of joint torque calculations can be achieved using this approach.     

Evaluating model reduction under parameter uncertainty

Background

The dynamics of biochemical networks can be modelled by systems of ordinary differential equations. However, these networks are typically large and contain many parameters. Therefore model reduction procedures, such as lumping, sensitivity analysis and time-scale separation, are used to simplify models. Although there are many different model reduction procedures, the evaluation of reduced models is difficult and depends on the parameter values of the full model. There is a lack of a criteria for evaluating reduced models when the model parameters are uncertain.

Results

We developed a method to compare reduced models and select the model that results in similar dynamics and uncertainty as the original model. We simulated different parameter sets from the assumed parameter distributions. Then, we compared all reduced models for all parameter sets using cluster analysis. The clusters revealed which of the reduced models that were similar to the original model in dynamics and variability. This allowed us to select the smallest reduced model that best approximated the full model. Through examples we showed that when parameter uncertainty was large, the model should be reduced further and when parameter uncertainty was small, models should not be reduced much.

Conclusions

A method to compare different models under parameter uncertainty is developed. It can be applied to any model reduction method. We also showed that the amount of parameter uncertainty influences the choice of reduced models.

     

Scapular correlates of muscle morphology in Papio cynocephalus.

The scapula responds to the compressive states of its surrounding matrix produced by muscle functioning and weight bearing in the upper extremity with discernible structural correlates. These structural correlates can be utilized to infer locomotor behavior. After dissection and drying, the weights of several muscles of the shoulder region of 11 adult female baboons (Papio cynocephalus) were statistically compared to various dimensions of the bony scapula. A significant correlation was obtained between the weights of the individual compressive muscles, the combined weights of the compressive muscles and a scapular dimension of width. A nonsignificant correlation between these muscles and a sscapular dimension of length was also found. The results of this study were compared to those of a previous study of the scapular musculature in Macaca and opposing conclusions were obtained. The advisability of lumping macaques and baboons into a single gross locomotor category is rejected.