The expression of the skeletal muscle force-length relationship in vivo: A simulation study

The force-length relationship is one of the most important mechanical characteristics of skeletal muscle in humans and animals. For a physiologically realistic joint range of motion and therefore range of muscle fibre lengths only part of the force-length curve may be used in vivo, i.e. only a section of the force-length curve is expressed. A generalised model of a mono-articular muscle-tendon complex was used to examine the effect of various muscle architecture parameters on the expressed section of the force-length relationship for a 90° joint range of motion. The parameters investigated were: the ratio of tendon resting length to muscle fibre optimum length (L_TR:L_F·OPT) (varied from 0.5 to 11.5), the ratio of muscle fibre optimum length to average moment arm (L_F·OPT:r) (varied from 0.5 to 5), the normalised tendon strain at maximum isometric force (c) (varied from 0 to 0.08), the muscle fibre pennation angle (θ) (varied from 0° to 45°) and the joint angle at which the optimum muscle fibre length occurred (φ). The range of values chosen for each parameter was based on values reported in the literature for five human mono-articular muscles with different functional roles. The ratios L_TR:L_F·OPT and L_F·OPT:r were important in determining the amount of variability in the expressed section of the force-length relationship. The modelled muscle operated over only one limb at intermediate values of these two ratios (L_TR:L_F·OPT=5; L_F·OPT:r=3), whether this was the ascending or descending limb was determined by the precise values of the other parameters. It was concluded that inter-individual variability in the expressed section of the force-length relationship is possible, particularly for muscles with intermediate values of L_TR:L_F·OPT and L_F·OPT:r such as the brachialis and vastus lateralis. Understanding the potential for inter-individual variability in the expressed section is important when using muscle models to simulate movement.     

Predicting human chronically paralyzed muscle force: a comparison of three mathematical models.

Chronic spinal cord injury (SCI) induces detrimental musculoskeletal adaptations that adversely affect health status, ranging from muscle paralysis and skin ulcerations to osteoporosis. SCI rehabilitative efforts may increasingly focus on preserving the integrity of paralyzed extremities to maximize health quality using electrical stimulation for isometric training and/or functional activities. Subject-specific mathematical muscle models could prove valuable for predicting the forces necessary to achieve therapeutic loading conditions in individuals with paralyzed limbs. Although numerous muscle models are available, three modeling approaches were chosen that can accommodate a variety of stimulation input patterns. To our knowledge, no direct comparisons between models using paralyzed muscle have been reported. The three models include 1) a simple second-order linear model with three parameters and 2) two six-parameter nonlinear models (a second-order nonlinear model and a Hill-derived nonlinear model). Soleus muscle forces from four individuals with complete, chronic SCI were used to optimize each model's parameters (using an increasing and decreasing frequency ramp) and to assess the models' predictive accuracies for constant and variable (doublet) stimulation trains at 5, 10, and 20 Hz in each individual. Despite the large differences in modeling approaches, the mean predicted force errors differed only moderately (8-15% error; P=0.0042), suggesting physiological force can be adequately represented by multiple mathematical constructs. The two nonlinear models predicted specific force characteristics better than the linear model in nearly all stimulation conditions, with minimal differences between the two nonlinear models. Either nonlinear mathematical model can provide reasonable force estimates; individual application needs may dictate the preferred modeling strategy.     

The muscle line of action in current models of the human cervical spine: a comparison with in vivo MRI data

Cervical muscles are commonly represented by line-of-action models. This investigation evaluates the performance of three types of model implementations, based on their ability to mimic geometric in-vivo aspects of muscles. Five prominent pairs of neck muscles were reconstructed in three head positions using magnetic resonance imaging. Based on the reconstructions, muscle approximations were created that represent the muscles with piecewise straight lines. Measured and modelled muscle approximations were compared with respect to their pulling directions at the attachment sites and the overall distance between the muscle paths. Muscle lengths were evaluated in two ways. First, length discrepancies were determined between measured and modelled muscles depending on the head position. Second, the difference of muscle lengths in neutral and deflected head positions for measurement and models were calculated. The results indicate considerable differences between models and measurements. Pulling directions, for instance, differed by up to 40°, depending on the chosen muscle and the type of muscle implementation.     

The muscle line of action in current models of the human cervical spine: a comparison with in vivo MRI data

Cervical muscles are commonly represented by line-of-action models. This investigation evaluates the performance of three types of model implementations, based on their ability to mimic geometric in-vivo aspects of muscles. Five prominent pairs of neck muscles were reconstructed in three head positions using magnetic resonance imaging. Based on the reconstructions, muscle approximations were created that represent the muscles with piecewise straight lines. Measured and modelled muscle approximations were compared with respect to their pulling directions at the attachment sites and the overall distance between the muscle paths. Muscle lengths were evaluated in two ways. First, length discrepancies were determined between measured and modelled muscles depending on the head position. Second, the difference of muscle lengths in neutral and deflected head positions for measurement and models were calculated. The results indicate considerable differences between models and measurements. Pulling directions, for instance, differed by up to 40°, depending on the chosen muscle and the type of muscle implementation.     

Estimating trees from filtered data: Identifiability of models for morphological phylogenetics

As an alternative to parsimony analyses, stochastic models have been proposed ( [Lewis, 2001] and [Nylander et al., 2004]) for morphological characters, so that maximum likelihood or Bayesian analyses may be used for phylogenetic inference. A key feature of these models is that they account for ascertainment bias, in that only varying, or parsimony-informative characters are observed. However, statistical consistency of such model-based inference requires that the model parameters be identifiable from the joint distribution they entail, and this issue has not been addressed.Here we prove that parameters for several such models, with finite state spaces of arbitrary size, are identifiable, provided the tree has at least eight leaves. If the tree topology is already known, then seven leaves suffice for identifiability of the numerical parameters. The method of proof involves first inferring a full distribution of both parsimony-informative and non-informative pattern joint probabilities from the parsimony-informative ones, using phylogenetic invariants. The failure of identifiability of the tree parameter for four-taxon trees is also investigated.     

The force-length relationship of a muscle-tendon complex: experimental results and model calculations

Models are useful when studying how architectural and physiological properties of muscle-tendon complexes are related to function, because they allow for the simulation of the behaviour of such complexes during natural movements. In the construction of these models, evaluation of their accuracy is an important step. In the present study, a model was constructed to calculate the isometric force-length relationship of the rat extensor digitorum longus muscle-tendon complex. The model is based on the assumption that a muscle-tendon complex is a collection of independent units, each consisting of a muscle fibre in series with a tendon fibre. By intention, values for model parameters were derived indirectly, using only the measured maximal isometric tetanic force, the distance between origin and insertion at which it occurred (optimum lOI) and an estimate of muscle fibre optimal length. The accuracy of the calculated force-length relationship was subsequently evaluated by comparing it to the relationship measured in isometric tetanic contractions of a real complex in the rat. When the length of distal muscle fibres, measured during isometric contraction at optimal lOI of the whole complex, was used as an estimate for muscle fibre optimal length of all muscle fibre-tendon fibre units in the model, the calculated relationship was too narrow. That is, both on the ascending limb and on the descending limb the calculated tetanic force was lower than the measured tetanic force.(ABSTRACT TRUNCATED AT 250 WORDS)     

Effect of muscle contraction levels on the force-length relationship of the human Achilles tendon during lengthening of the triceps surae muscle-tendon unit

Findings from animal experiments are sometimes contradictory to the idea that the tendon structure is a simple elastic spring in series with muscle fibers, and suggest influence of muscle contraction on the tendon mechanical properties. The purpose of the present study was to investigate the influence of muscle contraction levels on the force-length relationship of the human Achilles tendon during lengthening of the triceps surae muscle-tendon unit. For seven subjects, ankle dorsiflexion was performed without (passive condition) and with contraction of plantar flexor muscles (eccentric conditions, at 3 contraction levels) on an isokinetic dynamometer. Deformation of the Achilles tendon during each trial was measured using ultrasonography. The Achilles tendon force corresponding to the tendon elongation of 10mm in the passive condition was significantly smaller than those in the eccentric conditions (p<0.05 or p<0.01). Within the eccentric conditions, the Achilles tendon force corresponding to the tendon elongation of 10mm was significantly greater in the maximal contraction level than those in submaximal eccentric conditions (p<0.05 or p<0.01). In addition, the tendon stiffness was greater in higher contraction levels (p<0.05 or p<0.01). Present results suggest that the human tendon structure is not a simple elastic spring in series with muscle fibers.     

Computation of diagnostic data from coronary sinus pressure: a comparison between two possible models

We have evaluated two mathematical models to describe the increase in coronary sinus pressure (CSP) following pressure controlled intermittent coronary sinus occlusion (PICSO). The models are evaluated and compared on the basis of human and canine data. Both models were fitted by non-linear least squares algorithms. Next, derived quantities, such as plateau, rise-time and mean integral of the coronary sinus pressure were calculated from the model parameters. Corresponding quantities for the two models were compared with regard to mean values, rate of successful calculation and specific features characterizing the human or canine case. One model was found to be superior for investigational purposes. The other model was found to be more stable in critical situations and is therefore suggested for usage in closed loop regulation of PICSO. Physiologically, the differences in mean values of the derived quantities between the two models were found to be negligible. The formal statistical significance of the differences is but a consequence of the large number of PICSO cycles analysed.     

Estimating human long bone cross-sectional geometric properties: a comparison of noninvasive methods

Cross-sectional properties (areas, second moments of area) have been used extensively for reconstructing the mechanical loading history of long bone shafts. In the absence of a fortuitous break or available computed tomography (CT) facilities, the endosteal and/or periosteal boundaries of a bone may be approximated using alternative noninvasive methods. The present study tests whether cross-sectional geometric properties of human lower limb bones can be adequately estimated using two such techniques: the ellipse model method (EMM), which uses biplanar radiography alone, and the latex cast method (LCM), which involves molding of the subperiosteal contour in combination with biplanar radiography to estimate the contour of the medullary canal. Results of both methods are compared with "true" cross-sectional properties calculated by direct sectioning. The study sample includes matched femora and tibiae of 50 Pecos Pueblo Amerindians. Bone areas and second moments of area were calculated for the midshaft femur and tibia and proximal femoral diaphysis in each individual. Percent differences between methods were derived to evaluate directional (systematic) and absolute (random) error. Multiple regression was also used to investigate the sources of error associated with each method. The results indicate that while the LCM shows generally good correspondence to the true cross-sectional properties, the EMM generally overestimates true parameters. Regression equations are provided to correct this overestimation, and, when applied to another sample, are shown to significantly improve estimates for the femoral midshaft, although corrections are less successful for the other section locations. Our results suggest that the LCM is an adequate substitute for estimating cross-sectional properties when direct sectioning and CT are not feasible. The EMM is a reasonable alternative, although the bias inherent in the method should be corrected if possible, especially when the results of the study are to be compared with data collected using different methods.     

A comparison of human motoneuron data to simulated data using cat motoneuron models.

The response of repetitively firing human motoneurons to a composite excitatory input was evaluated. It was clearly shown that the response of the motoneurons to the transient input decreased with an increase in the background firing rate of the cell. The current model of repetitively firing human motoneurons could not account for this experimental result. Therefore, a compartmental modelling approach was used to simulate the repetitive firing properties of anaesthetised cat motoneurons under current clamp conditions. The modelled motoneurons were used in simulations similar to the experimental paradigms where the response to a composite excitatory input was evaluated at different background firing rates. The motoneuron models also showed a decrease in response to the excitatory input at faster background firing rates. The results suggest that human motoneurons are more comparable to motoneurons in the anaesthetised cat preparation than formerly thought. The results also demonstrate that the apparent efficacy of a synaptic input may be modulated by changes in background firing rate of the postsynaptic neuron.     

A comparison of muscle energy models for simulating human walking in three dimensions

The popular Hill model for muscle activation and contractile dynamics has been extended with several different formulations for predicting the metabolic energy expenditure of human muscle actions. These extended models differ considerably in their approach to computing energy expenditure, particularly in their treatment of active lengthening and eccentric work, but their predictive abilities have never been compared. In this study, we compared the predictions of five different Hill-based muscle energy models in 3D forward dynamics simulations of normal human walking. In a data-tracking simulation that minimized muscle fatigue, the energy models predicted metabolic costs that varied over a three-fold range (2.45–7.15 J/m/kg), with the distinction arising from whether or not eccentric work was subtracted from the net heat rate in the calculation of the muscle metabolic rate. In predictive simulations that optimized neuromuscular control to minimize the metabolic cost, all five models predicted similar speeds, step lengths, and stance phase durations. However, some of the models predicted a hip circumduction strategy to minimize metabolic cost, while others did not, and the accuracy of the predicted knee and ankle angles and ground reaction forces also depended on the energy model used. The results highlights the need to clarify how eccentric work should be treated when calculating muscle energy expenditure, the difficulty in predicting realistic metabolic costs in simulated walking even with a detailed 3D musculoskeletal model, the potential for using such models to predict energetically-optimal gait modifications, and the room for improvement in existing muscle energy models and locomotion simulation frameworks.     

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.     

Muscle spindles in the deep muscles of the human neck: a morphological and immunocytochemical study.

Muscle spindle density is extremely high in the deep muscles of the human neck. However, there is a paucity of information regarding the morphology and immunoreactivity of these muscle spindles. The objective of this study was to investigate the intrafusal fiber content and to assess the myosin heavy chain (MyHC) composition of muscle spindles from human deep neck muscles. In addition to the conventional spindles containing bag(1), bag(2), and chain fibers (b(1)b(2)c spindle), we observed a number of spindles lacking bag(1) (b(2)c spindle) or bag(2) (b(1)c spindle) fibers. Both bag(1) and bag(2) fibers contained slow tonic MyHCs along their entire fiber length and MyHCI, MyHCIIa, embryonic, and alpha-cardiac MyHC isoforms along a variable length of the fibers. Fetal MyHC was present in bag(2) fibers but not in bag(1) fibers. Nuclear chain fibers contained MyHCIIa, embryonic, and fetal isoforms with regional variations. We also compared the present data with our previous results obtained from muscle spindles in human biceps brachii and the first lumbrical muscles. The allotment of numbers of intrafusal fibers and the MyHC composition showed some muscle-related differences, suggesting functional specialization in the control of movement among different human muscles.     

The phylogenetic position of Brachiopoda—a comparison of morphological and molecular data

Analyses of rRNA and rDNA among Metazoa result in a hypothesis of a sistergroup relationship of Brachiopoda and certain spiralian taxa, whereas analyses of morphological data imply that Brachiopoda show affinities to Deuterostomia within the Radialia. Regarding Brachiopoda as a derived spiralian taxon must be followed by a reinterpretation of the evolution of distinct brachiopod morphological characters—like cleavage pattern, coelom or larva. The experimental insertion of a monophyletic taxon consisting of Brachiopoda and Phoronida into a widely accepted phylogenetic tree of Spiralia leads to the hypothesis that at least trimeric organization, mesosomal tentacular apparatus and heterogeneously assembled metanephridia are products of convergent evolution in Brachiopoda plus Phoronida and Deuterostomia. The hypothesis of a radialian nature of Brachiopoda and Phoronida, as implied by morphological data, remains as the most parsimonious possibility to explain the evolution of seven regarded characters (cleavage pattern, larva, tentacular apparatus, coelom, metameric segmentation, metanephridia and chaetae) in Brachiopoda. Due to the conflicting results of both methods a hitherto undetected systematical problem is discussed possibly hindering data comparability. If the course of evolution can principally be inferred from the information preserved in recent and fossil animals, the results should be congruent in the analyses of both, molecular and morphological data.     

Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors

Background

Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor.

Results

We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise.

Conclusions

The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling.

Reviewers

This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).     

Confronting different models of community structure to species-abundance data: a Bayesian model comparison

Species abundances are undoubtedly the most widely available macroecological data, but can we use them to distinguish among several models of community structure? Here we present a Bayesian analysis of species‐abundance data that yields a full joint probability distribution of each model's parameters plus a relatively parameter‐independent criterion, the posterior Bayes factor, to compare these models. We illustrate our approach by comparing three classical distributions: the zero‐sum multinomial (ZSM) distribution, based on Hubbell's neutral model, the multivariate Poisson lognormal distribution (MPLN), based on niche arguments, and the discrete broken stick (DBS) distribution, based on MacArthur's broken stick model. We give explicit formulas for the probability of observing a particular species‐abundance data set in each model, and argue that conditioning on both sample size and species count is needed to allow comparisons between the two distributions. We apply our approach to two neotropical communities (trees, fish). We find that DBS is largely inferior to ZSM and MPLN for both communities. The tree data do not allow discrimination between ZSM and MPLN, but for the fish data ZSM (neutral model) overwhelmingly outperforms MPLN (niche model), suggesting that dispersal plays a previously underestimated role in structuring tropical freshwater fish communities. We advocate this approach for identifying the relative importance of dispersal and niche‐partitioning in determining diversity of different ecological groups of species under different environmental conditions.