Corrigendum: Where have the dead gone?

[This corrects the article DOI: 10.3389/fmed.2022.837287.].

Three-dimensional spinal shape changes during daily activities.

MOTIVATION: Intuitive assessment of spinal motion poses a tremendous challenge to both physicians and computer modelers. On the one side, medically detailed analyses of spinal shapes, such as computer tomography or X-ray images, are usually subject to static boundary constraints, thereby omitting dynamic information. On the other side, complex computer simulations often lack proper calibration aside from few control points, particularly regarding the three-dimensional arrangement of the spinal column and its idiomotion. PURPOSE: Here, we investigate whether the full three-dimensional changes in spinal shape over time can be concisely detected and depicted. Further, we assess which parts of the spine undergo significant changes during various daily activities. METHODS: We utilize a set of previously published motion capture data from the spinous processes (sacrum up to vertebra C7) of 17 healthy individuals performing the daily tasks of standing, walking, stair climbing, sitting down, and lifting. These three-dimensional, time-dependent marker positions were approximated by a Bézier curve at each time instant. The curves' characteristics, i.e.curvature and torsion, were calculated and juxtaposed for each individual and each activity over time. A statistical parametric mapping revealed significant changes in spinal shape. RESULTS: We found the individual spinal shape characteristics being recognizably preserved during all activities. The walking task did not significantly alter the spinal curvature, while sitting and forward bending significantly altered the lumber and whole spine curvature, respectively. Torsion did not show any significant alterations. CONCLUSION: Based on these results, we suggest that individualized dynamic information on spinal shapes can improve (i) the evaluation of (healthy) motion characteristics, (ii) the detection of pathologies, and (iii) individualized computer simulation models.

Augmenting the Cobb angle: Three-dimensional analysis of whole spine shapes using Bézier curves.

BACKGROUND AND OBJECTIVE: The identification and classification of pathological spinal deformities poses a major challenge to any diagnostician. First, available medical images are usually two-dimensional projections, obscuring elaborated spatial information. Second, several measurement techniques with different thresholds for certain clinical syndromes make it difficult to classify measured results. Here, a method is presented to augment and standardize the analysis of spinal shapes in three dimensions. METHODS: Regarding the first limitation, (semi-)automatic, three-dimensional segmentation techniques of medical images have already been developed. To overcome the second, we propose here a representation of the whole spine by a Bézier curve using the vertebral centers as control points. After normalization, a differential-geometric approach yields information on curvature and torsion at each spinal level as well as in between. RESULTS: Based on literature data and multi-body simulations, we show how these quantities alter with individual posture and during motion. Robustness with respect to missing data is investigated. Approaches towards the identification of spinal disorders are motivated. CONCLUSION: Our results emphasize the need for individualizable identification and classification of spinal deformities and give an outlook on how it might be achieved. The presented methodology constitutes the first fully three-dimensional analysis of spinal shapes, i.e. without the requirement of certain physiological planes (e.g. the sagittal plane) or landmarks (e.g. the apex vertebra).

The impact of travelling on the COVID-19 infection cases in Germany.

BACKGROUND: COVID-19 continues to disrupt social lives and the economy of many countries and challenges their healthcare capacities. Looking back at the situation in Germany in 2020, the number of cases increased exponentially in early March. Social restrictions were imposed by closing e.g. schools, shops, cafés and restaurants, as well as borders for travellers. This reaped success as the infection rate descended significantly in early April. In mid July, however, the numbers started to rise again. Of particular reasons was that from mid June onwards, the travel ban has widely been cancelled or at least loosened. We aim to measure the impact of travellers on the overall infection dynamics for the case of (relatively) few infectives and no vaccinations available. We also want to analyse under which conditions political travelling measures are relevant, in particular in comparison to local measures. By travel restrictions in our model we mean all possible measures that equally reduce the possibility of infected returnees to further spread the disease in Germany, e.g. travel bans, lockdown, post-arrival tests and quarantines. METHODS: To analyse the impact of travellers, we present three variants of an susceptible-exposed-infected-recovered-deceased model to describe disease dynamics in Germany. Epidemiological parameters such as transmission rate, lethality, and detection rate of infected individuals are incorporated. We compare a model without inclusion of travellers and two models with a rate measuring the impact of travellers incorporating incidence data from the Johns Hopkins University. Parameter estimation was performed with the aid of the Monte-Carlo-based Metropolis algorithm. All models are compared in terms of validity and simplicity. Further, we perform sensitivity analyses of the model to observe on which of the model parameters show the largest influence the results. In particular, we compare local and international travelling measures and identify regions in which one of these shows larger relevance than the other. RESULTS: In the comparison of the three models, both models with the traveller impact rate yield significantly better results than the model without this rate. The model including a piecewise constant travel impact rate yields the best results in the sense of maximal likelihood and minimal Bayesian Information Criterion. We synthesize from model simulations and analyses that travellers had a strong impact on the overall infection cases in the considered time interval. By a comparison of the reproductive ratios of the models under traveller/no-traveller scenarios, we found that higher traveller numbers likely induce higher transmission rates and infection cases even in the further course, which is one possible explanation to the start of the second wave in Germany as of autumn 2020. The sensitivity analyses show that the travelling parameter, among others, shows a larger impact on the results. We also found that the relevance of travel measures depends on the value of the transmission parameter: In domains with a lower transmission parameter, caused either by the current variant or local measures, it is found that handling the travel parameters is more relevant than those with lower value of the transmission. CONCLUSIONS: We conclude that travellers is an important factor in controlling infection cases during pandemics. Depending on the current situation, travel restrictions can be part of a policy to reduce infection numbers, especially when case numbers and transmission rate are low. The results of the sensitivity analyses also show that travel measures are more effective when the local transmission is already reduced, so a combination of those two appears to be optimal. In any case, supervision of the influence of travellers should always be undertaken, as another pandemic or wave can happen in the upcoming years and vaccinations and basic hygiene rules alone might not be able to prevent further infection waves.

Muscle active force-length curve explained by an electrophysical model of interfilament spacing.

The active isometric force-length relation (FLR) of striated muscle sarcomeres is central to understanding and modeling muscle function. The mechanistic basis of the descending arm of the FLR is well explained by the decreasing thin:thick filament overlap that occurs at long sarcomere lengths. The mechanistic basis of the ascending arm of the FLR (the decrease in force that occurs at short sarcomere lengths), alternatively, has never been well explained. Because muscle is a constant-volume system, interfilament lattice distances must increase as sarcomere length shortens. This increase would decrease thin and thick-filament electrostatic interactions independently of thin:thick filament overlap. To examine this effect, we present here a fundamental, physics-based model of the sarcomere that includes filament molecular properties, calcium binding, sarcomere geometry including both thin:thick filament overlap and interfilament radial distance, and electrostatics. The model gives extremely good fits to existing FLR data from a large number of different muscles across their entire range of measured activity levels, with the optimized parameter values in all cases lying within anatomically and physically reasonable ranges. A local first-order sensitivity analysis (varying individual parameters while holding the values of all others constant) shows that model output is most sensitive to a subset of model parameters, most of which are related to sarcomere geometry, with model output being most sensitive to interfilament radial distance. This conclusion is supported by re-running the fits with only this parameter subset being allowed to vary, which increases fit errors only moderately. These results show that the model well reproduces existing experimental data, and indicate that changes in interfilament spacing play as central a role as changes in filament overlap in determining the FLR, particularly on its ascending arm.

Load Distribution in the Lumbar Spine During Modeled Compression Depends on Lordosis.

Excessive or incorrect loading of lumbar spinal structures is commonly assumed as one of the factors to accelerate degenerative processes, which may lead to lower back pain. Accordingly, the mechanics of the spine under medical conditions, such as scoliosis or spondylolisthesis, is well-investigated. Treatments via both conventional therapy and surgical methods alike aim at restoring a "healthy" (or at least pain-free) load distribution. Yet, surprisingly little is known about the inter-subject variability of load bearings within a "healthy" lumbar spine. Hence, we utilized computer tomography data from 28 trauma-room patients, whose lumbar spines showed no visible sign of degeneration, to construct simplified multi-body simulation models. The subject-specific geometries, measured by the corresponding lumbar lordosis (LL) between the endplates of vertebra L1 and the sacrum, served as ceteris paribus condition in a standardized forward dynamic compression procedure. Further, the influence of stimulating muscles from the M. multifidus group was assessed. For the range of available LL from 28 to 66°, changes in compressive and shear forces, bending moments, as well as facet joint forces between adjacent vertebrae were calculated. While compressive forces tended to decrease with increasing LL, facet forces were tendentiously increasing. Shear forces decreased between more cranial vertebrae and increased between more caudal ones, while bending moments remained constant. Our results suggest that there exist significant, LL-dependent variations in the loading of "healthy" spinal structures, which should be considered when striving for individually appropriate therapeutic measures.

Intuitive assessment of modeled lumbar spinal motion by clustering and visualization of finite helical axes.

A variety of medical imaging procedures, cadaver experiments, and computer models have been utilized to capture, depict, and understand the motion of the human lumbar spine. Particular interest lies in assessing the relative movement between two adjacent vertebrae, which can be represented by a temporal evolution of finite helical axes (FHA). Mathematically, this FHA evolution constitutes a seven-dimensional quantity: one dimension for the time, two for the (normalized) direction vector, another two for the (unique) position vector, as well as one for each the angle of rotation around and the amount of translation along the axis. Predominantly in the literature, however, movements are assumed to take place in certain physiological planes on which FHA are projected. The resulting three-dimensional quantity - the so-called centrode - is easily presentable but leaves out substantial pieces of available data. Here, we investigate and assess several possibilities to visualize subsets of FHA data of increasing dimensionality. Finally, we utilize an agglomerative hierarchical clustering algorithm and propose a novel visualization technique, namely the quiver principal axis plot (QPAP), to depict the entirety of information inherent to hundreds or thousands of FHA. The QPAP method is applied to flexion-extension, lateral bending, and axial rotation movements of a lumbar spine within both a reduced model as well as a complex upper body system.

Rules of nature's Formula Run: Muscle mechanics during late stance is the key to explaining maximum running speed.

The maximum running speed of legged animals is one evident factor for evolutionary selection-for predators and prey. Therefore, it has been studied across the entire size range of animals, from the smallest mites to the largest elephants, and even beyond to extinct dinosaurs. A recent analysis of the relation between animal mass (size) and maximum running speed showed that there seems to be an optimal range of body masses in which the highest terrestrial running speeds occur. However, the conclusion drawn from that analysis-namely, that maximum speed is limited by the fatigue of white muscle fibres in the acceleration of the body mass to some theoretically possible maximum speed-was based on coarse reasoning on metabolic grounds, which neglected important biomechanical factors and basic muscle-metabolic parameters. Here, we propose a generic biomechanical model to investigate the allometry of the maximum speed of legged running. The model incorporates biomechanically important concepts: the ground reaction force being counteracted by air drag, the leg with its gearing of both a muscle into a leg length change and the muscle into the ground reaction force, as well as the maximum muscle contraction velocity, which includes muscle-tendon dynamics, and the muscle inertia-with all of them scaling with body mass. Put together, these concepts' characteristics and their interactions provide a mechanistic explanation for the allometry of maximum legged running speed. This accompanies the offering of an explanation for the empirically found, overall maximum in speed: In animals bigger than a cheetah or pronghorn, the time that any leg-extending muscle needs to settle, starting from being isometric at about midstance, at the concentric contraction speed required for running at highest speeds becomes too long to be attainable within the time period of a leg moving from midstance to lift-off. Based on our biomechanical model, we, thus, suggest considering the overall speed maximum to indicate muscle inertia being functionally significant in animal locomotion. Furthermore, the model renders possible insights into biological design principles such as differences in the leg concept between cats and spiders, and the relevance of multi-leg (mammals: four, insects: six, spiders: eight) body designs and emerging gaits. Moreover, we expose a completely new consideration regarding the muscles' metabolic energy consumption, both during acceleration to maximum speed and in steady-state locomotion.

Muscle-driven and torque-driven centrodes during modeled flexion of individual lumbar spines are disparate.

Lumbar spine biomechanics during the forward-bending of the upper body (flexion) are well investigated by both in vivo and in vitro experiments. In both cases, the experimentally observed relative motion of vertebral bodies can be used to calculate the instantaneous center of rotation (ICR). The timely evolution of the ICR, the centrode, is widely utilized for validating computer models and is thought to serve as a criterion for distinguishing healthy and degenerative motion patterns. While in vivo motion can be induced by physiological active structures (muscles), in vitro spinal segments have to be driven by external torque-applying equipment such as spine testers. It is implicitly assumed that muscle-driven and torque-driven centrodes are similar. Here, however, we show that centrodes qualitatively depend on the impetus. Distinction is achieved by introducing confidence regions (ellipses) that comprise centrodes of seven individual multi-body simulation models, performing flexion with and without preload. Muscle-driven centrodes were generally directed superior-anterior and tail-shaped, while torque-driven centrodes were located in a comparably narrow region close to the center of mass of the caudal vertebrae. We thus argue that centrodes resulting from different experimental conditions ought to be compared with caution. Finally, the applicability of our method regarding the analysis of clinical syndromes and the assessment of surgical methods is discussed.

Exhaustion of Skeletal Muscle Fibers Within Seconds: Incorporating Phosphate Kinetics Into a Hill-Type Model.

Initiated by neural impulses and subsequent calcium release, skeletal muscle fibers contract (actively generate force) as a result of repetitive power strokes of acto-myosin cross-bridges. The energy required for performing these cross-bridge cycles is provided by the hydrolysis of adenosine triphosphate (ATP). The reaction products, adenosine diphosphate (ADP) and inorganic phosphate (P i ), are then used-among other reactants, such as creatine phosphate-to refuel the ATP energy storage. However, similar to yeasts that perish at the hands of their own waste, the hydrolysis reaction products diminish the chemical potential of ATP and thus inhibit the muscle's force generation as their concentration rises. We suggest to use the term "exhaustion" for force reduction (fatigue) that is caused by combined P i and ADP accumulation along with a possible reduction in ATP concentration. On the basis of bio-chemical kinetics, we present a model of muscle fiber exhaustion based on hydrolytic ATP-ADP-P i dynamics, which are assumed to be length- and calcium activity-dependent. Written in terms of differential-algebraic equations, the new sub-model allows to enhance existing Hill-type excitation-contraction models in a straightforward way. Measured time courses of force decay during isometric contractions of rabbit M. gastrocnemius and M. plantaris were employed for model verification, with the finding that our suggested model enhancement proved eminently promising. We discuss implications of our model approach for enhancing muscle models in general, as well as a few aspects regarding the significance of phosphate kinetics as one contributor to muscle fatigue.

Lumbar spinal ligament characteristics extracted from stepwise reduction experiments allow for preciser modeling than literature data.

Lumbar ligaments play a key role in stabilizing the spine, particularly assisting muscles at wide-range movements. Hence, valid ligament force-strain data are required to generate physiological model predictions. These data have been obtained by experiments on single ligaments or functional units throughout the literature. However, contrary to detailed spine geometries, gained, for instance, from CT data, ligament characteristics are often inattentively transferred to multi-body system (MBS) or finite element models. In this paper, we use an elaborated MBS model of the lumbar spine to demonstrate how individualized ligament characteristics can be obtained by reversely reenacting stepwise reduction experiments, where the range of motion (ROM) was measured. We additionally validated the extracted characteristics with physiological experiments on intradiscal pressure (IDP). Our results on a total of in each case 160 ROM and 49 IDP simulations indicated superiority of our procedure (seven and eight outliers) toward the incorporation of classical literature data (on average 71 and 31 outliers).

Inter-filament spacing mediates calcium binding to troponin: A simple geometric-mechanistic model explains the shift of force-length maxima with muscle activation.

The maximum of a muscle fiber's force-length curve (FLC) shifts to shorter lengths as muscle activation increases. State-of-the-art muscle models cannot explain the mechanistic basis for this shift, which is therefore either omitted or added ad hoc in a descriptive manner. A more theoretical approach developed by Hatze, who had particularly modeled the process of muscle activation, does predict this shift but can be shown to consist of multiple mathematical attempts that are all inconsistent with their common assertion: to represent local volume constancy. What mechanism may underlie the experimentally well-known shift has thus remained unclear. We work out here that the simple assumption of sarcomere volume constancy can, first of all, indeed explain the shift in the activity-Ca2+ relation as a function of sarcomere length by the enforcement of a decrease in inter-filament spacing that must occur as sarcomere length increases. We show that physiological data of this shift are consistent with a simply linear dependency of troponin (volumetric) density on sarcomere length. Further incorporating filament overlap as a second, independent mechanism, we can moreover reproduce, by means of a single master equation, an entire set of measured FLCs from literature, which testify shifts in their maxima at different levels of activation. We conclude that both phenomena, the shift in activity-Ca2+ relations with length and the shift in the maxima of FLCs with Ca2+, can be explained by the superposition of two mechanisms immediately connected to the same sarcomere state variable length: filament overlap and inter-filament spacing.

Corrigendum to "Comparative Sensitivity Analysis of Muscle Activation Dynamics".

[This corrects the article DOI: 10.1155/2015/585409.].

Comparative Sensitivity Analysis of Muscle Activation Dynamics.

We mathematically compared two models of mammalian striated muscle activation dynamics proposed by Hatze and Zajac. Both models are representative for a broad variety of biomechanical models formulated as ordinary differential equations (ODEs). These models incorporate parameters that directly represent known physiological properties. Other parameters have been introduced to reproduce empirical observations. We used sensitivity analysis to investigate the influence of model parameters on the ODE solutions. In addition, we expanded an existing approach to treating initial conditions as parameters and to calculating second-order sensitivities. Furthermore, we used a global sensitivity analysis approach to include finite ranges of parameter values. Hence, a theoretician striving for model reduction could use the method for identifying particularly low sensitivities to detect superfluous parameters. An experimenter could use it for identifying particularly high sensitivities to improve parameter estimation. Hatze's nonlinear model incorporates some parameters to which activation dynamics is clearly more sensitive than to any parameter in Zajac's linear model. Other than Zajac's model, Hatze's model can, however, reproduce measured shifts in optimal muscle length with varied muscle activity. Accordingly we extracted a specific parameter set for Hatze's model that combines best with a particular muscle force-length relation.