On a three-dimensional constitutive model for history effects in skeletal muscles

Impact of lengthening velocity on the generation of eccentric force by slow-twitch muscle fibers in long stretches

After an initial increase, isovelocity elongation of a muscle fiber can lead to diminishing (referred to as Give in the literature) and subsequently increasing force. How the stretch velocity affects this behavior in slow-twitch fibers remains largely unexplored. Here, we stretched fully activated individual rat soleus muscle fibers from 0.85 to 1.3 optimal fiber length at stretch velocities of 0.01, 0.1, and 1 maximum shortening velocity, vmax, and compared the results with those of rat EDL fast-twitch fibers obtained in similar experimental conditions. In soleus muscle fibers, Give was 7%, 18%, and 44% of maximum isometric force for 0.01, 0.1, and 1 vmax, respectively. As in EDL fibers, the force increased nearly linearly in the second half of the stretch, although the number of crossbridges decreased, and its slope increased with stretch velocity. Our findings are consistent with the concept of a forceful detachment and subsequent crossbridge reattachment in the stretch's first phase and a strong viscoelastic titin contribution to fiber force in the second phase of the stretch. Interestingly, we found interaction effects of stretch velocity and fiber type on force parameters in both stretch phases, hinting at fiber type-specific differences in crossbridge and titin contributions to eccentric force. Whether fiber type-specific combined XB and non-XB models can explain these effects or if they hint at some not fully understood properties of muscle contraction remains to be shown. These results may stimulate new optimization perspectives in sports training and provide a better understanding of structure-function relations of muscle proteins.

Determination of muscle shape deformations of the tibialis anterior during dynamic contractions using 3D ultrasound

Purpose

In this paper, we introduce a novel method for determining 3D deformations of the human tibialis anterior (TA) muscle during dynamic movements using 3D ultrasound.

Materials and Methods

An existing automated 3D ultrasound system is used for data acquisition, which consists of three moveable axes, along which the probe can move. While the subjects perform continuous plantar- and dorsiflexion movements in two different controlled velocities, the ultrasound probe sweeps cyclically from the ankle to the knee along the anterior shin. The ankle joint angle can be determined using reflective motion capture markers. Since we considered the movement direction of the foot, i.e., active or passive TA, four conditions occur: slow active, slow passive, fast active, fast passive. By employing an algorithm which defines ankle joint angle intervals, i.e., intervals of range of motion (ROM), 3D images of the volumes during movement can be reconstructed.

Results

We found constant muscle volumes between different muscle lengths, i.e., ROM intervals. The results show an increase in mean cross-sectional area (CSA) for TA muscle shortening. Furthermore, a shift in maximum CSA towards the proximal side of the muscle could be observed for muscle shortening. We found significantly different maximum CSA values between the fast active and all other conditions, which might be caused by higher muscle activation due to the faster velocity.

Conclusion

In summary, we present a method for determining muscle volume deformation during dynamic contraction using ultrasound, which will enable future empirical studies and 3D computational models of skeletal muscles.

The role of pennation angle and architectural gearing to rate of force development in dynamic and isometric muscle contractions

BACKGROUND

Associations between muscle architecture and rate of force development (RFD) have been largely studied during fixed-end (isometric) contractions. Fixed-end contractions may, however, limit muscle shape changes and thus alter the relationship between muscle architecture an RFD.

AIM

We compared the correlation between muscle architecture and architectural gearing and knee extensor RFD when assessed during dynamic versus fixed-end contractions.

METHODS

Twenty-two recreationally active male runners performed dynamic knee extensions at constant acceleration (2000°s-2) and isometric contractions at a fixed knee joint angle (fixed-end contractions). Torque, RFD, vastus lateralis muscle thickness, and fascicle dynamics were compared during 0-75 and 75-150 ms after contraction onset.

RESULTS

Resting fascicle angle was moderately and positively correlated with RFD during fixed-end contractions (r = 0.42 and 0.46 from 0-75 and 75-150 ms, respectively; p < 0.05), while more strongly (p < 0.05) correlated with RFD during dynamic contractions (r = 0.69 and 0.73 at 0-75 and 75-150 ms, respectively; p < 0.05). Resting fascicle angle was (very) strongly correlated with architectural gearing (r = 0.51 and 0.73 at 0-75 ms and 0.50 and 0.70 at 75-150 ms; p < 0.05), with gearing in turn also being moderately to strongly correlated with RFD in both contraction conditions (r = 0.38-0.68).

CONCLUSION

Resting fascicle angle was positively correlated with RFD, with a stronger relationship observed in dynamic than isometric contraction conditions. The stronger relationships observed during dynamic muscle actions likely result from different restrictions on the acute changes in muscle shape and architectural gearing imposed by isometric versus dynamic muscle contractions.

Influence of muscle length on the three-dimensional architecture and aponeurosis dimensions of rabbit calf muscles

The function of a muscle is highly dependent on its architecture, which is characterized by the length, pennation, and curvature of the fascicles, and the geometry of the aponeuroses. During in vivo function, muscles regularly undergo changes in length, thereby altering their architecture. During passive muscle lengthening, fascicle length (FL) generally increases and the angle of fascicle pennation (FP) and the fascicle curvature (FC) decrease, while the aponeuroses increase in length but decrease in width. Muscles are differently structured, making their change during muscle lengthening complex and multifaceted. To obtain comprehensive data on architectural changes in muscles during passive length, the present study determined the three-dimensional fascicle geometry of rabbit M. gastrocnemius medialis (GM), M. gastrocnemius lateralis (GL), and M. plantaris (PLA). For this purpose, the left and right legs of three rabbits were histologically fixed at targeted ankle joint angles of 95° (short muscle length [SML]) and 60° (long muscle length [LML]), respectively, and the fascicles were tracked by manual three-dimensional digitization. In a second set of experiments, the GM aponeurosis dimensions of ten legs from five rabbits were determined at varying muscle lengths via optical marker tracking. The GM consisted of a uni-pennated compartment, whereas the GL and PLA contained multiple compartments of differently pennated fascicles. In the LML compared to the SML, the GM, GL, and PLA had on average a 41%, 29%, and 41% increased fascicle length, and a 30%, 25%, and 33% decrease in fascicle pennation and a 32%, 11%, and 35% decrease in fascicle curvature, respectively. Architectural properties were also differentiated among the different compartments of the PLA and GL, allowing for a more detailed description of their fascicle structure and changes. It was shown that the compartments change differently with muscle length. It was also shown that for each degree of ankle joint angle reduction, the proximal GM aponeurosis length increased by 0.11%, the aponeurosis width decreased by 0.22%, and the area was decreased by 0.20%. The data provided improve our understanding of muscles and can be used to develop and validate muscle models.

3D ultrasound-based determination of skeletal muscle fascicle orientations

Architectural parameters of skeletal muscle such as pennation angle provide valuable information on muscle function, since they can be related to the muscle force generating capacity, fiber packing, and contraction velocity. In this paper, we introduce a 3D ultrasound-based workflow for determining 3D fascicle orientations of skeletal muscles. We used a custom-designed automated motor driven 3D ultrasound scanning system for obtaining 3D ultrasound images. From these, we applied a custom-developed multiscale-vessel enhancement filter-based fascicle detection algorithm and determined muscle volume and pennation angle. We conducted trials on a phantom and on the human tibialis anterior (TA) muscle of 10 healthy subjects in plantarflexion (157 ± 7∘ ), neutral position (109 ± 7∘ , corresponding to neutral standing), and one resting position in between (145 ± 6∘ ). The results of the phantom trials showed a high accuracy with a mean absolute error of 0.92 ± 0.59∘ . TA pennation angles were significantly different between all positions for the deep muscle compartment; for the superficial compartment, angles are significantly increased for neutral position compared to plantarflexion and resting position. Pennation angles were also significantly different between superficial and deep compartment. The results of constant muscle volumes across the 3 ankle joint angles indicate the suitability of the method for capturing 3D muscle geometry. Absolute pennation angles in our study were slightly lower than recent literature. Decreased pennation angles during plantarflexion are consistent with previous studies. The presented method demonstrates the possibility of determining 3D fascicle orientations of the TA muscle in vivo.

A biophysically guided constitutive law of the musculotendon-complex: modelling and numerical implementation in Abaqus

BACKGROUND AND OBJECTIVE

Many biomedical, clinical, and industrial applications may benefit from musculoskeletal simulations. Three-dimensional macroscopic muscle models (3D models) can more accurately represent muscle architecture than their 1D (line-segment) counterparts. Nevertheless, 3D models remain underutilised in academic, clinical, and commercial environments. Among the reasons for this is a lack of modelling and simulation standardisation, verification, and validation. Here, we strive towards a solution by providing an open-access, characterised, constitutive relation (CR) for 3D musculotendon models.

METHODS

The musculotendon complex is modelled following the state-of-the-art active stress approach and is treated as hyperelastic, transversely isotropic, and nearly incompressible. Furthermore, force-length and -velocity relationships are incorporated, and muscle activation is derived from motor-unit information. The CR was implemented within the commercial finite-element software package Abaqus as a user-subroutine. A masticatory system model with left and right masseters was used to demonstrate active and passive movement.

RESULTS

The CR was characterised by various experimental data sets and was able to capture a wide variety of passive and active behaviours. Furthermore, the masticatory simulations revealed that joint movement was sensitive to the muscle's in-fibre passive response.

CONCLUSIONS

This user-material provides a "plug and play" template for 3D neuro-musculoskeletal finite element modelling. We hope that this reduces modelling effort, fosters exchange, and contributes to the standardisation of such models.

How velocity impacts eccentric force generation of fully activated skinned skeletal muscle fibers in long stretches

Eccentric muscle contractions are fundamental to everyday life. They occur markedly in jumping, running, and accidents. Following an initial force rise, stretching of a fully activated muscle can result in a phase of decreasing force ('Give') followed by force redevelopment. However, how the stretch velocity affects 'Give' and force redevelopment remains largely unknown. We investigated the force produced by fully activated single skinned fibers of rat extensor digitorum longus muscles during long stretches. Fibers were pulled from length .85 to 1.3 optimal fiber length at a rate of 1, 10 and 100% of the estimated maximum shortening velocity. 'Give' was absent in slow stretches. Medium and fast stretches yielded a clear 'Give'. After the initial force peak, forces decreased by 11.2% and 27.8% relative to the initial peak force before rising again. During the last half of the stretch (from 1.07 to 1.3 optimal fiber length, which is within the range of the expected descending limb of the force-length relationship), the linear force slope tripled from slow to medium stretch and increased further by 60% from medium to fast stretch. These results are compatible with forcible cross-bridge detachment and re-development of a cross-bridge distribution, and a viscoelastic titin contribution to fiber force. Accounting for these results can improve muscle models and predictions of multi-body simulations.

A Physiology-Guided Classification of Active-Stress and Active-Strain Approaches for Continuum-Mechanical Modeling of Skeletal Muscle Tissue

The well-established sliding filament and cross-bridge theory explain the major biophysical mechanism responsible for a skeletal muscle's active behavior on a cellular level. However, the biomechanical function of skeletal muscles on the tissue scale, which is caused by the complex interplay of muscle fibers and extracellular connective tissue, is much less understood. Mathematical models provide one possibility to investigate physiological hypotheses. Continuum-mechanical models have hereby proven themselves to be very suitable to study the biomechanical behavior of whole muscles or entire limbs. Existing continuum-mechanical skeletal muscle models use either an active-stress or an active-strain approach to phenomenologically describe the mechanical behavior of active contractions. While any macroscopic constitutive model can be judged by it's ability to accurately replicate experimental data, the evaluation of muscle-specific material descriptions is difficult as suitable data is, unfortunately, currently not available. Thus, the discussions become more philosophical rather than following rigid methodological criteria. Within this work, we provide a extensive discussion on the underlying modeling assumptions of both the active-stress and the active-strain approach in the context of existing hypotheses of skeletal muscle physiology. We conclude that the active-stress approach resolves an idealized tissue transmitting active stresses through an independent pathway. In contrast, the active-strain approach reflects an idealized tissue employing an indirect, coupled pathway for active stress transmission. Finally the physiological hypothesis that skeletal muscles exhibit redundant pathways of intramuscular stress transmission represents the basis for considering a mixed-active-stress-active-strain constitutive framework.

Architectural model for muscle growth during maturation

Muscle architecture, which includes parameters like fascicle length, pennation angle, and physiological cross-sectional area, strongly influences skeletal muscles' mechanical properties. During maturation, the muscle architecture has to adapt to a growing organism. This study aimed to develop an architectural model capable of predicting the complete 3D fascicle architecture for primarily unipennate muscles of an arbitrary age, based on fascicle data for an initial age. For model development, we collected novel data on 3D muscle architecture of the rabbit (Oryctolagus cuniculus) M. plantaris of eight animals ranging in age from 29 to 106 days. Experimental results show that plantaris muscle belly length increases by 73%, whereas mean fascicle length and mean pennation angle increases by 39 and 14%, respectively. Those changes were incorporated into the model. In addition to the data collected for M. plantaris the predictions of the model were compared to existing literature data of rabbit M. soleus and M. gastrocnemius medialis. With an error of −1.0 ± 8.6% for relative differences in aponeurosis length, aponeurosis width, muscle height, and muscle mass, the model delivered good results matching interindividual differences. For future studies, the model could be utilized to generate realistic architectural data sets for simulation studies.

Power Amplification Increases With Contraction Velocity During Stretch-Shortening Cycles of Skinned Muscle Fibers

Muscle force, work, and power output during concentric contractions (active muscle shortening) are increased immediately following an eccentric contraction (active muscle lengthening). This increase in performance is known as the stretch-shortening cycle (SSC)-effect. Recent findings demonstrate that the SSC-effect is present in the sarcomere itself. More recently, it has been suggested that cross-bridge (XB) kinetics and non-cross-bridge (non-XB) structures (e.g., titin and nebulin) contribute to the SSC-effect. As XBs and non-XB structures are characterized by a velocity dependence, we investigated the impact of stretch-shortening velocity on the SSC-effect. Accordingly, we performed in vitro isovelocity ramp experiments with varying ramp velocities (30, 60, and 85% of maximum contraction velocity for both stretch and shortening) and constant stretch-shortening magnitudes (17% of the optimum sarcomere length) using single skinned fibers of rat soleus muscles. The different contributions of XB and non-XB structures to force production were identified using the XB-inhibitor Blebbistatin. We show that (i) the SSC-effect is velocity-dependent-since the power output increases with increasing SSC-velocity. (ii) The energy recovery (ratio of elastic energy storage and release in the SSC) is higher in the Blebbistatin condition compared with the control condition. The stored and released energy in the Blebbistatin condition can be explained by the viscoelastic properties of the non-XB structure titin. Consequently, our experimental findings suggest that the energy stored in titin during the eccentric phase contributes to the SSC-effect in a velocity-dependent manner.

3D finite element models from serial section histology of skeletal muscle tissue - The role of micro-architecture on mechanical behaviour

In this contribution we create three-dimensional (3D) finite element models from a series of histological sections of porcine skeletal muscle tissue. Image registration is performed on the stained sections by affinely aligning them using auxiliary markers, followed by image segmentation to determine muscle fibres and the extracellular matrix in each section, with particular regard to the continuity of the fibres through the stack. With this information, 3D virtual tissue samples are reconstructed, discretised, and associated with appropriate non-linear elastic anisotropic material models. While the gross anatomy is directly obtained from the images, the local directions of anisotropy were determined by the use of an analogy with steady state diffusion. The influence of the number of histological sections considered for reconstruction on the numerically simulated mechanical response of the virtual tissue samples is then studied. The results show that muscle tissue is fairly heterogeneous along the fascicles, and that transverse isotropy is inadequate in describing their material symmetry at the typical length scale of a fascicle. Numerical simulations of different load cases suggest that ignoring the undulations of fibres and their non-uniform cross-sections only moderately affects the passive response of the tissue in tensile and compressive modes, but can become crucial when predicting the response to generic loads and activation.

Modelling force-length-activation relationships of wrist and finger extensor muscles

The wrist and finger extensors play a crucial role in the muscle coordination during grasping tasks. Nevertheless, few data are available regarding their force-generating capacities. The objective of this study was to provide a model of the force-length-activation relationships of the hand extensors using non-invasive methods. The extensor carpi radialis (ECR) and the extensor digitorum communis (EDC) were studied as representative of wrist and finger extensors. Ten participants performed isometric extension force-varying contractions in different postures on an ergometer recording resultant moment. The joint angle, the myotendinous junction displacement and activation were synchronously tracked using motion capture, ultrasound and electromyography. Muscle force was estimated via a musculoskeletal model using the measured joint angle and moment. The force-length-activation relationship was then obtained by fitting a force-length model at different activation levels to the measured data. The obtained relationships agreed with previously reported data regarding muscle architecture, sarcomere length and activation-dependent shift of optimal length. Muscle forces estimated from kinematics and electromyography using the force-length-activation relationships were comparable, below 15% differences, to those estimated from moment via the musculoskeletal model. The obtained quantitative data provides a new insight into the different muscle mechanics of finger and wrist extensors. Graphical abstract By combining in vivo data (kinematics, dynamometry, electromyography, ultrasonography) during isometric force-varying contractions with musculoskeletal modelling, the force-length-activation relationships of both finger and wrist extensors were obtained. The results provided a new insight into the role of hand extensors in the generation and control of hand movements.

Microstructural analysis of skeletal muscle force generation during aging

Human aging results in a progressive decline in the active force generation capability of skeletal muscle. While many factors related to the changes of morphological and structural properties in muscle fibers and the extracellular matrix (ECM) have been considered as possible reasons for causing age-related force reduction, it is still not fully understood why the decrease in force generation under eccentric contraction (lengthening) is much less than that under concentric contraction (shortening). Biomechanically, it was observed that connective tissues (endomysium) stiffen as ages, and the volume ratio of connective tissues exhibits an age-related increase. However, limited skeletal muscle models take into account the microstructural characteristics as well as the volume fraction of tissue material. This study aims to provide a numerical investigation in which the muscle fibers and the ECM are explicitly represented to allow quantitative assessment of the age-related force reduction mechanism. To this end, a fiber-level honeycomb-like microstructure is constructed and modeled by a pixel-based Reproducing Kernel Particle Method (RKPM), which allows modeling of smooth transition in biomaterial properties across material interfaces. The numerical investigation reveals that the increased stiffness of the passive materials of muscle tissue reduces the force generation capability under concentric contraction while maintains the force generation capability under eccentric contraction. The proposed RKPM microscopic model provides effective means for the cellular-scale numerical investigation of skeletal muscle physiology. NOVELTY STATEMENT: A cellular-scale honeycomb-like microstructural muscle model constructed from a histological cross-sectional image of muscle is employed to study the causal relations between age-associated microstructural changes and age-related force loss using Reproducing Kernel Particle Method (RKPM). The employed RKPM offers an effective means for modeling biological materials based on pixel points in the medical images and allow modeling of smooth transition in the material properties across interfaces. The proposed microstructure-informed muscle model enables quantitative evaluation on how cellular-scale compositions contribute to muscle functionality and explain differences in age-related force changes during concentric, isometric and eccentric contractions.