A 3D-geometric model for the deformation of a transversally loaded muscle

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.

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.

Three-dimensional modelling of human quadriceps femoris forces

Quadriceps intramuscular anatomy is typically described in two dimensions. However, anatomical descriptions indicate fascicles in the quadriceps may have a three-dimensional orientation. The purpose of this investigation was to quantify the maximum force generating capacity of the individual quadriceps' muscles in three dimensions. Muscle architectural parameters were obtained from three cadaver specimens (two female) and input into a geometry-based multiple fascicle muscle force model. Vastus lateralis, vastus medialis, and rectus femoris had partitions which could be defined based on differences in the sense and direction of fascicles between partitions. Vastus lateralis and rectus femoris were bipennate due to partitions sharing an aponeurosis. Vastus lateralis deep and superficial partitions exerted posterior- (maximum: -29 ± 5 N) and anterior-directed (maximum: 58 ± 15 N) forces on their shared distal aponeurosis. Rectus femoris medial and lateral partitions exerted medial- (maximum: -38 ± 17 N) and lateral-directed (maximum: 19 ± 12 N) forces on their shared proximal aponeurosis. All vastus medialis fascicles ran along the proximal-distal axis. However, fascicles arising near the lesser trochanter also ran along the superficial-deep axis, while fascicles arising from the linea aspera ran along the medial-lateral axis. Thus, vastus medialis could be divided into longus and oblique partitions. Due to the large pennation angle, vastus medialis oblique could exert maximum medial-directed (-219 ± 93 N) and proximal-directed (279 ± 168 N) forces at approximately -40° and -70° knee flexion, respectively, indicating dual roles for vastus medialis oblique dependent on knee flexion angle.

The Energy of Muscle Contraction. II. Transverse Compression and Work

In this study we examined how the strain energies within a muscle are related to changes in longitudinal force when the muscle is exposed to an external transverse load. We implemented a three-dimensional (3D) finite element model of contracting muscle using the principle of minimum total energy and allowing the redistribution of energy through different strain energy-densities. This allowed us to determine the importance of the strain energy-densities to the transverse forces developed by the muscle. We ran a series of in silica experiments on muscle blocks varying in initial pennation angle, muscle length, and external transverse load. As muscle contracts it maintains a near constant volume. As such, any changes in muscle length are balanced by deformations in the transverse directions such as muscle thickness or muscle width. Muscle develops transverse forces as it expands. In many situations external forces act to counteract these transverse forces and the muscle responds to external transverse loads while both passive and active. The muscle blocks used in our simulations decreased in thickness and pennation angle when passively compressed and pushed back on the load when they were activated. Activation of the compressed muscle blocks led either to an increase or decrease in muscle thickness depending on whether the initial pennation angle was less than or greater than 15°, respectively. Furthermore, the strain energy increased and redistributed across the different strain-energy potentials during contraction. The volumetric strain energy-density varied with muscle length and pennation angle and was reduced with greater transverse load for most initial muscle lengths and pennation angles. External transverse load reduced the longitudinal muscle force for initial pennation angles of β₀ = 0°. Whereas for pennate muscle (β₀ > 0°) longitudinal force changed (increase or decrease) depending on the muscle length, pennation angle and the direction of the external load relative to the muscle fibres. For muscle blocks with initial pennation angles β₀ ≤ 20° the reduction in longitudinal muscle force coincided with a reduction in volumetric strain energy-density.

The Energy of Muscle Contraction. I. Tissue Force and Deformation During Fixed-End Contractions

During contraction the energy of muscle tissue increases due to energy from the hydrolysis of ATP. This energy is distributed across the tissue as strain-energy potentials in the contractile elements, strain-energy potential from the 3D deformation of the base-material tissue (containing cellular and extracellular matrix effects), energy related to changes in the muscle's nearly incompressible volume and external work done at the muscle surface. Thus, energy is redistributed through the muscle's tissue as it contracts, with only a component of this energy being used to do mechanical work and develop forces in the muscle's longitudinal direction. Understanding how the strain-energy potentials are redistributed through the muscle tissue will help enlighten why the mechanical performance of whole muscle in its longitudinal direction does not match the performance that would be expected from the contractile elements alone. Here we demonstrate these physical effects using a 3D muscle model based on the finite element method. The tissue deformations within contracting muscle are large, and so the mechanics of contraction were explained using the principles of continuum mechanics for large deformations. We present simulations of a contracting medial gastrocnemius muscle, showing tissue deformations that mirror observations from magnetic resonance imaging. This paper tracks the redistribution of strain-energy potentials through the muscle tissue during fixed-end contractions, and shows how fibre shortening, pennation angle, transverse bulging and anisotropy in the stress and strain of the muscle tissue are all related to the interaction between the material properties of the muscle and the action of the contractile elements.

The time course of calf muscle fluid volume during prolonged running

Muscle fluid is essential for the biochemistry and the biomechanics of muscle contraction. Here, we provide evidence that muscle fluid volumes undergo significant changes during 75 min of moderate intensity (2.7 ± 0.4 m/s) running. Using MRI measurements at baseline and after 2.5, 5, 10, 15, 45 and 75 min, we found that the volumes of calf muscles (quantified through average cross-sectional area) in 18 young recreational runners increase (up to 9% in the gastrocnemii) at the beginning and decrease (below baseline levels) at later stages of running. However, the intensity of changes varied between analyzed muscles. We speculate that these changes are induced by muscle activity and dehydration-related changes in osmotic pressure gradients between intramuscular and extramuscular spaces. These findings highlight the complex nature of muscle fluid shifts during prolonged running exercise.

Effect of Static Stretching, Dynamic Stretching, and Myofascial Foam Rolling on Range of Motion During Hip Flexion: A Randomized Crossover Trial

Siebert, T, Donath, L, Borsdorf, M, and Stutzig, N. Effect of static stretching, dynamic stretching, and myofascial foam rolling on range of motion during hip flexion: A randomized crossover trial. J Strength Cond Res XX(X): 000-000, 2020-Static and dynamic stretching (DS) are commonly used in sports and physical therapy to increase the range of motion (ROM). However, prolonged static stretching (SS) can deteriorate athletic performance. Alternative methods to increase ROM are thus needed. Foam rolling (FR) may initiate muscle relaxation, improve muscular function, physical performance, and ROM. Previous studies that examined effects of FR on ROM did not control for increased tissue compliance or shifted pain threshold. In this study, the isolated influence of altered tissue compliance on ROM after FR, SS, and DS was investigated using a randomized crossover design. Hip flexion ROM at given joint torques before and after SS, DS, and FR was randomly assessed in 14 young male adults (age: 23.7 +/- 1.3 years; height: 182 +/- 8 cm; body mass: 79.4 +/- 6.9 kg). Hip flexion ROM was measured in the sagittal plane with the subjects lying in a lateral position (no gravitational effects on ROM measurements). Surface electromyographic (EMG) analysis of 2 representative hip extensors (M. biceps femoris and M. semitendinosus) was applied to control for active muscle contribution during ROM measurements. Significant increases in ROM for SS (3.8 +/- 1.1[degrees]; p < 0.001) and DS (3.7 +/- 1.8[degrees]; p < 0.001) were observed, but not for FR (0.8 +/- 3.1[degrees]; p = 0.954). Because stretch forces on tendon and muscle tissue during SS and DS predominately act in longitudinal direction, FR induces mainly transversal forces in the muscle tissue. Thus, increased ROM after FR reported in the literature is more likely due to a shift in the pain threshold. These results provide a better understanding of differential loading conditions during SS, DS, and FR for coaches and practitioners.

A hill-type muscle model expansion accounting for effects of varying transverse muscle load

Recent studies demonstrated that uniaxial transverse loading (F_G) of a rat gastrocnemius medialis muscle resulted in a considerable reduction of maximum isometric muscle force (ΔF_im). A hill-type muscle model assuming an identical gearing G between both ΔF_im and F_G as well as lifting height of the load (Δh) and longitudinal muscle shortening (Δl_CC) reproduced experimental data for a single load. Here we tested if this model is able to reproduce experimental changes in ΔF_im and Δh for increasing transverse loads (0.64 N, 1.13 N, 1.62 N, 2.11 N, 2.60 N). Three different gearing ratios were tested: (I) constant G_c representing the idea of a muscle specific gearing parameter (e.g. predefined by the muscle geometry), (II) G_exp determined in experiments with varying transverse load, and (III) G_f that reproduced experimental ΔF_im for each transverse load. Simulations using G_c overestimated ΔF_im (up to 59%) and Δh (up to 136%) for increasing load. Although the model assumption (equal G for forces and length changes) held for the three lower loads using G_exp and G_f, simulations resulted in underestimation of ΔF_im by 38% and overestimation of Δh by 58% for the largest load, respectively. To simultaneously reproduce experimental ΔF_im and Δh for the two larger loads, it was necessary to reduce F_im by 1.9% and 4.6%, respectively. The model seems applicable to account for effects of muscle deformation within a range of transverse loading when using a linear load-dependent function for G.

Intermuscular pressure between synergistic muscles correlates with muscle force

The purpose of the study was to examine the relationship between muscle force generated during isometric contractions (i.e. at a constant muscle–tendon unit length) and the intermuscular (between adjacent muscles) pressure in synergistic muscles. Therefore, the pressure at the contact area of the gastrocnemius and plantaris muscle was measured synchronously to the force of the whole calf musculature in the rabbit species Oryctolagus cuniculus. Similar results were obtained when using a conductive pressure sensor, or a fibre-optic pressure transducer connected to a water-filled balloon. Both methods revealed a strong linear relationship between force and pressure in the ascending limb of the force-length relationship. The shape of the measured force–time and pressure–time traces was almost identical for each contraction (r=0.97). Intermuscular pressure ranged between 100 and 700 mbar (70,000 Pa) for forces up to 287 N. These pressures are similar to previous (intramuscular) recordings within skeletal muscles of different vertebrate species. Furthermore, our results suggest that the rise in intermuscular pressure during contraction may reduce the force production in muscle packages (compartments).

Force reduction induced by unidirectional transversal muscle loading is independent of local pressure

Transversal unidirectional compression applied to muscles via external loading affects muscle contraction dynamics in the longitudinal direction. A recent study reported decreasing longitudinal muscle forces with increasing transversal load applied with a constant contact area (i.e., leading to a simultaneous increase in local pressure). To shed light on these results, we examine whether the decrease in longitudinal force depends on the load, the local pressure, or both. To this end, we perform isometric experiments on rat M. gastrocnemius medialis without and with transversal loading (i) changing the local pressure from 1.1-3.2Ncm(-2) (n=9) at a constant transversal load (1.62N) and (ii) increasing the transversal load (1.15-3.45N) at a constant local pressure of 2.3Ncm(-2) (n=7). While we did not note changes in the decrease in longitudinal muscle force in the first experiment, the second experiment resulted in an almost-linear reduction of longitudinal force between 7.5±0.6% and 14.1±1.7%. We conclude that the observed longitudinal force reduction is not induced by local effects such as malfunction of single muscle compartments, but that similar internal stress conditions and myofilament configurations occur when the local pressure changes given a constant load. The decreased longitudinal force may be explained by increased internal pressure and a deformed myofilament lattice that is likely associated with the decomposition of cross-bridge forces on the one hand and the inhibition of cross-bridges on the other hand.

A multi-scale continuum model of skeletal muscle mechanics predicting force enhancement based on actin-titin interaction

Although recent research emphasises the possible role of titin in skeletal muscle force enhancement, this property is commonly ignored in current computational models. This work presents the first biophysically based continuum-mechanical model of skeletal muscle that considers, in addition to actin-myosin interactions, force enhancement based on actin-titin interactions. During activation, titin attaches to actin filaments, which results in a significant reduction in titin's free molecular spring length and therefore results in increased titin forces during a subsequent stretch. The mechanical behaviour of titin is included on the microscopic half-sarcomere level of a multi-scale chemo-electro-mechanical muscle model, which is based on the classic sliding-filament and cross-bridge theories. In addition to titin stress contributions in the muscle fibre direction, the continuum-mechanical constitutive relation accounts for geometrically motivated, titin-induced stresses acting in the muscle's cross-fibre directions. Representative simulations of active stretches under maximal and submaximal activation levels predict realistic magnitudes of force enhancement in fibre direction. For example, stretching the model by 20 % from optimal length increased the isometric force at the target length by about 30 %. Predicted titin-induced stresses in the muscle's cross-fibre directions are rather insignificant. Including the presented development in future continuum-mechanical models of muscle function in dynamic situations will lead to more accurate model predictions during and after lengthening contractions.

A physically motivated constitutive model for 3D numerical simulation of skeletal muscles

A detailed numerical implementation within the FEM is presented for a physically motivated three-dimensional constitutive model describing the passive and active mechanical behaviors of the skeletal muscle. The derivations for the Cauchy stress tensor and the consistent material tangent are provided. For nearly incompressible skeletal muscle tissue, the strain energy function may be represented either by a coupling or a decoupling of the distortional and volumetric material response. In the present paper, both functionally different formulations are introduced allowing for a direct comparison between the coupled and decoupled isochoric-volumetric approach. The numerical validation of both implementations revealed significant limitations for the decoupled approach. For an extensive characterization of the model response to different muscle contraction modes, a benchmark model is introduced. Finally, the proposed implementation is shown to provide a reliable tool for the analysis of complex and highly nonlinear problems through the example of the human mastication system by studying bite force and three-dimensional muscle shape changes during mastication.

Muscle force depends on the amount of transversal muscle loading

Skeletal muscles are embedded in an environment of other muscles, connective tissue, and bones, which may transfer transversal forces to the muscle tissue, thereby compressing it. In a recent study we demonstrated that transversal loading of a muscle with 1.3Ncm(-2) reduces maximum isometric force (Fim) and rate of force development by approximately 5% and 25%, respectively. The aim of the present study was to examine the influence of increasing transversal muscle loading on contraction dynamics. Therefore, we performed isometric experiments on rat M. gastrocnemius medialis (n=9) without and with five different transversal loads corresponding to increasing pressures of 1.3Ncm(-2) to 5.3Ncm(-2) at the contact area between muscle and load. Muscle loading was induced by a custom-made plunger which was able to move in transversal direction. Increasing transversal muscle loading resulted in an almost linear decrease in muscle force from 4.8±1.8% to 12.8±2% Fim. Compared to an unloaded isometric contraction, rate of force development decreased from 20.2±4.0% at 1.3Ncm(-2) muscle loading to 34.6±5.7% at 5.3Ncm(-2). Experimental observation of the impact of transversal muscle loading on contraction dynamics may help to better understand muscle tissue properties. Moreover, applying transversal loads to muscles opens a window to analyze three-dimensional muscle force generation. Data presented in this study may be important to develop and validate muscle models which enable simulation of muscle contractions under compression and enlighten the mechanisms behind.

Work partitioning of transversally loaded muscle: experimentation and simulation

Skeletal muscles are surrounded by other muscles, connective tissue and bones, which may transfer transversal forces to the muscle belly. Simple Hill-type muscle models do not consider transversal forces. Thus, the aim of this study was to examine and model the influence of transversal muscle loading on contraction dynamics, e.g. on the rate of force development and on the maximum isometric muscle force (Fim). Isometric experiments with and without transversal muscle loading were conducted on rat muscles. The muscles were loaded (1.3 N cm⁻²) by a custom-made plunger which was able to move in transversal direction. Then the muscle was fully stimulated, the isometric force was measured at the distal tendon and the movement of the plunger was captured with a high-speed camera. The interaction between the muscle and the transversal load was modelled based on energy balance between the (1) work done by the contractile component (CC) and (2) the work done to lift the load, to stretch the series elastic structures and to deform the muscle. Compared with the unloaded contraction, the force rate was reduced by about 25% and Fim was reduced by 5% both in the experiment and in the simulation. The reduction in Fim resulted from using part of the work done by the CC to lift the load and deform the muscle. The response of the muscle to transversal loading opens a window into the interdependence of contractile and deformation work, which can be used to specify and validate 3D muscle models.