The energy of muscle contraction. II. Transverse compression and work
D. S. Ryan, S. Domínguez, S. A. Ross, N. Nigam, J. M. Wakeling
TThe energy of muscle contraction. II. Transversecompression and work. ∗ David S. Ryan † Sebasti´an Dom´ınguez ‡ §
Stephanie A. Ross † Nilima Nigam ‡ James M. Wakeling † April 30, 2020
Abstract
In this study we reproduced this compression-induced reduction in muscle force throughthe use of a three-dimensional finite element model of contracting muscle. The model usedthe principle of minimum total energy and allowed for the redistribution of energy throughdifferent strain energy- densities; this allowed us to determine the importance of the strainenergy-densities to the transverse forces developed by the muscle. Furthermore, we wereable to study how external work done on the muscle by transverse compression affectsthe internal work and strain-energy distribution of the muscle. We ran a series of in sil-ica experiments on muscle blocks varying in initial pennation angle, muscle length, andcompressive load. As muscle contracts it maintains a near constant volume. As such, anychanges in muscle length are balanced by deformations in the transverse directions suchas muscle thickness or muscle width. Muscle develops transverse forces as it expands.In many situations external forces work to counteract these transverse forces. Muscleresponds to external transverse compression while both passive and active. Transversecompression leads to a reduction in muscle thickness and pennation angle when the muscleis passive, and a reduction to the longitudinal force in its line-of-action when the muscleis active. Greater transverse compression leads to greater force reduction. The muscleblocks used in our simulations decreased in thickness and pennation angle when passivelycompressed, and pushed back on the compression when they were activated. We show howthe longitudinal force from the muscle reduces with increased compressive load and thatthis reduction is dependent on the pennation angle and muscle length. The compression-induced reductions in the longitudinal muscle force were largely due to the volumetricstrain-energy density, which is function of the bulk modulus of the muscle tissue and thedilation of the tissue.
Keywords : muscle, energy, finite element model, compression, transverse, tissue, deforma-tion, 3D ∗ This work was partially supported by the Natural Sciences and Engineering Research Council of Canadaand the Comisi´on Nacional de Investigatici´on Cient´ıfica y Tecnol´ogica of Chile. † Department of Biomedical Physiology and Kinesiology, Simon Fraser University, Burnaby, BC, Canada ‡ Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada § Corresponding author: [email protected] a r X i v : . [ q - b i o . T O ] A p r nergy of Muscle Contractions Ryan et al.
Muscles change in length and develop longitudinal force when they contract, and these resultin internal work being done within the muscle. Muscles additionally expand and develop forcesin transverse directions, again resulting from internal work. However, the transverse action ofmuscle is rarely studied. In this paper we show how longitudinal and transverse forces anddeformations of the muscle are coupled via the internal energy of the muscle, and in particularthrough the redistribution of energy across different forms of strain-energy potentials.Shape changes and muscle forces occur in all three dimensions when muscles contract. As amuscle shortens, it must increase in girth or cross-sectional area in order to maintain its volume(Zuurbier and Huijing, 1993; B¨ol et al., 2013; Randhawa and Wakeling, 2015). Transverseexpansions of contracting muscle have been reported in both animal (Brainerd and Azizi, 2005;Azizi et al., 2008) and human studies (Maganaris et al. 1998; Randhawa et al., 2013; Dickand Wakeling, 2017), and transverse forces generated internally in the muscle can ‘lift’ weightsduring contraction (Siebert et al. 2014). Conversely, transverse loads that compress the musclein its cross-section should be transferred to forces and changes in length along the line-of-actionof the muscle.The force that a muscle develops in its line-of-action depends on pressure and external loadsapplied in the transverse direction. Various researchers have used models of fibre-wound helicaltubes (mimicking the endomysium and perimysium of the extracellular matrix) to explain thetransfer of radial to longitudinal forces and deformations in the muscle (Azizi et al. 2017;Sleboda et al. 2017, 2019). Loading the extracellular matrix by increasing the volume of thesemimembranosus muscle of the bullfrog, using osmotic pressure, increases the passive forcein the line-of-action of the muscle (Sleboda et al. 2017; 2019). Limiting radial expansion ofmuscle by more circumferentially oriented fibres in the helix reduces the extent to which musclecan shorten, and placing a stiff tube around contracting frog plantaris muscle reduces both howmuch the muscle shortens and work done in the line-of-action (Azizi et al. 2017).Transverse external loads act to compress passive muscle as they do mechanical work onthe tissue. Compression of passive muscle has been described for isolated medial gastrocnemiusin rats (Siebert et al. 2014, 2016) and for gluteus maximus (Linder-Ganz et al. 2007) andmedial gastrocnemius (Stutzig et al. 2019; Ryan et al. 2019) muscles in humans. Musclesbulge to resist the transverse loads when they activate and work is generated from forces thatdevelop in the transverse direction. The work generated from these transverse forces can bethought of as ‘lifting work’, especially if it is working against gravity (see the Methods for theformal definition). The muscle volume-specific energy involved in this ‘lifting work’ from themedial gastrocnemius has been approximately 1.1-1.2 x 10 3 J m -3 (Siebert et al. 2014) inthe rat, and 1.1 x 10 3 J m -3 in humans (Stutzig et al. 2019): it should be noted that inthese experiments the plungers that applied the transverse load covered only about 20is about2 orders of magnitude less than the work that could be done by the longitudinal muscle force(Weis-Fogh and Alexander, 1977). This force in the line-of-action during muscle contraction isreduced when the muscle does work to resist the transverse loads (Siebert et al. 2014; 2018;Stutzig et al. 2019; Ryan et al. 2019), and the extent of this force reduction depends onthe transverse force rather than the external load applied to the muscle (Siebert et al. 2016).Siebert and colleagues (2012) explained transverse muscle forces and bulging from previous datausing a hydraulically driven model that transfers load between the transverse to longitudinaldirections. They used an ellipsoidal geometry with constraints that governed anisotropy in the2nergy of Muscle Contractions
Ryan et al. deformations: their model indicated that anisotropy in the connective tissue was important forthe transfer of loads between the transverse and longitudinal directions.Muscles are additionally packaged together in anatomical compartments, and they squeezeon each other as they bulge during contraction. This caused a decrease in the force from thecombined quadriceps in the rabbit when compared to the sum of the individual muscle forcesif they were stimulated separately (de Brito Fontana et al. 2018, 2020), although the reasonsfor this were not clear. Not all muscles increase in thickness during isometric contractions, andthus we should not expect that every muscle will squeeze into neighbouring muscles when theyactivate. Muscles with lower pennation angles ( <
15 degrees) tend to bulge, whereas morepennate muscle may thin as they activate (Randhawa et al. 2013; Wakeling et al. 2020).The changes in muscle force with external loads are length-dependent. For example, theforce in bullfrog semimembranosus was decreased at short lengths and increased at long lengths,when compared to the resting length, when a pressure cuff was applied around the muscle toapply transverse force (Sleboda et al. 2019). Sleboda and colleagues (2019) explained thesefindings using a helically wound model in which the muscle acts to return to a length at whichthe angle of the helical fibres returns to their ideal pitch of 55 degrees (Wainwright et al.1976), and this pitch was assumed to occur at the muscle’s resting length. In contrast, greaterreductions in muscle force have been detected in human plantarflexor muscles when they arecompressed while at longer lengths (knee extended; Siebert et al. 2018) than at a shorter length(knee flexed: Ryan and Siebert observations).When muscles activate, they increase in their pennation angle both for shortening andfor isometric contractions. Internal deformations additionally occur within muscle when it iscompressed: external transverse loads cause a reduction in the mean fibre pennation angle(Wakeling et al. 2013), and a reduction to the extent the pennation angle increases when themuscle contracts (Ryan et al. 2019). We previously showed that the redistribution of strain-energy potentials within contracting muscle changes with the pennation angle (Wakeling etal. 2020), and it is likely that work done on and by the muscle generated by forces in thetransverse direction would also affect the strain-energy potentials within the muscle. Thus,we would expect that the external compression affects the strain-energy potentials within themuscle, that in turn could explain the changes in force in the line-of-action. The redistributionbetween the forms of energy also depends on the muscle length (Wakeling et al. 2020), and thismay drive an interaction between the muscle length and the force reduction that occurs withexternal load, however, this has not yet been examined. It is also likely that the strain-energypotentials and the transfer of external loads on the muscles will depend on the direction of theexternal load relative to the fibre pennation (for example, is the muscle compressed from itstop or from its side within the transverse plane).These recent studies have shown that muscle force changes when external transverse loadsare applied to the muscle, and that this effect is length dependent. Theories have discussed howtransverse forces are transferred to longitudinal forces through the properties of the connectivetissue in the muscle (Smith et al., 2011; Sleboda et al. 2019). However, these studies have notexplained how the defomation of muscle tissues due to external compression loads is affectedby the internal geometry or by the direction of the applied load relative to the muscle fibres.Our previous description of muscle (Wakeling et al. 2020), that quantifies how strain-energypotentials in the contractile elements are redistributed throughout the muscle volume andthe contractile force is redirected across the muscle tissue, is well suited to understand these3nergy of Muscle Contractions
Ryan et al. mechanics of muscle compression. The purpose of this study was to identify whether thealtered muscle forces that occur with compression can be explained in terms of the strain-energy potentials within the muscle, and in particular to account for the role of muscle length,pennation angle and the direction of the external load on the changes to muscle force.
In this paper we present simulations to compare the changes in the internal energy and internalpressure of muscle tissue during an external compression. We modelled the muscle as a three-dimensional and nearly incompressible fibre-reinforced composite biomaterial. The presenceof 1D fibres through the base-material, representing contractile elements, results in an overallanisotropic response of the muscle tissue. The formulation of our model used in all simulationsis based on the balance of strain-energy potentials as presented in Wakeling et al. 2020, andwas solved using the finite element method (FEM). The main change is that here we study theinfluence of external loading on muscle force output.The internal pressure is related to the muscle volume and the volumetric strain energy-density Ψ vol in the muscle:Ψ vol ( u , p, J ) := κ (cid:0) J − J ) − (cid:1) + p (cid:0) J − I ( F ) (cid:1) , (1)and can be calculated from the first variation in the volumetric strain energy-density withrespect to J : p − κ (cid:0) J − /J (cid:1) = 0 , where κ is the bulk modulus of the tissue, J the dilation, and I ( F ) the third invariant of thedeformation tensor F (Wakeling et al. 2020). Note that we use strain-energy potentials tocompare between blocks for the most of the discussion because the muscle blocks in this studyall had the same initial volume. Muscle can show small changes in volume when it contracts(Neering et al., 1991; Smith et al., 2011; Bolsterlee et al., 2017), and so we have modelled themuscle as a nearly incompressible tissue (Wakeling et al. 2020). We constructed a series of blocks of parallel-fibred and unipennate muscle with cuboid geome-tries (30x10x10 mm) and no aponeurosis. The origin of the coordinate system was centredwithin the blocks for their initial configuration V . The muscle blocks had faces in the positiveand negative x, y, and z sides. We defined the length of the blocks as the distance between thepositive and negative x-faces in the x-direction, and the thickness as the distance between thepositive and negative z-faces in the z-direction. The muscle fibres were parallel to each otherand the xz plane in V , but oriented at an initial pennation angle β (0-40 degrees) away fromthe x-direction. We set the initial length of the fibres to their optimal length ( λ iso = 1), and thenormalized muscle length ˆ l to 1 for the undeformed blocks in their initial configuration V . Thematerial properties for the muscle tissue were the same as for our previous study (Wakeling etal. 2020), and we continued to use a scaling factor s b ase of 1.5 for the base-material stiffness.In silico compression tests were conducted in a series of stages as shown in Fig. 1. Thedifferent steps in our compression tests can be listed as follows:4nergy of Muscle Contractions Ryan et al. (A) We initially fixed the -x face in all directions, fixed the -z face in the z axis only, and weapplied a traction to the +x face to either stretch or shorten the passive muscle. Thistraction was applied in the direction normal to the +x face in the initial configuration V .(B) Next, we changed the constraints on the -x face to fix it only in the x-direction, fixed the-z face in all directions and we compressed the passive muscle by applying a transverseload (traction) of 0, 5, 15 or 30 kPa, consistent with previous experimental loads (Ryan etal., 2019; Stutzig et al., 2019). This traction was either on the +z face for ‘top’ loading,or the traction was from both y-faces for ‘side’ loading.(C) Finally, we fixed both x-faces in the x-direction, maintained the z-constraint on the -zface, and maintained the transverse traction to compress the muscle. During this stagewe ramped the activation ˆ a from 0 to 100% over a series of 10 time-steps. The work done by the muscle tissues during deformation is defined in terms of the force devel-oped by the muscle, denoted by F , at a given point and the displacement u at the same point.The total work is then defined as W int := (cid:90) V F · u dV, where V is the current configuration of the muscle tissue, and the dot between the vectorsdenote the dot product. The external work done by the prescribed compression loads on partsof the surface of the muscle geometries, which we denote by S , are computed as W ext := (cid:90) S p ˆ n · u dS, where p ˆ n is the transverse force of the external compression and ˆ n is the normal unit vectoron the surface S . During activation of the muscle fibres the internal force may be greater thanthe force on the system from external loads on the surface S . In such cases, one can see thatthe muscle surface pushes back as the external compression load can no longer compress thetissues. The non-zero force that pushes back on the surface of the muscle, denoted by F lift ,defines a non-zero work which is done by the tissues. We refer to this work as the ‘lifting work’of the muscle. The ‘lifting work’ is defined as W lift := (cid:90) F lift · u dS. The FEM model calculates tissue properties across a set of 128,000 quadrature points withineach muscle block. We defined an orientation for the fibres at each quadrature point. Thepennation angle β in the undeformed and current β states were calculated as the angle be-tween the fibre orientations and the x-axis: this is an angle in 3D space, similar to the 3Dpennation angles defined by Rana et al. 2013. We calculated forces F as the magnitude of5nergy of Muscle Contractions Ryan et al. force perpendicular to a face on the muscle. The longitudinal muscle force in the line-of-actionis denoted by F x .The strain-energies are initially calculated as strain energy-densities y, which are the strain-energy for a given volume of tissue, in units J m − . We compute the total strain-energy ofthe tissue. The strain-energy potential U is the strain-energy in the tissue, in units of Joules.We calculated U at each given state by integrating y across the volume of muscle tissue atthat state. We computed volumetric, muscle base-material, muscle active-fibre, and musclepassive-fibre strain-energy potentials: U vol , U base , U act , U pas , respectively (see Appendix A inWakeling et al. 2020). When the external load was applied to the passive muscle blocks on the z-face (‘top’ loading),the blocks decreased in their thickness in the z-direction, and also in their pennation angle. Theextent of this compression increased with the external load. There was a minor effect of thepennation angle β on this passive compression, with blocks at intermediate pennation anglesof β = 20 or 30 degrees compressing slightly more than the parallel fibred block, or the β = 40degrees block. The increases in tissue compression with increases in external transverse loadare consistent with experimental measures using compression bandages (Wakeling et al. 2013)or weighted plungers on the medial gastrocnemius (Stutzig et al. 2019), and due to sitting onthe gluteus maximus in humans (Linder-Ganz et al. 2007). Note that the internal pressure inthe passive muscle block decreased with increasing external transverse load (Fig. 2b).The internal pressure increased as the muscle activated. The internal pressure for themaximum activation state of ˆ a = 1 was 37 ±
20 kPa (mean ± S.D., N = 60) across allgeometries and transverse loads, which is within the range of intramuscular pressures measuredfor muscle contractions: 13-40 kPa in the frog gastrocnemius, 27 kPa for the soleus (Aratowet al. 1993) and 30 kPa for the tibialis anterior in humans (Ates et al. 2018). However, theexternal transverse load was not directly proportional to the increase in the internal pressurein the muscle blocks (Fig. 2b and 3). Indeed, the coefficient of determination between externaltransverse load and the internal pressure was r = 0 .
052 for all states (N=600), and r = 0 . J changes, causing the volumetric6nergy of Muscle Contractions Ryan et al. strain energy-density to change (see Equation 1 in Methods section for the formal definitionof this strain energy-density in terms of the dilation). Changes in both muscle length andpennation cause changes in the volume and therefore local changes in the dilation. With thedefinition of the internal pressure, we see that it also varies with muscle length and pennationangle. However, we noted that internal pressure does not possess a direct relation with theexternal transverse load: this is illustrated in Fig. 3 where each given transverse load can causea range of different internal pressures depending on the length or pennation angle of the muscleblock being compressed. Note that this highlights important considerations for interpretingexperimental data (Wakeling et al. 2013; Siebert et al. 2016, de Brito Fontana et al. 2018;Sleboda et al. 2019) where muscle is compressed using transverse external loads, because theinternal pressure in the muscle may not be directly related to the extent of the external load.The compressed muscle blocks changed in thickness when they were activated (Fig. 5A).The least pennate muscle blocks ( β ≤
10 degrees) increased in thickness (bulged) when theywere activated, and the more pennate blocks ( β ≥
20 degrees) decreased in thickness. Thisdifference in the direction of bulging was consistent with previous experimental (Maganaris etal., 1998; Randhawa et al., 2013; Randhawa and Wakeling, 2013, 2018; Raiteri et al. 2016)and modeling results (Wakeling et al. 2020). The parallel fibred block of muscle ( β = 10degrees) bulged during activation, however there was minimal effect of the transverse load onthis bulging. By contrast, contracting against greater transverse loads increased the bulgingand the pennation angle of the pennate muscle blocks (Fig. 5B). However, the pennation anglenever returned to its undeformed values of β during these compressions and contractions. Allthe pennate blocks ‘lifted’ the external load when it was applied from the ‘top’, or in otherwords, the distance between the z-faces increased when the load was applied to the z-face. The‘lifting work’ against this external load was the lowest for the β = 10 degrees at 0.3 x 10 3J m − , and increased to 1.4 x 10 3 J m − for β = 40 degrees when measured in comparisonto the unloaded state. This range of ‘lifting work’ spans the values recorded in experimentalstudies: 1.1-1.2 x 10 3 J m − (Siebert et al. 2014) in rats, and 1.1 x10 3 J m − in humans(Stutzig et al. 2019), where this work is expressed as a muscle volume-specific energy density.The strain-energy increased in the muscle during contraction (Fig.4). The strain-energyredistributed across different strain-energy potentials (volumetric, base-material, active-fibreand passive-fibre) in a complex manner that depended on the muscle length, activation andpennation angle, similar to our previous study (Wakeling et al. 2020). Muscles with greaterinitial pennation angle β developed much larger base-material strain-energy potentials (Fig.4), due to the greater shortening of the muscle fibres, in a manner also seen in our previousstudy (Wakeling et al. 2020). The volumetric strain-energy potential was larger at longermuscle lengths ˆ l for β ≤
20 degrees as also shown in our previous study (Wakeling et al.2020), but the relation was more complex at β ≥
30 degrees (Fig. 4). The volumetric strain-energy potential was reduced with greater transverse external loads for most muscle lengthsand pennation angles β , apart from at β = 40 degrees and ˆ l = 1 .
0, and at β of 30-40 degreesand ˆ l = 0 . F x developed in the line-of-action of the muscle varied with muscle length, pen-nation angle and external load (Fig. 6). For comparative purposes, the change in force ∆ F x isexpressed relative to the maximum uncompressed force for that muscle at length ˆ l and penna-tion angle β , and ∆ F x was normalized to the maximum uncompressed force that occurred atthe resting length ˆ l = 1 . β . We also compared here the results for7nergy of Muscle Contractions Ryan et al. transverse loading from the ‘top’ (z-) and the ‘side’ (y-) direction. The force ∆ F x was reducedwith compression for muscle blocks with β ≤
20 degrees, with the force reduction becominggreater for increased transverse loads. The reduction in force was greater when the load wasfrom the top than from the side. Patterns of force reduction ∆ F x differed for the more pennateblocks: for the β = 40 degrees block the force actually increased with transverse loading fromthe top. For the transverse loading from the top, the muscle blocks showed the greatest forcereduction at their initial length ˆ l = 1 . β = 0 degrees. The greatest force reduction at theshort muscle length ˆ l = 0 . β = 10 degrees, and the greatest force reduction atthe longest length ˆ l = 1 . β = 30 degrees, with these patterns varying slightlywhen the load was applied to the sides of the blocks. Here we show that the force reductionfor muscle depends in a complex manner on the length, pennation angle and direction of thetransverse load, with this effect being due to the way in which the strain-energy potentials areredistributed across the muscle during these externally loaded contractions (Fig. 4).The contributions of the different strain-energy potentials to the force F x in the line -of-action of the muscle are shown in Fig. 7. For muscle with low to moderate initial pennation β ≤
20 degrees, the largest reduction in force that occurred with external transverse load wasfrom a reduced contribution from the volumetric strain-energy potential. Reductions in thevolumetric strain-energy potential were less pronounced with external transverse load for thehighest pennation angle of β = 40 degrees, however at this pennation angle the muscle forceactually increased at the highest transverse load and shortest length: this can be attributedto the substantial increase of the contribution from the base- material strain-energy potentialthat occurred for this state. In this study we used the finite element method (FEM) to evaluate a 3D model of skeletalmuscle, based on the principles of continuum mechanics, to probe the relation between externaltransverse load on the muscle and the force that it can develop in its line-of-action as well as theinternal work done by the tissues. The FEM model contained a series of constitutive relationsthat are based on phenomenological descriptions of contractile elements and tissue properties(see details of the model in Wakeling et al. 2020): none of these relations were specificallyderived from or optimized to the transverse response of muscle contractions, in contrast toprevious models (Randhawa and Wakeling, 2015; Siebert et al. 2012, 2014, 2018). Nonetheless,the model predicted many of the general features of the compression response that have beenpreviously reported, and so these general features emerge from the physical principles thatgovern 3D deformations in muscle tissue. We chose the main compression direction to be fromthe ‘top’ which is an external load acting parallel to the plane of the muscle fibres and is thedirection that was tested in previous uniaxial loading in both animal (Siebert et al. 2014, 2016)and human experiments (Ryan et al. 2019; Stutzig et al. 2019). With this compression fromthe top, our model predicted that passive muscle tissue would decrease in thickness (Fig. 2),and the fibres would decrease in pennation angle, supporting experimental results (Ryan et al.2019). When the compressed muscle was activated, the model predicted that it would increasein thickness to ‘lift’ the external load, generating lifting work (Siebert et al. 2014) in the samerange 0.3-1.4 x 10 3 J m − as experimental measures 1.1-1.2 x 10 3 J m − (Siebert et al. 2014;Stutzig et al. 2019), where this work is expressed as a muscle volume-specific energy density.8nergy of Muscle Contractions Ryan et al.
The model in this study highlights the length-dependency of reductions in muscle force withapplied transverse loads. Most of our simulations showed a force reduction with transverseload, and the extent of the force reduction was length-dependent at any given pennation angle.However, some of the conditions at the highest pennation angles β = 30 −
40 degrees showedincreases in force (Fig. 6). The length dependency derives from both the fibre and the baseproperties of the muscle model. The fibres are encoded as contractile elements that have length-dependent force properties for both the active-fibre and passive-fibre components, and the baseproperties are governed by both the volumetric and base-material relations (Rahemi et al. 2014;Wakeling et al. 2020). The combination of the volumetric and base-material properties resultsin a tissue that tends to return to its initial state (volume and shape) after it has been deformed,and this is a similar property to the helical-wound representation of connective tissue modelledby Sleboda and colleagues (2019). It should be noted that the initial undeformed state thatthese models return to is a discretionary choice between studies, and so it should not be expectedthat the exact same length-dependency of the compression-force reduction would occur acrossthe different models. Indeed, this is the case where the helical model predicts force increase atlonger lengths (Sleboda et al. 2019), whereas our model predicts these increases at the largestpennation angles (Fig. 6).The pennation angle of the muscle had a pronounced effect on the muscle response tocompression in terms of tissue deformation (Fig. 5), strain-energy potentials (Fig. 4), and thechanges in muscle force (Fig. 6). When pennate muscle contracts, the fibres rotate to greaterpennation angles as they shorten (Fukunaga et al. 1997; Maganaris et al. 1998). Muscle fibresact to draw the aponeuroses together (or for these simulations, the z-faces) as they shorten,which tends to decrease the muscle thickness. However, the fibres increase in girth duringshortening in order to maintain their volume (Rahemi et al., 2014, 2015). The increase in girthmay be in either the width or thickness direction, and indeed the relative deformations mayvary between muscles (Randhawa and Wakeling, 2015, 2018) due to stress asymmetries throughthe muscle (Wakeling et al. 2020). However, a general effect is for the muscle to increase inpennation angle to allow their fibres to fit within the enclosed volume of the muscle tissue(Zuurbier and Huijing, 1993; Fukunaga et al., 1997). This increase in pennation angle tendsto increase the muscle thickness (Randhawa and Wakeling, 2018), which in turn resists musclecompression acting from the ‘top’ direction and contributes to the lifting work of the muscle.The results from these simulations support this explanation. We additionally show how thestrain-energy potentials redistribute within the muscle in a pennation- dependent manner (Fig.4; Wakeling et al. 2020). Thus, the response to the compression and the internal work thatcan be done by the muscle will also be pennation dependent, due to the altered balance ofstrain-energy potentials within the muscle. We show here that the force reduction that occurswith transverse loading of the muscle seems particularly dependent on the volumetric strain-energy potential (Fig. 7), that in turn varies with pennation angle and the direction of theexternal load relative to the fibres (side or top: Fig. 4).Strain-energy potentials develop during contraction and are distributed through the muscle(Wakeling et al. 2020). When the muscle contracts it increases in its free energy, with this en-ergy being derived from the hydrolysis of ATP to ADP within the muscle fibres (Woledge et al.,1985; Aidley, 1998). The active-fibre strain-energy potentials are redistributed to passive-fibrestrain- energy potentials and then to the base material strain-energy potential that developsin the bulk muscle tissue within the muscle fibres (excluding the myofilament fraction), con-9nergy of Muscle Contractions
Ryan et al. nective tissue surrounding the muscle fibres such as the extracellular matrix, and in sheets ofconnective tissue that form the aponeuroses and internal and external tendons. Energy is alsoused to change the muscle volume. Whilst muscle is often assumed to be incompressible, smallchanges in volume can occur in fibres (Neering et al., 1991), bundles of fibres called fascicles(Smith et al., 2011), and in whole muscle (Bolsterlee et al., 2017): these changes in volumeare energy-consuming processes. The volumetric strain-energy potential, which accounts for anenergetic penalty to any changes in volume that occur, builds up as the muscle is activated andshows slight increases in volume (Wakeling et al. 2020). The transverse external loads in thisstudy act to compress the volume of the muscle (Fig. 2A). These changes in volume relate tochanges to the volumetric strain-energy potential as the muscle is compressed. The volumetricstrain-energy potential contributes to the contractile force F x in the line- of-action for all barthe β = 40 degrees condition at its shortest length. Thus, the compression-induced reductionsin volumetric strain-energy potential result in the reductions to force in the line-of-action duringthe muscle contractions (Fig. 7).The volumetric strain-energy potential is arguably the least-well characterized componentof the internal energy in the muscle in our simulations. The extent of the increase in volumeand the volumetric strain-energy potential is related to the choice of the bulk modulus κ ofthe tissue. A constitutive equation to calculate the volumetric strain-energy potential has notbeen defined for muscle tissue, and so we used a general form (equation 1) that is used forcompressible neo-Hookean material (see, e.g. Pelteret 2012). Here we used a value of κ = 10 Pa that was consistent with previous studies (Rahemi et al., 2014, 2015, Wakeling et al. 2020).We previously showed that this k resulted in volume changes of 2%-4% during contraction offully active parallel muscle fibres. Nonetheless, a previous study showed that k can be variedacross a wide range of magnitudes and still result in similar predictions of tissue deformation(Gardiner and Weiss, 2001). Given the apparent importance of the volumetric strain-energypotential to the modulation of contractile force in response to muscle compression, establishingmuscle-specific constitutive equations for the volumetric strain energy-density and values forthe bulk modulus will be an important area of future investigation.
Conflict of Interest
The authors declare that the research was conducted in the absenceof any commercial or financial relationships that could be construed as a potential conflict ofinterest.
Author Contributions
DR, NN, JW contributed to the study design. DR, SD, SR, NN,JW contributed to the model development. DR ran all the simulations for the paper and dataanalysis. DR and JW contributed to the first draft of the manuscript. DR, SD, SR, NN, JWcontributed to final manuscript preparation.
Funding
We thank the Natural Sciences and Engineering Research Council of Canada forDiscovery Grants to J.M.W. and N.N., and an Alexander Graham Bell Canada GraduateScholarship-Doctoral to S.A.R. We are also grateful for funding to S.D. from Comisi´on Nacionalde Investigaci´on Cient´ıfica y Tecnol´ogica of Chile through Becas-Chile.
Acknowledgments
We thank Tobias Siebert for extensive discussions and inspiration aboutthe mechanics of muscle compression. 10nergy of Muscle Contractions
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