Series elasticity
The concept of “series elasticity” is used in structural mechanics to describe ideal situations with the aim to understand the behaviour and properties of complex systems. When two elements are said to be arranged “in series”, it implies that the instantaneous internal forces in the two elements are always the same, or at least in constant proportion, independent of the loading history and independent of the material properties. For example, in Fig. (1), the force exerted in the idealized contractile element (CE), is always matched instantaneously by the elastic spring in series with CE, the series elastic element (SE). The term “elastic” implies that the strain is instantaneously given by the force applied to the SE element. Therefore, an elastic material has the same strain for a given force, independent of the history of force application (fast or slow; or increasing vs. decreasing force (Fig. 4a). The best known example of an elastic element is the idealized, linear spring, or Hooke’s law, where the elongation of the spring is always (and instantaneously) given by the force applied to the spring; i.e. F = kx, where F is the applied force, k the spring constant (stiffness), and x the deformation of the spring from its zero-strain, unloaded length.
In the natural world, there is no perfectly elastic element. Quartz fibres are found to approximate perfect elasticity the best. Rubber is also almost perfectly elastic while tendons are not. Tendons become stiffer when forces are applied faster, thereby exhibiting visco-elastic properties, and elongation is not given by the force exclusively. Tendons also have a distinct hysteresis of about 10%, as illustrated conceptually (but not in magnitude) in the example shown in Fig. (4B), which means that the energy applied in stretching a tendon exceeds the energy that is returned by the tendon by about 10% when force is removed. A perfectly elastic material, by definition, does not have a hysteresis, as its deformation is always the same for a given force and independent of the history of force application.
One might be tempted to stop any discussion on muscle series elasticity here, as perfectly elastic materials do not exist in nature, and materials often implicated to be elastic in muscles, such as tendons, cross-bridges, titin, and aponeuroses, are not elastic ([25,26,27]). However, in practice, it is sometimes useful, and is done frequently in biomechanics, to consider nearly elastic materials as elastic in order to gain an understanding of complex systems. So we will proceed.
Series elasticity in muscles
Muscles have a number of passive (non-contractile) “elastic” elements that have been implicated with series elasticity. Some elastic elements implicated in being “in series” with the “contractile element” are the free tendon, the muscle internal aponeuroses, the structural protein “titin”, the elastic elements in the cross-bridges (the S2 element in Huxley’s 1969 [28] notation, or the AB element in Huxley and Simmons’ 1971 [29] notation), and the Z-bands in sarcomeres. Here, I will focus primarily on series elasticity of the entire muscle. However, for completeness, I will also briefly discuss cross-bridge, titin, and Z-band elasticity, as they have been implicated as being in series with some molecular or sub-cellular component of muscle.
Briefly, cross-bridge elasticity, according to classic cross-bridge models, is in series with the cross-bridge head; that is, whatever force is transmitted from the cross-bridge head to the actin filament is thought to be transmitted by an elastic element that attaches the cross-bridge head to the myosin filament backbone (S2 in Huxley’s 1969 [28] notation). In fact, in the original cross-bridge theory (Huxley, 1957 [30]), the cross-bridge head is attached to the myosin backbone via a linearly elastic spring that is arranged in series with the cross-bridge head. The force of the cross-bridge was then assumed to be given by the elongation of that linear spring from its equilibrium position: Fcb = kcbx; where Fcb is the force in a cross-bridge, kcb is the (constant) cross-bridge spring stiffness, and x is the elongation of the cross-bridge spring element from its equilibrium position. The idea of a linear cross-bridge stiffness has been challenged [31] and is likely not correct. Nevertheless, the notion of linear elasticity in cross-bridges continues to persist. Furthermore, myofibrils and fibres of a muscle have complex (parallel) connections, and thus, cross-bridges in different myofibrils and fibres cannot be considered mechanically “in series” with each other.
The molecular spring titin is interesting to contemplate as a series elastic element. Titin spans the half-sarcomere from the M-line to the Z-band. It is thought to be rigidly attached to the myosin filament with no (or only very little) possibility for elongation in the A-band region of the sarcomere (Fig. 5). However, titin runs freely across the I-band region from the end of the myosin filament to approximately 50 nm away from the Z-band where it combines with the actin filament [32,33,34]. The I-band region of titin is known to be extensible, and is thought to be virtually elastic if elongation is small and no immunoglobulin domains of titin are unfolded [35, 36]. Because of this structural arrangement, titin filaments are in series with the myosin filament in the passive muscle, assuming the idealized case that there are no (cross-bridge) connections between actin and myosin in the passive state. In the active state, when cross-bridges are formed between actin and myosin, titin is not in series with the myosin filament anymore, as its force would not represent the force carried by the myosin filament, while in the idealized passive state, it would. In the active state, titin acts more like a spring that is in parallel to the cross-bridges; that is, its force adds algebraically with the forces of the cross-bridges interacting between an actin-myosin pair. Note, that in normal muscle, each half-myosin is associated with six titin filaments [37], so when attempting to calculate the forces in a titin filament in a passive muscle, this ratio needs to be kept in mind. Furthermore, in disuse atrophied muscles, or in spastic muscles of children with cerebral palsy, this 6:1 ratio of titin filaments vs. half myosin, becomes smaller and might be as low as 3:1 [38, 39]. However, like for the cross-bridge elasticity, titin elasticity is not in series with the entire muscle.
A single myofibril consists of sarcomeres arranged in series with one another. That is, each sarcomere transmits the same force at any given time as the next one. Therefore, the instantaneous forces measured at the end of a myofibril are the same as the instantaneous forces transmitted by each sarcomere in that myofibril (which is the primary reason why single myofibril mechanical experiments are so powerful). Similarly, the Z-band in a single myofibril is in series with its neighbouring sarcomeres, and any force transmitted across the Z-band will be the same as the force of the sarcomeres. However, multiple myofibrils in a muscle/fibre are structurally arranged in parallel, and sarcomeres of neighbouring myofibrils are connected by various structural proteins (desmin being the most acknowledged), and thus, the Z-bands in neighbouring myofibrils and fibres are not in series with each other. As a consequence of this highly connected and integrated arrangement of sarcomeres in myofibrils and fibres, the system of sarcomeres is mathematically redundant, and it is impossible to determine the force in a given sarcomere of a muscle, even when the muscle force and the target sarcomere length are known.
Tendon and aponeuroses
Returning to the discussion of series elasticity in entire muscles, tendons and aponeuroses have often been treated, implicitly or explicitly, as the series elastic elements of skeletal muscles. The argument frequently made is that since tendon and aponeurosis are structurally in series with the muscle fibres, as suggested in the schematic drawing by Ettema and Huijing (1990) [40] (Fig. 6), they are also mechanically in series. This thinking is exemplified by measurements of aponeuroses and tendon elongations, relating these elongations to muscle force, and then assuming that there is a relationship between muscle force and tendon/aponeurosis length that is governed solely by the constitutive equation of the aponeurotic/tendinous tissue. While this thinking is justified for the free tendon of a muscle [14, 23, 41], it is not for the internal aponeuroses of muscles, as has frequently been done [e.g .[42, 43]].
Implicitly, aponeuroses tissues have been assumed to be series elastic elements of muscles in studies where “series elasticity” is defined/obtained by subtracting fibre/fascicle length from the entire muscle-tendon unit length (e.g. [17, 21]. It has been shown theoretically that forces in aponeuroses are not the same as in the free muscle tendon [e.g.43], and that the pressure and shear rigidity of muscles play a crucial role in the relationship between tendon and aponeurosis forces [e.g .[24, 44]]. However, before conducting a detailed theoretical analysis of the relationship between tendon and aponeuroses forces, and sharing experimental observations of directly measured muscle forces and aponeurosis deformations, we would like to define what we mean by the (free) tendon and (inner) aponeuroses of muscles.
For simplicity, but without loss of generality, let us assume we are dealing with a unipennate muscle, for example, the cat medial gastrocnemius muscle (Fig. 7). The free tendon of the muscle is defined as the connective tissue, tendinous material that is external to the muscle belly, as indicated in Fig. (7). The cat medial gastrocnemius has two aponeuroses, one located proximally and the other distally on the muscle (Fig. 7). They are composed of connective tissues to which the muscle fibres insert. The aponeuroses, by virtue of their location, are exposed to the pressure and shear forces exerted by the muscle upon contraction, while the tendon is not. Pressure and shear forces need to be considered when calculating the forces transmitted by aponeuroses, while the tendon simply transmits whatever force is produced by the muscle’s contractile and passive structures [e.g.23]. Therefore, the tendon can safely be considered mechanically “in series” with the muscle, while the aponeuroses cannot.
Although illustrated on the example of a unipennate muscle, the general statement that the free tendon is always mechanically in series with the contractile part of the muscle, the muscle belly, is correct in general for fusiform and multi-pennate muscles. Similarly, aponeuroses, as defined above for a unipennate muscle, are never mechanically in series with the free tendon or the contractile part of the muscle, and in contrast to the free tendon, will always have a force that varies along its length. Again, this statement is generally correct for any muscle that has aponeuroses embedded within the contractile part of the muscle.
Aponeuroses are sometimes also referred to as the pearly white fibrous tissues that take the place of tendons in flat muscles having a wide area of attachment. For muscles with such wide areas of attachment, for example in the human abdominal area, the hand and feet, aponeuroses may lie outside the muscles and may be arranged in series with the contractile elements of muscles. However, for the sake of clarity (and also for its common use in biomechanics research), we consider aponeuroses here as shown in Fig. (7); that is, aponeuroses are internal to the muscle with the contractile fibres inserting into them.
Why aponeuroses cannot be considered “in series” with either the free tendon or the muscle: theoretical considerations
Let us assume we have a muscle with contractile fibres, purely elastic aponeuroses (A), and a purely elastic tendon (T) (Fig. 8a) [25]. We further assume that the muscle is incompressible. Incompressibility is enforced by an incompressible, elastic material (C) inside the borders formed by the aponeuroses and the contractile fibres. For this simple representation of a muscle, we can calculate the forces in T and A at any time for an assumed contraction/force of the fibres. Let us further assume we stretch the muscle first passively until a certain amount of passive force is developed, then activate the muscle isometrically, shorten it back to its original length while activated, and finally deactivate the muscle, so it has reached its initial passive configuration (Fig. 8b). When going through this dynamic contraction, the forces in the aponeurosis are always smaller than in the tendon, and the aponeurosis forces change when the elasticity, specifically the shear modulus of the incompressible muscle (C – Fig. 8a), is changed (not shown). When assuming the shear modulus to be zero (which is unrealistic for muscle tissue), the hysteresis observed in Fig. (8b) for the stretch-shortening cycle disappears (not shown). But even for this extreme case, the tendon and aponeurosis forces are not the same [24]. Furthermore, the result obtained here is not exclusive to an incompressible muscle, but would also be obtained with a compressible material.
The theoretical example discussed above has been taken from one of our previous publications, and details of the calculations and the model can be obtained from [24]. We conclude that for this representation of a unipennate muscle, the force in the aponeurosis is not related in a simple way to the force in the tendon; i.e., the muscle force. Even though in the example we only discuss conditions of muscle activation and deactivation, and an isometric and concentric contraction, the findings are independent of the contractile conditions and are also correct for an eccentric contraction or a stretch-shortening cycle.
In a further refinement of the model shown above, we can divide the muscle into multiple panels separated by contractile fibres (Fig. 9a), and repeat the stretch-shortening cycle from the previous example. When doing so, it can be shown that the aponeurosis force becomes smaller when going from the “attached” end (panel 1–3) to the “free” end (panel 7–9 – Fig. 9b – bottom aponeurosis). This result is consistent with the observed “thinning” of aponeuroses from the “attached” to the “free” end, as for example illustrated in the medial gastrocnemius of the cat (i.e. a thinning of the medial aponeurosis from the left “attached” to the right “free” end – Fig. 7). Furthermore, observe that the aponeurosis forces can be negative (corresponding to a shortening of the aponeurosis) in the presence of positive tendon forces (Fig. 9b – panel 7–9). A shortening of aponeurosis segments upon muscle activation, and associated increase in force, has been observed experimentally [44,45etc.]. For the details of this previously published analysis, please refer to Epstein et al. [24].
We conclude from these theoretical considerations that aponeuroses forces are not the same as tendon forces, that aponeuroses forces are not in a constant ratio to tendon forces, that aponeuroses forces are smaller than tendon forces, and that they can be negative (aponeuroses shortening) in the presence of positive tendon forces. Furthermore, aponeuroses forces vary along the aponeuroses and tend to be greater at the “attached” compared to the “free” end, in agreement with the generally observed tapering of the thickness of aponeuroses from the “attached” to the “free” end.
Why aponeuroses cannot be considered “in series” with either the free tendon or the muscle: experimental observations
Even though the muscle models developed above contain the essential elements of a real muscle: contractile fibres, “elastic” aponeuroses, an incompressible muscle substance, and an “elastic” tendon, its predictions might not reflect a real muscle. In particular, one might argue that there is no direct measurement of aponeurosis forces, and indeed, to our best knowledge, such forces have never been measured in an intact muscle. However, when measuring aponeuroses elongations for a variety of conditions, observations have been made that are incompatible with an “in series” arrangement of aponeuroses with either tendons or with muscle fibres.
For example, Lieber et al. [45] measured aponeuroses elongations as a function of tendon force in frog semitendinosus for passive and active muscle conditions. They found that aponeurosis elongations were significantly greater in the passive compared to the active muscle (Fig. 10). At corresponding force levels (50% of the maximal isometric force at optimal length), they found aponeurosis strains of about 5 and 23% for the active and passive conditions, respectively (Fig. 10). They concluded from this result that an “active contraction actually altered aponeurosis material properties”. It seems unlikely that a non-contractile material, like the aponeurosis of the frog semitendinosus muscle, could change its material properties upon muscle activation. Rather, one would suspect that the material properties of the aponeurosis remained the same but the forces acting on the aponeurosis, for a given muscle force, differ between the active and passive conditions, and were not related in a simple way to the tendon force. The error made in the interpretation by Lieber et al. (2000) was that they assumed that the tendon force, which they measured directly, was the same as the aponeuroses force, independent of the muscle length and independent of the muscle’s active state. Activation in muscles is associated with increases in internal pressure and changes in stiffness, including shear stiffness [23, 24, 44, 47, 48], thus assuming that the tendon force is equivalent to the aponeurosis force, and implying material properties based on such thinking, will lead to erroneous interpretations of aponeurosis function, mechanical properties, and energetic results. The experimental observations by Lieber et al. [45] are captured generically in our theoretical model above (Fig. 9b), where the relationship between tendon force and aponeurosis force changes when the muscle is activated, and an increase in tendon force with activation was associated with a decrease in aponeurosis force and aponeurosis length, agreeing with the experimental observations by Lieber et al. [45].
Significantly shorter aponeurosis length in active compared to passive muscle have been published prior to the Lieber et al. [45] paper. For example, Zuurbier et al. [46] reported that “aponeurosis length as a function of aponeurosis force was significantly shorter in the active compared to the passive … condition”, for the proximal aponeurosis of the unipennate medial gastrocnemius muscle of the rat. This statement reflects their observation of aponeurosis length in active and passive muscle for corresponding muscle forces, and they relate (again erroneously) the tendon force to the aponeurosis force, not accounting for the fact that the relationship between tendon and aponeurosis force changes with activation due to the increase in muscle pressure and shear stiffness upon muscle activation. This does not diminish their observation, merely the interpretation of their results, as activation of a muscle, and associated increase in tendon force, can lead to decreased aponeuroses forces, as shown in our theoretical considerations above (Fig. 9b).
Magnusson et al. [42] were among the first to claim that they quantified the mechanical properties of aponeuroses in intact human skeletal muscles. Their highly cited paper represents a careful attempt of quantifying the stiffness and Young’s modulus of human medial gastrocnemius tendon and aponeurosis. However, they estimated the aponeurosis force “…. by dividing the externally measured moment by the tendon moment arm.” While this is perfectly acceptable for tendon/muscle force estimates, this approach is not appropriate for estimating the variable forces in the aponeurosis, as it (typically vastly) overestimates the aponeurosis forces. They found similar elongations for the proximal and distal segments of the medial gastrocnemius aponeurosis and concluded that “…. the stiffness was similar for the two regions.” Their conclusion (again) is based on the assumption that equal elongation (of the distal and proximal aponeurosis segments) was associated with equal forces acting on these two segments, which is incorrect as the aponeurosis forces vary along the aponeurosis (and thus are likely substantially different for the distal and proximal segments), and the aponeurosis forces are not equivalent to the muscle/tendon force. Their calculation of aponeurosis stiffness, thus, is an (likely vast) overestimation of the true value, which is confirmed in studies where the true (isolated) aponeurosis material properties have been compared to the aponeurosis elongations and equivalent tendon forces in intact muscles [46]. Furthermore, their conclusion that proximal and distal aponeurosis stiffness are the same, is probably not correct. Rather, the similar elongations of these two segments likely reflects a continuous change in the stiffness of the aponeurosis along its length that matches the changing in vivo forces acting along the aponeurosis in such a manner that aponeurosis strains are “constant” along its length.
In the above examples, material properties of intact aponeurosis have been implied from the elongations of the aponeuroses and the corresponding forces in the muscles/tendons. The implicit assumption in these examples is that material properties, such as stiffness or the Young’s modulus, can be derived by assuming that the forces acting on the aponeuroses are those measured at the distal end of tendons. From a mechanical point of view this is incorrect, as shown in the theoretical considerations above. It leads (typically) to overestimations of the actual aponeurosis stiffness.
Aside from ill-fated attempts to measure the material properties of aponeurosis in intact human skeletal muscles [e.g.41], another frequently used mechanical concept is that of the storage and release of mechanical energy in muscle series elastic elements. This topic will be discussed in the following paragraphs.
Storage and release of energy in “series elastic” muscle elements
Many movements in animals, including humans, are cyclic in nature and are associated with a stretch-shortening cycle of the muscle-tendon unit complex [49]. It has been argued that many muscles are built to take mechanical and energetic advantage of the stretch-shortening cycle through their series elastic elements (i) by affecting the rate of change in the contractile elements of the muscle [1], (ii) by storing and releasing potential energy in the series elastic elements [20]; and (iii) by increasing force/work in the shortening phase of the stretch-shortening cycle through mechanisms of residual force enhancement [18, 22].
Muscle series elasticity, in this context, has frequently been defined, implicitly or explicitly, as the elements “obtained by subtracting muscle fiber length from origin to insertion distance” [e.g.20]. This definition has been applied to measure elastic energy storage and release in intact muscles of freely moving animals. For example, Roberts et al. [17] calculated the tendon energy recovery in the lateral gastrocnemius of Turkeys using the muscle force (measured at the calcified tendon) and the tendon/aponeurosis stiffness (calculated by the elongation of tendon and aponeurosis in isometric contractions and assuming muscle/tendon force to be equivalent to the variable forces acting along the aponeurosis). This procedure leads to overestimates of the actual aponeurosis stiffness as muscle pressure and shear forces created upon muscle activation are neglected, resulting in overestimations of the energy recovered by the aponeurosis.
We measured the force, muscle-tendon unit length, and a mid-belly fascicle length in the cat medial gastrocnemius muscle for a variety of locomotor conditions, including walking, trotting, galloping, and jumping (Fig. 11a). In analogy with van Ingen Schenau et al. [21] and Roberts et al. [17], we then subtracted the instantaneous fascicle lengths from the instantaneous muscle tendon unit length (Fig. 11b), and plotted this difference (assumed to represent the series elastic element of muscle) against the muscle force measured at the distal end of the gastrocnemius tendon (Fig. 11c). When doing this, we consistently observed a positive work loop for the assumed series elasticity. However, since a (visco-) elastic element can at best release the same amount of energy that was initially stored in it, and thus cannot create a positive work loop as shown in Fig. (11C), we must conclude that the muscle/tendon force measured is not related in a direct and simplistic manner to the aponeurosis elements of the muscle. In other words, subtracting the fascicle length from the total muscle-tendon unit length (and accounting for the angle of pennation) does not provide a series elastic element in the mechanical sense [53]. The forces acting on the aponeurosis are not related in a simple manner to the muscle/tendon forces. Assuming that they are can give results of work/energy production that are thermodynamically not possible. In order to demonstrate that aponeurosis elongations are not related to muscle/tendon force, we also measured segmental elongations of the lateral aponeurosis of the cat medial gastrocnemius muscle for multiple step cycles and various locomotor conditions. In all cases, the segmental aponeurosis elongations were not related to the muscle/tendon force in a unique manner (Fig. 12). Rather, the range of aponeurosis elongations was similar for the recovery phase of the step cycle (where muscle forces were small), and the active force producing stance phase (where forces were high). Note also that all “force-elongation loops” for the stance phase of locomotion in this example are counter-clockwise, that is, if we assumed that the aponeurosis was “in series” with the muscle/tendon (where the force was measured), we would obtain positive work loops, once more illustrating that the aponeurosis length changes cannot be related directly to the tendon force by assuming a mechanical “in series” arrangement.
