One of the first books addressing muscle as well as whole body fatigue was published at the beginning of the 20th century Mosso, and a multitude of research has followed since, see Gandevia for a thorough review. Commonly, muscle fatigue, i. The main part ca. Due to the broad scientific interest, many metabolic factors of force decay have been identified, for example adenosine triphosphate ATP consumption Sieck et al.
A summarizing flowchart can be found in Abbiss and Laursen , Figure 9. Although the role of some factors is controversial, e. Despite vast physiological findings, biomechanical models, including force decay on the time scale of seconds, focus on rather phenomenological descriptions of typical force decay patterns Liu et al.
Here, we develop a mathematically straightforward, yet physiology-based model that is able to explain the majority of early force decay while being computationally manageable. As validation, isometric contraction data at different muscle lengths from rabbit M. Our aim is to provide a basic enhancement of the well-known realm of Hill-type muscle models by describing bio-chemical kinetics that can be altered or extended, depending on the experimental resolution at hand.
I , The animals were anesthetized with Bupivacain Jenapharm, 1 ml, 0. In brief, the muscle to be examined was freed from its surrounding tissues and the rabbit was fixed with bilateral bone pins to a stereotaxic frame. The distal tendon of the muscle was attached horizontally to a muscle lever system Aurora Scientific B-LR. Because the muscle performance depends on temperature, the animal was heated during the complete experimental procedure using a heating pad Harvard Apparatus, For this study, we particularly investigated the performance of M.
Thus, for GAS, there are two additional isometric measurements at the beginning of the descending limb of the force length relation and mm muscle length. One additional measurement for the GAS at mm was conducted with 1, ms of stimulation. In this section, we work out an enhancement of an existing Hill-type muscle model see Appendix A , which describes the dynamics of the activation and contraction of muscle fiber material. The model is enhanced by an additional process phosphate dynamics , which is formulated in terms of a differential-algebraic equation.
Beforehand, in literature, we identified five distinguishable approaches of modeling short-term force decay patterns in skeletal muscle as a consequence of ongoing stimulation.
Arranged with respect to increasing physiological verisimilitude these are. Reviews of applying this method can be found in El ahrache et al. We aim at predicting the force decay of a biomechanical muscle computer model as response to a given muscle stimulus forward dynamic simulation.
Opposing, in approach 1 the force level is considered given and the stimulus is to be estimated in order to match the observed force decay inverse dynamic simulation.
Thus, here approach 1 can be ruled out. As well can approach 4 , because it does not incorporate inner-muscular properties. An advantage of approaches 2 and 3 is the computational and epistemological simplicity, whereas a disadvantage is the missing physiological interpretability of the model parameters. In contrast, the physiological backbone of approach 5 is beyond dispute, however, so is its computational complexity.
Our aim is to combine the strengths of approaches 2 , 3 , and 5 , simultaneously neglecting their weaknesses, by developing a computationally cheap model based on physiological knowledge cf. As mentioned before as well as in Shorten et al. Therefore, a single, linear ODE is added to the existing model see Equation 3 , describing the change in [ATP] and [P i ] within the sarcoplasmatic reticulum and its interplay with cross-bridge mechanics, based on known reaction kinetics, for a detailed description see Appendix B.
Following the iconic paper of Lymn and Taylor , ATP binds to the attached myosin head to allow its release from the actin filament.
The energy release in the hydrolysis is then used to re-configure the myosin head for re-attachment in order to undergo another power stroke. Although many physiological details remain uncertain, the core element of the cross-bridge cycle can be represented by a simple pseudo-first order equilibrium.
Herein, k con and k hyd denote the rates of condensation catalyzed by mitochondrial ATP synthase and hydrolysis catalyzed by ATPase within the myosin head , respectively, with the corresponding equilibrium constant. Note that although this dynamic is motivated by physiological considerations, it constitutes for a vastly oversimplified description of the bio-chemical processes underlying the full cross-bridge cycle.
However, Equation 3 can be enhanced if explicit measurements e. Appendix A. The condensation rate k con is assumed to be a constant. It has been shown that, within resting muscle fibers, there is ATP consumption, but far less than in activated ones Hilber et al. Summarizing, the following ansatz is obtained:. Note that the alterations of the rate constants with respect to external conditions, such as fiber type, pH value, or temperature, are not yet incorporated cf.
It is further assumed that in a sufficiently activated fiber there is enough calcium to simultaneously enable the ATPase within the myosin head De La Cruz and Ostap, ; MacIntosh et al.
Note that magnesium, which serves a co-factor, is not yet incorporated within our model, and assumed to be sufficiently available. Equations 22 — 24 as well as Carlson and Wilkie , Hilber et al. Appendix A from now on refers to the relative amount of force-producing cross-bridges formed at the available active sites, i.
Compare Appendix A. Certainly, our introduced model enhancement is far from capturing every physiological cause of force decay in skeletal muscle. Yet, given that the full mechanisms of the remaining factors are at best partially known MacIntosh and Rassier, ; Allen et al. Table 1 summarizes the parameters necessary to describe the full phosphate kinetics as explicitly explained in Appendix B. Table 1. Overview of the nine new model parameters, necessary to describe phosphate dynamics cf.
Appendix B. Following Figure 1 , each experiment had a runtime of 1. The control function is zero, i. Within the last 0. Valid estimations for static parameters, such as slack lengths and maximum force, are possible at the equilibrium states for zero and full stimulation, respectively.
Figure 1. Models were fit to data, where different lengths are indicated by different colors as labeled in the top row. Parameter values are summarized in Table 2. In order to assess the effect of the new state introduced in Equations 3 and 37 , as well as to increase validity, a comparative parameter estimation was conducted; once with the basic model as presented in Appendix A and once with the enhanced model. The formalization of the underlying optimization procedure can be found in Appendix C.
Table 2 lists the obtained parameter values as well as the pre-set boundaries see next section 4. Figure 1 shows the associated force-time model outputs.
These force-time results, obtained by least-square fits, are compared with respect to residues for both L 1 absolute and L 2 squared distances for the purpose of showing consistency throughout the model variants. Table 2. Bounds and optimized parameter values resulting from the estimation process, described in Appendix C. To further support the validity of the exhaustion model, we compared the model prediction with one additionally available experimental data set, namely an isometric contraction of GAS with longer stimulation time 1.
Thus, the model is able to predict isometric experiments with prolonged stimulation times. However, this statement only holds true for a single additional isometric contraction. Further long-stimulation experiments, also with PLA, should be conducted in order to strengthen this claim. As solely determined by isometric contractions, the herein obtained model parameters should in any case be cautiously considered when aiming at dynamic simulations with high shortening velocities. Figure 2.
Stimulation duration is 1. For more details see Figure A1 in Appendix B. It has been further possible to restrict the search area for the algorithm, i.
Naturally, deviations in parameter values for these different muscles occurred in slack lengths as well as forces. Further, a challenge occurred due to the redundancy in serial and parallel elasticities.
As is apparent from Equation 9 , serial and parallel elastic element are assumed to operate in series. Further, MTU length is held constant static in isometric experiments. In consequence, the individual contributions of parallel and elastic elasticities of the muscle in dynamic situations can not be clearly resolved. Hence, restrictions in either parameter set might influence the convergence property of the other and the herein presented bounds have to be considered with caution.
For the rest of the parameters, there is no systematic change detectable. Activation dynamics parameters remain almost constant. Figures 1E,F. This observation is in accordance with experiments Phillips et al. The optimized parameters are in good agreement with prior experiments Siebert et al. Hence, the shape of the descending branch of the force-length relationship as well as the results for the non-linear to linear transition of the SEE have generally to be taken with caution cf.
Figure 4. Mainly this difference is explained by A rel, 0 being 3. The force-length relation of GAS shows a broad plateau-like region with steep ascending and descending limbs, whereas PLA shows a shallow ascending limb see Figure 4. We already mentioned the lack of trustfulness of the descending limb's shape.
However, the difference in the force-length relation might additionally be influenced by pennation of the fibers. As the fiber shortens, its pennation angle increases Drazan et al. At higher forces and lower velocities, the force-length characteristic of the fiber is then transformed geared to an altered force-length characteristic of the whole muscle Azizi et al.
Figure 1 shows the force-time measurements compared to the model evaluations for GAS left column and PLA right column. The residues in the optimized least-square sense L 2 were for comparability scaled to data points and account for In order to give an alternative error estimate, independent of the objective function, the absolute L 1 deviation of model and measurement scaled to a single data point was calculated. These values are 1. In Figure 1B the aforementioned effect of the strong damper in terms of an early force overshoot can be observed, especially for short lengths.
Figure 3 shows the time courses of the sensitivity for all model parameters for two exemplary cases, one for a long GAS muscle and one for a short PLA. The local sensitivity values see again Appendix C indicate how small changes in the parameter value would affect the model output at the corresponding time instances.
Altogether, Figure 3 may help the reader to estimate the influence of parameters across time and magnitude as well as to acknowledge the setting of bounds in Table 2. Figure 3. Large absolute sensitivities indicate the local importance of parameter value accuracy.
For further explanation see text. On the one hand, they are properly determinable, but on the other hand they have to be known with high accuracy cf.
Likewise, switches in sign of the sensitivities can be observed for all SEE and most PEE parameters, indicating the different influences in passive and active muscle states. For the shorter muscle Figure 3B , parameters describing the ascending branch are of importance, as would have been expected.
The influence of eccentric parameters is visible for the regions of force decay, be it due to phosphate accumulation or end of stimulation. The influence of Hill parameters on the model output is clearly visible after start and end of stimulation. The former source omitted a functional dependency, whereas the latter gave a decreasing linear fit, although the shape of the altered sarcomere force-length relation was prominently shaped as described.
Figure 4 shows the force rates of our data, compared to the theoretically predicted relation. Color encoding is the same as in Figure 1. Data were smoothed by a moving average filter with 40 ms width for clarity of depiction, but not for calculations. Inlay shows the mean force rates solid lines in the 0. B Experimental force decay rates vs. CE length at different MTU lengths colored circles as extracted from A in comparison to theoretical modified CE force-length relation Equation 8, black line.
An additional data point diamond was taken from another GAS experiment see Figure 2. To predict the decay in muscle force during isometric contractions, we have added a third ODE Equation 3 , which describes the dynamics of [ATP], to an existing Hill-type muscle model which consisted of already two differential and several algebraic equations see Appendix A. Hill's hyperbolic force-velocity relation Hill, , empirically found for muscle fibers, is in the core of Hill-type models.
Therefore, such models are of empirical, macroscopic, and consequently reduced character. They often show deficiencies in their capabilities of reproducing the wealth of physiological and experimental conditions. For compensating one such deficiency, we have herein followed the methodological path of step-wise enhancing a Hill-type model. Furthermore, because the Hill relation originates in both mechanics and thermodynamics, it may already well represent basic properties of active muscle tissue during dynamic interactions with other body tissues.
We therefore conclude that the methodological path chosen has a sound basis. In this study, we have now enhanced the activation dynamics part of our initial model. Hatze had thoroughly described the effect of a neural impulse on calcium dynamics up to the binding to troponin and subsequent clearance of tropomyosin from the actin helix cf.
However, he assumed that each possible cross-bridge would instantaneously form and generate force in the wake of troponin activation , thereby ignoring any further mechanisms and processes delaying or interfering with a cross-bridge's force generation. For example, as already motivated and sketched in Figure 5 in more detail, ATP hydrolysis reaction kinetics and phosphate accumulation are well-known mechanisms to have a bearing on cross-bridge dynamics.
This new state can likewise be interpreted as an attachment-to-detachment ratio in the sense of Huxley Figure 5. Schematic cross-bridge Lymn-Taylor cycle For their explanations, see text. Although most processes are easily reversible, for the sake of clarity only one direction is depicted.
It does not seem expedient to aim here for a higher structural and parametric resolution of the underlying processes within our model of activation dynamics, given and solely based on our experimental set-up isometric whole muscle contractions.
Yet, the chosen methodological path of step-wise model enhancement by adding structure-based equations is perfectly designed to do so, namely, to factor in, on the basis of experiments, processes like receptor-ligand binding processes and state transitions Bagshaw and Trentham, ; Trentham et al. When trying to understand which process is effected by P i kinetics, it is required to implement it in an even further enhanced model of muscle activation-contraction dynamics.
The non-linear interactions of essential processes and mechanisms can only be understood in a cause-effect sense by mathematically formulating them in terms of at least algebraic but usually even differential equations and coupling these to state-of-the-art models formulated the same way. Eventually, this also applies to the very core of Hill-type models of muscle contraction: Hill's phenomenological, steady-state relation must then be replaced by a mechanistic model that represents both the basic force interactions and the thermodynamics on the cross-bridge level, formulated in terms of structure-based physical properties cf.
ODEs are used as standard method to model a characteristic evolution of an independent variable, which also depends on its own derivatives. Often, the time evolution of such an independent variable is studied in search for characteristic time constants of the modeled systems.
The main Equation 3 of this contribution represents the time evolution of [ATP] or [P i ] with the characteristic time constants for condensation k con and hydrolysis k hyd. Figures 1 , 4. As described in the methods section 3 , the constants themselves describe a system characteristic, which represents a snapshot of even more detailed underlying processes.
Typical underlying processes for ATP hydrolysis k hyd are the filling and draining of the ATP reservoirs including the actual type of flow during filling and draining: laminar or turbulent, the mixing of materials within the reservoir, and the residence time of the ATP on the myosin, i. Additionally, recent discoveries hint at two different types of ATP-myosin binding Amrute-Nayak et al.
The authors proposed that each myosin head has one site for ATP switching between two conformers Tesi et al. Admittedly, our model provides a rather phenomenological description of phosphate-induced force decay.
Table 1 , were a priori fixed according to literature. Force decay along with phosphate accumulation in the sarcoplasma is surely the physiological process that has a functional effect in natural contraction conditions. Yet, phosphate dilution can have the reciprocal impact of increasing isometric force Phillips et al. Phosphate dynamics is thus rather a boundary condition for the Lymn-Taylor cycle than strongly and non-linearly interacting with cross-bridge dynamics.
Therefore, the choice of our methodological path see section 5. Other chemical factors that are known to have an effect on cross-bridge cycling, such as magnesium contributions, calcium precipitation, or glycolysis, have so far not been incorporated into our enhanced model of activation dynamics. Figure 5 gives a schematic overview of this cycle in accordance with our current interpretations cf. Our scheme is further consistent with crystallography measurements of geometric configurations conformational states of the myosin S1 part Geeves and Holmes, , pp.
Hence, myosin heads might be bound in a rigor formation, cf. The gray continuation to the left of the actin filament accounts for the post-power-stroke translation after one full cycle 1-…-7 , i. This process might be slowed down or hindered by higher [P i ] in the surrounding Tesi et al. Just as step 5, this process can also be interfered with by a higher [P i ] Kerrick and Xu, Note that ATP hydrolysis is commonly assumed to take place before re-attachment, but might also take place afterwards Tonomura et al.
In all, the presented model Equation 3 subsumes several even more detailed processes by assuming k hyd and k con. It remains open to integrate these into the equation, additionally. As was shown in this contribution, prerequisites are data of very good quality and a first guess of the system dynamics.
Until then, the presented approach integrates phenomenologically-based molecular kinetics into macroscopic muscle models, enhancing them tremendously. The presented model considers the simulation of the early phase of fatigue. The first component is associated with elevated [P i ] due to the breakdown of PCr. The effect of elevated [P i ] on force might decrease with increasing temperature to physiological relevant conditions Debold et al.
Nocella et al. Focusing on the time course of force decay Nocella et al. Furthermore, they showed that the decrease in tetanic force mainly results from depressing the individual cross-bridge force and accelerated cross-bridge kinetics. Nelson and Fitts observed at a pH of 6. The role of acidosis in acute fatigue remains controversial and a major unresolved issue is whether the force-reducing effects of elevated [P i ] in fatigue are amplified by the concomitant acidosis Cheng et al.
In our experiments, we can almost exclude the possibility that the pH decreased to values of 6. Thus, from an experimental point of view, the herein applied model-based parameter estimation fitting method Wagner et al.
In general, their results are in good agreement with muscle model parameters determined in the present study. Maximum shortening velocity was overestimated GAS: factor 1.
This can be explained by the limited parameter range available in the fitting method. Shortening velocity of the contractile component reaches maximally 0. Thus, uncertainty in parameter estimation is higher for v max and P max compared to classic methods, in which v max can be approximated by contractions against low loads or unloaded contractions Edman, In contrast, parameters of the force-length relation are almost similar between classic and fitting method.
Muscular fatigue plays an important role in the assessment of work-place ergonomics in order to accurately predict demands on workers with respect to the muscular forces required for their work tasks. To this end, endurance time is a measure used to characterize muscular loading situations.
It was introduced to quantify the time a subject can hold a specific load by muscular contraction Rohmert, and has been used to study many different postures and muscles e. In general, measurements reveal that endurance time is shorter for higher muscular forces. Furthermore, experiments show that below a certain load, endurance time becomes very long indicating a muscular load where the normal ATP resynthesis rate is sufficient to compensate the static energy requirement for the muscles.
The model presented here also shows this behavior Figure 6 , although we prefer the term exhaustion time when talking about the inability to maintain a certain, length-dependent force. For stimulation values around 0. Figure 6. Colors and line-styles in both sub-figures are coherent, indicating MTU lengths [as specified in A ] and stimulation levels [as specified in B ]. Two observations should be highlighted. First, the force level in B at which theoretically infinite exhaustion time occurs settles at around 0.
Second, the force at the longest muscle length [blue line in B ] settles at significantly higher values, which is due to passive PEE forces that were herein not modeled to show any exhaustion.
In addition, the model predicts a length dependence of the exhaustion time Figure 7. One core assumption of the model is that the hydrolysis rate decreases with increasing muscle fiber length see Equation 4. This assumption was derived from the experimental observation that the force decay due to exhaustion scales with muscle length, see experimental data in Figure 4.
This characteristic is, thus, immediately reproduced by the model, as shown in Figure 4 and Equation 8. More precisely, our model strongly suggests that the mechanism s that govern the exhaustion process are the same as those governing the length dependency of ATP hydrolysis. Figure 7. A Simulated force-length-exhaustion time diagram for variations in MTU lengths and stimulation levels as given in Figure 6.
B—D Contour plots through the three different planes in A. In B , the force-length diagram for various exhaustion times is shown. The active and passive force-length characteristic of the CE is clearly visible.
In C , the force-exhaustion time courses for various CE length are displayed. The longer the CE, the longer the exhaustion time. The typical exponential or hyperbolic characteristic Rohmert, ; Frey-Law and Avin, is visible.
D Shows CE length-exhaustion time curves for various force levels. The higher the force, the shorter the exhaustion time at the same length. For longer muscles, the passive force determines the exhaustion time.
This is an issue also discussed in the literature with respect to endurance time. Interestingly, the findings are ambiguous. Some experiments also found that fatiguability is reduced for longer muscle lengths, i. Figure 7D Sacco et al. Another important source of energy is glycogen which is stored in the muscle and therefore readily accessible. Glycogen can be broken down anaerobically and used in this way would last only min. The sources of energy so far mentioned do not require oxygen anaerobic pathways and are the only sources of energy in very brief maximal activities or when the oxygen supply is not available.
Alternatively glycogen can be broken down aerobically. Although this process is markedly slower than the anaerobic breakdown, it produces enough ATP to keep the muscle contracting at a near maximal rate for 30—60 min.
ATP can also be produced from the aerobic metabolism of fat stores, which are very large but can only be metabolised relatively slowly. Once the glycogen stores are depleted, muscles must rely on fat metabolism. The running speed during a m sprint is much higher than during longer runs; the short duration means that fatigue is less of a problem. However, even during short sprints there is some fatigue and the maximum running speed in a m sprint occurs after about 60 m.
Why does the running speed decline during such a short sprint? The answer to this is not entirely clear. It is, however, clear that lactic acid has little to do with it. Relatively little lactic acid is formed during such a short activity. Instead most of the energy comes from breakdown of phosphocreatine. Breakdown of phosphocreatine consumes hydrogen ions so the net effect is that myoplasmic pH is not significantly altered during the sprint. Thus phosphate ion accumulation is probably an important contributor to fatigue during a m sprint.
Without a rapid ATP supply, ADP will accumulate and this will slow down crossbridge cycling and ion pumping and hence decrease the power output of the muscle. In recent years it has become increasingly popular for sprinters to ingest enormous amounts of creatine and this seems to have a small beneficial effect on the performance in sprint running — but for elite athletes even a very small improvement can make the difference between winning and losing.
Excessive creatine intake results in increased levels of phosphocreatine in muscles so that the period of close to maximal performance is prolonged. A continuous maximal contraction is needed when lifting something very heavy, like a piano. Everyone will be aware of how rapidly fatigue can set in during such activities. In this situation the muscle machinery is going at full speed and energy is consumed at a rapid rate.
In addition, the blood flow to the active muscle s is stopped during maximal contractions so that no delivery of oxygen to support muscle contraction or removal of metabolites or ions will occur. Thus severe fatigue develops within seconds and the muscle becomes rapidly weaker. Changes of the ionic distribution over the cell membrane probably contribute to this type of fatigue.
Each action potential is associated with entry of sodium ions into the cell and exit of potassium ions from the cell; consequently potassium ions tend to accumulate outside of the fibres and this results in depolarization and impaired electrical activation of muscle cells. This extracellular accumulation of potassium is likely to be larger in the narrow lumen of the t-tubules see Fig. Figure 2. Single muscle fibre approach to studies of muscle fatigue.
Panel A. Single fibre dissected from the mouse flexor brevis muscle. Note metal clips attach to the tendons provide connection to the force transducer. Panel B. Fluorescent image of a single fibre and a microelectrode just after the fibre has been pressure-injected with the calcium sensitive indicator indo Panel C.
Force record from a single fibre subjected to repeated tetani. Panel D. Intracellular calcium records from various stages of the force record in Panel C. C, Gonzalez-Serratos, H. Muscle Res. Cell Motil. Godt, R. Gordon, A. Physiol Rev. Helander, I. Inesi, G. Roles of back-inhibition, leakage, and slippage of the calcium pump. Kabbara, A. Lamb, G. Laver, D. Ma, J. M, Campbell, K. Science : 99— Millar, N. A steady-state and transient kinetic study.
Owen, V. Pate, E. Phillips, S. C, and Kushmerick, M. Posterino, G. L, and Lamb, G. Ranatunga, K.
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