Effect of strain rates on the mechanical response of whole muscle bundle

Muscle injuries frequently happen during sports activities and exercise, which could have serious consequences if not diagnosed and treated promptly. This research aims to investigate the quasi-static and dynamic responses of over 30 fresh frog semitendinosus muscles utilizing Split Hopkinson Pressure Bars (SHPB) and a material testing system under strain rates between 0.001 ~ 200 s−1. To accommodate the special shape of muscle–tendon-bone samples, PLA clampers were produced by the 3D printer to properly hold and prevent slipping during the testing process. The mechanical characteristics of the whole muscle bundle, including Young’s modulus and stress–strain curve, are illustrated at various strain rates. The findings showed that the muscle properties were sensitive to strain rate when under passive deformation. Both maximum stress and Young’s modulus increased with the rise of strain rate, and modulus at 200 s−1 can be as high as 10 times compared with quasi-static conditions.


Introduction
Hamstring muscle strain injury is one of the most common sports injuries. It accounted for 11% of the injuries in preseason training and 12% of the injuries in competition soccer seasons in England. Hamstring muscle strain injury can be caused by shortened optimum muscle length, lack of muscle flexibility, hamstring strength imbalance, insufficient warmup, fatigue, lower back injury, and increased muscle neural tension [1]. The treatment of hamstring muscle injuries, in general, includes three main phases: the resolution of inflammation, angiogenesis, and the repair of the muscle tissue itself [2]. Serious muscle injuries, if not diagnosed and treated promptly, can have serious consequences [3,4].
Understanding the biomechanics of muscles can not only predict and prevent injury, but it can also help to develop diagnosis and treatment methods. A whole skeletal muscle is a complex structure consisting of skeletal muscle tissue, connective tissue, nerve tissue, blood, or vascular tissue. However, the critical component generating the loading capacity of muscle fascicles in the muscle-loading model is called the sarcomere [5]. The muscle fiber is comprised of sarcomeres bound by Z-disks, and the collision between the myosin and the Z-disk is significant in increasing the passive tensile force [5,6]. The collision in this region is formed by titin, a protein greater than 1 µm that functions as a spring. Titin consists of 244 individually folded protein domains. These domains unfold when the protein is stretched and refold when the tension is released [7,8]. The mechanical behavior of muscle tissues has been investigated in past decades, including the scale of fiber, bundle, fascicle, or whole muscle-tendon architecture. In a study by Sun et al., which performed rabbit muscle-tendon test at stretch rates between 0.5 to 100 cm/min, results showed that responses of muscular tissues depend non-linearly on both strain and strain rate where the effect of strain is more than strain rate [9]. Another study with rabbit muscle-tendon specimens also indicated a similar response of samples as failure load increased at a high stretch rate while maximum strain remained at approximately 0.6 mm/mm [10]. The strain rate is an essential aspect of the loading response of muscle tissue; therefore, some other authors investigated its effect on the muscle of some species in compressive and tensile biomechanics [11][12][13][14]. In a tensile test by Safran et al. at a rate of 10 cm/min, the authors illustrated that increased temperature of the muscular bundle improved the failure stress and length of the muscle [15]. On the other hand, some results were found that because of the complex structure, there was a difference in Young's modulus of muscle at scales from fiber, bundle, and fascicle to whole muscle [16,17]. Thus, it is necessary to construct the property curves at different scales and strain rates in muscle studies.
In this study, we investigated the mechanical responses of the whole muscle-tendonbone structure collected from frog legs under quasi-static and dynamic loading conditions in a strain rate range from 0.001 to 200 s −1 . Experiments with the whole muscle-tendonbone structure ensured that the architecture of the muscular bundle did not change, and mechanical properties also represented the properties between the tendon and muscle. This experiment more closely illustrated the response of the whole human muscle-tendon bundle as it was extended by realistic load conditions.

Materials and methods
The fresh frog legs were collected from the local market and immediately transferred to the laboratory. The frog legs were put into 0.9% physiological saline to prevent water loss. The frog semitendinosus muscle-tendon fascicles used in this research were 40 ± 3 mm in length and roughly 4 ± 0.5 mm in diameter, measured at the center of the muscle bundle. To test the mechanical tensile properties of specimens, we utilized the Lloyd friction testing machine made by AMETEK for experiments at low strain rates from 0.001 to 0.5 s −1 and Hopkinson bar for strain rates between 50 and 200 s −1 . A total of over 30 frog semitendinosus muscle-tendon-bone structures were tested in this research, and at least five specimens were studied in each strain rate range. Each specimen was only tested once to reduce the influence of the previous experiment process and environmental aspects. Specially designed clampers were utilized to ensure that samples were centered on the machine shaft and to eliminate specimen slippage during inspection (Fig. 1). Samples tested at low strain rates were extended to 100% strain. Before starting the test, the samples were pre-stretched by pulling to a force of 0.2 N to align the muscle tissues, which could be dramatically influenced by titin and the length of the sarcomere [5,6,18,19]. The Hopkinson bar system combines 3 bars: 400 mm striker bar, 2000 mm incident bar, and 1000 mm transmitted bar. All bars of the Hopkinson system are made of 6061 alloy aluminum with a 20 mm diameter. However, the transmission signal of soft biological materials is feeble. To solve this problem, the transmitted bar is a hollow rod with an inner diameter of 18 mm [19,20]. The striker bar is connected to a pneumatic cylinder, and its movement generates a collision between the striker and incident bars. The collision wave of this process propagates in the incident bar until it transits through the face between the specimen and the incident bar (incident pulse ɛ i ). This creates the reflected and transmitted pulses (ɛ r , ɛ t ) in the incident and transmitted bars, respectively [19,[21][22][23]. To measure the deformation generated by the pulses in the incident and transmitted bars, four sensors and two Wheatstone bridges are used. The signal is amplified by the amplifier before transferring to an oscilloscope, and the reflected and transmitted pulses (ɛ r , ɛ t ) are calculated from the oscilloscope data by MATLAB software. The muscle-tendon samples are clamped at the ends of the bars, as shown in Fig. 2 (setup of Hopkinson tensile bar). Two clampers made of PLA material by the 3D printing method were also used to clamp the bone sections of the samples (Fig. 1b). The clampers could be moved to accommodate different Fig. 1 The structure of the clampers in the experiment at quasi-static strain rate a and dynamic strain rate b Fig. 2 Tendon-muscle tensile test set up the experiment and clamping condition bone sizes to ensure pulse transmission through the sample. In addition, a hollow transmitted bar solved the aforementioned problem that the waves on the transmitted bar were too weak. The transmitted pulse passing through the hollow bar created an amplified strain signal, which the sensors could measure easily [19,20]. It was noted that the recorded strain ranges were smaller compared to experiments at a quasi-static strain rate.
The strain, stress, and strain rate s , s , s of test samples are calculated from the following equations based on one-dimensional stress wave theory [24][25][26][27]: where c 0 : the wave velocity in the incident and transmission bar l s : the initial length of the specimens E: Elastic Modulus of the pressure bar A, A s : area of pressure bars and area of specimens

Results
The stress-strain curves in Figs. 3 and 4 were collected from 5 similar tests in a quasi-static range and 6 tests over a dynamic range. This study tested the mechanical response of the whole muscle-tendon bundle in a total of six different strain rate values. In a study by Bing Yu et al., the maximum muscle eccentric contraction velocity of about 20 male athletes' The stress-strain curves of 5 muscle-tendon specimens at the 0.001 1/s strain rate hamstring semitendinosus muscles during sprinting was measured from 19.3 to 68 m/s, while the peak positive muscle-tendon elongation velocity was from 1 to 1.5 m/s [28][29][30]. The results showed that a strain rate of 200 s −1 can properly represent most muscle strain rates in sports activities. The average stress vs strain curves of 4 lower strain rates was calculated and illustrated in Fig. 5. The profile of curves was parallel to the findings of some research at lower strain rates, consisting of two regions (slack and linear regions) [31]. Curves observed at the dynamic strain rate (Fig. 4) only showed a linear region, but it was also similar in shape to other reports at higher strain rates [10,13,14,32]. The tests demonstrated that the mechanical property of the tendon-muscle bundle was sensitive to strain rate. Young's modulus and the break stress increased at the higher strain rate. However, the maximum stress happened at approximately 0.4 strain in all the quasi-static experiments (Fig. 5). The strain corresponding to the maximum stress was not affected by the strain rate. In contrast, strain at maximum stress was about 0.015 (at a strain rate of 90 s −1 ) to 0.035 (at a strain rate of 200 s −1 ) at dynamic conditions. However, this was not the deformation that caused the rupture in the muscle bundles. Because the strains in experiments with the Hopkinson bar system were generated by the deformation pulse, the strain in the sample was not large enough to cause major fractures in the structure of the muscle bundle. The strain rate's influences on muscle mechanics were confirmed by plotting the magnitude of Young's modulus and maximum stress (Fig. 6) corresponding to the various strain rates. The curve fitting for Young's modulus in Fig. 6 was generated from logarithmic and polynomial equations. The equation y = 0.2667ln(x) + 4.8809 was used in the strain rate scope from 0.001 to 0.1, while the fitted curve of strain rate from 0.1 to 200 s −1 was provided by a polynomial equation y = − 0.0003x 2 + 0.1966x + 4.5647. Two equations constructed a curve being most suitable to experimental data. Both the maximum stress and the elastic modulus of the muscle bundle changed significantly under the strain rate changes. So the muscle was found to be significantly sensitive to strain rate. The tearing stress of the muscle-tendon bundle increased from 0.67 to 1.25 MPa (about 1.9 times), corresponding to a change of strain rate from 0.001 to 0.5 s −1 . At the quasi-static strain rate range, the change in maximum stress could be represented by the function y = 0.0961ln(x) + 1.334. The maximum stress value shifted in an extensive range at the strain rate from 180 to 200 s −1 . However, the average maximum stress was insignificantly different from the strain rate of 0.5 s −1 , and there was no rupture pointed out in the samples. A rise was observed in Young's modulus under the dynamic strain rate range, and it was higher about ten times as compared to quasi-static loading conditions.

Discussion
At the quasi-static strain rate, curves exhibited an initial non-linear (slack region) at strains lower than 0.15. A linear region was observed at the next strain range, followed by a nonlinear region prior to the gradual failure. This profile of stress-strain curves of the whole muscle-tendon structure was parallel at scales of muscle architecture of some species [5,6,33,34]. The responses of skeletal muscle bundles to passive traction at a low strain rate, as shown in areas of the stress-strain curve, are determined primarily by the property of titin, a large protein that tethered the myosin and the Z-disc [5,6,16,35]. On the other hand, research about sliding theory illustrated that the myosin was almost not deformed during contraction [36][37][38]. This supported a prediction that myosin did not play a significant role in muscle deformation as titin at low strain rates and minor traction. The slack area was assumed to be formed by straightening the large folded state in the immunoglobulin domains, Fig. 6 Maximum stress and Young's modulus at various strain rates. Young's modulus fit curve constructed from logarithmic and polynomial equations y = 0.2667ln(x) + 4.8809, y = − 0. 0003x. 2 + 0.1966x + 4.5647 and the linker regions between the domains adopt a bent configuration in the titin molecule. The small folded regions in the titin molecule would be straightened as the passive stress increases, and this created a linear region in the stress-strain curve [39]. The modification of titin has been demonstrated at the molecular level [40][41][42]. The responses of skeletal muscle bundles to passive stress were shown to be quite similar to those of titin molecular.
At the high strain rates, the toe region vanished because the helical and folded structures of titin did not have enough time to release and create a large strain [5,10,13,14,[31][32][33][34]. This was similar to materials with large molecular structures as polymers. The sensitivity of the biomechanical properties of the whole muscle-tendon bundle to strain rate was clearly shown in Fig. 6. The load-bearing capacity of the sample markedly logarithmically increased even at maximum stress and elastic modulus in the quasi-static strain rate region. These factors continuously sharply increased at the dynamic strain rate from 80-200 s −1 but have not yet reached the tear limit of the samples. Although the response of muscle at three scales (fiber, bundle, and whole muscle bundle) was quite similar in the contour of the stress-strain curves, the difference was prominent in the value. The previous findings by Winters et al. in rabbits also showed that the passive tension modulus increase 1.24 times from fiber scale to bundle and 1.14 times from bundle to fascicle, and the whole muscle bundle level has the highest value [16,17]. The increase of modulus across different scales was assumed because of the titin molecular mass, myosin heavy chain isoform distribution, and collagen length. The responses of the whole muscle-tendon bundle to passive tensile in this study showed that the muscle-tendon bundle was sensitive to strain rate (the strainrate sensitivity coefficient: m = ( ln ∕ ln � )| ,T = 0.10 ± 0.007 in strain rate from 0.001 to 0.5 s −1 [43,44]). This coefficient was analogous to the human muscle (m = 0.12) in the strain range from 0.01 to 100 s −1 [45].
To provide an assessment of the strain rate's effect on the passive tension mechanics of the whole muscle-tendon bundle, this study was compared to some previous research performed only with the muscle bundle [13,14,32]. The maximum passive stress of the muscle-tendon structure in this study was higher from 2 to 5 times than active stress created by the stimulus, while Young's modulus was approximately 20 times larger [46][47][48][49][50]. Passive tensile properties varied widely across species, muscle scale, and strain rates. The previous research in Table 1 showed that the strain rate significantly influenced the elastic modulus more than the maximum stress. Our study showed a similar maximum stress rate at the whole muscle scale. While the elastic modulus, which was pointed out from previous research, increased about 8 to 15 times at the high strain rate as the maximum stress altered only about two times. These results supported that strain rate generated a similar alteration ratio of maximum stress at species and scales. However, Young's modulus value in this research was 2 to 30 times higher than other research performed at the fascicle scale. This result was caused by two reasons: species difference and muscle-bundle scale. In recent research, Ward et al. experimented with three rabbit muscles (tibialis anterior, extensor digitorum longus, and extensor digitorum of the second toe) and found that Young's modulus of whole muscle was about 25 times larger than the fascicle [51]. Moreover, the modulus of the whole muscle was from 2 to 7.5 MPa, and this value was the same as Young's modulus at the low strain rate in this research.
The whole muscle passive response was dominated by extracellular structures creating the large modulus [51,54,55]. The study of Ward et al. also showed that collagen content plays a significant role in increasing Young's modulus and stress of the whole muscle bundle compared with the muscle fiber. Studies of muscle properties should not be confined to small-size scales. However, titin was the primary determinant of Young's modulus of muscle at small scales [56]. Deformation of the whole muscle bundle occurred in the tendon This study and the muscle structures at the same time. Therefore, the results of current studies are not enough to evaluate the specific percentage of the effect of titin and collagen on the change of the passive property of the whole muscle bundle when the strain rate changes. On the other hand, it reflected the synthesis property of both parts in the whole muscle-tendon bundle. While tendons and ligaments are also shown to be sensitive to strain rates, Young's modulus of the tendon is much larger than muscle, and it can reach about 800 Mpa [57][58][59]. If the strain in the tendon area is considered negligible, the deformation (ΔL) and stress (δ) in the muscle area are similar for the same tensile force. Then Young's modulus (E) calculated by the formula E=δ*L 0 /ΔL has a more significant value because of the larger sample length. That can also be one of the factors that increased Young's modulus value.

Conclusion
The purpose of this study is to determine the tensile properties of the whole muscle-tendon fascicles at the quasi-static and dynamic strain rates. The results show the alternation of stress and Young's modulus under various strain rates from 0.001 to 200 s −1 . Both stress and modulus are sensitive to strain rate. Yield stress increases 1.9 times within the change of strain rate from 0.001 to 0.5 s −1 , while Young's modulus changes 1.6 times. A dramatic rise of Young's modulus is observed at dynamic impact and reaches about 30 MPa at 194 s −1 . At the quasi-static strain rate, the stress-strain curves comprise three evident regions because the proteins in the sarcomere have enough time to straighten. But the property of the sample only indicates a linear region at high strain rates, similar to soft tissue. The findings also demonstrated a difference in the passive mechanical property of the whole muscle bundle and muscle bundle in an impact. However, it seems to be different about the value while there is no change in the contour of the stress-strain curves. Whole muscle bundles have a larger strength, which is predicted because the extracellular structures of muscle are not affected by sample isolation. Because of those factors, it is necessary to add the influence factor of strain rate and muscle-bundle scale in constructing muscle mechanisms.
Funding This work was supported by the Ministry of Science and Technology, Taiwan, MOST 109-2637-E-992-011.

Competing interests
The authors declare no competing interests.