Skip to main content
PMC home page
NIHPA Author Manuscripts logoLink to NIHPA Author Manuscripts
. Author manuscript; available in PMC: 2015 Apr 1.
Published in final edited form as: Med Sci Sports Exerc. 2014 Apr;46(4):776–786. doi: 10.1249/MSS.0000000000000172

Computational Models Predict Larger Muscle Tissue Strains at Faster Sprinting Speeds

Niccolo M Fiorentino 1, Michael R Rehorn 2, Elizabeth S Chumanov 3, Darryl G Thelen 3,4, Silvia S Blemker 1,2
PMCID: PMC3960352  NIHMSID: NIHMS542367  PMID: 24145724

Abstract

Introduction:

Proximal biceps femoris musculotendon strain injury has been well established as a common injury among athletes participating in sports that require sprinting near or at maximum speed; however, little is known about the mechanisms that make this muscle tissue more susceptible to injury at faster speeds.

Purpose:

Quantify localized tissue strain during sprinting at a range of speeds.

Methods:

Biceps femoris long head (BFlh) musculotendon dimensions of 14 athletes were measured on magnetic resonance (MR) images and used to generate a finite element computational model. The model was first validated through comparison with previous dynamic MR experiments. After validation, muscle activation and muscle-tendon unit length change were derived from forward dynamic simulations of sprinting at 70%, 85% and 100% maximum speed and used as input to the computational model simulations. Simulations ran from mid-swing to foot contact.

Results:

The model predictions of local muscle tissue strain magnitude compared favorably with in vivo tissue strain measurements determined from dynamic MR experiments of the BFlh. For simulations of sprinting, local fiber strain was non-uniform at all speeds, with the highest muscle tissue strain where injury is often observed (proximal myotendinous junction). At faster sprinting speeds, increases were observed in fiber strain non-uniformity and peak local fiber strain (0.56, 0.67 and 0.72, for sprinting at 70%, 85% and 100% maximum speed). A histogram of local fiber strains showed that more of the BFlh reached larger local fiber strains at faster speeds.

Conclusions:

At faster sprinting speeds, peak local fiber strain, fiber strain non-uniformity and the amount of muscle undergoing larger strains are predicted to increase, likely contributing to the BFlh muscle’s higher injury susceptibility at faster speeds.

Keywords: acute strain injury, hamstrings, athletes, finite element model, active lengthening

INTRODUCTION

Muscle strain injury remains a prevalent problem in recreational, collegiate and professional sports, consuming a significant portion of sports medicine practice (24). Athletes participating in high-speed sports, which require sprinting near or at maximum speed, are especially susceptible to acute muscle strain injury (29). While it has been well documented that the hamstring muscles are highly susceptible to acute injury while sprinting, the amount of local strain experienced by muscle tissue during sprinting and the mechanisms that lead to increased injury risk with speed remain unclear.

Animal models of muscle damage following eccentric contractions have established a strong connection between the magnitude of mechanical strain experienced by muscle and the extent of damage (28). Furthermore, animal experiments have demonstrated that the site of failure in muscle corresponds to the region of highest localized tissue strain (2). Experiments in animal tissue have also shown directly that muscle is most susceptible to damage when it is actively generating force and lengthening rather than passively lengthening (5, 27). In vivo experiments during active lengthening in a running guinea fowl found a regionally dependent amount of tissue strain and an increase in local strain with running speed (9). The ideas set forth by animal models of contraction-induced muscle damage provide a strong basis for understanding mechanisms that potentially lead to clinical muscle strain injury; however, the direct application and relation of the results to hamstring muscle mechanics during sprinting at high speeds is relatively limited. One of the goals of the current study was to create computational models of the human hamstrings that allow us to predict localized muscle tissue strain while sprinting at high speeds.

Previous studies of the hamstring muscles during running or sprinting generally acquire external measurements of surface electromyography (EMG), ground reaction forces and/or marker-based joint kinematics (40, 48). Multiple studies have confirmed that the biceps femoris long head muscle, the most commonly injured hamstring muscle (25), is active and lengthening during the late swing phase of sprinting (13, 47, 50). As sprinting speed increases, previous studies have also shown that the BF muscle activity increases (19) and it does more negative work (i.e., absorbs more energy) (11). Counter to intuition, however, peak musculotendon strain remained invariant at top-end sprinting speeds (11). While marker-based sprinting studies answer questions about potential injury conditions on the global muscle-tendon unit level, injury occurs at the local muscle tissue level; therefore, the behavior of muscle tissue while sprinting may help explain muscle’s increased injury susceptibility at faster speeds.

Dynamic imaging and computational modeling have the potential to bridge the gap in knowledge between musculotendon dynamics derived from whole body mechanical models and measurements in animal models of contraction-induced muscle damage. For example, recent advances in dynamic magnetic resonance (MR) imaging permit strain measurement in muscle tissue during joint motion (52). Dynamic MRI experiments in the BFlh muscle have shown that tissue strain is larger during active compared to passive lengthening (16) and is highest near the BFlh’s myotendinous junction (MTJ) (44), which is where injury is frequently observed (1, 46). The drawback of dynamic MR experiments is that limitations in scanner bore size and the number of repetitions necessary for dynamic sequences constrain the amount of activation and lengthening that can be studied, which make testing hypotheses about increased strain and higher strain non-uniformity during high-load lengthening contraction infeasible. A recent computational modeling study demonstrated the importance of the BFlh’s architecture to its muscle tissue strain distribution and strain injury susceptibility (35); however, the model was not validated against in vivo strain measurement and the simulations were based on non-specific, low levels of activation and muscle-tendon length change. No studies currently exist that show how the behavior of muscle tissue changes with sprinting speed. Understanding how muscle tissue behavior changes with sprinting speed will provide direct insight into muscle’s increased injury susceptibility at faster sprinting speeds.

The goals of this study were to (i) generate a finite-element model of the BFlh based on musculotendon dimensions of athletes participating in high-speed sports, (ii) validate model predictions with in vivo dynamic MR experiments, and (iii) use the model to predict how localized tissue strains in the BFlh are affected by sprinting speed.

METHODS

Finite-element mesh generation and fiber mapping

The dimensions of the finite-element computational mesh were based on the average musculotendon dimensions of University of Virginia Track and Field athletes. Fourteen athletes (seven male, height: 177 +/− 9 cm, mass: 71 +/− 10 kg) provided informed consent to participate in a University Internal Review Board approved study. All athletes participated in high-speed events and had no history of acute hamstring strain injury. Magnetic resonance (MR) images were acquired with a Siemens Trio 3T MR scanner (Erlangen, Germany) using a Dixon sequence (TR 7 ms, TE 2.45 ms, flip angle 9 degrees, 5-mm slice thickness, field-of-view 375 mm × 500 mm, imaging matrix 504 × 672) with a high spatial resolution (0.74 × 0.74 mm2). Axial plane images were acquired from the muscle-tendon unit’s origin at the hip to the insertion at the knee. The Dixon imaging sequence acquires separately an image when the signal from water and fat are in-phase and when the signal from water and fat are out-of-phase, and the in-phase and out-of-phase images were combined to provide high contrast between muscle, fat and connective tissue but not suffer from chemical shift artifact (15).

Biceps femoris long head musculotendon dimensions were measured directly on MR images using OsiriX Imaging Software (37) (Figure 1A) or were derived from full segmentations of the muscle using custom in-house MATLAB (The Mathworks, Inc., Natick, MA, USA) software (Table 1). Width and thickness measurements were acquired directly on axial-plane images, whereas muscle length, which runs in the superior-inferior direction perpendicular to the axial imaging plane, was derived from full segmentations. Average musculotendon measurements were used to define the dimensions of the finite-element (FE) mesh, which included three materials (proximal tendon and aponeurosis, muscle, and distal tendon and aponeurosis) connected directly by coincident nodes. A previous modeling study found that a simplified FE computational model based on musculotendon measurements produced the same results as a segmentation-based subject-specific model (35). The FE computational mesh was generated using TrueGrid software (XYZ Scientific Applications) and included 2960 hexahedral elements and 2112 nodes. A convergence study of peak local strain was conducted to determine the number of elements to include in the computational mesh. Representative muscle fibers were mapped through the finite element mesh from the fibers’ origin on the proximal aponeurosis to the fibers’ insertion on the distal aponeurosis (4). Muscle fibers were used to define the initial fiber direction at each element, which was an input to the constitutive model, and to calculate spatially varying local tissue strain.

Figure 1.

Figure 1

Open in a new tab

Computational model. Musculotendon measurements of collegiate track and field athletes were acquired from axial plane MR images and were used to generate a finite-element (FE) computational mesh (A). Muscle fiber direction was defined at each element by mapping fibers through the FE mesh from the fibers’ origin on the proximal aponeurosis to the insertion on the distal aponeurosis (4). To compare model predictions and in vivo measurements, a separate computational mesh was generated based on MR imaging subjects in a previous study (16). The muscle-tendon length change boundary condition was based on the dynamic MR experiment, and muscle activation was based on experiments in the exercise device (45) (B). For sprinting simulations, forward dynamic simulations of measured sprinting kinematics yielded BFlh muscle activation and muscle-tendon length change at 70%, 85% and 100% of maximum sprinting speed (11). Muscle activation was applied to muscle tissue and muscle-tendon length change was applied to the distal end of the distal tendon while holding the proximal end fixed. Simulations were performed from mid-swing to foot contact.

Table 1.

Model inputs. Musculotendon measurements were taken from high-resolution axial plane MR images and averaged across 14 University of Virginia Track and Field athletes. Musculotendon measurements were used to define the geometry of the computational mesh. Material parameters were defined for implementation in the transversely isotropic, hyperelastic and quasi-incompressible material model detailed in (3). AP: anterior-posterior. ML: mediallateral. σmax: peak isometric stress. P1: along-fiber extension multiplicative modulus. P2: along-fiber extension exponential modulus. λofl: along-fiber stretch at optimal fiber length. λ*: stretch at which stress-strain relationship becomes nonlinear. Ao: exponential shear modulus (42). Gφ: along-fiber shear modulus. Gϕ: cross-fiber shear modulus. K: bulk modulus.

Musculotendon measurements [cm]
muscle connective tissue
AP ML proximal
tendon		 proximal aponeurosis distal aponeurosis distal
tendon
width width length length length width thickness length width thickness length
4.78 3.93 28.55 5.96 17.32 1.13 0.23 20.89 4.87 0.13 9.61
(0.58) (0.24) (3.78) (0.14) (1.87) (0.24) (0.06) (1.98) (0.55) (0.02) (1.29)
Material parameters
muscle connective tissue
σ max P1 P2 λ ofl λ* Ao Gϕ Gφ K P1 P2 λ ofl λ* Ao Gϕ Gφ K
0.125 0.04 6.6 1.0 1.06 2.0 3.87*10−3 2.24*10−2 5.0*102 1.2 50 1.0 1.03 2.0 3.0 15.0 5.0*103
MPa dimensionless MPa MPa MPa MPa dimensionless MPa MPa MPa
Open in a new tab

Constitutive model of muscle and connective tissue

A full description of the constitutive model has been published previously (3). Briefly, muscle and connective tissue were modeled as transversely isotropic, hyperelastic and quasi-incompressible (3, 14, 49). For a hyperelastic material, stress and strain are related through the following expression:

S=2∗∂ϕ∕∂C

where S is the 2nd Piola-Kirchoff stress tensor

Φ is the strain-energy density function

C is the right-Cauchy-Green strain tensor

To employ transverse isotropy, a preferred direction is defined at each finite element. For tendon, the preferred direction is defined along the length of the connective tissue, or in the superior-inferior direction of the element. For muscle, this is defined by the direction of the muscle fiber and specifies the direction along which active stress is generated. The amount of activation linearly modulates muscle fiber’s force-length relationship (51). The amount of stretch in the direction of the fiber is referred to as along-fiber stretch (λ). For muscle and connective tissue, the relationship between passive along-fiber stretch and stress is characterized by a nonlinear toe region followed by a linear increase (51). Shear deformation along the fiber and across the fiber are represented by an exponential relationship between shear strain and stress (42). The materials are quasi-incompressible, which is enforced by highly penalizing changes in volume. Simulations were run with the nonlinear implicit FE solver Nike3D (34). Material constants for muscle (Table 1) were defined based on a recent study of along-fiber extension, cross-fiber extension, and along-fiber shear experiments in muscle tissue (32). Peak isometric stress (i.e., specific tension) was taken from a study on the maximum force generating capacity of fibers in human vastus lateralis muscle (43). Connective tissue parameters (Table 1) were derived from linear moduli and stress-strain values reported from experiments in tendon (10).

Comparison to dynamic imaging

We previously measured local tissue strains in the BFlh muscle using a dynamic MRI technique (16). The study reported first principal strain in muscle tissue adjacent to the proximal MTJ during passive lengthening and during active lengthening of the BFlh (for N = 13 healthy subjects). To compare in vivo measurements and model predictions, a separate finite-element computational mesh was generated based on musculotendon dimensions of the subjects in the dynamic imaging study (Figure 1B). Simulation boundary conditions were defined based on measurements while subjects were exercising in a MR-compatible device that induced lengthening contractions in the hamstrings muscles (45). Because the dynamic MR technique measured displacement in muscle tissue, the muscle-tendon length change boundary condition was applied to the distal MTJ while the proximal end was held fixed. Forward dynamic simulations of subjects exercising in the MR-compatible device were used to define the temporal variation in muscle activation (45, 48).

Simulations of sprinting

Muscle activation and muscle-tendon length change for sprinting were defined based on measured joint kinematics and forward dynamic simulations of sprinting at 70%, 85% and 100% of maximum speed (Figure 1B) (11). In a previous study the forward dynamic simulation framework has been shown to predict similar timing between EMG activity and model predictions for the lateral hamstring muscles (12). A computed muscle control (CMC) algorithm (48) was used to determine muscle excitation patterns such that the forward dynamic model closely matched joint kinematics measured in 19 athletes during treadmill sprinting (11). To account for muscle redundancy, numerical optimization was used to minimize the sum of squared weighted contractile element forces. Constraints on the timing of muscle excitation were added to ensure that excitations were minimal when EMG data indicated low levels of activation. Muscle-tendon length change and muscle activations were averaged across all subjects. Average activation and length change trajectories at each speed were then applied as boundary conditions to the muscle and the distal end of the computational mesh, respectively. To apply the appropriate amount of muscle-tendon unit length change relative to the length of the muscle-tendon unit, the finite-element simulation began at the muscle-tendon length measured on MR images, which corresponded to the mid-swing, or 65% of the gait cycle, and proceeded to foot contact, or 100% of the gait cycle.

Simulation analysis

For each simulation, model predictions of whole muscle fiber length change and local fiber strain were analyzed. To determine whole-fiber length change from simulations, representative muscle fibers were mapped through the computational mesh and tracked throughout the simulation (4). To find local fiber strain, sixty-one points along each fiber were tracked throughout the finite-element simulation, which permitted values for local fiber strain to be calculated at 60 intra-fiber segments along each representative muscle fiber. Local fiber strain was calculated as the difference in intra-fiber segment length at each time in the simulation and the original segment length divided by the original length. Local fiber strain is a measure of spatially varying engineering strain. For lengthening, strain is greater than zero. For shortening, strain is less than zero. Based on the local fiber strain data, the peak local strain for each fiber was found at each time step in the simulation, and the peak local fiber strain for each fiber was averaged at each sprinting speed. In addition, to assess the amount of strain non-uniformity, peak local fiber strain for each fiber was plotted against strain of the fiber, where values further away from unity represent more non-uniform strain distributions. Spatially varying fiber strain was analyzed to assess which region(s) of the muscle experienced the largest fiber strain. Fiber strain was averaged in three evenly spaced regions to quantify fiber strain in muscle tissue near the proximal MTJ, in the middle of the muscle and near the distal MTJ. At the overall peak strain for each speed, a local fiber strain histogram plot demonstrates how much of the muscle was undergoing relatively larger strains.

RESULTS

The model simulation of dynamic MR experiments compared favorably to experimental measures of tissue strain (Figure 2). The model predicted first principal strains in muscle tissue adjacent to the proximal MTJ that were within a standard deviation of measurements during both passive lengthening (0.13 vs. 0.13 +/− 0.06 for MR measurements) and active lengthening (0.22 vs. 0.19 +/− 0.09 for MR measurements) (standard deviation represents variability across imaging subjects). The similar magnitudes of 1st principal strain demonstrate the computational model’s ability to replicate muscle tissue strains experienced in vivo.

Figure 2.

Figure 2

Open in a new tab

Model-imaging comparison. Model simulation results for 1st principal strain were compared to in vivo measurements in a recent dynamic MR imaging study (16). Average 1st principal strain in the muscle tissue adjacent to the proximal MTJ was reported as 0.13 +/− 0.06 during passive lengthening experiments and 0.19 +/− 0.09 during active lengthening experiments (standard deviation represents variability across N = 13 imaging subjects). Model simulations results found an average of 0.13 and 0.22 for passive lengthening and active lengthening, respectively.

Sprinting simulations showed that whole-fiber length change was non-uniform throughout the BFlh muscle and peaked prior to foot contact (Figure 3A). Maximum whole-fiber length change averaged across all representative fibers was 2.7 +/− 0.1 cm, 3.2 +/− 0.1 cm and 3.3 +/− 0.2 cm for maximum muscle-tendon unit lengthening of 3.9 cm, 4.6 cm and 4.8 cm at sprinting speeds of 70%, 85% and 100% of maximum, respectively. Maximum length change of muscle fibers relative to the length change of the muscle-tendon unit was 0.68, 0.70 and 0.68 at 70%, 85% and 100% of maximum speed. Strain of the entire MTU was 0.09, 0.10 and 0.11 at 70%, 85% and 100% of maximum speed. Maximum whole-fiber strain averaged over all fibers was 0.24 +/− 0.01, 0.29 +/− 0.01 and 0.29 +/− 0.2 and was at most 0.27, 0.32 and 0.34 for 70%, 85% and 100% of maximum speed, respectively.

Figure 3.

Figure 3

Open in a new tab

Fiber length change, local peak strain and local peak strain relative to whole-fiber strain. Whole-fiber length change in the FE computational model (muscle fibers) was non-uniform throughout the BFlh muscle and increased with sprinting speed (A) (representative fibers are plotted as individual lines). The boundary condition for muscle-tendon unit length change was plotted for comparison (muscle-tendon unit). Peak local along-fiber strain was found for each representative muscle fiber (B). To assess the uniformity of fiber strain while sprinting and during dynamic MR experiments, local peak fiber strain was plotted as a function of whole-fiber strain for each representative fiber (C). Values on the unity line represent perfectly uniform strain distribution along the fiber.

Peak local fiber strain was non-uniform throughout the BFlh muscle, reached a maximum before foot contact, and increased with sprinting speed (Figure 3B). Averaged over all fibers, peak local fiber strain was 0.46 +/− 0.08, 0.55 +/− 0.09 and 0.59 +/− 0.09, with a maximum of 0.56, 0.67 and 0.72, for sprinting at 70%, 85% and 100% of maximum speed, respectively. Average peak local fiber strain also increased after normalization by strain of the MTU (5.20, 5.31 and 5.48 for sprinting at 70%, 85% and 100% of max. speed) and was highest at the fastest sprinting speed after normalization by average strain of the whole muscle fibers (1.93, 1.91 and 2.01 for sprinting at 70%, 85% and 100% of max. speed). Maximum peak local strain also increased with sprinting speed after normalization by MTU strain (6.41, 6.49 and 6.67 for sprinting at 70%, 85% and 100% of max. speed) and was highest at the fastest speed after normalization by the maximum whole-fiber strain (2.08, 2.06 and 2.15 for sprinting at 70%, 85% and 100% of max. speed).

To assess the uniformity of local fiber strain and the dependency on strain of the whole fiber, peak local fiber strain was plotted as a function of whole-fiber strain (Figure 3C). The average distance to the unity line was 0.17 +/− 0.05, 0.20 +/− 0.07 and 0.22 +/− 0.06 for sprinting at 70%, 85% and 100% maximum speed, indicating that local fiber strain becomes more non-uniform with increasing sprinting speed. For comparison, distance to the unity line was also calculated for simulations of passive and active lengthening MR experiments and was found to be 0.02 +/− 0.01 and 0.05 +/− 0.01 for passive and active lengthening, respectively.

Fiber strain distribution was found to be non-uniform at each sprinting speed and for MR-based simulations, with the largest fiber strain along the proximal MTJ and decreasing with distance toward the distal MTJ (Figure 4). Average fiber strain near the proximal myotendinous junction, in the middle of the muscle fibers and near the distal myotendinous junction increased with sprinting speed, with the largest increases in muscle tissue near the proximal MTJ. In addition, as sprinting speed increased, a larger portion of the muscle experienced larger local fiber strains, as demonstrated by bigger bins at larger strains in a histogram plot of local fiber strain (Figure 5).

Figure 4.

Figure 4

Open in a new tab

Along-fiber strain distribution. Along-fiber strain distribution was analyzed at the time of maximum local strain and was found to be non-uniform throughout the BFlh muscle (shown in longitudinal cross-section). Color maps of along-fiber strain distribution are shown for simulations of passive and active lengthening during dynamic MR experiments and for sprinting at 70%, 85% and 100% of maximum speed. Along-fiber strain was averaged in three evenly spaced regions along the muscle fibers from the proximal myotendinous junction (MTJ) to the distal MTJ. Average along-fiber strain was found to decrease as a function of distance from the proximal MTJ at each speed and to increase as a function of speed in each region.

Figure 5.

Figure 5

Open in a new tab

Histogram of along-fiber strain. Local along-fiber strain was calculated for 60 intra-fiber segments along 180 fibers and was binned according to amount of strain. Histograms are shown for simulations of passive and active lengthening during dynamic MR experiments (top row) and for sprinting at 70%, 85% and 100% of maximum speed (bottom row). Bigger bins to the right indicate a higher portion of the muscle undergoing larger strains.

DISCUSSION

Finite-element model simulations accurately predicted local muscle tissue strains when compared to measurements in dynamic MR experiments, demonstrating the model’s ability to predict muscle tissue strains experienced in vivo. Simulations of sprinting demonstrated increased BFlh peak local fiber strain and higher fiber strain non-uniformity at faster sprinting speeds. Whole-fiber length change relative to muscle-tendon unit (MTU) length change remained relatively constant with increasing speed; however, peak local fiber strain relative to strain of the MTU increased with speed and peak local fiber strain relative to strain of whole muscle fibers was highest at the fastest speed. These results offer new insights into the deformation of muscle tissue during sprinting and provide a possible explanation for muscle tissue’s increased strain injury susceptibility at faster speeds.

This is the first study to provide direct insight into localized muscle tissue strain in the oft-injured hamstring muscles during sprinting, because measuring local muscle tissue mechanics while sprinting is not yet possible and marker-based studies of sprinting gait only provide information about the behavior of the entire muscle-tendon unit. In the current study, we combined forward dynamic simulation output with finite-element modeling to study the effect of increased sprinting speed on local muscle fiber strain. Previous forward dynamic simulations of sprinting matched marker measurements of joint kinematics by driving a lumped parameter muscle-tendon model with activations (48), or an inverse dynamics approach has been used along with static optimization to derive MTU results from calculated joint torques (40). Marker-based studies of sprinting have shown that the BFlh MTU does more work with increasing sprinting speed (11, 39), however, insights into the behavior of the MTU only provide a more global mechanism for muscle injury. The power of the modeling approach described in this paper is that the output of the computational model is spatially varying tissue strain, which can identify regions of large localized tissue strain and test hypotheses about what factors lead to increases in strain injury susceptibility.

At all speeds the computational model found that local fiber strain was non-uniform throughout the BFlh muscle and highest along the proximal MTJ (Figure 4), which corresponds to the most frequent injury location in the BFlh (1, 46). Large localized fiber strain where injury is frequently observed supports the hypothesis that large localized tissue strain is the injury mechanism in lengthening contractions during sprinting. Experimental studies in humans have also measured regionally varying muscle tissue strain (52), and, in the BFlh, higher strain closer to the proximal MTJ during lengthening contractions (44). In all three evenly spaced regions along the muscle fiber, sprinting simulations showed that the average magnitude of fiber strain increased with increasing speed, with the largest increases in the muscle tissue near the proximal MTJ. These results are in agreement with sonomicrometry experiments in guinea fowl that found local tissue strain increased with running speed and increases were the largest in regions that experienced the largest strains at slower speeds (9). The current study’s results are the first to demonstrate increased muscle tissue strain with sprinting speed in humans, which is important because the BFlh’s increased injury susceptibility with sprinting speed has yet to be explained on the local, muscle tissue level.

Model simulations found that peak local fiber strain was nonuniform throughout the BFlh and occurred during the late swing phase of sprinting (prior to foot contact) (Figure 3B), which corresponds well with the timing of acute strain injury observed during sprinting kinematics studies (18, 38). The amount of strain non-uniformity increased with sprinting speed, as shown by the larger in difference normalized peak local fiber strain and whole fiber strain in Figure 3C (0.17, 0.20 and 0.22 for sprinting at 70%, 85% and 100% maximum speed). Furthermore, as sprinting speed increased, more of the muscle experienced higher fiber strain, which is shown by the relatively larger bins at larger fiber strains in the histogram in Figure 5. If acute muscle strain injury is sustained by tissue strain crossing an injury threshold, as suggested by injury after single active stretches in animal experiments (17), more of the muscle will be susceptible to injury at faster speeds. Similarly, if muscle injury occurs after accumulation of damage to muscle tissue during an activity, as suggested by stretch-shorten cycles in animals over physiological ranges (8), more muscle tissue undergoing larger tissue strain will increase muscle’s strain injury susceptibility at faster sprinting speeds.

The observed increase in strain non-uniformity and peak local strain was a result of higher muscle activations at faster speeds. The primary factor that leads to this phenomenon is the converging of the BFlh’s longitudinal cross-sectional area near the proximal MTJ. The region of muscle tissue near the proximal MTJ is smaller in cross-sectional area than neighboring regions in the middle of the muscle and near the distal myotendinous junction. The relatively larger cross-sectional areas of adjacent tissue generate more force for a given amount of muscle activation, and the larger forces must be balanced by forces generated in tissue near the proximal MTJ. Given the smaller cross-sectional area, muscle tissue near the proximal MTJ will experience higher stresses and as a result undergo greater strain than neighboring tissue. Higher levels of activation at faster sprinting speeds will exacerbate this effect because of muscle’s nonlinear force-length relationship, and tissue near the proximal MTJ will strain even more. More specifically, at higher activation levels the active force-length curve will influence the total force-length curve more, and, given that strains in these simulations occur on the descending limb of the force-curve, relatively more strain will be necessary for a given increase in stress.

Animal experiments are able to relate directly the amount of strain and damage to muscle tissue. A study in rabbit extensor digitorum longus (EDL) muscle-tendon units found that an active strain of 0.15 (of the entire MTU) was necessary to detect damage (17), though the threshold for muscle damage from imposed strain has not been consistent in the literature. Other studies found much higher amounts of active MTU strain are necessary to produce damage (e.g., 0.30 strain in Brooks et al. (5)). Both thresholds are larger than the amount of MTU strain in this study (0.11 for sprinting at 100% maximum speed). On the local level, a previous study measured local mechanical strain during passive extension of animal muscle-tendon units, and muscle tissue was shown to suffer damage at a local mechanical strain of 0.61 (2). The maximum peak local strain in the current modeling study was 0.72 at 100% maximum sprinting speed. It should be noted that the reference configuration for the current study is mid-swing, which is not passively unloaded like animal MTU studies, and therefore caution should be taken when directly comparing the results of the current modeling study to those of animal studies.

The relative amount of muscle fiber strain to MTU strain in sprinting simulations was slightly larger than observations in active lengthening of intact rabbit MTUs. Sprinting simulations had 2.7 times more fiber strain than MTU strain for 100% maximum speed, and rabbit MTU lengthening experiments showed about 2.6 times more fiber strain than MTU strain averaged over the course of many strain shorten cycles (7). As demonstrated by the same study, the relative amount of muscle fiber strain in animal MTU experiments depends on the reference length (7). In addition, the muscle-tendon architecture, magnitude of length change and amount of activation will alter the amount of muscle fiber strain relative to MTU strain. It would also be expected that, relative to the submaximal activations in the current study, maximally activated muscle in animal studies will generate more force and cause more tendon stretch for a given MTU lengthening and result in relatively less muscle fiber strain. In any case, the results presented here demonstrate that the magnitude of MTU strain (0.11) is not representative of the amount of muscle fiber strain (0.29 +/− 0.02 for sprinting at 100% max.), which is also not representative of the amount of local fiber strain (peak 0.59 +/− 0.09 for sprinting at 100% maximum speed).

While the magnitudes differed, the shape of the curves for MTU length change, whole fiber length change and peak local along-fiber strain were qualitatively similar with only a shift of peak local along-fiber strain relative to MTU and whole fiber length change (Figure 3A-B). The shift was speed dependent, with maximum local along-fiber strain occurring slightly later for faster speeds, suggesting that the timing between activation and MTU length could have an effect on the timing of peak local strain. Sonomicrometry experiments in animals (6, 30, 36) and ultrasound experiments in humans (21) have observed a much larger timing difference between MTU and fiber dynamics. These experiments, however, were performed during different tasks, on different muscles, and with different relative timings between muscle activation and MTU length change. Moreover, the shape of the MTU length change and activation curves likely differed from the current study, making a direct comparison between studies more difficult.

The computational model’s material parameters incorporated recent measurements in muscle (32) and connective tissue (10). A sensitivity analysis of material parameters showed that the shear properties of tendon had the least effect on muscle fiber strain and peak local tissue strain, with twice as stiff and twice as compliant tendon shear moduli resulting in less than a 1% change in peak local fiber strain. On the other hand, doubling the along-fiber stiffness of tendon had a larger influence on muscle tissue strain, with twice as stiff along-fiber extension moduli (P1 and P2) yielding an 11% increase in peak along-fiber strain. The dependence of muscle fiber strain on tendon stiffness has also been observed in forward dynamic simulations of running (47). It should be noted that the choice of material model affects the strain magnitude and distribution of model predictions. Specifically, the anisotropy of intramuscular tissues can introduce anisotropy in the transverse direction (41), which has been suggested in a previous two-dimensional modeling study of a pennate MTU to influence strain (20). With respect to the current study’s results, introducing transverse anisotropy would likely not influence our conclusions, since our sensitivity analysis varied along-fiber shear modulus—which would be affected directly by inducing transverse anisotropy—and the strain distribution and magnitude was not changed.

Of all the parameters, peak isometric stress had the biggest influence on peak local fiber strain, with a 50% increase in peak isometric stress resulting in an increase of 13% in peak along-fiber strain. It should be noted that peak isometric stress is an intrinsic property of muscle fibers that may not necessarily change with training. Training likely alters neuromuscular coordination (i.e., the magnitude and timing of muscle activation and muscle-tendon length change) and the amount of muscle tissue available for generating force (i.e., muscle hypertrophy) rather than muscle’s intrinsic ability to generate force. Muscle’s intrinsic ability to generate force could be changed by altering fiber type composition, given that different fiber types exhibit different levels of peak isometric stress (43). While the model’s results were sensitive to perturbations in certain material parameters, the values that were used yielded results that match experimental data, which inspires confidence in the model’s ability to replicate tissue strains experienced in vivo. Furthermore, the conclusions of this study regarding increases in tissue strain at faster sprinting speeds were not altered by changes in the model’s material parameters.

The current study’s model results help fill the gap in knowledge between whole-body kinematics studies and animal muscle fiber experiments by predicting local fiber strain during physiological muscle activations and muscle-tendon length changes. This approach can be used in future studies of tissue-level muscle function during movements that cannot be imaged directly or measured with joint-level kinematic studies. A future application of this modeling framework would be to generate finite-element models based on the range of musculotendon dimensions rather than using one mesh of the average dimensions of all subjects, as a previous modeling study (35) and imaging experiments (16, 44) have found a connection between musculotendon morphology and internal muscle tissue strains. Additionally, an anatomical study has found variable thickness of the BFlh muscle along its length (23), which could influence strain experienced by muscle. However, a previous modeling study that included variable muscle thickness simulated active lengthening of the BFlh and found a similar strain distribution as the current study (35). In the present study, our goal was to address the effects of increased running speed and not the influence of subject musculotendon dimensions. An additional consideration in future studies would be to collect dynamic MR images, musculotendon dimensions, motion capture and EMG data on the same subjects rather than incorporate data from different sources like in the current study. This approach would also permit further validation of the modeling approach, where subject-specific FE model simulations with individualized boundary conditions could be performed for each subject and validated against subject-specific strain measurements.

In addition the reference configuration for the current study was in the middle of the swing phase of sprinting, because the muscle-tendon length measured on MR images corresponded to the MTU length at 65% of the sprinting gait cycle in forward dynamic simulations; future simulations starting from an earlier point in the gait cycle might find a different absolute magnitude of fiber strain and relative change with speed. Moreover, similar changes might accompany simulations that continue until later points in the gait cycle, such as through the stance phase, where high amounts of BFlh activation have been measured (22, 26, 31) and it has been suggested as a time when the hamstring muscles may be vulnerable to injury (33). Future simulation efforts should also investigate a wider range of speeds to further elucidate the connection between muscle tissue strain and sprinting speed, as a couple of normalized values (average peak local fiber strain normalized by average strain of whole muscle fibers and maximum peak local strain normalized by maximum whole fiber strain) were lower at 85% than 70% (though all values were highest at 100% maximum speed).

An additional limitation to the current approach is that the effects of contraction velocity and passive visco-elasticity were not included in the constitutive formulation. Future model developments include adding these effects in the constitutive equation, which will possibly alter changes with speed, including the relative magnitude strain and the timing between activation, MTU lengthening and local fiber strain. Lastly, the displacement boundary condition for the simulations in this study was applied to the distal end while holding the proximal end fixed, which was based on the length change output from forward dynamic simulations. Applying separate length changes to the MTU at the knee and hip in order to incorporate simultaneous joint angle changes would result in the same strain distributions, because the computational model does not take into account inertial effects and the MTU is not mechanically coupled with neighboring muscles.

To summarize, we developed a unique approach to reveal the tissue-level behavior of the oft-injured biceps femoris long head muscle during sprinting. Forward dynamic simulations output of muscle activation and muscle-tendon length change were incorporated into a finite-element model of sprinting at faster speeds. The models showed that as sprinting speed increases so does peak local fiber strain, fiber strain non-uniformity and the amount of muscle tissue that undergoes larger strain. Increases in peak local fiber strain are attributed to the converging of the BFlh cross-sectional area near the proximal MTJ and increases in muscle activation at faster sprinting speed. Increased peak local fiber strain in the muscle tissue where injury is often found provides additional evidence for the hypothesis that localized fiber strain is the injury mechanism during lengthening contractions. Larger peak fiber strain, higher fiber strain non-uniformity, and a higher percentage of the muscle undergoing larger strain provide evidence that may explain muscle’s increased strain injury susceptibility at faster sprinting speeds and in high-speed sports.

With the current modeling approach, future model simulations can aid in the design and vetting of potential neuromuscular coordination strategies for reducing strain injury incidence by testing which strategies reduce localized muscle tissue strain. Furthermore, with a better understanding of the connection between neuromuscular strategies and increased strain injury susceptibility at faster sprinting speeds, future training programs can be designed to limit the variables that lead to increased injury susceptibility and reduce the number of acute muscle strain injuries.

Acknowledgments

This work was supported by funding from the National Institutes of Health R01 AR056201. The authors also thank the Achievement Rewards for College Scientists Foundation for their support. The authors thank Amy Silder and Christopher Westphal for their assistance with dynamic MRI testing and forward dynamic simulations of MR experiments. Natalie Kramer and Christopher Herb are thanked for their assistance in recruiting track and field athletes at the University of Virginia. The results of the present study do not constitute an endorsement by ACSM.

Footnotes

Funding Disclosure: This work was funded by the National Institutes of Health R01 AR056201.

The authors have no conflicts of interest to disclose.

Conflict of Interest

The authors have no conflicts of interest to disclose.

References

  • 1.Askling CM, Tengvar M, Saartok T, Thorstensson A. Acute first-time hamstring strains during high-speed running: a longitudinal study including clinical and magnetic resonance imaging findings. Am J Sports Med. 2007;35(2):197–206. doi: 10.1177/0363546506294679. [DOI] [PubMed] [Google Scholar]
  • 2.Best T, McElhaney J, Garrett W, Myers B. Axial strain measurements in skeletal muscle at various strain rates. J Biomech Eng. 1995;117(3):262–5. doi: 10.1115/1.2794179. [DOI] [PubMed] [Google Scholar]
  • 3.Blemker SS, Pinsky PM, Delp SL. A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii. J Biomech. 2005;38(4):657–65. doi: 10.1016/j.jbiomech.2004.04.009. [DOI] [PubMed] [Google Scholar]
  • 4.Blemker SS, Delp SL. Three-dimensional representation of complex muscle architectures and geometries. Ann Biomed Eng. 2005;33(5):661–73. doi: 10.1007/s10439-005-1433-7. [DOI] [PubMed] [Google Scholar]
  • 5.Brooks SV, Zerba E, Faulkner JA. Injury to muscle fibers after single stretches of passive and maximally stimulated muscles in mice. J Physiol. 1995;488(2):459–69. doi: 10.1113/jphysiol.1995.sp020980. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 6.Butterfield TA, Herzog W. Effect of altering starting length and activation timing of muscle on fiber strain and muscle damage. J Appl Physiol. 2006;100(5):1489–98. doi: 10.1152/japplphysiol.00524.2005. [DOI] [PubMed] [Google Scholar]
  • 7.Butterfield TA, Herzog W. Quantification of muscle fiber strain during in vivo repetitive stretch-shortening cycles. J Appl Physiol. 2005;99(2):593–602. doi: 10.1152/japplphysiol.01128.2004. [DOI] [PubMed] [Google Scholar]
  • 8.Butterfield TA. Eccentric exercise in vivo: strain-induced muscle damage and adaptation in a stable system. Exerc Sport Sci Rev. 2010;38(2):51–60. doi: 10.1097/JES.0b013e3181d496eb. [DOI] [PubMed] [Google Scholar]
  • 9.Carr JA, Ellerby DJ, Marsh RL. Differential segmental strain during active lengthening in a large biarticular thigh muscle during running. J Exp Biol. 2011;214(20):3386–95. doi: 10.1242/jeb.050252. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 10.Chandrashekar N, Slauterbeck J, Hashemi J. Effects of cyclic loading on the tensile properties of human patellar tendon. Knee. 2012;19(1):65–8. doi: 10.1016/j.knee.2010.11.014. [DOI] [PubMed] [Google Scholar]
  • 11.Chumanov ES, Heiderscheit BC, Thelen DG. The effect of speed and influence of individual muscles on hamstring mechanics during the swing phase of sprinting. J Biomech. 2007;40(16):3555–62. doi: 10.1016/j.jbiomech.2007.05.026. [DOI] [PubMed] [Google Scholar]
  • 12.Chumanov ES, Heiderscheit BC, Thelen DG. Hamstring Musculotendon Dynamics during Stance and Swing Phases of High-Speed Running. Med Sci Sports Exerc. 2011;43(3):525–32. doi: 10.1249/MSS.0b013e3181f23fe8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 13.Chumanov ES, Schache AG, Heiderscheit BC, Thelen DG. Hamstrings are most susceptible to injury during the late swing phase of sprinting. Br J Sports Med. 2012;46(2):90. doi: 10.1136/bjsports-2011-090176. [DOI] [PubMed] [Google Scholar]
  • 14.Criscione JC, Douglas AS, Hunter WC. Physically based strain invariant set for materials exhibiting transversely isotropic behavior. J Mech Phys Solids. 2001;49(4):871–97. [Google Scholar]
  • 15.Dixon W. Simple Proton Spectroscopic Imaging. Radiology. 1984;153(1):189–94. doi: 10.1148/radiology.153.1.6089263. [DOI] [PubMed] [Google Scholar]
  • 16.Fiorentino NM, Epstein FH, Blemker SS. Activation and aponeurosis morphology affect in vivo muscle tissue strains near the myotendinous junction. J Biomech. 2012;45(4):647–52. doi: 10.1016/j.jbiomech.2011.12.015. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 17.Hasselman CT, Best TM, Seaber AV, Garrett WE. A threshold and continuum of injury during active stretch of rabbit skeletal muscle. Am J Sports Med. 1995;23(1):65–73. doi: 10.1177/036354659502300111. [DOI] [PubMed] [Google Scholar]
  • 18.Heiderscheit BC, Hoerth DM, Chumanov ES, Swanson SC, Thelen BJ, Thelen DG. Identifying the time of occurrence of a hamstring strain injury during treadmill running: a case study. Clin Biomech (Bristol, Avon) 2005;20(10):1072–8. doi: 10.1016/j.clinbiomech.2005.07.005. [DOI] [PubMed] [Google Scholar]
  • 19.Higashihara A, Ono T, Kubota J, Okuwaki T, Fukubayashi T. Functional differences in the activity of the hamstring muscles with increasing running speed. J Sports Sci. 2010;28(10):1085–92. doi: 10.1080/02640414.2010.494308. [DOI] [PubMed] [Google Scholar]
  • 20.Hodgson JA, Chi S, Yang JP, Chen J, Edgerton VR, Sinha S. Finite element modeling of passive material influence on the deformation and force output of skeletal muscle. Journal of the Mechanical Behavior of Biomedical Materials. 2012;9:163–83. doi: 10.1016/j.jmbbm.2012.01.010. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 21.Ishikawa M, Pakaslahti J, Komi PV. Medial gastrocnemius muscle behavior during human running and walking. Gait Posture. 2007;25(3):380–4. doi: 10.1016/j.gaitpost.2006.05.002. [DOI] [PubMed] [Google Scholar]
  • 22.Jonhagen S, Ericson M, Nemeth G, Eriksson E. Amplitude and timing of electromyographic activity during sprinting. Scand J Med Sci Sports. 1996;6(1):15–21. doi: 10.1111/j.1600-0838.1996.tb00064.x. [DOI] [PubMed] [Google Scholar]
  • 23.Kellis E, Galanis N, Natsis K, Kapetanos G. Muscle architecture variations along the human semitendinosus and biceps femoris (long head) length. J Electromyogr Kinesiol. 2010;20(6):1237–43. doi: 10.1016/j.jelekin.2010.07.012. [DOI] [PubMed] [Google Scholar]
  • 24.Kibler WB. Clinical aspects of muscle injury. Med Sci Sports Exerc. 1990;22(4):450–2. [PubMed] [Google Scholar]
  • 25.Koulouris G, Connell D. Evaluation of the hamstring muscle complex following acute injury. Skeletal Radiol. 2003;32(10):582–9. doi: 10.1007/s00256-003-0674-5. [DOI] [PubMed] [Google Scholar]
  • 26.Kyrolainen H, Avela J, Komi P. Changes in muscle activity with increasing running speed. J Sports Sci. 2005;23(10):1101–9. doi: 10.1080/02640410400021575. [DOI] [PubMed] [Google Scholar]
  • 27.Lieber RL, Woodburn TM, Friden J. Muscle damage induced by eccentric contractions of 25% strain. J Appl Physiol. 1991;70(6):2498–507. doi: 10.1152/jappl.1991.70.6.2498. [DOI] [PubMed] [Google Scholar]
  • 28.Lieber RL, Friden J. Muscle damage is not a function of muscle force but active muscle strain. J Appl Physiol. 1993;74(2):520–6. doi: 10.1152/jappl.1993.74.2.520. [DOI] [PubMed] [Google Scholar]
  • 29.Lysholm J, Wiklander J. Injuries in Runners. Am J Sports Med. 1987;15(2):168–71. doi: 10.1177/036354658701500213. [DOI] [PubMed] [Google Scholar]
  • 30.Maas H, Gregor RJ, Hodson-Tole EF, Farrell BJ, Prilutsky BI. Distinct muscle fascicle length changes in feline medial gastrocnemius and soleus muscles during slope walking. J Appl Physiol. 2009;106(4):1169–80. doi: 10.1152/japplphysiol.01306.2007. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 31.Mero A, Komi PV. Electromyographic activity in sprinting at speeds ranging from sub-maximal to supra-maximal. Med Sci Sports Exerc. 1987;19(3):266–74. [PubMed] [Google Scholar]
  • 32.Morrow DA, Donahue TLH, Odegard GM, Kaufman KR. Transversely isotropic tensile material properties of skeletal muscle tissue. J Mech Behav Biomed Mater. 2010;3(1):124–9. doi: 10.1016/j.jmbbm.2009.03.004. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 33.Orchard JW. Hamstrings are most susceptible to injury during the early stance phase of sprinting. Br J Sports Med. 2012;46(2):88–9. doi: 10.1136/bjsports-2011-090127. [DOI] [PubMed] [Google Scholar]
  • 34.Puso MA, Maker BN, Ferencz RM, Hallquist JO. NIKE3D: A nonlinear, implicit, three-dimensional finite element code for solid and structural mechanics. User's Manual. 2007 [Google Scholar]
  • 35.Rehorn MR, Blemker SS. The effects of aponeurosis geometry on strain injury susceptibility explored with a 3D muscle model. J Biomech. 2010;43(13):2574–81. doi: 10.1016/j.jbiomech.2010.05.011. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 36.Roberts TJ, Azizi E. The series-elastic shock absorber: tendons attenuate muscle power during eccentric actions. J Appl Physiol. 2010;109(2):396–404. doi: 10.1152/japplphysiol.01272.2009. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 37.Rosset A, Spadola L, Ratib O. OsiriX: An open-source software for navigating in multidimensional DICOM images. J Digital Imaging. 2004;17(3):205–16. doi: 10.1007/s10278-004-1014-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 38.Schache AG, Wrigley TV, Baker R, Pandy MG. Biomechanical response to hamstring muscle strain injury. Gait Posture. 2009;29(2):332–8. doi: 10.1016/j.gaitpost.2008.10.054. [DOI] [PubMed] [Google Scholar]
  • 39.Schache AG, Blanch PD, Dorn TW, Brown NAT, Rosemond D, Pandy MG. Effect of running speed on lower limb joint kinetics. Med Sci Sports Exerc. 2011;43(7):1260–71. doi: 10.1249/MSS.0b013e3182084929. [DOI] [PubMed] [Google Scholar]
  • 40.Schache AG, Dorn TW, Blanch PD, Brown NAT, Pandy MG. Mechanics of the human hamstring muscles during sprinting. Med Sci Sports Exerc. 2012;44(4):647–58. doi: 10.1249/MSS.0b013e318236a3d2. [DOI] [PubMed] [Google Scholar]
  • 41.Sharafi B, Blemker SS. A micromechanical model of skeletal muscle to explore the effects of fiber and fascicle geometry. J Biomech. 2010;43(16):3207–13. doi: 10.1016/j.jbiomech.2010.07.020. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 42.Sharafi B, Ames EG, Holmes JW, Blemker SS. Strains at the myotendinous junction predicted by a micromechanical model. J Biomech. 2011;44(16):2795–801. doi: 10.1016/j.jbiomech.2011.08.025. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 43.Shoepe TC, Stelzer JE, Garner DP, Widrick JJ. Functional adaptability of muscle fibers to long-term resistance exercise. Med Sci Sports Exerc. 2003;35(6):944–51. doi: 10.1249/01.MSS.0000069756.17841.9E. [DOI] [PubMed] [Google Scholar]
  • 44.Silder A, Reeder SB, Thelen DG. The influence of prior hamstring injury on lengthening muscle tissue mechanics. J Biomech. 2010;43(12):2254–60. doi: 10.1016/j.jbiomech.2010.02.038. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 45.Silder A, Westphal CJ, Thelen DG. A magnetic resonance-compatible loading device for dynamically imaging shortening and lengthening muscle contraction mechanics. J Med Devices. 2009;3(3):1–5. doi: 10.1115/1.3212559. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 46.Silder A, Heiderscheit BC, Thelen DG, Enright T, Tuite MJ. MR observations of long-term musculotendon remodeling following a hamstring strain injury. Skeletal Radiol. 2008;37(12):1101–9. doi: 10.1007/s00256-008-0546-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 47.Thelen DG, Chumanov ES, Best TM, Swanson SC, Heiderscheit BC. Simulation of biceps femoris musculotendon mechanics during the swing phase of sprinting. Med Sci Sports Exerc. 2005;37(11):1931–8. doi: 10.1249/01.mss.0000176674.42929.de. [DOI] [PubMed] [Google Scholar]
  • 48.Thelen DG, Anderson FC. Using computed muscle control to generate forward dynamic simulations of human walking from experimental data. J Biomech. 2006;39(6):1107–15. doi: 10.1016/j.jbiomech.2005.02.010. [DOI] [PubMed] [Google Scholar]
  • 49.Weiss JA, Maker BN, Govindjee S. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng. 1996;135(1-2):107–28. [Google Scholar]
  • 50.Yu B, Queen RM, Abbey AN, Liu Y, Moorman CT, Garrett WE. Hamstring muscle kinematics and activation during overground sprinting. J Biomech. 2008;41(15):3121–6. doi: 10.1016/j.jbiomech.2008.09.005. [DOI] [PubMed] [Google Scholar]
  • 51.Zajac FE. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit Rev Biomed Eng. 1989;17:359–411. [PubMed] [Google Scholar]
  • 52.Zhong X, Epstein FH, Spottiswoode BS, Helm PA, Blemker SS. Imaging two-dimensional displacements and strains in skeletal muscle during joint motion by cine DENSE MR. J Biomech. 2008;41(3):532–40. doi: 10.1016/j.jbiomech.2007.10.026. [DOI] [PMC free article] [PubMed] [Google Scholar]

RESOURCES