A three filament mechanistic model of musculotendon force and impedance

Figures

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  Figure 1.. Overview of the VEXAT model and its components. The name of the VEXAT model comes from the viscoelastic crossbridge and active titin elements (A) in the model. Active tension generated by the lumped crossbridge flows through actin, myosin, and the adjacent sarcomeres to the attached tendon (B). Titin is modeled as two springs of length l1 and l2 in series with the rigid segments LT12 and LIgD. Viscous forces act between titin and actin in proportion to the activation of the muscle (C), which reduces to negligible values in a purely passive muscle (D). We modeled actin and myosin as rigid elements; the XE, titin, and the tendon as viscoelastic elements; and the ECM as an elastic element.

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  Figure 2.. The force-length and force-velocity relations of the cat soleus model. The model relies on Bézier curves to model the nonlinear effects of the active-force-length curve, the passive-force-length curve (A), and the force-velocity curve (B). Since nearly all of the reference experiments used in the ‘Biological benchmark simulations’ section have used cat soleus, we have fit the active-force-length curve (fL(⋅)) and passive-force-length curves (fPE(⋅)) to the cat soleus data of Herzog and Leonard, 2002. The concentric side of the force-velocity curve (fV(⋅)) has been fitted to the cat soleus data of Herzog and Leonard, 1997.

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  Figure 3.. The passive force-length relations of the ECM, titin, and titin’s segments. The passive force-length curve has been decomposed such that 56% of it comes from the ECM while 44% comes from titin to match the average of ECM-titin passive force distribution (which ranges from 43% to 76%) reported by Prado et al., 2005 (A). The elasticity of the titin segment has been further decomposed into two serially connected sections: the proximal section consisting of the T12, proximal IgP segment and part of the PEVK segment, and the distal section consisting of the remaining PEVK section and the distal Ig segment (B) The stiffness of the IgP and PEVK segments has been chosen so that the model can reproduce the movements of IgP/PEVK and PEVK/IgD boundaries that Trombitás et al., 1998b (C) observed in their experiments. The curves that appear in subplots A and B come from scaling the two-segmented human soleus titin model to cat soleus muscle. The curves that appear in subplot C compare the human soleus titin model’s IgP, PEVK, and IgD force-length relations to the data of Trombitás et al., 1998b (see in Appendix 2 for details).

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  Figure 4.. A graphical overview of a system’s response in the time-domain, frequency-domain, its gain-response, and its phase-response. Evaluating a system’s gain and phase response begins by applying a pseudo-random input signal to the system and measuring its output (A). Both the input and output signals (A) are transformed into the frequency domain by expressing these signals as an equivalent sum of scaled and shifted sinusoids (simple example shown in B and C). Each individual input sinusoid is compared with the output sinusoid of the same frequency to evaluate how the system scales and shifts the input to the output (D). This process is repeated across all sinusoid pairs to produce a function that describes how an input sinusoid is scaled (E) and shifted (F) to an output sinusoid using only the measured data (A).

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  Figure 5.. The time-domain and power-spectrum of the bandwidth-limited stochastic perturbation signals. The perturbation waveforms are constructed by generating a series of pseudo-random numbers, padding the ends with zeros, by filtering the signal using a 2nd order low-pass filter (wave forms with –3 dB cut-off frequencies of 90 Hz, 35 Hz, and 15 Hz appear in A) and finally by scaling the range to the desired limit (1.6 mm in A). Although the power spectrum of the resulting signals is highly variable, the filter ensures that the frequencies beyond the –3 dB point have less than half their original power (B).

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  Figure 6.. The response of the elastic-tendon models (VEXAT and Hill) to the 15Hz and 90Hz perturbations. The 15 Hz perturbations show that the VEXAT model’s performance is mixed: in the time-domain (A.) the VAF is lower than the 78–99% analyzed by Kirsch et al., 1994; the gain response (B.) follows the profile in Figure 3 of Kirsch et al., 1994, while the phase response differs (C.). The response of the VEXAT model to the 90 Hz perturbations is much better: a VAF of 91% is reached in the time-domain (D.), the gain response follows the response of the cat soleus analyzed by Kirsch et al., 1994, while the phase-response follows biological muscle closely for frequencies higher than 30 Hz. Although the Hill’s time-domain response to the 15 Hz signal has a higher VAF than the VEXAT model (G.), the RMSE of the Hill model’s gain response (H.) and phase response (I.) shows it to be more in error than the VEXAT model. While the VEXAT model’s response improved in response to the 90 Hz perturbation, the Hill model’s response does not: the VAF of the time-domain response remains low (J.), neither the gain (K.) nor phase responses (L.) follow the data of Kirsch et al., 1994. Note that the Hill model’s…

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  Figure 7.. The stifffness-force and damping-force (impedance-force) relations of the models when coupled with elastic and rigid tendons. When coupled with an elastic-tendon, the stiffness (A) and damping (B) coefficients of best fit of both the VEXAT model and a Hill model increase with the tension developed by the MTU. However, both the stiffness and damping of the elastic-tendon Hill model are larger than Kirsch et al.’s coefficients (from Figure 12 of Kirsch et al., 1994), particularly at higher tensions. When coupled with rigid-tendon the stiffness (C) and damping (D) coefficients of the VEXAT model remain similar, as the values for koX and βoX have been calculated to take the tendon model into account (see Appendix 2.5 for details). In contrast, the stiffness and damping coefficients of the rigid-tendon Hill model differ dramatically from the elastic-tendon Hill model: while the elastic-tendon Hill model is too stiff and damped, the rigid-tendon Hill model is not stiff enough (compare A. to C.) and far too damped (compare B. to D.). Coupling the Hill model with a rigid-tendon increases the VAF from 30–51% to 86% but this improved accuracy is made using stiffness and damping that…

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  Figure 8.. The VAF of each model varies systematically with the type of model, perturbation amplitude, frequency, and the nominal tension. Kirsch et al., 1994 noted that the spring-damper model of best fit has a VAF of between 78–99% across all experiments. We have repeated the perturbation experiments to evaluate the VAF across a range of conditions: two different tendon models, three perturbation bandwidths (15 Hz, 35 Hz, and 90 Hz), three perturbation magnitudes (0.4 mm, 0.8 mm, and 1.6 mm), and ten nominal force values (spaced evenly between 2.5 N and 11.5 N). Each bar in the plot shows the mean VAF across all 10 nominal force values, with the whiskers extending to the minimum and maximum value that occurred in each set. The mean VAF of the VEXAT model changes by up to 36% depending on the condition, with the lowest mean VAF occurring in response to the 0.4mm 15 Hz perturbation with an elastic-tendon (A), and the highest mean VAF occurring in response to the 90 Hz perturbations with the rigid-tendon (C). In contrast, the mean VAF of the Hill model varies by up to 67% depending on the condition, with the lowest VAF occurring in the 15 Hz 0.4 mm trial with the elastic-tendon (A), and the…

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  Figure 9.. A comparison of the tension developed by the VEXAT and Hill models when actively lengthened at 9 mm/s on the descending limb of the force-length relation. Herzog and Leonard, 2002 actively lengthened (A) cat soleus muscles on the descending limb of the force-length curve (where l~M>1 in Figure 2A) and measured the force response of the MTU (B). After the initial transient at 2.4s the Hill model’s output force drops (B) because of the small region of negative stiffness (C) created by the force-length curve. In contrast, the VEXAT model develops steadily increasing forces between 2.4 and 3.4s and has a consistent level of stiffness (C) and damping (D).

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  Figure 10.. The passive force-length relations of two diffferent models of titin: a linear extrapolation and a WLC model. We consider two different versions of the force-length relation for each titin segment of the VEXAT model (A): a linear extrapolation, and a WLC model extrapolation. Leonard et al., 2010 observed that active myofibrils continue to develop increasing amounts of tension beyond actin-myosin overlap (B, grey lines with ±1 standard deviation shown). When this experiment is replicated using the VEXAT model (B, blue & magenta lines) and a Hill model (C red lines), only the VEXAT model with the linear extrapolated titin model is able to replicate the experiment with the titin-actin bond slipping off of the actin filament at 3.38 loM.

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  Figure 11.. The force-velocity relation of the VEXAT and Hill models. When the experiments of Hill, 1938 are simulated (A), the VEXAT model produces a force-velocity profile (blue dots) that approaches zero more rapidly during shortening than the embedded profile fV(⋅) (red lines). By scaling vmaxM by 0.95 the VEXAT model (magenta squares) is able to closely follow the force-velocity curve of the Hill model. While the force-velocity curves between the two models are similar, the time-domain force response of the two models differs substantially (B). The rigid-tendon Hill model exhibits a sharp nonlinear change in force at the beginning (0.1 s) and ending (0.21 s) of the ramp stretch.

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  Figure 12.. The passive and active force-length relation of the VEXAT and Hill models. When the passive and active force-length experiments of Gordon et al., 1966 are simulated, the VEXAT model (blue dots) and the Hill model (red lines) produce slightly different force-length curves (A) and force responses in the time-domain (B). The VEXAT model produces a right shifted active force-length curve, when compared to the Hill model due to the series elasticity of the XE element. By shifting the underlying curve by 2koX to the left the VEXAT model (magenta squares) can be made to exactly match the force-length characteristic of the Hill model.