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Research Papers

A Nonlinear Model of Passive Muscle Viscosity

[+] Author and Article Information
G. A. Meyer, A. D. McCulloch

Department of Bioengineering,  University of California, San Diego La Jolla, CA 92093

R. L. Lieber1

Department of Orthopaedic Surgery,  University of California, San Diego and Veterans Affairs Medical Center, La Jolla, Ca 92093 e-mail: rlieber@ucsd.edu

1

Address for correspondence: Richard L. Lieber, Department of Orthopaedic Surgery (9151) Ph.D, V.A. Medical Center and U.C. San Diego, 3525 John Hopkins Court, San Diego, CA 92121.

J Biomech Eng 133(9), 091007 (Oct 11, 2011) (9 pages) doi:10.1115/1.4004993 History: Received March 01, 2011; Accepted August 17, 2011; Published October 11, 2011; Online October 11, 2011

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of a pseudoplastic model of passive muscle mechanics. Figure 1 Modified version of the Hill 3-element model [6]. Spring elements are linear due to the linear stress-strain behavior of mouse muscle fibers and the dashpot is a nonlinear element whose behavior is a function of time, strain and strain rate. Figure 1 Schematic of two step strain inputs for a stress relaxation test, one at high strain rate (ɛ·1) and one a low strain rate (ɛ·2). Viscosity in this model is a function of time, strain and strain rate η(t,ɛ,ɛ·)) illustrated graphically in (Fig 1). The shape of the resulting stress relaxation curves are shown in (Fig 1) where the stretch at the higher strain rate results in the higher peak stress (σ(ɛ·1,η)).

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Figure 2

The pseudoplastic model better represents stress relaxation data from a mouse muscle fiber compared to the 3rd order Hill structural model. The fiber was stretched to 30% FL at 20 FL/s to approximate an instantaneous length change. The pseudoplastic model (red) is a better fit to the raw data (black) than the 3rd order Hill model (blue) during the phase of fast relaxation. Inset shows the data magnified over the first 0.2 s of stress relaxation.

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Figure 3

Change in viscosity over time during fiber stress relaxation. Individual data points (black circles) were derived from 1st order Hill fits to the stress relaxation of a fiber at discrete time points. Eq. 3 provided a good fit to the data with an r-squared value of 0.998.

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Figure 4

The pseudoplastic model better represents the strain rate sensitivity of a mouse muscle fiber than the 3rd order Hill structural model. A single mouse muscle fiber was strained to 50% FL at 20 FL/s (Fig. 4 and at 2 FL/s (Fig. 4). The 3rd order Hill model (blue) and the pseudoplastic model (red) were fit to raw data (black) at 20 FL/s and then used to predict behavior during a 2 FL/s stretch. The 3rd order Hill model underestimates stress at 2 FL/s, while the pseudoplastic model accurately represents peak stress.

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Figure 5

Mouse muscle fibers exhibit pseudoplasticity. Figure 5 single fiber was strained to 40% FL at three different rates. The fastest rate stretch resulted in the fastest stress decay indicating the lowest viscosity. All strain ramps had the same time interval and the same peak stress. Figure 5 single fiber was strained from slack length to 20% FL at three different rates. Again, the fastest stretch had the fastest stress decay, indicative of pseudoplastic behavior. Noise on the traces represents less than 1 mV noise at these low stresses.

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Figure 6

Mouse muscle fibers do not obey superposition. Figure 66(a) Peak (closed circles) and relaxed (open circles) stresses from a single fiber strained incrementally at 10% FL increments (blue) and 20% FL increments (turquoise). These data were used to predict stresses from a stretch from slack to 40% FL based on the principle of superposition (asterisks). Predictions match experimental data for the relaxed stress (open green circle), but overestimate the peak stress by as much as 95% (closed green circle). Figure 66(b) raw stress relaxation data from a fiber stretched in 10% FL increments (black). The 3rd order Hill model is fit to stretches of the same fiber from slack but provides a poor prediction of incremental stress (blue). The pseudoplastic model overcomes the limitations of superposition by adding strain dependent viscous parameters and provides a much better fit (red). Relaxation plots are truncated at 0.3 s to show the initial fast phase of relaxation. Both models converge with the raw data at the fully relaxed stress.

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Figure 7

Viscosity is a function of strain in mouse muscle fibers. Figure 7 Normalized stress relaxation data from a single fiber subjected to an incremental stress relaxation test from slack length to 50% FL at 20 FL/s. Stress is decay is slower with increasing strain, indicating fiber viscosity is increasing as a function of resting strain. Figure 7 The pseudoplastic model was used to locally fit incremental stress relaxation data from six fibers from slack length to 100% FL at 20 FL/s. Parameters and both increase linearly as a function of resting strain, but remains relatively constant.

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Figure 8

Schematics of the classic Hill model of muscle viscoelasticity (A) and the 3rd order Hill model (generalized Maxwell model) (B). The contractile element is represented by a dashpot with damping constant c since only passive muscle mechanics are considered here. The series and parallel springs are represented by linear spring constants of modulus ks and kp respectively. The 3rd order Hill model includes two additional branches of dashpot and series spring in parallel, which add two additional time constants to the stress decay during relaxation.

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Figure 9

The 3rd order Hill model better represents stress relaxation data from a mouse muscle fiber than the 1st order Hill structural model. The fiber was stretched to 30% FL at 20 FL/sec to approximate an instantaneous length change. The 3rd order Hill model (blue) provides a better fit to the raw data (black) than the 1st order Hill model (green) during the phase of fast relaxation. Inset shows the data magnified over the first 0.2 seconds of stress relaxation.

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