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Annals of Biomedical Engineering

, Volume 47, Issue 11, pp 2168–2177 | Cite as

The Contributions of Individual Muscle–Tendon Units to the Plantarflexor Group Force–Length Properties

  • Mehrdad Javidi
  • Craig P. McGowan
  • David C. LinEmail author
Article

Abstract

The combined force–length (F–L) properties of a muscle group acting synergistically at a joint are determined by several aspects of the F–L properties of the individual musculotendon units. Namely, misalignment of the optimal lengths of the individual muscles will affect the group F–L properties. This misalignment, which we named \(M_{\text{opt}}^{\text{MT}}\), arises from the properties of the muscles (i.e., optimum fiber length and pennation angle) and of their tendons (i.e., compliance and slack length). The aim of this study was to measure the F–L properties of kangaroo rat plantarflexors as a group and individually and determine the effects of \(M_{\text{opt}}^{\text{MT}}\) on the group F–L properties. Specifically, we performed a sensitivity analysis to quantify how \(M_{\text{opt}}^{\text{MT}}\) influences the tradeoff between maximizing the peak force vs. having a wider group F–L curve. In the kangaroo rat, we found that the optimal lengths of two bi-articular musculotendon units, the plantaris and the gastrocnemius, were misaligned by 1.8 mm, but this amount favored maximal peak force rather than increasing F–L curve width. Because we measured the misalignment in situ, we could directly assess the tradeoff between maximizing peak force vs. a wider F–L curve without making modeling assumptions about the individual muscle or tendon properties.

Keywords

Kangaroo rat Muscle optimal length Force–length properties 

Abbreviations

F

Force measured by servomotor (N)

\(L^{\text{MT}}\)

Musculotendon length equal to distance between origin of muscles and motor arm (mm)

\(\Delta L^{\text{MT}}\)

Change in musculotendon length (mm)

FGRP

Musculotendon force of the muscle group (N)

\(F_{\text{a}}^{\text{GRP}}\)

Active force of muscle group (N)

\(F_{\text{p}}^{\text{GRP}}\)

Passive force of muscle group (N)

\(F_{\text{a}}^{\text{M}}\)

Active force of muscle M (either gastrocnemius (GAS) or plantaris (PL)) (N)

\(F_{\text{p}}^{\text{M}}\)

Passive force of muscle M (N)

\(F_{\text{o}}^{\text{GRP}}\)

Maximum isometric force of the muscle group (N)

\(F_{\text{o}}^{\text{M}}\)

Maximum isometric force of the muscle M (N)

\(L_{\text{o - GRP}}^{\text{MT}}\)

Musculotendon length at maximum isometric force of muscle group (mm)

\(L_{\text{o}}^{{{\text{MT}}^{ *} }}\)

Initial musculotendon length, which is an estimation of \(L_{\text{o - GRP}}^{\text{MT}}\) (mm)

\(L_{\text{o - M}}^{\text{MT}}\)

Musculotendon length at maximum isometric force of muscle M (mm)

\(L_{\text{o}}^{\text{M}}\)

The length of muscle belly M at maximum isometric force of muscle M (mm)

\(L_{\text{c}}^{\text{M}}\)

Distance between the pair of sonometric crystals inserted into muscle M (mm)

\(L_{\text{o - c}}^{\text{M}}\)

Distance between crystal pair at maximum isometric force of muscle M (mm)

\(\theta_{\text{f}}^{\text{M}}\)

Pennation angle of muscle M (degrees)

\(L_{\text{f}}^{\text{M}}\)

Fiber length of muscle M (mm)

\({\text{CSA}}^{\text{M}}\)

Functional cross sectional area of muscle M (mm2)

\(\sigma_{\max}\)

Maximum isometric stress (kPa)

\(L_{\text{s - M}}^{\text{T}}\)

Slack length of the tendon of muscle M (mm)

\(\varepsilon_{\text{max - M}}^{\text{T}}\)

Tendon strain of muscle M at its maximum isometric force (%\(L_{\text{s - M}}^{\text{T}}\))

\({\text{L}}^{\text{ac}}\)

Relative displacement between the two muscle–tendon units (mm)

\(M_{\text{opt}}^{\text{MT}}\)

Distance between \(L_{\text{o - M}}^{\text{MT}}\) of two different muscles (mm)

\(F_{\max}\)

Maximum force of the F–L curve of muscle group calculated by model (N)

W

Width of the F–L curve of the muscle group calculated by model (mm)

Notes

Acknowledgments

Work supported by Army Research Office #66554-EG (DCL and CPM) and National Science Foundation #1553550 (CPM).

Supplementary material

10439_2019_2288_MOESM1_ESM.pdf (37 kb)
Supplementary material 1 (PDF 42 kb)

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Copyright information

© Biomedical Engineering Society 2019

Authors and Affiliations

  1. 1.Voiland School of Chemical Engineering and BioengineeringWashington State UniversityPullmanUSA
  2. 2.Department of Biological SciencesUniversity of IdahoMoscowUSA
  3. 3.WWAMI Medical Education ProgramUniversity of IdahoMoscowUSA
  4. 4.Washington Center for Muscle BiologyWashington State UniversityPullmanUSA
  5. 5.Department of Integrative Physiology and NeuroscienceWashington State UniversityPullmanUSA

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