Research Papers

Calculating Individual and Total Muscular Translational Stiffness: A Knee Example

[+] Author and Article Information
Joshua G. A. Cashaback

Department of Kinesiology,
McMaster University,
Hamilton, ON, L8S 2K1, Canada
e-mail: cashabjg@mcmaster.ca

Michael R. Pierrynowski

School of Rehabilitation Science,
McMaster University,
Hamilton, ON, L8S 2K1, Canada

Jim R. Potvin

Department of Kinesiology,
McMaster University,
Hamilton, ON, L8S 2K1, Canada

1Corresponding author. Present address: McMaster University, 1280 Main Street West, Hamilton, ON, L8S 2K1, Canada.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received October 2, 2012; final manuscript received March 30, 2013; accepted manuscript posted April 8, 2013; published online May 9, 2013. Assoc. Editor: Kenneth Fischer.

J Biomech Eng 135(6), 061006 (May 09, 2013) (7 pages) Paper No: BIO-12-1461; doi: 10.1115/1.4024162 History: Received October 02, 2012; Revised March 30, 2013; Accepted April 08, 2013

Research suggests that the knee joint may be dependent on an individual muscle's translational stiffness (KT) of the surrounding musculature to prevent or compensate for ligament tearing. Our primary goal was to develop an equation that calculates KT. We successfully derived such an equation that requires as input: a muscle's coordinates, force, and stiffness acting along its line of action. This equation can also be used to estimate the total joint muscular KT, in three orthogonal axes (AP: anterior-posterior; SI: superior-inferior; ML: medial-lateral), by summating individual muscle KT contributions for each axis. We then compared the estimates of our equation, using a commonly used knee model as input, to experimental data. Our total muscular KT predictions (44.0 N/mm), along the anterior/posterior axis (AP), matched the experimental data (52.2 N/mm) and was well within the expected variability (22.6 N/mm). We then estimated the total and individual muscular KT in two postures (0 deg and 90 deg of knee flexion), with muscles mathematically set to full activation. For both postures, total muscular KT was greatest along the SI-axis. The extensors provided the greatest KT for each posture and axis. Finally, we performed a sensitivity analysis to explore the influence of each input on the equation. It was found that pennation angle had the largest effect on SI KT, while muscle line of action coordinates largely influenced AP and ML muscular KT. This equation can be easily embedded within biomechanical models to calculate the individual and total muscular KT for any joint.

Copyright © 2013 by ASME
Topics: Muscle , Stiffness , Knee
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Grahic Jump Location
Fig. 1

Musculotendon coordinates of the insertion or node before (Bx, By, Bz) and after (Bx + δx, By, Bz) an infinitesimal perturbation (δx) along the x-axis. The (Ax, Ay, and Az) coordinates represent the origin of the muscle. Coordinates are taken from the joint center (0,0,0). l0 and l1 are the muscle length before and following the perturbation.

Grahic Jump Location
Fig. 2

Joint translation stiffness for the three orthogonal axes of the knee joint at 0 deg and 90 deg of knee flexion. The upper error bars represent the calculated standard deviation from the sensitivity analysis due to inputted variable uncertainty. The lower error bars exclude the uncertainty from pennation angle in order to demonstrate their large influence when estimating translational stiffness.

Grahic Jump Location
Fig. 3

Individual muscle contributions (%) to total muscular translational stiffness at (a) 0 deg and (b) 90 deg of knee flexion along the anterior/posterior (AP), superior/inferior (SI), and medial/lateral (ML) directions. The following muscles are included: vastus lateralis (VL), intermedius (VI), and medialis (VM), rectus femoris (RF), semimembranosus (SM), semitendinosus (ST), biceps femoris long (BFL) and short (BFS), sartorius (SA), tensor fascia latae (TFL), gracilis (GR), and gastrocnemius lateral (LG) and medial (MG).




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