Biological Cybernetics

, Volume 97, Issue 1, pp 63–79

High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models

Original Paper

DOI: 10.1007/s00422-007-0160-6

Cite this article as:
Günther, M., Schmitt, S. & Wank, V. Biol Cybern (2007) 97: 63. doi:10.1007/s00422-007-0160-6

Abstract

High-frequency vibrations e.g., induced by legs impacting with the ground during terrestrial locomotion can provoke damage within tendons even leading to ruptures. So far, macroscopic Hill-type muscle models do not account for the observed high-frequency damping at low-amplitudes. Therefore, former studies proposed that protective damping might be explained by modelling the contractile machinery of the muscles in more detail, i.e., taking the microscopic processes of the actin–myosin coupling into account. In contrast, this study formulates an alternative hypothesis: low but significant damping of the passive material in series to the contractile machinery—e.g., tendons, aponeuroses, titin—may well suffice to damp these hazardous vibrations. Thereto, we measured the contraction dynamics of a piglet muscle–tendon complex (MTC) in three contraction modes at varying loads and muscle–tendon lengths. We simulated all three respective load situations on a computer: a Hill-type muscle model including a contractile element (CE) and each an elastic element in parallel (PEE) and in series (SEE) to the CE pulled on a loading mass. By comparing the model to the measured output of the MTC, we extracted a consistent set of muscle parameters. We varied the model by introducing either linear damping in parallel or in series to the CE leading to accordant re-formulations of the contraction dynamics of the CE. The comparison of the three cases (no additional damping, parallel damping, serial damping) revealed that serial damping at a physiological magnitude suffices to explain damping of high-frequency vibrations of low amplitudes. The simulation demonstrates that any undamped serial structure within the MTC enforces SEE-load eigenoscillations. Consequently, damping must be spread all over the MTC, i.e., rather has to be de-localised than localised within just the active muscle material. Additionally, due to suppressed eigenoscillations Hill-type muscle models taking into account serial damping are numerically more efficient when used in macroscopic biomechanical neuro-musculo-skeletal models.

List of symbols

MTC

Muscle–tendon complex

CE

Contractile element

PE

Parallel element

PEE

Parallel elastic element

SE

Serial element

SEE

Serial elastic element

SOL

M. soleus

FDS

M. flexor digitorum superficialis

GM

M. gastrocnemius medialis

GL

M. gastrocnemius lateralis

q

normalised muscle activation

q0

minimum value of q

\(\dot{q}\)

time derivative of q

τq

time constant of rising activation

βq

ratio between τq and time constant of falling activation

STIM

Muscle stimulation

lCE

length of CE

\(\dot{l}_{{\rm CE}} = v_{{\rm CE}}\)

contraction velocity of CE

l0

mean anatomical length of MTC

lm

length of model MTC

lm,0

typical length of model MTC

\(\dot{l}_{{\rm m}}\)

velocity of model MTC

Fm

force of model MTC

FSEE

force of SEE

FPEE

force of PEE

FCE

force of CE

lSE

length of SE

lSEE,0

rest length of SEE

lSEE,nll

length of SEE at non-linear-linear transition in FSEE(lSE)

ΔUSEE,nll

relative stretch at non-linear-linear transition in FSEE(lSE)

ΔFSEE,0

force at non-linear-linear transition in FSEE(lSE)

ΔUSEE,l

relative stretch in linear part for force increase ΔFSEE,0

KSEE,l

stiffness of the linear part of FSEE(lSE)

νSEE

exponent of FSEE(lSE) in the non-linear part

KSEE,nl

factor of non-linearity in FSEE(lSE)

Fisom

normalised isometric force–length relation of CE

lCE,opt

optimal fibre length

νCE,limb

exponent of Fisom(lCE) on either ascending or descending limb

ΔWlimb

width of Fisom(lCE) on either ascending or descending limb

Fmax

maximum isometric force

Arel

coordinate of pole in \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) normalised to current isometric force Fmax qFisom(lCE)

Arel,0

maximum value of Arel

Brel

coordinate of pole in \(F_{{\rm CE}}(\dot{l}_{CE})\) normalised to lCE,opt

Brel,0

maximum value of Brel

vmax

concentric contraction velocity at FCE = 0

vmax,0

maximum concentric contraction velocity

\(L_{A_{{\rm rel}}}\)

length dependency of Arel

\(L_{B_{{\rm rel}}}\)

length dependency of Brel

\(Q_{A_{{\rm rel}}}\)

activation dependency of Arel

\(Q_{B_{{\rm rel}}}\)

activation dependency of Brel

lPEE,0

rest length of PEE

νPEE

exponent of FPEE(lCE)

KPEE

factor of non-linearity in FPEE(lCE)

\({\mathcal{F}}_{{\rm PEE}}\)

force of PEE if lCE is stretched to ΔWlimb=des

\({\mathcal{L}}_{{\rm PEE},0}\)

rest length of PEE normalised to lCE,opt

FEPS

numerical limit for defining zero Fisom(lCE)

d V d Fcon

inclination of linear concentric continuation of \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) for FCE < 0

Secc

step in inclination of \(F_{{\rm CE}}(\dot{l}_{{\rm CE}} = 0)\) between eccentric and concentric force–velocity relation(s)

\({\mathcal{F}}_{{\rm ecc}}\)

coordinate of pole in \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) normalised to Fmax q Fisom(lCE) for \(\dot{l}_{{\rm CE}} > 0\)

Ftrans

force where linear continuation of eccentric \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) relation starts

vtrans

velocity where linear continuation of eccentric \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) relation starts

d V d Fecc

inclination of linear eccentric continuation of \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) for FCE > Ftrans

lPE

length of PEE ( = lCE)

dPE

(constant) damping coefficient of PE

dSE

damping coefficient of SE

dSE,max

maximum value in dSE(lCE, q)

RSE

minimum value of dSE normalised to dSE,max

DSE

dimensionless factor to scale dSE,max

\(\dot{l}_{{\rm SE}} = v_{{\rm SE}}\)

contraction velocity of SE

t

time

g

vector of gravitational acceleration

dext

modelled external damping

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut für Astronomie und Astrophysik, Abteilung Theoretische Astrophysik, Biomechanik-GruppeEberhard-Karls-UniversitätTübingenGermany
  2. 2.Orthopädische Klink, BiomechaniklaborEberhard-Karls-UniversitätTübingenGermany
  3. 3.Institut für Sportwissenschaft, Lehrstuhl für BewegungswissenschaftFriedrich-Schiller-UniversitätJenaGermany
  4. 4.Institut für Sportwissenschaft, Arbeitsbereich IIIEberhard-Karls-UniversitätTübingenGermany
  5. 5.Institut für Sport und SportwissenschaftAlbert-Ludwigs-UniversitätFreiburgGermany