Schwinger function

2013-11-18T07:46:38Z

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In [[biomechanics]], '''Hill's muscle model''' refers to either Hill's equations for [[Tetanic contraction|tetanized]] [[muscle contraction]] or to the 3-element model. They were derived by the famous [[physiologist]] [[Archibald Vivian Hill]].

==Equation to tetanized muscle==
This is a popular [[state equation]] applicable to [[skeletal muscle]] that has been stimulated to show [[Tetanic contraction]]. It relates [[stress (mechanics)|tension]] to velocity with regard to the internal [[thermodynamics]]. The equation is

:<math>\left(v+b\right)(F+a) = b(F_0+a)</math>

where
* <math>F</math> is the tension (or load) in the muscle
* <math>v</math> is the velocity of contraction
* <math>F_0</math> is the maximum isometric tension (or load) generated in the muscle
* <math>a</math> coefficient of shortening heat
* <math>b=a\cdot v_0/F_0</math>
* <math>v_0</math> is the maximum velocity, when <math>F=0</math>

Although Hill's equation looks very much like the [[van der Waals equation]], the former has units of energy [[dissipation]], while the latter has units of [[energy]]. Hill's equation demonstrates that the relationship between F and v is [[Hyperbolic growth|hyperbolic]]. Therefore, the higher the load applied to the muscle, the lower the contraction velocity. Similarly, the higher the contraction velocity, the lower the tension in the muscle. This hyperbolic form has been found to fit the empirical constant only during [[Isotonic (exercise physiology)|isotonic contractions]] near resting length.<ref name="Hill_1938">{{cite journal
|author=Hill, A.V.
|title=The heat of shortening and dynamics constants of muscles
|journal= Proc. R. Soc. Lond. B
|publisher=Royal Society
|location=London
|date=October 1938
|volume=126
|issue=843
|pages= 136–195
|doi=10.1098/rspb.1938.0050 }}</ref>

The muscle tension decreases as the shortening velocity increases. This feature has been attributed to two main causes. The major appears to be the loss in tension as the cross bridges in the [[Sarcomere|contractile element]] and then reform in a shortened condition. The second cause appears to be the fluid viscosity in both the contractile element and the connective tissue. Whichever the cause of loss of tension, it is a [[viscous friction]] and can therefore be modeled as a fluid [[dashpot|damper]]
.<ref name="Fung">{{cite book
|author=Fung, Y.-C.
|title=Biomechanics: Mechanical Properties of Living Tissues
|publisher=Springer-Verlag
|location=New York
|year=1993
|pages= 568
|isbn=0-387-97947-6}}</ref>

==Three-element model==
[[Image:Lengthtension.jpg|thumb|right|400px|Muscle length vs Force. In Hill's muscle model the active and passive forces are respectively <math>F_{CE}</math> and <math>F_{PE}</math>.]]
[[Image:Hill muscle model.svg|150px|right|thumb|Hill's elastic muscle model. F: Force; CE: Contractile Element; SE: Series Element; PE: Parallel Element.]]

The '''three-element Hill muscle model''' is a representation of the muscle mechanical response. The model is constituted by a contractile element ('''CE''') and two [[non-linear]] [[Spring (device)|spring elements]], one in [[Series and parallel circuits|series]] ('''SE''') and another in parallel ('''PE'''). The active [[force]] of the contractile element comes from the force generated by the [[actin]] and [[myosin]] cross-bridges at the [[sarcomere]] level. It is fully extensible when inactive but capable of shortening when activated. The [[connective tissue]]s ([[fascia]], [[epimysium]], [[perimysium]] and [[endomysium]]) that surround the contractile element influences the muscle's force-length curve. The parallel element represents the passive force of these connective tissues and has a [[soft tissue]] mechanical behavior. The parallel element is responsible for the muscle passive behavior when it is [[stretched]], even when the contractile element is not activated. The series element represents the [[tendon]] and the intrinsic elasticity of the myofilaments. It also has a soft tissue response and provides energy storing mechanism.<ref name="Fung"/><ref name="Martins">{{cite journal
|author=Martins, J.A.C.; Pires, E.B; Salvado, R.; Dinis, P.B.
|title=Numerical model of passive and active behavior of skeletal muscles
|journal=Computer methods in applied mechanics and engineering
|publisher=Elsevier
|year=1998
|volume=151
|pages= 419–433
|doi=10.1016/S0045-7825(97)00162-X}}</ref>

The net force-length characteristics of a muscle is a combination of the force-length characteristics of both active and passive elements. The forces in the contractile element, in the series element and in the parallel element, <math>F_{CE}</math>, <math>F_{SE}</math> and <math>F_{PE}</math>, respectively, satisfy
:<math>F = F_{PE}+F_{SE} \quad \mathrm{and} \quad F_{CE}=F_{SE} \;.</math>

On the other hand, the muscle length <math>L</math> and the lengths <math>L_{CE}</math>, <math>L_{SE}</math> and <math>L_{PE}</math> of those elements satisfy
:<math>L = L_{PE} \quad \mathrm{and} \quad L = L_{CE}+L_{SE} \;.</math>

During [[Isometric exercise|isometric contractions]] the series elastic component is under tension and therefore is stretched a finite amount. Because the overall length of the muscle is kept constant, the stretching of the series element can only occur if there is an equal shortening of the contractile element itself.<ref name="Fung"/>

===Viscoelasticity===
Muscles present viscoelasticity, therefore a viscous damper may be included in the model, when the [[dynamics (mechanics)|dynamics]] of the second-order critically damped twitch is regarded. One common model for muscular viscosity is an [[Exponentiation|exponential]] form damper, where
:<math>F_{D} = k(\dot{L}_{D})^a</math>
is added to the model's global equation, whose <math>k</math> and <math>a</math> are constants.<ref name="Fung"/>
==Matlab Model==
From the following website, http://youngmok.com/hill-type-muscle-model-with-matlab-code/ it is possible to download Matlab codes for the Hill type muscle model (M-file and Simulink Block).

==See also==
* [[Muscle_contraction#Force-length_and_force-velocity_relationships|Muscle contraction]]

==References==
<references/>

{{DEFAULTSORT:Hill's Muscle Model}}
[[Category:Biomechanics]]
[[Category:Equations]]
[[Category:Exercise physiology]]

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Schwinger function