Editing Hill's muscle model

{{Short description|Biomechanical paradigm explaining how muscles work}}
In [[biomechanics]], '''Hill's muscle model''' refers to the 3-element model consisting of a contractile element (CE) in series with a lightly-damped elastic spring element (SE) and in parallel with lightly-damped elastic parallel element (PE).  Within this model, the estimated force-velocity relation for the CE element is usually modeled by what is commonly called Hill's equation, which was based on careful experiments involving [[Tetanic contraction|tetanized]] [[muscle contraction]] where various muscle loads and associated velocities were measured.  They were derived by the famous [[physiologist]] [[Archibald Vivian Hill]], who by 1938 when he introduced this model and equation had already won the Nobel Prize for Physiology. He continued to publish in this area through 1970. There are many forms of the basic "Hill-based" or "Hill-type" models, with hundreds of publications having used this model structure for experimental and simulation studies.  Most major musculoskeletal simulation packages make use of this model.   

==AV Hill's force-velocity equation for 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), \qquad (1)</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 |doi-access=free
}}</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 
|author1=Martins, J.A.C. |author2=Pires, E.B |author3=Salvado, R. |author4=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
|issue=3–4 |pages= 419–433
|doi=10.1016/S0045-7825(97)00162-X|bibcode=1998CMAME.151..419M }}</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}, \qquad F^{CE}=F^{SE}, \qquad (2)</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}, \qquad L = L^{CE}+L^{SE}, \qquad (3)</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"/>

The forces in the parallel, series and contractile elements are defined by:<math display="block">F^{PE}(\lambda_f) = F_{0}f^{PE}(\lambda_f), \qquad F^{SE}(\lambda^{SE},\lambda^{CE}) = F_0f^{SE}(\lambda^{SE},\lambda^{CE}), \qquad F^{CE}(\lambda^{CE},\dot{\lambda}^{CE},a) = F_0f_{L}^{CE}(\lambda^{CE})f_{V}^{CE}(\dot{\lambda}^{CE})a, \qquad (4)</math>where <math display="inline">\lambda_f, \lambda_{CE}, \lambda_{SE}</math> are strain measures for the different elements defined by:<math display="block">\lambda_{f} = \frac{L}{L_0}, \quad \lambda^{CE} = \frac{L^{CE}}{L_0}, \quad \lambda^{SE} = \frac{L}{L^{CE}}, \qquad (5)</math>where <math display="inline">L</math> is the deformed muscle length and <math display="inline">L^{CE}</math> is the deformed muscle length due to motion of the contractile element, both from equation (3). <math display="inline">L_0</math> is the rest length of the muscle. <math>\lambda_{f}</math> can be split as <math display="inline">\lambda_{f} = \lambda^{SE}\lambda^{CE}</math>. The force term, <math>F_{0}</math>, is the peak isometric muscle force and the functions <math display="inline">f^{PE}, f^{SE}, f_L^{CE}, f_V^{CE}</math> are given by:<math display="block">\begin{array}{lcr}
f^{PE}(\lambda_f) = \begin{cases} 2cA(\lambda_f-1)e^{c(\lambda_f-1)^2}, & \lambda_f>1 \\ \text{0}, & \text{otherwise} \end{cases}, & (6) \\ [4pt]
f^{SE}(\lambda^{SE},\lambda^{CE}) = \begin{cases} 0.1(e^{100\lambda^{CE}(\lambda^{SE}-1)}-1), & \lambda^{SE}\geq1 \\ \text{0}, & \text{otherwise} \end{cases}, & (7) \\ [4pt]
f_L^{CE}(\lambda^{CE}) = \begin{cases} -4(\lambda^{CE}-1)^2+1, & 0.5\leq\lambda^{CE}\leq1.5 \\ \text{0}, & \text{otherwise} \end{cases}, & (8) \\ [4pt]
f_V^{CE}(\dot{\lambda}^{CE}) = \begin{cases} \text{0}, & \dot{\lambda}^{CE}<-10s^{-1} \\ -\frac{1}{\arctan(5)}\arctan(-0.5\dot{\lambda}^{CE})+1, & -10s^{-1}\leq\dot{\lambda}^{CE}\leq2s^{-1}\\ \frac{\pi}{4\arctan(5)}+1, & \dot{\lambda}^{CE}>2s^{-1} \end{cases}, & (9) 
\end{array}</math>

where <math>c, A</math> are empirical constants. The function <math>a(t)</math> from equation (4) represents the muscle activation. It is defined based on the ordinary differential equation:<math display="block">\frac{da(t)}{dt} = \frac{1}{\tau_{rise}}(1-a(t)u(t)+\frac{1}{\tau_{fall}}(a_{min}-a(t))(1-u(t))), \qquad (10)</math>where <math>\tau_{rise}, \tau_{fall}</math> are time constants related to rise and decay for muscle activation and <math>a_{min}</math> is a minimum bound, all determined from experiments. <math>u(t)</math> is the neural excitation that leads to muscle contraction.<ref>{{Cite journal|date=1990-01-01|title=An optimal control model for maximum-height human jumping|url=https://www.sciencedirect.com/science/article/abs/pii/002192909090376E|journal=Journal of Biomechanics|language=en|volume=23|issue=12|pages=1185–1198|doi=10.1016/0021-9290(90)90376-E|issn=0021-9290|last1=Pandy|first1=Marcus G.|last2=Zajac|first2=Felix E.|last3=Sim|first3=Eunsup|last4=Levine|first4=William S.|pmid=2292598|doi-access=free}}</ref><ref>{{Cite journal|last1=Martins|first1=J. A. C.|last2=Pato|first2=M. P. M.|last3=Pires|first3=E. B.|date=2006-09-01|title=A finite element model of skeletal muscles|url=https://doi.org/10.1080/17452750601040626|journal=Virtual and Physical Prototyping|volume=1|issue=3|pages=159–170|doi=10.1080/17452750601040626|s2cid=137665181|issn=1745-2759|url-access=subscription}}</ref>

===Viscoelasticity===
Muscles present [[viscoelasticity]], therefore a viscous damper may be included in the model, when the [[dynamics (mechanics)|dynamics]] of the [[Order (differential equation)|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, \qquad (11)</math>
is added to the model's global equation, whose <math>k</math> and <math>a</math> are constants.<ref name="Fung"/>

==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]]