Hill's model
Encyclopedia
Hill's muscle model refers to either Hill's equations for tetanized muscle contraction
or to the 3-element model. They were derived by a famous physiologist named Archibald Vivian Hill.
that has been stimulated to show Tetanic contraction
. It relates tension to velocity with regard to the internal thermodynamics
. The equation is
where
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
. 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 contractions
near resting length.
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 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 damper
.
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 elements
, one in 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.
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,
, 
and 
, respectively, satisfy
On the other hand, the muscle length
and the lengths 
, 
and 
of those elements satisfy
During 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.
of the second-order critically damped twitch is regarded. One common model for muscular viscosity is an exponential
form damper, where
is added to the model's global equation, whose
and 
are constants.
Muscle contraction
Muscle fiber generates tension through the action of actin and myosin cross-bridge cycling. While under tension, the muscle may lengthen, shorten, or remain the same...
or to the 3-element model. They were derived by a famous physiologist named Archibald Vivian Hill.
Equation to tetanized muscle
This is a popular state equation applicable to skeletal muscleSkeletal muscle
Skeletal muscle is a form of striated muscle tissue existing under control of the somatic nervous system- i.e. it is voluntarily controlled. It is one of three major muscle types, the others being cardiac and smooth muscle...
that has been stimulated to show Tetanic contraction
Tetanic contraction
A tetanic contraction occurs when a motor unit has been maximally stimulated by its motor neuron. This occurs when a muscle's motor unit is stimulated at a sufficiently high frequency of multiple impulses. Each stimulus causes a twitch. If stimuli are delivered slow enough, the tension in the...
. It relates tension to velocity with regard to the internal thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
. The equation is
where
-
is the tension (or load) in the muscle -
is the velocity of contraction -
is the maximum isometric tension (or load) generated in the muscle -
coefficient of shortening heat -
-
is the maximum velocity, when 
Although Hill's equation looks very much like the van der Waals equation
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...
, the former has units of energy dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
, while the latter has units of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
. Hill's equation demonstrates that the relationship between F and v is hyperbolic
Hyperbolic growth
When a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth.More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as x \to 0 is infinity: any similar graph is said to exhibit...
. 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 contractions
Isotonic (exercise physiology)
In an isotonic contraction, tension remains unchanged and the muscle's length changes. Lifting an object at a constant speed is an example of isotonic contractions. A near isotonic contraction is known as Auxotonic contraction....
near resting length.
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 contractile element
Sarcomere
A sarcomere is the basic unit of a muscle. Muscles are composed of tubular muscle cells . Muscle cells are composed of tubular myofibrils. Myofibrils are composed of repeating sections of sarcomeres, which appear under the microscope as dark and light bands...
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 damper
Dashpot
A dashpot is a mechanical device, a damper which resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly used in conjunction with a spring...
.
Three-element model
Spring (device)
A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...
, one in series
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...
(SE) and another in parallel (PE). The active force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
of the contractile element comes from the force generated by the actin
Actin
Actin is a globular, roughly 42-kDa moonlighting protein found in all eukaryotic cells where it may be present at concentrations of over 100 μM. It is also one of the most highly-conserved proteins, differing by no more than 20% in species as diverse as algae and humans...
and myosin
Myosin
Myosins comprise a family of ATP-dependent motor proteins and are best known for their role in muscle contraction and their involvement in a wide range of other eukaryotic motility processes. They are responsible for actin-based motility. The term was originally used to describe a group of similar...
cross-bridges at the sarcomere
Sarcomere
A sarcomere is the basic unit of a muscle. Muscles are composed of tubular muscle cells . Muscle cells are composed of tubular myofibrils. Myofibrils are composed of repeating sections of sarcomeres, which appear under the microscope as dark and light bands...
level. It is fully extensible when inactive but capable of shortening when activated. The connective tissue
Connective tissue
"Connective tissue" is a fibrous tissue. It is one of the four traditional classes of tissues . Connective Tissue is found throughout the body.In fact the whole framework of the skeleton and the different specialized connective tissues from the crown of the head to the toes determine the form of...
s (fascia
Fascia
A fascia is a layer of fibrous tissue that permeates the human body. A fascia is a connective tissue that surrounds muscles, groups of muscles, blood vessels, and nerves, binding those structures together in much the same manner as plastic wrap can be used to hold the contents of sandwiches...
, epimysium
Epimysium
Epimysium is a layer of connective tissue, which ensheaths the entire muscle. It is composed of dense irregular connective tissue. It is continuous with fascia and other connective tissue wrappings of muscle including the endomysium, and perimysium. It is also continuous with tendons where it...
, perimysium
Perimysium
Perimysium is a sheath of connective tissue that groups muscle fibers into bundles or fascicles.Recent advances in muscle physiology suggest that the perimysium plays a role in transmitting lateral contractile movements...
and endomysium
Endomysium
The endomysium, meaning within the muscle, is a layer of connective tissue that ensheaths a muscle fiber and is composed mostly from reticular fibers. It also contains capillaries, nerves, and lymphatics...
) 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
Soft tissue
In anatomy, the term soft tissue refers to tissues that connect, support, or surround other structures and organs of the body, not being bone. Soft tissue includes tendons, ligaments, fascia, skin, fibrous tissues, fat, and synovial membranes , and muscles, nerves and blood vessels .It is sometimes...
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
Tendon
A tendon is a tough band of fibrous connective tissue that usually connects muscle to bone and is capable of withstanding tension. Tendons are similar to ligaments and fasciae as they are all made of collagen except that ligaments join one bone to another bone, and fasciae connect muscles to other...
and the intrinsic elasticity of the myofilaments. It also has a soft tissue response and provides energy storing mechanism.
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,
, and , respectively, satisfyOn the other hand, the muscle length
and the lengths , and of those elements satisfyDuring isometric contractions
Isometric exercise
Isometric exercise or isometrics are a type of strength training in which the joint angle and muscle length do not change during contraction...
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.
Viscoelasticity
Muscles present viscoelasticity, therefore a viscous damper may be included in the model, when the dynamicsDynamics
Dynamics may refer to:-Physics and engineering:* Dynamics , the time evolution of physical processes** Aerodynamics, the study of gases in motion...
of the second-order critically damped twitch is regarded. One common model for muscular viscosity is an exponential
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
form damper, where
is added to the model's global equation, whose
and are constants.