1.1 The Muscle
Before any mechanical models of the muscle can be discussed, an understanding of the underlying biology and physiology is required. A large muscle, such as the quadriceps is shown in Fig. 1.1. This muscle is connected to the joints by tendons at the ends of the muscle. In most muscles, the tendon and the muscle fibers are not in direct alignment; that is, the orientation of the muscle fibers is not parallel to the orientation of the tendons. The angle between the muscle fibers and tendon is called the pennation angle. The pennation angle affects the amount of force the muscle applies to the joints between which it is connected. For muscles with large pennation angles, if the muscle generates a large force, very little of this force can be harnessed to cause motion in the joint limbs; instead, most of the force generated in the muscle is wasted in deforming the shape of the muscle or tendon.
Figure 1.1: Schematic of a muscle/tendon unit, such as the quadriceps [Kaufman 89]
A detailed breakdown of a large muscle’s architecture, such as the quadriceps of Fig. 1.1, is shown in Fig. 1.2. The top muscle in the figure is identical to that of Fig. 1.1, except the tendons have been removed, leaving only the muscle behind. This large muscle is divided into groups of muscle fibers which in turn are divided into myofibrils.
Myofibrils are long slender parallel cylinders of muscle protein. These myofibril bundles are sectioned along their axial length into series of contractile units called sarcomeres. Each sarcomere is approximately 3 μm in length. The section of myofibril shown in Fig. 1.2 contains two sarcomeres, one of which is circled to make it easier to identify. The sarcomeres of the myofibril are the force generating units of the muscle.
The myofibrils are composed of myofilaments which are groupings of proteins. The proteins composing the myofilaments, shown in the lowest levels of the figure, are known as myosin and actin filaments. The myosin filaments are much thicker than the actin filaments, with six actin filaments surrounding each myosin filament. Force is generated when the myosin heads, the tear-dropped shaped objects hanging off the myosin, bind to the actin filament and pull the actin fibers along the length of the myosin fibers. When the myosin head binds with the actin filament, a cross-bridge is formed. This is known as the sliding filament theory of muscle contraction and was developed by the Huxleys’ in the 1950s with the aide of electron microscopy. The actual mechanisms allowing the myosin heads to bind with and then pull on the actin fibers will be discussed in greater detail in a later section of this chapter. To give a feel for the dimensions upon which contractions occur, the myosin heads are a few nanometers in diameter, and the spacing between the myosin heads is between 10 nanometers and 100 nanometers.
1.2 Motor Unit Recruitment during a Contraction
Although the actual force development mechanism occurs at the sarcomere level, a motor unit, defined as a nerve axon and the set of muscle fibers the axon is capable of exciting, is the smallest functional unit associated with force generation. A muscle fiber is comprised of a grouping of homogeneous sarcomeres of equal length that are simultaneously neurally excited to generate force [Zajac, 89]. The size of the motor units relative to the other components of the muscle is shown in the top part of Fig. 1.2.
A muscle is then composed of n motor units controlled by n separate nerve axons, Fig. 1.3. In order for force to be generated, the brain sends an electrical signal through the nerves to the motor, causing the muscle to contract. A pictorial depiction of the electrical impulses, u1(t), u2(t),… traveling from the brain along the nerve to stimulate the motor units into generating force is shown in Fig. 1.3. Since the motor units are aligned in parallel, the force generated by the muscle fibers of the individual motor units, F1M, F2M…, sum together to produce the net muscle force. The properties of a motor unit are then just scaled versions of the muscle fibers which in turn are scaled versions of the sarcomere [Zajac 89].
Three distinctive types of motor units exist: fatigue resistant motor units, slow fatiguing motor units, and fast fatiguing motor units. Fatigue resistant motor units generate little force, but can continually generate that force almost indefinitely. Slow fatiguing motor units generate larger forces than fatigue resistant motor units, but they can not generate force indefinitely; they will tire after a time and stop producing force. The final set of motor units, the fast fatiguing motor units, is capable of generating large force. These large forces however, can not be maintained for long periods of time and these motor units quickly fatigue and produce no more force.
Figure 1.2: Architecture of a muscle [Warwick 73]
Figure
1.3: Collection of muscle fibers into the motor
units which comprise a single muscle [Zajac 89]
To prevent rapid fatigue of a muscle and to ensure the muscles are efficient in generating force, the brain recruits motor units in a well-established manner. This order is shown in Fig. 1.4. When the muscle needs to generate a force, the brain begins by sending the electrical signal to the smaller, fatigue resistant motor units first. The brain actives the fewest number of motor units needed in order to generate enough force to perform the demanded task. When motor units are recruited, the brain starts with the smallest motor unit first. As more force is needed, the brain sends electrical signals to the motor unit more quickly. The shorter the time between electrical pulses from the brain, the more force a muscle generates. In Fig. 1.4, these pulses are shown schematically in the lower section of the figure. As shown in the figure, the pulses are at first sent slowly to the first motor unit (M.U.1). As more force is needed, the resting time between pulses decreases. This continues until the limit or threshold force of the motor unit is reached. At this point, the motor unit excitation rate becomes constant, and another motor unit is recruited to help generate force.
Figure 1.4: Recruitment order of motor units (M.U.) [Maurel 98]
The threshold forces, shown in the top section of Fig. 1.4, are the forces at which the current number of motor units can no longer generate enough force and more motor units must be recruited. Recruiting more motor units requires the activation of the larger motor units in order to produce the required force. These motor units are no longer the fatigue resistant units and are susceptible to fatigue. The brain again begins stimulating these motor units at a slow rate just large enough to generate adequate force. As more force is needed, the brain sends signals more rapidly to this motor unit. Once this motor unit’s threshold has been reached, the brain recruits the next motor unit. This process continues until a sufficient number of motor units have been recruited for equilibrium to be reached between the force demanded and force generated. The brain attempts to distribute the demand on the fatigable motor units across motor units with similar force generating capabilities. The brain does this so a single motor unit is not constantly activated, resulting in slower muscle fatigue and by allowing the motor units to have some rest, the motor units can begin to recover from fatigue. By switching between similar sized motor units, the brain is able to minimize muscle fatigue while at the same time generate the required force. When the force demanded by a muscle begins to decline, the brain releases the motor units in the reverse order; the large force producing, but fast fatiguing motor units are released first, then the next largest motor unit until finally only the fatigue resistant units are generating force.
1.3 Triggering Muscle Contractions
To trigger a muscle contraction, the brain sends an electrical signal through a nerve to the muscle. Fig. 1.5 will aide in describing the mechanisms responsible for triggering a contraction. Between the muscle and the nerve ending, however, there exists a gap, known as the synaptic gap, which prevents the electrical signal from reaching the muscle. Since the electrical signal cannot span this gap, the axon ending at the end of the nerve converts the electrical signal into a chemical signal (Acetylcholine (ACh)), called a neurotransmitter. The chemical signal is able to traverse the synaptic gap and reach the muscle membrane of the myofibril called the sarcoplasmic reticulum where it binds to protein and is once again converted into an electrical signal. This electrical signal then travels along the length of the myofibrils searching for the location where the myofibrils are divided along their length into sarcomeres. At these divisions, there exist T tubules into which the electrical signal travels.
Figure 1.5: The triggering of a contraction [Freudenrich 01]
T tubules contain the sarcomere’s storage reserves of calcium ions (Ca2+). The calcium is prevented from entering the sarcoplasm, which is a fluid surrounding the myosin and actin filaments, by a calcium gate. Once the electrical signal enters the T tubules, the electrical signal causes the calcium gates to open for a short period of time (on the order of ms). During the time period in which the calcium gate is open, the calcium floods out of the T tubules into the sarcoplasm fluid. After the electrical signal leaves, the calcium gate will once again close, preventing more calcium from entering the sarcoplasm. Once calcium enters the sacroplasm, it interacts with the myosin and actin filament, causes the myosin heads to form cross-bridges to the actin filaments, resulting in the generation of force. How calcium causes force to be generated is examined in the next section of this chapter. As soon as calcium enters the sarcoplasm, calcium pumps within the sarcoplasmic reticulum become active and remove calcium from the sarcoplasm. This prevents calcium from building up in the sarcoplasm. Once the calcium has been removed from the sacroplasm, cross-bridges stop forming and force can no longer be generated in the sarcomere.
1.4 Muscle Contraction and Force Generation
Fig. 1.6 contains a series of sequential plots depicting the mechanics by which a muscle contracts. Fig. 1.6a shows the muscle before the introduction of calcium into the sarcoplasm. When a muscle is not generating force, this is the typical layout of the myosin and actin filaments. In the figure, the thick filament is the myosin fiber while the thin filament is the actin fiber. The myosin binding sites on the actin filaments a covered with by a thin filament known as tropomyosin. Since the tropomyosin covers the myosin biding sites, the myosin heads cannot attach to the actin and form cross-bridges. Attached to the tropomyosin filament is a troponin-complex which will become important once calcium enters the sarcoplasm.
Fig. 1.6b depicts the introduction of calcium ions into the sarcoplasm. Once calcium enters the sarcoplasm, it binds with the troponin-complex to form a calcium-troponin complex. Upon binding with calcium, troponin changes shape and slides the tropomyosin off the actin, allowing binding sites between the actin and myosin to be exposed, as shown in Fig. 1.6c. Once the myosin heads have bonded to the actin fibers, a cross-bridge has formed, making it possible for a contraction to occur. When the cross-bridges form, two chemicals, adenosine diphosphate (ADP) and inorganic phosphate (Pi), are released into the sarcoplasm. The release of these chemicals result in the myosin heads bending, causing the actin fibers to be pulled past the myosin fibers, as shown in Figs. 1.6d and e. The sliding of the actin filament past the myosin filament results in a shrinking of the sarcomere. The shrinking of the sacromeres is what cause force to be generated in the muscle.
At the end of the power stroke, Fig. 1.6e, when the myosin head has cocked back to its furthest position, a new chemical, adenosine triphoshate (ATP), is released to the sarcomplasm. The ATP binds to the myosin head, causing the myosin head to release the actin binding site, Fig. 1.6f. This allows the myosin head to swing back to its original position and prepare for another contraction. The ATP meanwhile is broken down into ADP and Pi by mitochondria in the myosin fibers, Fig. 1.6f. The reintroduction of the ADP and Pi into the sarcoplasm causes the myosin head to cock back and prepare to once again bind with the actin fiber.
The binding of the myosin heads throughout the sarcomere occur asynchronously; that is some myosin heads are binding while other heads are releasing the actin filaments. By binding in an asynchronous manner, the muscle is able to generate a smooth force since cross-bridges formation and the contraction of the sarcomere occur in a continuous manner. This process must be performed repeatedly during a muscle contraction as a single power stroke of the myosin head results in a shortening of the muscle by about 1%. Since muscles contract 35% to 50% of their resting length, the cross-bridge formations must form repeatedly during a single muscle contraction [Freudenrich 01].
Figure 1.6: Sequential figures of cross-bridge formations, attachments, and the chemical processes required for the formation of cross-bridges [Freudenrich 01]
The interaction of the actin and myosin filaments when forming cross-bridges gives rise to the large-scale, macroscopic physical properties of the muscle. The large-scale properties of the muscle have been investigated by many over the past several decades. The investigators have created mechanical muscle models (spring-dashpot system) which can capture and explain the physical properties of the muscle. The large-scale properties of the muscle and the mechanical model used to capture these properties are discussed in the following chapter.
2.1 Hill’s Model
In efforts to capture the complex actions performed by muscles, many simplified models have been created. One of the earliest and perhaps most popularly employed muscle model is the Hill model, developed by Archibald Hill in 1938. Hill performed experiments on frog muscles in an attempt to capture the energy relations between muscle shortening (lengthening) and the positive (negative) work performed. In this manner, Hill hoped to capture the effects load has on the shortening or lengthen velocity of the muscle [Hill 38].
In his experiments, Hill hung a weight from the frog muscle, stimulated the muscle with an electrical current, and recorded the weight, and the muscle’s contractile velocity, length and heat production during the contraction. He then used these measurements to determine the force-length-velocity properties of the muscle. Hill performed these experiments on single fibers within a sarcomere. He then assumed the properties of these fibers could be scaled to the entire sarcomere which in turn could be scaled to the overall muscle. To capture these force-length-velocity properties of a large muscle, a mechanical muscle model was created.
The Hill model, shown in Fig. 2.1, is composed of three elements: two which are arranged in series which, in turn, are in parallel with the third element. The contractile element is freely extendable when at rest, but capable of shortening when activated by an electrical stimuli. The contractile element is connected to an elastic serial element. The serial element accounts for the muscle elasticity during isometric (constant muscle length) force conditions. These two elements are then joined in parallel with another elastic element used to account for the elasticity of the muscle at rest. The parallel element accounts for the inter-muscular connective tissues surrounding the muscle fibers while the series elastic element accounts for the elasticity of the cross-bridges within the muscle [Fung 93].
Figure 2.1: Hill’s three element mechanical model [Fung 93]
2.1.1 Force-length properties
The properties of the three elements in the Hill model which are responsible for force generation are defined in terms of force-length properties and force-velocity properties. The force-length property is based on isometric muscle contraction, that is, the force generated within the muscle as the length of the muscle remains constant. During an isometric contraction, the series element lengthens while the contractile element shortens. The lengthening of the serial element must be equal to the shortening of the contractile element for the overall length of the muscle to remain unchanged. As the serial element lengthens, the parallel element no longer remains slack and develops tension in a non-linear manner [Winter 90]. The total force developed within the muscle is then the sum of the forces in both the active and passive muscle tissue. An example of the force-length curve common to skeletal muscles is shown in Fig. 2.2a.
As can be seen from the Fig. 2.2a, the force developed in the contractile element, PCE, has an optimal muscle length, loM, at which peak force can be developed. At lengths both shorter and longer than this optimal length, the force developed within the contractile element diminishes. An explanation for this decline in force can be explained by examining the cross-bridges in the muscle. At lengths significantly less than the optimal length, the actin and myosin filaments are grouped close together. The closeness of these fibers impedes the binding of the actin and myosin when cross-bridges interfere and block the other cross-bridge binding sites. As the muscle lengthens, the fibers are not restricted to as tight an area and the actin/myosin binding process is able to proceed with fewer fibers impeding the cross-bridge formations. With less impedance for cross-bridge formation, larger force can be developed. When the muscle length is larger than the optimal length, the actin and myosin fibers have too much separation, resulting in the formation of too few cross-bridges. As the muscle shortens, more of the actin filament can be reached by the myosin heads, allowing more myosin heads to span the gap and form cross-bridges, resulting in larger force generation. Fig. 2.3 contains actin and myosin filaments which pictorially depicts how the various muscle lengths affect the magnitude of force generated. When a muscle is not fully activated, the active force-length curve is a scaled version of the fully activated curve, Fig. 2.2b. The passive force-length curve however remains unchanged as it is independent of activation [Winter 90].
Figure 2.3: Cross-bridge clusterings during a) short, b) optimal, and c) long muscle lengths
2.1.2 Force-velocity properties
The dependence of muscle force on contractile velocity is captured by the force-velocity curve shown in Fig. 2.4. When a tensile force is applied to a muscle, the muscle will shorten and then stop. The length at which the shortening stops is the length at which the muscle can support the force in steady-state [Zajac 89]. The velocity at which the muscle contracts to the steady-state length is determined by the force-velocity properties of Fig. 2.4. As a muscle shortens during contraction, muscle tension decreases. [Winter 90] contributes this loss of force to the cross-bridges of the contractile element breaking and reforming in a shortened condition, as well as passive damping due to viscous fluid in the contractile element and the surrounding connective tissue. At optimal muscle length and full activation, the muscle’s maximum shortening velocity can be determined. At this velocity, the muscle is incapable of supporting a load. By using this force-velocity curve, [Zajac 89] was able to describe the mechanical power output delivered by an active muscle (see Fig. 2.5).
Figure 2.4: Force-velocity property of muscle [Pandy 90]
Figure 2.5: Power output by an active muscle [Winter 90]
2.2 Zajac’s Modifications to the Hill Model
Since the introduction of Hill’s model, various modifications have been made to more accurately incorporate further complexities and increase the model’s accuracy; the most notable of which were made by Zajac.
Zajac based his model on the Hill muscle, but extended the Hill model to include the tendon connection and to account for muscle fiber pennation angles. The pennation angle is the angle made between the muscle and tendon at the point where they connect, as shown by the muscle schematic of Fig. 2.6a. Based on these modifications, the Hill model evolved into the more complex model shown in Fig. 2.6b.
In Fig. 2.6b, α is pennation angle, the l’s are the lengths of the serial element (SE), contractile element (CE), tendon (T), the muscle (M), and the muscle tendon system, (MT). In a similar manner, the k’s in the figure are the spring stiffnesses of the various elements.
Based on geometry, the musculotendon actuator force-length-velocity properties can be defined with Eq. 2.1.
(2.1)
with 
and 
where PT tendon tension
km, kT muscle and tendon stiffness
kpe, kse parallel and series elements stiffnesses
lm, α muscle length and pennation angle
vmt, vce musculotendon and contractile element velocities
The complete derivation of the output force, PT, from Zajac’s musculotendon model is given in Appendix A. In Zajac’s model, five parameters, kPE, kT, kM, kSE, and vCE need be determined before solving Eq. 2.1.
The passive stiffness of the parallel element kPE
is defined by the experimental force-length curves; kPE =
. The stiffness of the serial spring element kSE
is given by Eq. 2.2.
with Po =
(2.2)
In Eq. 2.2, PSE is the force in the serial
element, Po is the maximum isometric contraction force of the fully
activated muscle at optimal length lom.
is the fully activated muscle force of the contractile
element. The constants 100 and 10 in Eq. 2.2 come from the slope and intercept
of the line relating muscle force to muscle stiffness, shown in Fig. 2.7a. The
force values in Fig. 2.7a have been normalized by the optimal or maximum force
the muscle can generate, Po, in order to allow the plot to be
generalized for all muscles instead of needing a plot for every muscle of the
body. The stiffness of the muscle, kM, can then be determined by
summing the stiffnesses of the series and parallel elements.
The stiffness of the tendon, kT, is given by Eq. 2.3 which was determined with the help of the tendon force vs. tendon strain plot of Fig 2.7b.
(2.3)
where lOT is the optimal length of the tendon at
which peak force can be generated in the muscle and delivered to the joint
after passing through the tendon connection, εOT is the strain
in the muscle at tendon force 
. For Eq. 2.3 to be valid, it is assumed strain in the tendon
remains below 4%. In Fig. 2.7b, the tendon force, PT, has again been
normalized by the optimal force, PO, in order to allow the
equation to be generalized for any tendon in the body.
The final unknown variable needed to solve Eq. 2.1 is the
shortening velocity of the contractile element, vCE. By assuming the
contraction and activation mechanisms of the muscle are uncoupled, Zajac
expressed the isometric contraction force, 
, as the experimental isometric contraction force of the
fully activated muscle 
scaled by the
effective activation rate a(t), resulting in Eq. 2.4.
(2.4)
To determine the activation of the muscle, Zajac then wrote a first order differential equation relating the activation rate a(t) of the muscular contraction to the neural excitation signal, u(t), of the muscle as [Zajac 89]:
(2.5)
with β, τrise, τfall
constants and 
. In most practical
cases however, the excitation signal can be considered as an on-off signal
allowing Eq. (2.5) to be reduced to:
for u(t) = 1, and
for u(t) = 0 (2.6)
in which amin is the lower bound on muscle
activation which will allow the inversion of the force-velocity properties used
in later in Eq. 2.8. By further assuming the isometric force-length and
force-velocity properties were uncoupled, Zajac normalized the isometric
contraction force 
for the scaling the
force-velocity curve, allowing effective contraction force to be expressed as:
(2.7)
Based on Eq. 2.7, the muscle contractile force, PCE, appears as a function of the length of the muscle, lM, the shortening velocity, vCE, and the activation, a(t), of the contractile element. By inverting Eq. 2.7, the shortening velocity of the contractile element can be expressed as a function of the musculotendon tension, PT, length lMT, and activation a(t) as:
vCE = vCE[PT, lMT, a(t)] (8)
With the five unknown variables of Eq. 2.1 known, Eq. 2.1 can now be solved to determine the force in the muscle as a function of time. By normalizing Eq. 2.1, the normalized force-time plot of Fig. 2.8 was created.
In Fig. 2.8, the force in the tendon, PT, which is
the solution to the differential equation of Eq. 2.1, has been plotted for
various tendon slack lengths. The more slack in the tendon, the longer it takes
for peak force to be reached, but eventually, no matter the tendon slack
length, the peak force generated by the muscle reaches a constant value. In the
force-time plot of Fig. 2.8, τ is normalized time, 
is the initial
non-zero normalized tendon force, and 
is the steady state
normalized tendon force. 
shown in the plot, is
the solution sought from solving Eq. 2.1.
Figure 2.8: Isometric musculotendon actuator force generated by fully-activated muscle [Zajac 86]
2.3 Comments on the Zajac Model
Although the Zajac model has been used by many researchers investigating human motion and biomechanics such as the European CHARM project [Maurel 98], the Danish Anybody researchers [http://anybody.auc.dk/publications.htm, Rasmussen 01, Christensen 02], Musculographics researchers, and universities, including Northwestern University, University of Texas at Austin, and Stanford University [Anderson 99, Anderson 01a, Anderson 01b Anderson 02, Arnold 00, Delp 98, Herrmann 99, Murray 00, Murray 02, Piazza 01, Thelen 03, Bhargava 03, Lemay 96, Llyod 03, Finni 01, Ledoux 01, Jacobs 96, Klute 99, Shelburne 97], the Zajac model does not appear to have a well founded physiologically based interpretation; due to this, in this paper, a different model, the Wexler model, described next, will be used to capture fatigue and in investigating human performance and motion. The Wexler model contains a similar spring representation for muscle stiffness, but its activation, calcium kinetics, is based on the same principles responsible for activation in real humans. The calcium rates in the human muscle can be measures and corresponded to a muscle activation rate, giving muscle activation a physically interpretable meaning, unlike the Zajac model. Wexler took an approach opposite of Hill in his experimental work. Where Hill worked with individual muscle fibers and assumed the properties of a single fiber are scaled versions of the motor unit which in turn were scaled versions of the muscle, Wexler worked with a full muscle. Wexler’s experimental work involved stimulating an entire muscle and recording the properties for the entire muscle, not individual muscle fibers. He then also accounted for how force is generated at the nano-scale level, described in Chapter I, with a calcium kinetics model. The work of Wexler, Ding, and Binder-Macleod is described in the following chapters.
3.1 A Predictive Mathematical Model for Skeletal Muscle Forces
Beginning in the 1997, a series of papers, published by Welxer, Ding, and Binder-Macleod, began laying the ground work for a skeletal muscle force prediction model. The initial investigation and development of the model was based on experiments with rats, but since the publication of the first paper, the force prediction model has been extended to include humans. In addition to extending the model to include humans, Wexler, Ding, and Binder-Macleod (in the remainder of this paper, for simplicity, only Wexler’s name will be used when referring to Wexler, Ding, and Binder-Macleod) further increased its utility by incorporating an ability to predict fatigue, which will be discussed in great detail in the following chapter.
To develop the mathematical model, Wexler used both analytical mean to model physiological aspect of the muscle, as well as experimental methods in the form of functional electrical stimuli to both parameterize and validate the model. Functional electrical stimulation is the activation of a muscle with a brief frequency train by an external stimulus [Wexler 97]. In this case, an electrical pulse is applied to a muscle, causing to contract and generate force. The external stimulation is used in place of the electrical pulses from the brain since the external pulses allowed for better control and more exact knowledge about the frequency of pulses stimulating the muscle. In the development of the model that follows, all parameters and values discussed were obtained for rats. The extension of the model to human can be made, but to avoid confusion, in this chapter, all values discussed are for rats.
In the analytical development of the model, Wexler broke the contraction stage into three distinct physiological parts: calcium release and reabsorbsion by the sarcoplasmic reticulum, the rate at which calcium binds and unbinds to troponin, and the generation of force due to cross-bridge cycling and the friction and elasticity of the muscle fibers.
3.1.1 Balancing the calcium flux
When an electrical stimulation (either external or from the brain) excites a muscle, calcium from the sarcoplasmic reticulum (SR) floods into the sarcoplasm (SP). Once the calcium enters the sarcoplasm, it is then reactively reabsorbed back into the sarcoplasmic reticulum by a Ca2+-ATPase. Calcium pumps within the sarcoplasmic reticulum remove calcium from the sacroplasm. By balancing the rate at which calcium enters and leaves the sarcoplasm, the calcium flux, J, of the muscle can be written as:
(3.1)
where ko is the rate of Ca2+ -ATPase reabsorbtion from the sarcoplasm to the sarcoplasmic reticulum and k is the permeability of the sarcoplasmic reticulum membrane to Ca2+ when the calcium gate channels are open.
Although there are no published results for the length of time the calcium gates remain open, by comparing fitting results for different opening times with experimental results, Wexler determined the gate opening time, k, to be as follows:
(3.2)
where it is noted here once more that these gate opening times apply only to rats [Wexler 97]. The time interval for which the calcium gates in humans are open is often considerable longer than the 4 ms listed in Eq. 3.2.
3.1.2 Calcium-troponin complex binding and unbinding
Once calcium is released into the sarcoplasm, it reacts with troponin (Tn) to for a calcium-troponin complex (Ta). Eq. 3.3 [Wexler 97] balances the rate at which calcium and troponin combine to form the calcium-troponin complex.
(3.3)
where k1 and k2 are the forward and backward rates at which calcium and troponin combine to form the calcium-troponin complex, and the rate at which the calcium-troponin complex is broke back down into calcium and troponin. From chemical kinetics and membrane transport, Eqs. 3.1 and 3.3 can be written as two differential equations as shown in Eqs. 3.4 and 3.5 [Wexler 97].
(3.4)
and
(3.5)
In Eqs. 3.4 and 3.5, [To] = [Ta] + [Tn]. The differential equation of Eq. 3.4 describes the calcium transients in the sarcoplasmic reticulum while the differential equation of Eq. 3.5 describes the calcium-troponin binding process. The first two terms in Eq. 3.4 account for the dissociation of the troponin and the binding of the calcium to the troponin [Wexler 97]. The third term is the rate of calcium concentration increase due to the diffusion of calcium from the sarcoplasmic reticulum while the final term accounts for the diffusion and reabsorbsion of calcium back into the sarcoplasmic reticulum [Wexler 97]. The first term in Eq. 3.5 accounts for the binding of calcium to the troponin while the second term represents the dissociation of the calcium-troponin complex [Wexler 97].
3.1.3 Mechanics of force generation
When calcium combines with troponin, a change occurs in the troponin which causes it to change shape and shift its position. Since the troponin is attached to actin filaments, when the troponin shifts position, the tropomyosin filaments covering the actin are also shifted. The shifting causes the tropomyosin to slide off the actin binding sites, exposing the binding sites to the myosin heads. The myosin heads can then attach to the actin and form cross-bridges. These cross-bridges then pull the actin filaments toward the center of the myosin filaments resulting macroscopically in the generation of force. The force generated by the muscle is modeled with the spring-damper-motor series alignment shown in Fig. 3.1. The damper accounts for the viscous resistance of the contractile element and the connective tissue while the motor captures the contractile component or the sliding of the actin and myosin filaments of muscle fibers and the spring represents the tendonous portion and the series elastic component of the muscle [Wexler 97]. The complexity of the muscle model can be increased by adding additional springs and dampers in series and in parallel as needed.
The force exerted in the damper is given by:
(3.6)
Figure 3.1: Mechanical model for isometric force generation [Wexler 97]
where b is the damping coefficient, x is the length of the spring, and V is the contractile velocity of the motor, given by:
(3.7)
where B is a constant of proportionality and Fm is the peak force of the motor. Assuming the stiffness of the tendon and serial elastic components of the muscle to be represented by a linear spring of stiffness K, the force in the spring is given by:
(3.8)
By
differentiating Eq. 3.8 with time and combining with Eq. 3.6 to eliminate 
, and then combining the result with Eq. 3.7 to eliminate V, the force in the muscle takes on the
form of the following differential equation:
(3.9)
where
is a time constant
over which force decays. Wexler expected the friction between the actin and
myosin fibers to be higher due to cross-bridge recycling and so set 
where τ1
is the time constant in the absence of cross-bridges and τ2 is
the additional friction due to the cross-bridge chemical bonds. Using this
value for 
and replacing KB with a new constant A results in the final form of the force
differential equation given by Eq. 3.10 [Wexler 97].
(3.10)
Eqs. 3.4, 3.5, and 3.10 then govern the force dynamics of the muscle. The three coupled differential equations which describe the transients of the calcium, troponin, and force require knowledge about nine parameters before the three differential equations can be solved. A summary of the variables, their units, and definitions is given in Table 3.1.
3.2 Determination of Parameter Values
To determine the nine parameters needed in solving the three differential equations governing the generation of force, Wexler performed a series of experiments on the gastrocnemius muscles of adult Sprague-Dawley rats. In the development of the model, the rat sample size, N, was 14. The rats were deeply anesthetized and were then mounted to a rigid frame that immobilized the test leg. An electrode was then positioned at the nerve leading to the muscle; the nerve leading to the muscle was then clipped close to the hip to prevent any reflex activation of the muscle is response to the electrical stimulation [Wexler 97]. The tendon and the muscle were then removed from the rat and
Table 3.1: Summary of dependent variable solved for in the three differential equations [Wexler 97]
attached to a force transducer with a heavy suture material to allow the isometric muscle force to be recorded [Wexler 97]. Muscle temperate was maintained at a constant value of 35oC by radiant heat while the body of the rat was maintained at 38oC by a heating pad [Wexler 97]. Force measurements were then digitally recorded at a rate of 500 samples per second.
Before any really measurements were taken, single pulses were applied to the muscle to determine the muscle resting length which would produce the greatest force when stimulated. Once this length was determined, a series of ten six pulse frequency trains were applied to the muscles. Six of the trains were constant frequency trains (CFT), meaning all pulses in the train (a total of six pulses per train) were applied with a constant spacing between the pulses. The constant interval pulses used in the parameterization of the model were 10, 20, 30, 40, 50, and 100 ms trains. The remaining four pulses trains were variable frequency trains (VFT). This meant a pulse was applied, followed by a second pulse 10 ms later, and then the remaining four pulses occurring at regular intervals, with separations occurring at either 20, 30, 40, or 50 ms periods. The various frequency train stimulations were applied in random sequences with each train separated by a minimum of 10 s.
From the frequency train data, various plots could be created from which the parameters needed to solve Eqs. 3.4, 3.5, and 3.10 could be determined. The parameters used in the model were the averaged values obtained for the gastrocnemius muscle from the 15 rat test subjects. The average values and standard deviations for the parameters used in solving Eqs. 3.4, 3.5, and 3.10 are listed in Table 3.2.
3.3 Predicting Muscle Forces: Calculation Results
Once the parameters needed by Eqs. 3.4, 3.5, and 3.10 were determined, the predicative capabilities of the model could be analyzed. This was accomplished by comparing the force-time plot predicted by the model with force-time plots obtained from experiments. Fig. 3.2 is the force-time plot predicted by the model when the calcium gate is opened once every 40 ms and then remains open for 4 ms. The 40 ms stimulation frequency was used to parameterize the model and then used in predicting forces for other stimulation frequencies.