Biomechanics Principles and Applications - Donald R. Peterson & Joseph D. Bronzino

aData from Lieber et al., 1990, 1992.

bBR: brachioradialis; EDC I, EDC M, EDC R, and EDC S: extensor digitorum communis to the index, middle, ring, and small fingers, respectively; EDQ: extensor digiti quinti; EIP: extensor indicis proprious; EPL: extensor pollicis longus; FDP I, FDP M, FDP R, and FDP S: flexor digitorum profundus muscles; FDS I, FDS M, FDS R, and FDS S: flexor digitorum superficialis muscles; FDS I(P) and FDS I(D): proximal and distal bellies of the FDS I; FDS I(C): the combined properties of the two bellies as if they were a single muscle; FPL: flexor pollicis longus; PQ: pronator quadratus; PS: palmaris longus; PT: pronator teres.

to the charged organic matrix. The total stress acting on the cartilage is separated into independent solid and fluid phases:

σ T = σ s + σ f

where s denotes the solid phase and f the fluid phase. The relative motion of the phases defines the equilibrium equations

where α is tissue solid content and k the tissue permeability coefficient. In addition to the equilibrium equations, each phase is subject to separate constitutive relations:

σ f = − pa I and σ s = −αpa I + De

where pa is the apparent tissue stress, D is the material property tensor, and e is the strain tensor. For a hyperelastic solid phase

De = λTr(e )I + 2μe

where λ and μ are the Lame´ constants.

Series elastic element

Parallel elastic element

Contractile

element

FIGURE 2.3 The Hill model of muscle separates the active properties of muscle into a contractile element, in series with a purely elastic element. The properties of the passive muscle are represented by the parallel elastic element.

Both the elastic response and the relaxation function are empirically determined. The common form for the relaxation function is a sum of exponentials

G (t) = A + Bi e−t/τi

i

The form of the elastic response varies, but usually includes a power or exponential term to accommodate the toe region.

2.3.3 Muscle

2.3.3.1 Types of Muscle Models

There are three general classes of models for predicting muscle force: biochemical, or crossbridge, models; constitutive models; and phenomenological, or Hill, models. Crossbridge models [Huxley, 1957; Huxley and Simmons, 1971] attempt to determine force from the chemical reactions of the crossbridge cycle. Though accurate at the cross-bridge cycle, it is generally computationally prohibitive to model a whole muscle in this manner. Constitutive models, such as that described by Zahalak and Ma [1990], generally attempt to determine muscle behavior by describing populations of cross-bridges. A potentially powerful approach, this technique has not yet been widely adopted. The vanguard of muscle modeling remains the phenomenological model first described by Hill [1939], that describes the viscoelastic behavior of skeletal muscle using a framework analogous to the standard linear solid (Figure 2.3). Although the series elastic element represents primarily tendon, some series elasticity is found even in muscles lacking any external tendon or in segments of single fibers. The parallel elastic element represents the passive properties of the muscle, currently thought to reside primarily in titin. The contractile component is described by independent isometric force–length (Figure 2.4) and isotonic force–velocity relations (Figure 2.5) and an activation function [Zajac, 1989].

2.3.3.2 Muscle Force----Length Relationship

Under conditions of constant length, muscle force generated is proportional to the magnitude of the interaction between the actin and myosin contractile filaments. Myosin filament length in all species is approximately 1.6 μm, but actin filament length varies (Table 2.7). Optimal sarcomere length and maximum sarcomere length can be calculated using these filament lengths. For optimal force generation, each half

2.3.3.3 Muscle Force----Velocity Relationship

Under conditions of constant load the relationship between force and velocity is nearly hyperbolic [Hill, 1938]. The shortening force–velocity relation can be described by:

(P + a)v = b(P0 − P )

while the lengthening relation can be described by:

vmax + v

F = 1.8 − 0.8 Vmax − 7.6 V

The dynamic parameters (a, b, and Vmax) vary across species and fiber types (Table 2.3).

It should be noted that this formulation omits several potentially important force-generating phenomena. Notable among these are the persistent extra tension obtained following stretch [Edman et al., 1982], the exaggerated short range stiffness [Rack and Westbury, 1974], and changes in the force–length relation associated with activation level [Rack and Westbury, 1969]. Some of these features can be accommodated by considering series elasticity and sarcomere length inhomogeneity [Morgan, 1990], and each represents a nonlinearity that substantially complicates modeling and may not be necessary for first approximations of muscle function.

Common applications of muscle modeling include forward simulation to predict output forces or motions and inverse analysis to estimate the muscle forces that produced an observed motion. In neither of these cases is it necessarily practical to determine muscle contractile properties empirically, and it is frequently necessary to resort to estimation of the force–length and force–velocity relations from muscle structure. If a muscle is considered to be a composition of uniform sarcomeres in series and in parallel, then the deformation of single sarcomeres can be estimated from whole muscle length changes. A simplified view of a muscle is an array of identical fibers of uniform length arranged at a common pennation angle to the line of force. Peak isometric tension can be estimated from PCSA. Pennation angle determines the relationship between muscle and fiber length changes:

If sarcomere length is known at any muscle length, it is then possible to scale the sarcomere length– tension and velocity–tension relations to the whole muscle. When reporting architectural data (Tables), muscle and fiber lengths should be normalized to optimal sarcomere length.

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