A mathematical model of force transmission from intrafascicularly terminating muscle fibers

Abstract

Many long skeletal muscles are comprised of fibers that terminate intrafascicularly. Force from terminating fibers can be transmitted through shear within the endomysium that surrounds fibers or through tension within the endomysium that extends from fibers to the tendon; however, it is unclear which pathway dominates in force transmission from terminating fibers. The purpose of this work was to develop mathematical models to (i) compare the efficacy of lateral (through shear) and longitudinal (through tension) force transmission in intrafascicularly terminating fibers, and (ii) determine how force transmission is affected by variations in the structure and properties of fibers and the endomysium. The models demonstrated that even though the amount of force that can be transmitted from an intrafascicularly terminating fiber is dependent on fiber resting length (the unstretched length at which passive stress is zero), endomysium shear modulus, and fiber volume fraction (the fraction of the muscle cross-sectional area that is occupied by fibers), fibers that have values of resting length, shear modulus, and volume fraction within physiologic ranges can transmit nearly all of their peak isometric force laterally through shearing of the endomysium. By contrast, the models predicted only limited force transmission ability through tension within the endomysium that extends from the fiber to the tendon. Moreover, when fiber volume fraction decreases to unhealthy ranges (less than 50%), the force-transmitting potential of terminating fibers through shearing of the endomysium decreases significantly. The models presented here support the hypothesis that lateral force transmission through shearing of the endomysium is an effective mode of force transmission in terminating fibers.

Figure 1

Schematic of the analytical shear model in the undeformed configuration (A) and deformed configuration (B). The model assumes that the terminating fiber has a circular cross-section and is surrounded by an annular layer of endomysium (A) which transmits the force generated in the fiber to the surrounding muscle tissue. The surrounding muscle tissue is held at a constant length, and its presence was simulated by constraining the outer surface of the endomysium from moving in the fiber direction (z). This constraint was imposed based on the assumption that the surrounding muscle tissue is sufficiently stiff so that its deformation in response to the contractile force of the terminating fiber is negligible. One end of the fiber was constrained from moving in the z direction to represent an end attached to a tendon. L₀, r₀ and h₀ are fiber resting length, initial fiber radius and endomysium thickness, respectively (A). The thickness of endomysium, h₀, is shared by neighboring fibers, therefore fiber volume fraction was calculated based on half the thickness of the endomysium: $V_{\text{f}}=r_0^2/{\left(r_0+\frac{h_0}{2}\right)}^2=1/{(1+\kappa /2)}^2$. l is the fiber length at equilibrium, and l = λL₀ (B). r(λ) and h(λ) are the fiber radius and the endomysium thickness at fiber length l, respectively (B). δ(λ, z) is the displacement of a point on the periphery of the fiber at distance z from the fixed end of the fiber (B).

Figure 2

Schematic of analytical tensile model, in the undeformed (A) and deformed (B) conditions. The fascicle is assumed to be held at a fixed length ( $L_0^{\text{fascicle}}$), and the force transmitted from the fiber is transmitted to the tendon at the other end by stretching the endomysium.

Figure 3

Finite element models of a single fiber with a circular cross-section (A) and a terminating fiber (red) within a fiber bundle (B), both shown in the activated condition. The FE models with circular cross-section (A) have a fiber diameter of 80μm, a volume fraction of 75% and have lengths varying from 120μm (L₀/2r₀ = 1.5) and 1.6mm (L₀/2r₀ = 20) at 80μm increments. To create a model based on a histological cross-section (B), we obtained an image at 20X magnification that contained several fibers, and outlined the boundaries of nine adjacent fibers that formed a bundle. The resulting fiber polygons were scaled to create a fiber volume fraction of 75%. The cross-sectional geometry remains constant along the length of the fibers. Hexahedral meshes were created based on the outlines in Ansys 11 (Ansys, Inc. Canonsburg, PA). The model is 120μm long (L₀/2r₀ = 1.5). The terminating (middle) fiber has the same cross-sectional area as the fiber with the circular geometry ( $\frac{\pi }{4}{(80\mu \text{m})}^2$). The middle fiber was constrained in the z direction at one end. All other fibers were constrained in the z direction at both ends.

Figure 4

The ratio of force transmitted laterally through shearing of the endomysium from a terminating fiber to peak isometric force that can be generated in a fiber of the same cross-sectional area, plotted against the ratio of fiber resting length to fiber diameter (A). λ, the ratio of fiber length at equilibrium to fiber resting length (B). We normalized the force transmitted from the fiber (equation 2.1.2), by the fiber’s peak isometric force, F_iso = Aσ_iso. The quantity plotted on the vertical axis (A) is therefore F_fiber/F_iso = σ_fiber(λ)/λσ_iso. From equation (2.1.1.2) it is clear that λ is a function of L₀/2r₀, the ratio of fiber resting length to initial fiber diameter, G_end, endomysium shear modulus and κ, and thereby V_f, fiber volume fraction. For values of λ smaller than one, F_fiber/F_iso is a monotonically increasing function of λ. In these plots fiber volume fraction (V_f) was held constant at 90% (Typical for a healthy muscle (Table 1)) and endomysium shear modulus (G_end) was varied from 5Pa which is lower than any estimates based on the literature (Table 1) to 3000Pa which is on the same order as the highest available estimates based on the literature (Table 1). The shaded region highlights the physiologically relevant range of values for L₀/2r₀ based on literature and cited in Table 1.

Figure 5

The ratio of force transmitted laterally through shearing of the endomysium from a terminating fiber to peak isometric force that can be generated in a fiber of the same cross-sectional area, plotted against the ratio of fiber resting length to fiber diameter (L₀/2r₀ is varied by increments of 1) (A). λ, the ratio of fiber length at equilibrium to fiber resting length (B). Endomysium shear modulus was held constant at 5Pa and fiber volume fraction was varied from 95% to 40%. This ranged was chosen based on values reported in the literature for healthy and spastic muscle tissue (Table 1). The shaded region highlights the physiologically relevant range of values for L₀/2r₀ based on the literature and cited in Table 1.

Figure 6

The ratio of force transmitted longitudinally through tension within the endomysium from a terminating fiber to peak isometric force that can be generated in a fiber of the same cross-sectional area, plotted against the ratio of fiber resting length to fascicle length (A). λ, the ratio of fiber length at equilibrium to fiber resting length (B). From equation (2.1.2.1) it is clear that λ is a function of $L_0/L_0^{\text{fascicle}}$, the ratio of fiber resting length to fascicle length, and C_end, endomysium tensile modulus. Endomysium tensile modulus (C_end) was varied from 1kPa to 100kPa. This range was chosen to encompass the range of estimates for the tensile modulus of the endomysium available in the literature (Table 1). The shaded region highlights the physiologically relevant range of values for $L_0/L_0^{\text{fascicle}}$ based on the literature, cited in Table 1.

Figure 7

The ratio of force transmitted from a terminating fiber to peak isometric force when force is only transmitted laterally through shearing of the endomysium compared to when it is transmitted only longitudinally through tension in the endomysium. The fiber was assumed to have a diameter of 30μm and was within a 50cm long fascicle (e.g. human sartorius (Heron and Richmond, 1993)). We used a value of C_end = 40kPa for the tensile modulus of endomysium (the highest estimate based on the literature (Table 1)) and an endomysium shear modulus of 5Pa (the lowest estimate based on the literature (Table 1)). The shaded area highlights the range of lengths for intrafascicular terminating fibers reported for the human sartorius (Heron and Richmond, 1993).

Figure 8

Comparison between the forces predicted by the analytical shear model ( ), FE models with a circular fiber cross-section ( ) for different fiber resting lengths lengths ranging from 120μm (L₀/2r₀ = 1.5) to 1.6mm (L₀/2r₀ = 20), and the FE model of a terminating fiber surrounded by a bundle of fibers created based on a histological cross-section ( ) with a resting length of 120μm (L₀/2r₀ = 1.5). In all models the fiber diameter is 80μm, and fiber volume fraction is 75%.

Figure 9

Explanation of the effects of fiber volume fraction. For the same shear angle γ in the endomysium, the fiber with the lower fiber volume fraction (A) will shorten more than the fiber with higher volume fraction (B) l₁ < l₂.