Effects of Age on Mechanical Properties of Dorsiflexor and Plantarflexor Muscles

Abstract

Redundancy in the human muscular system makes it challenging to assess age-related changes in muscle mechanical properties in vivo, as ethical considerations prohibit direct muscle force measurement. We overcame this by using a hybrid approach that combined magnetic resonance and ultrasound imaging, dynamometer measurements, muscle modeling, and numerical optimization to obtain subject-specific estimates of the mechanical properties of tibialis anterior, gastrocnemius, and soleus muscles from young and older adults. We hypothesized that older subjects would have lower maximal isometric forces, slower contractile and stiffer elastic characteristics, and that subject-specific muscle properties would give more accurate joint torque predictions compared to generic properties. Unknown muscle model parameters were obtained by minimizing the difference between simulated and actual subject torque-time histories under both isometric and isovelocity conditions. The resulting subject-specific models showed age- and gender-related differences, with older adults displaying reduced maximal isometric forces, slower force–velocity and altered force–length properties and stiffer elasticity. Tibialis anterior was least affected by aging. Subject-specific models gave good predictions of experimental concentric torque-time histories (10–14% error), but were less accurate for eccentric conditions. With generic muscle properties prediction errors were about twice as large. For maximum predictive power, musculoskeletal models should be tailored to individual subjects.

Appendix

Each muscle–tendon unit was represented by a two-component Hill-type32 model. This phenomenological lumped-parameter model incorporated a contractile element (CE) in series with an elastic element (SEE). The behavior of the CE was defined by excitation–activation, force–length, and force–velocity relations. The behavior of the SEE was defined by a force–extension relation. Both force–length and force–velocity relations were linearly scaled with activation.

Excitation–Activation Relationship

An exponential characterized the relationship between the excitation input to the muscle model and the activation of the CE.7 Upon receiving an excitatory input μ, the time-course for rising CE activation λ was

$$ \lambda_{i} = \lambda_{i - 1} + \left[ {\mu_{i} \left( {1 - e^{{\left( {{{ - \Updelta t} \mathord{\left/ {\vphantom {{ - \Updelta t} \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)}} } \right)\left( {\mu_{i} - \lambda_{i - 1} } \right)} \right] $$

(A1)

where i denotes the sample number, Δt is the time-step, and τ is a time constant specifying the rate of activation.

Force–Length Relationship

The isometric force producing potential of the CE (FP) depended on the maximal isometric force capability of the CE (P ₀), the activation (λ), and normalized CE length (L _CE/L ₀). The latter specifies the position on the force length relation, which is defined as an inverted parabola with width coefficient W, such that

$$ FP = P_{0} \lambda \left[ {100\;W\left( {\frac{{L_{\text{CE}} }}{{L_{0} }} - 1} \right)^{2} + 1} \right] $$

(A2)

Force–Velocity Relationship

The force–velocity relation was defined by a rectangular hyperbola based on Hill,33 which has been shown in many experimental preparations.4,13,14 The shape of this relation is determined by the constants a and b, which can be expressed as normalized values a/P ₀ and b/L ₀.33 If the instantaneous force generated by the CE (P) is less than FP the CE must be shortening, such that

$$ v_{\text{CE}} = - \left[ {\frac{{\left( {FP + a} \right)b}}{{\left( {P + a} \right)}} - b} \right] $$

(A3)

where v _CE is the CE velocity. If P is greater than FP, the CE must be lengthening. Therefore, based on FitzHugh22

$$ v_{\text{CE}} = \frac{{b\left[ {\left( {FP \cdot \varepsilon } \right) - FP} \right]\left( {FP - P} \right)}}{{\left( {FP + a} \right)\left[ {P - \left( {FP \cdot \varepsilon } \right)} \right]}} $$

(A4)

where ε is the saturation force for an eccentric contraction (eccentric plateau).

Force–Extension Relationship

The amount of SEE extension for a given force relative to the SEE slack length L _S, i.e., the stiffness, was defined by a second-order polynomial. The length of the SEE (L _SEE) was given by

$$ L_{\text{SEE}} = \frac{{L_{\text{S}} }}{{2P_{0} \alpha }}\left( {2P_{0} \alpha - P_{0} \beta + \sqrt {P_{0}^{2} \beta^{2} + 4P_{0} \alpha P} } \right) $$

(A5)

where α and β are coefficients defining the shape of the polynomial.

Muscle Model Force Change

The rate of change of muscle force with respect to time is given by

$$ \frac{dP}{dt} = \frac{{\sqrt {P_{0}^{2} \beta^{2} + 4P_{0} \alpha P} }}{{L_{\text{S}} }} \cdot v_{\text{SEE}} $$

(A6)

where v _SEE is the velocity of the SEE, given by

$$ v_{\text{SEE}} = v_{\text{MT}} - v_{\text{CE}} $$

(A7)

where v _MT is the velocity of the musculotendon complex. During model simulation, this force change was integrated to give the muscle force at the next time step.