The Viscoelastic Properties of Passive Eye Muscle in Primates. I: Static Forces and Step Responses

The viscoelastic properties of passive eye muscles are prime determinants of the deficits observed following eye muscle paralysis, the root cause of several types of strabismus. Our limited knowledge about such properties is hindering the ability of eye plant models to assist in formulating a patient's diagnosis and prognosis. To investigate these properties we conducted an extensive in vivo study of the mechanics of passive eye muscles in deeply anesthetized and paralyzed monkeys. We describe here the static length-tension relationship and the transient forces elicited by small step-like elongations. We found that the static force increases nonlinearly with length, as previously shown. As expected, an elongation step induces a fast rise in force, followed by a prolonged decay. The time course of the decay is however considerably more complex than previously thought, indicating the presence of several relaxation processes, with time constants ranging from 1 ms to at least 40 s. The mechanical properties of passive eye muscles are thus similar to those of many other biological passive tissues. Eye plant models, which for lack of data had to rely on (erroneous) assumptions, will have to be updated to incorporate these properties

Relaxation spectrum in m3LR (0.5 mm steps from different initial lengths).

<p>Relaxation spectrum in m3LR (0.5 mm steps from different initial lengths).</p

Force measured during a 0.5 mm step.

<p>All data shown are from the superior rectus (SR) of the second monkey (m3). Each panel shows force (red) and normalized speed (blue) as a function of change in muscle length, for different initial lengths spanning the entire range tested. (The speed trace is used only to indicate how it varies during the elongation; its magnitude has no meaning).</p

Passive force as a function of length in five eye muscles from three monkeys.

<p>A–E: Length-Tension curve for a single muscle. Blue points: average of the force measured during 200 ms before each step. Red points: estimate of the force at the final elongation after each step using the steady-state value from the spectral fit (see text). Black line: fit based on Eq. 1. F: Fits from panels B–E. The first muscle was excluded because the muscle was “cleaned” too extensively and it slipped around the globe (gray arrow in panel A). Muscles: lateral rectus (LR), superior rectus (SR).</p

Sources of noise in our force measurements.

<p>Data are from the lateral rectus (LR) of the second monkey (m3). A: Heartbeat noise templates at different muscle lengths. B: Magnitude of the heartbeat noise (average over the time interval indicated by the red bar in A) as a function of muscle length. A least-squares cubic fit to the data is shown in blue. C: Respiration noise templates at different muscle lengths. D: Magnitude of the respiration noise (average in the time interval indicated by the red bar in C) as a function of muscle length. A least-squares cubic fit to the data is shown in blue. E: Red: Part of the relaxation response measured in the same muscle after a quick step. Blue: Same as red trace, but after template-based removal of the heartbeat noise (shifted down by 1 gf for clarity). Green: same as blue trace, but after template-based removal of the respiration noise (shifted down by 2 gf for clarity). The green trace is the denoised data used in all subsequent analyses.</p

Relationship between fit quality and line spectrum spacing.

<p>A: The E-T algorithm was run on several simulated relaxation responses, each obtained by adding noise to the fit to an actual relaxation response (see text). As the spacing between spectrum lines (abscissa) decreases, the variability in the moduli of the fits to the simulated responses increases, because moduli for neighboring time constants are traded off against each other. As the spacing is increased this trade off is not possible anymore, and the variability in the fit reflects only the noise in the fitted responses. For each spacing, we computed, for each time constant, the standard deviation of the moduli over the set of fits to the simulated responses. We plot the mean of this measure over all the time constants against the spacing, separately for each muscle. B: As the spacing between spectrum lines (abscissa) increases, the fit deteriorates. For a given spacing, we computed the sum squared error for the fit to each simulated response. We plot the mean of this measure as a function of spacing, separately for each muscle.</p

Relaxation spectrum in m2SR (0.5 mm steps from different initial lengths).

<p>Relaxation spectrum in m2SR (0.5 mm steps from different initial lengths).</p

Passive force-length relationships reported in the literature.

<p>A: Data from the lateral rectus in cats, pooled across studies. The red fit represents original data, whereas the other fits have been scaled along both axes (see text) in an (obviously failed) attempt to reconcile the various data sets. B: The passive force-length relationship in human horizontal recti, as measured in studies on strabismic subjects (see text). The elongation is referred to the straight ahead position.</p

Relaxation responses from the superior rectus in m3.

<p>A: Data (black) and fits (green) for five different steps (blue numbers are the final length in mm). The steps at the smallest final length are not plotted for clarity, but the fits were just as good. B: Same as A, but using a logarithmically spaced abscissa to improve visualization of the force at short times. C: Moduli associated with each time constant in the fit; one line for each final length. D: Normalized moduli as a function of muscle length after the step; one line for each time constant. Note that in this panel the shortest lengths are also represented.</p

Relaxation spectrum in m4LR (0.5 mm steps from different initial lengths).

<p>Relaxation spectrum in m4LR (0.5 mm steps from different initial lengths).</p