Fig 1.
The mechanical response of tendon is constructed by aggregating the mechanical behavior of strings with different slack lengths, resulting in a non-linear tendon stress-strain curve.
Fig 2.
Tendon adaptation modeling cycle.
a) discretized tendon model, b) force-extension response of the tendon model, c) three-component Hill musculotendon model, d) metabolic rate calculation model, e) model to scale contractile force from muscle, f) collagen fiber mechanical damage model and, g) collagen fiber proteolytic damage and repair model, h) updated tendon fiber length distribution results in a new tendon configuration and completion of a remodeling cycle of tendon adaptation model. Where PDF is probability density function, LS represents tendon slack length that is equal to the slack length of the shortest fiber, SD represents fiber length standard deviation, a measure of fiber dispersion, and Li represents fiber slack length.
Fig 3.
Normalized fatigue curve for whole tendon.
The fatigue curve is constructed by rescaling the empirical data from [38] so the ultimate tensile strength equals the in vivo failure stress reported for young adults, 100 MPa [64–66]. Cumulative damage to collagen fibers is based on fitting a commonly adopted exponential failure function to typical fatigue test data on human Achilles tendon.
Fig 4.
a) remodeling of collagen fibers by mechanical damage and repair, (I) shorter fibers are subject to higher strains, (II) fiber focal damage under mechanical strain forming a gap, (III) fiber repair at a longer length by filling in the gap with new collagen, b) repair probability density function following mechanical damage, quantifies the bias toward fiber lengthening, c) remodeling of collagen fiber by proteolytic damage and repair, (IV) longer fibers are subject to lower strains, thus more likely to be degraded by proteases, (V) new collagen forms across the gap while excess fiber is degraded resulting in a shorter fiber (VI), c) repair probability density function following proteolytic damage quantifies the bias towards fiber shortening.
Fig 5.
Probability function of fiber proteolytic damage as function of fiber strain.
Fibers are completely shielded from proteolytic damage for peak fiber strains εmax ≥ 1.5%.
Fig 6.
a) ankle torque over a gait cycle, b) ankle angle over a gait cycle, c) force-extension response of tendon from our discrete tendon model, d) required muscle force to produce the ankle torques in part (a), e) muscle length-force relation, f) muscle force-velocity relation, g) metabolic cost rate as a combination of generated muscle heat and mechanical work, h) ankle torque adjustment according to the ratio of the current total metabolic cost to the minimum total metabolic cost for a range of mean tendon lengths and fiber length standard deviations (SDs). Total metabolic cost results in part (h) are plotted for mean tendon lengths between 250mm to 275mm and fiber length SD = 1.5 mm.
Table 1.
Parameter names, symbols and values.
Fig 7.
Mechanical response of tendons with different fiber length distributions.
Force-extension curves for tendons with identical mean fiber length and different length standard deviations (SDs) i.e. SDs = 0.5% and 1.5% of mean fiber length.
Fig 8.
Tendon remodeling by mechanical damage and repair.
a) tendon remodeling by mechanical damage alone over 15 days, b) tendon remodeling by mechanical damage and repair over 90 days. Tendon peak force and number of loading cycles are kept constant for all simulated days.
Fig 9.
Tendon remodeling by proteolytic damage and repair.
(a) tendon remodeling by proteolytic damage alone over 15 days, b) tendon remodeling by proteolytic damage and repair over 90 days. Tendon peak force and number of loading cycles are kept constant for all simulated days.
Fig 10.
Tendon remodeling in response to metabolic cost.
Tracking geometric changes in four sample tendons with initial geometries corresponding with points A, B, C and D over 720 days of simulation. a-d) Total metabolic cost, mean tendon length, fiber length standard deviation, mechanical damage rate and proteolytic damage rate plots over 720 days for paths A-D respectively.
Fig 11.
Collagen fiber turnover time during tendon remodeling.
(a) Collagen fiber synthesis turnover time and (b) degradation turnover time for selected tendons of (Fig 10A–10D) during the remodeling process. Collagen fiber turnover time at equilibrium reaches around 180 years.
Fig 12.
Tendon remodeling exploration.
a) 55 equally spaced initial tendon geometries selected across the metabolic cost region, b) final geometries after 720 days of remodeling. White points indicate unstable initial geometries, dark points indicate initial geometries converging to an equilibrium state.
Table 2.
Sensitivity to changes of model parameters (% change Y/% change P, where Y is tendon length or dispersion, and P are model parameters).
Fig 13.
Relative relation of mechanical and proteolytic damages at different mean tendon lengths.
Intersection at point A denotes an unstable state where remodeling lengths diverge. Point B denotes a stable state where remodeling lengths converge. Arrows signify the direction of tendon length change driven by damage and repair processes.