Rapid predictive simulations with complex musculoskeletal models suggest that diverse healthy and pathological human gaits can emerge from similar control strategies

Abstract

Physics-based predictive simulations of human movement have the potential to support personalized medicine, but large computational costs and difficulties to model control strategies have limited their use. We have developed a computationally efficient optimal control framework to predict human gaits based on optimization of a performance criterion without relying on experimental data. The framework generates three-dimensional muscle-driven simulations in 36 min on average-more than 20 times faster than existing simulations-by using direct collocation, implicit differential equations and algorithmic differentiation. Using this framework, we identified a multi-objective performance criterion combining energy and effort considerations that produces physiologically realistic walking gaits. The same criterion also predicted the walk-to-run transition and clinical gait deficiencies caused by muscle weakness and prosthesis use, suggesting that diverse healthy and pathological gaits can emerge from the same control strategy. The ability to predict the mechanics and energetics of a broad range of gaits with complex three-dimensional musculoskeletal models will allow testing novel hypotheses about gait control and hasten the development of optimal treatments for neuro-musculoskeletal disorders.

Keywords: biomechanics; locomotion; optimal control; three-dimensional.

Figure 1.

Simulated walking gaits with nominal and alternative cost functions. (a) Joint angles (add, adduction). The nominal cost function predicted an extended knee during mid-stance and limited ankle plantarflexion at push-off. Not squaring or removing the metabolic energy rate term from the cost function increased knee flexion but also trunk sway (e) and step width (g). (b) Joint torques. An extended knee resulted in small knee torques but limited ankle plantarflexion did not result in reduced ankle torques. (c) Joint powers. Limited ankle plantarflexion resulted in reduced ankle powers. (d) Ground reaction forces (BW, body weight; GC, gait cycle). (e) Trunk sway (i.e. trunk rotation in frontal plane). (f) Muscle activations (gluteus med, gluteus medius; min, minimus; semiten, semitendinosus; bic, biceps; fem, femoris; sh, short head; lat, lateralis; gastroc med, gastrocnemius medialis; ant, anterior). Removing the muscle activity term from the cost function resulted in unrealistically high muscle activations. The experimental electromyography data (grey curves) were normalized to peak nominal activations (black curves). (g) Metabolic cost of transport (COT), step width and stride length. The nominal COT matched experimental data [8]. (h) Resultant walking pattern with nominal cost function (electronic supplementary material, movie S1). Experimental data are shown as mean ± 2 s.d. (Online version in colour.)

Figure 2.

Alterations in gait features with speed. (a) Quadratic and linear regressions (black curves) based on simulation results (coloured markers) between metabolic cost of transport (COT) and speed for walking (0.73–2.23 m s⁻¹; $R^2=0.98$) and running (2.23–2.73 m s⁻¹; $R^2=0.66$), respectively. (b) Linear regression (black curve) based on simulation results from walking (coloured markers) between stride frequency and speed ($R^2=0.99$). The regression line is compared with the one obtained from experimental data [39]. (c) Vertical ground reaction forces (BW, body weight) at walking speeds less than the preferred walking speed of 1.33 m s⁻¹ (i), at walking speeds greater than the preferred walking speed (ii) and at running speeds (iii). Each coloured curve represents a simulation result for a different gait speed. (Online version in colour.)

Figure 3.

Effect of muscle weakness on walking pattern. (a) Hip muscle weakness. Reducing hip muscle strength by 50, 75 and 90% resulted in increased trunk sway and step width and decreased hip torques. (b) Ankle plantarflexor weakness. Reducing ankle plantarflexor strength by 50, 75 and 90% resulted in increased knee flexion and ankle dorsiflexion and decreased stride lengths that reduced ankle torques. Experimental data of the healthy subject are shown as mean ± 2 s.d. The simulations minimized the nominal cost function at the subject's preferred walking speed (1.33 m s⁻¹). (Online version in colour.)

Figure 4.

Simulated walking gait of an amputee with a transtibial passive prosthesis. Simulated ankle torques (red curves) matched the average ankle torques of six transtibial amputees [43]. The metabolic cost of transport (COT) for healthy and amputee walking was similar. Experimental data (grey envelopes) are shown as mean ± 2 s.d. The simulations minimized the nominal cost function at an imposed speed of 1.33 m s⁻¹. The prosthesis geometry is for visualization only.