Predictive simulation of post-stroke gait with functional electrical stimulation

Post-stroke patients present various gait abnormalities such as drop foot, stiff-knee gait (SKG), and knee hyperextension. Functional electrical stimulation (FES) improves drop foot gait although the mechanistic basis for this effect is not well understood. To answer this question, we evaluated the gait of a post-stroke patient walking with and without FES by inverse dynamics analysis and compared the results to an optimal control framework. The effect of FES and cause-effect relationship of changes in knee and ankle muscle strength were investigated; personalized muscle–tendon parameters allowed the prediction of pathologic gait. We also predicted healthy gait patterns at different speeds to simulate the subject walking without impairment. The passive moment of the knee played an important role in the estimation of muscle force with knee hyperextension, which was decreased during FES and knee extensor strengthening. Weakening the knee extensors and strengthening the flexors improved SKG. During FES, weak ankle plantarflexors and strong ankle dorsiflexors resulted in increased ankle dorsiflexion, which reduced drop foot. FES also improved gait speed and reduced circumduction. These findings provide insight into compensatory strategies adopted by post-stroke patients that can guide the design of individualized rehabilitation and treatment programs.

where ti and t f are initial and final times, respectively; a and Fm are muscle activation and tendon force, respectively; Re is the reserve actuator; and W E1−3 are weight factors (W E1 = 0.00750, W E2 = 0.99240, and W E3 = 0.00005).
The muscle-tendon parameters were bound between 50% and 200% of the initial values. The number of mesh intervals was 100 and the tolerance was 10 -4 . Parameters for all muscles with the exception of those spanning the lumbar joint were estimated. The parameter estimation converged in 5400 iterations after 19.95 h of central processing unit (CPU) time. Generic and personalized muscle-tendon parameters resulting from the parameter estimation are presented in Table S2.

Tracking and predictive simulations
Each tracking simulation result presented in this work comprised four simulations, i.e., one for each trial performed during the gait analysis. The initial guess, bounds, and scaling of joint kinematics were based on the gait trial that was tracked. Details of the calculation of these parameters have been previously published 3 . Thus, the gait speed and stride time were the same as in the ID result. The number of mesh intervals in the optimization was 100 and the tolerance was 10 -4 . The objective function for tracking was as follows: where q and M j are the joint angle and moment, respectively; Fr and Mr are the ground reaction force (GRF) and moment, respectively; the subscript R represents the experimental data; and W T 1−6 are the weight factors.
For the predictive simulation, the initial guess for joint kinematics was based on the data of a healthy subject walking used by Falisse et al. 3 , representing a normal gait. The bounds of stride time varied between 2 and 2.5 s in the simulations. The bounds and scaling for joint kinematics were calculated based on one trial in the DF condition, which allowed the model to achieve the ROM observed in the pathologic gait pattern. Details of this calculation and the description of other parameters used in the problem formulation can be found elsewhere 3 . The parameters for the contact spheres used in the GRF prediction were the mean of values optimized in Track-DF PM-High and Track-FES. The objective function was determined with the following equation: whereĖ is the metabolic energy rate, Dist is the distance traveled by the pelvis in the forward direction, and W P1−4 are the weight factors. The metabolic energy rate used in Eq. S4 was modeled as a smooth approximation of a model developed by Bhargava et al. 3,4 . The number of mesh intervals was 400 and the tolerance was 10 -4 . Although the large number of mesh intervals increased the computational cost, the resultant predictive simulations were more robust against changes in the settings.
The weight factors used in the tracking and predictive simulations are presented in Table S3. The resultant number of iterations, CPU time, and stride time of each simulation are presented in Table S4. The function in the sagittal plane of the major muscles and muscles spanning the knee and ankle joints is presented in Table S5.

Sensitivity analysis of predictive simulations
The settings used during the predictive simulation were individually altered, and the effects of these changes on gait parameters were analyzed. The bound of stride time varied between the simulations. Figure S9 shows the gait abnormality metrics for all settings. All other results presented in the main document and in the Supplementary Information used Setting 1.

Supplementary videos
We created three supplementary videos of gait patterns showing the views in the sagittal plane and frontal plane. Two complete gait cycles are presented in the predictive simulations.  Table S1. Passive knee moment parameters.

Set
K K K p p pa a as s ss s s1 1 1 K K K p p pa a as s ss s s2 2 2 K K K p p pa a as s ss s s3 3 3 K K K p p pa a as s ss s s4 4 4 θ θ θ p p pa a as s ss s s1 1 1 θ θ θ p p pa a as s ss s s2 2 2 (Nm) ( The passive joint moment function is presented in Eq. S1. For PM-Def, the default values of the passive moment parameters used by Falisse et al. 3 were applied. For PM-High, the parameters were changed to increase the passive knee flexion moment that can be attained. For PM-None, the extension angle limit was increased beyond the knee range of motion, resulting in no passive moment being generated.      Track-DF PM-None Figure S1. Relationship between passive knee moment and angle for the function in Eq. S1 and for the tracking of gait in the DF condition (ipsilateral knee joint).