Bayesian inference of physiologically meaningful parameters from body sway measurements

Figure 1. Sway model21. Human body is modelled as a rigid, weightless inverted pendulum with a point mass, m, at the height, h, of the body’s center-of-mass (com), rotating around a single link (ankle). The perturbations or noise, σ, due to gravitation and internal and external disturbances, such as blood circulation and respiration, cause the body to be unstable. Passive (stiffness K, damping B) and active (stiffness P, damping D) control (here proportional and derivative controllers) calculate a control torque, T c, that is needed to counteract the torques caused by disturbances and gravitation (T d and T g). Passive control describes continuous muscle tone, and active control –that acts with time delay, Δ,– describes CNS action. Active control is ‘ON’ only when needed, according to the magnitude and direction of body angle, θ, and angular velocity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{{\boldsymbol{\theta }}}$$\end{document}θ˙. The level of control, C ON, models this amount or fraction of…

Figure 2. Manifestation of measured COP and COM signals, and of a simulated COM signal. The measured COM is calculated from the COP signal using Eq. (9).

Figure 3. Simulated COM sway signals (top panel) and corresponding summary statistics (four lowest panels). The three columns present three simulated test subjects. The blue COM signals are the original three signals that were simulated with known parameter values. The red COM signals were simulated using parameters that were sampled from the joint posterior PDFs that were inferred from the original COM signals by the SMC-ABC algorithm (see Section Methods: Statistical inference of the model parameters). The lower panels show the summary statistics: amplitude -, velocity -, and acceleration histograms and spectra. In each panel, the blue line is the true summary statistic calculated from the original COM signals (average of the three), and the blue shadowed region presents 95% CIs that were calculated from the COM signals simulated using parameters that were sampled from the inferred marginal posterior PDFs. Leftmost column: true parameters: m = 54 kg, h = 0.85 m, P = 124 Nm/rad, D = 8.7 Nms/rad, Δ = 0.16 s, σ = 0.22 Nm, C ON = 0.63; estimated parameters: P = 119 Nm/rad, D = 24 Nms/rad, Δ = 0.18 s, σ = 0.23 Nm, C ON = 0.67. Mid column: true parameters: m = 80 kg, h = 0.78 m, P = 128…

Figure 4. Marginal posterior probability density functions of the five parameters: (a) Stiffness, P; (b) Damping, D; (c) Time delay, Δ; (d) Noise, σ; and (e) Level of control, C ON. Vertical lines present true parameter values (green, thick), estimated parameter values (green, dotted), 50% CIs (black, solid), and 95% CIs (red, dashed). These results are from the same simulated test subject as in the rightmost panel in Fig. 3. The ranges on the x-axes correspond to the ranges of the prior distribution.

Figure 5. Estimated parameters (posterior mean values) against true parameters. The equation for the estimated parameters against the true parameters is presented with a blue thin line. The equation should ideally be y = x, as indicated with a red thick line. The corresponding adjusted R 2 values are shown in the figures.

Figure 6. Sensitivity analysis. (a) The results are averaged (mean discrepancy and 95% CIs) across the 10 simulated subjects and 100 simulation rounds per subject. All summary statistics are included. (b–f) Amplitude-, velocity -, acceleration- histograms, and spectrum used one at the time to form the summary statistics. The results are averaged across 1000 simulation rounds of one representative test subject, the subject presented in the rightmost panel in Fig. 3. and in Fig. 4. The parameters are (b) stiffness, P, (c) damping, D (please note the wider x-axis scale, from 0.1 to 5), (d) time delay, Δ, (e) noise, σ, and (f) level of control, C ON. Briefly, the steeper the curve the more effectively the summary statistics detects changes in model parameters.

Figure 7. Real COM sway signals (top panel) and corresponding summary statistics (lower panels). The three columns present three real subjects. The blue COM curves correspond to the measured signals. The red COM signals represent values simulated using parameters that were sampled from the joint posterior PDFs that were inferred from the measured COM signals by the SMC-ABC algorithm. The lower panels show the summary statistics: amplitude -, velocity -, and acceleration histograms and spectra (see Section Methods: Statistical inference of the model parameters). In each figure, the blue line is the true summary statistic calculated from the original COM signals, and the blue shadowed regions present 95% CIs that were calculated using the COM signals that were simulated using parameters that were sampled from the inferred marginal posterior PDFs. The mass and estimated height of COM (see Section Methods: The control model) are: leftmost column: m = 68 kg and h = 0.80 m; mid column: m = 66 kg and h = 0.89 m; rightmost column: m = 68 kg and h = 0.87 m.

Figure 8. Marginal posterior probability density functions for the five parameters (real subject, same one as in Fig. 7, mid panel): (a) Stiffness, P; (b) Damping, D; (c) Time delay, Δ; (d) Noise, σ; and (e) Level of control, C ON. Vertical lines present estimated parameter values (green, dotted), 50% CIs (black, solid), and 95% CIs (red, dashed). The ranges on the x-axes correspond to the ranges of the prior distribution.