A new measure for quantifying the bilateral coordination of human gait: effects of aging and Parkinson’s disease

Abstract

The bilateral coordination of locomotion has been described in detail in animal studies and to some degree in man; however, the mechanisms that contribute to the bilateral coordination of gait in humans are not fully understood. The objective of the present study was to develop a measure for quantifying the bilateral coordination of gait and to evaluate the effects of aging and Parkinson’s disease (PD) on this new metric. To this end, we compared the gait of healthy older adults to that of healthy young adults and patients with PD. Specifically, we defined the stride duration of one foot as a gait cycle or 360°, determined the relative timing of contra-lateral heel-strikes, and defined this as the phase, ϕ (ideally, ϕ = 180° for every step). The sum of the coefficient of variation of ϕ and the mean absolute difference between ϕ and 180° was defined as the phase coordination index (PCI), representing variability and inaccuracy, respectively, in phase generation. PCI values were higher (poorer bilateral coordination) in patients with PD in comparison to the healthy older adults (P < 0.006). Although gait speed and stride time variability were similar in the healthy young and older adults, PCI values were significantly higher among the healthy elderly subjects compared to the young adults (P < 0.001). Regression analysis suggests that only about 40% of the variance in the values of PCI can be explained by the combination of gait asymmetry (as defined by the differences in each leg’s swing times), gait speed and stride time variability, pointing to the independent nature of this new metric. This study demonstrates that bilateral coordination of gait deteriorates with aging, further deteriorates in PD, and is not strongly associated with other spatio-temporal features of gait.

Appendix: Analytical and numerical analyses of the relationship between the parameters ϕ_CV and ϕ_ABS

In this appendix, we present analytical and numerical analyses that describe the relationship between the two components of the PCI defined to incorporate the estimates by which a subject is constantly and accurately generating left–right stepping phases while walking.

Analytical analysis

Denote by ϕ = ϕ₁,ϕ₂, …, ϕ_N, a set of N measurements of the phase ϕ.

Let $\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }$ be the mean value of ϕ, i.e.

$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi } = \frac{1} {N}{\sum\limits_{i = 1}^N {\varphi _{i} } }, $$

(2)

and δ the standard deviation of ϕ given by

$$ \delta = {\sqrt {\frac{1} {N}{\sum\limits_{i = 1}^N {(\varphi - } }{\mathop \varphi \nolimits_i })^{2} } } $$

(3)

The coefficient of variation of ϕ is defined by

$$ \varphi \_{\text{CV}} = \frac{\delta } {{\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }}} $$

(4)

Define:

$$ \varphi \_{\text{ABS}} = \frac{1} {N}{\sum\limits_{i = 1}^N {|{\mathop \varphi \nolimits_i }} } - {\mathop {180}\nolimits^ \circ }| $$

(5)

From (4) we obtain:

$$ {\left( {\varphi \_{\text{CV}}} \right)}^{2} = \frac{{{\sum\nolimits_{i = 1}^N {({\mathop \varphi \nolimits_i } - \ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi })} }^{2} }} {{N{\mathop {\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }}\nolimits^2 }}}, $$

(6)

and (5) yields:

$$\begin{aligned}{}(\varphi \_{\text{ABS}})^{2} = & \frac{1} {{{\mathop N\nolimits^2 }}}{\left( {{\sum\limits_{i = 1}^N {|{\mathop \varphi \nolimits_i } - 180^{ \circ } |} }} \right)}^{2} \\ = & \frac{1} {{{\mathop N\nolimits^2 }}}{\left[ {{\sum\limits_{i = 1}^N {({\mathop \varphi \nolimits_i } - 180^{ \circ } )^{2} + 2} }{\sum\limits_{i = 1}^N {{\sum\limits_{j = i + 1}^N | }({\mathop \varphi \nolimits_i } - 180^{ \circ } )(\varphi _{j} - 180^{ \circ } )|} }} \right]} \\ \end{aligned} $$

(7)

But the left term in the square parentacese of Eq. 7 can be written as:

$$ \begin{aligned}{} \frac{1} {{N^{2} }}{\sum\limits_{i = 1}^N {(\varphi _{i} - 180^{ \circ } )} }^{2} & = \frac{1} {{N^{2} }}{\sum\limits_{i = 1}^N {{\left[ {|(\varphi _{i} - \overline{\varphi } ) + (\overline{\varphi } - 180^{ \circ } )|} \right]}^{2} } } \\ & = \frac{1} {{N^{2} }}{\sum\limits_{i = 1}^N {(\varphi _{i} - \overline{\varphi } )^{2} + } }\frac{2} {{N^{2} }}{\sum\limits_{i = 1}^N {(\varphi _{i} - \overline{\varphi } )[\Delta \varphi ]} } + \frac{1} {{N^{2} }}N[\Delta \varphi ]^{2} \\ \end{aligned} $$

(8)

where [Δϕ] is defined as follows: $[\Delta \varphi ] = \overline{\varphi } - 180^{^\circ }$

Note that the middle term in Eq. 8 can be written as:

$$ \frac{2} {{N^{2} }}{\sum\limits_{i = 1}^N {(\varphi _{i} - \overline{\varphi } )} }[\Delta \varphi ] = \frac{2} {{N^{2} }}[\Delta \varphi ]\,\,(N\overline{\varphi } - N\overline{\varphi } ) = 0 $$

(9)

Returning to Eq. 7 and using the relation described in Eq. 6, ABS² (ϕ) can be now written as:

$$ (\varphi \_{\text{ABS}})^{2} = \frac{{\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }^{2} }} {N}(\varphi \_{\text{CV}})^{2} + \frac{{[\Delta \varphi ]^{2} }} {N} + \frac{2} {{N\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }^{2} }}{\sum\limits_{i = 1}^{N - 1} {{\sum\limits_{j = i + 1}^N {{\left[ {|({\mathop \varphi \nolimits_i } - 180^{ \circ } )({\mathop \varphi \nolimits_j } - 180^{ \circ } )|} \right]}} }} } $$

(10)

or alternatively:

$$ (\varphi \_{\text{CV}})^{2} = \frac{N} {{\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }^{2} }}(\varphi \_{\text{ABS}})^{2} - \frac{{[\Delta \varphi ]^{2} }} {{\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }^{2} }} - \frac{2} {{N\ifmmode\expandafter\bar\else\expandafter\=\fi{\varphi }^{2} }}{\sum\limits_{i = 1}^{N - 1} {{\sum\limits_{j = i + 1}^N {{\left[ {|({\mathop \varphi \nolimits_i } - 180^{ \circ } )({\mathop \varphi \nolimits_j } - 180^{ \circ } )|} \right]}} }} } $$

(11)

Equation 11 suggests that quadratic–quadratic relation describes in part the analytic dependency between ϕ_CV and ϕ_ABS. At the same time this dependency may vary substantially, given a series of ϕ _i as expressed by the middle and the left terms of the right part of the equation. In the following paragraphs, a numerical analysis describes how the relation between ϕ_CV and ϕ_ABS is altered with accordance to different series of ϕ _i .

Numerical analysis

To explore a priori relations between ϕ_CV and ϕ_ABS, three separate artificial data sets were established. For each data set 31 left–right stepping phase series of 267 strides each were established. Each of the 31 series represents data obtained from a different walking human subject. Across all series in each data set, the mean values of ϕ ranged from 165° to 195° by increments of 1° (total of 31 mean values). The distribution of ϕ values around the mean value was identical for all 31 series, i.e. the standard deviation of the mean (SD) was identical for all 31 series. SD values were assigned to be 1.1, 4.4, 17.6 in the first (A), second (B) and thirds (C), data sets, respectively. Thus for each data set, 31 pairs of ϕ_CV and ϕ_ABS could be generated. Figure 4, depicts the relationships between ϕ_ABS and ϕ_CV for each of the data sets. On the abscissa the increase in the values of ϕ_CV corresponds to the decrease in the mean value of ϕ from 195° to 165° (a trivial outcome from the fact that SD was maintained constant). It can be seen that the corresponding ϕ_ABS values decrease as the mean value of ϕ approaches 180° and increase again as the mean value of ϕ continues to decrease towards 165°. It may be concluded that in a given value of SD there would be a monotonous change in the values of ϕ_ABS as a function of ϕ_CV, in half of the range of ϕ_CV, and opposite monotonous change in the second range of ϕ_CV. As can be seen by the different range of variation which is covered by ordinate axes in all three panels, we may conclude that actual experimental data collected may in part be more influenced by the direct dependency between ϕ_CV and ϕ_ABS, while for other terms this direct dependency is less obvious. Therefore, a priori for critical statistical tests based on this analysis it was determined to treat ϕ_CV and ϕ_ABS as inter-correlated parameters.

Fig. 4

Numerical analyses depicting the relation between ϕ_ABS and ϕ_CV. The trace in each panel shows calculated values of ϕ_ABS and ϕ_CV when the standard deviation of the mean value of ϕ is kept constant while the mean value of ϕ varies between 165° and 195° (left most and right most pairs of values, respectively, in all panels)