I. Introduction

Prevention of fall-related injuries is a major and urgent clinical problems related to functional ability and quality of life of elderly people^［1-5］ . Although various neurological diseases have already been identified as major risk factors of falling^［5,6］, other risks prevailing among the subclinical population, including increased age, disturbed gait, or poor balance, also need to properly evaluated^［2,5］. Non-invasive, non-expensive and easy-to-use method to monitor the daily changes of functional abilities to control posture will play a complementary role in improving clinical diagnosis for risk to fall. The displacement of the center of foot pressure （COP） measured by stabilometry is one method already being used^［7-10］, but its detection of dynamic properties of human postural control is insufficient out^{［3,7,10-12］}. Changes in COP relative to the balance control system in children have also been studied^［13-15］. However, few studies have focused on the dynamic aspect of postural control by spectral analysis of ground reaction force^［11］. Therefore, the present study focuses on the vertical ground reaction force （VGRF）, the unique external controllable force that can support the human body against gravity during a quiet stance. We extracted VGRF data from conventional stabilometry outputs of a previous study consisting of more than 2000 volunteers ranging from 3 to 93 years of age^［16］, as VGRF reflects the dynamic properties of the postural control. We analyzed VGRF by frequency domain to characterize human balance control as a function within a dynamic system. We then evaluated changes in spectral features relative to age to clarify the relationship between the human balance function and human aging and/or development.

II. Subjects and Methods

Among^［16］, 2201 healthy volunteers （age range, 3 to 94） from our previous study, 1745 （784 males 961 females） were analyzed. Data with obvious artifacts were excluded by visual inspection of the original waveforms and power spectra. The distribution of age includes 9 groups: I, 3-10 years （n＝412）; II, 11-20 （n＝358）; III, 21-30 （n＝339）; IV, 31-40 （n＝152）; V, 41-50 （n＝88）; VI, 51-60 （n＝89）; VII, 61-70 （n＝160）; VIII, 71-80 （n＝119）; and IX, 81-94 （n＝28）. Informed consent was obtained from every participant.
All subjects were asked not to move their head or limbs and to stand quietly on the force plate with feet together for 60 s with their eyes open （EO） and then for 60 s with their eyes closed （EC）. Body sway was recorded using a stabilometer （G5500, Anima, Japan）.
The load cell signals were sampled at 20 samples per second and filtered with a 12-bit analog to digital （A/D） converter at 0.05-50 Hz. The three calibrated load cell signals Fi（t）（i＝1,2,3） from stabilometry are summed to give the time series VGRF（t）≡ΣiFi（t）^{(Fig. 1A and 1B)}.

Individual body mass W is calculated by averaging VGRF over time W≡（ΣiFi（t））/G, where constant G denotes acceleration of gravity. The vertical acceleration a（t） is derived from the equation of motion of the body: a(t)=VGRF)T/W-G^{(Fig. 1C)}. An example of the a（t） time course of a 51 year-old man subject for the initial 30 s is shown in Fig. 1D. The magnitude of the vertical vibration is represented by the secondary moment （variance） of the acceleration. The average frequency f is computed as an indicator of trend in the frequency domain using the power-weighted mean, where p（f） denotes the power at frequency f f≡∫ f.p(f)df/∫p(f)df.
Each time series VGRF（t） and a（t） was divided into 3 data epochs with the length of 600 time points （30 s） lagged for half the length and for each epoch, a fast Fourier transform is then applied. The periodograms were smoothed across epochs and averaged with moving 15 frequency point （0.5 Hz） windows to estimate the power spectral density function （PSD）.
To examine the precise age effects on the parameters, polynomial regression models were also examined by age group. The maximum significant degree of the polynomial models were determined （P＜0.01） for both EO and EC according to the likelihood ratio tests （LRT）, step by step between degrees k and k+1 （k＝1, 2, 3...）.
For the power profiles of vertical vibration, we performed ANOVA for every frequency point to specify the bands with continual frequency points, where the PSD levels have significant difference （P＜0.01） among age groups. Multiple comparison tests were performed using the pairwise t-test at the specified frequency bands after adjusting the p-values by Holm’s method. The power ratio of EC to EO （EC/EO） was calculated in decibels （dB） for all subjects and averaged for each age group to give dB of geometric mean.

III. Results

The relationship between magnitude of vibration indicated by the variance of vertical acceleration in dB and age was analyzed for EO ^{(Fig. 2A)} and EC ^{(Fig. 2B)}. LRT showed the maximum significant degree （P＜0.01） of 4 for the polynomial regression models. The coefficient estimation and ANOVA of each regression model are shown in Table 1. Adjusted R2 was 0.457 for EO and 0.472 for EC. Each of the fitted quartic curves had a unique local minimum and the ages where the local minimum occurred were 32.3 years for EO and 29.7 years for EC. The bottom trend （which is defined as the slope of the common tangent） and the center （the midpoint between these points of tangency） of the curve were 0.084 dB/year and 53.8 years for EO and 0.091 dB/year and 52 years for EC, respectively. The average frequency of vibration （Fig. 2C and 2D） constantly increased from 5 Hz in group I to 6 Hz in group VII: in this age range （3-70 years）, a significant shift of frequency was found between every adjacent age group （P＜ 0.01）.

Figure 3A presents VGRF power for each group with EO. The 1.5-2.5 Hz component declined with age by 10 years of age, which was still high in young children （I）, and the 5.0-8.0 Hz component increased from group III to VIII. Two significant bands were found at 1.5-2.5 Hz and 5.0-8.0 Hz. Bands 1.5-2.5 Hz and 5.0-8.0 Hz for EC were similar to those for EO; a peak at 3 Hz in group II was found only for EO ^{(Fig. 3B)}. The acceleration power for each age group is shown in Fig. 3C and 3D. While ANOVA showed significant differences among age groups, younger adults （group III-VI） shared similar power profiles with a gradual increase in the range 2-5 Hz, a peak at around 6.5 Hz and a steep decrease above 6.5 Hz. Again, only group II showed a peak at 3 Hz for EO.

The results of the multiple comparison tests at significant bands are shown in Fig. 4. Significant differences at 2.0 Hz and 6.5 Hz were found between the elderly groups （VIII-IX） and younger adult groups （III-VI） and between the youngest groups （I-II） and younger adult groups （III-VI） ^{(Fig. 4A and 4C)}. Significant differences at 3 Hz were detected between the oldest group IX and groups VI-VII and between groups II and VI-VII, while significant differences at 7.0 Hz were between groups VIII and III-VI.

The effects of EC on PSD of vertical vibration are shown in Fig. 5^{(Fig. 5)}. The increase in overall magnitude of vibration was large for both young children and elderly people. The peak at 3 Hz for groups I and II, in addition to 7-Hz peak for group IX, was remarkable.

IV. Discussion

Our results of VGRF and vertical acceleration revealed that vibration of the human body in the vertical direction is affected by aging but not constantly throughout the life cycle. For subjects aged 21 to 60 years （group III-VI）, the magnitude of vertical vibration reached a minimum at 32.3 （EO） or 29.7 （EC） years old with a comparatively low flat bottom centered at 52 or 54, respectively, and increase rate less than 1 dB per decade. Frequency analysis revealed similarly low power spectra for this age range, and vertical vibration showed no significant effect relative to aging ^{(Fig. 4)}.
These same subjects （III-IV） also showed deviations in their power profiles from both the young subjects （I-II） and the elderly subjects （VIII-IX）. In fact, the multiple comparison test at 6.5 Hz showed significant difference between III-VI and I-II and between III-VI and VIII-IX. These overall tendencies were also observable for the magnitude of vibration ^{(Fig. 2A and 2B)}.
The patterns of these age-related changes were comparable with those of the path length on the horizontal plane. The profiles shared a local minimum around 30 years and a low flat bottom centered at around 50 years with a slight increase at a rate about 1 dB/decade, while having steep deviations for both young and elderly subjects. The result patterns were a well-fitted quartic curve.
Assuming that the increase in vibration power represents an increase in energy expenditure needed to control a quiet stance, the low stable power observed in subjects aged 20-60 years suggests stability and/or efficacy in the human postural control system. Conversely, the higher profiles in the younger （3-10 years） and elderly subjects （61-93 years） suggest either immature or declined stability and/or efficacy of the system.
The human postural control is achieved by a complex control system composed of multiple subsystems, such as visual, proprioceptive and vestibular^［3,10］ . To characterize the complex dynamic properties of the system, analysis of output signals in the frequency domain is useful^{［10,11,17,18］}. Our study of the EC/EO power ratio ^{(Fig. 5)} indicates an enhanced magnitude of vibration, specifically a peak at 3 Hz in younger subjects （3-20 years old）. Whether these peaks are valid, that is, whether they reflect either substantial PSD rise in EC or substantial PSD fall in EO, was investigated. Investigating acceleration PSD of younger subjects revealed a real peak at 3 Hz for EC ^{(Fig. 3A)}, but no peaks for EO ^{(Fig. 3B)}; although PSD of group I had a less prevalent peak than that of II, both differed from the neighboring frequency bins. This 3-Hz component in young subjects might represent the instability derived from immature compensation for the visual system, indicating the importance of visual input in the stability of postural vibration in the vertical direction at 3 Hz in children. In addition, vestibular-proprioceptive system may be immature in children. Of note, a 7-Hz positive peak in the group IX series are also observed, but inspection of PSD for EO and EC revealed no substantial peak.
Studies have focused on the horizontal fluctuation in COP to estimate individual abilities to control posture^{［7-9,13-15］}. COP is primarily the point of action on the support plane of the resultant VGRF that, assuming a quasi-static state, balances gravity; in this situation, the body achieves equilibrium at every moment, i.e., both momentum and angular momentum remain constant. As COP approximates well the projection of the center of mass at the surface of support in the proposed models for human postural control^［19-23］, changes in COP have often been analyzed to indicate the degree of postural stability. Conversely, COP inclines to represent quasi-static, i.e., less dynamic, properties of postural stability. In fact, studies have questioned whether COP measurements are sensitive enough to detect early changes or dynamic features in standing balance^{［3,11-13］}.
In this study, acceleration was obtained based on measurement of real forces acting between the sole and support platform in the vertical direction （VGRF）, which was actually mobilized during maintaining an upright stance while damping the perturbation from various origins^［24,25］. Thus, we suggest that estimates of VGRF record the dynamic events eliciting the act of postural control and thus indicate dynamic properties of the system.
In conclusion, measuring vertical acceleration of the human body is expected to play a substantial role in assessing developmental and/or age-related changes in human postural control functions.