Investigation of Age-Related Changes in Physiological Kinetic Tremor

Abstract

Tremor is the most common movement disorder characterized by repetitive and stereotyped movements. Most investigations on tremor attempt to understand its relation to neuromuscular dysfunctions. Therefore, there is a lack of studies that aim to investigate the complex relation between the physiological tremor and aging, especially in kinetic conditions. In this context, the main motivation of this research was to quantify age-related changes in the kinetic tremor of clinically healthy individuals. For this, a number of features extracted from tremor activity, obtained from digitized drawings of Archimedes’ spirals, were analyzed. In total, 59 subjects participated in the experiments. These individuals were divided into seven groups according to their ages and two types of analysis were carried out. First, the tremor activity of young and elderly adults was investigated by means of traditional features commonly used in tremor analysis, and secondly, linear discriminant analysis (LDA) was employed for the study of the correlation between age and tremor. The results showed significant statistical differences between the kinetic tremor activity of the young and elderly groups. Furthermore, it was found that LDA allowed for the estimate of a unique feature, so-called LDA-value, which showed to be linearly correlated with age.

Appendix

This appendix describes the traditional set of features employed in the data analysis.

Frequency Domain Features

From the power spectrum S _x of the signal, obtained from the Fourier transform, the following features were estimated:

• 1. Mean frequency: it is the expected value of S _x as defined in Eq. (19).

  $$ f_{\text{mean}} = {\frac{{\sum\nolimits_{i = 1}^{N} {\left( {S_{x} \left( i \right)*f\left( i \right)} \right)} }}{{\sum\nolimits_{i = 1}^{N} {S_{x} \left( i \right)} }}} $$

  (19)

  where f_mean is the mean frequency, N the number of frequency bins in the spectrum, f(i) the frequency of the spectrum at bin i, and S _x (i) is the intensity of the spectrum at bin i.

• 2. Peak frequency: it is the frequency for which S _x is maximal.

• 3. Frequency of 50%: it is also known as the median frequency. It is the frequency that divides the area under S _x into two equal parts.

• 4. Frequency of 80%: it is the frequency for which 80% of the total energy of S _x is below it.

Detrended Fluctuation Analysis

DFA is a tool for analysis of random signals that estimates the α exponent which may characterize the nature of the time-series.9,26 The steps to estimate the value of the α exponent are provided below.

The discrete time-series S(n), n = 0,…,N − 1, where N is the number of samples, is divided into K intervals, of τ samples each, without overlapping. For each interval K, K = 0,…,N − 1, the mean value $\overline{{S_{k} }}$ (Eq. 20), the parameter y _k (Eq. 21) and the linear model z _k (i) (Eq. 22) are calculated, where a and b are the angular and linear coefficients of the linear model, respectively, for kth interval.

$$ \overline{{S_{k} }} = {\frac{1}{\tau }}\sum\limits_{i = k\tau }^{k\tau - 1} {S(i)} $$

(20)

$$ y_{k} = \sum\limits_{i = k\tau }^{k\tau - 1} {\left( {S(i) - \overline{{S_{k} }} } \right)} $$

(21)

$$ z_{k} (i) = ai + b,\quad k\tau \le i \le k\tau - 1 $$

(22)

The fluctuation function FF _k for each interval k is calculated by Eq. (23). Then the mean value of FF _k for all intervals is estimated as in Eq. (24).

$$ {\text{FF}}_{k} = \sqrt {{\frac{1}{\tau }}\sum\limits_{i = k\tau }^{k\tau - 1} {\left| {y_{k} - z_{k} \left( i \right)} \right|^{2} } } $$

(23)

$$ F_{k} = \frac{1}{K}\sum\limits_{k = 0}^{K - 1} {{\text{FF}}_{k} } $$

(24)

A behavior F _k  ~ τ ^α is expected, where the characteristic exponent (α) can be extracted through the inclination of the straightline in the graph log(F _k ) vs. log(τ). The interpretation of the exponent α is as follows: if the exponent (α) is less than 0.5 it characterizes an anti-persistent signal; α larger than 0.5 characterizes a persistent signal; and a white noise has an exponent (α) equal to 0.5.

Mean Speed

The MS of the tremor activity is the average of the instantaneous velocity v(n) as given in Eqs. (25) and (26), where, n is the discrete time and T the period, which is the inverse of the sampling frequency.

$$ {\text{MS}} = {\frac{1}{N - 1}}\sum\limits_{i = 0}^{N - 1} {v\left( i \right)} $$

(25)

$$ v(n) = {\frac{{\left| {S\left( {n + 1} \right) - S(n)} \right|}}{T}} $$

(26)

Total Displacement

The TD of the tremor activity is calculated by summing up all the distances from two consecutive samples, as shown in Eq. (27).

$$ {\text{TD}} = \sum\limits_{i = 0}^{N - 1} {\left| {S\left( {i + 1} \right) - S(i)} \right|} $$

(27)

RMS Mean

Also known as quadratic mean, the RMS is a statistical measure of the magnitude of a varying quantity, and it can be calculated by using Eq. (28).

$$ {\text{RMS}} = \sqrt {{\frac{{\sum\nolimits_{i = 0}^{N - 1} {\left( {S\left( i \right)} \right)^{2} } }}{N - 1}}} $$

(28)

Approximate Entropy

Approximate entropy (ApEn) is a tool used to quantify the regularity of a signal.27 It is usually normalized between 0 and 2, where 0 represents a predictable signal (e.g., sinusoidal signal) and 2 an unpredictable signal (e.g., a white noise).

In order to calculate the approximate entropy of the time-series S(n) it is necessary to select values for the parameters w and μ, where w is the length of a pattern, and μ is the criterion of similarity or the tolerance for pattern comparison. If a signal window of w samples beginning at sample i is denoted by p _w (i), then two patterns p _w (i) and p _w (j) will be similar if the difference between any pair of corresponding measures of the patterns is less than μ, therefore, [S(i + l) − S(j + l)] < μ, for 0 ≤ l < w.

Being P _w the set of all patterns of length w in S(n), C _iw (μ) is the number of patterns in P _w that are similar to p _w (i). In this case, C _iw (μ) can be calculated for each pattern in P _w , setting up C _w (μ) as the average of these values. C _w (μ) measures the regularity or the frequency of similar patterns to a certain pattern in S(n), with a window length equal to m, obeying the tolerance μ. Therefore, the approximate entropy of S(n) can be defined as in Eq. (29).

$$ ApEn\left( {w,\mu ,S(n)} \right) = \ln \left[ {{\frac{{C_{w} \left( \mu \right)}}{{C_{w + 1} \left( \mu \right)}}}} \right] $$

(29)

The ApEn measures the similarity between patterns p _w (i) and p _w (j). This technique was applied to the tremor signal with a value of w equal to 2 and the value of μ equal to 0.2SD(S(n)), where 0.2SD(S(n)) is the standard deviation of S(n), as suggested by Pincus.27

First-Order Smoothness

This tool can characterize imperfections in spirals drawn by the subjects. The calculation of this feature is based on the overall deviation of the spiral, in such a way that an ideal spiral results in a value of the first-order smoothness equal to zero. It happens because, in this case, there is a constant rate of change in the values of the spiral. Thus, it can be concluded that the larger the value of the first-order smoothness, the larger will be the deviation from the drawn spiral with respect to a given template.28

The first-order smoothness can be calculated through the divergence of all Δr/Δθ, where Δr is the difference between the ideal spiral’s radius and that resultant from the spiral drawn by subjects and Δθ is the difference between the ideal spiral’s angle and that resultant from the spiral drawn by subjects. This feature is calculated by Eq. (30).

$$ {\text{First-order smoothness}} = \ln \left[ {{\frac{1}{\theta }}\sum {\left( {{\frac{\Updelta r}{\Updelta \theta }} - \overline{{r_{\theta } }} } \right)}^{2} } \right] $$

(30)

where θ is the total angle over which the spiral is drawn and $\overline{{r_{\theta } }}$ is the RMS value of Δr/Δθ.

Second-Order Smoothness

The second-order smoothness can be defined as the rate of change of first-order smoothness. This feature can be calculated through the first derivative of first-order smoothness. Equation (31) shows how second-order smoothness is calculated.

$$ {\text{Second-order smoothness}} = \ln \left[ {{\frac{1}{\theta }}\sum {\left( {{\frac{{\Updelta {\frac{\Updelta r}{\Updelta \theta }}}}{\Updelta \theta }} - d\overline{{r_{\theta } }} } \right)^{2} } } \right] $$

(31)

where $d\overline{{r_{\theta } }}$ is the derivative of the RMS Δr/Δθ value.28

First-Order Zero Crossing Rate

This feature is a measure of irregularity of the signal and shows how frequently Δr/Δθ values cross their own RMS value. The first-order zero crossing rate (FOZCR) is more sensitive to small or frequent fluctuations. Equation (32) shows the calculation of this feature, where N is the total number of data points in the time series, i is a specific data point in time series, $\overline{{r_{\theta } }}$ is the RMS value of Δr/Δθ.28

The sign(x) function works in such a way that:

• If x > 0, then sign(x) = 1

• If x = 0, then sign(x) = 0

• If x < 0, then sign(x) = −1

$$ {\text{FOZCR}} = \left[ {{\frac{1}{{2\left( {N - 1} \right)}}}\sum\limits_{i = 1}^{N - 1} {\left[ {{\text{sign}} \left\{ {\left({{\frac{\Updelta r}{\Updelta \theta }}} \right)_{i + 1} - \left({{\frac{\Updelta r}{\Updelta \theta }}}\right)_{{\overline{{r}_{\theta}} }} } \right\}} \right]} - \left[{{\text{sign}}\left\{ {\left( {{\frac{\Updelta r}{\Updelta \theta }}}\right)_{i} - \left( {{\frac{\Updelta r}{\Updelta \theta }}}\right)_{{\overline{{r}_{\theta}} }} } \right\}} \right]}\right]*100\% $$

(32)

Second-Order Zero Crossing Rate

This feature is the rate of change of the first-order zero crossing rate, i.e., first derivative. The second-order zero crossing rate (SOZCR) provides valuable additional information concerning how regularly irregular or irregularly irregular a given spiral is. This feature can be calculated by Eq. (33), where $\overline{{r_{\theta } }}$ is the RMS value of change in radius vs. angle.28

$$ {\text{SOZCR}} = \left[ {{\frac{1}{{2\left( {N - 1} \right)}}}\sum\limits_{i = 1}^{N - 1} {\left[ {{\text{sign}}\left. {\left( {{\frac{{\Updelta {\frac{\Updelta r}{\Updelta \theta }}}}{\Updelta \theta }}} \right)} \right|\left( {i + 1} \right) - \left( {d\overline{{r_{\theta } }} } \right)} \right]} - \left[ {{\text{sign}}\left. {\left( {{\frac{{\Updelta {\frac{\Updelta r}{\Updelta \theta }}}}{\Updelta \theta }}} \right)} \right|\left( i \right) - \left( {d\overline{{r_{\theta } }} } \right)} \right]} \right]*100\% $$

(33)

Residual

This feature reflects the total distance between the spiral after the process of linearization and a line of best fit on the radius vs. angle graph. The larger this value is the more spiral changes its shape in an irregular way. This can be called RMS of r (radius) and it can be calculated by using Eq. (34), where Δr is the radius divergence of each point of the linearized digitizing tablet signal from the line of best fit, that represents the ideal spiral.28

$$ {\text{Residual}} = \sqrt {\sum\limits_{i = 1}^{N} {{\frac{{\left( {\Updelta r_{1} } \right)^{2} + \left( {\Updelta r_{2} } \right)^{2} + \left( {\Updelta r_{3} } \right)^{2} + \cdots + \left( {\Updelta r_{N} } \right)^{2} }}{N - 1}}} } $$

(34)