1 Introduction

1.1 Background and Clinical Diagnosis of Parkinson’s Disease

Japan has a rapidly aging population: as of October 1, 2015, the population aged 65 and over has reached about 34 million, accounting for nearly 1/3 of the total population, and this percentage has kept rising since 1950 and is expected to increase in the future [1]. As a result, incidence of various aging-related diseases is becoming increasingly frequent. Parkinson’s disease is one of them [24].

Parkinson’s disease is caused by the disruption of dopaminergic neurotransmission in the basal ganglia [9]. Concerning the clinical diagnosis, assessment is generally based on the patient’s motor features - Tremor at rest, Rigidity, Akinesia (or bradykinesia) and Postural instability, under the well-known acronym “TRAP” as four primary symptoms [5].

In Japan, following a medical interview and a neurological examination, the potential patient will often receive an image inspection, and it is common to use a SPECT (single-photon emission computed tomography) gamma scanner to directly observe the amount of dopaminergic neurons in the subject’s brain. As the following figure shows, a Parkinson’s disease patient has significantly less amount of dopaminergic neurons than a healthy individual (Nishi-Niigata Chuo National Hospital [8]) (Fig. 1).

Fig. 1.
figure 1

SPECT images of dopaminergic neurons of a healthy individual (left) and a Parkinson’s patient (right).

Full size image

1.2 Possibility of a Simple Test Using Pulse Waves

The SPECT imaging is widely applied in Japan’s hospitals in diagnosing Parkinson’s disease, but for the patients, the SPECT scan procedure seems not a comfortable experience. Injection of gamma-emitting radioisotope (RI) is required. After the injection, the patient normally has to wait 3 to 6 h in order for the RI to take effect in the brain. Then, the SPECT scan itself needs approximately 30 min to complete (Nihon Medi-Physics Co., Ltd. [7]). In addition, such inspection is relatively expensive: an insured patient under 70 years old (the applicable coinsurance rate is 30%) has to pay about 25,000 JPY (about 200 EUR under the exchange rate 1 EUR = 125.44 JPY).

It would be beneficial for the potential patients if a simple – less time-consuming and expensive – test can help distinguish Parkinson’s disease sufferers from healthy individuals. However, little literature, if any, is available on this subject.

Studies have shown that non-motor symptoms are also common in Parkinson’s disease. Two categories of these symptoms are autonomic dysfunction [25] and cognitive and neurobehavioural abnormalities. The latter includes depression [6] and dementia [2].

Meanwhile, in our recent studies, we have discovered indicators – the largest Lyapunov exponent (LLE) and the autonomic nerve balance (ANB), both computed from pulse wave data – for identifying mental status changes [13, 23] and mental disorders, including dementia [11, 13, 17] and depression [4, 13, 18]. A comprehensive explanation is available in Oyama’s 2012 book [10] and a review article [15].

Inspired by the relevance of Parkinson’s disease to mental disorders and the effectiveness of the pulse wave analysis in detecting mental disorders, we have made an attempt to observe if any characteristic patterns of Parkinson’s disease sufferers exist in their pulse waves.

This study has succeeded in discovering such characteristic patterns, by comparing the sample entropy computed from the pulse wave data. More precisely, what we applied is the sample entropy with two parameters – the length of subsequences of the data sequence and the tolerance – set to certain fixed values. We define this indicator as “border of Parkinson Entropy (BPE)”. Besides, in addition to BPE, statistically significant difference is also in the LLE values from the same pulse wave data.

Furthermore, we have incorporated the function of BPE computation and result display into “Alys”, an application installed on an Android tablet that we developed for real-time mental health check-up [16]. With “Alys”, not only status of mental health, but also risk of Parkinson’s disease can be checked in a convenient and economical way. In particular, it takes only 5 s to show the test result on the screen.

2 Computational Methods

In this study, we mainly propose two indicators – the border of Parkinson entropy (BPE) and the largest Lyapunov exponent (LLE). We begin with the introduction of sample entropy.

2.1 Sample Entropy

As a conventional method for studying the complexity in biological time series, the sample entropy is defined as the opposite of the natural logarithm of the conditional probability that two sequences that are similar for certain points within a given tolerance still remain similar when one consecutive point is included [19].

To begin with, given a time-series sequence

$$ \left\{ {x\left( 1 \right), \ldots ,x\left( N \right)} \right\}, $$
(1)

its subsequence with a length of m can form a vector

$$ X_{m} \left( i \right)\, = \,\left( {x\left( i \right),x\left( {i + 1} \right), \ldots ,x\left( {i + m - 1} \right)} \right) $$
(2)

and, in the same fashion, an (m + 1) subsequence can be denoted as

$$ X_{m + 1} \left( i \right) = \left( {x\left( i \right),x\left( {i + 1} \right), \ldots ,x\left( {i + m} \right)} \right). $$
(3)

Here, the range of i is from 1 to N − m so that both (2) and (3) are well-defined.

Next, the distance between two m-long subsequences Xm(i) and Xm(j) is defined as

$$ \left| {X_{m} \left( i \right) - X_{m} \left( j \right)} \right| = \mathop {\hbox{max} }\limits_{0 \le k \le m - 1} \left| {x\left( {i + k} \right) - x\left( {j + k} \right)} \right| $$
(4)

For a given Xm(i), its r-neighbourhood is

$$ \{ X_{m} \left( j \right): \left| {X_{m} \left( i \right) - X_{m} \left( j \right)} \right| < r\} $$
(5)

Let \( B_{i}^{m} \left( r \right) \) denote the probability that another subsequence is in its r-neighbourhood. Thus,

$$ B_{i}^{m} \left( r \right) = \frac{{\# \{ X_{m} \left( j \right): \left| {X_{m} \left( i \right) - X_{m} \left( j \right)} \right| < r, 1 \le j \le N - m,j \ne i\} }}{N - m - 1} $$
(6)

Note that when counting the number of such subsequences in the numerator of (6), since Xm(i) itself should be excluded, there are a total of N − m − 1 candidates. Hence the denominator N − m − 1. Regarding Xm+1(i), we use a different notation \( A_{i}^{m} \left( r \right) \) to denote the probability that another (m + 1)-long subsequence is in its r-neighbourhood:

$$ A_{i}^{m} \left( r \right) = \frac{{\# \{ X_{m + 1} \left( j \right): \left| {X_{m + 1} \left( i \right) - X_{m + 1} \left( j \right)} \right| < r, 1 \le j \le N - m,j \ne i\} }}{N - m - 1} $$
(7)

For the whole time-series sequence (1), the probability corresponding to (6) or (7) can be given as an average taken over all subsequences, from i = 1 to i = N − m, as follows.

$$ B^{m} \left( r \right) = \frac{1}{N - m}\mathop \sum \limits_{i = 1}^{N - m} B_{i}^{m} \left( r \right) $$
(8)
$$ A_{{}}^{m} \left( r \right) = \frac{1}{N - m}\mathop \sum \limits_{i = 1}^{N - m} A_{i}^{m} \left( r \right) $$
(9)

with tolerance r for m-long subsequences of an N-point time-series sequence is therefore computed by the following formula.

$$ SampEn\left( {m, r,N} \right) = - \ln \frac{{A^{m} \left( r \right)}}{{B^{m} \left( r \right)}} $$
(10)

In our recent studies on the indication of mental health from pulse waves, the device “Lyspect” (developed by Chaos Technology Research Laboratory) has been frequently applied [12]. We have upgraded the device to make the computation of sample entropy possible. The following shows the value of sample entropy (vertical axis) as a function of the tolerance r (horizontal axis), with the length of subsequence m fixed. A total of 9 graphs are displayed, for m = 2 to 10 (Fig. 2).

Fig. 2.
figure 2

Display of sample entropy with “Lyspect”.

Full size image

2.2 Border of Parkinson Entropy

We define the border of Parkinson entropy (BPE) as the sample entropy with m = 2 and r = 10%, namely,

$$ BPE = SampEn\left( {2,10\% } \right) $$
(11)

(The length of the time series sequence, N, is dropped for convenience.) These two parameters were decided this way after trials and errors in search for an ideal indicator that shows statistically significant difference between Parkinson’s disease sufferers and healthy individuals. This will be explained in Sect. 4.1.

As mentioned at the end of Sect. 1, we have imbedded the function of BPE computation in our device “Alys”. A normalized result display, in a range of 0–10, is applied with a semi-circular graph, in consistency with the display of largest Lyapunov exponent and autonomic nerve balance. We will introduce this new performance in Sect. 5.

2.3 Largest Lyapunov Exponent

The mathematical definition and computation of the largest Lyapunov exponent (LLE) is elaborated in almost each of our papers on the indication of mental health from pulse waves (for the most updated work, refer to [16] and [15]). In this article, although we mainly study the BPE, we still examine whether a relevant result holds from the viewpoint of the LLE behaviour. The following is a brief explanation on the definition and computational method of the LLE.

Consider the same time-series sequence (1). If we let τ denote a constant delay and d be the embedding dimension, then the phase space can be reconstructed with vectors represented as

$$ \begin{aligned} X\left( i \right) = & (x\left( i \right),x(i - \tau ), \ldots ,x(i - (d - 1)\tau )) \\ = & \{ x_{k} \left( i \right)\}_{k = 1, \ldots ,d} \\ \end{aligned} $$
(12)

where

$$ x_{k} \left( i \right) = x(i - \left( {k - 1} \right)\tau ),k = 1, \ldots ,d. $$
(13)

Regarding the fingertip pulse waves, the optimal choices for the constant delay and the embedding dimension are determined from the autocorrelation coefficient of the wave [21, 22]:

$$ \tau = 50\,\text{ms} $$
(14)

and

$$ d = 4. $$
(15)

Then the LLE is defined by the following formula.

$$ LLE = \mathop {\lim }\limits_{t \to \infty } \mathop { \lim }\limits_{ \epsilon \to 0} \frac{1}{t}\log \frac{{\left| {\delta X_{ \epsilon } \left( t \right)} \right|}}{\left| \epsilon \right|} $$
(16)

where

$$ \delta X_{ \epsilon } \left( t \right) = X\left( t \right) - X_{ \epsilon } \left( t \right) $$
(17)

represents the divergence of the trajectories, with initial condition of

$$ \epsilon = X\left( 0 \right) - X_{ \epsilon} \left( 0 \right) $$
(18)

In our studies, the method proposed by [20] is applied for estimating the LLE.

In our devices “Lyspect” [12] and “Alys” [16], the value of LLE is normalized to a range of 0–10 in the result display.

Our previous studies have shown important results concerning the LLE as a mental health indicator. The values of LLE of a mentally healthy individual fluctuate from 2 to 7, centred at 5. When LLE is abnormally high, the mental immunity of the individual is so strong that he or she is likely to go to extremes: such individual can be easily irritated and take unexpected actions. On the other hand, when it is abnormally low, the mental immunity is so weak that the individual is prone to mental illnesses. In other words, a high LLE indicates a mental status of adapting to the external environment (we simply called it “external adaptation” in some of our previous articles), while a low LLE indicates a status of “internal focusing”.

2.4 Autonomic Nerve Balance

The autonomic nerve balance (ANB) is another important indicator in our recent studies [16] and [15].

We consider the high frequency (HF, 0.15–0.40 Hz) component and the low frequency (LF, 0.04–0.15 Hz) component, which represents parasympathetic nerve activity and sympathetic nerve activity, respectively. Then, we define the autonomic nerve balance (ANB) as a normalized index ranging from 0 to 10 as follows.

$$ ANB = 10\,B/3.5, $$
(19)

where

$$ B = \text{ln}\,\left( {LF} \right)\,/\,\text{ln}\,\left( {HF} \right). $$
(20)

From the ranges of LF and HF, we can clearly observe that B ranges from 1, when both LF and HF take the value 0.15, to approximately 3.5, when LF reaches its minimum while HF is at its maximum. Therefore, ANB goes from approximately 2.86 to 10.

In our devices, like LLE, we apply a 0–10 valued graph to display the result of ANB. ANB < 5 indicates predominance of parasympathetic nerve while ANB > 5 indicates sympathetic predominance.

The computation method has also been embedded in the devices “Lyspect” and “Alys”, and it has been registered as a U.S. patent [3].

3 Experiment

3.1 Devices

We apply an infrared sensor (UBIX Corporation) to take in pulse waves from the subjects, and computer software “Lyspect” (Chaos Technology Research Laboratory) to analyse the data.

The pulse waves are taken in as 200 Hz analogue data, saved as text file, and then input to “Lyspect” for analysis. To reduce noise from the external environment (such as the power supply), the fast Fourier transform is applied in order that only data with frequency less than 30 Hz (It has been shown by additional trials that 8 Hz will suffice to produce the same analytical results) is to be analysed.

3.2 Subjects

Two groups of subjects, the Parkinson’s disease patients and healthy individuals, are studied.

The former group consists of 45 patients diagnosed as Parkinson’s disease, aged from 40 to 65. The latter group consists of 113 healthy university students, aged from 19 to 20.

3.3 Process of Measurement

Informed consent was obtained from all subjects in the measurement.

For each subject, a 2-min measurement was performed for 2 to 3 times in a relaxed condition at room temperature (25 ℃) and the average result of measurement was used for analyse. Specifically, for the healthy students, it was sufficient to take 2 times because their results were stable, while for each of the Parkinson’s disease sufferers, measurement was performed 3 times at intervals.

For a part of the Parkinson’s disease sufferers, in order to reduce measurement errors due to tremor, a common symptom of the disease, the sensor was attached to the subject’s earlobe instead of fingertip.

4 Analysis and Result

4.1 Comparison of Sample Entropy

As introduced at the end of Sect. 2.1, “Lyspect” can display the sample entropy values SampEn (m, r) as a function of r, for different m’s. We observed that as m increases, the range of SampEn (m, r) tends to concentrate and less sensitive to r, so we decided to apply m = 2. In the following, SampEn (2, r) is compared between the two groups.

The following graph shows SampEn (2, r) for the group of 113 healthy individuals. We observe that when the tolerance r changes from a small value over 0 to a little more than 40%, the sample entropy value with m = 2 monotonically decreases and the range of SampEn (2, r) is bounded in (0, 0.4) for each subject of this group.

Similarly, SampEn (2, r) for the group of Parkinson’s disease suffers is shown in the following graph. The tolerance changes in the same way as the above. SampEn (2, r) is monotonically decreasing, but the range of SampEn (2, r) is remarkably wider than the healthy individuals’ group.

In hopes of finding an ideal indicator to distinguish Parkinson’s disease sufferers from healthy individuals, based on the data from our measurement, we have performed analysis of variance (ANOVA) for various r’s. Consequently, through trial and error, we found that when r = 10%, the result of ANOVA shows highly statistically significant difference in SampEn (2, 10%) between Parkinson’s disease sufferers and healthy individuals. The basic information of SampEn (2, 10%) values for the analysis are given in the following Table 1.

Table 1. SampEn (2, 10%) data information.
Full size table

The ANOVA for the difference in SampEn (2, 10%) between the two groups produces the following result (Table 2).

Table 2. ANOVA for the difference in SampEn (2, 10%).
Full size table

Since the p value is less than 0.0001, the SampEn (2, 10%) values between the two groups are statistically different at 0.01% significance level, or at 99.99% confidence level. This is why we call SampEn (2, 10%) border of Parkinson’s entropy, or BPE. The distribution of BPE values for the two groups can also be compared in the following figure. One can clearly observe that the Parkinson’s disease sufferers exhibit a significantly higher BPE than the healthy students.

4.2 Sample Entropy and Progression of Parkinson’s Disease

Another observation made is that the sample entropy value tends to increase as the Parkinson’s disease sufferer deteriorates.

The following shows the status of SampEn (2, r) for a same Parkinson’s disease sufferer on two different dates of measurement. On July 31, 2016, there was no particular problem reported, but after 3 months, on November 1, 2016, the patient reported difficulty to move and occurrence of drooling, which interfered the patient’s daily life. We clearly observe that for each tolerance r, SampEn (2, r) (and thus BPE in particular) on the latter date is higher than that on the former date.

Therefore, for a same patient, BPE may be a potential indicator for checking the progression of Parkinson’s disease. Doctors may refer to the BPE value when they conduct medical examination by interview.

4.3 Comparison of LLE and ANB

Since LLE has played a leading role in our studies on the indication of mental health from pulse waves, LLE values computed by “Lyspect” between the two groups are also compared and analysed.

The basic information of LLE values for the analysis are given in the following Table 3.

Table 3. LLE data information.
Full size table

Recall from Sect. 2.3 that the LLE value is normalized to range from 0 to 10. Next, the result of ANOVA for the difference in LLE between the two groups is stated in the following Table 4.

Table 4. ANOVA for the difference in LLE.
Full size table

Since the p value is less than 0.0001, the LLE values between the two groups are statistically different at 0.01% significance level, or at 99.99% confidence level. The following figure compares the distribution of LLE values between the two groups. Obviously, the LLE of the group of Parkinson’s disease patients is significantly lower than that of the healthy individuals’ group.

The above result is consistent with the fact that depression is a common symptom of Parkinson’s disease [6] and the result we have obtained in our recent studies that a low LLE indicates weakness in mental immunity which leads to depression [10].

In addition, we have also looked over ANB computed from the same data. Like in BPE and LLE, we have obtained statistically significant difference in the ANB values between the two groups. However, since medicine that the patients are taking can affect the nervous system and thus influence the result of ANB, we withhold further analysis.

4.4 Discriminant Analysis of BPE

As presented in Sect. 4.1, the BPE can provide as an indicator for identifying Parkinson’s disease sufferers. Next, discriminant analysis is carried out, with the help of statistical software, in order to determine critical values of BPE to distinguish Parkinson’s disease sufferers from healthy individuals. The process and result of the discriminant analysis are shown below (Table 5).

Table 5. Result of discriminant analysis of BPE.
Full size table

From the result, we conclude that our pulse wave data infer that if BPE ≥ 0.3017, the probability of suffering Parkinson’s disease is 94.65%, and if BPE < 0.2189, the probability of not suffering Parkinson’s disease is 97.48%.

5 Checking BPE with “Alys”

In this section, we introduce our upgraded version of “Alys”, with which the analysis and result display of BPE have become possible. We explain the procedure of visualizing BPE with “Alys”.

  1. 1.

    Start “Alys”

  2. 2.

    Connect the sensor to the tablet through a USB connector

  3. 3.

    Click the tool mark on the upper right, select “Set Properties” and then select “Compute BPE” from the “Execution of Analysis Mode”

    The 4 options, from top to bottom, are “Standard Execution Mode”, “Demonstration Mode”, “Live Mode” and “BPE Computation”. The “BPE Computation” comes at last as it is a newly added function.

  4. 4.

    Back to the “Set Properties” menu, set the measurement time (in second) and determine the critical value of BPE that is to be normalised to 5.0 in the result display. When this setting is done once, it will be saved so users need not set each time

    In the above figure, the items shown in “Set Properties” Menu are interpreted, from top to bottom, in the following.

    • Moving average of real-time Lyapunov (Here Lyapunov simply means the LLE.)

    • Real-time sound recording

    • Measurement time (in second)

    • Critical value of BPE that is to be normalized to 5.0 in the result display

    • Quality check for pulse waves.

    Regarding the “Measurement time (in second)”, we have improved the system so that analytical result of BPE can be obtained with as short as 5 s of measurement.

    Concerning “Critical value of BPE that is to be normalized to 5.0 in the result display”, from the result of discriminant analysis in Sect. 4.4, we may use 0.31 (slightly higher than 0.3017) as the critical value corresponding to 5.0, the central value of the normalized BPE.

  5. 5.

    Start to take the pulse from a fingertip

    When the measurement time set in the previous step has elapsed, the measurement will end and a semi-circular graph will be displayed.

    The BPE is normalized to range from 0 to 10, centred at 5.0, which corresponds to the critical BPE value set at the previous step. From the above figure we observe that the subject’s normalized BPE is 2.4, which is less than 5.0, so this subject may not be a Parkinson’s disease sufferer.

  6. 6.

    Other options

    Users may view their records of BPE values taken in the past in both “List Mode” and “Graph Mode”. The former makes a list of all recent records, while the latter displays all results on the same semi-circular graph.

    Moreover, the data saved in the tablet can be attached to email or sent to Cloud.

6 Conclusion and Remark

In this study, we firstly reviewed the background of the occurrence of Parkinson’s disease in Japan, taking the demographic state into consideration. Following that, we presented the common approach of its clinical diagnosis. Observing the role of mental disorders as a bridge – the relevance of Parkinson’s disease to mental disorders, and the effectiveness of the pulse wave analysis in detecting mental disorders – we proposed a new indicator for identifying Parkinson’s disease, the border of Parkinson’s entropy (BPE). We collected a considerable number of pulse wave data, computed the BPE values with our device, and performed statistical analysis to obtain persuasive result. The analysis reached a conclusion that the BPE can provide as a potentially effective indicator of Parkinson’s disease.

However, since this indicator is newly proposed, there is much room for improvement: there may exist better choice for the parameters of the sample entropy, and statistical analysis needs to be performed based on larger samples in order to obtain more convincing result. We will strive to collect and analyse more data in the future.

In regard to the upgraded “Alys”, since 5 s will suffice to produce analytical result, we believe it can enable users to conduct self-check in a convenient and economical way, without time and space limitation. Test results of BPE may provide a reference for doctors conducting a medical examination for a potential patient. Moreover, the result of LLE, together with BPE, may also be a worthwhile reference.

Declaration

This article is a revised and extended version of a previous paper [14]. Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14, Formulae 1 to 11 and all Tables are borrowed from the original paper.

Fig. 3.
figure 3

Graph of SampEn (2, r) for healthy individuals.

Full size image
Fig. 4.
figure 4

Graph of SampEn (2, r) for Parkinson’s sufferers.

Full size image
Fig. 5.
figure 5

Comparison of distribution of BPE values.

Full size image
Fig. 6.
figure 6

Graph of SampEn (2, r) for a Parkinson’s sufferer in different condition of disease progression.

Full size image
Fig. 7.
figure 7

Comparison of distribution of LLE values.

Full size image
Fig. 8.
figure 8

Process of discriminant analysis of BPE.

Full size image
Fig. 9.
figure 9

The welcoming window of “Alys”.

Full size image
Fig. 10.
figure 10

Connection of the sensor and the tablet.

Full size image
Fig. 11.
figure 11

Option list of “Execution of Analysis Mode”.

Full size image
Fig. 12.
figure 12

Option list of “Set Properties”.

Full size image
Fig. 13.
figure 13

(Left) Display of waveform during a measurement; (Right) graph for normalized BPE.

Full size image
Fig. 14.
figure 14

Display of past records in “List Mode” (left) and “Graph Mode” (right).

Full size image