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國立臺灣大學電機資訊學院電信工程學研究所 博士論文
Graduate Institute of Communication Engineering College of Electrical Engineering and Computer Science
National Taiwan University
Doctoral Dissertation
具有強健性之非線性特徵提取法應用於臨床醫學 Robust Methods for Nonlinear Behavior Identification in
Clinical Applications
張儀中 Yi-Chung Chang
指導教授：曹建和 博士 Advisor : Jenho Tsao, Ph.D.
中華民國 104 年 1 月
Jan, 2015
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致 謝
念博班對我而言向來不是意料中的事，自從碩班受到曹老師啟發，接觸了醫用 超音波領域，也跟著高中同學楊智傑，與尹彙文與許淑霞醫師接觸了心率變異相 關領域，畢業後因緣際會地從較擅長的電子產業做到挑戰性高的醫療儀器領域，
一路走來跌跌撞撞，過程中也慢慢了解到自己在醫療領域的不足之處，因而決定 攻讀博士。
在念博班的過程中，因為興趣所以研究越做越深，卻往往難以收斂，多虧了我 的學長羅孟宗老師，亦師亦友的指導兼鼓勵，還有一起研究努力的多個團隊，包 括了東華的吳賢財老師與其實驗室成員，台大何亦倫主任與林彥宏醫師、黃惠君 醫師，林亮宇醫師與葉惠敏醫師，北榮陳適安主任與林彥璋醫師、鍾法博醫師，
國泰王拔群主任與其團隊，終能將這些過程轉換成具體研究。
研究中所用到的許多方法也都仰賴中央大學數據中心黃鍔主任與彭仲康教授的 指導，還有中心許多同事於論文上的協助包括王淵弘博士、王政嚴博士、楊緒文 博士、葉家榮博士、林澂博士。特別感謝林澂在許多方面尤其是生理上的意見，
給了我很多啟發，還有感謝所有中心的研究助理，在許多研究上都幫忙了不少忙，
還有中心的行政助理們，有你們的默默支持，讓許多研究能順利進行。
最後，將本篇論文獻給我的家人，沒有你們的支持就沒有現在的成果。
張儀中 於民國 104 年 1 月
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Contents
中文摘要 . . . 5
Abstract . . . 6
1 Introduction ………. 8
1.1 Homeostasis and Correlations ……… 8
1.2 Heart Rate Variability and Autonomic Nervous System ……… 9
1.3 Dynamical System and Orbits ……… 10
1.4 Reconstruct the Dynamics ……… 12
1.5 Multiple Time Scale Dynamics ……… 14
1.6 Attracting Orbit and Discrete Dynamical System ……… 16
2 Quantization of Multi-scale correlation ……….. 21
2.1 Application of a Modified Entropy Computational Method in Assessing the Complexity of Pulse Wave Velocity Signals in Healthy and Diabetic Subjects ……… 21
2.2 Outlier-resilient complexity analysis of heart beat dynamics … 33 3 Quantification of Attracting Orbit ……… 48
3.1 New Method to Noninvasively Monitor Fetal Heart Rate during Cesarean Section ……… 48
3.2 Nonlinear Analysis of Fibrillatory Electrogram Similarity to Optimize the Detection of Complex Fractionated Electrograms During Persistent Atrial Fibrillation ……… 63
4 Conclusion ……….. 81
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中文摘要
近年來，許多領域持續發現及探討身體系統的複雜性，透過分子生物與病理機 轉的研究，這些複雜機轉最終很可能會形成一個完整理論。另一方面，生醫信號 分析也從生理系統的複雜性與調控理論之中發展出一些方法，能提取複雜特質和 數據之下的訊息。這些非線性特徵提取方法，能夠從模糊不清的訊息下找出非線 性的特徵作為疾病的判斷依據，其中一些方式也被證實效果優於傳統方法。然而 非線性特徵卻容易受到臨床條件和環境的限制，如有限的數據長度，資料偶有會 有參差不齊或是雜訊干擾。除此之外，提取方法也會造成失真，所使用之方法也 可能無法有效濾除其他因素影響，有時甚至會大大增加後續分析的困難。本研究 提出數種強健性的改良方法用來識別臨床數據的非線性特徵，以滿足臨床要求。
第一個探討的非線性特徵是訊息在多尺度的相關性，衡量方式是用資訊理論中 的熵及渾沌碎形理論中的尺度，以不同時間尺度下的相關性，衡量系統的複雜特 質。其中一個應用是透過計算血管動脈脈搏波速資訊的多尺度相關性，於大尺度 的計算中增加計算準確度的方法，病患的量測時間能大幅減少到 12 分鐘。本方法 能以較小的樣本大小(即 600 個連續信號)，在區分健康、中年、糖尿病患之間，
達到與傳統的方法(即 1000 個連續信號)同樣的靈敏度。
另一個應用是在心率變異度分析中多尺度相關性的計算，透過改良的方法來抵 抗心律不整因素的干擾，用於辨別安裝葉克膜病患的存活率。這項研究中提出了 一種新的方法，通過分析在不同時間尺度的符號時間序列的不規則性來估計信號 的複雜性，能有效避免葉克膜病患頻繁發生的心律不整所造成的干擾，該方法能 夠檢測心臟調節功能的降低，並避免治療充血性心臟衰竭和葉克膜重症患者更加 惡化。研究結果顯示，在嚴重干擾又同時有大量異常數值的心跳序列中，本方法 能夠可靠地評估其多尺度的複雜性，因此可以作為一個有效的臨床工具，用於監 控重症患者的心率調節功能。
第二個探討的非線性特徵是動態系統軌道的特質，這是透過相位空間軌跡計算 而得。其中一個應用是在剖腹分娩過程中，以非侵方式從母體腹部體表取得心電 圖，再透過幾種強健方法的處理，得出胎兒心電圖。最後透過類週期特性將胎兒 心跳辨識出來，並使用心率變異參數量化軌跡，以獲取剖腹產對胎兒心跳與神經 系統的作用。這項研究結果顯示，麻醉前，麻醉後，和分娩前 5 分鐘心率變異都 明顯上升，該方法能夠可靠地評估胎兒對手術的反應，未來可以作為一個臨床工 具，用於監控剖腹分娩過程中胎兒的狀態。
另一個應用是利用非線性波形相似度分析方法用於心房電圖，以找出重要的複 雜碎裂心房電圖區域供心房電燒手術之用。該方法首先利用軌道的特徵找出每段 週期，然後計算相空間這些軌跡的統計特性(相似性指數)。研究結果顯示，相似 性指數在電燒成功病患的複雜碎裂心房電圖區域上較高，此類病患的預後也較好，
這暗示了複雜碎裂心房電圖區域中相似性指數高的部分跟心房振顫的產生與維持 有關聯。
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Abstract
In recent years, the complexity of human body has been continuous revealed and discussed in many fields, it may eventually lead to a complete theory through the studies on pathogenesis and molecular biology of disease. On the other hand, the complex theory combined with the homeostasis mechanism has been used for biomedical signal analysis trying to identify such complex phenomena and underlying information behind the clinic data. These methods can help to extract non-linear feature from ambiguous information as the disease assessments, some of them have been accepted to have more advantages than traditional ones. However, such refining procedure are subject to many restrictions in clinical conditions and environments, such as limited data length, information may be occasional uneven or noise interfered. In addition, the extraction itself can also lead to distortions, the interference from other mechanism may not be effectively removed which raised the difficulty on the subsequent analysis. Therefore, this thesis proposes several robust methods to identify the specific nonlinear features in clinic data series and try to fulfill the clinical requirements.
The first portion of nonlinear feature is quantization of multi-scale correlation. It was derived from the entropy in information theory as well as the coarse-graining in chaos-fractal theory to quantify the complexity of a system through the correlations at different time scale. In the first study, a novel approach has been proposed to decrease the length of data in complexity calculation of pulse wave velocity (PWV) such that the time for data acquisition can be substantially reduced to 12 minutes. It utilized a smaller sample size (i.e. 600 consecutive signals) with remarkable preservation of sensitivity in differentiating among the healthy, aged, and diabetic populations compared with the conventional method (i.e. 1000 consecutive signals).
The second study utilized the multi-scale correlation of heart beat intervals (RRI) on critical patients whose life continuation relies on extracorporeal membrane oxygenator (ECMO). This study propose a new approach to estimate the complexity in a signal by analyzing the irregularity of the sign time series of coarse-grained time series at different time scales. Without removing any outliers due to ectopic beats, the method is able to detect a degradation of cardiac control in patients with congestive heart failure and a more degradation. Moreover, the derived complexity measures can predict the
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mortality of ECMO patients. These results indicate that the proposed method may serve as a promising tool for monitoring cardiac function of patients in clinical settings.
In the second portion of nonlinear feature, the trajectories on phase space have been used for calculating statistical properties of the orbits in a dynamic system. In the first study, a novel method been proposed to noninvasively derive the fetus ECG signals from the maternal abdominal ECG during the cesarean section (CS). The heart beat series derived from the noisy signal were then quantified by several heart rate variability (HRV) methods. Moat parameters tell that the HRV increased 5 minutes after anesthesia and 5 minutes before delivery. These results shows that the proposed method may serve as a promising tool to obtain significant information about the fetal condition during labor.
In the second study, a nonlinear-based waveform similarity analysis of the local electrograms has been proposed, aiming to detect crucial complex fractionated atrial electrograms (CFEs) in atrial fibrillation (AF) ablation. This method firstly identify each cycle of orbits in the dynamic system and then calculate the statistical properties (similarity index, SI) of these trajectories on phase space. The result shows the average SI of the targeted CFEs was higher in termination patients, and they had a better outcome. This study suggested that sites with a high level of fibrillation electrogram similarity at the CFE sites were important for AF maintenance.
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Chapter 1 Introduction
1.1 Homeostasis and Correlations
Living creatures can accept environmental stimulus, generate the appropriate responses, and automatically keep the cells, tissues and organs maintained in an optimal state that allow them to adjust its performance to the varying internal and external demands. In the short term, the mechanisms in human body involved the control of cardiac and respiratory rate, blood glucose concentration and body temperature. In the long term, the control of circadian rhythms, regulation of inflammatory processes and control of the immune system were also involved. As a result, our body contains different control loops to stay alive and this phenomenon is called "homeostasis" which is vital to life and important for disease assessment [1, 2].
All homeostatic control mechanisms have three main components for regulation: a receptor that monitors and responds to environmental changes. A control center that determines an appropriate response to the stimulus. A effector that can receive signals from the control center [3]. Through these components and pathways, a change will occur on the effector to correct the deviation by depressing it with negative feedback [2, 4]. When the stimulus occurred and followed by a change on the effector, a time-lagged causal relation between them has been built. The states of the control system may be represented by a time series so that data in the series will fluctuates between the stimulus and response. The time-lagged correlations might arise in such time series.
Conceptually, the time-lagged correlations existing in a physiological data series implies regulation in the homeostatic system, the time scale can range from several milliseconds (e.g. for neurons) to several days (e.g. for immune system response to vaccination) [4- 6].
Based on causality and feedback, the early studies on homeostasis model is almost equivalent to that on the regulatory system, i.e., the control theory. Although the application of traditional control theory helps people to predict the behavior of simple systems in the body, the handling of a complex system such as immunologic network is difficult and sometimes does not lead to satisfactory solutions [6]. As a result, based on the time-lagged correlate properties, some studies try to apply the autoregressive (AR) to model the homeostatic process and estimate the system response. The AR modeling
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assume that any stationary process can be expressed as a sum of two components: a stochastic component that a linear combination of lags of a white noise process and a deterministic component which is uncorrelated with the latter stochastic component.
The noise source is reminiscent of the external stimulates while the deterministic component is reminiscent of the set-points. By applying this model, one can mathematically estimations the response of a system through the multiple realizations [7].
However, stationary model still relies on the assumptions of simple stationary process which may not be able to provide a comprehensive view for disorders or diseases.
Under normal healthy conditions, the physiological fluctuations are usually neither random nor too regular that system is under control and behaves stationary. On the contrary, when multiple factors interact to produce the imbalance, particularly under serious disease, the physiological fluctuations may become unpredictable [8]. The system is out of control and becomes non-stationary, there is no numerical solution in such condition. As a result, it would be preferable to identify system states from correlations rather than solve the underlying equations in clinical applications.
1.2 Heart Rate Variability and Autonomic Nervous System
As mentioned earlier, homeostatic regulations in human body involved the control of cardiac rate [3], therefore, quantifying the physiological fluctuations through the easily accessible heart beat series has becomes popular in recent years [9-11]. The fluctuations on cardiac rate is measured through the variation in the beat-to-beat intervals and represented by the heart rate variability (HRV). The heart beats are originally trigger by the sinoatrial node (SA node) such that most variations are the results from different inputs of SA node. The main inputs are the autonomic nervous system and humoral factors. Other inputs includes the respiratory arrhythmia and the low-frequency oscillations associated with Mayer waves of blood pressure [12].
The electrical impulse from SA node may be delayed or blocked on an unhealthy heart tissues that leads to irregular heart beat and causes errors in the calculation of the HRV [13]. Therefore, traditional HRV analysis only calculate the normal sinus rhythms, i.e. N-N intervals, such as SDNN (standard deviation of NN intervals), pNN50 (proportion of successive NNs that differ by more than 50ms), power spectral bands of
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the interpolated NN intervals. The high frequency (HF) band ranges from 0.15 to 0.4 Hz, low frequency (LF) ranges from 0.04 to 0.15 Hz, and the very low frequency (VLF) ranges from 0.0033 to 0.04 Hz [14].
Although cardiac automaticity is intrinsic to SA node, heart rate and rhythm are largely under the control of autonomic nervous system (ANS). The ANS is responsible for maintaining homeostasis and regulates the function of all innervated tissues and organs throughout the vertebrate body [15]. Since homeostatic regulation are important and required for survival, the actions of the ANS usually occur independent of our consciousness as the name suggests. The ANS has two divisions that work to counteract each other and keep the body in balance, the sympathetic nervous system (SNS) and the parasympathetic nervous system (PSNS). Decreased PSNS activity or increased SNS activity will result in reduced HRV. And the high frequency activity has been linked to PSNS activity [16].
Typically, reduction of HRV is associated with ill state, and such symptoms has been reported in several cardiovascular and non-cardiovascular diseases, such as myocardial infarction [9], heart failure [10], diabetes and hypertension [17,18]. On the contrary, several heart related disease may cause heart-rate turbulence which increase HRV [19].
As a result, a proper complexity in heart beat fluctuations has been accepted as a hallmark of healthy in physiology and is believed to reflect system adaptability in response to constant changes in internal and external inputs [20]. However, such statement needs more evidences and theory to explain the underlying phenomenon, otherwise, identifying the healthy and diseased state through the physiological fluctuations is more like a "blind men and elephant" approach.
1.3 Dynamical System and Orbits
From an engineer's perspective, the concept of the homeostasis can be represented by a complex feedback system and mathematically modeled as a dynamical system [21]. It can model the nonlinear feedback loops but not limited to feedback loops, actually, it has been widely used in many fields including biology and physiology. Often, the dynamic system model is in the form of a set of differential equations depicting the mechanistic interactions between components of the system are constructed:
) , (u P f u
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Here u
is an N dimensional vector describing the state of the system at any given time.
f
is a vector field describing the dynamics of the system. Parameter vectorP
and the vector field function f
together determine evolution rule of the dynamical system that describes what future states follow from the current state. The output of the model is the orbit (state function or trajectory), the path that state follows through space as a function of time. The orbits under the same model (the same vector field f
) will be different if started from different position. Among these possible orbits (realizations), different initial condition may leads to different outcome, therefore, the features on the vector field f
are more interest. Often, in a given system all orbits may tend to a point or a closed curve which constitutes an attractor for the system. By examining the state space in the neighborhood of a given attractor we can determine the basin of attraction for that attractor. For example, to simulation a ventricular cell one can use numerical integration of the Hodgkin-Huxley-type ionic model using a forward Euler scheme, with V at time
t
t calculated as:
) ( ) ( / ) ( )
(t t V t t C I t
V m i ,
where C is membrane capacitance and m I represents the individual ionic currents i [22]. Lower figures are the simulation results in time domain (Figure 1, left) and in the phase plane (Figure 1, right).
Figure 1 The voltage data and correspond phase-plane trajectory in the model The orbits behave as a typical limit cycle nonlinear oscillator along with a closed curve attractor, the homeostasis on such model is dynamic equilibrium. If the parametersP
in the model is time independent, i.e.P t P
)
( , the system is referred to as non-autonomous, otherwise, the system is autonomous. The time vary parameters P(t) thus can be used to model the environmental stimulus. As the autonomous system
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disturbed by external stimulus, the orbit may shift from the closed curve or equilibrium point, and it may go back to the initial state after a period of time [23].
Recall the "set-point" concept of the control theory, as the effector correct the deviation by depressing it, the state variable will return to the baseline. In the phase plane, it acts like the orbit disturbed by a stimulus and finally converges to the equilibrium point (attractor or set-point). As a result, the concepts in homeostasis thus are related to the probability distribution of the orbits in the dynamical system model.
In short, if paths of orbits are similar to each others, there will exist a attractor around them, and the nonlinear correlation between them are high. In other words, the similar but different path of orbits in the phase space implies the existing of governing rules of homeostatic control that "attract' the orbits". It also explained why a proper complexity in physiological fluctuations are neither random nor too regular under normal healthy conditions [20].
1.4 Reconstruct the Dynamics
Most control mechanisms are involved with the ANS which carries signals from the central nervous system to all organs of the body, in addition, the ANS regulates the function of most tissues and organs in the body [15]. As a result, it needs huge number of state variables involved in the regulations to describe the system completely. Without enough variables, it's really hard to know the whole picture describes the interactions between each components. Therefore, this raise a problems of knowing the properties of a dynamic system with limit information.
Figure 2 The Lorenz attractor and correspond differential equations
From Takens’ Embedding theorem, “if we measure any single variable with
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sufficient accuracy for a long period of time, it is possible to reconstruct the underlying dynamic structure of the entire system from the behavior of that single variable using delay coordinates and the embedding procedure” [25]. Figure 2 shows a well-known Lorenz attractor with the trajectory as a function of time [x(t), y(t), z(t)] derived from the differential equations. By applying the embedding method on the x(t), the time delayed series [x(t),x(t−τ),x(t−2τ)] are plotted as figure 3 and the topological structure of the Lorenz attractor is preserved by the reconstruction.
Figure 3 The time domain signal and the trajectory in reconstructed phase space In 1983, Procaccia and Hentschel [26] described a numerical procedure to introduce a characteristic known now as the correlation dimension. As mention above, if paths of orbits are similar to each others, the nonlinear correlation between them are high. After that, sample-entropy and approximate-entropy have been proposed [27]. Because the correlations in the phase space can distinguish colored noise from deterministic chaotic behavior while the autocorrelation functions cannot [24], and the algorithm for correlation calculation is relatively simple and fast, such method has become one of the most popular characteristics of time series analysis.
However, the time scale of the physiological applications can range from several milliseconds to several days as mentioned earlier. The time lag L during the embedding procedure need to be defined first which is a trade-off between the small delay (L = 1) and the large delay (L = 100), as figure.4 shows. The Embedding theorem or correlation
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dimension on certain time delay L fail to account for the multiple time scales inherent in time series (such as homeostatic system).
Figure 4 (a) the original phase plot and reconstructed phase space with different embedding delay L: (b) L = 1, (c) L = 10 and (d) L = 100 respectively. The delay parameter L determines the dimension of the reconstructed space and also the scales of fluctuations that can be seen on the reconstructed space.
1.5 Multiple Time Scale Dynamics
To solve this, the famous approach is the Multiscale Entropy Analysis (MSE) proposed by C.K. Peng in 2002 [11]. Through this method, complexity of a system could be derived from nonlinear correlations of variables at multiple time scales. The term "Entropy" has been defined as inversely related to energy (in the form of heat:
D k
entropy log ) in the "Second Law" of classical thermodynamics, and commonly understood as a measure of disorder. Latter, Shannon and Weaver proposed the famous H measure: H
pilnpi, commonly understood as a measurement of uncertainty.However, neither the second law nor Shannon Entropy make distinction between living and non-living things.
In the book "What is Life?", Erwin Schrödinger proposed a controversial concept that life decreases or maintains its entropy by feeding on negative entropy [28]. The concept solve the conflict that life's dynamics may be argued to go against the tendency of second law due to the closed system. In other words, life tend to be highly ordered rather than unpredictable or random, the interactions between them should be correlated and meaningful which implies the entropy of life is low. But to maintain such order, life needs multiple scale structures and complex feedback paths [29]. Life thus become order and complex which may sound like an oxymoron, but this is due to mistaking the
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meaning of complexity and randomness [21]. Intuitively, complexity is associated with
“meaningful structural richness”, which, in contrast to the outputs of random phenomena, exhibits relatively higher regularity [20].
It thus raises a practical problem to distinguish complex structure from order and disordered systems. For this, MSE utilizes the entropy as a measuring tool for quantify the correlations in different time scales rather to directly use the entropy to "measure"
the complexity. This turns the calculation into structural quantification which qualify the complexity in the way that more correlations in different time scale corresponds to more complex structures and more control mechanisms. As a result, the MSE method shows that correlated random signals (colored noise) are more complex than uncorrelated random signals (white noise) since the former has the correlations while the latter has not. The MSE use the averaging method to form new data series in coarse-graining steps (as figure 5 shows) and finally calculates the sample-entropy of each new series [20].
Figure 5 Coarse-graining steps for multiple scale entropy calculation
For MSE calculations, if time-lagged correlation exists in multiple time scales, it reflects that the system has complexity in structures with more adaptability in response to constant changes in internal and external inputs. Such concepts has been accepted as a hallmark of healthy physiological control and have been applied in many applications.
However, most signal analysis in clinical applications are difficult to deal with, which always need modified methods to solve them, Chapter 2 will give two examples.
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1.6 Attracting Orbit and Discrete Dynamical System
Although the complexity of a system can be reconstructed and ranked from a time series through MSE method, the underlying dynamics are still hard to figure out without the topological information. Furthermore, some clinical application such as surgery guiding needs spatial-temporal information for assessments, which is out of the scope of statistics analysis such as entropy based calculations.
Back to the nonlinear oscillator and attracting orbits in phase space mentioned above.
There are several known periodic biological fluctuations that also fulfill the assumptions of attracting orbits in phase space, such as heart cycle, respiration cycle and circadian rhythm [30-32]. There are strict definitions of attracting orbits or periodic orbit in mathematics, but here we take the idea of the Poincaré map of a periodic orbit [33].
The stability of a periodic orbit in a dynamic system are usually calculated by the Poincaré map which replaces the n-dimensional continuous vector field with an (n−1) dimensional map, as figure 6 shows. Such map can be interpreted as a discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many topological properties of periodic orbits of the original system and has a lower-dimensional state space it is often used for analyzing the original system [34].
Figure 6 The discrete dynamical system derived from the intersection of the periodic orbits in a dynamic system.
Take the heart cycle as an example: the discrete dynamical system was firstly start on the R-wave peaks derived from the ECG signal. The cycle length dynamics then can be reconstructed through the n and (n+1) intervals in the Poincaré plot, as figure 7(b)
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shows.
Second, the amplitude dynamics can be reconstructed directly through the time delayed embedded as shows in figure 7(c). The R-peaks in the reconstructed phase space thus become the discrete dynamical system that represented the amplitude variations of R-wave due to the respiratory effect (EDR), as shows in figure 7(d).
Since the cycle length of the heart beats can be modulated by the respiratory, there are similarities in the distribution of these two different approaches (as figure 7(b) and (d) shows). The distribution on the Poincaré plot can be characterized by fitting an ellipse to it [35]. The length of axis 1 is defined as the SD of the plot data in that direction which describes the instantaneous beat-to-beat variability of the data, SD1. The length of axis 2 is defined as the SD of the plot data in the perpendicular direction, SD2, as figure 7(b) shows.
Figure 7 (a)ECG wave and correspond R-peak markers. (b) Poincaré plot of RR intervals. (c) The ECG signal and correspond R-peak markers on the reconstructed {x(t), x(t+L)} space, the area of the markers is partially enlarged in (d).
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This shows the attracting orbits can be reconstructed through different approaches, and each of them corresponds to different physiological meanings (cycle length or the amplitude of the peaks in ECG wave). Hence, the way to identify the feature points and the approach for reconstructed the discrete dynamical system would be the key.
Chapter 3 will give two examples to explain how to identify the feature points from noisy data and quantify the attracting orbit from these points while the spatial-temporal information are still preserved.
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Chapter 2
Quantization of Multi-scale correlation
2.1 Application of a Refined Entropy Computational Method in Assessing Complexity of Pulse Wave Velocity Signals in Healthy and Diabetic Subjects
2.1.1 INTRODUCTION
Atherosclerosis, which is the major pathological change underlying most cardiovascular diseases, has been reported to be associated with advanced age, history of stroke, diabetes, hypertension, and cerebrovascular disease. Pulse wave velocity(PWV) is one of the most popular non-invasive parameter for the assessment of atherosclerosis. Despite different equipment used for data acquisition, a mean value is usually obtained from the examinee for evaluating the severity of the condition(Blacher, Asmar et al. 1999; Laurent, Boutouyrie et al. 2001; Yamashina, Tomiyama et al. 2002;
Mitchell, Parise et al. 2004; Tsai, Chen et al. 2005; Wu, Hsu et al. 2012). On the other hand, Costa et al. found healthy subjects and those with heart conditions can be reliably differentiated by a simple measure based on the thermodynamical concept of “entropy”
(Costa, Goldberger et al. 2002). “Multi-scale entropy (MSE)” is a non-linear means of assessing the complexity of physiological signals (Costa, Goldberger et al. 2002; Costa and Healey 2003; Costa, Peng et al. 2003).Compared to the traditional complexity measures, MSE has the advantage of being applicable to both physiologic and physiologic signals of finite length. MSE, which was first reported by Costa et al. to compare the differences in R-R interval (RRI) among healthy subjects, patients with atrial fibrillation and those with congestive heart failure (CHF) (Costa, Goldberger et al.
2002), has been successfully applied to the interpretation of physiological series and data from patients with various diseases. In previous studies, MSE provided the best prognostic prediction in patients with congestive heart failure (CHF) (Ho YL, Lin C, Lin YH, Lo MT. The prognostic value of non-linear analysis of heart rate variability in patients with congestive heart failure--a pilot study of multiscale entropy. PloS one
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2011;6:e18699.) and patients receiving unilateral primary carotid angioplasty and stenting were reported to exhibit acute increase of complexity in the neurocardiovascular dynamics (H. K. Yuan, C. Lin, P. H. Tsai, F. C. Chang, K. P. Lin, H.
H. Hu, M. C. Su, and M. T. Lo, "Acute increase of complexity in the neurocardiovascular dynamics following carotid stenting," Acta Neurologica Scandinavica, vol. 123, pp. 187-192, 2011.). In 2006, Escudero et al. reported significant difference in entropy values from signals of electroencephalograms (EEG) between healthy individuals and those with Alzheimer’s disease after data processing with MSE(Escudero, Abásolo et al. 2006). Accordingly, we have previously shown that healthy, aged, and diabetic subjects can be distinguished with MSE using 1000 successive PWV signals with a scale factor of 10(Wu, Hsu et al. 2011). Despite being reliable, the whole recording process takes up to 30 minutes that is usually not well tolerated by aged or diseased subjects (Wu, Hsu et al. 2011).
To refine the assessment approach, the present study proposes a novel means of computation, “short time multiscale entropy (sMSE)”, in an attempt to reduce the time for data acquisition through refined computation of the harvested data. To compare between MSE and sMSE in terms of their sensitivity and validity in differentiating signals of small sample size and among healthy, aged, and diabetic subjects, both simulation signals and PWV data from testing subjects were used for the current study.
2.1.2 METHODS
Subject Population and Grouping
The testing subjects were divided into four groups, including healthy young individuals of age between 20 and 40 (Group 1, n =24), healthy aged subjects of age between 20 to 40 (Group 2, n =30), middle-aged patients with well-controlled diabetes mellitus type 2 [Defined as age between 41 to 80 and 6.5% < glycosylated hemoglobin (HbA1c) level < 8.0%] (Group 3, n =18), and middle-aged patients with poorly-controlled diabetes mellitus type 2 (Defined as age between 41 to 80 with HbA1c level ≥ 8.0%) (Group 4, n = 22). All participants were volunteers. Diabetic patients, who were recruited from the diabetic outpatient clinic of Hualien Hospital from July 2009 to October 2010, fit all the three criteria of (1) Fasting blood sugar
>126mg/dL, (2) HbA1c level > 6.5%, and (3) Established diagnosis of diabetes mellitus
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type 2 with a follow-up period > 2 years. On the other hand, healthy subjects, who were recruited from the health screening clinic of Hualien hospital during the same period, had to fill out a questionnaire declaring the absence of medical history of cardiovascular diseases (i.e. Stroke, hypertension, diabetes). The whole study has been approved by the Institutional Review Board (IRB) of Hualien Hospital and National Dong Hwa University. Informed consents were signed by all testing subjects.
Short Time Multiscale Entropy (sMSE)
The original MSE comprises of two steps: 1) coarse-graining the signals using different time scales; 2) quantifying the degree of irregularity in each coarse-grained time series using sample entropy (SampEn). However, the major challenge of of MSE in clinical application is the need of massive data for the reliability.
Short time multiscale entropy (sMSE) is a novel approach of computation that enables the use of large scale factor for analysis on data acquired through a shortened time period. The basic concept is the creation of different time series through removing a small number of recordings from the beginning without affecting the overall trend and complexity of the acquired signals. The acquired time series then undergo Sample Entropy (SE)(Richman and Moorman 2000) computation with steady values of entropy obtained (Figure.1).
Through altering the number of Lag from 0 to L (where L = τ– 1, τ = coarse-grained scale factor) on the native time series (1), a new time series,T(P), can be obtained (2).
Thus, the number of new time series generated is L+1.
TN={X1，X2，. . .，XN-1，XN} (1)
T(P)={Xk，Xk+1，Xk+2，. . .，XN-1，XN}，k=p+1，p=0,1,2,. . .,L (2) The L+1 time series acquired then undergo coarse-grained processing with a scale factor τ (3), giving the time series of y(p)(τ). Hence
yj(p)(τ)=1τ∑jτ+pk=(j−1)τ+1+pXk，1 ≤ j ≤⌊N−Pτ ⌋，p=0,1,2,…..,L (3) The L+1y(p)(τ) are then subjected to Sample Entropycomputation and averaged, giving sMSEτ of scale factor τ(4)
sMSEτ=L+11 ∑Lp=0SE(y(p)(τ)) (4)
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Figure 1. Method of short time multi-scale entropy (sMSE) computation Short Time Multi-scale Entropy Index (sMEI) Using PWV Series
The results of MSE from 1000 successive PWV signals were compared with those of sMSE acquired from computation on the first 600 PWV signals using the novel computation approach in the current study. Utilizing a scale factor of 10, the present study categorized scale factors into short time multi-scale entropy index with small scale (sMEISS, scale1 to scale5) (5) and short time multi-scale entropy index with large scale (sMEILS, scale 6 to scale 10) (6)that were used to compare with the respective values of MEISS and MEILS from our previous study using MSE (Wu, Hsu et al. 2011).
sMEISS=10(∑5τ=1sMSEτ) (5) sMEILS=10(∑10τ=6sMSEτ) (6)
Study Design
The study comprised two parts. The first part involved the design of the sMSE method with simulation signals of white noise and 1/f noise using the MATLAB R2008b package (MathWorks, Natick, Massachusetts, U.S.A.). The second part focused
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on computation of PWV-based multiscale entropy index in study subjects with small scale and that with large scale using MSE method on 1000 successive PWV signals that are referred to as MEISS (PWV1000) and MEILS (PWV1000), respectively. The computation has been previously described (Wu, Hsu et al. 2011). Utilizing the same approach, 600 successive PWV signals were obtained for the calculation of MEISS
(PWV600) and MEILS (PWV600). Comparisons were first made between MEISS (PWV1000) and MEISS (PWV600) as well as between MEILS (PWV1000) and MEILS (PWV600) to study if a reduction in available data would affect the ability of differentiation among different groups. In addition, MEISS (PWV600) and MEILS (PWV600) were compared with sMEISS (PWV600) and sMEILS (PWV600), respectively, to investigate possible enhancement in sensitivities using the novel method for data processing.
Statistical Analysis
Average values are expressed as mean±SD. Statistical Package for the Social Science (SPSS, version 14.0 for Windows, SPSS Inc., Chicago, IL) was used for statistical analysis. Independent t-test was adopted for the determination of the significance of difference in study parameters among different groups. A probability value, p, of<0.05 represents statistical significance.
2.1.3 RESULTS
Computation of Sample Entropy Using Multi-Scale Entropy (MSE) and Short Time Multi-Scale Entropy (sMSE) methods on Simulation Signals
Values of sample entropy were acquired through multi-scale entropy (MSE) (Figure 2a) and short time multi-scale entropy (sMSE) (Figure 2b) methods using simulation white noise and 1/f noise with different scale factors on 30 sets of 1000 successive signals. The results showed that the values of sample entropy decreased with an increase in values of the coarse grained scale factor regardless of the method used. On the other hand, computation with 1/f noise eliminated the effect of scale factor, giving a value of around 2 for both methods (Figure 2a & 2b). Comparison of changes in values of sample entropy using multi-scale entropy (MSE) and short time multi-scale entropy (sMSE) approaches with different scale factors on 600 successive white noise signals (Figure 3) showed a steady drop in sample entropy as the scale factor increased from 1 to 4. From the scale factor 5 onwards, sample entropy from MSE began to exhibit
26
remarkable fluctuations, while that from sMSE showed a relatively steady decrease.
Figure 2. Simulation signals. (a)Values of sample entropy acquired through multi-scale
entropy (MSE) computation using white noise and 1/noise with different scale factors on 30 sets of 1000 successive signals. (b)Values of sample entropy acquired through short-time multi-scale entropy (sMSE) computation using white noise and 1/noise with different scale factors on 30 sets of 1000 successive signalsFigure 3. Comparison of changes in values of sample entropy using multi-scale entropy
(MSE) and short time multi-scale entropy (sMSE) methods with different scale factors on 600 successive white noise signals27
Table 1
Comparison of anthropometric, hemodynamic, serum biochemical, arterial stiffness, and multiple entropy parameters among the testing subjects
Parameter600 points Group 1 Group 2 Group 3 Group 4
Number 24 30 18 22
Ages (years) 25.8±5.6 52.6±6.6** 56.5±9.3 57.9±9.5 Duration of
Diabetes(years)
0 0 6.8±3.8 11.7±6.8++
circumference(cm) 79.9±10.8 84.3±10.1 92.2±10.1ε 96.7±12.8
BMI (kg/m2) 22.6±3.5 24.2±3.9 26.9±3.7 28.4±5.2
SBP (mmHg) 115.5±9.8 115.5±14.4 129.8±22.0ε 125.7±19.4
DBP(mmHg) 70.1±6.6 73.9±10.0 78.5±13.6 75.5±10.8
PP(mmHg) 44.5±6.6 41.1±9.4 51.2±12.3ε 45.1±6.7
HbA1c(%) 5.5±0.2 5.8±0.4* 6.8±0.7εε 9.53±1.9++
HDL(mg/dL) 41.7±11.5 49.4±14.1 39.9±11.4 43.2±14.9
Triglyceride (mg/dL)
100.6±74.0 106.0±54.9 107.0±51.7 156.9±74.3+ Fasting Blood
Sugar(mg/dL)
92.4±8.4 96.1±9.9 128.5±28.1εε 182.8±61.9+
PWV1000(m/s) 4.4±0.3 4.7±0.4* 5.0±0.3ε 5.1±0.6
MEISS (PWV1000) 96.5±4.4 97.4±4.3 98.4±6.7 91.5±12.5+ MEILS (PWV1000) 89.4±7.3 84.3±6.3* 79.6±9.2ε 71.9±12.6+ MEISS (PWV600) 97.0±7.6 99.1±4.3 100.9±8.3 93.3±12.4+ MEILS (PWV600) 88.3±10.8 86.1±12.8 85.2±11.0 82.9±11.6 sMEISS(PWV600) 95.9±10.0 96.8±7.1 96.9±11.3 89.2±12.1+ sMEILS (PWV600) 92.2±8.9 86.8±11.3* 80.5±6.2ε 73.7±11.4+ Group 1: Healthy young subjects without known cardiovascular disease; Group 2: Healthy middle-aged subjects without known cardiovascular disease; Group 3: Middle-aged individuals with well-controlled diabetes mellitus type 2; Group 4: Middle-aged patients with poorly-controlled diabetes mellitus type 2. Values are expressed as mean+SD. BMI=body mass index; SBP=systolic blood pressure; DBP=diastolic blood pressure; PP=pulse pressure;
HbA1c=glycosylated hemoglobin; HDL=high-density lipoprotein; PWV1000=1000 successive pulse wave velocity using the distance from the sternal to the second toe divided by the time difference between R wave on LeadⅡof ECG to the corresponding foot point of pulse wave of second toe; MEISS (PWV1000)= 1000 successive PWV-based multiscale entropy index with small scale; MEILS (PWV1000) = 1000 successive PWV-based multiscale entropy index with large scale; MEISS (PWV600)= 600 successive PWV-based multiscale entropy index with small
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scale; MEILS (PWV600)= 600 successive PWV-based multiscale entropy index with large scale;
sMEISS (PWV600)= 600 successive PWV-based short time multiscale entropy index with small scale; sMEILS (PWV600)= 600 successive PWV-based short time multiscale entropy index with large scale.
*p<0.05 Group 1 ＆ Group 2, εp<0.05:Group 2 ＆ Group 3, +p<0.05:Group 3 ＆ Group 4,
**p<0.001 Group 1＆Group 2, εεp<0.001:Group 2＆Group 3, ++p<0.001:Group 3＆Group 4
Demographic and Biochemical Parameters
Subjects in Group 3 was significantly older than those in Group 2 who, in turn, were significantly older than those in Group 1 (all p<0.001) (Table 1).The duration of diagnosed diabetes was significantly longer in Group 4 than that in Group 3(p<0.001).
Although there was no significant difference in body mass index (BMI) among the four groups, the waist circumference was significantly larger with systolic blood pressure higher in individuals in Group 3 compared to those in Group 2(both p=0.005). Besides, the pulse pressure was also substantially higher in Group 3 than that in Group 2 (p=0.001). Moreover, the levels of HbA1c were significantly different among the four groups with Group 4 being the highest, followed by Group 3, Group 2, and Group 1 (Group 1 vs. Group 2, p=0.007; Group 2 vs. Group 3 & Group 3 vs. Group 4, p<0.001), although the parameter was within normal range (i.e. <6.0%) in Group 1 and Group 2.
Serum triglyceride was also significantly higher in Group 4 than in Group 3 (p=0.037).
Furthermore, fasting blood sugar level was highest in Group 4, followed by Group 3 and Group 2, while there was no notable difference between Group 1 and Group 2 (Group 2 vs. Group 3, p<0.001; Group 3 vs. Group 4, p=0.003).
Comparisons among PWV1000, MEISS (PWV1000), MEILS (PWV1000), MEISS (PWV600), MEILS (PWV600), sMEISS (PWV600) and sMEILS (PWV600)
PWV1000 was lowest in Group 1, followed by that of Group 2 and Group 3, while there was no remarkable difference in this parameter between Group 3 and Group 4 (Group 1 vs. Group 2, p=0.007; Group 2 vs. Group 3, p=0.009). MEISS (PWV1000) wassignificantly higher in Group 3 than that in Group 4 (p=0.02). On the other hand, MEILS
(PWV1000) was higher in Group 1 than that in Group 2 (p=0.03), significantly higher in Group 2 than that in Group 3 (p=0.016), and higher in Group 3 compared to that in Group 4 (p=0.04). Although MEISS (PWV600) was significantly higher in Group 3 than that in Group 4 (p=0.005), there was no significant difference in MEISS (PWV600) between Group 1 and Group 2. Failure in differentiation was noted between Group 1
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and Group 2 using MEISS (PWV600) as well as among Groups 2, 3, and 4 using MEILS
(PWV600). Similar to that of MEISS (PWV600), although sMEISS (PWV600) was
significantly higher in Group 3 than that in Group 4 (p=0.011), it failed to differentiate among Groups 1, 2, and 3. On the other hand, sMEILS (PWV600) was significantly higher in Group 1 than that in Group 2 (p=0.029), higher in Group 2 than that in Group 3 (p=0.045), and higher in Group 3 than that in Group 4 (p=0.045).
Figure 4. Values of sample entropy obtained through computation using short time
multi-scale entropy (sMSE) method on 600 successive pulse wave velocity (PWV) signals. Group 1: Healthy young subjects without known cardiovascular disease; Group 2: Healthy middle-aged subjects without known cardiovascular disease; Group 3:Middle-aged individuals with well-controlled diabetes mellitus type 2; Group 4:
Middle-aged patients with poorly-controlled diabetes mellitus type 2
There was an overall reduction in sample entropy with an increase in scale factors (Figure 4). While no significant difference among the four groups was noted on a scale factor less than 6, significant differences began to emerge when the scale factor was 6 or above. The value of sample entropy was highest in Group 1, followed by that of Group 2, Group 3, and Group 4.
2.1.4 DISCUSSION
PWV is one of the most popular noninvasive assessment tools for the assessment of atherosclerosis (Laurent, Boutouyrie et al. 2001; Tsai, Chen et al. 2005; Cecelja and Chowienczyk 2009) that operates on the assumption that PWV is a stationary parameter.
However, after analyzing the data on PWV over 1000 cardiac cycles within 30 minutes, our previous study (Wu, Hsu et al. 2011) demonstrated that PWV is a non-stationary
30
parameter, the variability of which may reflect subtle atherosclerotic change that was missed by taking only the mean value for analysis. That study explored the possibility of combining MSE and PWV in assessing sugar control and progression of vascular pathology in diabetic patients and elderly to allow timely therapeutic intervention. Other than MSE, various tools for non-linear data analysis are available, including Ensemble empirical Mode Decomposition (EEMD) (Wu and Huang 2009; Chang and Liu 2011), linguistic (Yang, Hseu et al. 2003; Lei, Li et al. 2007) and fractal (Goldberger, Amaral et al. 2002; Tapanainen, Thomsen et al. 2002) analyses.
Albeit sensitive in differentiating healthy, aged, and diabetic subjects, one of the pitfalls of applying MSE for PWV signal analysis is the relatively long time for data collection that involves the acquisition of 1000 successive signals in 30 minutes (Wu, Hsu et al. 2011). Our experience showed that, although a scale factor of 10 can be used for analyzing 1000 successive PWV signals to produce significant outcomes, the use of scale factor 10 on a smaller sample size acquired within a shorter time period would give aberrant results (Figure 3). In an attempt to solve the problem, the current study introduced a novel non-linear computational method, sMSE, that gave values of sample entropy comparable to those obtained through MSE from a relatively long period of simulation signals (Figure 2a & 2b). The results, therefore, are consistent with those from the study of Peng et al. that also demonstrated similar results in simulation study on healthy subjects and those with cardiac diseases (Costa, Goldberger et al. 2005).
Using a relatively small simulation sample size of 600, the changes in sample entropy acquired with MSE and sMSE were compared (Figure 3). The results showed spiking increases in entropy at a scale factor of 6, 9, and 10 using the MSE method, while sample entropy from sMSE exhibited a relatively steady reduction throughout the elevation in scale factor from 1 to 10. Compared to traditional MSE, the significantly reduced standard deviation of sMSE indicates the reduction of the cost of the experimentation. Therefore, despite a smaller sample size, sMSE could still produce results similar to that of MSE on a large sample (Figure 2a). The results highlight the applicability of sMSE in the analysis of signals acquired through a long time period and also those from a relatively short period (i.e. 600 consecutive signals) using a scale factor of 10 to produce steady results that could not be obtained through the original MSE approach. The results from simulation studies are consistent with those from human subjects. Although MEISS (PWV600) and MEILS (PWV600) failed to reproduce the
31
significant results from MEISS (PWV1000) and MEILS (PWV1000) after curtailing the sampling size, sMEISS (PWV600) and sMEILS (PWV600) were found to be as sensitive as MEISS (PWV1000) and MEILS (PWV1000) in differentiating among the four groups.
Failure in differentiation among the four groups using MEILS (PWV600)(Table 1) may be due to the marked fluctuations in sample entropy at large scale factors (Figure 3).
Furthermore, consistent with the findings of previous studies (Wu, Hsu et al. 2011), the results of the present study also demonstrated a reduction in signal complexity with age and the severity of diabetes (Figure 4).
The present study has its limitations. First, compared with MSE, the method of sMSE requires a larger volume of computation. Second, although we have established a signal-to-scale factor ratio of 100 (i.e. 1000 successive signals/ scale factor 10) as a minimal requirement for successful computation using the MSE approach and a reduced ratio of 60 for sMSE in this study, whether aberrancy would arise from sMSE using a ratio below 60 remains to be elucidated.
In conclusion, the present study demonstrated that, using a novel sMSE approach for PWV signal analysis, the time for data acquisition can be substantially reduced from 30 minutes to 10 minutes with remarkable preservation of sensitivity in differentiating among the healthy, aged, and diabetic populations compared with the conventional MSE method.
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2.2 Outlier-Resilient Complexity Analysis of Heartbeat Dynamics
2.2.1 INTRODUCTION
Complexity in physiological outputs is believed to be a hallmark of healthy physiological control. How to accurately quantify the degree of complexity in physiological signals with outliers remains a major barrier for translating this novel concept of nonlinear dynamic theory to clinical practice. Here we propose a new approach to estimate the complexity in a signal by analyzing the irregularity of the sign time series of its coarse-grained time series at different time scales. Using surrogate data, we show that the method can reliably assess the complexity in noisy data while being highly resilient to outliers. We further apply this method on human heartbeat recordings.
Without removing any outliers due to ectopic beats, the method is able to detect a degradation of cardiac control in patients with congestive heart failure and a more degradation in critically ill patients whose life continuation relies on extracorporeal membrane oxygenator (ECMO). Moreover, the derived complexity measures can predict the mortality of ECMO patients. These results indicate that the proposed method may serve as a promising tool for monitoring cardiac function of patients in clinical settings. Many physiological variables such as motor activity and heart rate display seemingly irregular fluctuations over a wide range of time scales1,2. Under normal healthy conditions, these physiological fluctuations are neither random nor too regular, possessing robust, multi-scale dynamic patterns that are independent of external influences 3-5. Such a complexity in physiological fluctuations has been accepted as a hallmark of healthy physiology and is believed to reflect system adaptability in response to constant changes in internal and external inputs. Numerous studies have supported this theory of complexity by showing that physiological fluctuations become either too random or too regular with aging and under pathological breakdowns 6-10.
Despite the physiological importance of the complexity theory, its application to clinical studies has been hindered by the lack of algorithms that can be easily implemented for accurate estimation of the degree of complexity in physiological fluctuations3,4. One generic challenge for algorithm design is to account for the effects of “outliers”, which often exist in clinical recordings due to not only external random
