Non-Gaussian characteristics of heart rate variability in health and disease Ken Kiyono Division of...

Non-Gaussian characteristics of heart rate variability in health and disease

Ken KiyonoDivision of Bioengineering

Graduate School of Engineering Science Osaka University

Multifractal Analysis: From Theory to Applications and Back

Outline

1. Heart rate variability (HRV)

2. Relation between multifractality and non-GaussiantyMultiplicative random cascade model

3. Multiplicative decomposition of non-Gaussian noiseCharacterization method of non-Gaussian distribution

4. Non-Gaussian properties of HRVPrediction of mortality risk

5. Summary

Heart rate variability (HRV)

HRV is the temporal fluctuation of heart rhythm. The times series is derived from the QRS to QRS (RR) interval sequence of

the ECG, by extracting only normal-to-normal interbeat intervals. The normal ECG is composed of a P wave, a QRS complex and a T wave.

RR intervals fluctuate beat by beat.

Electrocardiogram (ECG)

[Kleiger et al., Am. J. Cardiol., 59 (1987) 256]

HRV as a predictor of mortality

Reduced heart-rate variability has been shown to be a risk factor for increased mortality after myocardial infarction

[Reprinted from Kleiger et al., Am. J. Cardiol., 59 (1987) 256]

SD of RRINo. of

PatientsMortality rate

> 100 ms 211 9.0%

50-100 ms 472 13.8%

< 50 ms 125 34.4%

Lower HRV was associated with a higher risk.

Mortality rate of patients after myocardial infarction

Physiological cause of heart rate variability

HRV is mainly controlled by autonomic nervous system

(ANS). Parasympathetic blockade reduces heart rate fluctuation.

[Reprinted from 井上博編，循環器疾患と自律神経機能 ( 第２版 ) ， 2001, 医学書院 ]

Pharmacological blockade experiment Intravenous administration of propranolol and atropine.

[Kleiger et al., Am. J. Cardiol., 59 (1987) 256]

Frequency characteristics of HRV

Through the Fourier transform, observed signals can be decomposed into the

superposition of sinusoidal signal with different frequencies and amplitudes.

HF and LF components of HRV

short-term HRV ( ～ 5 min)

High frequency (HF; 0.15-0.4 Hz) band ： Synchronization between respiration (~ 4 s) and HRV

Low frequency (LF; 0.04-0.15 Hz) band ：Mayer wave (~10 s) , Blood pressure oscillations

sympathetic control

parasympathetic control

1/f fluctuation of HRV

Healthy HRV power spectrum shows a 1/f-type power-law scaling

[Kobayashi, Musha, IEEE Trans. Biomed. Eng. 29, 456 (1982)]

1/f noise and fractal (1)

A simple repeating rule can produce 1/f noise.

Deterministic fractality Stochastic fractality

Koch curve

1/f noise

1/f noise and fractal (2)

Random variables lie on a dyadic grid.

function.floor theis where

,1

)(

12/)1(

・

　

m

j

ji jmi

WX

Multifractality of Heart Rate Variability

Analogy between HRV and cascade model

Outline





5. Summary

Multiplicative random cascade (1)

(cf. Additive random cascade can generate 1/f noise)

Multifractal time series can be generated by multiplicative cascade.

Multiplicative random cascade (2)

where is the floor function.

Non-Gaussian PDF of cascade model

Let us consider a multiplicative cascade-type model,

where is the floor function. When and , the probability density function of is given by

Multiscaling property of structure function

The structure function is defined as,

,

where

Multiscaling property of cascade-type model

Using a Gaussian approximation of the partial sums, we obtain

where ΨY is the cumulant-generating function of Y(j)

(If Y(j) is a Gaussian variable, ΨY is a quadratic function.)

white Gaussian

Multiscale PDF analysis

i

kki

s

jjiisiis

UZ

UZZZ

1

1

. where

,

Fine resolution

Coarse resolution

Partial sum process {DsZi}

Deformation of PDFs across scales

Convergence to a Gaussian

[Castaing et al., Physica D, 46, 177 (1990); Kiyono et al., IEEE TBME 53, 95-102 (2006)]

Convergence process to a Gaussian

Log-normal cascade model iid sequence

u iu i

s = 1, 2, 4, 8, 16, 32from top to bottom

Outline





5. Summary

Parameter estimation problem of non-Gaussian processes

Conventional models of non-Gaussain distributions ■ Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)]

This model is involved with turbulent cascade picture.

■ Superstatistics [Beck & Cohen, Physica A, 322, 267 (2003)]

Superstatistics considers a driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature).

■ Heavy tailed distributions (independently and identically distributed process)

◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (0 < a < 2) for large |x|

◇ stretched exponential distribution, P(x) exp(-∝ g|x|a)

PL(x): local equilibrium distribution

f (b): fluctuations of intensive parameter

PL(x): PDF at integral scale L

G(s): fluctuations through energy cascade

Models for non-Gaussian fluctuations





■ Heavy tailed distributions




f(b): fluctuations of intensive parameter


G(ln s): fluctuations through energy cascade

The Mellin convolution of f and g :

X ~ PL(x) sX s ~ G(ln s)

Parameter estimation problem of non-Gaussian processes





■ Heavy tailed distributions (independently and identically distributed process)




f (b): fluctuations of intensive parameter


G(s): fluctuations through energy cascade

Ut = Xt exp Yt

observed non-Gaussian noise

Multiplicative decomposition of non-Gaussian noise

u t

Assume that U has a unimodal symmetric distribution.

(cf. Multifractal random walk [Bacry et al., Phys. Rev. E 64, 026103 (2001)])

Ut = Xt exp Yt

Gaussian noise

log-amplitude fluctuationobserved non-Gaussian noise

Multiplicative decomposition of non-Gaussian noise

u t

yt

x t

amplitude fluctuation

Log-amplitude cumulants (1)

cumulant of Yt = cumulant of ln|Ut| - cumulant of ln|Xt | U is observable X is a Gaussian

By assuming Ut = Xt exp Yt , we can obtain the following relation,

Log-amplitude cumulants Ck (cumulants of Y) can be estimated from {Ut}.

Log-amplitude cumulants (2)

■ Definition of log-amplitude cumulants Consider a process {Ut} described by a multiplication of random variables,

where Xt and Yt are random variables independent of each other and Xt is a standard

Gaussian random variable with zero mean and unit variance.

In this process, log-amplitude cumulants are defined as cumulants of Yt .

Ut = Xt exp Yt

Gaussian distribution on (C2 , C3 ) plane

Third log-amplitude cumulant C3 vs. second log-amplitude cumulant C2

X ~ N(0, ),(Y = const.)

Castaing’s model on (C2 , C3 ) plane


→ deviation from a Gaussian shape

X ~ Gaussian×log-normal

Castaing’s model using log-normal distribution

[Castaing, Gagne & Hopfinger, Physica D, 46 (1990) 177]

log-normal x Gaussian

lGaussian (l → 0):

log10 P(x) ~ -x 2

non-Gaussian PDFwith fat tails

Superstatistics on (C2 , C3 ) plane


power-law tails

exponential tails(n = 2)

Superstatistical distributions

[Beck & Cohen, Physica A, 322, 267 (2003)]

Local equilibrium distribution Intensity parameter fluctuation

Marginal distribution

Superstatistical distributions

[Beck & Cohen, Physica A, 322, 267 (2003)]

b

q-Gaussian distribution (t distribution)

b

Bessel function distribution of the second kind

Stable distributions on (C2 , C3 ) plane


All of Cn have closed-form expressions.

(C2 , C3 ) plane


Scale dependence of log-amplitude cumulants

Log-normal cascade 　 model(multifracrtal noise) iid variables

ssmsC log~)log()( 2022

C2(s

)

If {Xi} is a white noise process, the autocovariance of {Yi} can be estimated by

where t > 0.

Autocovariance of the log-amplitude(magnitude correlation)

Log-normal cascade 　 model(multifracrtal noise) iid variables

[A. Arneodo et al., Phys. Rev. Lett. 80, 708 (1998); K. Kiyono et al., Phys. Rev. Lett. 95, 058101 (2005)]

Outline





5. Summary

Detrending procedure of non-stationary time series

Local detrending by fitting and subtracting a polynomial(e.g. detrended fluctuation analysis [C.-K. Peng et al., Chaos 5, 82-87 (1995)])

Log-returns,

Using a wavelet with vanishing moments

High-pass filtering

Detrended time series of HRV

[J. Hayano et al, Frontiers in Physiology. 2, 65 (2011)]

heart rate variability

← Heart rate variabilityafter acute myocardial infarction

← Detrended serieswith zero mean and unit variance

← Probability distribution of the detrended series (on log-linear plot)

survivor nonsurvivor nonsurvivor

Healthy subjects vs heart failure patients

Healthy subjects vs heart failure patients

CHF patients (n = 108)Healthy subjects (n = 123)

Non-Gaussianity at the time scale of 25 sec is important for risk stratification [J. Hayano et al, Frontiers in Physiology. 2, 65 (2011); K. Kiyono et al., Heart Rhythm, 5, 261-268, (2008)]

Scale dependence of non-Gaussianity

25 sec 25 sec

(n = 69)

(n = 39) (n = 45)

(n = 625)

(follow-up of mean of 33 months) (follow-up for median of 25 months)

Non-Gaussian heart rate as a risk factor for mortality

scale = 40, 100, 400, 1000 beats

Non-Gaussian index of HRV was a significant and independent mortality predictor in patients with congestive heart failure (CHF) [Kiyono et al., Heart Rhythm, 5, 261-268, (2008)].

C2 = λ2,when Y is a Gaussian

Non-Gaussian heart rate as a risk factor for cardiac mortality

Increased non-Gaussianity of heart rate variability predicts cardiac mortality after an acute myocardial infarction. [J. Hayano et al, Frontiers in Physiology. 2, 65 (2011)].

survivor nonsurvivor nonsurvivor

Non-Gaussian index λ

■ Multiplicative Stochastic Process

■ q th non-Gaussian index [Kiyono et al., Phys. Rev. E, 76, 041113 (2007)]

Assumption: Y is a Gaussian (multiplicative log-normal process)

■ Log-amplitude cumulants

[Kiyono, Konno, Phys. Rev. E., 87, 052104 (2013)]

ttt YWX expW: Gaussian variable Y: log-amplitude

If Y is a Gaussian,

lq = const,

C2 = lq2.

If X is a Gaussian,

C2 = lq2 = 0

Prediction of sudden cardiac death

[Heart Rhythm, 5, 269-270 (2008)]

Outline





5. Summary

Summary

Multiscale PDF analysis is applicable to a variety of real-world time series.[K. Kiyono et al. Phys. Rev. Lett. 96, 068701 (2006); Phys. Rev. Lett. 95, 058101; Phys. Rev. Lett. 93, 178103 (2004)]

Non-Gauusain PDF can be characterized by log-amplitude cumulants.[K. Kiyono, Phys. Rev. E 79, 031129 (2009); K. Kiyono, H. Konno, Phys. Rev. E 87, 052104 (2013)]

Increased non-Gaussianity of heart rate variability predicts mortality.[K. Kiyono et al., Heart Rhythm 5, 261-268, (2008); J. Hayano et al. Front Physiol. 2, 65 (2011)] 　

CollaboratorsJunichiro Hayano (Nagoya City University)

Eiichi Watanabe (Fujita Health University)

Hidetoshi Konno (Tsukuba University)

Yosiharu Yamamoto (University of Tokyo)

Taishin Nomura (Osaka University)

Akihiro Azuma (Osaka University)Tetsutaro Endo (Osaka University)Koichi Takeuchi (Osaka University)Syota Fujii (Osaka University)Ysutaka Moriwaki (Osaka University)Yauyuki Suzuki (Osaka University)

Non-Gaussian characteristics of heart rate variability in health and disease Ken Kiyono Division of...

Documents

Transcript of Non-Gaussian characteristics of heart rate variability in health and disease Ken Kiyono Division of...