heart rate variability: challenge for both experiment and modelling

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Heart rate variability: challenge Heart rate variability: challenge for both experiment and for both experiment and

modellingmodelling

Heart rate variability: challenge Heart rate variability: challenge for both experiment and for both experiment and

modellingmodelling

I. Khovanov,

N. Khovanova, P. McClintock, A. StefanovskaPhysics Department,

Lancaster University

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Heart rate variability (HRV)Heart rate variability (HRV)Heart rate variability (HRV)Heart rate variability (HRV)

Outline

● Motivations

● Experiment

● Modelling

● Summary

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The object of investigation is heart rate


Heart Rate Variability

ElectroCardioGramSinoAtrial Node

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24h RR-intervals


RR

-int

erva

ls,

sec

Number of interval

Medical people:Average over one hour rhythm

Physicists:Entropies,Dimensions,Long-range correlation,Scaling,Multifractal etc

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HRV is the product of the integrative control system


Parasympathetic branch Vagus nerve fibres

Fast and -

Sympathetic branch Postganglionic fibres

Slow and +

From receptor afferents: baro-receptors, chemo-receptors, stretch receptors etc

Vagal SympInput

Nucleous

Medulla

Hypothalamus

Cortex (higher centres)

--+

SAN

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RespirationRespiration masks other rhythms masks other rhythmsRespirationRespiration masks other rhythms masks other rhythms

Circles corresponds to RR-intervals,

Dashed line corresponds to respiration

respiration

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Use of apnoea (breath holding)?Use of apnoea (breath holding)?Use of apnoea (breath holding)?Use of apnoea (breath holding)?

1) The use of breath holding as longer as possible, BUT physiologists discussed a long breath holding as one of unsolved problems with a specific dynamics (Parkers, Exp. Phys. 2006)

2) We can notice: In spontaneous respiration there are apnoea intervals (not very long)

An idea is to prolong by

keeping normocapnia

Physiology literature said

30 sec is fine(!)

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Specific form of respirationSpecific form of respiration Specific form of respirationSpecific form of respiration

Intermittent type (intervals of fixed duration, 30 sec)

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The idea is to consider HRV without dominate external perturbation, but keeping all internal perturbation and without modification of a net of regulatory networks

The task is to study intrinsic dynamics on

short-time scales

●Special design of experiments: relaxed, supine position,records of 45-60 minutes

● Time-series analysis of sets of short time-series (appr. 40Apn.int X 35RR-int)

Intrinsic dynamics of regulatory systemIntrinsic dynamics of regulatory systemIntrinsic dynamics of regulatory systemIntrinsic dynamics of regulatory system

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Decomposition (nonlinear transformation) of Decomposition (nonlinear transformation) of heart rate by specific forms of respirationheart rate by specific forms of respiration Decomposition (nonlinear transformation) of Decomposition (nonlinear transformation) of heart rate by specific forms of respirationheart rate by specific forms of respiration

Circles corresponds to RR-intervals,

Dashed line corresponds to respiration

respiration

Object of analysis: a set of RR-intervals {RRi}j corresponded to apnoea intervals

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RR-intervals

Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals.

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

East

West

North Number of interval, i

RR

i-RR

1 [

sec]

RR

i-R

R1 [

sec] Increments RR

RRi=RRi-RRi-1

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RR-intervals during apnoea intervals is non-stationary.

The use of random walk framework.

DFA (detrended fluctuational analysis): scaling exponent (Peng’95)

Aggregation analysis: scaling exponent b (West’05)

Both methods for the considered time-series estimate a diffusion velocity

Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals. Non-stationary dynamics of RR-intervals.

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

East

West

North

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Scaling exponent Scaling exponent by DFA by DFA Scaling exponent Scaling exponent by DFA by DFA

DFA method (C.-K. Peng, Chaos,1995)

(1) Integration of RR-intervals:

n

(2) Calculation of linear trend yn(k) for time window of length n(3) Calculation of scaling function for set of n

4 15n

(4) Determination of

=1,5 corresponds to Brown noise

(free Brownian motion)

)log()(log nnF

NkRRkyk

ii ...1)(

1

2)()(1

1)( kyky

NnF n

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Scaling exponent Scaling exponent bbby aggregation analysis by aggregation analysis Scaling exponent Scaling exponent bbby aggregation analysis by aggregation analysis

Aggregation method (B. J. West, Complexity, 2006)

Invention by L. R. Taylor, Nature, 1961(1) Creating a set of aggregated time-series:

(2) Calculation of the variance and mean for each m=1,2…:

(3) Determination of bb=2 corresponds to Brown noise

(free Brownian motion) ))(log()(log mbm

mk

kiim RRky )(

M

km mky

Mm

1

2)()(

1

1)(

M

km ky

Mm

1

)(1

)(

10:1m

))(log( m

)(log m

The aggregation method is close to the stability test for the

increments RR

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Scaling exponents Scaling exponents and and bbon the base of 24h RR-on the base of 24h RR-intervalsintervals

Scaling exponents Scaling exponents and and bbon the base of 24h RR-on the base of 24h RR-intervalsintervals

DFA and the aggregation method in the presence respiration (the previous published results)

Brown noise, Brownian motion

)8.1:4.1()2.1:0.1( b

0.25.1 BB b

White noise

0.5 1.0B Bb

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Scaling exponents Scaling exponents and and bbRR-intervals during apnoeaRR-intervals during apnoeaScaling exponents Scaling exponents and and bbRR-intervals during apnoeaRR-intervals during apnoea

DFA and the aggregation method without respiration

Brown noise, Brownian motion

(1.3 :1.7) (1.8 : 2.0)b

0.25.1 BB bWhite noise

0.5 1.0B Bb

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RR-intervals during apnoea intervals is non-stationary.

So let us use stationary increments RRi=RRi-RRi-1

then use the modified definition of ACF () to use non-overlapped windows corresponding apnoea intervals

Dynamics of increments Dynamics of increments RR. ACFRR. ACFDynamics of increments Dynamics of increments RR. ACFRR. ACF

di jRR

1i i ij jRR RR RR

- time delay kj – number of RR-intervals in each apnoeaN –total number of apnoea intervals

jk

iii

N

j

RRRRM 111

1)(

Finally use fitting by the function

2cosexp)( aappr

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i jRR di j

RRCrosses corresponds to calculations using RRThe solid line corresponds to approximation by

ACF of ACF of RR-intervalsRR-intervalsACF of ACF of RR-intervalsRR-intervals

e cosa

Fast decay of ACF with weak oscillations near 0.1 Hz

Oscillations are on-off nature and observed for parts of apnoea intervals and, not in all measurements.

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=2=1.5=1=0.5

Distribution of increments of RR-intervals Distribution of increments of RR-intervals P(P(RR)RR)Distribution of increments of RR-intervals Distribution of increments of RR-intervals P(P(RR)RR)

Calculate histogram and fit by -stable distribution.

A random variable X is stable, if for X1 and X2 independent copies of X, the following equality holds:

2

221e

2

x

]2:0(),,( XP1 2

d

aX bX cX g d

Means equality in distribution

is a stability index defines the weight of tails

=2 Normal (Gaussian) distribution

=1 Cauchy (Lorentz) distribution

22 )(

1

x

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Yellow areas and cycles correspond to histograms The solid lines is fit by the normal distribution (=2)The dashed lines corresponds to the -stable distributions

Distribution of increments of RR-intervalsDistribution of increments of RR-intervalsDistribution of increments of RR-intervalsDistribution of increments of RR-intervals

2

221e

2

x

80.1

86.1

The previous published results for 24h RR-intervals )7.1:5.1(Apnoea intervals )2:8.1(

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HRV intrinsic dynamics HRV intrinsic dynamics HRV intrinsic dynamics HRV intrinsic dynamics

Summary of experimental results:

RR-intervals show stochastic diffusive dynamics.HRV during apnoea can be described as a stochastic process with stationary increments

Conclusion: Intrinsic dynamics is a result of integrative action of many weakly interacting components

Increments RR describes by -stable process with a weak correlation

In zero approximation RR corresponds to uncorrelated normal random process and RR-intervals show classical free Brownian motion.

1i i iRR RR RR

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Modelling Modelling Modelling Modelling

Heart beat is initiated in SANSinoatrial node

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Isolated heart (e.g. in case of brain dead)

ModellingModellingModellingModelling

No signal from nervous system

Nearly periodic oscillations, butheart rate is 200 beats/min

whereas in normal state60-80 beats/min

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Vagal activation


Parasympathetic branch Vagus nerve fibres

Fast and -

Threshold potential

Potential of hyperpolarizationSlope of depolarization

●Decreasing depolarization slope● Increasing hyperpolarization potential

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Sympathetic activation


Sympathetic branch Postganglionic fibres

Slow and +

● Increasing depolarization slope● Decreasing hyperpolarization potential

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Integrate & Fire model

ti ti+1 ti+2

Threshold potential

Ut

Ur

Hyperpolarizationpotential

Integration slope1/

1i i iRR t t

1( ) ( )t i t i iU t U t Random numbers having the stable distribution

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FitzHugh-Nagumo model

3 / 3 ( )

( )u

v

u u u v t

v u a t

( )

( )

( )

( )

u

v

t

t

t

a a t

● Additive versus multiplicative noise● Noise properties

What kind of noise will produce non-Gaussianity of increments RR

heart rate variability: challenge for both experiment and modelling

Documents

set of rr

integration of rr

considered timeseries

set of aggregated timeseries

short timeseries appr

specific dynamics parkers

exponent b west05both

respiration masks