Multifractality in Detrended Human Heart Beat Increment Yue-Kin - - PowerPoint PPT Presentation

Multifractality in Detrended Human Heart Beat Increment

Yue-Kin Tsang

Scripps Institution of Oceanography University of California, San Diego Emily S. C. Ching Department of Physics The Chinese University of Hong Kong

The Cardiac Pump

Right atrium

◗ ◗ ◗ ◗

Right ventricle

❈ ❈ ❈ ❈ ❈ ❈

Left atrium Left ventricle

◗ ◗ ◗ ◗ ◗ ✘ ✘ ✘ ✘ ✾ ✘ ✘ ✘ ✘ ✾ ✻ ❄

The Cardiac Pump

SA node

◗ ◗ ◗ ❅ ❅ ❅ ❘ ❄

electric impulse initiated in the sinoatrial (SA) node spreads to the atria — atrial contraction

The Cardiac Pump

AV node

❩❩❩❩❩❩ ❩ ❅ ❅ ❅ ■ ✄ ✄ ✄✄ ✗

electric impulse initiated in the sinoatrial (SA) node spreads to the atria — atrial contraction impulse reaches the atrioventricular (AV) node and is conducted to the ventricles — ventricular contraction delay in the passage of the impulse occurs in the AV node

Heart Beat Intervals

Electrocardiogram (ECG)

55 56 57 58 59 60 61

time (second)

RRi

P-wave : atrial activation QRS complex : ventricular activation T-wave : recovery phase

• f ventricular

RR-intervals (RRi) — measure of heart rate

Heart Rate Variability (HRV)

Time series of RRi : b(i)

10000 20000 30000 40000 50000

i

0.4 0.6 0.8 1.0 1.2

b(i) 0.5 0.6 0.7 0.8 0.9 1.0 1.1

b

1 2 3

PDF of b

Reasons to study HRV

To understand how the autonomic nervous system (ANS) control heart rate

sympathetic branch of the ANS increases heart rate and parasympathetic branch decreases heart rate both branches are active and interacts, parasympathetic effects usually dominate parasympathetic branch affects heart rate with a much shorter delay

Abnormalities in HRV have prognostic significance

healthy human RRi shows multifractality multifractality lost in congestive heart failure condition

P . Ch. Ivanov et al., Multifractality in human heartbeat dynamics, Nature 399, 461 (1999)

Multifractal

A timeseries X(i) has different statistical properties at different scales.

∆nX(i) ≡ X(i + n) − X(i)

(1) Probability density function of ∆nX(i)

Pn(∆nX) changes shape with n

(2) Structure function

Sq(n) ≡ |∆nX|q ∼ nζq ζq is a nonlinear function of q

Multifractal

(1) ⇔ (2) Let Y = ∆nX |∆nX|21/2 Pn(∆nX) has same shape for different n ⇔ ¯ Pn(Y ) is independent of n |Y |q =

• |Y |q ¯

Pn(Y ) dY is independent of n |Y |q = |δnX|q |δnX|2q/2 = Sq(n) [S2(n)]q/2 ∼ nζq−qζ2/2 ζq = q 2ζ2

scale-invariant Pn(∆nX) ⇐

⇒ ζq ∝ q

An Example from Fluid Turbulence

Multifractal scaling in temperature increments ∆nθ from turbulent thermal convective experiments:

• 6 -4 -2

2 4 6

∆nθ

• 6
• 4
• 2

log Pn(∆nθ)

n=4 n=32 n=256 n=4096

1 2 3 4

q

1 2

ζq / ζ2

[ data from B. Castaing et al., J. Fluid Mech. 204,1 (1989) ]

Multifractality in Healthy Heart Rate

Healthy heart rate (RRi) increments ∆nb from data of daytime normal sinus rhythm:

• 4
• 2

2 4

∆nb

• 6
• 4
• 2

log Pn(∆nb)

n=4 n=32 n=256 n=4096

1 2 3

q

1

ζq / ζ2

[ data from http://physionet.org ]

Scale-invariant Detrended RRi ?

“scale-invariance in the PDF of detrended healthy human heart rate increments” A detrend procedure for non-stationary timeseries:

• 1. B(i) = i

j=1 b(j)

• 2. divide B(i) into segments of size 2n
• 3. fit B(i) in each segment with the best d-th order

polynomial, p(n)

d (i)

• 4. B∗(i) = B(i) − p(n)

d (i)

(“trend” removed)

• 5. ∆nB∗(i) = B∗(i + n) − B∗(i)

Pn(∆nB∗) is scale-invariant

• K. Kiyono et al., Phys. Rev. Lett. 93,17 (2004)

PDF of ∆nB∗ in Healthy Heart Rate

• 4
• 2

2 4

∆nB*

• 6
• 5
• 4
• 3
• 2
• 1

log Pn(∆nB*)

n=16 n=64 n=256 n=1024

But isn’t healthy heart rate multifractal !?

Detrended Heart Rate b∗(i)

∆nB∗(i) is actually related to the sum of b(i)

∆nB∗(i) = B∗(i + n) − B∗(i) = B(i + n) − B(i) − [p(n)

d (i + n) − p(n) d (i)]

=

i+n

• j=i+1

b(j) − ∆np(n)

d (i)

A natural definition of detrended heart rate b∗(i) is

B∗(i) ≡

i

• j=1

b∗(j) ⇒ ∆nB∗(i) =

i+n

• j=i+1

b∗(j) b∗(i) = B∗(i) − B∗(i − 1)

Structure function for ∆nb∗(i)

∆nb∗(i) = ∆nb(i) − [∆np(n)

d (i) − ∆np(n) d (i − 1)]

2 4 6 8 10 12

log2 n

2 4 6 8 10 12

log2 Sq(n)

0.2 0.6 1.0 1.6 2.0 2.6 3.0

ζq for Healthy Heart Rate

1 2 3

q

1

ζq / ζ2

∆nb* ∆nB* ∆nb

The detrended healthy heart rate b∗(i) is indeed multifractal

ζq for Pathological Heart Rate

Data from congested heart failure patients:

1 2 3

q

1

ζq / ζ2

∆nb* ∆nB* ∆nb

Scale-invariance of Pn(∆nB∗) is a characteristic independent of the multifractality of HRV

Detrend Analysis in Turbulence

Follow the ideas of detrended analysis of HRV:

θ(ti) = temperature measurement from thermal

convective experiments:

Θ(ti) ≡

i

• j=1

θ(tj) Θ∗(ti) ≡ Θ(ti) − p(n)

d (ti)

∆nΘ∗(ti) = Θ∗(ti+n) − Θ∗(ti) Θ∗(ti) ≡

i

• j=1

θ∗(tj)

Detrend Analysis in Turbulence

1 2 3

q

1

ζq / ζ2

∆nθ∗ ∆nΘ∗ ∆nθ

PDF of ∆nΘ∗ is not scale-invariant.

Detrend Analysis in Turbulence

• 6
• 4
• 2

2 4 6

∆nΘ

∗

• 6
• 4
• 2

log Pn(∆nΘ

∗)

n=4 n=32 n=256 n=4096

Detrend Analysis in Turbulence

Recall :

B(i) ≡

i

• j=1

b(j) ∆nB∗(i) =

i+n

• j=i+1

b(j) − ∆np(n)

d (i)

= ⇒ ∆nB∗(i) = ∆nB(i) − ∆np(n)

d (i)

∆nΘ∗(i) = ∆nΘ(i) − ∆np(n)

d (i)

Now,

|∆nB(i)|q ∼ nq |∆nΘ(i)|q ∼ nq

So p(n)

d (i) is responsible for the different scaling behav-

ior in |∆nB∗(i)|q and |∆nΘ∗(i)|q

Summary

We clarify that the scale-invariant of Pn(∆nB∗) in healthy heart rate is for the sum of detrended heart rate b∗ Pn(∆nb∗) for healthy heart rate increments is indeed scale dependent, as expected from the multifractality of b. Pn(∆nB∗) is scale-invariant in pathological heart rate in patients suffering from congestive heart failure Pn(∆nΘ∗) is scale dependent in the multifractal temperature measurements in turbulent thermal convective flows.