Nonlinear dynamic Nonlinear dynamic system analyzing for system analyzing for

heart rate heart rate variability variability

mathematical modelmathematical model A.Martynenko & M. YabluchanskyA.Martynenko & M. Yabluchansky

Kharkov National University Kharkov National University (Ukraine)(Ukraine)

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HRV – naturally observing HRV – naturally observing phenomenon of nonlinear dynamic phenomenon of nonlinear dynamic behavior of the cardiovascular behavior of the cardiovascular systemsystem

Non Linear Mathematical Model (NL Non Linear Mathematical Model (NL MM) of HRVMM) of HRV Why do we need this model?Why do we need this model? What is the advantage of NL MM for What is the advantage of NL MM for

investigation of ANS?investigation of ANS?

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Mathematical model Mathematical model differential equationsdifferential equations

Regulatory (ANS) group: Regulatory (ANS) group: Nonlinear dynamic quasi-Nonlinear dynamic quasi-periodical processes in humoral periodical processes in humoral (1)(1), sympathetic , sympathetic (2)(2), , parasympathetic parasympathetic (3)(3) branches of nervous system branches of nervous system

d2(ANSd2(ANSii)/dt2 = F)/dt2 = Fii(R(Ri i d(ANSd(ANSii)/dt, A)/dt, Aii ANSANSii, S, Sjj ANSANSjj, , BioMechanics), i,j = 1,2,3,BioMechanics), i,j = 1,2,3,

BioMechanics group: BioMechanics group: equations that describe function equations that describe function of biomechanical parameters forming ANS activityof biomechanical parameters forming ANS activity

d2(BMd2(BMii)/dt2 = B)/dt2 = Bii(R(Rii d(BMd(BMii)/dt, A)/dt, Aii BMBMii, HR, HRññ, ANS), i = , ANS), i = 4,5,6,4,5,6,

HR equation - HR equation - describe HR changes from cycle to cycle describe HR changes from cycle to cycle

d2(HRd2(HRññ)/dt2 = )/dt2 = ffii(R (R d(HRd(HRññ)/dt, A )/dt, A HRHRññ,S,Sjj ANSANSjj), j = 1,2,3.), j = 1,2,3.

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Scheme of cardiovascular Scheme of cardiovascular regulationregulation

(cross-linkage in mathematical (cross-linkage in mathematical model)model)

Symp

HRV

Gumr.

Parasmph.

PR

ABP

SV

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HRV (corr=0.993) and HRV (corr=0.993) and spectrum (corr=0.999) spectrum (corr=0.999) (registration vs. model)(registration vs. model)

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First result of NL MM – new First result of NL MM – new technique of spectral domain technique of spectral domain

separationseparation

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‘‘Nlyzer’ by TU DarmstadtNlyzer’ by TU Darmstadt

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Standard nonlinear Standard nonlinear analyzesanalyzes

Fractal Dimension Fractal Dimension (D2)(D2) HRV – 4.74 – 5.3HRV – 4.74 – 5.3 ECG – 2.65 – 3.5ECG – 2.65 – 3.5

N points for N points for embeddingembedding N=10N=102+0.4D22+0.4D2=10000=10000

Autocorrelation for Autocorrelation for time delaytime delay

Entropy and mutual Entropy and mutual informationinformation

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Lorentz attractor Lorentz attractor (D2=2.06)(D2=2.06)

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Attractor reconstruction Attractor reconstruction (20 min)(20 min)

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Attractor reconstruction (2500 Attractor reconstruction (2500 heartbeat)heartbeat)

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Attractor reconstruction (5000 Attractor reconstruction (5000 heartbeat)heartbeat)

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Attractor reconstruction Attractor reconstruction (10000 heartbeat)(10000 heartbeat)

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Attractor reconstruction Attractor reconstruction (15000 heartbeat)(15000 heartbeat)

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Attractor reconstruction Attractor reconstruction (20000 heartbeat or about 5 (20000 heartbeat or about 5

hours)hours)

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Attractor reconstruction Attractor reconstruction (3min+MM)(3min+MM)

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Attractor reconstruction Attractor reconstruction (15000 heartbeat)(15000 heartbeat)

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Poincare map (3 min+ Poincare map (3 min+ MM)MM)

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Attractor reconstruction (3 Attractor reconstruction (3 min + MM)min + MM)

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Attractor reconstruction (3 Attractor reconstruction (3 min + MM)min + MM)

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HRV attractor and its P-HRV attractor and its P-map map

x y z( ) a b c( )

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ConclusionConclusion Cardiovascular regulation is nonlinear Cardiovascular regulation is nonlinear

dynamic system, and then we need nonlinear dynamic system, and then we need nonlinear mathematical modeling for their mathematical modeling for their investigation.investigation.

Advantages of NL MM:Advantages of NL MM: New technique of spectra domain separationNew technique of spectra domain separation Great time compression: We don’t need 4-5 hours Great time compression: We don’t need 4-5 hours

of registration for attractor reconstruction – only of registration for attractor reconstruction – only 3 min of registration and NL MM3 min of registration and NL MM

Attractor visualization in Humoral - Sympathetic Attractor visualization in Humoral - Sympathetic - Parasympathetic phase space is very good- Parasympathetic phase space is very good