perfect matrices of lagrange differences for the...
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IntroductionMethods and results
Summary
Perfect matrices of Lagrange differences for theinterpretation of dynamics of the cardio-vascular
system
P. Palevičius1 M. Ragulskis1 V. Šiaučiūnaitė1 A. Vainoras2
1Research Group for Mathematical and Numerical Analysis of Dynamical Systems,Kaunas University of Technology
2Institute of Cardiology, Lithuanian University of Health Sciences
NASCA, 2018
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Outline
1 IntroductionBackground informationMotivationThe objective of the research
2 Methods and resultsData retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
3 Summary
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
Electrocardiography
Electrocardiography (ECG) analysis is the basic, the primary,and the most studied noninvasive technique used for thecontemporary investigation of the functionality of thecardiovascular system.Cardiac time intervals are sensitive markers of cardiacdysfunction.Interrelations between ECG parameters is still an active areaof research (e.g. functional relations among RR, JT, and QTintervals).
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
Cardiac time intervals
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
Challenge #1
Insufficient identification of the specificity of relationshipsbetween ECG parameters.QT is the sum of the depolarization and the repolarization of heart’smyocardium. However, depolarization and repolarization may havecompletely different variation tendencies. For example, depolarization andrepolarization may both become shorter during the load—but sometimesdepolarization becomes longer and repolarization becomes shorter (whatis a rather common phenomenon during the extreme loads). This means,that QT-RR relationship cannot describe the behavior of the humancardiovascular system in general. Similar problems exist for otherrelationships between ECG parameters (for example PQ-RR, etc.).
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
Challenge #2
A fixed model of relationships between ECG parameterscannot always hold even for a particular person.Relationships between ECG parameters do vary according to variousphysiological and pathological reasons. Holistic models interpret a humanbeing as a complex system - where nonlinear chaotic processes play animportant role in the relationships between different subsystems and ingenerating reactions of these subsystems. Therefore, it is probablyillogical to seek a unified deterministic model which could describe therelationships between ECG parameters. It makes sense to observedynamical processes, dynamical relationships - which could exhibitcomplex chaotic behavior.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
Challenge #3
Time-averaged relationships between ECG parameters are notable to represent the complexity of these relationships indifferent time scale lengths.The relationships between ECG parameters do depend on different timescales and other factors. The width of the observation window mayseriously impede the computational results.The complexity of therelationships, the chaotic nature of the processes, the fractality of timescales - all that yields a necessity to develop such computationaltechniques which could assess the dynamism of these relationships.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Background informationMotivationThe objective of the research
The objective of the research
The objective of this research is to propose a visualizationtechnique of relationships between RR and JT intervals whichcould reveal the evolution of complex dynamical processes inthe self-organization of the heart system during the load andthe recovery processes.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
ECG analysis system “Kaunas-Load”
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
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Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Data
The bicycle ergometry system is used for generating stepwiseincreasing physical loads.The bicycle ergometry exercise is initiated at 50W load - andthe load is increased by 50W every consecutive minute.The patient is asked to maintain a constant 60 revolutions perminute bicycle pedals spinning rate during the whole exercise.The load is increased up to 250W and the exercise iscontinued until the first clinical indications for the limitationof the load according to AHA are observed.Sequences of RR and JT intervals are recorded during thebicycle ergometry experiment and denoted as vectorsx = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn).
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Definition of the perfect matrices of Lagrange differences
We will consider second order square matrices and assume thatevery element of a matrix can be either a single element of thetime series x or y - or a difference between elements of time seriesx and y:
a ∈ {±x,±y,± (x − y)}
Definition. A perfect matrix of Lagrange differences is a secondorder square matrix whereas elements of that matrix do satisfy thefollowing requirements:
All elements of the matrix are different.Zeroth order differences are located on the main diagonal.First order differences are located on the secondary diagonal.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Definition of the perfect matrices of Lagrange differences
Indexes of x and y can take one of the three possible values:i ∈ {n − δ, n, n + δ} , where n is the current time moment andδ is the time lag; δ ∈ N.The perfect matrix of Lagrange differences is lexicographicallybalanced - the number of symbols of x and y in theexpressions of all elements of the matrix is the same.The perfect matrix of Lagrange differences is balanced inrespect of time - the number of indices with subscripts −δ and+δ in the expressions of all elements of the matrix is the same.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
An example of the perfect matrix of Lagrange differences
1
(0 xn+δ − yn+δ
xn−δ − yn−δ yn
)2
(yn xn+δ − yn+δ
xn−δ − yn−δ yn
)3
(xn xn+δ − yn+δ
xn+δ − yn+δ yn
)4
(xn xn+δ − yn+δ
xn−δ − yn−δ yn
)
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
The classification of perfect matrices of Lagrangedifferences
There are 18 types of perfect matrices of Lagrange differences.One of the main characteristics of any square matrix are itseigenvalues (or the spectrum of the matrix).It is clear that elements on the diagonal and the anti-diagonalcan be interchanged and their signs can be switched withoutaffecting the maximal absolute eigenvalue.It means that every graphical representation of a matrix canbe a result of 24 = 16 distinct perfect matrices of Lagrangedifferences in terms of the maximal absolute eigenvalue.Overall, 18 different representations yield 288 distinct perfectmatrices of Lagrange differences.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
The construction of the optimization problem
We have to select a scalar parameter representing localrelationships described by perfect Lagrange matrices.The parameters under the consideration are: max |λ|, min |λ|,structural coefficient str = max |λ| /min |λ|, and discriminantdsk = (a11 − a22)2 + 4a12a21.The values of the particular parameter computed for theperfect matrix of Lagrange differences constructed from theRR (x-time series) and JT (y-time series) are denoted as pk,k = 1, 2, . . . , 1199.Let us denote the scaled inverse values of the load as lk,k = 1, 2, . . . , 1199.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Load target function values for the person #1
Opposite values of the load are normed to the y-axis and fit it to the interval[0; 1]. Therefore, 0W load is mapped to 1; 50W - to 0.8; 100W - to 0.6; 150W
- to 0.4; 200W - to 0.2 and 250W - to 0. In other words, we construct thetarget function - and the parameter representing local relationships described byperfect Lagrange matrices should follow this target function as close as possible.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
The construction of the optimization problem
The optimization problem is formulated as follows - minimizeRMSE (root mean square error) between lk and pk by selecting themost appropriate parameter:
argmin
√√√√ 1
N
N∑k=1
(lk − pk)2
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
The averaged RMSE values for all perfect matrices
We fix max |λ| as the best parameter representing the local relationshipsdescribed by perfect Lagrange matrices.
Note: We do not preselect a single perfect matrix of Lagrange differences - wedo average RSME for all 18 perfect matrices.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Internal and external smoothing
Computational experiments are continued withmax
∣∣∣λ(L(s)δ,k
)∣∣∣, where L(s)δ,k denotes a perfect matrix of
Lagrange differences centered around time moment k.We introduce internal smoothing combined with movingaveraging (MA):
pk (s,△,m) =1
(2m + 1) △
k+m∑j=k−m
△∑δ=1
max∣∣∣λ(L(s)
δ,k
)∣∣∣Here △ is the radius of internal smoothing and m is the radius ofexternal smoothing.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Internal and external smoothing
It was observed that a moderate internal and external smoothing(m =△= 3) helps to minimize RMSE for almost all perfectmatrices of Lagrange differences. In fact, the average value ofRMSE (averaged for s = 1, 2, . . . , 18) is minimal at m =△= 3.Therefore, m =△= 3 is fixed for further computations.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Load target function and pk (3, 3, 3) for the person #1
The lowest RMSE value was achieved at m =△= 3 and s = 3. This figureshows the variation of optimal parameter pk (3, 3, 3) (normalized into interval[0; 1], reconstructed using the optimal matrix of Lagrange differences and
optimal smoothing.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Phase plane of optimal parameter series pk during the load
Time-delay embedding (τ = 4RR intervals) is used tovisualize temporarystabilization and thesubsequent loss of the stabilityof attractors in the phaseplane.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Why?
A natural question is whether such complex computations arerequired - maybe straightforward visualization of RR and JJ in a
phase plane would reveal similar relationships?
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Data retrievalPerfect matrices of Lagrange differencesVisualization of the load and the recovery processes
Phase plane of the RR and JT interval series
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
Summary
Summary
Novel visualization technique of relationships between RR andJT intervals is proposed.An important aspect of the presented computationaltechnique is that an optimal set of parameters (s,m,△) canbe reconstructed for every individual person.This technique is able to reveal the complexity of theself-organization of the heart system during the load and therecovery processes.A physician can observe the “collapse of complexity” at theend of the bicycle stress test, temporary stabilization oftransient attractors during the load, rich dynamical behaviorof the heart system during the recovery process.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
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Future research
We doubt if it is possible to quantity such complex transientdynamics by a single biomarker. However, development ofpattern classification algorithms for automatic analysis oftransient orbits generated by p (s,△,m) (and relating thesepatterns to novel markers for early disease diagnosis) remainsa definite objective of future research.
Palevičius et al. Perfect matrices of Lagrange differences for the CVS
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IntroductionMethods and results
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Thank you for your attention!
Palevičius et al. Perfect matrices of Lagrange differences for the CVS