ECG Analysis

Pattern recognition basics I: time domain methods

PATTERN: time domain

Example: Automatic detection of pathological ECG-pattern: depressed ST-segment: e. g. subendocardial ischemia

classification

normal beat infarction

pattern recognition

normal template

(feature extraction)

STD

QRSD

TD

TA

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Pattern recognition: Detection of characteristic points in time series

detection of characteristic points

QRS detection

detection of the start of P,Q,T

detection of the end of P,S,T

detection of max. pressure (systoly)

detection of the pulse wave

detection of the start of a pulse wave

detection of the start of a sound

detection of the start of inspiration

Generation of timeseries of the intervals

betweencharacteristic points

Detection of characteristic

patterns

etc...

PATTERN: time domain

Preprocessingx

decision rulexp pos

filtering

derivation

non-linear function

correlation

time frequency transforms

other transforms

...

threshold

template correlation

triangulation

extrapolation

curve fitting

(1) Determination of the occurrence and position of a wave

PATTERN: time domain

Detection of characteristic waves

Aim: detect the position of a wave which occurs repetitively in a biomedical signal.

PATTERN: time domain (1.1) Detection by threshold

Decision rule: The wave is considered as detected if it exceeds a certain threshold.

Preprocessing condition: all other waves, which are of no interest must remain below the threshold

Determination of the position: The position must be assigned to a characteristic point (fiducial point) among those which exceed the threshold (maximum of the wave, maximum of an interpolation of the wave, midpoint between the two threshold crossings, etc...)

possible fiducial points

PATTERN: time domain

wave

slow, elevated amplitude

fast, elevated amplitude

low amplitude

Elimination of baseline-offset

lowpass filtering

bandpass filtering

Detection of event-correlated other characteristic waves,elimination of this ‘trigger wave’

Thresholding methods usually require preprocessing:

PATTERN: time domain Example 1: Detection of the pulse wave

The elimination of the baseline-offset (maybe also detrending)allows the utilization of a constant threshold

raw data

Step 1: Detrending, baseline removal

PATTERN: time domain

Reduction

ofnoiselow

pasfiltering

10 Hz

Step2: Noise elimination by lowpass filtering

PATTERN: time domain

Be k={k(1)....k(M)} the series of positions for which holds x(k(i))>threshold (∀ i∈(1,M))

We need two series: The one with upstroke threshold crossings and the one with downstroke threshold crossings. In this way we can determine the wave location from those two characteristic points.

k=find(signal>threshold) %According MATLAB command

Step3: Thresholding

threshold

PATTERN: time domain

function [kpp,kpn]=prok(k,sep)%k: indices of the signal values above the threshold%sep: separation condition between waves

kpp(1)=k(1); %the first entry of k is the first pointkpn=[];for i=1:length(k)-1

if k(i+1)-k(i)>sep %two waves must be separated at least by sepkpp=[kpp,k(i+1)]; %The point before sep is the last one of thekpn=[kpn,k(i)]; %wave, the next one is the starting point of the next

waveend

endkpn=[kpn,k(length(k))]; %The last point of k is the last one of the last wave

Now we have to assign a position to the wave. The alignment point for wave i lies between kpp(i) and kpn(i)

MATLAB implementation for the detection of a pulse wave

PATTERN: time domain

kpn(i)

wave(kpp(i):kpn(i))

kpp(i)pos(i)?

for i=1:length(kpp)[m,n(i)]=max(wave(kpp(i):kpn(i)));pos1(i)=kpp(i)+n(i)-1;

end

Possibility 1: maximum of the wave

Possibility 2: midpoint in between kpp(i) and kpn(i)

pos2=(kpp+kpn)/2;

Possibility 2 is only valid for symmetric waves (not true in the shown case)

Step 4: Assignment of the fiducial point PATTERN: time domain

wave=wave(:); %Convert wave to column vectorfor i=1:length(pos1)-1

wavel=[wavel (wave(pos1(i)-matr:pos1(i)+matl))];end

mean(wave)

averaging of the signal

time series of intervals

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iem

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s)

diff(pos)/fs

Step 5: Further processingExamples: averaging and shape analysis,interval analysis

intervals before and after fiducial point

PATTERN: time domainfunction [pos,sal]=prondpul(wave,facd,fs,matr,madel)%wave: input signal%facd: threshold factor(between 0 y 1)%fs: sampling frequency%matr: samples backwards for alignment of waves%madel: samples forward for alignment of waves%pos: vector with the positions (maxima) of the waves%sal: wave matrix, the columns are detected individual wavessal=[];wavef=lbmed(wave,40*fs/1000,fs); %baseline offset elimination[b,c]=butter(2,20/fs); %filter coefficientswavef2=filtfilt(b,c,wavef); %bidirectional filteringk=find(wavef2>max(wavef2)*facd); %calculate positions above threshold[kpp,kpn]=prok(k,200*fs/1000); %starting and end points of wavesfor i=1:length(kpp)

[m,n(i)]=max(wavef2(kpp(i):kpn(i)));endpos=kpp+n-1; %Calculate positionif pos(length(pos))+madel>length(wave)

pos=pos(1:length(pos)-1); %avoid errors endwavef2=wavef2(:);for i=1:length(pos)

sal=[sal wavef2(pos(i)-matr:pos(i)+madel)]; %assemble wave matrixend DEMO ECG-P-extraction : synchronous averaging.m

Complete MATLAB-example for wave extraction PATTERN: time domain

Example 2: QRS - detectionfunction r=detqrs(y,fs,fac)% detection of qrs% r= position of R (samples)% y: ECG% fs: sampling frequency (Hz)% fac: empirical factor (between 0,7 y 1,5) %Use 0,7 if obtained RR has many false positives[b,a]=butter(2, [10*2 30*2]/fs); %Bandpass filtering between 10 and 30 Hzyf=(filtfilt(b,a,y)); %zero phase filteringyfd=(diff(yf)); %Differentiation[b,a]=butter(2,30/fs); %lowpass filtering at 15 Hzyfd=filtfilt(b,a,yfd); %zero phase filteringyfds=(round(yfd/max(yfd)*fac)); %normalizing and thresholding signalyfdds=diff(yfds); %nonzero if thresh. signal changesk=find(yfdds); %find changes in the tresh signalsep_time=0.6; %min. expect. interval [s]sep=sep_time*fs; %convert to samples

[kpp,kpn]=prok(k,sep); %reference points for wavelets

for i=1:length(kpp)[m,n(i)]=max(y(kpp(i):kpn(i))); %calc of position (maximum)

endr=kpp+n-1;

PATTERN: time domain

Illustration of the QRS-detection original

bandpass filtered (yf)

PATTERN: time domain

lowpass filtered (yf)

thresholding (yfds)

PATTERN: time domain

differentiation (yfdds)

detected R peaks

PATTERN: time domain

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number of beat

RR

(s)

Possible application: analysis of the time series of RR-intervals

PATTERN: time domain Heartrate: sampling intervals are not equidistant

4456 4458 4460 4462 4464 4466 4468 4470 4472 4474

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raw ECG with heavy 50 Hz interference

raw ECG signal, contaminated withnoise and interferences

classical R-peak-detector with integrated filters

R-peak

50 Hz interference

T-wave (comp. large)

50-Hz-filter

z. B. fS = 800 HzH(z) = 1 + z-8

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gain

of

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filt

er

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phas

e of

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pass

filt

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This filter eliminates all harmonics of 50 Hz

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ECG after 50 Hz removal

filtered ECG signal, contaminated withnoise

R-peak

noise

T-wave (comp. large)

FIR-lowpass,linear phase

z. B. cutoff ~ 100 HzH(z) = 1 + 4z-1 + 6z-2 + 4z-3 + 1z-4

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PATTERN: time domain

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differentiated and low-pass filtered ECG after 50 Hz removal

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low-pass filtered ECG after 50 Hz removal

filtered ECG signalDifferentiation for determinationof the maxima: search for zero-crossings

discrete differentiatory[n]= x[n] - x[n-1]

H(z) = 1 - z-1

T-wave (comp. large)R-peak

zero crossing

PATTERN: time domain

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differentiated and low-pass filtered ECG after 50 Hz removal

if neither a) nor b)

criterion c):no crossing of S2within 160 ms after P3(time of 2nd crossingwith S1)

criterion a):no crossing of S2within 160 ms after P1here violated

160 ms

P1

criterion b):no crossing of S1within 160 ms after P2here fulfilled

P2

160 ms

S1

S2

recognition of a valid QRS-complex

Determine first zero-crossing after P 1

P3

PATTERN: time domain

Example 3: P-wave detection

Problem: Frequency components of QRS and P-wave overlap (if the threshold is lowered to detect the P-wave, also the QRS complex is detected)

Solution: Detection of the QRS and then search backwards for finding the P-wave

possearchwindow

QRS

P

PATTERN: time domain function [ppos,rpos]=searchp(ecg,fs,trev,tw, factor1)%ecg: input ecg%fs: sampling frequency%trev: time backwards from QRS which defines end of the search window%tw: duration of search window%factor1:

rpos=detqrs(ecg_act,fs,factor1); %Detection of R-peak

if rpos(1)-tw-trev<1rpos=rpos(2:length(rpos)); %Exclusion of first QRS if period

end;

%------------filtering and baseline elimination--------------[b,a]=butter(12,30/fs); %15Hz lowpass filterecg_act2=filtfilt(b,a,ecg_act);

%ecg_act3=ecg_act2-medfilt1(ecg_act2,80); %baseline elimination with median filter

[b,a]=butter(2,6/fs,'high'); %baseline elimination with 3Hz highpass filter

ecg_act3=filtfilt(b,a,ecg_act2);%----------------------------------------------

for i=1:length(rpos) %set window and find maximumx=ecg_act3(rpos(i)-tw-trev:rpos(i)-trev);[m,n]=max(x);ppos(i)=n+rpos(i)-tw-trev-1;

end

MATLAB implementation: P-wave detectionPATTERN: time domain

ppos(1)

Illustration of the result: detected P-wave peaksPATTERN: time domain

Selection of a template

similarity functionSearch for the maximumof the similarity

excessive computationtime in case of large

data records

requires zero meanand lowpass

noise reduction

The waves are aligned in such a way that the similarity function

reaches a maximum

Realization of a wave with position marker assigned by the user (e. g. maximum)

e. g. cross correlation same as in threshold detection

(1.2) Detection by template correlation

PATTERN: time domain

Original wave

Template correlation (here with the R-wave-template)

Cross correlation function

Thresholding

Maxima of Xcorr represent exact locations of R-waves

PATTERN: time domain

function pp=pospl(x,plan,fs)%x: input sequence%plan: template%fs: sampling frequency%pp: location indices of the waves similar to the template

x=x-mean(x); %Elimination of DC components

xc=xcorr(plan,x); %Cross correlation

xc=xc(1:length(xc)/2); %Elimination of the second half (zeroes)

k=find(xc>0.7*max(xc)); %Detect approximate locations of high similarity

[kpp,kpn]=prok(k,300*fs/1000); %determine upstroke and downastroke crossingsfor i=1:length(kpp)

[m,n(i)]=max(xc(kpp(i):kpn(i))); %Detect indices for maximum similarityendpp=kpp+n-1+fix(length(plan)/2); %refer detected positions to positions

in original signal

MATLAB implementation

PATTERN: time domain

The template method is more robust versus noise than a threshold method but needs much more operations (computation time)

Previous detection by threshold

correct the position by template

select window which includes the wave

Select template

The shorter the window the lower the computational cost but, on the other handthe worse the possible correction of the wave’s position

PATTERN: time domainMATLAB-example: correlation with template from the signal itself

function rcor=cdet(r,c1,pref,lba,lfor)%r: approximate wave location, c1: signal%pref: index of the wave which should be taken as template,%lba: samples backwards (window), lfor: samples forwards%rcor: corrected wave position indexini=1;while r(ini)-lba<1

rcor(ini)=r(ini);ini=ini+1;endfin=length(r);while r(fin)+lfor>length(c1)

rcor(fin)=r(fin);fin=fin-1;endqref=c1(r(pref+ini)-lba:r(pref+ini)+lfor); %select templatefor i=ini:fin

qi=c1(r(i)-lba:r(i)+lfor); %select windowx=xcorr((detrend((qi))),detrend((qref)),'coeff'); %correlation[m,n]=max(x); %maximum similarity

rcor(i)=r(i)+length(qi)-n; %update position

end

Illustration: improved P-wave location in ECG

QRS detection

cdet.m

x Detection of P wave(threshold)

searchp.m

r Correctionof P wave(template)

p pc

PATTERN: time domain

Differences in the location by threshold and by template correlation, refinement of the location

Assignment of the positionby wave characteristics(e. g. maximum)

refin

ement

Position determinantionwith a global template

Threshold detection

Detection (correction) by template

alignment by thresholdand maximumtemplate

sample wave

alignmentby templatetemplate

sample wave

PATTERN: time domain

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Application 1: time series of the P-R-interval

beat number

P-R interval

(ms)

PATTERN: time domainMATLAB-Example

Application 2: Synchronized averaging

If we can assume that a biosignal contains repetitive patternswhich stem from the same generating process (mostly assumed as a random process), The SNR can be improved by averaging of subsequent patterns. Example: ECG, evoked responses

example: average P-wavefor template generation

mean P-wave

Classical averaging requires the following conditions to be met:•accurate detection of a fiducial point (alignment)•stationarity of the signal and of the noise•no correlation between signal and noise

Inaccuracies in the alignment (‚trigger jitter‘) cause additional and usually unwanted low-pass filtering of the signal. Assume M signalsxm, m = 1... M to be averaged whereby each realization xm is derivedfrom a prototype signal s by delaying it by a random number θm.

∑∑==

+==M

mm

M

mm ks

Mkx

Mkx

11

][1

][1

][ θ

Fourier transform of a single realization yields a spectrum

mjfm efSfX θπ2)()( =

The average spectrum has the expectation:

{ } { } { }mm jfjf eEfSefSEfXE θπθπ 22 )()()( ==

This means that the signal S(f) is filtered with a filter function

{ } θθ θπθπ depeEfH jfjf m ∫∞

∞−== 22 )()(

p(θ) is the pdf of θ and consequently H(f) is the FT of p(θ), a functionwhich is known as the ‚characteristic function‘ in random theory. For a normal distribution of the jitter with mean 0 and STD σθ H(f) is:

2)(2)( θθσπfefH −=

The 3 dB cutoff frequency isθσ

1325.0=cf

If the distribution is uniform between 0 and c we obtain

θσ13.045.0 ==

cfc

The cutoff frequency does not depend much on the type of distribution, the only important parameter is the STD of the jitter.

Example: Assume a sampling frequency of 1000 Hz we obtain a cutoff frequency of 450 Hz which lies below the Nyquist frequencyof 500 Hz !

In practice the sampling interval itself introduces a minimum triggerjitter which is uniformly distributed between 0 and 1/fs. Hence weobtain a ‚natural‘ cutoff frequency of

sc ff 45.0=

Implications: If for the alignment of a wave (e. g. the P-wave) thefiducial point of another, time-locked wave is used, the variabilityof the delay between both waves enters into the trigger jitter. Example: alignment of P-waves with the R-peak as fiducial point � variability of the P_R interval affects P-wave averaging.

Hence the algorithm for the determinantion of the fiducial point mustbe chosen such as to minimize the trigger jitter. The best performance is obtained with template correlation methods (matchedfilter) but this holds strictly only for Gaussian white noise.

Example: alignment of EPs with a certain scatter in the latencytimes

The latency time can vary between different recurrences (τ01, τ02, τ03).In contrast to ECG (natural trigger, forms part of the signal, high signal energy of the QRS-complex) a good template is not availablebecause the individual realizations are too noisy� iterative algorithm

Step 1: calculate a zeroth approximation of the template

∑=

=N

ii nx

Nn

1

0 ][1

][ˆ0S

step 2: cross correlate each 0x i with and estimate the 0τ0i0Sstep 3: shift each 0xi[n] � 1xi[n]= 0xi[n-0τ0i/ts ]

τ01

0x1

τ020x2

...

τ030x3

step 4: calculate new estimate from the 1xi[n]1S ∑=

=N

ii nx

Nn

1

1 ][1

][ˆ1S

iterate

single realization of an EP-recording

result after 200 iterations

result after 800 iterations

Increase of the SNR with thenumber of iterations

For a comparison of some alignment methods see: Jané, R. et al. (1991). Alignmentmethods for averaging of highresolution cardiac signals: a comparative study of performance. IEEE Trans Biomed Eng 38: 571-579.

Alternatively the alignment can also be achieved from the analysisof the cross spectrum between template and signal. If the templatey and the signal x deviate only by an attenuation factor α then weobtain:

02)( τπα jfxxxy eGfG =

Hence the delay time τ0 (misalignment) of the signal with respect to the template is visible as a phase shift and can be evaluated at every frequency

ff

fπ

ϕτ2

)()(0 =

Additionally a group delay can be calculated as the derivative of thephase with respect to the frequency. This method, however, is verysensitive to noise, especially in those parts of the spectrum, wherethe noise predominates over the signal. In order to reduce theuncertainty of this estimator a weighted averaged delay can becalulated:

( )∫

∫∞

∞

=o xy

o xyxy

dffGf

dfffGf

)(2

)()(2ˆ

20

π

ϕπτ

The phase readings are weighted with the modulus of the cross spectrum in order to minimize the effect of contributions from thoseparts of the spectrum which contain low signal energy (and henceessentially noise !)

A third estimator for the misalignment can be built with multichannelAR-methods as estimators for the cross spectra. This method will not be treated in this lecture.

Performance of correlation (Rxy), cross spectral estimator (CCG) and AR method for the P-wave alignment in ECG signals

Correlation versuscross spectrum: no pronounced difference.The jitter increasesconsiderably above a certain noise level.

noise [mVeff]

noise [mVeff]

AR-method: The jitter ishigher than that of Rxy and CCG at low noise levels, butminor at high noise levels. Results are shown for 3 different signal bandwidths(25, 50 100 Hz).

Application 3: Detection of abnormal beats in ECG

This algorithm is a special case of template correlation. Here insteadof the correlation function the correlation coefficient ρ is applied as a measure of similarity. [SOURCE: SCHREIER 1998]

step 1: QRS-detection

step 3: calculate the ρ between the template and the next x i of the seriesof beats

][][ˆ 0 nxn i=0S

step 2: chose one beat which is considered as normal and create a template set iteration counter i = 10S

step 4: if ρ >= ρtresh then accept x i as normal beat and update the template(weighted averaging):

∑

∑

=

−

=

+= i

kk

ii

i

kk

0

1

0

ˆˆ

ρ

ρρ xSS

1i-

ieach beat is weightedaccording to its correlationwith the template

iterate for i = 1 to N, N number of beatsMATLAB-Example

Alignment in case of nonstationary noise

The noise reduction approaches only the 1/N rule if the noise isstationary. In practical cases (e. g. ECG) this condition is notnecessarily met. The reasons can be physiological (respiration, movements, electrode artifacts). Because of this the improvement in SNR by averaging can be much worse than 1/N. A very prominent method for considering non-stationarities is weighted averaging, theweights depending inversely on the variance of the noise in therespective signal segment:

∑ ∑= =

==M

m

M

mmmm wkxw

Mkx

1 1

1,][1

][

∑=

= M

i i

mmw

12

2 111

σσ

with

segments with higher error variance contribute less to the final result

(2) Determination of the starting and end points of waves

Pi Pf Qi Sf Ti Tf

start ofpuls wave

isoelectric segment

wave

tstart end

e. g.: ECG:

We need algorithms for the classification which samples form part of the isoelectricsegment and which ones belong to the wave. The start or end of the wave would then be the separation point for samples which enter into the classification and which one not.Common problems: noise, interferences, baseline drifts

Detection of start and end

threshold in the derivative

intersection of straight lines

triangulation

wavelet analysis

time frequency analysis

require theprevious detection

of the wave

(2.1) Detection by thresholding the derivative

Principle: An isoelectric segment has (theoretically) a derivative zero (at least close to zero). In contrast the wave contributes with a significant derivative, except around its extrema

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derivative

Procedure: start at the maximum of the wave and search for the maximum derivative (in modulus) towards the start (end) of the wave. Then follow this direction until the absolute derivative falls below a certain fraction of the maximum.

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-0.09

end

wav

ede

rivat

ive

filtering of signal

reduction of noiseand interference

(lowpass)

baseline driftelimination(highpass)

detection of maximain the wave

by threshold or template

windowing

must include that part of the wave

which lies betweenmaximum and

isoelectric segment

detection of startand end

the wave must bemonotonic for correct functionthe lower the thresholdthe farther away is the separation pointfrom the wave’s maximum

schematic of the processing flow

function pin=inic(x,pma,l,fs,pen,fac);%x: input signal%pma: positions of the maxima%l: window length in ms%fs: sampling frequency%pen: 1 if wave is positive, -1 else%fac: threshold factor%pin: starting point positions of the waves

l=l*fs/1000; %conversion ms � samplesfor i=1:length(pma) %from first to last detected wavexmu=x(pma(i)-l:pma(i)); %select a window of the wavexmud=diff(xmu); %differentiate[m,n]=max(xmud*pen); %find maximum of derivativexmud=xmud(1:n); %cut the windowk=find(pen*xmud<m/fac); %find points below thresholdk=k(length(k)); %the separation point lies at the crossingpin(i)=pma(i)-l+k-1; %calculate positionend

MATLAB implementation, find start of wave

function pfi=final(x,pma,l,fs,pen,fac);%x: input signal%pma: positions of the maxima%l: window length in ms%fs: sampling frequency%pen: 1 if wave is positive, -1 else%fac: threshold factor%pfi: end point positions of the waves

l=l*fs/1000; %conversion ms � samplesfor i=1:length(pma) %from first to last detected wavexmu=x(pma(i):pma(i)+l); %select a window of the wavexmud=diff(xmu); %differentiate[m,n]=max(xmud*pen); %find maximum of derivativexmud=xmud(n:length(xmud)); %cut the windowk=find(pen*xmud<m/fac); %find points below thresholdk=k(1); %the separation point lies at the crossingpfi(i)=pma(i)+n+k-1; %calculate positionend

MATLAB implementation, find end of wavePATTERN: time domain

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pin=inic(signal,202,100,1000,1,5)» pin =143pfi=final(signal,202,length(signal)-203,1000,-1,5)» pfi =249

wave duration: 249-143=106 ms

[m,n]=max(signal)m =1219n =265» pin=inic(signal,n,100,1000,1,5)pin =181» pfi=final(signal,530,length(signal)-531,1000,-1,5)pfi =645wave duration: 645-181= 464 ms

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reference point for calculating the end of the wave

Result thresholdingPATTERN: time domain

(2.2) Detection by the intersection of tangents

Principle: the starting and end point is determined by the intersection of 2 straight lines. One of them is a tangent of the wave, the other one is a linear approximation of the isoelectric segment.

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Result depends on the locatopn of the tangent

PATTERN: time domain

Proceeding: Starting from the maximum slope of the wave a point of the wave is selectet at which the slope is a certain fraction of the maximum slope. Then the tangent in this point is calculated. On the other hand another straight line is adjusted to the isoelectric segment from several recurrecnes (regression line with minimum quadratic error criterion). Then the intersection of both lines is carried out.

prerequisites: •Same preprocessing as for threshold detection of waves•the respective wave segment must be monotone

problem: the position of the intersection point depends on the slope� flatter waves yield higher uncertainty

PATTERN: time domainfunction pin=inicr(x,pos,l,fs,fac,pen,miso)%x: input signal%pma: positions of the maxima%l: window length in ms%fs: sampling frequency%pen: 1 if wave is positive, -1 else%fac: threshold factor%pin: starting point positions of the wavesl=l*fs/1000; %ms � samplesfor i=1:length(pos) % from first to last detected wave

xm=x(pos(i)-l:pos(i)); %signal windowxmd=pen*diff(xm); %calculate derivative[m,n]=max(xmd); %maximum of the derivativexmd=xmd(1:n); %second windowk=find(xmd<m/fac); %find threshold of derivativek=k(length(k));r1=xmd(k)*pen*((1:length(xm))-k)+xm(k); %tangent on the wavexiso=xm(1:miso); %window for the isoelectric segmentxiso=xiso(:)';p=polyfit((1:miso),xiso,1); %estimate regression liner2=polyval(p,(1:length(xm))); %estimate regression linek=find(r2<r1); %find intersectionk=k(1);pin(i)=pos(i)-l+k-1; %calculate position

end

MATLAB implementation, find start of wave PATTERN: time domain

function pfin=finalr(x,pos,l,fs,fac,pen,miso)%x: input signal%pma: positions of the maxima%l: window length in ms%fs: sampling frequency%pen: 1 if wave is positive, -1 else%fac: threshold factor%pfin: end point positions of the wavesl=l*fs/1000; %ms � samplesfor i=1:length(pos) % from first to last detected wave

xm=x(pos(i):pos(i)+l); %signal windowxmd=pen*diff(xm); %calculate derivative[m,n]=max(xmd); % maximum of the derivativexmd=xmd(n:length(xmd)); %second windowk=find(xmd<m/fac); %locate thresholdk=k(1); %find tangent pointr1=xmd(k)*pen*((1:length(xm))-(k+n-1))+xm(k+n-1); %tangentxiso=xm(length(xm)-miso:length(xm)); %isoelectric segmentxiso=xiso(:)';p=polyfit(length(xm)-miso:length(xm),xiso,1); %estimate regression liner2=polyval(p,(1:length(xm))); %estimate regression linek=find(r2>r1); %Intersect

k=k(1); %intersection pointpfin(i)=k+pos(i); %Calculate position

end

MATLAB implementation, find end of wave PATTERN: time domain

0 50 100 150 200 250 300

pin=inicr(mu,187,186,1000,2,1,20)» pin =132pfin=finalr(mu,187,length(mu)-188,1000,2,-1,20)» pfin =240wave duration: 108 ms

Result intersectionPATTERN: time domain

(2.3) Detection by triangulation

Principle: The area which is determined by 3 points (one on the wave, the second in the isoelectric segment and the third in between) is maximum if the third point is located at the beginning of the wave

0 50 100 150 200 250 300-8

-6

-4

-2

0

2

4

6

8

10

A1A2

A1 < A2

PATTERN: time domain

Proceeding: Select an analysis window (which most probably contains the third point) the ends of which are formed by one point of the isoelectricsegment (with constant distance to the maximum of the wave) and the maximum of the slope of the wave.

maximum area

area for different locations

max !

PATTERN: time domain

function [pfin,are]=fintr(x,pos,l,fs,pen)%x: input signal%pos: positions of the maxima%l: window length between maximum and isoelectric segment in ms%fs: sampling frequency%pen: 1 if wave is positive, -1 else%fac: threshold factor%pfin: end position of wave%are: triangulation area

l=l*fs/1000; %ms � samplesfor i=1:length(pos) %from first to lastxm=x(pos(i):pos(i)+l); %windowingxd=diff(xm)*pen; %differentiate[m,n]=max(xd); %maximum slopexm=xm(n:length(xm)); %new window (from isoelectric point to max. slope)for j=1:length(xm)are(j)=abs(0.5*(j*(-xm(length(xm))+xm(j))-(-j+length(xm))*(xm(1)-xm(j))));end %calculate area[m,nn]=max(are); %maximum areapfin(i)=nn+n-1+pos(i); %wave positionend

MATLAB implementation, find start of wave PATTERN: time domain

1340 1360 1380 1400 1420 1440 1460 1480 1500 1520 1540

pini=initr(onda,1450,150,1000,1)>>pini =1403pfin=fintr(ecgf,1450,90,1000,-1)>>pfin = 1516

wave duration: 113 ms

Result triangulation

The triangulation algorithm is very robust

PATTERN: time domain

Conclusions

algorithms for thedetection of start

and end

localization of the wave

knowledge of an appropriate analysis window

knowledge about the wave shape

previous lowpass filtering required

detection by threshold or

intersection

the result depends on anempirically chosen parameter

Deteccion bytriangulation

the result depends only on thetwo (mostly uncritical) end

points of the analysis window

PATTERN: time domain

Methodology of HRV-analysis(source: www.hrvcongress.org/courses)

HRV ANA

• Analysis in the time domain (Time domain methods)

• Analysis in the frequency domain (Spectral domain methods)

• non-linear methods

• mathematical modelling

(1) Analysis in the time domain(Time domain methods)

Basis of the analysis is either the HR in any time instant(‚instantaneous HR‘) or the interval between two subsequent QRS complexes of normal heart actions (‚Normal-normal‘, abbrev. NN). For this purpose a detection of the R-peaks must be carried out first. Then a series of inter-beat intervals is generated.

(1.1) Statistical methods

The measures obtained after analyses of a series of instantaneous heart rates or NN intervals can be divided into 2 classes:

1. those derived from direct measurements of the instantaneousheart rate or NN intervals

2. those derived from the differences between NN intervals

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The following indices have been defined so far:

• Mean heart rate (HR, int/min)• Mean NN interval (mNN or mRR, ms)• Standard deviation of the NN interval (sdNN or sdRR, ms) - the square root of variance between the NN intervals. Since variance is mathematically equal to total power of spectral analysis, sdNN reflects all the cyclic components responsible for variability in theperiod of recording. • Standard deviation of the average NN interval calculated o ver 5-min periods withinthe 24 - hours recording (SDANN, ms). This parameter is an estimate of the changes in heart rate due to cycles longer than 5 minutes• Mean of the standard deviations of all NN intervals for all 5 min segments of entirerecording (SDNN index, ms) . This index reflects the variability due to cycles shorter than 5 minutes• The square root of the mean squared differences of success ive NN intervals(RMSSD, ms).• The number of pairs of adjacent NN intervals differing by more th an 50 ms in theentire recording (NN50) . NN50 count divided by the total number of all NN intervals(pNN50, %)

The last three measurements reflect high frequency variations in the HRV and thus are highly correlated.

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Frequently applied index: RMSSD

RMSSD: Root Mean Square of Successive RR-

DifferencesRR1 RR2 RR3 RR4 RR5 .......

D1 D2 D3 D4 D5 .......

N

DN

ii∑

= =1

2

RMSSD

„Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology“, Circulation 93, pp: 1043-1065, 1996

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(1.2) Geometrical Methods

Conversion of the series of NN intervals into geometric patterns:

(a) interval tachogram is a graph of the NN intervals complexes

time [min]0 3

60

80

HR

„Tachogram“

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(b) scaterogram: Abscissa of the point is the length of theprevious NN interval. Ordinate of the point it the length of thesubsequent NN interval. The decreased HRV is reflected in a greater density and smaller size of the cloud. The type of thepoint distribution is determined by origin of QRS complexes.

(2) Spectral Domain Methods

The important characteristics of the spectrum are the power density of the spectrum (power spectral density, PSD) and the powers of its separate zones.

time [min]0 3

60

80

HR

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

20

40

60

80

100

120

140

160

PSD [n.u.]

LF HF

frequency [Hz]

„Tachogram“

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VLF

typical HRV-spectrum for short-term analysis

HF

LF

4 spectral zones are distinguished:• ULF, ultra low frequencies, <0,003 Hz• VLF, very low frequencies, 0,003 - 0,04 Hz• LF, low frequencies, 0,04 - 0,15 Hz• HF, high frequencies, 0,15 - 0,4 Hz

HRV ANA

Indices derived from short-term recordings (5 min)

• Total Power [ms2] - Variance of all NN intervals• VLF, [ms2] - power in the very low frequency range • LF, [ms2] - power in the low frequency range• LFnorm, n.u. - LF power in normalized units: • LF/(Total Power - VLF)*100 • HF, [ms2] - power in the high frequency range • HFnorm, n.u. - HF power in normalized units:• HF/(Total Power - VLF)*100 • LF/HF - ratio LF /HF

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Indices derived from long-term recordings (24 h)

• Total Power [ms2] - Variance of all NN intervals• ULF, [ms2] - power in the ultra low frequency range• VLF, [ms2] - power in the very low frequency range • LF, [ms2] - power in the low frequency range• LFnorm, n.u. - LF power in normalized units: • LF/(Total Power - VLF)*100 • HF, [ms2] - power in the high frequency range

HRV ANA Physiological interpretation of the different spectralcomponents

ULF-band: origin is unknown, however the prognosis of a sudden death according to its power is the most accurate.

VLF-band: still a matter of investigation. Most probable corre-lates: thermoregulation, renin-angiotensin system activation, changes in physical activity.

LF-band: to some extent generated by baroreceptormodulations of sympathetic and vagus nervous tone. Accordingto other references it is mostly dependent on the sympatheticactivation.

HF-band: respiratory modulation of the vagus nerve activity.

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(3) Non - Linear Methods

The stochastic indexes of the regulatory systems functioningmay be obtained by chaos analyses of the heart rate variability. Such indexes are thought to reflect the stressresistance of regulatory systems.

(4) Mathematic Modeling Methods

Mathematic modeling is a special tool that gives the possibilityto evaluate those properties of regulatory systems and processes that can't be obtained by direct measurements.Sensitivity and specificity of the effective systems to humoralsympathetic and parasympathetic influences can be obtainedwith the help of these methods.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

20

40

60

80

100

120

LF HF

LF/HF = 2,53

LF-band (0.04÷0.15 Hz): power is influenced by the sympathetic efferent activity

HF-band (0.15÷0.4 Hz): power mainly determined by the vagal efferent activity

LF/HF - typical:

LF-power

HF-powerLF/HF =

Rest

Orthostasis

Sleeping

≥≥≥≥ 1

÷÷÷÷ 20

< 1

Index of „sympatho-vagal“ balance

PSD [n.u.]

Frequency [Hz]

HRV ANA

Beispiel: Herzrate

4456 4458 4460 4462 4464 4466 4468 4470 4472 4474

0

0.5

1

1.5

2

2.5

3

Abtastung nicht äquidistant

Solution: Interpolation and resampling of the heart rate at a frequency sufficiently higher than half of the maximumfrequency of interest.(1) definition of a resampling frequency fr (recommended 4 Hz)(2) Define a local window at each sampling time tk between

tk-1 and tk+1(3) calculate a fractional R rk = fr nk/2 (see figure), e. g.

(b/I3+c/I4)/(fr/2) average beats per windowlength

HR, resampled

(4) Deconvolution with local window spectrum

1/I1 1/I2 1/I3 1/I4

t

HR [1/s]

ffr/2

original HR-spectrum warped HR-spectrum

window spectrum W

f

restauration of originalspectrum by multiplicationwith 1/W

1/W

T=2/fr

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