heart sounds, ecg & fractals. 2 ecg wave ecg wave heart sounds heart sounds audicor’s solution...

Heart Sounds, ECG & Fractals

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ECG WaveECG Wave

Heart SoundsHeart Sounds

Audicor’s SolutionAudicor’s Solution

Abnormal SoundsAbnormal Sounds

The HeartThe Heart

Fractal DimensionFractal Dimension

Sound AnalysisSound Analysis

The Heart

The heart is a 2-step mechanical The heart is a 2-step mechanical pump that is coordinated by pump that is coordinated by

precisely timed electrical impulsesprecisely timed electrical impulses..

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The ENDThe END

Fractal ResultsFractal Results

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ECG WaveECG Wave

Heart soundsHeart sounds



The HeartThe Heart



The Heart

The heart is a pulsating pump that The heart is a pulsating pump that composes of four chambers and composes of four chambers and

four heart valvesfour heart valves . .The upper chambers are the right The upper chambers are the right atrium (RA) and left atrium (LA)atrium (RA) and left atrium (LA)

The lower chambers are the right The lower chambers are the right ventricle (RV) and left ventricle ventricle (RV) and left ventricle

(LV)(LV)..

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The ENDThe END


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The HeartThe Heart



Electrophysiology of cardiac conduction

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Heart Valves

The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Events occurring during the cardiac cycle

The cardiac cycle consists of two basic The cardiac cycle consists of two basic

componentscomponents : :

A period of ventricular diastole during A period of ventricular diastole during which the ventricles are filled with which the ventricles are filled with blood.blood.

A period of ventricular systole during A period of ventricular systole during which blood is propelled out of the which blood is propelled out of the heart. heart.

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Events occurring during the cardiac cycle

Clinically, systole is taken as the Clinically, systole is taken as the interval between the first and the interval between the first and the second heart sound. second heart sound.

Diastole is considered to be the interval Diastole is considered to be the interval between second heart sound and the between second heart sound and the first heart sound.first heart sound.

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



The Electrical system

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The ENDThe END


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ECG ECG WaveWave




The HeartThe Heart



What is measured on the ECG

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Rate and rhythm of the heart.Rate and rhythm of the heart.

Evidence of heart enlargement.Evidence of heart enlargement.

Evidence of damage to the heartEvidence of damage to the heart

Impaired blood flow to the heartImpaired blood flow to the heart

Heart rhythm problemsHeart rhythm problems

Electrolyte imbalanceElectrolyte imbalance

The ENDThe END


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ECG WaveECG Wave




ElectrophysiologyElectrophysiology



What are the limitations of the ECG

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The ECG is a static pictureThe ECG is a static picture

Many heart attacks cannot be Many heart attacks cannot be detected by ECG.detected by ECG.

Many abnormal patterns on an ECG Many abnormal patterns on an ECG may be non-specific.may be non-specific.

The ECG may be normal despite the The ECG may be normal despite the presence of a cardiac conditionpresence of a cardiac condition

The ENDThe END


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ECG WaveECG Wave







ECG wave

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ECG tracings show a pattern of electrical ECG tracings show a pattern of electrical impulses that are generated by the heart.impulses that are generated by the heart.

P wave: the P wave: the sequentialsequential activation activation (depolarization) of the right and left atria(depolarization) of the right and left atria

QRS complex: right and left ventricular QRS complex: right and left ventricular depolarizationdepolarization

ST segmet: ventricular repolarization.ST segmet: ventricular repolarization.

The T wave corresponds to electrical The T wave corresponds to electrical relaxation and preparation for their next relaxation and preparation for their next muscle contraction.muscle contraction.The ENDThe END


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ECG WaveECG Wave







ECG wave

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ML ModelML Model


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ECG WaveECG Wave




The HeartThe Heart



Heart Sounds

The auscultation of the heart may reveal The auscultation of the heart may reveal different phenomena called heart sounds different phenomena called heart sounds

and murmursand murmurs..

Heart sounds are a prolonged series of Heart sounds are a prolonged series of vibrations of both high and low frequencyvibrations of both high and low frequency

The murmurs are a longer series of The murmurs are a longer series of vibrations, mostly of either high or low vibrations, mostly of either high or low

frequencyfrequency..

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Heart Sounds

The sounds heard during auscultation are The sounds heard during auscultation are called the first (S1) and second (S2) heart called the first (S1) and second (S2) heart

sounds respectively, sounds respectively, with respect to their temporal relationship, with respect to their temporal relationship,

and are systolic sounds.and are systolic sounds.

Phonocardiography often yields third (S3) and Phonocardiography often yields third (S3) and fourth (S4) heart sounds especially in children fourth (S4) heart sounds especially in children

and in cases of heart disease. These are and in cases of heart disease. These are diastolic soundsdiastolic sounds . .

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Heart Sounds Genesis

Many hypotheses have been suggested to Many hypotheses have been suggested to explain the origin of these sounds.explain the origin of these sounds.

Some being controversial at the time. Some being controversial at the time.

With the advent of echocardiography the With the advent of echocardiography the movement of intracardiac structures could movement of intracardiac structures could be monitored with virtually no time delay.be monitored with virtually no time delay.

Concerning S1, S2, and S4 these Concerning S1, S2, and S4 these controversies have largely been resolved. controversies have largely been resolved. However there still exists controversy However there still exists controversy regarding S3.regarding S3.

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The ENDThe END


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ECG MeasurementsECG Measurements







S1 & S2

S1 occurs when the mitral andS1 occurs when the mitral andtricuspid valves close at the beginning of tricuspid valves close at the beginning of

systolesystole . .

S2 results from closure of the aortic and S2 results from closure of the aortic and pulmonic valves at the end of systolepulmonic valves at the end of systole..

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The ENDThe END


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ECG WaveECG Wave


Abnormal soundsAbnormal sounds

The HeartThe Heart



S3

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low frequency soundlow frequency sound: 0 - 70 Hz: 0 - 70 Hz50% in the 0-15 Hz band50% in the 0-15 Hz band

occurring in early ventricular diastoleoccurring in early ventricular diastole

due to due to over-distention of the ventricle over-distention of the ventricle during the rapid early filling phaseduring the rapid early filling phase

occurs 0.12 – 0.20 secs after S2occurs 0.12 – 0.20 secs after S2

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The physiological cause and effect of S3

The third heart sound (S3) occurs 0.12 to 0.20 seconds after S2 in early Diastole.

Of the many proposed theories, the most likely explanation is that excessive rapid filling of the ventricle is suddenly halted, causing vibrations that are audible as S3.

Pathologic states where an S3 is encountered include anemia, thyrotoxicosis, mitral regurgitation, hypertrophic cardiomyopathy, aortic and tricuspid regurgitation and left ventricular dysfunction.

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The ENDThe END


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ECG WaveECG Wave



The HeartThe Heart



S4

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low frequency soundlow frequency sound: 0 - 70 Hz: 0 - 70 Hz

occuring at the late diastolic filling occuring at the late diastolic filling phase at when the atria contract phase at when the atria contract

Ventricles have decreased Ventricles have decreased compliance, or receive increased compliance, or receive increased

diastolic volume diastolic volume occurs just before S1 occurs just before S1

70 ms after onset of ECG P wave70 ms after onset of ECG P wave

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The physiological cause and effect of S4

The S4 occurs just before the first heart sound in the cardiac cycle.

It is produced in late diastole as a result of atrial contraction causing vibrations of the LV muscle, mitral valve apparatus, and LV blood mass.

Disease processes that produce an S4 include hypertension, aortic stenosis and regurgitation, severe mitral regurgitation, cardiomyopathy, and ischemic heart disease.

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The ENDThe END


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The physiological cause and effect of S3 & S4

In 1856, In 1856, PotainPotain first described "gallop rhythm" first described "gallop rhythm" as an audible phenomenon in which a tripling as an audible phenomenon in which a tripling or quadrupling of heart sounds resembles the or quadrupling of heart sounds resembles the

canter of a horse. canter of a horse.

That term is still used to describe a third or That term is still used to describe a third or fourth heart sound. fourth heart sound.

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The ENDThe END


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Gallops are diastolic events and seem to beGallops are diastolic events and seem to berelated to 2 periods of filling of the ventricles:related to 2 periods of filling of the ventricles:

1.1. The rapid filling phase (ventricular diastolic The rapid filling phase (ventricular diastolic gallop or S3)gallop or S3)

2.2. The presystolic filling phase related to The presystolic filling phase related to atrial systole (atrial gallop or S4) atrial systole (atrial gallop or S4)

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The ENDThe END


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Experimental evidence in both humans and Experimental evidence in both humans and animal models suggests that abnormal animal models suggests that abnormal compliance of the left ventricle is often compliance of the left ventricle is often associated with an S4 and/or a pathological associated with an S4 and/or a pathological S3. S3.

In the early diastolic phase of the cardiac In the early diastolic phase of the cardiac cycle, the left ventricle relaxes and the cycle, the left ventricle relaxes and the intraventricular blood pressure falls below intraventricular blood pressure falls below that of the left atrium. that of the left atrium.

Therefore, blood flows from the atrium into Therefore, blood flows from the atrium into the ventricle. the ventricle.

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The ENDThe END


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This continues until the intraventricular This continues until the intraventricular pressure equals the pressure in the atrium pressure equals the pressure in the atrium and the flow of blood into the ventricle and the flow of blood into the ventricle therefore stops. therefore stops.

This deceleration of the blood early in This deceleration of the blood early in diastole produces vibrations inside the diastole produces vibrations inside the ventricle, which can result in an S3 if the ventricle, which can result in an S3 if the vibrations have sufficient energy. vibrations have sufficient energy.

The steep left ventricular pressure increase The steep left ventricular pressure increase in early diastole causes a reversal of the in early diastole causes a reversal of the transmitral pressure gradient and hence a transmitral pressure gradient and hence a more rapid deceleration of inflow.more rapid deceleration of inflow.

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The ENDThe END


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Since the vibrations of S3 occur during Since the vibrations of S3 occur during deceleration of inflow, a conversion of deceleration of inflow, a conversion of kinetic into vibratory energy is likely. kinetic into vibratory energy is likely.

These vibrations are audible if transmitted These vibrations are audible if transmitted with enough intensity. with enough intensity.

The higher the inflow rate (valve The higher the inflow rate (valve regurgitation) and the steeper the rapid regurgitation) and the steeper the rapid filling wave (high filling rates), the greater filling wave (high filling rates), the greater the deceleration and more likely an S3 will the deceleration and more likely an S3 will occur. occur.

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The ENDThe END


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S3 is produced when the rapidly distending S3 is produced when the rapidly distending ventricle reaches a point when its distention ventricle reaches a point when its distention is checked by the resistance of its wall and is checked by the resistance of its wall and the ensuing vibrations are audible as the the ensuing vibrations are audible as the third heart sound.third heart sound.

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The remainder of the filling of the ventricle The remainder of the filling of the ventricle occurs late in diastole because of active occurs late in diastole because of active contraction of the atrium. contraction of the atrium.

The deceleration of the blood later in The deceleration of the blood later in diastole diastole also produces vibrations inside the also produces vibrations inside the ventricle. ventricle.

If the atrial contraction that produced the If the atrial contraction that produced the late diastolic filling was sufficiently strong late diastolic filling was sufficiently strong and the ventricle is relatively stiff, these and the ventricle is relatively stiff, these vibrations may have enough energy to vibrations may have enough energy to produce an S4. produce an S4.

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The ENDThe END


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ECG WaveECG Wave






Heart Sounds Characteristics

The ENDThe END


The HeartThe Heart

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http://www.cardiologysite.com/auscultation/html/s3_gallop.html

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ECG WaveECG Wave






Heart Sounds

The ENDThe END


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http://depts.washington.edu/physdx/heart/tech2.htmlThe HeartThe Heart

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ECG WaveECG Wave






PCG against ECG

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The HeartThe Heart

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ECG WaveECG Wave






PCG against ECG

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The HeartThe Heart

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ECG WaveECG Wave






The relationship between heart sounds and cardiac events

The ENDThe END


The HeartThe Heart

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ECG WaveECG Wave




The HeartThe Heart



How do S3 & S4 Help?

The ENDThe END


Experimental evidence suggests that Experimental evidence suggests that abnormal compliance of the left ventricle is abnormal compliance of the left ventricle is

often associated with an S4 and/or a often associated with an S4 and/or a pathological S3.pathological S3.

The S3 may be a normal finding in patients The S3 may be a normal finding in patients less than 30 years old.less than 30 years old.

However, in older patients, the S3 is usually However, in older patients, the S3 is usually evidence of impaired ability of the ventricle to evidence of impaired ability of the ventricle to

contract during systole.contract during systole.

The prevalence of the S4 increases with age The prevalence of the S4 increases with age and usually indicates an abnormal increase in and usually indicates an abnormal increase in

ventricular stiffnessventricular stiffness Lets Lets GoGo!!

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ECG WaveECG Wave




The HeartThe Heart



Clinical significance of S3

The ENDThe END


The presence of S3 may be the earliest clue The presence of S3 may be the earliest clue to left ventricular failure. to left ventricular failure.

The presence of heart disease and may offer The presence of heart disease and may offer valuable information about diagnosis, valuable information about diagnosis,

prognosis, and treatment.prognosis, and treatment.

The most useful clinical importance of S3 is in The most useful clinical importance of S3 is in detecting left-sided heart failure, especially in detecting left-sided heart failure, especially in

the early stages when other signs may be the early stages when other signs may be normal. normal.

More recently, S3 was the best predictor of More recently, S3 was the best predictor of response to digoxin in CHF patients. response to digoxin in CHF patients.

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ECG WaveECG Wave




The HeartThe Heart



Usefulness of S3, S4 & ECG in assisting early detection

The ENDThe END


Several types of cardiac disease have Several types of cardiac disease have characteristic electrical and hemodynamic characteristic electrical and hemodynamic manifestations. manifestations.

For example, acute myocardial ischemia is For example, acute myocardial ischemia is typically associated both with displacement typically associated both with displacement of the ST segments of the ECG and with of the ST segments of the ECG and with alterations of the mechanical properties of alterations of the mechanical properties of the left ventricle. the left ventricle.

The latter changes may produce pathological The latter changes may produce pathological heart sounds – S3 and/or S4. heart sounds – S3 and/or S4.

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ECG WaveECG Wave




The HeartThe Heart



S3 In Children

The ENDThe END


The genesis of S3 has been clearly The genesis of S3 has been clearly associated associated with the rapid filling phase of diastole. with the rapid filling phase of diastole.

Present work has shown that S3 occurs Present work has shown that S3 occurs earlier in the cardiac cycle with increase in earlier in the cardiac cycle with increase in age of child subjects. age of child subjects.

This supports the hypothesis that S3 is due This supports the hypothesis that S3 is due to L.V. reaching it’s elastic limit during to L.V. reaching it’s elastic limit during diastole. diastole.

This notion is supported further by the This notion is supported further by the finding of the spectral energy of S3 is finding of the spectral energy of S3 is distributed more towards the high distributed more towards the high frequency frequency of the end of the spectrum with age. of the end of the spectrum with age. Lets Lets

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ECG WaveECG Wave




The HeartThe Heart



S3 In Children

The ENDThe END


This is consistent with an increase in This is consistent with an increase in stiffness of the L.V. with age. The resonant stiffness of the L.V. with age. The resonant frequencies of L.V. increase with stiffness. frequencies of L.V. increase with stiffness.

higher frequencies are more attenuated by higher frequencies are more attenuated by passage through body tissue than lower passage through body tissue than lower

frequencies. frequencies.

As the frequency distribution of S3 is shifted As the frequency distribution of S3 is shifted to higher frequencies as the child becomes to higher frequencies as the child becomes

older, it would be expected that the energy in older, it would be expected that the energy in S3 would decrease with age. S3 would decrease with age.

Thus S3 usually disappears around Thus S3 usually disappears around adulthood, but may reoccur with cardiac adulthood, but may reoccur with cardiac

pathology. pathology. Lets Lets GoGo!!

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ECG WaveECG Wave



The HeartThe Heart



Combining ECG & Heart Sounds

Audicor uses an advanced new technology Audicor uses an advanced new technology called correlated audioelectric called correlated audioelectric

cardiography (COR)cardiography (COR) . .

This technology builds on the traditional This technology builds on the traditional findings of the standard, 12-lead resting findings of the standard, 12-lead resting ECG, augmenting it by simultaneously ECG, augmenting it by simultaneously

acquiring acoustical signals from both the acquiring acoustical signals from both the V3 and V4 lead positionsV3 and V4 lead positions..

((Two acoustic sensors replace the V3 and Two acoustic sensors replace the V3 and V4 ECG electrodes of a standard 12-Lead V4 ECG electrodes of a standard 12-Lead

ECGECG ) )

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The ENDThe END


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ECG WaveECG Wave



The HeartThe Heart




AudicorAudicor CE combines the detection of CE combines the detection of heart sounds with ECG data in order to heart sounds with ECG data in order to provide physicians with additional provide physicians with additional information that is valuable in information that is valuable in assessingassessing::

S3 and S4 heart sounds that may be S3 and S4 heart sounds that may be indicative of acute indicative of acute coronary coronary syndrome or heart failure syndrome or heart failure

Acute and prior (age-undetermined) Acute and prior (age-undetermined) myocardial infarction (MI) myocardial infarction (MI)

Ischemia Ischemia Left ventricular hypertrophy (LVH)Left ventricular hypertrophy (LVH)

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart




The S3 heart sound is often very difficult to The S3 heart sound is often very difficult to detect by auscultation due to its low detect by auscultation due to its low frequency and intensityfrequency and intensity . .

Noisy clinical environments further complicate Noisy clinical environments further complicate this difficultythis difficulty..

To improve the detection of the S3, Inovise To improve the detection of the S3, Inovise Medical, Inc. has developed AUDICOR®, a Medical, Inc. has developed AUDICOR®, a device that records and algorithmically device that records and algorithmically interprets simultaneous 12-lead ECG and interprets simultaneous 12-lead ECG and electronic cardiac sound recordingelectronic cardiac sound recording

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Audicor Heart Sound Algorithm

The AUDICOR heart sounds algorithm receives The AUDICOR heart sounds algorithm receives three synchronous inputsthree synchronous inputs : :

11 . .A standard ECG signalA standard ECG signal 22 . .Two single-channel sound signalsTwo single-channel sound signals..

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The ENDThe END


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ECG WaveECG Wave




The HeartThe Heart



Processing The Sound Data

The ENDThe END


The sound data for each channel is then The sound data for each channel is then processed by removing offsets, prescaling, and processed by removing offsets, prescaling, and filtering it into narrow frequency bands to filtering it into narrow frequency bands to optimize the detection of each S1 through S4 optimize the detection of each S1 through S4 heart sound. heart sound.

Using the ECG as a reference, the S1 and S2 Using the ECG as a reference, the S1 and S2 detection time windows are identified for each detection time windows are identified for each beat.beat. Utilizing a threshold adaptively computed from Utilizing a threshold adaptively computed from a moving window root mean square for each a moving window root mean square for each frequency band, the location of each S1 and S2 frequency band, the location of each S1 and S2 is determined within the computed detection is determined within the computed detection window.window.

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ECG WaveECG Wave




The HeartThe Heart



Detecting S3

The ENDThe END


The S3 detection time windows are located The S3 detection time windows are located using information within the ECG and the using information within the ECG and the computed position of the S2 offsetcomputed position of the S2 offset . .

The energy content is determined within the The energy content is determined within the S3 detection time windowS3 detection time window . .

Using a set of rules based on frequency and Using a set of rules based on frequency and amplitude measurements, possible S3s are amplitude measurements, possible S3s are detected within the S3 windowsdetected within the S3 windows..

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ECG WaveECG Wave




The HeartThe Heart



Detecting S4

The ENDThe END


The S4 detection time windows are located The S4 detection time windows are located based on PQ intervals and Q-wave onset based on PQ intervals and Q-wave onset positionspositions . .

Further processing on S4s is similar to that Further processing on S4s is similar to that described before for S3sdescribed before for S3s . .

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ECG WaveECG Wave




The HeartThe Heart



ECG Diagnostic Algorithm

The ENDThe END


Superior diagnostic performance up to 84% Superior diagnostic performance up to 84% more sensitive than current systems in acute more sensitive than current systems in acute MI detection, particularly in women was MI detection, particularly in women was

achieved in the following waysachieved in the following ways: : •• Developed the computerized ECG diagnosticDeveloped the computerized ECG diagnostic

algorithms using very large clinicallyalgorithms using very large clinically

correlated databases of over 100,000 ECGscorrelated databases of over 100,000 ECGs..

• •Divided the data into demographicallyDivided the data into demographically balanced learning and test sets tobalanced learning and test sets to

help ensure that it was not overtraining the help ensure that it was not overtraining the

algorithms using limited sets of data algorithms using limited sets of data..

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ECG WaveECG Wave




The HeartThe Heart




The ENDThe END


• •Accounted for differences in gender and ageAccounted for differences in gender and age , , emphasizing features that discriminateemphasizing features that discriminate

between prior and acute MIbetween prior and acute MI..

• •Avoided circularity in developing and testingAvoided circularity in developing and testing the algorithms by selecting all cases andthe algorithms by selecting all cases and

non-cases of various diseases using criterianon-cases of various diseases using criteria

independent of the ECGindependent of the ECG

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The HeartThe Heart




The ENDThe END


• •Criteria for IMI is based uponCriteria for IMI is based upon thethe relationships between portions of therelationships between portions of the vectorcardiographic (VCG) QRS loop invectorcardiographic (VCG) QRS loop in thethe

frontal plane and the corresponding portionsfrontal plane and the corresponding portions of the ECG QRS complexesof the ECG QRS complexes recorded in leadsrecorded in leads

II and IIIII and III . .

• •Commercial ECG algorithms for detection ofCommercial ECG algorithms for detection of prior myocardial infarction (MI)prior myocardial infarction (MI)

predominantly rely on QRS criteria and onpredominantly rely on QRS criteria and on established qualitative ST and T changesestablished qualitative ST and T changes . .

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The HeartThe Heart




The ENDThe END


• •Two distinct new approaches for quantifying Two distinct new approaches for quantifying ST and T changes to assist with the ST and T changes to assist with the

detection of prior MIdetection of prior MI . .

11 . .The first method uses the mean axes ofThe first method uses the mean axes of vectorcardiographic T-loops taken fromvectorcardiographic T-loops taken from

the inverse Dower transform of the 12the inverse Dower transform of the 12 - - lead ECG to indicate ischemic regions oflead ECG to indicate ischemic regions of

the left ventricular wallthe left ventricular wall . .

22 . .The second method establishes regionalThe second method establishes regional scores for residual ST elevationscores for residual ST elevation

supportive of ischemia or infarctionsupportive of ischemia or infarction..

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The HeartThe Heart




The ENDThe END


• •These 2 ST-T measures qualify borderlineThese 2 ST-T measures qualify borderline QRS infarct criteria, resulting in compositeQRS infarct criteria, resulting in composite

criteria having higher sensitivities andcriteria having higher sensitivities and

specificities than QRS criteria alonespecificities than QRS criteria alone..

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The HeartThe Heart




The ENDThe END


LVH is defined as values of the ECG leftLVH is defined as values of the ECG left ventricular mass index (LVMI) >116 g/m(2)ventricular mass index (LVMI) >116 g/m(2)

in men or >104 g/m(2) in womenin men or >104 g/m(2) in women . .

Univariate linear regression was performedUnivariate linear regression was performed separately on the male and female subjectsseparately on the male and female subjects

in the Learning Set to find all the ECGin the Learning Set to find all the ECG parameters that correlated significantly withparameters that correlated significantly with

LVMILVMI . .

Multivariate linear regression (MLR) wasMultivariate linear regression (MLR) was applied to these parameters to identify the 4applied to these parameters to identify the 4

variables for each sex that discriminatedvariables for each sex that discriminated best between the subjects with and withoutbest between the subjects with and without

LVHLVH . .Lets Lets GoGo!!

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ECG WaveECG Wave




The HeartThe Heart



Diagnostic Performance of a Computerized Algorithm for Augmenting the ECG with Acoustical Data

The eNDThe eND


Results:Results:

The following data show the ability of the The following data show the ability of the computerized acoustical algorithm to detect an computerized acoustical algorithm to detect an

S3 or an S4 in patients in a variety of clinical S3 or an S4 in patients in a variety of clinical settings.settings.

The performance of the algorithm is compared The performance of the algorithm is compared to a consensus of 2 experienced cardiologists to a consensus of 2 experienced cardiologists concerning the audibility of the recorded S3 or concerning the audibility of the recorded S3 or

the S4 in the each of the same patients.the S4 in the each of the same patients.

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The HeartThe Heart




Diagnostic Performance of a Computerized Algorithm for Augmenting the ECG with Acoustical Data

The ENDThe END

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The HeartThe Heart




Audicor Analysis

The ENDThe END

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The HeartThe Heart




Audicor Analysis

The ENDThe END

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The HeartThe Heart




The AUDICOR Decision Pathway – EMS

The ENDThe END

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The HeartThe Heart




The ENDThe END

Acute myocardial ischemia often displaces the Acute myocardial ischemia often displaces the ST segments in the ECG. ST segments in the ECG.

However, since the specificity and sensitivity However, since the specificity and sensitivity for ischemia of ST segment displacement for ischemia of ST segment displacement

are imperfectare imperfect

Echocardiography and radionuclide studies are Echocardiography and radionuclide studies are often used to augment the ECG in often used to augment the ECG in evaluating patients for ischemia.evaluating patients for ischemia.

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ECG, Hemodynamic & Acoustical Findings: Experimental Model of Myocardial Ischemia

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The HeartThe Heart





The ENDThe END

Ischemia also has hemodynamic effects that Ischemia also has hemodynamic effects that include reduced left ventricular (LV) include reduced left ventricular (LV) contractility and compliancecontractility and compliance . .

These hemodynamic changes are typically These hemodynamic changes are typically associated with a third and fourth heart associated with a third and fourth heart sound (S3 and S4), respectivelysound (S3 and S4), respectively..

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The HeartThe Heart





The ENDThe END

Conclusion:Conclusion:

Detecting and recording heart sounds may Detecting and recording heart sounds may improve the identification of acute improve the identification of acute

myocardial ischemia as the cause of ST myocardial ischemia as the cause of ST segment abnormalities.segment abnormalities.

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The HeartThe Heart




ECG & Acoustical Data in the detection of

Left Ventricular Enlargement

The ENDThe END

The presence of 3The presence of 3rdrd and 4 and 4thth heart sounds was heart sounds was associated mainly with relative prolongation of associated mainly with relative prolongation of the PR interval and with flattening or the PR interval and with flattening or negativity of T waves in multiple leads.negativity of T waves in multiple leads.

Conversely these sounds were not associated Conversely these sounds were not associated with the abnormalities of QRS voltage with the abnormalities of QRS voltage traditionally attributed to increased left traditionally attributed to increased left ventricular mass.ventricular mass.

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The HeartThe Heart




ECG & Acoustical Data in the detection of

Left Ventricular Enlargement

The ENDThe END

Conclusion:Conclusion:

ECG and acoustical data can detect ECG and acoustical data can detect abnormalities of ventricular function that abnormalities of ventricular function that the cardiac diseases responsible for LVE the cardiac diseases responsible for LVE produce.produce.

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The HeartThe Heart




Detecting Hemodynamic Abnormalities Using ECG and Cardiac Acoustical Data

The ENDThe END

BackgroundBackground:: Hemodynamic abnormalities can produce ECG Hemodynamic abnormalities can produce ECG changeschanges . .

For example, the ECG evidence of left For example, the ECG evidence of left ventricular hypertrophy (LVH) is a ventricular hypertrophy (LVH) is a consequence of the hemodynamic consequence of the hemodynamic abnormalities that produced the LVHabnormalities that produced the LVH . .

However they hypothesized that abnormal However they hypothesized that abnormal hemodynamics are more likely to predict hemodynamics are more likely to predict the presence of a third heart sound (S3) the presence of a third heart sound (S3) than of ECG findingsthan of ECG findings..

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The ENDThe END

ConclusionsConclusions::

The electronically recorded S3 is The electronically recorded S3 is associated with a wider range of associated with a wider range of hemodynamic abnormalities than is hemodynamic abnormalities than is ECG evidence of LVHECG evidence of LVH , ,

ST-T or prior MI and can therefore ST-T or prior MI and can therefore augment the diagnostic capabilities of augment the diagnostic capabilities of the ECGthe ECG..

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Detecting Hemodynamic Abnormalities Using ECG and Cardiac Acoustical Data

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Using Simultaneous ECG and Acoustical Data to Evaluate and Monitor Patients with Cardiac Disease

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BackgroundBackground:: Acute myocardial ischemia is associated with Acute myocardial ischemia is associated with hemodynamic as well as ECG abnormalitieshemodynamic as well as ECG abnormalities . .

For example, impaired left ventricular (LV) For example, impaired left ventricular (LV) systolic function can produce a third heart systolic function can produce a third heart sound (S3) that previous research, as sound (S3) that previous research, as reflected in the ACC/AHA Practice reflected in the ACC/AHA Practice Guidelines, has shown to be associated Guidelines, has shown to be associated with increased clinical riskwith increased clinical risk..

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The ENDThe END

ResultsResults:: In the 89 pre-cath patients, the recorded S3 In the 89 pre-cath patients, the recorded S3 had a sensitivity /specificity for detecting had a sensitivity /specificity for detecting an LV ejection fraction <50% and LV an LV ejection fraction <50% and LV enddiastolic pressure >15mmHg of 13/21 enddiastolic pressure >15mmHg of 13/21 (sens, 62%); 60/68 (spec, 88%)(sens, 62%); 60/68 (spec, 88%)..

In the acute MI patient, acoustical changes In the acute MI patient, acoustical changes preceded ECG changes and a new S3 preceded ECG changes and a new S3 appeared shortly after the onset of the appeared shortly after the onset of the acute MIacute MI..

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The ENDThe END

ConclusionsConclusions:: Electronically recorded S3 identifies patients Electronically recorded S3 identifies patients with impaired LV systolic function and with impaired LV systolic function and recorded heart sounds can be added torecorded heart sounds can be added tomulti-parameter monitoring of patients with multi-parameter monitoring of patients with suspected acute MIsuspected acute MI..

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HMM ModelsHMM Models

Automatic Segmentation of Heart Sound Signals Using Hidden Markov Models

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Segmentation of heart sounds into their Segmentation of heart sounds into their component segments, using Hidden Markov component segments, using Hidden Markov Models. Models.

The heart sounds data is preprocessed into The heart sounds data is preprocessed into feature vectors, where the feature vectors are feature vectors, where the feature vectors are comprised of the average Shannon energy of comprised of the average Shannon energy of the heart sound signal, the delta Shannon the heart sound signal, the delta Shannon energy, and the delta-delta Shannon energy.energy, and the delta-delta Shannon energy.

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Pre-processing

The ENDThe END

The system filters the original heart sound The system filters the original heart sound signal using a band-pass filter with cutoff signal using a band-pass filter with cutoff frequencies at 30 Hz and 200 Hz. Next, the frequencies at 30 Hz and 200 Hz. Next, the signal is normalized according to:signal is normalized according to:

Then, it calculates the average Shannon Then, it calculates the average Shannon energy energy

In continuous 0.04-second segments, with In continuous 0.04-second segments, with 0.02 seconds of overlap per segment.0.02 seconds of overlap per segment.

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))((max

)()(

ix

kxkx

i

norm

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Pre-processing

The ENDThe END

Shannon energy emphasizes the medium Shannon energy emphasizes the medium intensity signals and attenuates the high intensity signals and attenuates the high intensity signals. intensity signals.

This tends to make medium This tends to make medium and high intensity signals similar in amplitude.and high intensity signals similar in amplitude.

The system calculates the average Shannon The system calculates the average Shannon energy of each frame, where Xenergy of each frame, where Xnorm norm is the is the normalized heart signal, using:normalized heart signal, using:

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)(log)(/1 2

1

2 ixixNE norm

N

inorms

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Pre-processing

The ENDThe END

Then, the system normalizes the average Then, the system normalizes the average Shannon energy over all of the frames, whereShannon energy over all of the frames, where is the average Shannon energy foris the average Shannon energy for frame tframe t

the mean valuethe mean value

the standard deviationthe standard deviation

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)(tEs

))(( tEs

))(( tEs

))((

))(()()(

tE

tEtEtP

s

ssa

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Mel-spaced filterbanks

The ENDThe END

Next, the system extracts the spectralNext, the system extracts the spectralCharacteristics from the heart sound signal.Characteristics from the heart sound signal. Since the average duration of the S1 sound is Since the average duration of the S1 sound is 0.16 seconds (empirical), the system divides 0.16 seconds (empirical), the system divides the signal into 0.15-second frames, with 0.02 the signal into 0.15-second frames, with 0.02 seconds of overlap for each frame.seconds of overlap for each frame.

The frequency spectrum of S1 contains peaks The frequency spectrum of S1 contains peaks in the 10 to 50 Hz range and the 50 to 140in the 10 to 50 Hz range and the 50 to 140Hz range, while the frequency spectrum of S2 Hz range, while the frequency spectrum of S2 Contains peaks in the 10 to 80 Hz range, the Contains peaks in the 10 to 80 Hz range, the

80 80 to 200 Hz range, and the 220 to 400 Hz range. to 200 Hz range, and the 220 to 400 Hz range. As a result, this study limits the spectral As a result, this study limits the spectral feature extraction between the frequencies offeature extraction between the frequencies of10 Hz and 430 Hz.10 Hz and 430 Hz. Lets Lets

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The ENDThe END

Mel-Spaced filter banks provide a simple Mel-Spaced filter banks provide a simple method for extracting spectral characteristics method for extracting spectral characteristics from an acoustic signal.from an acoustic signal.

This method involves creating a set ofThis method involves creating a set oftriangular filter banks across the spectrum.triangular filter banks across the spectrum.

The filterbanks are equally spaced along the The filterbanks are equally spaced along the mel-scale, as defined in:mel-scale, as defined in:

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1(log2595)( 10

jfMel

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The ENDThe END

Equal spacing on the mel-scale provides non-Equal spacing on the mel-scale provides non-Linear spacing on the normal frequency axis. Linear spacing on the normal frequency axis.

This non-linear spacing means that there are This non-linear spacing means that there are numerous, small banks at the lower numerous, small banks at the lower frequencies and sparse, large banks at the frequencies and sparse, large banks at the Higher frequencies. Higher frequencies.

Since most of the energy of the heart sounds Since most of the energy of the heart sounds is is

in the lower frequency ranges, using a mel-in the lower frequency ranges, using a mel-scale matches the frequency spectrum of the scale matches the frequency spectrum of the heart sounds.heart sounds.

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The ENDThe END

Each triangular filter is multiplied by the Each triangular filter is multiplied by the Discrete Fourier transfer of the heart sound Discrete Fourier transfer of the heart sound frame and summed. frame and summed.

This creates a set of frequency bins, whereThis creates a set of frequency bins, whereeach bin represents a portion of the frequency each bin represents a portion of the frequency spectrum.spectrum.

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Regression coefficients

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The final feature extraction step is to calculate The final feature extraction step is to calculate a set of regression coefficients. Regression a set of regression coefficients. Regression coefficients are used to represent the changes coefficients are used to represent the changes in each feature over time. in each feature over time.

The system computes the first order The system computes the first order regression regression

(delta coefficients) and the second order (delta coefficients) and the second order Coefficients (delta-delta coefficients) using the Coefficients (delta-delta coefficients) using the following regression formula:following regression formula:

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2

1

2

)(

tt

t

ccd

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Regression coefficients

The ENDThe END

The system combines the Shannon energy, the The system combines the Shannon energy, the Spectral features, and the regression Spectral features, and the regression coefficients into a single feature vector per coefficients into a single feature vector per frame. frame.

It stores these feature vectors for later use in It stores these feature vectors for later use in the training and testing of the heart sound the training and testing of the heart sound HMM.HMM.

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Heart sound Hidden Markov Model

The ENDThe END

One can model the phonocardiogram signal asOne can model the phonocardiogram signal asa four state HMM:a four state HMM:

The first state corresponds to S1.The first state corresponds to S1.

The second state corresponds to the The second state corresponds to the silence silence

during the systolic period. during the systolic period.

The third state corresponds to S2.The third state corresponds to S2.

The fourth state corresponds to the silence The fourth state corresponds to the silence during the diastolic period.during the diastolic period.

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The ENDThe END

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The ENDThe END

This model ignores the possibility of the S3 This model ignores the possibility of the S3 and and

S4 heart sounds, because these sounds are S4 heart sounds, because these sounds are difficult to hear and record; therefore, they are difficult to hear and record; therefore, they are most likely not noticeable in the heart sound most likely not noticeable in the heart sound data.data.

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The ENDThe END

This four state HMM is useful for modeling theThis four state HMM is useful for modeling thesequence of symbols (or labels) of the sequence of symbols (or labels) of the phonocardiogram;phonocardiogram;

However, it is too simple to accurately modelHowever, it is too simple to accurately modelThe transitions between sound and silence. The transitions between sound and silence.

One solution is to embed another HMM inside One solution is to embed another HMM inside of each of the heart sound symbol states. of each of the heart sound symbol states.

The embedded HMM models the signal as it The embedded HMM models the signal as it traverses a specific labeled region of the traverses a specific labeled region of the signal.signal.

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The ENDThe END

Using this combined approach, we can model Using this combined approach, we can model both the high-level state sequence of both the high-level state sequence of our signal (S1-sil-S2-sil) and the continuous our signal (S1-sil-S2-sil) and the continuous transitions of the signal. transitions of the signal.

This type of model is similar to how a speech This type of model is similar to how a speech processing system has a high-level processing system has a high-level probabilistic grammar to model the transition probabilistic grammar to model the transition of words or phonemes, and an embedded HMM of words or phonemes, and an embedded HMM for each Phoneme.for each Phoneme.

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Discussion

The ENDThe END

Shannon Energy features with and without the Shannon Energy features with and without the Melspaced filterbank features are nearly Melspaced filterbank features are nearly identical in performance. identical in performance.

Shannon Energy features are better suited for Shannon Energy features are better suited for lowering the frame error rate while Mel-spacedlowering the frame error rate while Mel-spacedfilterbanks are better suited for lowering the filterbanks are better suited for lowering the model error rate. model error rate.

Mel-spaced filterbanks are marginally better as Mel-spaced filterbanks are marginally better as features for noisy PCGs than features for noisy PCGs than Clean PCGs.Clean PCGs.

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FFTFFT

FFT S3 Analysis

The ENDThe END

The PCG and ECG data were sampled at a The PCG and ECG data were sampled at a rate of 2042 Hz (this gave a Nyquist rate of 2042 Hz (this gave a Nyquist frequency of about 1000 Hz).frequency of about 1000 Hz).

The signals are then digitized by means of The signals are then digitized by means of a two channel, 8 bit analogue to digital a two channel, 8 bit analogue to digital converter controlled by an Intel 8085 converter controlled by an Intel 8085 microprocessor based system (sdk85) with microprocessor based system (sdk85) with 8k memory.8k memory.

Each sampled datum was represented as Each sampled datum was represented as an unsigned hexadecimal number.an unsigned hexadecimal number.

These files are then simultaneously plotted These files are then simultaneously plotted by means of a graphics terminal. by means of a graphics terminal.

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FFTFFT

FFT S3 Analysis

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The ECG was used as a time reference for The ECG was used as a time reference for the PCG plot, which aided in obtaining the the PCG plot, which aided in obtaining the starting and end points of S3.starting and end points of S3.

Each PCG file was multiplied by a file Each PCG file was multiplied by a file containing a hamming window (0.54 + containing a hamming window (0.54 + 0.46cosθ) co-positioned with the S3, but 0.46cosθ) co-positioned with the S3, but zero everywhere else. This had the effect of zero everywhere else. This had the effect of extracting the S3 from the PCG file and extracting the S3 from the PCG file and multiplying it by a window function.multiplying it by a window function.

A conventional FFT is then applied to these A conventional FFT is then applied to these files to produce the S3 spectra. files to produce the S3 spectra.

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FFTFFT

Why FFT does not work for S3

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When applying it to the short duration S3, the When applying it to the short duration S3, the FFT suffers from a fundamental limitation in FFT suffers from a fundamental limitation in frequency resolution determined by the frequency resolution determined by the window size.window size.

The FFT gives poor resolution for S3 spectral The FFT gives poor resolution for S3 spectral analysis. The time duration of S3 is relatively analysis. The time duration of S3 is relatively short (50 ms). short (50 ms).

This short observation time, combined with the This short observation time, combined with the spectral blurring effects of the window spectral blurring effects of the window function accounts for the poor resolution of function accounts for the poor resolution of

the the FFT method. FFT method.

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FFTFFT


The ENDThe END

The major limitation of the FFT approach to The major limitation of the FFT approach to spectral analysis is that of frequency spectral analysis is that of frequency resolution, i.e. the capability of distinguishing resolution, i.e. the capability of distinguishing between closely spaced spectral peaks. between closely spaced spectral peaks.

The FFT resolution is about 1/T, where T is the The FFT resolution is about 1/T, where T is the available data time. Hence, when dealing with available data time. Hence, when dealing with short data lengths the resolution is restricted. short data lengths the resolution is restricted.

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FFTFFT


The ENDThe END

Another problem inherent to the FFT method is Another problem inherent to the FFT method is the effect of spectral “leakage”.the effect of spectral “leakage”. In FFT analysis a real signal represents a In FFT analysis a real signal represents a truncated function, which is equivalent to truncated function, which is equivalent to multiplying it by a “window” function. multiplying it by a “window” function.

The resultant FFT spectrum contains energy The resultant FFT spectrum contains energy due to both the signal itself and the window due to both the signal itself and the window function. function.

The result is the spectrum of the convolution The result is the spectrum of the convolution of of

the signal and window functions. This leakage the signal and window functions. This leakage can be reduced by appropriate design of the can be reduced by appropriate design of the window function, but this always results in window function, but this always results in reduced frequency resolution. reduced frequency resolution. Lets Lets

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MESAMESA

Maximum Entropy Spectral Analysis

The ENDThe END

Classical spectral analysis requires the Classical spectral analysis requires the assumptions, about the signal under analysis, assumptions, about the signal under analysis, of long samples of data and of stationarity. of long samples of data and of stationarity.

However in real applications of biomedical However in real applications of biomedical spectral analysis both these assumptions are spectral analysis both these assumptions are violated. violated.

In the case of the spectral analysis of S3, the In the case of the spectral analysis of S3, the time duration is short enough to consider it time duration is short enough to consider it stationary; but the assumption of a long signal stationary; but the assumption of a long signal history is obviously erroneous. history is obviously erroneous.

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MESAMESA

Maximum Entropy Spectral Analysis

The ENDThe END

The MESA technique has been demonstrated to The MESA technique has been demonstrated to produce superior spectral resolution when produce superior spectral resolution when compared with more traditional methods, compared with more traditional methods, especially for short data lengths [Burg 1967, especially for short data lengths [Burg 1967, Kay 1981, Ulrych 1975].Kay 1981, Ulrych 1975].

Another advantage of MESA is that one can Another advantage of MESA is that one can use use

a simple rectangular window as there is no a simple rectangular window as there is no spectral “leakage”.spectral “leakage”.

Studies have shown that FFT is incapable of Studies have shown that FFT is incapable of satisfactorily resolving the frequency peaks of satisfactorily resolving the frequency peaks of in S3 and introduces unwanted leakage. in S3 and introduces unwanted leakage.

However the Maximum Entropy Method is However the Maximum Entropy Method is capable of satisfactory resolution with no capable of satisfactory resolution with no leakage. leakage.

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Future ResearchFuture Research

Conclusions

The ENDThe END

Research has shown that S3 can Research has shown that S3 can significantly enhance heart disease significantly enhance heart disease analysis.analysis.

The normal range of human hearing lies The normal range of human hearing lies within the range of 20 Hz – 20000 Hz, with within the range of 20 Hz – 20000 Hz, with maximum sensitivity lying in the speech maximum sensitivity lying in the speech range; about 1000 Hz to 3000 Hz.range; about 1000 Hz to 3000 Hz.

In order to be heard, low frequency sounds In order to be heard, low frequency sounds such as S3, must attain energy levels such as S3, must attain energy levels thousands of times greater than those thousands of times greater than those needed by vibrations within the speech needed by vibrations within the speech range. range.

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Conclusions

The ENDThe END

Thus the extraction of S3 & S4 is almost Thus the extraction of S3 & S4 is almost entirely dependant on the development of entirely dependant on the development of accurate automated methods.accurate automated methods.

Following a review of the literature it is Following a review of the literature it is apparent that a truly successful S3 & S4 apparent that a truly successful S3 & S4 detection algorithm has yet to be detection algorithm has yet to be developed.developed.

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Future Research

The ENDThe END

My future research may be focused on My future research may be focused on developing an accurate S3 & S4 detection developing an accurate S3 & S4 detection algorithm.algorithm.

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The End

Thank YouThank You

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The ENDThe END


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Fractal Definition

The ENDThe END


An object which appears self-similar under An object which appears self-similar under varying degrees of magnificationvarying degrees of magnification..

In effect, possessing symmetry across scale, In effect, possessing symmetry across scale, with each small part of the object replicating with each small part of the object replicating the structure of the wholethe structure of the whole . .

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Self-similarity

The ENDThe END


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Fractal Dimension

The ENDThe END


All fractals are characterized by their own All fractals are characterized by their own dimension, which is usually a non-integer dimension, which is usually a non-integer

dimension,dimension,that is greater than their topological that is greater than their topological

dimension and less than their Euclidean dimension and less than their Euclidean dimension.dimension.

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Fractal Dimension

The ENDThe END


This definition of fractal dimension is often This definition of fractal dimension is often used as an alternative definition of fractal used as an alternative definition of fractal objects. objects.

However, the fractal dimension may be However, the fractal dimension may be estimated in numerous ways, such as the box-estimated in numerous ways, such as the box-counting dimension and the variance counting dimension and the variance dimension.dimension.

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Chaotic dynamics

The ENDThe END


A chaotic system refers to a system that never A chaotic system refers to a system that never exactly repeats its behavior. exactly repeats its behavior.

Regardless of how long we let the model run Regardless of how long we let the model run for, we would never come across a repetition for, we would never come across a repetition in the waveform due to its aperiodic feature. in the waveform due to its aperiodic feature.

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Chaos

The ENDThe END


This behavior is known as chaos. This behavior is known as chaos.

By plotting the models variables against each By plotting the models variables against each other we receive a visualization of the other we receive a visualization of the dynamics of the system. dynamics of the system.

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Phase Portrait

The ENDThe END


The phase portrait will have the same form The phase portrait will have the same form although the model is started with different although the model is started with different

initial conditions, the system will be attracted initial conditions, the system will be attracted to this type of final solution.to this type of final solution.

In two dimensions these plots are known as In two dimensions these plots are known as phase portraits.phase portraits.

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Strange Attractors

The ENDThe END


This plot is known as the attractor of the This plot is known as the attractor of the system. system.

The attractors for chaotic systems are termed The attractors for chaotic systems are termed strange attractors. strange attractors.

The fractal structures of these strange The fractal structures of these strange attractors may be classified by calculating attractors may be classified by calculating their fractal dimensions.their fractal dimensions.

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The Cantor Set

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The cantor set consists of an infinite set of The cantor set consists of an infinite set of disappearing line segments in the unit disappearing line segments in the unit interval. interval.

The set is generated by iteratively removing The set is generated by iteratively removing the middle third of line segments, resulting the middle third of line segments, resulting in a collection of infinitely many in a collection of infinitely many disappearing line segments lying on the unit disappearing line segments lying on the unit interval. interval.

Both the line segments individual and Both the line segments individual and combined length approach zero as the combined length approach zero as the number of line segments approach infinity. number of line segments approach infinity.

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The Cantor Set

The ENDThe END


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The Koch curve

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The iterative procedure used to construct The iterative procedure used to construct the the Koch curve begins, similar to the Cantor set, Koch curve begins, similar to the Cantor set, with the initiator of the set as the unit line with the initiator of the set as the unit line segment. segment.

The generator is constructed by removing The generator is constructed by removing the middle third of the line segment and the middle third of the line segment and then replace it with two equal segments then replace it with two equal segments formed as two sides of a triangle. formed as two sides of a triangle.

The process is repeated an infinite number The process is repeated an infinite number of of times to produce the Koch curve. times to produce the Koch curve.

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The Koch curve

The ENDThe END


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Regular Fractals

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Regular fractals are fractal objects that Regular fractals are fractal objects that possess exact self-similarity, objects with possess exact self-similarity, objects with structures comprising of exact copies of structures comprising of exact copies of

themselves at all magnifications. themselves at all magnifications.

The most commonly known regular fractals are The most commonly known regular fractals are possibly the Cantor set and the Koch curve, possibly the Cantor set and the Koch curve, both simply constructed using an iterative both simply constructed using an iterative

procedure. procedure.

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Random Fractals

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Random fractals are statistically self-similar. Random fractals are statistically self-similar.

Each small part of a random fractal has the Each small part of a random fractal has the same statistical properties as the whole. same statistical properties as the whole.

Random fractals may be constructed Random fractals may be constructed mathematically by introducing a random mathematically by introducing a random feature in the generating process of a feature in the generating process of a regular regular fractal. fractal.

For instance, when generating the Cantor set For instance, when generating the Cantor set any third of the line segment is removed any third of the line segment is removed instead of the middle third. instead of the middle third.

Many properties of natural objects and Many properties of natural objects and phenomena may be described using random phenomena may be described using random fractals.fractals.

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Fractal boundaries

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In order for a fractal curve to be classified as aIn order for a fractal curve to be classified as afractal boundary it must meet two conditions:fractal boundary it must meet two conditions:

1.1. The curve must be non-crossing, The curve must be non-crossing, meaning that the fractal curve does not meaning that the fractal curve does not intersect itself intersect itself

2. As the fractal curve is zoomed in it 2. As the fractal curve is zoomed in it reveals more structure (details).reveals more structure (details).

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Fractal boundaries

The ENDThe END


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Fractal boundaries

The ENDThe END


Dimension measurements are suitable in order Dimension measurements are suitable in order to characterize and quantify the statistical to characterize and quantify the statistical self-similarity property of random fractal self-similarity property of random fractal boundaries. boundaries.

As random fractals do not possess exact self-As random fractals do not possess exact self-similarity the similarity dimension may not be similarity the similarity dimension may not be used. Instead we define estimates of the used. Instead we define estimates of the fractal dimension of random fractals, which do fractal dimension of random fractals, which do not require the exact self-similar property.not require the exact self-similar property.

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The box counting dimension

The ENDThe END


The box counting dimension, enables non-The box counting dimension, enables non-integer dimensions to be found for fractal integer dimensions to be found for fractal curves.curves.

covers the object in self-similar boxes.covers the object in self-similar boxes.

In order to determine the box counting In order to determine the box counting dimension of a fractal object, the object is dimension of a fractal object, the object is covered with elements or boxes of side length covered with elements or boxes of side length ε. The number of boxes, ε. The number of boxes, NN, required to cover , required to cover the object together with the side length ε is the object together with the side length ε is then used to determine the dimension. then used to determine the dimension.

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BD

BD

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The ENDThe END


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The ENDThe END


The straight line is covered with elements of The straight line is covered with elements of length ε, for simplicity we assume that the line length ε, for simplicity we assume that the line is of unit length. is of unit length.

In order to cover the line In order to cover the line NN elements are elements are required regardless of the dimension of the required regardless of the dimension of the elements, here illustrated as cubes. elements, here illustrated as cubes.

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The ENDThe END


To cover the unit line segment any elements To cover the unit line segment any elements with a dimension greater than or equal to the with a dimension greater than or equal to the dimension of the line itself may be used, and dimension of the line itself may be used, and still only require still only require NN of them. of them.

This leads to the following expression:This leads to the following expression:

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11 NL

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The ENDThe END


If, in stead, the same procedure would be If, in stead, the same procedure would be applied to a plane of unit area, the expression applied to a plane of unit area, the expression received would be:received would be:

Similar reasoning with a 3-dimensional object Similar reasoning with a 3-dimensional object would lead to:would lead to:

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12 NA

13 NV

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The ENDThe END


In general, in order to cover an object of unit In general, in order to cover an object of unit hypervolume the number of elements reuired hypervolume the number of elements reuired are:are:

In logarithmic form:In logarithmic form:

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BDN /1

)/1log(/)log( NDB

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By disregarding the assumption of unit By disregarding the assumption of unit hypervolume, a general expression of the box hypervolume, a general expression of the box counting dimension may be received: counting dimension may be received:

where V is the hypervolume of the fractal where V is the hypervolume of the fractal object. object.

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)/1log(/))log()(log( VNDB

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By rearranging the expression it is easy to see By rearranging the expression it is easy to see that it is an equation of the straight line,that it is an equation of the straight line,where the gradient of the line is the box where the gradient of the line is the box counting dimension of the object, counting dimension of the object, and by plotting and by plotting log(N) log(N) againstagainst log(1/ log(1/εε) for ) for various elements with different side lengths d various elements with different side lengths d the box counting dimension may be the box counting dimension may be determined:determined:

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)log()/1log()log( VDN B

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To obtain a measure of the box counting To obtain a measure of the box counting dimension there are different methods of dimension there are different methods of covering the fractal object. Three of them are covering the fractal object. Three of them are illustrated below.illustrated below.

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The first method illustrated is covering the The first method illustrated is covering the curve by placing boxes against each other in curve by placing boxes against each other in a way that a minimum number of boxes are a way that a minimum number of boxes are being used. being used.

Another method is to cover the fractal object Another method is to cover the fractal object with a grid of boxes and count the number with a grid of boxes and count the number of of boxes that contain a part of the curve. boxes that contain a part of the curve.

The last method illustrated is covering the The last method illustrated is covering the curve with circles instead of boxes, placed in curve with circles instead of boxes, placed in a similar way as with the boxes in the first a similar way as with the boxes in the first method.method. Lets Lets

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Regardless of which method being used, the Regardless of which method being used, the box counting dimension is still obtained from box counting dimension is still obtained from the derivate of the plot of the derivate of the plot of log(N) log(N) againstagainst log(1/log(1/εε):):

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)/1log(

)log(lim

0

NDB

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Normally, in practical applications, the box Normally, in practical applications, the box counting dimension is estimated by selecting counting dimension is estimated by selecting two points at small values of ε in the plot, two points at small values of ε in the plot, resulting in an estimation given by: resulting in an estimation given by:

To receive a more accurate estimate of the box To receive a more accurate estimate of the box counting dimension a best fitted line may be counting dimension a best fitted line may be drawn through the points at small values of ε. drawn through the points at small values of ε. The slope, and consequently the box counting The slope, and consequently the box counting dimension, is then calculated from this best dimension, is then calculated from this best fitted line.fitted line.

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))/1log()/1/(log())log()(log( 1212 NNDB

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Regular Brownian motion

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Regular Brownian motion, or Brownian motion, Regular Brownian motion, or Brownian motion, is named after its discoverer Robert Brown.is named after its discoverer Robert Brown.

He observed that small particles floating in He observed that small particles floating in water underwent rapid irregular motions due water underwent rapid irregular motions due to their bombardment by water molecules. to their bombardment by water molecules.

If a group of particles is released at a certain If a group of particles is released at a certain location the bombarding molecules will cause location the bombarding molecules will cause the particles to spread out, diffuse, through the particles to spread out, diffuse, through time. time.

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The trajectories of particles undergoing The trajectories of particles undergoing Brownian motion in the plane may cross over Brownian motion in the plane may cross over themselves. themselves.

Hence, Brownian motion may not be classified Hence, Brownian motion may not be classified as a fractal boundary. as a fractal boundary.

Furthermore, as the Brownian trajectory is Furthermore, as the Brownian trajectory is zoomed into, more structure is revealed, zoomed into, more structure is revealed, indicating that the statistical self-similarity indicating that the statistical self-similarity features of the Brownian trajectory extends features of the Brownian trajectory extends over all scales of magnification over all scales of magnification

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If the Brownian motion is sampled at intervals If the Brownian motion is sampled at intervals of and the positions of the sampled points at of and the positions of the sampled points at time are denoted by , then the time are denoted by , then the observed steps taken in the two coordinate observed steps taken in the two coordinate directions directions and both follows a and both follows a Gaussian probability distribution. Gaussian probability distribution.

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tit ),( ii yx

1 iii xxx1 iii yyy

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Consequently, the step lengths between Consequently, the step lengths between observed points also follows a Gaussian observed points also follows a Gaussian distribution:distribution:

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22 )()( iii yxr

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The methods generally used to construct a The methods generally used to construct a Brownian motion in the plane are derived from Brownian motion in the plane are derived from these features. these features.

Hence, the motion is constructed by using Hence, the motion is constructed by using steps in the two coordinate directions, steps in the two coordinate directions, ∆x∆x and and ∆∆y y randomly selected from a Gaussian randomly selected from a Gaussian distribution.distribution.

The step length r randomly selected from a The step length r randomly selected from a Gaussian distributionGaussian distributionTThe step angel is randomly selected from a he step angel is randomly selected from a uniform distribution between 0 and .uniform distribution between 0 and .

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The time trace of Brownian motion, The time trace of Brownian motion, B(t)B(t), is , is equal to the time history of the coordinates of equal to the time history of the coordinates of a Brownian trajectory, illustrating how the a Brownian trajectory, illustrating how the coordinate values vary in time. coordinate values vary in time.

The construction of a Brownian motion trace is The construction of a Brownian motion trace is derived from the property that successively derived from the property that successively increments the trace following a Guassian increments the trace following a Guassian distribution:distribution:

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))()(( ttBtB

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Thus, by sampling the Brownian motion trace Thus, by sampling the Brownian motion trace at discrete times at discrete times

a discrete approximation may be constructed, a discrete approximation may be constructed, by summarizing a series of random by summarizing a series of random incremental steps, incremental steps,

Thus , is built up as an accumulated sum, Thus , is built up as an accumulated sum,

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titi

)( jtR

)( itB

)()()()( 11

ii

i

jji tRtBtRtB

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For a continuous Brownian motion trace, For a continuous Brownian motion trace, B(t)B(t), , the self-similar properties are apparent as the self-similar properties are apparent as zooming into it. zooming into it.

Both the original trace and zoomed in traces Both the original trace and zoomed in traces displays the similar irregularity, as they are displays the similar irregularity, as they are statistically self-similar. statistically self-similar.

However, in order to retain the self-similar However, in order to retain the self-similar properties of the original trace the axes need properties of the original trace the axes need to be scaled differently. to be scaled differently.

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By considering pairs of points on a Brownian By considering pairs of points on a Brownian motion trace separated by a time motion trace separated by a time it is possible to state a relationship between it is possible to state a relationship between the mean absolute separation in the mean absolute separation in B(t)B(t) between between these points, and the time these points, and the time separation.separation.

The obtained expression is given by The obtained expression is given by where the exponent, here equal to ½, is where the exponent, here equal to ½, is denoted the Hurst exponent, denoted the Hurst exponent, HH::

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sT

)()( tBTtBB s

2/1STB

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The Hurst exponent is the reason why a The Hurst exponent is the reason why a Brownian motion trace only remains Brownian motion trace only remains statistically self-similar under scaling when the statistically self-similar under scaling when the axes axes B(t) B(t) and and tt are scaled differently. are scaled differently.

Consequently, if the time is scaled by a factor Consequently, if the time is scaled by a factor AA, , B(t) B(t) must be scaled by a factor in order must be scaled by a factor in order to retain the similar relationship between and . to retain the similar relationship between and .

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HA

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This is illustrated in the following example This is illustrated in the following example where the time where the time tt is scaled by the factor is scaled by the factor A:A:

This property of non-uniform scaling is known This property of non-uniform scaling is known as self-affinity, and is the reason for the two as self-affinity, and is the reason for the two scaling factors needed to retain the statistical scaling factors needed to retain the statistical self-similar properties self-similar properties

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BAtAtA HHHH )(

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All particles undergoing Brownian motion All particles undergoing Brownian motion diffuse through time in an average sense, diffuse through time in an average sense, the traces start at the traces start at t=0t=0 where where B(t)=0B(t)=0 and and continue to spread out from the origin as continue to spread out from the origin as time increases.time increases.

If a large number of particles are spreading If a large number of particles are spreading out from the origin through time, the out from the origin through time, the spreading process may be characterized spreading process may be characterized using averaged statistical properties. using averaged statistical properties.

When considering diffusion related When considering diffusion related problems, problems, it is more natural to use the standard it is more natural to use the standard deviation, as a measure of the deviation, as a measure of the spreading. spreading. Lets Lets

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C

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As the standard deviation and are As the standard deviation and are proportional the scaling relationship is given proportional the scaling relationship is given by :by :

The expression is commonly re-expressed as The expression is commonly re-expressed as

where K is denoted the diffusion coefficient.where K is denoted the diffusion coefficient.

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B

2/1tC

tKC 2

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It is possible to generate a Brownian motion It is possible to generate a Brownian motion trace with a certain diffusion coefficient trace with a certain diffusion coefficient KK, by , by selecting the incremental steps, selecting the incremental steps, from a Gaussian distribution where the from a Gaussian distribution where the standard deviation, is given by:standard deviation, is given by:

Where is the time interval between each Where is the time interval between each sample. sample. Hence, after number of time steps, the time Hence, after number of time steps, the time t t equals:equals:

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)( jtR

PtKP 2

tit

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Combining the above equations will result in Combining the above equations will result in that the standard deviation of diffusing that the standard deviation of diffusing particles may be expressed as:particles may be expressed as:

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2/1iPc

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Fractional Brownian motion

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The fractional Brownian motions, abbreviated The fractional Brownian motions, abbreviated fBms, are a generalisation of the regular fBms, are a generalisation of the regular Brownian motion. Brownian motion.

The Hurst exponents for fBms range from The Hurst exponents for fBms range from 0 < 0 < HH < 1 < 1where the special case of where the special case of HH=0.5 results in =0.5 results in regular Brownian motion. regular Brownian motion.

Normally, fBms are denoted where the Normally, fBms are denoted where the subscript subscript HH equals the Hurst exponent that is equals the Hurst exponent that is classifying the motion. classifying the motion.

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)(tBH

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Similar to regular Brownian motion traces the Similar to regular Brownian motion traces the fBm traces are self-affine processes. fBm traces are self-affine processes.

In addition, a scaled up part of an fBm requires In addition, a scaled up part of an fBm requires different scaling factors for the different scaling factors for the tt and and axes in order to retain its statistical self-axes in order to retain its statistical self-similar properties.similar properties.

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)(tBH

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The scaling relationship between the mean The scaling relationship between the mean absolute separation along the fBm trace and absolute separation along the fBm trace and the time of the separation is expressed as:the time of the separation is expressed as:

and similarly the standard deviation of and similarly the standard deviation of diffusing particles scales as:diffusing particles scales as: ~~

Where is the fractional diffusion Where is the fractional diffusion coefficient. coefficient.

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HSH TB

HC t H

FC tK )2(

FK

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The fractional diffusion coefficient is obtained The fractional diffusion coefficient is obtained by plotting against time since release. by plotting against time since release.

This results in a linear relationship, where the This results in a linear relationship, where the slope of the plot corresponds to twice the slope of the plot corresponds to twice the fractional diffusion coefficient. fractional diffusion coefficient.

However, this requires that is known, which is However, this requires that is known, which is not always the case in practical applications. not always the case in practical applications.

Instead a logarithmic plot of against time may Instead a logarithmic plot of against time may be used, where the gradient of the best-fitted be used, where the gradient of the best-fitted line through the experimental data equals and line through the experimental data equals and the crossing point on the axis equals the crossing point on the axis equals

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HC

/1)(

C)2log( FKH

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Heart Sound Analysis with Time Dependent Fractal Dimensions

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First I will explain the construction and use First I will explain the construction and use of fractal dimension trajectories, and how of fractal dimension trajectories, and how the selection of windows affects the the selection of windows affects the appearance of the trajectory. appearance of the trajectory.

Thereafter follows a description of the Thereafter follows a description of the different methods used to calculate the different methods used to calculate the dimension trajectories. dimension trajectories.

The description consists of two parts, a brief The description consists of two parts, a brief introduction to the methods containing the introduction to the methods containing the necessary theory, followed by some necessary theory, followed by some practical practical considerations that have to be accounted for considerations that have to be accounted for when applying them to discrete signals. when applying them to discrete signals. Lets Lets

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Heart Sound Analysis with Time Dependent Fractal Dimensions

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First I will explain the construction and use First I will explain the construction and use of fractal dimension trajectories, and how of fractal dimension trajectories, and how the selection of windows affects the the selection of windows affects the appearance of the trajectory. appearance of the trajectory.

Thereafter follows a description of the Thereafter follows a description of the different methods used to calculate the different methods used to calculate the dimension trajectories. dimension trajectories.

The description consists of two parts, a brief The description consists of two parts, a brief introduction to the methods containing the introduction to the methods containing the necessary theory, followed by some necessary theory, followed by some practical practical considerations that have to be accounted for considerations that have to be accounted for when applying them to discrete signals. when applying them to discrete signals. Lets Lets

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The End

Thank YouThank You

Lets Lets StopStop!!

The ENDThe END


heart sounds, ecg & fractals. 2 ecg wave ecg wave heart sounds heart sounds audicor’s solution...

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