Electronic Detection and Diagnosis of Health and Illness of Premature Infants
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Electronic Detection and Diagnosis of Health and Illness of Premature Infants. Co-Workers. Medical Folks Randall Moorman Pamela Griffin John Kattwinkel Alix Paget-Brown Brooke Vergales Kelley Zagols Andy Bowman Terri Smoot. Quants Statisticians Doug Lake George Stukenborg

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Electronic detection and diagnosis of health and illness of premature infants

Electronic Detection and Diagnosis of Health and Illness of Premature Infants


Co workers

Co-Workers

Medical Folks

Randall Moorman

Pamela Griffin

John Kattwinkel

Alix Paget-Brown

Brooke Vergales

Kelley Zagols

Andy Bowman

Terri Smoot

Quants

Statisticians

Doug Lake

George Stukenborg

Chemical Engineers

John Hudson

Matthew Clark

Craig Rusin

Lauren Guin

Physicists

Abigail Flower

Hoshik Lee

John Delos

Supported by NIH.


Overview

Overview

What can we Quants contribute?

Statistical measures

Signal Analysis

Pattern recognition methods

Dynamical theories

Medical Issues:

Apnea of Prematurity

Neonatal Sepsis

Observations:

Electronic Monitoring

of Heart, Respiration


Electronic detection and diagnosis of health and illness of premature infants

Outline

Sepsis

medical issue

heart rate monitoring

statistical measures

randomized clinical trial

pattern recognition

physiological modeling

Apnea

medical issue

chest impedance – cardiac artifact

filtering, signal analysis

examples

future work

Conclusion


Electronic detection and diagnosis of health and illness of premature infants

Sepsis

the presence of bacteria, virus, fungus, or other organism in the blood or other tissues and the toxins associated with the invasion.

Of 4 million births each year, 56,000 are VLBW (<1.5 Kg). For them the risk of sepsis is high (25-40%)

Significant mortality and morbidity (doubled risk of death in VLBW infants; increased length of NICU stay; high cost).

The diagnosis of neonatal sepsis is difficult, with a high rate of false negatives

Physicians administer antibiotics early and often.

Can heart rate monitoring give early warning of sepsis?(The invading organisms, or the immune response, may affect the pacemaking system.)

Randall Moorman


Does heart rate give warning of illness plot time between beats vs beat number

Does heart rate give warning of illness?Plot (Time Between Beats) vs (Beat Number)

ms

ms

beat number

beat number


Does heart rate give warning of illness plot time between beats vs beat number1

Does heart rate give warning of illness?Plot (Time Between Beats) vs (Beat Number)

Reduced variability

Repeated decelerations

pathologic

normal

vs.

pathologic

pathologic

expand


Statistical measures of rr interval data

Statistical Measures of RR Interval Data

  • Standard Deviation and Sample Entropy: Variability in the signal.

  • Sample Asymmetry: Prevalence of decelerations over accelerations implies a skew, or asymmetry, in the data which we can detect statistically.


Electronic detection and diagnosis of health and illness of premature infants

Find correlation of those measures with illness, and report correlation in terms of “fold increase of risk of sepsis”:

Take any random moment.

Examine a window of 24 hours (plus 18 hours or minus 6 hours) around that moment.

On average, 1.8% of the infants in our first study had a sepsis event within that 24 hour window.

If we report a “five-fold increase of risk of sepsis”, it means that about 10% of the infants showing those heart rate characteristics had a sepsis event within that 24 hour window.


Electronic detection and diagnosis of health and illness of premature infants

Medical Predictive Science Corporation

has developed and is marketing a system for

Heart Rate Characteristics monitoring in the NICU.

A computer beside each NICU bed continuously collects ECG data, extracts times of R peaks,

tracks RR intervals, and provides the following

Heart Rate Observation (HeRO).

This system is installed in several NICUs in the US,

and a large randomized clinical trial has just been completed.


Electronic detection and diagnosis of health and illness of premature infants

HRC rises before illness score


Electronic detection and diagnosis of health and illness of premature infants

Conclusion 1:

New quantitative analysis of noninvasive, electronically-measured heart rate characteristics – standard deviation, asymmetry, and sample entropy –

provides an early noninvasive warning

of sepsis events.


Electronic detection and diagnosis of health and illness of premature infants

Randomized Clinical Trial

8 HospitalsUVa, Wake Forest, UAl (birmingham), Vanderbilt, Umiami, Greenville SC, Palmer (Orlando) Penn State

Control

Save HeRO data

but do not display it

152 deaths/1489, 10.2%

133 deaths/757, 17.6%

Sample

Display HeRO Score

(but do not tell clinicians

what to do)

2989 VLBW (<1.5 Kg)

1513 ELBW (<1 Kg)


Electronic detection and diagnosis of health and illness of premature infants

Randomized Clinical Trial

8 HospitalsUVa, Wake Forest, UAl (birmingham), Vanderbilt, Umiami, Greenville SC, Palmer (Orlando) Penn State

Control

Save HeRO data

but do not display it

152 deaths/1489, 10.2%

133 deaths/757, 17.6%

Sample

Display HeRO Score

(but do not tell clinicians

what to do)

122 deaths/1500, 8.1%

100 deaths/756, 13.2%

2989 VLBW

Δ = 2.1% absolute, 22% relative

1513 ELBW

Δ = 4.4% absolute, 33% relative


Electronic detection and diagnosis of health and illness of premature infants

Conclusion 2:

New quantitative analysis of noninvasive, electronically-measured heart rate characteristics – standard deviation, asymmetry, and sample entropy –

saves lives.


Electronic detection and diagnosis of health and illness of premature infants

New Question:

Would direct measures of decelerations

provide additional information?


Pattern recognition for detecting decelerations abigail flower

Pattern Recognition

for detecting decelerations

Abigail Flower


Result storms of decelerations are highly predictive of sepsis

Result: “Storms” of Decelerations are Highly Predictive of Sepsis


Electronic detection and diagnosis of health and illness of premature infants

Statistical measures


Electronic detection and diagnosis of health and illness of premature infants

Conclusion 2

Counting and measuring decelerations gives a second method for early warning of sepsis.

Also an important finding was that HR decelerations

are surprisingly similar in infants.


Discovery

Discovery

We found that, within extended clusters of decelerations, there were sometimes shorter intervals of time, lasting up to several hours, in which the decelerations showed remarkable periodicity.


Electronic detection and diagnosis of health and illness of premature infants

For six infants in our study population we identified deceleration clusters in which periodicity was maintained for at least ten minutes.

In these periodic bursts of decelerations, the time to the next deceleration was about 15 sec.


Electronic detection and diagnosis of health and illness of premature infants

New Question:What Causes Periodic Decelerations?Can we develop a model capable of producing decelerations like those observed in neonatal HR records? Such models can guide future observations or experiments on animals.


Electronic detection and diagnosis of health and illness of premature infants

Partial Answers

A. A mathematical model with minimal physiological assumptions: a noisy Hopf bifurcation

  • Physiologically-based models

    The pacemaker?

    The baroreflex loop?

    Periodic apneas?


Electronic detection and diagnosis of health and illness of premature infants

HR jumps from small fluctuations [“steady-state”] to periodic decelerations [above-threshold oscillations].


Hopf bifurcations the most common way that a rest point changes to a cycle

Hopf Bifurcationsthe most common way that a rest point changes to a cycle

Hard (Subcritical) Hopf bifurcation:


Small excerpt of real data and simulation

Small excerpt of real data and simulation.

large periodic decels

data

small periodic fluctuations

simulation

Hard Hopf oscillations

noise-induced subthreshold oscillations


Electronic detection and diagnosis of health and illness of premature infants

Conclusion

Output of a Noisy Hopf Bifurcation Model

Resembles

Observed Periodic Decelerations

Thus we have a mathematical model with minimal physiological assumptions: a noisy Hopf bifurcation.

Is there a physiologically-based model?


Electronic detection and diagnosis of health and illness of premature infants

  • Pacemaker cells

  • Sino-Atrial node cells are the natural pacemaker of the heart. Can they go into “FM” mode with period ~15 seconds?

  • Baroreflex loop

  • The feedback loop connecting blood pressure and heart rate can go into oscillation with period ~14 seconds in infants. Does this resemble the observations?

  • Periodic apneas

  • Apneas produce decelerations of the heart. Sometimes they occur periodically, and the period is commonly ~15 seconds.


Electronic detection and diagnosis of health and illness of premature infants

Mayer (low freq) arrhythmia

RR

HR

BP

BP

  • Oscillation with period ~14 s in infants.

  • Feedback loop with time delay.

  • Many models connect Mayer waves with a Hopf bifurcation.

  • However, measurements indicate that under physiological conditions in adults, the feedback loop is stable (no oscillations)

  • Two studies (DeBoer et al. and Chapuis et al.) propose that observed Mayer waves are noise-driven subthreshold oscillations


  • Hypothesis noisy precursors are familiar mayer waves

    Hypothesis: Noisy Precursors are Familiar Mayer Waves.

    There MUST be a nearby threshold.

    Decelerations are result of a parameter

    going above the bifurcation.


    Electronic detection and diagnosis of health and illness of premature infants

    Rate of change of =Rate in from heart –

    blood volume in arteries Rate out through capillaries to veins

    dV / dt =heart rate x blood volume per beat 

    - [P(arteries) - P(veins) ] / [Resistance]

    P(arteries) =V(arteries) / [compliance = ]

    Similar formulas for veins

    Rate of change of = Neg(t) a negative term caused by high BP

    heart rate + Pos(t -τ ) a positive term caused by low BP

    which however is delayed

    A model of the baroreflex loop

    J. Ottesen, Roskilde University, Denmark


    Electronic detection and diagnosis of health and illness of premature infants

    A problem in nonlinear dynamics

    Do these differential equations have a bifurcation that causes heart rate and blood pressure to oscillate?


    Electronic detection and diagnosis of health and illness of premature infants

    A problem in nonlinear dynamics

    Do these differential equations have a bifurcation that causes heart rate and blood pressure to oscillate?

    YES!!

    What is the period ??


    Electronic detection and diagnosis of health and illness of premature infants

    A problem in nonlinear dynamics

    Do these differential equations have a bifurcation that causes heart rate and blood pressure to oscillate?

    YES!!

    What is the period ??

    Depending on parameters, can be ~ 15 seconds.

    Do they resemble the observed oscillations?


    Electronic detection and diagnosis of health and illness of premature infants

    A problem in nonlinear dynamics

    Do these differential equations have a bifurcation that causes heart rate and blood pressure to oscillate?

    YES!!

    What is the period ??

    Depending on parameters, can be ~ 15 seconds.

    Do they resemble the observed oscillations?

    Nope.

    Do measurements of HR and BP in infants show the large correlated oscillations?


    Electronic detection and diagnosis of health and illness of premature infants

    A problem in nonlinear dynamics

    Do these differential equations have a bifurcation that causes heart rate and blood pressure to oscillate?

    YES!!

    What is the period ??

    Depending on parameters, can be ~ 15 seconds.

    Do they resemble the observed oscillations?

    Nope.

    Do measurements of HR and BP in infants show the large correlated oscillations?

    Nope.


    Electronic detection and diagnosis of health and illness of premature infants

    • The Future: Incoming Data Streams

    • 1.Electronic diagnosis of infectious disease? Can we identify invading organisms by HR monitoring?

    • Preliminary evidence:

    • Reduced variability Gram-positive bacteria (e.g. Streptococci, Staphylococci)

    • vancomycin

    • Clusters of decels Gram-negative bacteria (e.g. E. coli, Pseudomonas)

    • gentamicin, cefotaxime

    • If this preliminary result holds up, we have the first example of continuous, noninvasive, purely electronic monitoring that gives early warning of infectious disease and also gives partial diagnosis, thereby identifying the recommended therapy.

    • (A medical tricorder)


    Electronic detection and diagnosis of health and illness of premature infants

    The Future: Experiments proposed or discussed

    2.Quadriplegics are subject to

    infections that they do not feel.

    Can heart rate monitoring

    provide early warning?

    3. Can large decelerations be induced in animals?

    Models suggest that increasing the time delay in the baroreflex loop pushes the system above bifurcation.

    Can this be done by chemical means (beta blockers)?

    Or electrical means (cut the sympathetic nerve

    and use an electrical time delay circuit)?

    What happens when corresponding experiments are done on newborn or premature animals?


    Electronic detection and diagnosis of health and illness of premature infants

    Conclusions

    • Decelerations are correlated with illness, and can be used as a new, noninvasive monitor for illness of premature infants.

    • Periodic decelerations resemble the output of a model based on Hopf bifurcation theory.

    • Study of periodic apneas is the next step.

    • The search for corresponding phenomena in animals has clinical, developmental, and dynamical significance.

      5. When we quants work together with physicians, and overcome the knowledge barrier and communication barriers between us, important and unexpected advances in health care can be made.


    Apnea of prematurity aop

    Apnea of Prematurity (AOP)

    Apnea(cessation of breathing) is very common for premature infants.

    • > half of babies whose birth weight < 1500 g (VLBW)

    • Almost all babies whose birth weight < 1000 g (ELBW)

    Definition of (clinical) AOP

    Cessation of breathing > 20sOR

    Cessation of breathing > 10s + Bradycardia (Heart Rate < 100 bpm) or O2 Desaturation (SpO2 < 80%)

    and

    VLBW : Very Low Birth Weight, < 1500g

    ELBW : Extremely Low Birth Weight, < 1000g


    Electronic detection and diagnosis of health and illness of premature infants

    Three types of apnea are common in premature infants.

    • Obstructive apnea : a blockage of the airway, typically accompanied by struggling or thrashing movements of the infant.

    • Central apnea : cessation of respiratory drive, and the infant makes no effort to breathe.

    • Mixed apneas : obstructive central.

      Central apnea is taken to indicate immaturity of control of respiration, and discharge from UVaNICU is delayed until apneas have been absent for 8 days.

      Apnea may be cause or an effect of many other clinical illnesses including sepsis or abnormal neurologic development. It is a serious clinical event which needs immediate medical attention.

      However, the current generation of apnea monitors is unsatisfactory.


    Data collection

    Data Collection

    • Since January 2009, we have collected all waveform and vital sign data from the bedside electronic monitors in the UVa NICU.

      • Waveforms: ECG, Chest Impedance, Pulse Ox

      • Vital signs:Heart & respiration rates, oxygen saturation of hemoglobin

      • ~ 1 TB / Baby-Year

    • About 1300 admissions for over two years, more than 300 VLBW infants.

    • Monitor alarms are stored (e.g. brady, desat, and apnea etc..).

    • Clinical data relevant to respiratory support are entered manually into an connected clinical database, including presence and type of ventilatory support such as mechanical ventilation.

    • Administration of medication (e.g. caffeine)


    Chest impedance measurement

    Chest Impedance Measurement

    • Easiest way to monitor the respiration of baby

    • Impedance between two electrodes placed at

    • the chest.

    • Use two electrodes already in use for the

    • measurement of the EKG signal, but use a

    • frequency (52 kHz) far outside the EKG signal.

    • Basic impedance (static) : several hundred ohms: muscles, tissue, blood + electrode-skin transitions, and wires.

    • Respiration : ~ 2 ohm

    • - Air has poor conductivity

    • - More air in the lung, higher impedance

    • Heart activity : ~ 0.5 ohm

    • - Blood is more conductive than air

    • - Pumping blood out of thorax, less impedance


    Electronic detection and diagnosis of health and illness of premature infants

    200

    100

    200

    100

    0

    Heart Rate

    ECG

    Chest Impedance

    Respiration Rate

    ECG & Chest Impedance


    Electronic detection and diagnosis of health and illness of premature infants

    200

    100

    200

    100

    0

    Heart Rate

    ECG

    Chest Impedance

    Respiration Rate

    ECG & Chest Impedance during Apnea event


    Electronic detection and diagnosis of health and illness of premature infants

    200

    100

    200

    100

    0

    Heart Rate

    ECG

    Chest Impedance

    Respiration Rate

    ECG & Chest Impedance during Apnea event


    Electronic detection and diagnosis of health and illness of premature infants

    How do we remove the cardiac artifact from chest impedance signal?

    Filtering, signal analysis, pattern recognition

    Hoshik Lee


    Electronic detection and diagnosis of health and illness of premature infants

    • Goal :

  • Filter the heart signal from chest impedance.

  • Simple Fourier filter fails. Heart beat band is too broad. Especially, the heart beat slows during apnea.

  • Fourier Transform of

    chest impedance

    Use the Heart as the Clock !

    Cardiac artifact in chest impedance

    contract/stretch

    RR intervals are evenly spaced.

    Heart Clock

    1RR

    1RR


    Electronic detection and diagnosis of health and illness of premature infants

    • Goal :

  • Filter the heart signal from chest impedance.

  • Simple Fourier filter fails. Heart beat band is too broad. Especially, the heart beat slows during apnea.

  • Fourier Transform of

    chest impedance

    Use the Heart as the Clock !

    Cardiac artifact in chest impedance

    contract/stretch

    RR intervals are evenly spaced.

    Heart Clock

    1RR

    1RR


    Electronic detection and diagnosis of health and illness of premature infants

    • Goal :

  • Filter the heart signal from chest impedance.

  • Simple Fourier filter fails. Heart beat band is too broad. Especially, the heart beat slows during apnea.

  • Fourier Transform of

    chest impedance

    Use the Heart as the Clock !

    Cardiac artifact in chest impedance

    contract/stretch

    RR intervals are evenly spaced.

    Heart Clock

    1RR

    1RR


    Electronic detection and diagnosis of health and illness of premature infants

    Fourier Transform of CI

    Fourier Transform of CI

    Using Heart Clock


    Electronic detection and diagnosis of health and illness of premature infants

    Fourier Transform of CI

    Fourier Transform of CI

    Using Heart Clock

    Cardiac Artifact


    Electronic detection and diagnosis of health and illness of premature infants

    Fourier Transform of CI

    Fourier Transform of CI

    Using Heart Clock

    Cardiac Artifact

    Breathing


    Electronic detection and diagnosis of health and illness of premature infants

    Fourier Transform of CI

    Fourier Transform of CI

    Using Heart Clock

    Cardiac Artifact

    Slow change : movement or unknown but not breathing

    Breathing


    Electronic detection and diagnosis of health and illness of premature infants

    `

    HR

    EKG

    Chest Impedance

    Chest impedance

    Cardiac Artifact Removed

    Filtered

    Chest Impedance

    There remain some small fluctuations in filtered CI. Compute the residual variance of the signal on 2 sec intervals, spaced by ¼ sec, and get a probability of apnea.


    Electronic detection and diagnosis of health and illness of premature infants

    Probability of Apnea

    ‘Energy’ in fluctuations (variance)


    Electronic detection and diagnosis of health and illness of premature infants

    Probability of Apnea

    ‘Energy’ in fluctuations (variance)

    Thresholding function looks like the Fermi distribution function. We obtain fitting function with two parameters.


    Electronic detection and diagnosis of health and illness of premature infants

    Summary: from Chest impedance to probability of apnea

    CI

    Remove cardiac artifact

    Remove slow variations and renormalize

    Compute

    2s variance

    P(apnea)


    Electronic detection and diagnosis of health and illness of premature infants

    Examples

    Philips Research North America 07/13/2011


    Electronic detection and diagnosis of health and illness of premature infants

    Philips Research North America 07/13/2011

    200

    100

    100

    1

    100

    0

    50

    0


    Very long apnea undetected by monitor

    Very Long Apnea Undetected by monitor

    We detected 782 apneas > 60 s in two years data


    Electronic detection and diagnosis of health and illness of premature infants

    Periodic Breathing = Periodic Apneas

    1

    0

    Philips Research North America 07/13/2011

    200

    100

    100

    100

    50

    0


    Electronic detection and diagnosis of health and illness of premature infants

    • Current Studies

    • What fraction of apneas are recorded by nurses?(~1/3)

    • How does the apnea rate change with age? ^

    • Does caffeine reduce apnea?(?)

    • Do transfusions reduce apnea?(Yes)

    • Can we get early warning of serious apneas?(Maybe)

    • What is the significance of long apneas?

    • Are short but periodic apneas benign?

    • Future Work

    • Convert to real-time monitor

    • -The system is set up for retrospective work : collecting statistics of apnea events and correlation with other clinical events.

    • Combined with (wearable) Automatic Stimulation system.

    • - e.g. vibrate shirt, mattress and so on.

    • Improve sepsis detection? (HeRO monitoring)

    • Add-on system which provides a new way to analyze data

    • that had previously been discarded


    Electronic detection and diagnosis of health and illness of premature infants

    Conclusions

    • Decelerations are correlated with illness, and can be used as a new, noninvasive monitor for illness of premature infants.

    • Periodic decelerations resemble the output of a model based on Hopf bifurcation theory.

    • Study of periodic apneas is the next step.

    • The search for corresponding phenomena in animals has clinical, developmental, and dynamical significance.

      5. When we quants work together with physicians, and overcome the knowledge barrier and communication barriers between us, important and unexpected advances in health care can be made.


    Electronic detection and diagnosis of health and illness of premature infants

    ASSUME:

    the heart rate is governed by some set of feedback loops that can be modeled by some big set of differential equations

    dx/dt = F(x;p)

    x = (x1,x2,…)(dynamical variables)

    p = (p1,p2,…)(parameters)

    Example:

    x1 = heart rate, x2 = blood pressure, x3 = …


    Electronic detection and diagnosis of health and illness of premature infants

    • The equations have a “rest point” at which

    • all variables are constant (including HR)

    • F(xrest(p);p) = 0.

    • Move origin of coordinates so rest point = 0.

    • F(x;p) can be expanded in a Taylor series

    • in x’s.

    • F(x;p) = Mx + xNx +….

    • Analyze first at linear level.


    Electronic detection and diagnosis of health and illness of premature infants

    Almost all matrices can be transformed to diagonal form

    At linear level

    Equations are real, so eigenvalues are real

    or come in complex conjugate pairs


    Electronic detection and diagnosis of health and illness of premature infants

    • All except two eigenvalues have negative

    • real parts. Two eigenvalues are complex,

    • and have real part near zero.

    is near zero, and as parameters p change, it can pass through zero.

    THEN:


    Electronic detection and diagnosis of health and illness of premature infants

    • [Center Manifold Theorem]

    • In the many-dimensional y-space, there is a

    • smooth two-dimensional surface called the

    • center manifold, in which all the interesting

    • behavior occurs:

    • If the system starts on that surface, it

    • stays on that surface

    • b. If it starts elsewhere, it decays to that

    • surface

    • The equations in that surface have a

    • Taylor expansion


    Electronic detection and diagnosis of health and illness of premature infants

    AND FURTHERMORE:

    [Normal Form Theorem]

    There exists a smooth change of coordinates

    such that the diff eqs can be put into a “Normal

    Form”; using polar coords in the center

    manifold, that form is:

    and this is easy to analyze !!!!


    Electronic detection and diagnosis of health and illness of premature infants

    YOU can easily show:

    As increases through zero,

    The stable rest point goes unstable, and

    EITHER

    A stable cycle appears

    OR

    An unstable cycle disappears

    That is a Hopf bifurcation.


    Electronic detection and diagnosis of health and illness of premature infants

    Summary: In the many-dimensional space, there is a smooth two-dimensional attractor in which all the interesting behavior occurs, and there exists a smooth change of coordinates such that the diff eqs can be put into a “Normal Form”:

    As increases through zero, the stable rest point goes unstable, and either a stable cycle appears,

    or an unstable cycle disappears

    That is a Hopf bifurcation.


    Electronic detection and diagnosis of health and illness of premature infants

    In the new variables the motion is a sinusoidal oscillation,

    We can connect to time between beats RR(t)

    by assuming RR = RR(u(t)) = RR(r cos ).

    But we know the shape of the decelerations, and we can represent that shape by a Fourier cosine series.

    Then the sinusoidal oscillation is converted into a sequence of decelerations.


    Electronic detection and diagnosis of health and illness of premature infants

    Small time delay

    Heart rate

    Arterial BP

    Venous BP


    Electronic detection and diagnosis of health and illness of premature infants

    These oscillations are “noisy precursors” :

    With no noise the system goes to steady state

    Subthreshold oscillations arise because of noise and a nearby bifurcation


    Electronic detection and diagnosis of health and illness of premature infants

    Peter Andriessen

    High freq fluctuations: RR ~ opposite to BP respiratory arrythmia

    Low freq fluctuations: BP leads RR Mayer waves


    Electronic detection and diagnosis of health and illness of premature infants

    Above the bifurcation (longer time delay)

    the system oscillates even with no noise.


    Electronic detection and diagnosis of health and illness of premature infants

    Above bif, the oscillations are large. The shape is (maybe) a passable imitation of periodic decels seen in infants. Period 10 s for adults.

    RR

    time


    Electronic detection and diagnosis of health and illness of premature infants

    • Comparison of model (based on adult data) with observations:

    • There is a bifurcation which produces large regular oscillations in HR. Increase in delay time of sympathetic action gives the bif.

    • Shape of decels – could be better.

    • Liapunov exponent is too small. (Slow approach to steady oscillations.)

    • It’s a soft Hopfbif instead of a hard Hopfbif.

    • (I think by playing with parameters we might convert it to hard Hopf.)

    • Continuing research:

    • More fun with baroreflex models

    • Begin study of respiration models


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