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How the heart beats: A mathematical model. Minh Tran and Wendy Cimbora Summer 2004 Math Biology Workshop. Anatomy of the Heart. The heart is a muscle: functions as a pump (circulates nourishment and oxygen to, and CO 2 and waste away)

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How the heart beats a mathematical model l.jpg

How the heart beats: A mathematical model

Minh Tran and Wendy Cimbora

Summer 2004 Math Biology Workshop


Anatomy of the heart l.jpg
Anatomy of the Heart

  • The heart is a muscle: functions as a pump (circulates nourishment and oxygen to, and CO2 and waste away)

  • 4 chambers: atria (input) and ventricles (output), upper and lower separate by valves

  • SA node: groups of cells on upper right atrium

  • AV node: between the atria and ventricles w/ in right atrial septum


Control via the sa node pacemaker l.jpg
Control via the SA node (pacemaker)

  • Contractions of heart controlled by electrical impulses (generated primarily by SA node, pacemaker cells)

  • Fires at a rate which controls the heart beat

  • Naturally discharge action potentials 70-80 per m

  • Input to the AV node comes from the A.P. propagating through atria from SA node

  • Then travels to the Bundle of His and Purkinje fibers, causing heart to contract


Simplified heart beat process l.jpg
Simplified Heart Beat Process

  • SA node fires

  • Electrical potential travels to AV node

  • We are concerned primarily with the AV node

  • It tells the heart when to beat based on condition of heart


Goal model electrical potential of the av node l.jpg

Assumptions for our model:

1) Potential decreases exponentially during the time between signals from SA node

2) Potential too high: no heart beat (heart hasn’t recovered), otherwise beat

3) If AV node accepts signal, tells heart to beat and electrical potential increases as a constant

Goal: Model Electrical Potential of the AV node


Model of the electrical potential of av node l.jpg
Model of the electrical potential of AV node

[Pt + S] e-DT Pt < P*

  • Pt+1 =

    Pt e-DT Pt > P*

    P = electrical potential of AV node

    S = constant increase of electrical potential of AV node

    D = rate of decrease (recovery rate of heart)

    T = time interval between firing from SA node

    P* = threshold (determines normal/abnormal beats)


Burning questions l.jpg
Burning Questions

  • What are some different patterns of heart beats?

  • Parameters: How many? Which could be varied? What does varying them mean? What are the ranges?

  • How does this piecewise function behave as we vary the parameters? Under what conditions does the model produce regular heart beats? Irregular?


Plot of p vs t normal heart rate l.jpg
Plot of P vs. t Normal heart rate

S=3, e-DT=1, Po=1, P*= 2

Potential is steady at 1.7459

beat = 1, no beat = 0


Plot of p vs t second degree block l.jpg
Plot of P vs. tSecond-degree block

S=2.5, e-DT=1, Po=.4, P*= 1

Potential bounces between 2 values

beat=1, no beat=0


Plot of p vs t wenckebach phenomenon l.jpg
Plot of P vs. tWenckebach Phenomenon

S=3, e-DT=1, Po=1, P*= 1.66

Potential bounces between 4 values (3 below threshold)

The heart beats 3 and skips 1 : beat=1, no beat=0


Slide11 l.jpg

S=3 e-DT=1 P* = 2 Po = 1

Cobwebbing (visualizing orbits and long term behavior)right: normal (stable fixed point) left bottom: 2nd deg. block (2 cycle)right bottom: Wenckebach (4 cycle)

P = S e-DT /( 1- e-DT )

S=2.5 e-DT=1 P* = 1 Po = .4

S=3 e-DT=1 P* = 1.66 Po = 1

P = S e-DT /( 1- e-2DT )

P = 3S e-3DT /( 1- e-4DT )


Bifurcation of a e dt what happens when lower s decrease in potential l.jpg
Bifurcation of a = e-DTWhat happens when lower S (decrease in potential)?

S = 2.5 P*=2

S = 1.0 P*=2

P<2 = beat & P>2 = no beat ( Heart beats less as we increase S)


Bifurcation of s what happens when we increase a e dt l.jpg
Bifurcation of SWhat happens when we increase a = e-DT?

e-DT= 0.2

e-DT = 0.8, DT ↓

more skipped beats

P<2 = beat & P>2 = no beat (heart beats less if we increase a)


3 d plot of 2 par vs p l.jpg
3-D plot of 2-par vs. P

For small S and a more beats occur & for large S and a more skips occur

P* = 2

Below the threshold, beats occur

Above the threshold, no beats occur


Fraction of skipped beats l.jpg
Fraction of Skipped Beats

irregular heart beats

irregular heart beats

regular heart beats

regular heart beats


Conclusion l.jpg
Conclusion

  • Our model did produce the several different beating patterns given assumptions

  • We were able to show how varying the parameters changes the beating patterns

  • However, this is a very simple model, only taking into account AV node as regulator of heart beating. This model does not take into account values of actual parameters of heart (e.g. S not a constant increase in potential), or other parts of the heart that might influence the beating (e.g. if the SA node fails)


Acknowledgements l.jpg
Acknowledgements

  • Frithjof Lutscher

  • Gerda De Vries

  • Alex Potapov

  • Andrew Beltaos

  • PIMS

We’re done!!!! On to the barbeque!!!!


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