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How the heart beats: A mathematical modelPowerPoint Presentation

How the heart beats: A mathematical model

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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

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 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)

- 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

- 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

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 nodeModel 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

- 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

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

Potential is steady at 1.7459

beat = 1, no beat = 0

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. 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

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-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 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

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

irregular heart beats

irregular heart beats

regular heart beats

regular heart beats

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

- Frithjof Lutscher
- Gerda De Vries
- Alex Potapov
- Andrew Beltaos
- PIMS

We’re done!!!! On to the barbeque!!!!

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