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# How the heart beats: A mathematical model - PowerPoint PPT Presentation

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

• 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

• 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

• 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 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)

• 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

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)

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

irregular heart beats

irregular heart beats

regular heart beats

regular heart beats

• 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)

• Frithjof Lutscher

• Gerda De Vries

• Alex Potapov

• Andrew Beltaos

• PIMS

We’re done!!!! On to the barbeque!!!!