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) The heart is a muscle: functions as a pump (circulates nourishment and oxygen to, and CO 2 and waste away) 4 chambers: atria (input) and ventricles (output), upper and lower separate by valves 4 chambers: atria (input) and ventricles (output), upper and lower separate by valves SA node: groups of cells on upper right atrium SA node: groups of cells on upper right atrium AV node: between the atria and ventricles w/ in right atrial septum 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) Contractions of heart controlled by electrical impulses (generated primarily by SA node, pacemaker cells) Fires at a rate which controls the heart beat Fires at a rate which controls the heart beat Naturally discharge action potentials per m Naturally discharge action potentials per m Input to the AV node comes from the A.P. propagating through atria from SA node 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 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

Goal: Model Electrical Potential of the AV node 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

Model of the electrical potential of AV node [P t + S] e -DT P t < P * [P t + S] e -DT P t < P * P t+1 = P t+1 = P t e -DT P t > P * P t e -DT P t > 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? What are some different patterns of heart beats? Parameters: How many? Which could be varied? What does varying them mean? What are the ranges? 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? 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 beat = 1, no beat = 0Potential is steady at S=3, e -DT =1, Po=1, P*= 2

Plot of P vs. t Second-degree block beat=1, no beat=0Potential bounces between 2 values S=2.5, e -DT =1, Po=.4, P*= 1

Plot of P vs. t Wenckebach Phenomenon The heart beats 3 and skips 1 : beat=1, no beat=0 Potential bounces between 4 values (3 below threshold) S=3, e -DT =1, Po=1, P*= 1.66

Cobwebbing (visualizing orbits and long term behavior) right: normal (stable fixed point) left bottom: 2 nd deg. block (2 cycle) right bottom: Wenckebach (4 cycle) S=3 e -DT =1 P* = 2 Po = 1 S=3 e -DT =1 P* = 1.66 Po = 1 S=2.5 e -DT =1 P* = 1 Po =.4 P = S e -DT /( 1- e -DT ) 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)? S = 1.0 P*=2S = 2.5 P*=2 P 2 = no beat ( Heart beats less as we increase S)

Bifurcation of S What happens when we increase a = e -DT ? e -DT = 0.8, DT ↓ more skipped beats e -DT = 0.2 P 2 = no beat (heart beats less if we increase a)

3-D plot of 2-par vs. P P* = 2 Below the threshold, beats occur Above the threshold, no beats occur For small S and a more beats occur & for large S and a more skips occur

Fraction of Skipped Beats regular heart beats irregular heart beats regular heart beats irregular heart beats

Conclusion Our model did produce the several different beating patterns given assumptions Our model did produce the several different beating patterns given assumptions We were able to show how varying the parameters changes the beating patterns 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) 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!!!!