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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

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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**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**[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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