How the heart How the heart beats: A beats: A

mathematical mathematical modelmodelMinh Tran and Wendy Minh Tran and Wendy

CimboraCimbora

Summer 2004 Math Biology Summer 2004 Math Biology WorkshopWorkshop

Anatomy of the HeartAnatomy of the Heart The heart is a muscle: The heart is a muscle:

functions as a pump functions as a pump (circulates nourishment (circulates nourishment and oxygen to, and COand oxygen to, and CO22 and waste away)and waste away)

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

SA node: groups of cells SA node: groups of cells on upper right atriumon upper right atrium

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

Control via the SA node Control via the SA node (pacemaker)(pacemaker)

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

Fires at a rate which controls the heart beatFires at a rate which controls the heart beat Naturally discharge action potentials 70-80 Naturally discharge action potentials 70-80

per mper m Input to the AV node comes from the A.P. Input to the AV node comes from the A.P.

propagating through atria from SA nodepropagating through atria from SA node Then travels to the Bundle of His and Then travels to the Bundle of His and

Purkinje fibers, causing heart to contractPurkinje fibers, causing heart to contract

Simplified Heart Beat Simplified Heart Beat ProcessProcess

SA node firesSA node fires Electrical potential Electrical potential

travels to AV nodetravels to AV node We are concerned We are concerned

primarily with the primarily with the AV nodeAV node

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

Goal: Model Electrical Goal: Model Electrical Potential of the AV nodePotential of the AV node

Assumptions for our modelAssumptions for our model: :

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

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

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

Model of the electrical Model of the electrical potential of AV nodepotential of AV node

[P[Ptt + S] e + S] e-DT -DT PPt t << P P**

PPt+1 t+1 = = PPt t ee-DT -DT PPt t > P> P**

P P = electrical potential of AV node= electrical potential of AV nodeS S = constant increase of electrical potential of AV node= constant increase of electrical potential of AV nodeD D = rate of decrease (recovery rate of heart) = rate of decrease (recovery rate of heart) T T = time interval between firing from SA node= time interval between firing from SA nodePP** = threshold (determines normal/abnormal = threshold (determines normal/abnormal

beats) beats)

Burning QuestionsBurning Questions

What are some different patterns of heart What are some different patterns of heart beats?beats?

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

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

Plot of P vs. t Plot of P vs. t Normal heart rateNormal heart rate

beat = 1, no beat = 0

Potential is steady at 1.7459

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

Plot of P vs. tPlot of P vs. tSecond-degree blockSecond-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. tPlot of P vs. tWenckebach PhenomenonWenckebach 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 Cobwebbing (visualizing (visualizing orbits and long orbits and long term behavior)term behavior)right: normal (stable fixed right: normal (stable fixed point) point) left bottom: 2left bottom: 2ndnd deg. block (2 deg. block (2 cycle)cycle)right bottom: Wenckebach (4 right bottom: Wenckebach (4 cycle)cycle)

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

S=3 e-DT=1 P* = 1.66 Po = 1S=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 Bifurcation of a = = ee-DT-DT

What happens when lower S What happens when lower S (decrease in potential)?(decrease in potential)?

S = 1.0 P*=2

S = 2.5 P*=2

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

Bifurcation of SBifurcation of SWhat happens when we increase What happens when we increase

a = ea = e-DT-DT??

e-DT = 0.8, DT ↓

more skipped beats

e-DT= 0.2

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

3-D plot of 2-par vs. P3-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 Fraction of Skipped BeatsBeats

regular heart beats

irregular heart beats

regular heart beats

irregular heart beats

ConclusionConclusion Our model did produce the several different Our model did produce the several different

beating patterns given assumptionsbeating patterns given assumptions We were able to show how varying the We were able to show how varying the

parameters changes the beating patternsparameters changes the beating patterns However, this is a very simple model, only However, this is a very simple model, only

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

AcknowledgementsAcknowledgements

Frithjof LutscherFrithjof Lutscher Gerda De VriesGerda De Vries Alex PotapovAlex Potapov Andrew BeltaosAndrew Beltaos PIMSPIMS

We’re done!!!! On to the

barbeque!!!!