Download HowTheHeartBeats

How the heart beats:
A mathematical model
Minh Tran and Wendy Cimbora
Summer 2004 Math Biology Workshop
Anatomy of the Heart
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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)
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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
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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
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Pt+1 =
[Pt + S] e-DT
Pt < P*
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
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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. t
Second-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. t
Wenckebach 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)
P = S e-DT /( 1- e-DT )
right: normal (stable fixed point)
left bottom: 2nd deg. block (2 cycle)
right bottom: Wenckebach (4 cycle)
S=2.5 e-DT=1 P* = 1 Po = .4
P = S e-DT /( 1- e-2DT )
S=3 e-DT=1 P* = 1.66 Po = 1
P = 3S e-3DT /( 1- e-4DT )
Bifurcation of a = e-DT
What 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 S
What 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
regular heart
beats
irregular heart
beats
regular heart beats
Conclusion
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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
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Frithjof Lutscher
Gerda De Vries
Alex Potapov
Andrew Beltaos
PIMS
We’re done!!!! On to
the barbeque!!!!