Download Will This Heart Skip a Beat Activity

Will this Heart Skip a Beat?
Background Information
The heart acts as a pump which converts
electrical energy to mechanical energy.
The heart is divided into four chambers,
two atria and two ventricles. The atria
receive blood from throughout the body.
Blood is then pumped through the
ventricles and out of the heart to the
rest of the body. The Sinoatrial (SA)
node is the pacemaker of the heart and
sends signals to the Atrioventricular
(AV) node. The AV node receives the
signal from the SA node and sends a
signal to the ventricles if the heart
should beat.
Symbols for use in the model:
𝑉 = potential of the AV node
𝑉𝑐 = threshold potential / critical potential of the AV node
𝑢 = signal arriving from the SA node to the AV node, every τ seconds.
For units, we will us mV (millivolts) and seconds.
We can use an equation to model the activity of the AV node:
𝑉𝑡+1 {
𝑐 ∙ 𝑉𝑡
∶ 𝑉𝑡 > 𝑉𝑐 (𝑛𝑜 𝑏𝑒𝑎𝑡)
𝑐 ∙ 𝑉𝑡 + 𝑢 ∶ 𝑉𝑡 ≤ 𝑉𝑐 (𝑏𝑒𝑎𝑡)
Adopted from:
Champanerkar, J., & Eladdadi, A. (2011). Mathematics and the Heart Workshop for High School Teachers. 2011 NHLBI-VCU World Conference on
Mathematical Modeling and Computer Simulation in Cardiovascular and Cardiopulmonary Dynamics. Williamsburg, PA.
Situation 1:
O The threshold voltage of the AV nodes is 20 mV.
O The potential at the AV node is 30 mV now.
O The potential at the AV node reduces to half its original between the two
signals arriving from the SA node.
O The SA node sends signals of strength 10 mV.
In Symbols:
𝑉𝑐 = 20.0 𝑚𝑉
𝑉𝑡 = 30.0 𝑚𝑉
𝑐 = 0.5
𝑢 = 10 𝑚𝑉
Will the heart beat? (Show your reasoning below) ___________
Continue modeling the heart, will the heart beat a second time? __________
Assume what we had before, except we will vary our c value.
1. 𝑐 = 0.6—The electric potential at the AV node dissipates to 60% of its original
value between 2 signals from the SA node.
2. 𝑐 = 0.7
3. 𝑐 = 0.8
Further exploration:
Create a Microsoft Excel Spreadsheet that will perform the calculations automatically
modeling the heart’s activity.
Developing Further:
Equilibrium
Equilibrium is a value of the potential of the AV node where the decay (by a factor of c) is
exactly balanced by the response to the signal (increase by u). This will mean that the
heart will beat steadily.
Doesn’t this mean our potential (V) will remain the same each time? So…
𝑉 = 𝑉𝑡 = 𝑉𝑡+1 = ⋯
Generalizing, 𝑉 = 𝑐 ∙ 𝑉 + 𝑢
We are aiming to find our equilibrium potential, so can we solve for V? (Hint: You will
need to factor!)
Return to our previous first example. Does equilibrium exist? (Show using the model you
just created)
If it exists, what is the equilibrium value? __________
Did your Excel Spreadsheet suggest this result? Explain.