This document describes the development of a dynamic model of blood circulation and pressure regulation. It involves modeling the heart as a flow source, modeling the vasculature as a linear graph with nodes for major arteries and veins, and deriving equations to describe pressure at each node based on compliance, resistance, and flow. The model does not include breathing effects and simulates a closed-loop system to study how the heart rate and arterial blood pressure respond to delayed signals from pressure receptors.

1. Sung Hoon (Steven) Yoon

2. Problem: blood
pressure = MAP
“Mean” Arterial
Pressure
Static value
Solution: blood
circulation dynamic
model
MAP(t)
Dynamic value
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)
Closed-Loop
Control System
Circulation
Loop
Dynamic System Vasculature
Control System
Input
Arterial
blood pressure
Controller Central
Nervous System
Sensor Baroreceptors
Actuator Aorta
Dynamic System
Input
Blood
Flow
Dynamic System
Output
Blood
Pressure

3. 𝑄 𝑎𝑜 𝑡 is the flow
source for the left
ventricle
Stroke Volume
70.5 mL
Basal Heart Rate
80 beats per minute
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)
Duration of heartbeat (s) 𝑇 0.75
Duration of systole (s) 𝑇𝑠 0.4𝑇
Time @ Maximum Flow (s) 𝑇 𝑚𝑎𝑥 0.16𝑇
Flow @ Maximum flow (L/s) 𝑄 𝑚𝑎𝑥 0.47
function [Qao,tt] = genQao(t)
T = 0.75; % duration of heartbeat (s)
tt = mod(t,T); % tt cycle from 0 to T
Ts = 0.4*T; % duration of systole (s)
Tmx = 0.16*T; % time @maximum flow (s)
Qmx = 0.47*10^3; % flow @maximum (L/s)
if tt >=0 && tt < Tmx,
Qao = Qmx*tt/Tmx;
elseif tt >= Tmx && tt < Ts,
Qao = Qmx*(Ts-tt)/(Ts-Tmx);
else
Qao = 0; % third section
end

4. Dynamic modeling
using linear graph:
Pressure Output
Flow Input
Defined 5 nodes,
including:
Systemic artery
Splanchnic vein
Right atrial
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)

5. Derived equations
from linear graph
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)
𝐹𝑠𝑎 =
𝑃𝑠𝑎 − 𝑃𝑠𝑝
𝐿 𝑠𝑎
−
𝑅 𝑠𝑎
𝐿 𝑠𝑎
𝐹𝑠𝑎
𝑃𝑠𝑎 =
1
𝐶𝑠𝑎
(𝑄 𝑎𝑜 − 𝐹𝑠𝑎)
𝑃𝑠𝑝 =
1
𝐶𝑠𝑎 + 𝐶𝑒𝑝
(𝐹𝑠𝑎 −
𝑃𝑠𝑝 − 𝑃𝑠𝑣
𝑅 𝑠𝑝
−
𝑃𝑠𝑝 − 𝑃𝑒𝑣
𝑅 𝑒𝑝
)
𝑃𝑠𝑣 =
1
𝐶𝑠𝑣
(
𝑃𝑠𝑝 − 𝑃𝑠𝑣
𝑅 𝑠𝑝
−
𝑃𝑠𝑣 − 𝑃𝑟𝑎
𝑅 𝑠𝑣
)
𝑃𝑒𝑣 =
1
𝐶𝑒𝑣
(
𝑃𝑠𝑝 − 𝑃𝑒𝑣
𝑅 𝑒𝑝
−
𝑃𝑒𝑣 − 𝑃𝑟𝑎
𝑅 𝑒𝑣
)

6. Compliance and
Resistance of veins
and arteries
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)
Signal Profile
Aortic Valve Highly compliant
Systemic Artery Compliant
Splanchnic Vein Non-compliant (constant) ; 3.45 – 3.55 mmHg
Right Atrium Slightly compliant (nearly constant); 1.5 – 3 mmHg

7. No breathing
effects
Closed-Loop
System
heart rate (bpm)
arterial blood
pressure (mmHg)
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)

8. Heart Rate
Mean Arterial
Pressure Regulation
Delayed Signals from
Pressure Receptors
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science 321.5887 (2008)