1. ENT 318/3 Artificial Organs
Modeling of cardiovascular system and VAD
Lecturer
Dr. Nashrul Fazli Mohd Nasir

2. What is modeling and why we need it?
In designing product, sometimes we have to make sure that the device we designed will works theoretically before applied to real situation.
Modeling is where we use a physical, mathematical or logical representation of a system of entities, phenomena or processes.
In case of modeling a VAD, we also need to model the cardiovascular system since VAD is designed to work with the heart.
Simulation is where we run our model to study how our model works and examine the result of any parameters that we desire.
For example, if we want to evaluate any closed-loop control system, we model our system with Simulink and run it to simulate the result.
Usually, mathematical and electrical analogy is used to model physiological system including our heart.

3. Electrical Analogy
Kirchoff’s law of current and potential can be applied:
1) Sum of currents entering any junction equal sum of currents leaving that junction (conservation of blood mass)
2) Sum of all voltages around a loop equals zero (pressure is potential difference)

4. Electrical Analogy

5. Electrical Analogy
Blood flowing from wider arteries into smaller arterioles encounters a certain resistance.
Consider an ideal segment of a cylindrical vessel. The pressure difference between its two ends and the flow through the vessel depend on each other.
It can be accurately approximated by a linear relation.
If we indicate by Rc the proportionality constant between the pressure difference P and the flow F then we can write

6. Electrical Analogy
Similarly, a resistor is an electronic component that resists an electric current by producing a potential difference between its end points.
According to with Ohm’s law, the electrical resistance Re is equal to the potential difference V across the resistor divided by the current I through the resistor.

7. Electrical Analogy
The walls of the vessels are surrounded by muscles that can change the volume and pressure in the vessel.
Consider the blood flow into such an elastic (compliant) vessel.
We denote the flow into the vessel by Fi and the flow out of the vessel by Fo.
Then the difference F = Fi − Fo which corresponds to the rate of change of blood volume in the vessel is related to a change of pressure P inside the vessel.
Assuming a linear relation, we have that

8. Electrical Analogy
A capacitor is an electrical device that can store energy between a pair of closely-spaced conductors.
When a potential difference is applied to the capacitor, electrical charges of equal magnitude but opposite polarity build up on each plate.
This process causes an electrical field to develop between the plates of the capacitor.
It gives rise to a growing potential difference across the plates.
This potential difference V is directly proportional to the amount of separated charge Q (e.g. Q = CeV ).
Since the current, I through the capacitor is the rate at which the charge Q is forced onto the capacitor (e.g. I = dQ/dt), this can be expressed mathematically as

9. Electrical Analogy
Blood is inert, when a pressure difference is applied between the two ends of a long vessel that is filled with blood, the mass of the blood resists the tendency to move due to the pressure difference.
Once more assuming a linear relation between the change of the blood flow (dF/dt ) and the pressure difference P we can write
 inductance is the property of a conductor by which a change in current in the conductor "induces" (creates) a voltage (electromotive force) in both the conductor itself (self-inductance)

10. Electrical Analogy
The inertia of blood can be modeled by a coil (also known as an ’inductor’)
The current in a coil cannot change instantaneously.
This effect causes the relationship between the potential difference V across a coil with inductance Le and the current I passing through it, which can be modeled by the differential equation

11. Electrical analog A segmented, arbitrary length of vessel.
The equivalent RLC circuit.

12. Modeling of Cardiovascular System and VAD
Windkessel Model
Models consist of ordinary differential equations that relate dynamics of aortic pressure and blood flow to various parameters such as arterial compliance, resistance to blood flow and inertia of blood.
2-Element Windkessel Model

13. Modeling of Cardiovascular System and VAD
P and F represents aortic pressure and blood flow rate in aorta.
C is arterial compliance.
R corresponds to resistance to blood as it passes from aorta to arterioles.
Rearranging we get,

14. Modeling of Cardiovascular System and VAD
Solving the equation by considering diastole period where F=0, we obtain,
Where P(td) is blood pressure in aorta at starting time of diastole, td
3-Element Windkessel Model

15. Modeling of Cardiovascular System and VAD
Addition of Ra which represents resistance encountered by blood as it enters aortic or pulmonary valve.
Applying Kirchoff law to the model we obtain,
Solving this equation when F=0 we get the same as previous model.
4-Element Windkessel Model

16. Modeling of Cardiovascular System and VAD
Coil Lc is added to represent inertia of blood.
Again applying Kirchoff law to the model we obtain,

17. Model 1 Electrical Analog Models Reference :
Y.-C. Yu, J.R. Boston, M. Simaan, and J.F. Antaki, “Estimation of systemic vascular bed parameters for artificial heart control”, IEEE Trans. Automat. Contr., vol. 43, pp , 1998.

19. Left ventricle as time varying capacitance.
Systemic circulation as four-element modified Windkessel model.
Pulmonary circulation and left atrium as single capacitance.
Heart valves as diode in series with resistors.
Ejection – DA on and DM off. Blood rapidly ejected to aorta.
Isovolumic Relaxation – DA and DM off. Ventricle relax and pressure falls.
Passive Filling – DA off and DM on. Blood from venous circulation returns to ventricle.
Isovolumic Contraction – DA and DM off. Ventricular increases rapidly without change in volume.

21. Model 2

22. A time varying nonlinear circuit model of a combined cardiovascular system and pump used to test feedback controller.
Pressure = v, flow = I.
Left ventricle is modeled as a time varying E (t). Mitral valve and aortic valve modeled as pressure dependent diodes (or switches) DM and DA.
Suction is modeled as a nonlinear resistor Rk and SVR as a linear resistor Rs.
Other components are constant capacitors and inductors.

23. Model 3 Electrical Analog Models Reference :
Y. Wu, P. Allaire, G. Tao, and D. Olsen, “Modeling, Estimation and Control of Cardiovascular System with A Left Ventricular Assist Device”, 2005 American Control Conference, pp , June 8-10, 2005.

24. Modeling of Cardiovascular System and VAD
Model Element
Physiological Meaning
CLV
Left ventricle CA
Aorta CV
Systemic vein and right atrium D1
Aortic valve D2
Mitral valve
TPR
Total peripheral resistance L
Blood inertia
RLV
Resistance of aortic valve RV
Resistance of vein and left atrium δm
Mean pressure disturbance generated by muscle pump δp
Pressure disturbance by pulmonary circulation

25. Modeling of Cardiovascular System and VAD
Variables
Physiological Meaning
PLV
Left ventricular pressure PA
Aortic pressure PV
Central venous pressure Q
Pump flow rate
TPF
Total peripheral flow rate

26. Model 4 Electrical Analog Models Reference :
J. Porter and Y.-C. Yu, “Pressure-Flow Modeling of A Rotary Ventricular Assist Device”, Bioengineering Conference 2006 Proc. of the IEEE 32nd Annual Northeast , pp

27. Modeling of Cardiovascular System and VAD
Electric circuit model of left heart with VAD.
CLV represents pumping action of heart.
RPin and LPin is VAD inflow resistance and inertance.
RPout and LPout is VAD outflow resistance and inertance.
Voltage source represents hydrostatic pressure generated by pump.

28. Mock Circulation Loop (MCL)
In vitro tool for evaluation of cardiovascular device design including the ventricular assist device.
Simulates human blood circulatory system.
Usually designed in form of hydraulic analogy of the blood circulation system.
Comprehensive knowledge in cardiovascular anatomy and hemodynamics is required to develop an accurate MCL.

29. Modeling of Cardiovascular System and VAD

30. Modeling of Cardiovascular System and VAD
MCL contains heart and vascular components of systemic and pulmonary circulation.
Systemic and pulmonary loop are in series.
Easily variable vascular parameters to dictate natural hemodynamic values.
Functional parameters from natural cardiovascular system to reproduce expected hemodynamic characteristic for each physiological condition.
Pneumatically actuated ventricular chambers representing the heart.
Open-to-atmosphere atria to replicate atrial compliance.
Atrial chambers change fluid volume in response to venous return.
Compressed air applied to ventricular chambers during systole and vented during diastole.

31. Modeling of Cardiovascular System and VAD
There is controller board to control ventricular contraction.
3/2 solenoid valve used in the air driveline.
When solenoid ‘on’, compressed air allowed to the chamber for systole.
When solenoid ‘off’, air pressure vented for diastole.
Solenoid switching determine heart rate.
Reference:
D. Timms, M. Hayne, K. McNeil and A. Galbraith, “A Complete Mock
Circulation Loop for the Evaluation of Left, Right, and Biventricular
Assist Devices”, Artificial Organs , Vol. 29 No. 7, pp , 2005.

32. Modeling of Cardiovascular System and VAD

33. Two check valves to simulate mitral valve and aortic valve.
Two cardiac simulators to simulate pulsatile functions of both ventrilces.
Two check valves to simulate mitral valve and aortic valve.
Pneumatic control box controlled air pressure inside the air chamber of cardiac simulator.
Pneumatic control box can be used to adjust heart rate and systolic ratio.
Three airtight tanks represents systemic arterial, systemic venous and pulmonary compliances.
Reference :
Y. Liu, P. Allaire, H. Wood and D. Olsen, “Design and Initial Testing of a Mock Human Circulatory Loop for Left Ventricular Assist Device Performance Testing”, Artificial Organs , Vol. 29 No. 4, pp , 2005.

34. Modeling of Cardiovascular System and VAD

35. Modeling of Cardiovascular System and VAD
Mock simulator of the left side of the cardiovascular system.
The ventricle pumps the saline solution in the aorta through the aortic valve (AV).
Aorta is connected to a thin tube, simulating the arterial resistance (APR) and then to the venous reservoir system and to the atrium.
This one is then linked to the mitral valve (MV) and to the ventricle.
Ventricular and aortic pressures (VP, AoP), aortic flow (AoF) are measured by dedicated transducers inserted in the simulator
Reference :
R. Zannoli, I. Corazza, and A. Branzi, “Mechanical simulator of the cardiovascular system”, Physica Medica , Vol. 25, pp ,