CARDIOVASCULAR FLUID MECHANICS LABORATORY
1. The Windkessel Model
The Windkessel lumped parameter model is a simplified way to simulate the pressure
waves in the cardiovascular system. It does so by “lumping” the total effects of the
compliance and resistance of the natural system into a set of mechanical components
that reproduce the pulsatile flow. In general, a mechanical system that uses the
Windkessel model is composed of a pump that simulates the heart, a compliance
chamber simulating the elastic response of the aorta and main arteries, and a resistive
element simulating the resistance to flow caused by arterioles and capillaries. These
three elements together can approximate the pressure and flow waves observed in a
natural cardiovascular system.
Figure 1 shows the main mechanical components of the Windkessel model applied to
a left heart simulator setup. In the lower region of the figure is the left ventricle
labeled as LV. The contraction movement of the ventricle (systole) produces the
mitral valve (MV) to close at the same time the aortic valve (AV) opens. The fluid
pushed by the left heart simulator goes to a compliance chamber (C) partially filled
with air. The air in this chamber allows the fluid pushed by the left ventricular pump
to fill part of the compliance chamber while other portion of the fluid is directed
through the resistive element (R) that simulates the peripheral resistance caused by
capillaries in the natural system.
C
R
MV
AV
LV
Figure 1 Mechanical components of the Windkessel model. LV –
Left ventricle. C – Compliance Chamber. R – Resistive
element. MV and AV are the Mitral and Aortic valves
respectively.
The same mechanical system can be represented with an analogous electric circuit
consisting of a source of current (flow), a capacitor (compliance) and a resistor (flow
resistance). Figure 2 shows the schematic of the circuit.
P
Qc
LV
Q(t)
QR
C
R
Figure 2 Electric analog to the Windkessel model. Pressure (P) is analogous
to voltage and Flow (Q) is analogous to current. The flow source
resembling the heart is an alternating square wave source.
The capacitance and resistance of the circuit can be expressed in their mathematical
form, and by using kirchhoff’s laws for current and voltages, the differential equation
modeling the system can be obtained.
For the resistance
P  QR  R
Equation 1
For the Capacitance
dP
QC  C 
dt
Equation 2
Kirchhoff’s current law
Qt   QC  QR
Equation 3
Substituting Eqn. 1 and 2 in Eqn. 3, the differential equation of the system is obtained
dP
P
C
 Q(t ) 
dt
R
Equation 4
The general solution for the differential equation is
 t /( RC )
t
1
P(t )  e t /( RC )  Q(t )e /( RC ) d  Pn e
0
C
Equation 5
Q(t) can be approximated as a discontinuous function (square wave) composed of two
regions, the general solution for the differential equation (Eqn. 5) can be broken down
in two equations each one describing one portion of the cardiac cycle. Top graph in
Figure 3 shows the flow waveform.
Flow
Qo
Time
Pressure
Eqn. 9
Pts
Eqn. 6
PT
Po
Systole
Diastole
Time
ts
T
Figure 3 Flow waveform top and Pressure waveform bottom. Both
graphs have been plotted against time. Times from 0 to ts
correspond to systole, and times from ts to T correspond to
diastole.
During Systole, for times 0 ≤ t ≤ ts the flow is constant with a value of Qo. Replacing
Qo in the general solution (Eqn. 5) the pressure wave describing the systolic pressure
is obtained as a function of time.
P(t )  RQo  ( RQo  Po )e t / RC
Equation 6
During Diastole, for times ts ≤ t ≤ T the flow is zero. After substituting Q(t)= 0 in
equation 5 its first term is cancelled leaving
P(t )  Pn e  t /( RC )
Equation 7
In Equation 7 Pn is constant. Its value can be found by substituting into the equation
any of the two boundary conditions defined either at ts or T. Using the boundary
condition at T, Pn is found to be equal to
Pn  PT eT /(RC )
Equation 8
Substituting Eqn. 8 into Eqn. 7 the pressure wave during diastole is obtained as a
function of time.
P(t )  PT e(T t ) /( RC )
Equation 9
Note: Alternatively to Equation 9 another equation describing diastole could have
been obtained by using the boundary conditions given by the point (Pts, ts) to obtain a
different value for Pn in equation 8.
Equations 6 and 9 together represent a single cardiac cycle occurring every T seconds.
Figure 3 (bottom graph) shows the general pressure wave generated by equations 6
and 9 during each of their corresponding time domains.
2. Left Heart Simulator
Introduction
The left heart simulator (Vivitro Systems) in the Cardiovascular Engineering Center is
a complete system for the laboratory assessment of replacement heart valves. The
system includes a model left heart, fluid pump, pressure and flow instrumentation,
and computer hardware and software for the monitoring acquisition and analysis of
test data.
The components of the system
1. Superpump and Pump regulator.
Figure 4 Superpump
The Superpump system consists of a piston-in-cylinder pump head driven by low
inertia electric motor. A linear actuator converts rotary motion of the motor to linear
displacement of the piston using a lead screw. The motor is driven by a power
amplifier. Position and velocity transducers provide feedback to the amplifier.
The system includes amplifier, linear actuator, motor, pump head, motor cooling fan,
and interconnecting cables.
The desired flows are generated by input of the appropriate waveform to the power
amplifier which can be switched to either position or velocity control.
2. Left heart simulation system
It is a compact model of the left heart system designed for assessing heart valve
function. The system can be rapidly filled and debubbled and drained. Transparent
materials allow visualization of valve action. Aortic root and characteristic
compliance are simulated. The system includes a variable peripheral resistance.
e
a
d
f
c
b
a.
b.
c.
d.
e.
f.
Figure 5 Left heart system
Atrium chamber
Ventricular chamber
Mitral valve position
Aortic valve position
Aortic chamber and compliance elements
Transducer ports
Peripheral resistance regulator
Figure 6 Peripheral resistance regulator
The peripheral resistance knob is a device that increases or reduces obstruction to
flow depending on the direction that the knob is turned.
General compliance chamber
Figure 7 Compliance chambers
Two compliance chambers simulate the whole compliance of the cardiovascular
system, the one on the left is connected to the left heart system right above the aortic
valve; and the one on the right is connected to the aortic chamber.
3. Pressure Transducers and control unit
Figure 8 Transducer and control unit
 Pressure transducer: Millar Micro-tip catheter transducers, model: SPC-320
 Control unit: TC-510 Pressure Control Unit
The TC-510 Pressure Control Unit is a passive interface between the Millar MikroTip catheter pressure transducer and strain gauge pressure monitors which supply
bridge excitation voltage and have balance and calibration controls for full-bridge
strain gauge pressure transducers. The TC-510 has an internal calibration circuit to
provide an electrical zero as well as a 100 mmHg calibration signal.
6. Electromagnetic flowmeter and transducer
Figure 9 Flowmeter
The flow measuring system consists of a single channel square wave flowmeter
(FM501 Carolina Medical) and an electromagnetic flow transducer. The standard
transducer has internal diameter 28mm for use in the Vivitro Systems model heart and
load. Five switch selectable low pass output filters are available. The system is
supplied with a calibration performed under dynamic flow conditions.
7. Programmed waveform generator
Figure 10 Screen of waveform generator computer
The Vivitro systems programmed waveform generator is capable of generating
complex waveforms at several different frequencies, ranging from 30- to 200 cycles
per minute. The unit is designed to provide input of physiological waveforms to the
Vivitro system’s Superpump system.
8. Main acquisition computer
 Acqknowledge 3.7.3 is the software used to collect data
Figure 11 Screen of data acquisition system (Biopac)
3. Experiment Preparation
1. Make Solution: blood analogue fluid
(35% Glycerin + 65% De-ion water+ 9 gram NaCl)/ Liter; percentage by Volume.
Density 1.133 g/ml, viscosity 3.2 cP at 37ºC.
2. Turn on Left Heart Simulator System
 Flow meter need to be warmed up for 30 minutes before experiment
 Caution – do not turn the flow meter to the “+” or “-“ position unless there is
flow through the system
3. Fill the atrium and ventricular chamber
 No air bubble in every chamber and tubing
4. Place pressure transducers into position: through ventricular and aortic ports.
4. Protocol
1. Open Acqknowledge 3.7.1 and set up the data acquisition channels:
 Channel 1: Aortic pressure
 Channel 2: Flow meter
2. Calibrate the pressure transducer and the flow meter – see operator manual for each
3. Measure and record the baseline number of each channel
4. Testing condition:
 Pump rate: 70 beats/Min; flow rate: 6 L/Min
 Set the pulse generator to 70 beats/min
 Adjust Peripheral resistance gauge to reach 80-120 mmHg
 Adjust the Amplitude of Superpump to reach 6 L/Min
1. Set the system to physiological
conditions: 120 / 80 mmHg & 70 BPM.
Record the resistance: R1
Collect Data:
R2-C1
Collect Data:
R1-C1
2. Turn the peripheral resistance knob to
increase the resistance. Record the
change in resistance: R2
3. Turn the peripheral resistance knob
again to obtain a resistance lower
than the original. Record the
resistance: R3
Collect Data:
R3-C1
4. Add 30cc of DI water to the General
Compliance Chamber (# 16). This will
serve as compliance C2. Collect data
for the same three resistance values as
before: R1, R2 and R3.
5. Add an additional 60cc of DI water to the
General Compliance Chamber (#16).
(Total volume 90cc more than original)
This will serve as compliance C3.
Collect the data for the same three
resistance values as before: R1, R2, and
R3.
Collect Data:
R1-C2
R2-C2
R3-C2
Collect Data:
R1-C3
R2-C3
R3-C3
5. Windkessel modeling exercise
In this exercise you will use the data collected during the lab to optimize the
Windkessel model of the aorta. The Simulink model numerically solves the
differential equation directly (equation 4 in the handout) rather than using the
analytical solution (equations 6 and 9 in the handout). To model your data collected
from the Vivitro system, you must copy three files (WindkesselOp_lab.m,
Windkessel_lsq_lab.m and simulink_Windkessel.mdl) into your Matlab workspace.
You must also copy the data files that you collected during the lab into the same
workspace. Note that the filenames for the data files must be valid Matlab variable
names (with .txt or .dat extensions). You will need to open the model
simulink_Windkessel.mdl in Simulink. Open the two Matlab programs into the editor
window of Matlab. Change the text on line 3 of WindkesselOp_lab.m and line 4 of
Windkessel_lsq_lab.m to the name of your data file. The data file must be a tabdelimited text file. The first column must contain the pressure data. The data file must
contain at least five complete cardiac cycles starting at the beginning of systole of the
first pulse. The sample rate must be 200/s. Input into the pulse generator block of the
Simulink_Windkessel.mdl model the cardiac period, systolic period, and flow rate
that were used for each experimental run that you recorded during the lab. Run the
Matlab program WindkesselOp_lab.m which will load the data file, initialize the
variables, and call Windkessel_laq_lab.m and the Simulink model to iteratively solve
the model and optimize on the model coefficients R and C. Repeat this for each run
from the lab exercise. For each run, record the optimized R and C coefficients and
print the plot of the arterial pressure data and fit curves. With the R and C values from
each run, solve equations 6 and 9 in the lab handout and compare the analytical
solutions to the solutions in which the differential equation was solved directly.
Explain any differences that you observe.
6. Lab reports
 Please read the general directions for writing lab reports
1. In the background section of the report don’t forget to list and explain the
assumptions used in the windkessel model.
2. From each file containing the specific information for an R-C obtain the
average aortic pressure waveform that represents such R-C combination, and
determine the mean flow.
3. Use the Matlab program provided (See instructions on section 5) to fit the
average waveforms to the windkessel model equation and obtain their
corresponding values for R and C.
4. Show your results with graphs comparing the experimental data with the fitted
model and use tables to display the values for R, C and mean flow.
5. Analyze your results and evaluate the accuracy of the model (Hint: use the
assumptions made for the model to support your analysis.) How could the
Simulink model be modified to better simulate the actual experiment?
6. If a patient has an aneurysm, how would that change the curve and the
coefficients R and C?
7. If a patient has severe atherosclerosis that affects the root of the aorta, how
would that change the curve and the coefficients R and C?
8. If a patient has an artificial aortic valve that contains a stent that extends into
the root of the aorta, how would that change the curve and the coefficients R
and C?
7. Note
1. The resistance values recorded during the experiment have no relevance in the
Windkessel model. When using the Matlab program, do not input these
values. The point of recording these values was to ensure the same resistance
measurements were repeated for each of the different compliances.
2. The numerical data does not provide the time of the measurement, you need to
add this. Remember that the sample rate is 200 measurements per second.