Arterial Resistance as a Function of Compliance
BE 310 Final Project
Group T1
Mike Dolan
Sigmund Kulessa
David Thakker
Bryan Wells
BACKGROUND
Biological Relevance
Basic physiological changes occur in the circulatory system as a result of
increased age. The large arteries gradually dilate and stiffen. These changes come about
because of progressive degeneration of the vascular wall, a process quite independent of
the disease, atherosclerosis. Collagen and ground substance increase, the elastic fibers of
the media become fragmented, and the collagen/elastin ratio rises. As a result, the
compliance of the aorta decreases by more then a factor of 2 between the ages of 20 and
60 years of age; the same change occurs in other large systemic and pulmonary arteries.
The diameter of the ascending aorta increases (from 1.0 to 1.8 cm), but wall thickness
enlarges proportionately more than vessel diameter [1]. It is evident that the characteristic
impedance of the aorta is affected more by the decrease in compliance and thickening of
the vessel walls than by vessel dilation. Thus, peripheral resistance rises with age.
Cardiac output also declines with age, about 7% per decade. The declining output is
apparently due to the greater load imposed by the increase in peripheral resistance and
not to the depressed myocardial contractility. The decreased cardiac output and increased
sluggishness of the blood, which have been shown to be effects of increased vascular
resistance, put older patients at a higher risk for stroke. For this reason, we have decided
to study the relationship between the compliance of a vessel and its peripheral resistance.
To do this we have employed the Windkessel model of arterial flow, where the parameter
that defines arterial compliance is known as the modulus of volumetric elasticity (E’).
The expected trend for arteries in the body with age is that an increase in modulus
of volumetric elasticity correlates to an increase in peripheral resistance (Rs). Younger
people have a lower E’ and have more compliant arteries, in contrast, elderly people have
arteries that are more rigid with a higher E’. Increasing the resistance to flow in the
arteries, can be fatal such is the case with arteriosclerosis [2]. Arteries throughout the body
may be affected by hardening, which causes symptoms because hardened arteries cannot
carry enough blood to the body. Narrowing or hardening of the arteries that feed the heart
(the coronary arteries) can lead to a heart attack. Narrowing and hardening of the arteries
in the legs can cause pain and difficulty walking. Arteries that lead to the brain may
cause a stroke if they become narrow and hard. Understanding the degree to which
peripheral resistance is affected by the hardening of the arteries therefore becomes an
important point in the pathology of arterial diseases.
Windkessel Model
The Windkessel theory models the arteries as a system of interconnected tubes
with fluid storage capacity. The length of artery is roughly approximated to operate as a
reservoir, which receives blood in an intermittent fashion and issues blood in a different
time dependent fashion [3]. The property of the system enabling it to store blood is
defined by its modulus of volume elasticity E’ as E’ = dp/dVo.
Figure 1: Model of the Windkessel concept.
The following mathematical model (see Equation 1), relating flow rate, pressure,
resistance, and the elastic properties of the artery define the Windkessel concept.
Q(t )  p / Rs  (1 / E ' )
dp
dt
Equation 1: Windkessel equation
When the assumption is made that inflow during systole is constant and equal to Qo,
pressure during systole equals:
pt   Rs Qo  Rs Qo  po e  E 't / Rs
Equation 2: Pressure during systole
Equation 2 is for 0 < t < ts, where ts is the duration of systole and po is the pressure at the
onset of systole. During diastole, when Q=0
pt   pT e ( E '/ RS )(T t )
Equation 3: Pressure during diastole
Equation 3 is for ts < t < T, where T is the duration of the cycle and pT is the arterial
pressure at the end of diastole. The total peripheral resistance can be calculated if the
stroke volume Vs and the pressure trace from 0 to T are known.
Rs 
1
Vs
T
 p(t )dt
0
Equation 4: Peripheral resistance
MATERIALS

C-FLEX L/S 18 E-06424-18 (5/16” inner diameter) (55 cm in length)

TYGON R-3603 (5/16” inner diameter) (55 cm in length)

Long balloon

T-connector (1/4” inner diameter)

Tube clamp

50-cc syringe

Pressure transducer

Water tank with flow cutoff valve

Threaded tube connector (used at the end of tube as a downstream resistor)
APPARATUS
12
12
Figure 2: Experimental apparatus modeling the Windkessel concept
Legend
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Water tank used to fill syringe and experimental artery with water.
Clamp used to block H2O flow to the syringe and tube after they have been filled.
50 cc syringe with expanded opening to enable a 50mL injection of water into the
artery within ~ 1 second.
Pressure transducer
Output from pressure transducer sent to A/D converter into LabVIEW.
Experimental artery (silicon tube, tygon tube or balloon), 55 cm in length
Resistive valve placed at the end of the artery to provide peripheral resistance.
Wood block used to keep the artery horizontally aligned with the pressure
transducer.
Collection bucket used to collect water ejected from the artery after injection.
Plastic T-connector
Brass T-connector
Water supply tubing
The water supply tubing (12) was connected into a plastic T-connector (10) that
joined the 50cc syringe (3) to the brass T-connector. The 50cc syringe was joined to the
plastic T-connector with a small piece of rigid tubing that fit securely around both the
opening of the syringe and the connector. The brass T-connector (11) was screwed into
the pressure transducer (4) and joined the plastic transducer to the experimental tubing
(6) (artery). The plastic T-connector was threaded so it screwed into the brass Tconnector. The experimental tubing was connected to the brass connector via a threaded
tube connector. Using the apparatus shown above, ten trials were conducted for each
type of tubing. The tank (1) was used to fill the tubing and the syringe with water taking
care not to allow any air bubbles into the setup. The water supply from the tank was then
turned off and the tube was clamped (2) to prevent hydrostatic pressure from interfering
with the trial. LabVIEW was configured to take 200 points per second with a total of
2000 points. The program was set to ‘Run’ and the bolus injection was sent through the
tubing and collected in the bucket at the other end. The same procedure was repeated for
remainder of the trials.
In the experimental set-up three tubes of varying compliance were used as part of
our arterial model. The most compliant tube consisted of a stretched balloon, the tube of
intermediate compliance was a silicon tube while the least compliant tube was made of
tygon plastic. The tubes were selected in order to provide the broadest range of
compliance.
RESULTS and DISCUSSION
Experimental Data
Using the pressure transducer, pressure vs. time traces were collected for three
different types of tubing of varying compliance. In each trial, the same procedure was
followed: a bolus injection of 50 ml of water was made into the tube while the pressure
transducer recorded the pressure data. Ten sets of pressure vs. time data were collected
for each of the three types of tubing. Using equations 3 and 4 above, values for Rs and E’
were calculated for each type of tubing.
First, a value for Rs was determined by
integrating over the duration of the pressure trace and dividing by the stroke volume (see
equation 4). An extrapolation of the exponential region of the pressure trace (i.e. diastole
region) was made. The resulting exponential fit was compared to equation 3 and values
for E’ were computed. Figures 3, 4 and 5 show representative pressure traces for each of
the three tubes.
Pressure vs. Time
(Soft) Silicon tubing
5
y = 4E+07e -6.2126x
R 2 = 0.9707
Pressure (psi)
4
3
2
1
0
-1
0
1
2
3
-2
Tim e (s)
Figure 3: Pressure vs. Time trace for soft silicon tubing
4
P ressure vs. Time
(R igid) Tygon tubing
5
P ressure (psi)
4
3
y = 9E +19e -12.435x
2
R 2 = 0.9902
1
0
0
-1
1
2
3
4
5
-2
Time (s)
Figure 4: Pressure vs. Time trace for rigid tygon tubing
Pressure vs. Time (Balloon)
5
y = 4.7306e -0.3573x
R2 = 0.7035
Pressure (psi)
4
3
2
1
0
2
2.5
3
-1
Time (s)
Figure 5: Pressure vs. Time trace for the balloon
3.5
For the soft and rigid tubing shown in Figures 3 and 4, there is a noticeable
oscillation that occurs during diastole. The oscillations are attributed to the standing
wave phenomenon, a characteristic that is ignored in the Windkessel model. In order to
compute the exponential decay during diastole (using equation 3), the peaks of the
damped sinusoid were fit to an exponential regression.
During experimentation, the only set of trials where a noticeable bulge was seen
(characteristic of the Windkessel concept, Figure 1) were the trials using the balloon
tubing.
Specifically, a radial expansion of the balloon could be seen immediately
following the bolus injection. The bulge acted like a capacitor and discharged water in a
time dependent fashion as can be seen by the regression analysis of Figure 5. The graph
shows that the peak pressure was reached at the end of injection (i.e. end of systole),
followed by a sharp dip at which time the pressure remaining in the tube was due to the
radial bulge. Finally, after the bulge discharged the liquid, the pressure inside the tube
returned to atmospheric pressure.
The diastole region of the pressure traces obtained from the hard and soft tube
trials can be seen to follow the Windkessel plot; where the envelope of the decaying
sinusoid of the experimental data is taken to reflect the exponential decay displayed in the
theoretical model. As can be seen in Figure 5, the diastolic regions for the balloon trials
differed significantly from the hard and soft tube data trends. The balloon diastoles
demonstrated a slight exponential decay and a nominal rise in pressure immediately
followed by an abrupt pressure drop.
The slight exponential decay in the balloon trials was attributed to the beginning
of the diastolic portion of the graph (as indicated by the pink exponential fit on Figure 5).
This diastolic region reflects the emptying of water from the balloon due to the elastic
properties and response of the balloon. The beginning of diastole was identified on the
pressure trace through comparison of time measurements (using a stopwatch) with the
time scale of the pressure trace itself. Once the diastolic portion of the trace was
identified, the curve was fit with an exponential regression as demanded by the
Windkessel model. Even though the diastolic pressure traces inherent in the balloon
trials did not conform to the typical trace of the Windkessel, the fitted exponential was
nonetheless used to define the relationship between peripheral resistance and the modulus
of volumetric elasticity (equation 3). Again, it must be stressed that the use of the
exponential fit on this particular portion of the trace is justified based on the results of the
time measurements taken and compared to the time domain of the LabVIEW pressure
trace. This comparison led the experimenters to conclude that the diastolic region of the
graph must correspond to the above mentioned region and therefore, must be described
by an exponential regression.
A summary of all major results of the project are displayed in the table 1 and in
graphical format (see figure 6).
Balloon
Soft
Hard
Resistance
(3.33E+08) + 3.45%
(3.61E+08) + 5.46%
(3.81E+08) + 3.70%
Mod. of Vol. Elasticity
(1.15E+08) + 8.30%
(2.55E+09) + 12.35%
(4.30E+09) + 9.79%
R e s is t a n c e ( P a *s /m ^ 3 )
Table 1: The following table displays the means and the standard deviations for the balloon, soft tube, and
hard tube.
R e s is t a n c e v s . M o d . o f V o l. E la s t ic it y
( s h o w in g g e n e r a l t r e n d o f a ll 3 t u b e s )
4 . 0 0 E+ 0 8
3 . 8 0 E+ 0 8
3 . 6 0 E+ 0 8
y = 0 . 0 1 1 5 x + 3 E+ 0 8
R2 = 1
3 . 4 0 E+ 0 8
3 . 2 0 E+ 0 8
1 . 0 0 E+ 0 8
2 . 1 0 E+ 0 9
4 . 1 0 E+ 0 9
6 . 1 0 E+ 0 9
M o d . o f V o l . El a s t i c i t y ( P a / m ^ 3 )
Figure 6: The following figure displays graphically the data found in Table 1.
Overall, the individual trials for the balloon, soft tube, and hard tube are
consistent within their respective sets of data. The data are precise since the standard
deviations are small (averaging 7.2%) as compared to the means.
Ten trials were
performed for each type of vessel. In Figure 6, the trend that resistance increases with the
modulus of volumetric elasticity is shown. The Windkessel model predicts that peripheral
resistance would decrease as compliance was increased.
In order to justify the existence of the observed trend the data points had to be
shown to be statistically distinguishable.
A statistical analysis involving the 95%
confidence limits of all the averaged points was carried out on the data. The results of
this statistical analysis are displayed in tables 2 and 3 below. All of the final mean values
of E’ and Rs were shown to be statistically different, with one exception. The peripheral
resistance values between the hard and soft tubes are statistically the same. This result
indicates that the material properties between the tygon and silicon tubing are too similar
in order for their calculated resistances to be distinguishable (outside of the given range
of uncertainty). As future consideration the tubing used should be chosen such that
material properties vary significantly in order to fully validate the hypothesis.
Resistances
Balloon
Soft
Hard
Mean
Mean + 95% Mean - 95%
3.33E+08
3.41E+08
3.25E+08
3.61E+08
3.75E+08
3.47E+08
3.81E+08
3.91E+08
3.71E+08
Table 2: Statistical analysis of the Rs values between tubes
Elasticity
Balloon
Soft
Hard
Mean
Mean + 95% Mean - 95%
1.15E+08
1.22E+08
1.08E+08
2.55E+09
2.77E+09
2.32E+09
4.30E+09
4.60E+09
4.00E+09
Table 3: Statistical analysis of the E’ values between tubes
Further Experimentation
A potential variation on the lab would include doing multiple, periodic pulses. A
second variation may involve putting tubes (of varying compliance) in series. This would
allow for additional types of analysis, specifically, incorporating electrical analogs into
the analysis for the tubes in series. If more time were allotted, one could place several
tubes together using y-connections as a model for an arterial tree. Lastly, a model could
be developed that would account for the standing wave phenomena observed in the data.
REFERENCES
1. Wilson, Robert Francis M.D., Critical Care Manual F. A. Davis Company,
Philadelphia., 1992.
2. United Care-Cepts, Inc., Hardening of the Arteries
http://www.mediconsult.com/ucc/cardiac/300_04.html
3. Noordergraaf, Abraham., Circulatory System Dynamics Academic Press, New
York., 1978