Vol. 282, Issue 2, R611-R622, February 2002
Dynamics of cerebral blood flow regulation explained using a
lumped parameter model
Mette S.
Olufsen1,
Ali
Nadim2, and
Lewis A.
Lipsitz3
1 Department of Mathematics, North Carolina State
University, Raleigh, North Carolina 27695; 2 Keck Graduate
Institute, Claremont Graduate University, Claremont, California
91711; and 3 Hebrew Rehabilitation Center for Aged, Beth
Israel Deaconess Medical Center, and Harvard Medical School,
Boston, Massachusetts 02131
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ABSTRACT |
The dynamic
cerebral blood flow response to sudden hypotension during posture
change is poorly understood. To better understand the cardiovascular
response to hypotension, we used a windkessel model with two resistors
and a capacitor to reproduce beat-to-beat changes in middle cerebral
artery blood flow velocity (transcranial Doppler measurements) in
response to arterial pressure changes measured in the finger
(Finapres). The resistors represent lumped systemic and peripheral
resistances in the cerebral vasculature, whereas the capacitor
represents a lumped systemic compliance. Ten healthy young subjects
were studied during posture change from sitting to standing. Dynamic
variations of the peripheral and systemic resistances were extracted
from the data on a beat-to-beat basis. The model shows an initial
increase, followed approximately 10 s later by a decline in
cerebrovascular resistance. The model also suggests that the initial
increase in cerebrovascular resistance can explain the widening of the
cerebral blood flow pulse observed in young subjects. This biphasic
change in cerebrovascular resistance is consistent with an initial
vasoconstriction, followed by cerebral autoregulatory vasodilation.
cerebral autoregulation; arterial modeling
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INTRODUCTION |
THE DYNAMIC CEREBRAL BLOOD
FLOW RESPONSE to sudden hypotension during posture change is
poorly understood. The aim of this work is to use a lumped parameter
model of cerebral blood flow to analyze changes in key parameters
(systemic and peripheral cerebrovascular resistances) during posture
change from sitting to standing. Such a model sheds light on vascular
adaptation to hypotensive stress and could ultimately help determine
the changes in cerebral autoregulation that occur in aging,
hypertension, and other clinical conditions.
The present work focuses on the middle cerebral artery (MCA) and its
peripheral vascular bed. The MCA is considered a conduit vessel, and
the smaller arteries and arterioles that branch off from the MCA are
modeled as resistance vessels that dilate or constrict to restore blood
flow when the perfusion pressure decreases or increases, respectively.
The cerebral autoregulatory response to pressure reduction during
posture change from sitting to standing is vasodilation of the
arterioles. However, the dynamic cerebral blood flow response to
posture change reflects not only the relative contributions from the
peripheral cerebrovascular resistance, but also changes in systemic
resistance, compliance, and heart rate.
To understand the regulatory response, transcranial Doppler (TCD)
measurements of cerebral blood flow velocity in the MCA and Finapres
measurements of arterial pressure in the finger from 10 healthy young
subjects were analyzed using a three-element windkessel model. Blood
flow in the MCA was obtained by multiplying the blood flow velocity by
the area of the MCA (assumed constant throughout the study). As shown
in Fig. 1, when the subject stands up
(after 60 s of sitting) the pressure falls and the blood flow pulse widens (systolic minus diastolic value is increased). After subject stands for 20 s (i.e., 80 s into the study), both arterial pressure and flow reach a new "steady" state.
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Fig. 1.
Measured arterial pressure p (A)
and flow q (B) during posture change. Immediately
after standing (marked with vertical lines at t = 60 s)
the pressure drops while the flow pulsatility widens (systolic minus
diastolic flow increases). After standing for 20 s (at t  80 s, marked with another set of vertical lines) the flow and
pressure return to new steady state values.
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The three-element windkessel model is comprised of two resistors and a
capacitor (see Fig. 2). We hypothesize
that the windkessel model represents the MCA as a constant-diameter
conduit vessel attached to a branching arbor of vessels representing
the cerebrovascular bed. The capacitor and one of the resistors are
considered lumped parameters, representing the vessels leading to (and
including) the MCA, whereas the other resistor represents the
resistance of the peripheral cerebrovascular bed.
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Fig. 2.
The circuit representing the middle cerebral artery (MCA)
and its peripheral cerebrovascular bed (subscript P). The
capacitor CS and resistor
RS are lumped parameters including the MCA and
systemic arteries leading to the MCA. We assume that the pressure of
the finger pF is approximately the same as the
pressure into the MCA. Finally, pV is the
pressure of the venous bed and pI is the
intracranial pressure.
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We present an analysis that demonstrates that by allowing the
windkessel model parameters to vary in time, one can successfully relate the pressure measured in the finger to the blood flow velocity measured in the MCA. Our model shows that the cerebrovascular resistance initially increases, possibly due to baroreflex-mediated regulatory responses to falling pressure, before the cerebral autoregulatory response sets in to dilate the vessels and decrease the
cerebrovascular resistance. In addition, our work indicates that the
initial increase in cerebrovascular resistance is responsible for the
widening of the blood flow pulse.
We have chosen to work with the simple three-element windkessel model,
because one aim of this work is to demonstrate a new methodology for
data analysis rather than to develop a complex lumped model that can
capture all details of the individual pulse profiles. Even with this
simple model, we were able to reproduce dynamic changes in the
pulsatile blood flow of the MCA during transient changes in arterial pressure.
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METHODS |
Modeling.
The model used in this work is a three-element windkessel model
often used in cardiovascular studies (27, 35, 48). The windkessel model can be represented by a circuit consisting of two
resistors RS and RP (mmHg
· s/cm3) and a capacitor CS
(cm3/mmHg) (see Fig. 2). We assume that
CS and RS represent the
systemic compliance and resistance of the arteries leading to (and
including) the MCA, whereas RP represents the
resistance associated with the peripheral cerebrovascular bed. Because
it is difficult to make measurements of pressure directly in the MCA
the input to the model is pressure in the finger
(pF, mmHg; alternatively one could use pressure
measurements from the earlobe). The output from the model is the
volumetric flow rate (qMCA, cm3/s)
in the MCA, which can be validated by comparison with corresponding measured data. In addition to the above elements, the circuit includes
intermediate flow and pressures: the flow and pressure of the
peripheral cerebrovascular bed (qP,
pP) the venous pressure (pV), and the intracranial pressure
(pI). However, these will not be determined
explicitly. If we assume that flow and pressure are correlated
(20) we can use an electrical circuit analogy and derive
equations for pressure and flow in the MCA. Assuming that the pressure
and flow can be described as sums of harmonic components of the form
p(t) = Pei
t and
q(t) = Qeit where
(s
1) is frequency, t (s)
is time, and P, Q are pressure and flow in the
frequency domain. Using these definitions, the equations for the
circuit in Fig. 2 are
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Assuming that PV = PI = 0 the above equations yield
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(1)
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where, as is common for electrical circuits, the impedance is
defined as Z = P/Q (mmHg · s/cm3).
More generally, for time-periodic signals of period T (s)
(the length of the cardiac cycle), the theory of Fourier series gives
with
where k = 2
k/T and 
T/2 
t T/2, with the relation in Eq. 1 valid for each of
the frequency components in the signal.
Using the above relationship between flow and pressure in the frequency
domain, the windkessel Eq. 1 can also be written as the
following ordinary differential equation in the time domain
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(2)
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Note that the combination of parameters
CSRP and
CSRSRP/(RS + RP), which appear as coefficients on the two sides
of Eq. 2, can be regarded as the characteristic relaxation
times in this model. Furthermore, if pressure
p(t) is regarded as input, this equation can be
integrated to find q(t) provided that the parameters are known.
The above interpretation of the lumped model is based on the location
of the blood flow velocity measurements. Effects due to changes in
venous and intracranial pressure are not specifically included in the
model. However, even though the model is simple, it is still able to
capture dynamic effects arising due to posture change. Another
important factor is that having a simple model with a small number of
parameters makes it easy to extract the dynamic variation of each of
the parameters from the measured data.
As mentioned earlier, the measurements provide data for pressure
in the finger and for velocities in the MCA. The windkessel model
provides a relation between blood flow (volumetric flow rate) and
pressure, not velocity and pressure. To obtain values for the flow, we
assume that the MCA has a constant radius of r = 2 mm
and the flow is q = r2v
(cm3/s), where v (cm/s) is the blood flow
velocity. The radius of the MCA varies among the different subjects;
however, because direct measurements of the radius are not available,
we simply assume it to be constant. All data shown in the results
section are based on the flow q (rather than the velocity).
Let ZW be the impedance obtained from the
windkessel model (Eq. 1) and let Zm
be the impedance obtained from the measurements by taking the ratio of
the discrete Fourier transform of the pressure and flow data. Then, the
parameters for the windkessel model can be determined by fitting the
impedance of the windkessel model to the impedance of the data.
The zero- and large-frequency limits of the windkessel model (Eq. 1) yield relations that only involve the resistances
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(3)
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For the measured impedances, the limit 
0 corresponds to the direct current (DC) value of the impedance, and
the limit 
corresponds to the highest
meaningful frequency in the measured data. The data contain a
significant amount of noise and hence the high-frequency limit is
better approximated by taking the mean value of the measured impedance
over a number of high frequencies.
Once RS and RP have been
determined, CS can be computed from analyzing
the modulus of the impedance |Z| as a function of
frequency. This can be done by estimating the measured impedance at a
point where the impedance curve obtained by the windkessel model passes through the impedance obtained by the measured data (e.g., between the
2nd and 3rd data points in Fig. 3). The
magnitude of the measured impedance at this point is
and the frequency at this point is
From (Eq. 1) we have
Inserting the averaged values for
2,3 and
(2,3) and solving
for CS yields
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(4)
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We have estimated the parameters of the windkessel model by
analyzing the data in two ways. First, we estimated parameters representing three periods: sitting (0-60 s), transition from sitting to standing (60-80 s), and standing (80-120 s). This
was done by means of the windowed Fourier transform. Second, we
estimated the parameters on a beat-to-beat basis, thus enabling us to
monitor how they change during posture change, and hence, how
autoregulation affects the peripheral cerebrovascular bed.
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Fig. 3.
Modulus of the impedance
|Z()| obtained using a windowed
Fourier transform of the data. The window was a box window with 50%
overlap and length 480. The solid black line with dots represents the
measured data, and the gray line represents the results obtained using
the windkessel model.
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The windowed Fourier transform (WFT) or the sliding Fourier transform
is widely used for extracting time-dependent spectra from a finite
length time series. It divides the time series into a finite number of
smaller series, which is individually Fourier transformed. Each section
is analyzed for its frequency content and then averaged over the finite
number of sections. The advantage of this technique is that significant
frequencies do not vanish, as would often occur if the full sequence
were analyzed using a conventional Fourier transform (FFT) method.
Because a single window is used for all frequencies in the WFT, the
resolution of the analysis is the same (equally spaced) at all
locations in the time-frequency domain.
FFT works well for signals with smooth or uniform frequencies, but it
has been found that the windowed Fourier transform works better with
signals having pulse-type characteristics, time-varying (nonstationary)
frequencies, or odd shapes. The FFT does not distinguish sequence or
timing information. For example, if a signal has two frequencies (a
high followed by a low or vice versa), the Fourier transform only
reveals the frequencies and relative amplitude, not the order in which
they occurred. So Fourier analysis works well with stationary,
continuous, periodic, differentiable signals, but other methods such as
the WFT are needed to deal with nonperiodic or nonstationary signals.
Figure 3 shows a comparison of the impedances from the data and the
windkessel model during the sitting period for a window size of 480. The data were sampled at 50 Hz; hence, a window containing 480 data
points covers 9.6 s. Thus, the data points in Fig. 3 are in
frequency increments of 1/9.6 = 0.104 Hz. The window type that
gave the most consistent results was a box window with a 50% overlap.
This analysis was carried out using a function from Matlab's signal
processing toolbox (21). The zero-frequency limit
Zm(0) = RS + RP of the WFT data can be obtained directly by
picking out the DC value of Zm. However, for
high frequencies the data are noisy so it is more difficult to find
RS. We investigated several options and found
the most stable results when RS is chosen as the
mean over frequencies <8 Hz (or an angular frequency <40 radians/s).
After the resistances have been estimated, CS
was obtained using Eq. 4. It may be noted that the drop of
|Z| occurs over the first few points that have
frequencies of order 0.1 Hz or periods of order 10 s. Thus, when
we analyze the data on a beat-to-beat basis, there is not enough
resolution in the frequency intervals to estimate the compliance.
The same criteria were used when the parameters where determined on a
beat-to-beat basis. However, to find the parameters on a beat-to-beat
basis, the time series must be separated at the beginning of each
cardiac cycle. This can be done by searching the pressure data for
local minima. Each period will have two minima: one representing the
beginning of a cardiac cycle and the other representing the beginning
of the dicrotic notch (see Fig. 4). The
stars in Fig. 4 represent the beginning of the cardiac cycle. A similar
analysis cannot be performed using the blood flow velocity data,
because the levels of noise are significantly higher and as a result
there will be a large number of local minima. Because pressures and
flows are recorded simultaneously, knowing the length of the cardiac
cycle and the local minima for the pressure enables us to find the
local minima for the flow. The only difference is that there could be a
small phase lag between the flow and pressure minima because the data
were not measured at the same location.
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Fig. 4.
Consecutive flow q (A) and
pressure p (B) profiles for each cardiac cycle
during a 100-s trial in 1 subject. Each cardiac cycle has 2 minima, 1 at the beginning of the cardiac cycle (*) and one at the dicrotic
notch. Note, the blood flow velocity and pressure are measured at 2 different locations and due to differences in the distance, the pulse
wave will not reach the 2 locations at the same time. Hence, there is a
constant phase lag (6 ms for this subject) between the 2 sites. This
has been corrected for in the graphs.
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The impedances can be found using the same method as for the WFT. The
only difference is that instead of calculating
Zm over the full interval (sitting, transition,
or standing) it was calculated for each cardiac cycle. However, as
discussed above, the CS cannot be calculated on
a beat-by-beat basis because the main variation in the impedance occurs
at very low frequencies (corresponding to periods that are longer than
a single cardiac cycle). This can be seen from Fig.
5: the impedance curve obtained by the
windkessel model is already flat when the first data point has reached
the level of RS. The measured data are acquired
at 50 Hz, so for a cardiac cycle of 1.2 s we get 60 data points
with a difference between discrete frequencies of 1/1.2 = 0.8
Hz. For the WFT, the interval with a window size of 480 data
points comprises approximately eight cardiac cycles or 8 × 1.2 = 9.6 s. The interval between two discrete frequencies is
0.1 Hz. So to calculate the compliance from the data only one mean
value can be obtained for each of the three states: sitting,
transition, and standing.
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Fig. 5.
Fourier transforms of pressure
P() (A), flow
Q() (B), and their ratio
(impedance) (C). The graphs show the absolute value as a
function of frequency using a single period during sitting (the period
starting at 29.12 s). In C, the impedance obtained from the
data (solid line with circles) is compared with that obtained by the
windkessel model (solid line) and the windowed Fourier transform
(dashed line). Note, the compliance cannot be found from the Fourier
transform of a single period because the frequency resolution is too
low. Thus the value of CS used is based on the
windowed Fourier transform shown in Fig. 3.
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Comparing RS and RP
obtained by the WFT during sitting with the average of those obtained
on a beat-to-beat basis, the results are consistent (see Fig. 5C).
Similar results can be obtained during the period of standing. However,
during the transitional period from sitting to standing, the values
obtained on a beat-to-beat basis vary significantly from the average
values obtained from the windowed Fourier transform.
In addition to the resistances and the compliance, the heart rate also
varies on a beat-by-beat basis. By separating the time series at the
beginning of each cardiac cycle, the heart rate is simply obtained by
HR = 1/T (beats/min) where T is the duration of the cardiac cycle.
Assuming that pressure is known (see Fig. 1) and that the parameters in
the windkessel model have been determined as describe above, the flow
can be computed by integrating the differential Eq. 2 and
comparing the results with the measured values (see Fig. 9).
Simulations were carried out for ten healthy young subjects. The
computed pulsatile flow profiles were compared with the actual transcranial Doppler measurements. Comparisons were made both with
respect to quantitative agreement over the entire dynamic range as well
as with mean flow, and systolic and diastolic flows for the two
steady-state situations (before and after standing). Statistical
significance was determined using two-way analysis of variance
(22) and P values <0.05 were considered
statistically significant.
Experimental methods.
The subjects of this study are ten carefully screened healthy young
volunteers aged 20-39 years, whose mean cerebral autoregulatory responses to posture change and carbon dioxide are described elsewhere (20).
During the protocol, heart rate was measured continuously from a
three-lead electrocardiogram, and beat-to-beat arterial pressure was
determined noninvasively from the middle finger of the nondominant hand
using a photoplethysmographic noninvasive pressure monitor (Finapres)
supported by a sling at the level of the right atrium to eliminate
hydrostatic pressure effects. To keep end-tidal CO2 constant, respiration was measured continuously using an inductive plethysmograph (Respitrace) and subjects breathed at 0.25 Hz (15 breaths/min) throughout each standing procedure by following
tape-recorded cues. All subjects underwent Doppler ultrasonography by a
trained technician to measure the changes in blood flow velocity within the MCA in response to active standing. The 2-MHz probe of a Nicolet Companion portable Doppler system was strapped over the temporal bone
and locked in position with a Mueller-Moll probe fixation device to
image the MCA. The MCA blood flow velocity was identified according to
the criteria of Aaslid (1) and recorded at a depth of
50-65 mm. The envelope of the blood flow velocity waveform, derived from a Fast-Fourier analysis of the Doppler frequency signal,
and continuous pressure and electrocardiogram signals were digitized at
250 Hz and stored in the computer for later offline analysis.
After instrumentation, subjects sat in a straight-backed chair with
their legs elevated at 90 degrees in front of them on a stool. For each
of two active stands, subjects rested in the sitting position for 5 min
then stood upright for 1 min. The initiation of standing was timed from
the moment both feet touched the floor. Data were collected
continuously during the final minute of sitting and the first minute of
standing during both trials.
The study was approved by the Institutional Review Board at the Hebrew
Rehabilitation Center for Aged and all subjects provided written
informed consent.
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RESULTS |
Figure 1 shows the measured pressure and MCA blood flow during the
transition from sitting to standing for one subject. While the pressure
and flow levels vary among the subjects, the characteristics of the
profiles are similar for all subjects. Immediately after the subjects
stood up their pressure dropped and flow pulse widened. Approximately
10 s after standing (at t 70 s) the
pressure started to increase and the flow pulse narrowed, reaching a
new steady state at t 80 s.
Table 1 shows mean values ± SD for
blood flow and pressure during sitting, transition, and standing. There
was no statistically significant difference between sitting and
standing values for pressure although the flows were significantly
decreased during standing. During the transition there were
statistically significant changes in both pressure and flow. The mean,
systolic, and diastolic pressures and the mean and diastolic flows
decrease, whereas the systolic flow increases. The heart rate increases
during standing and remains slightly increased until the end of the
study.
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Table 1.
Mean, systolic, and diastolic arterial pressures p and flows q, and
heart rate (beats/min) during sitting (0-60 s), transition
(60-80 s), and standing (80-120 s)
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Using the measured pressure and flow, we were able to extract the
dynamic changes of the windkessel model parameters during posture
change. These parameter changes are shown for one subject in Fig.
6. The results show that during posture
change, the mean flow and pressure decrease for about 10 s and
then increase for about 10 s, while the ratio of pressure to flow
(the total resistance RP + RS) increases slightly, then falls for about
10 s, and then increases for about 10 s to a new steady
state. The peripheral cerebrovascular resistance
RP increases significantly (for a few seconds)
in the beginning before it falls. As expected, the heart rate increases
for 10-15 s, and then it falls to the new steady state.
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Fig. 6.
The traces (top to bottom) show pressure
p (mmHg) (A), flow q
(cm3/s) (B), mean pressure (p mean)
(mmHg) (C), mean flow (q mean)
(cm3/s) (D), total resistance (p
mean/q mean) or RP + RS (mmHg s/cm3) (E),
systemic resistance (including the MCA) RS (mmHg
s/cm3) (F), peripheral cerebrovascular
resistance RP (mmHg s/cm3)
(G), and heart rate (beats/s) (H) all as
functions of time (s).
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The total resistance is the most one can determine by simply studying
the measured data. However, as shown in Fig. 6, we have been able to
obtain more information using the windkessel model by decomposing the
resistance into a systemic and a peripheral (cerebrovascular) part.
Figures 6, and 8-11 all contain data for one subject; but, data
for the other subjects are very similar. The results for the entire
study cohort are summarized in Table 2
and Fig. 7. The table shows that the
changes in resistances and compliance are all statistically
significant. However, as shown in Fig. 7, interindividual variation is
fairly large. During sitting and standing, the standard deviation is
30%, while during the transition it is nearly 40%. Comparisons of the
two trials within subjects shows good intraindividual reproducibility
for most subjects.
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Table 2.
Average values, standard deviation, and the P value (with a
5% significance level) for each of the parameters
RS, RP, and
RS +
RP, and CS for all 10 subjects
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Fig. 7.
Average values for the parameters
RS (A), RP
(B), and CS (C) during
each of the 3 periods. The averages for the resistances are found by
averaging the data obtained on a beat-to-beat basis for 2 trials for
each subject. The horizontal axis plots each subject (circles are used
for trial 1 and plusses for trial 2). The
vertical axis shows the average resistances or compliance. The solid
line represents the mean over all subjects and the dotted lines the
standard deviation.
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As shown in Fig. 8 the compliance has not
been analyzed on a beat-to-beat basis. Steady-state compliance values
were obtained during each of the three periods: sitting, transition,
and standing. To illustrate the change in compliance, we have
interpolated linearly between the two steady states and the transition
state. The compliance decreases and then increases to a new steady
state that is slightly lower than its original value.
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Fig. 8.
The compliance CS is based on the
values obtained using window-based Fourier analysis. Hence, each of the
3 periods is represented by an average compliance. Linear interpolation
is used to account for variation of compliance during the transition
period.
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Figure 9 illustrates the result of
applying the parameters obtained in Fig. 6 for the full duration of the
measurements. The top trace shows the computed flow and the bottom
trace shows the measured flow. The vertical lines at the 60-s mark
where the subject stands up, and the vertical lines at 80-s mark
approximately when flow and pressure have returned to the new steady
state during standing. The figure shows a very good correspondence
between the measured and computed flows. However, Fig. 9 does not show how well-computed and measured flows compare on a beat-to-beat basis.
To make this comparison we have computed beat-to-beat profiles at three
different times, at approximately 17 s, 70 s, and 85 s.
These traces shown in Fig. 10 exhibit
good correspondence between the measured and the computed data. Both
the profiles in Fig. 10 and the impedance plots in Figs. 3 and 5 show
some extra "bumps" that are not captured by the model. In Fig. 10,
the bumps appear immediately before the reflected wave. In the
impedance curves in Fig. 3, the extra bump occurs just after the
initial drop from the DC value. These bumps could be due to the
presence of inertia which is not included in the model. Lumped models
studied by Toy et al. (38) have shown that features
similar to the ones seen in our data can be captured by including more
elements (e.g., inertances or more resistors and capacitors) into the
model.
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Fig. 9.
Computed (top trace) and measured
(bottom trace) flow q for the entire duration of
the measurements. The vertical lines at the 60-s mark indicate where
the subject stands up, and the vertical lines at the 80-s mark indicate
where we estimate that the flow and pressure have returned to the new
steady state during standing.
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Fig. 10.
Simultaneous recordings of blood flow q
(A) in the MCA and arterial pressure p
(B) in the finger (meas, solid line with filled circles)
compared with computed MCA blood flow (wk, solid gray line) for 1 subject. The flow is obtained by multiplying the velocity with the
constant cross-sectional area of the MCA (radius = 2 mm). The
profiles are plotted during sitting (at t 17 s),
transition (at t 70 s), and standing (at
t 85 s).
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DISCUSSION |
Blood flow in the cerebral circulation is controlled by a dynamic
regulatory system. The regulation is active over a range of time scales
lasting from a few seconds to several minutes. The purpose of control
is to maintain a constant flow over a wide range of pressures. This is
done by regulating the diameters of vessels in the peripheral
cerebrovascular bed as well as by changing the heart rate and cardiac
output. The control appears to be active within upper and lower limits
of pressure (e.g., in young subjects, 50-150 mmHg) that shift
toward higher pressures in hypertensive subjects (36). The
autoregulatory control process in the cerebral vasculature is most
likely mediated by a combination of myogenic and metabolic mechanisms,
as well as changes in the activity of the autonomic nervous system
(29), but it is not yet fully understood. This paper
focuses on modeling the dynamics and control of cerebral blood flow
with the goal of better understanding the regulatory mechanisms in
young adults. The three-element windkessel model used in this work
included only very basic mechanisms, but we have shown that it is still
able to reproduce the measured data. We have kept the model simple,
because as discussed previously, we were interested in tracking how the
windkessel parameters changed during posture change. Including more
elements, and hence more parameters, would have made it much more
difficult to develop a procedure for automatically monitoring the
change of the parameters such that they could still be understood from
basic principles.
There are several published models of cerebral autoregulation, but to
our knowledge there are currently no other models that include both
cerebral autoregulation and pulsatility. Pulsatile models have been
developed for studying atherosclerosis in the carotid (30,
47) or for studying the dynamics of blood flow in the circle of
Willis (15, 17, 46). The models by Viedma et al.
(46) and Kufahl and Clark (17) are
distributed, i.e., they include time and one spatial dimension, whereas
the model by Hillen (15) is based on lumped parameters.
The models were developed to study the large anatomical variation of
the communicating arteries and the effect of various pathological
situations. The latter were verified by variation of the model
parameters to represent alterations in flow distribution due to
occlusions. Work by Ursino (42, 45) discusses the
importance of pulsatility in carotid baroreflex regulation. Modeling
pulsatility is important when studying perturbations such as mild
hemorrhage, carotid occlusion maneuvres, or genesis of self-sustained
arterial pressure waves.
Ursino and Lodi (39, 40, 42, 44) also made significant
contributions to modeling cerebral blood flow velocity and its
autoregulation. Their work is based on a steady-state model that mimics
the behavior of the intracranial arterial vascular bed, intracranial
venous vascular bed, cerebrospinal fluid absorption, and production.
Model parameters were computed using physiological considerations and
anatomical data from normal subjects. The cerebral circulation is
represented by a steady-state lumped parameter model. Each of the
elements in the model comprises a capacitor and a resistor, some of the
capacitors being passive and some active. The elements represent the
MCA, large and small pial arteries, the venous bed, the intracranial
pressure, and the cerebrospinal fluid circulation. A regulation loop is
applied at the large and small pial arteries. The large pial arteries
respond actively to changes in cerebral perfusion pressure, whereas the
small pial arteries are sensitive to percent changes in cerebral blood
flow velocity. Dynamics of each mechanism are simulated by means of a
gain factor and a first order low-pass filter with a time constant. Finally, autoregulation interacts nonlinearly through a sigmoidal static relationship.
Ursino et al. (43) used the above model for studying
effects of changes in intracranial pressure, systemic arterial
pressure, autoregulation, and intracranial compliance using
transcranial Doppler (TCD) waveforms (systolic, mean, and diastolic
blood flow velocity, peak-to-peak amplitude, and pulsatility index).
They concluded that information contained in the TCD waveform is
affected by many factors, including intracranial pressure, systolic
arterial pressure, and autoregulation. These factors can also be
incorporated within the lumped parameter models which we have
introduced here. Ursino and Gimmarco (41) studied the
interaction of cerebral blood flow velocity and cerebral plateau waves,
the effect of acute arterial hypotension on intracranial pressure, and
the role of cerebral hemodynamics during pressure volume index tests.
Recently, Ursino and Lodi (44) expanded the model to also
account for CO2 reactivity. To avoid the potential effect
of changes in CO2, our study controlled breathing at 0.25 Hz and maintained a constant end-tidal CO2.
Other groups have also modeled the cerebral circulation, e.g., Gaffie
et al. and Fincham and Tehrani (12, 13) who studied steady-state relationships between cardiac output and cerebral blood
flow velocity. This model described cerebral autoregulation in terms of
arterial blood levels of oxygen and carbon dioxide and the metabolic
rate ratio. Another example is a seven- compartment model developed by
Bekker et al. (3), who investigated the effects of various
vasodilatory/constrictive drugs on the intracranial pressure. The
effects of drugs were modeled using a variable arteriolar-capillary resistance. Models reproducing pulsatile flow and pressure phenomena (2, 27, 28, 32, 35, 48) do not include dynamics of the
regulation. These models were used to study dynamic changes in the flow
and pressure as the pulse wave propagates along the arterial tree. All
of the above models are distributed models (including one spatial
dimension) determining flow and pressure at any given location of a
vessel at all times. The models differ in the way they treat shear
stresses, the relation between pressure and cross-sectional area, and
the boundary conditions.
Our model is based on TCD methodology, which measures blood blow
velocity in the MCA. Because blood flow through an artery can be
estimated as the product of the true mean velocity and cross-sectional
area, changes in velocity represent changes in cerebral volumetric
flow, only if vessel diameter does not change. Several studies have
validated TCD for the measurement of cerebral blood flow velocity using
both invasive and noninvasive methods. Newell et al. (25)
compared changes in MCA velocities measured by TCD with invasive
internal carotid blood flow measurements during rapid deflation of
thigh blood pressure cuffs while subjects were undergoing carotid
artery surgery. They found a strong linear correlation (R = 0.995). Other groups have also shown excellent correlations
between TCD and invasive determinations of cerebral blood flow velocity
in humans (18, 19). With the use of either the noninvasive
xenon-133 method or single photon emission computerized tomography,
flow velocities from the middle, anterior, and posterior cerebral
arteries have been shown to correlate with simultaneously measured
velocity values from corresponding cerebral regions (4, 5, 7,
34). The correlation coefficient is highest for the MCA
(34), the vessel in which extensive data are available for
this study. In humans it is likely that changes in velocity within the
MCA correlate with changes in flow and constitute an acceptable index
of flow in this arterial segment.
As described above our model assumes that the MCA does not change its
diameter. This assumption is supported by several lines of evidence
suggesting that the MCA diameter does not change during hypotensive
stress induced by head-up tilt (6), lower body negative
pressure (33), and a number of other stimuli (11, 13, 16, 18, 19, 25). Because our model has one resistive term
(RS) representing the systemic arterial
circulation, which does change during the transition from sitting to
standing (see Fig. 6), we assume that this change represents total
systemic resistance rather than the resistance of the MCA. To address
this question in more detail a more elaborate model is called for, where the MCA is modeled explicitly as a vessel with a given diameter.
As shown in the previous figures, we were able to obtain excellent
agreement between the model and the measured results. Our results are
consistent with the following physiological mechanism: immediately
after standing up arterial pressure falls because of blood pooling in
the legs and splanchnic circulation. Our model suggests that there is
also an initial increase in cerebrovascular resistance. This may be due
to unloading of baroreceptors in the carotid arteries, aortic arch, and
cardiopulmonary circulation during the initial blood pressure decline,
causing reflex cardioacceleration and both systemic and cerebral
vasoconstriction. However, recent studies in normal and spinal
cord-injured subjects suggest that the cerebral circulation is weakly
innervated by the sympathetic nervous system. Other work examining the
response of spinal cord-injured patients to head-up tilt has found
cerebral blood flow decreased (14, 49) or remain unchanged
(23, 24) during arterial hypotension. Furthermore, direct
infusions of norepinepherine in both anesthetized (37) and
conscious able-bodied patients (26) have not been shown to
affect cerebral blood flow or vascular resistance. Therefore, the
initial increase in cerebrovascular resistance we observed may be a
passive, rather than baroreflex-mediated, response, due to a greater
reduction in cerebral blood flow relative to mean arterial blood
pressure during posture change.
When cerebral autoregulation becomes engaged approximately 10 s
after the initial fall in pressure, cerebrovascular resistance decreases as expected, to restore cerebral blood flow back to baseline.
The initial increase in cerebrovascular resistance may explain the
widening of the cerebral blood flow pulse velocity which was observed
in the young subjects. Figure
11A
shows that if we modify the cerebrovascular resistance to prevent its
increase (while keeping the total resistance constant) the blood flow
pulse does not widen (see Fig. 11B). In our previous study
of elderly subjects (20), flow pulses did not widen during
posture change. Although further research is needed, this may now be
explained by the absence of the initial cerebral vasoconstriction.
View larger version (26K):
[in this window]
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|
Fig. 11.
A: both the modified (solid line) and the
original (dashed line) parameters (RS,
RP, and RS + RP) without and with the initial increase in
cerebrovascular resistance RP. Note that the
total resistance is kept constant by modifying both
RS and RP. B:
finger pressure p (input to the model) and the modified
(middle) and original (lower) computed flows
q. Without the initial increase in the cerebrovascular
resistance RP, the blood flow pulse does not
widen (middle trace in B).
|
|
One important limitation of our work is the assumption that the finger
pressure can be used as input to the model. The forearm responds to the
unloading of baroreceptors with vasoconstriction that may uncouple the
central blood pressure from the finger pressure. However, even using
the finger pressure in the simple model adopted here we were able to
achieve good comparisons between the measured and computed data. If
future studies can measure cerebral perfusion pressure more directly,
our model should be even more relevant. We agree with the conclusion in
the recent work by Quick et al. (31) that even such simple
models "can play a vital role in solving aspects of the inverse
problem." [By the inverse problem, it is meant inferring properties
of the arterial system from measured pressure and flow. Quick et al.
(31) point out that the solution to such inverse problems
is not generally unique.] Our main conclusion is that there is a
biphasic response to orthostatic hypotension during the transition from
sitting to standing. Cerebral vascular resistance increases first,
before an autoregulatory decrease in resistance begins to dominate the control.
Perspectives
The lumped parameter model presented in this paper demonstrates a
biphasic cerebrovascular response to acute posture change in healthy
young subjects. This is characterized by initial peripheral cerebral
vasoconstriction (manifested by increased pulsatility), followed by
autoregulatory cerebral vasodilation. The unique finding of
initial vasoconstriction remains unexplained, but may represent rapidly
acting baroreflex control of the cerebral circulation, or passive
mechanisms due to greater initial reduction in cerebral blood flow
relative to the reduction in mean arterial pressure during standing.
Given its success in reproducing the dynamic changes in cerebral blood
flow seen during posture change, this model may be helpful in
elucidating mechanisms of abnormal cerebral autoregulation in a
variety of pathological conditions. For example, the
"paradoxical" increase in pulsatility reported during orthostatic
stress in patients with vasovagal syncope (9, 10) might be
explained by the early vasoconstriction revealed by our model. Because
earlier studies did not examine the dynamics of cerebral blood flow
response, they may have failed to recognize a later vasodilation after
posture change that reflects normal autoregulation. The model might
also be useful in the study of autoregulatory changes with aging,
hypertension, and cerebrovascular disease.
| |
ACKNOWLEDGEMENTS |
The modeling was supported by a Group Infrastructure Grant No.
DMS-9631755 from the National Science Foundation. The data collection
and analysis was supported by a Joseph Paresky Men's Associates grant
from the Hebrew Rehabilitation Center for Aged, a Research Nursing Home
Grant #AG04390, and an Alzheimers Disease Research Center Grant
#AG05134 from the National Institute on Aging. Dr. Lipsitz holds the
Irving and Edyth S. Usen and Family Chair in Geriatric Medicine at the
Hebrew Rehabilitation Center for Aged.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: M. S. Olufsen, Dept. of Mathematics, North Carolina State University, Box
8205, Raleigh, NC 27695 (E-mail:msolufse{at}unity.ncsu.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
10.1152/ajpregu.00285.2001
Received 22 May 2001; accepted in final form 10 October 2001.
| |
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ABSTRACT
INTRODUCTION
