Vol. 274, Issue 3, H1023-H1031, March 1998
MODELING IN PHYSIOLOGY
Mathematical considerations for modeling cerebral blood flow
autoregulation to systemic arterial pressure
Erzhen
Gao1,
William L.
Young2,3,4,
John
Pile-Spellman3,4,
Eugene
Ornstein2, and
Qiyuan
Ma1
1 Department of Electrical
Engineering, Columbia University, and
2 Departments of Anesthesiology,
3 Neurological Surgery, and
4 Radiology, College of
Physicians and Surgeons of Columbia University, New York, New York
10032
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ABSTRACT |
The shape of the autoregulation curve for
cerebral blood flow (CBF) vs. pressure is depicted in a variety of ways
to fit experimentally derived data. However, there is no general
empirical description to reproduce CBF changes resulting from systemic
arterial pressure variations that is consistent with the reported data.
We analyzed previously reported experimental data used to construct
autoregulation curves. To improve on existing portrayals of the fitting
of the observed data, a compartmental model was developed for synthesis of the autoregulation curve. The resistive arterial and arteriolar network was simplified as an autoregulation device (ARD), which consists of four compartments in series controlling CBF. Each compartment consists of a group of identical vessels in parallel. The
response of each vessel category to changes in perfusion pressure was
simulated using reported experimental data. The CBF-pressure curve was
calculated from the resistance of the ARD. The predicted autoregulation
curve was consistent with reported experimental data. The lower and
upper limits of autoregulation (LLA and ULA) were predicted as 69 and
153 mmHg, respectively. The average value of the slope of the
CBF-pressure curve below LLA and beyond ULA was predicted as 1.3 and
3.3% change in CBF per mmHg, respectively. Our four-compartment ARD
model, which simulated small arteries and arterioles, predicted an
autoregulation function similar to experimental data with respect to
the LLA, ULA, and average slopes of the autoregulation curve below LLA
and above ULA.
cerebral hemodynamics; cerebral circulation; limits of
autoregulation; compartmental flow model; simulation
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INTRODUCTION |
AUTOREGULATION is the intrinsic ability of an organ or
a vascular bed to maintain constant perfusion in the face of blood pressure changes. The precise molecular mechanisms of a cerebrovascular autoregulation bed are incompletely understood (5). Previous studies
have shown that, besides the most important factor, arterial blood
pressure, many other factors such as intracranial pressure (ICP) and
cerebral venous pressure affect cerebral blood flow (CBF)
autoregulation (2, 32, 41). The effects of changing arterial blood
pressure have been experimentally investigated for humans (18, 27, 33)
and a variety of animals (6, 16, 19, 20, 24). However, the data have
not been well summarized. The effects of changing ICP and cerebral
venous pressure on CBF autoregulation have been investigated
experimentally (2, 32, 41) and theoretically (8). Some groups (41) have reported that the cerebral vascular bed responds to changes in the
perfusion pressure gradient in a similar fashion, whether they result
from decreasing mean arterial pressure, increasing jugular venous
pressure, or increasing ICP. However, others (2) have reported that the
effect of changing ICP on the vessel inner radius may be quite
different from the effect of changing arterial blood pressure.
Previously published modeling efforts have focused primarily on
understanding mechanisms of autoregulation (4, 8, 39). However, the
autoregulation curve itself has not been precisely described in these
models by a simple formula (4, 8, 39).
This study was undertaken to develop an empirical formula of
autoregulation modeled on basic hemodynamic principles and experimental data. The modeling was done for a normal circulation. In our model, with the assumption that cerebral venous pressure and ICP were zero,
variations in perfusion pressure were induced solely by changes in
systemic arterial pressure. A microvascular network served as an
autoregulation device (ARD) to control CBF. A new compartmental model
simulating the autoregulatory function of an ARD was constructed to
synthesize the autoregulation curve. Our model was used to analyze the
contribution of different hierarchical levels of vessel sizes to
autoregulation. In addition to providing a mathematical description of
the autoregulation curve, this study attempts to place into an
interpretable framework experimental studies that describe hierarchies
of vasoactive behavior in resistive vessels between 50 and 300 µm.
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METHODS |
Overview
After reviewing a number of previous studies of cerebral
autoregulation, we summarized the autoregulation curves used by other authors into three general types. The disagreement between these curves
and experimental data is discussed. On the basis of the reviewed
experimental data, we constructed a compartmental model to simulate
autoregulatory function. An ideal microvascular network served as an
ARD to control CBF (Fig. 1). This model
consists of four compartments in series (A, B,
C, and D). Each
compartment contains a group of identical vessels in parallel. It was
assumed that these vessels determine autoregulation.
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Fig. 1.
Compartmental structure of an autoregulation device (ARD) representing
small arterial and arteriolar network in compartmental model for
control of cerebral blood flow (CBF). This model consists of 4 compartments: A, B, C, and
D. Number, diameter, and length of
vessels are listed in Table 2.
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Literature Review
A number of representative studies of autoregulation were reviewed with
attention to the lower and upper limits of autoregulation (LLA and ULA,
respectively; Table 1). The description of
autoregulation given by the various authors differs to some extent.
Many are limited in scope, with the range of the experimental data
(especially in humans) usually not exceeding the ULA. In addition, the
values of "normal" CBF, LLA, and ULA given by the various authors
differ in animal experiments (6, 16, 19, 20, 24) and human observations
(18, 27, 33) because of species differences, CBF methodology (direct or
indirect), and the presence of anesthetic agents. Another disagreement
among previous reports is that some authors (18, 27, 33) describe the
CBF-pressure curve below the ULA as a combination of two straight
lines, whereas other groups (6, 16, 19, 20, 24, 43) indicate that the autoregulation curve near the LLA is curved rather than straight.
Using a cranial window preparation, Kontos et al. (16) studied the
responses of cerebral arteries and arterioles in cats to acute
hypotension and hypertension (40-200 mmHg pressure). They
demonstrated that the vascular responses of cerebral arteries and
arterioles to arterial pressure were caliber dependent; e.g., as blood
pressure decreased, larger vessels (150-173 µm diameter) began
to dilate at a higher pressure, whereas their maximum response, which
occurred at a lower pressure, was smaller than that of the smaller
vessels (37-59 µm). We utilized the data reported by Kontos et
al. to construct a compartmental model for the purpose of this report.
Synthesis of Reviewed Autoregulation Curves
We summarized the previously described autoregulation curves into three
types (types 1-3).
Types 1 and 2 (fixed and variable maximal vasoreactivity).
The simplest type of autoregulation curve (type
1) is based on the premise that once the LLA or ULA
is reached, either maximal vasodilation or vasoconstriction of the
resistive bed has occurred; hence, flow becomes pressure passive (3, 9)
(Fig. 2). The CBF-pressure curve is made up
of three straight lines: one horizontal line between the LLA and the
ULA and two sloped lines below the LLA and above the ULA, respectively.
Importantly, this description of autoregulation implies that the slope
of the line above the ULA
(slopeupper) is smaller than the
slope of the line below the LLA
(slopelower).
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Fig. 2.
CBF, cerebrovascular resistance, and arteriolar diameter for fixed
maximal vasoreactivity type of autoregulation. Between lower and upper
limit of autoregulation (LLA and ULA), CBF is autoregulated by changing
vessel diameter. Vessel dilates as pressure decreases and reaches its
maximal size when pressure falls below LLA. Similarly, vessel
constricts as pressure increases and maintains its minimal size when
pressure rises above ULA. Two sloped lines are not in parallel with
each other (cf. variable maximal vasoreactivity type in Fig. 3).
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The second type of autoregulation (type
2) is similar to type
1, except the flow-pressure relationship above the ULA
has the same slope as, and is thus parallel to, the relationship below the LLA (Fig. 3). Type
2 is based on the fact that the CBF autoregulation curve most frequently described in the experimental literature shows
such a parallel pattern (13, 29).
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Fig. 3.
CBF, cerebrovascular resistance, and arteriolar diameter for variable
maximal vasoreactivity type of autoregulation. At pressure below ULA,
this type of autoregulation is the same as fixed maximal vasoreactivity
type of autoregulation. However, when pressure rises above ULA, CBF
increases at same rate as when pressure is below LLA. This implies that
arteriole dilates when pressure rises above ULA. Two sloped lines are
in parallel with each other (cf. fixed maximal vasoreactivity type in
Fig. 2).
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Type 3 (third-order polynomial fit).
Direct fitting of the observed autoregulation curve is one way to
obtain a mathematical function to describe autoregulation. Dirnagl and
Pulsinelli (6) used a third-order polynomial to fit the autoregulation
data of rats, although the coefficients of their fitted function were
not reported. In the present study, third-order polynomials were used
to fit previously reported autoregulation curves from the rat reported
by Dirnagl and Pulsinelli and human data reported by Olsen et al. (27)
(see RESULTS and Fig.
4). Because Olsen et al. presented only the
autoregulation curves below the ULA, these curves were extended to the
range above their ULA (assumed to be 150 mmHg) by assuming that each
curve was center symmetrical (before fitting), with the center at a
mean blood pressure of 100 mmHg and CBF of 50 ml · 100 g
1 · min1.
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Fig. 4.
CBF of type 3 autoregulation curves.
CBF curve was obtained by curve fitting to 3rd-order polynomial of data
reported by Dirnagl and Pulsinelli (6) (dashed line) and Olsen et al.
(27) (solid line). Prediction that blood flow ceases if pressure is
<30 (dashed line) or 20 (solid line) mmHg conflicts with experimental
observations.
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Construction of Compartmental Model
We constructed a new compartmental model of autoregulation by assuming
that the CBF is regulated by the response of resistive arteries and
arterioles to the change in perfusion pressure. We further assumed that
ICP and cerebral venous pressure were constant and set them at zero, so
that the net perfusion pressure was equal to arterial pressure. The
observed vascular diameters were used to calculate cerebrovascular
resistance and CBF as follows
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(1)
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where
signifies CBF, d
is vessel diameter, P is perfusion pressure, 
is blood viscosity
(3.5 cP), L is vessel length, and
R is vascular resistance.
An ARD as a compartmental structure was developed to control the CBF.
The model of an ARD was constructed with four series compartments
(A, B, C, and
D in Fig. 1). The detailed structural parameters, such as numbers, diameters, and lengths of the vessels within each compartment of the ARD, are listed in Table
2. In each compartment the vessel diameter
was selected to be similar to the classification of Kontos et al. (16)
for pial arterioles found in their experiment, as shown in Table 2. The
lengths of the vessels were based on the experimental data in a 20-kg
dog (21). The numbers of the vessels in each compartment were based on
Murray's law (17)
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(2)
|
where
rX and
NX are the
radius and number of the vessel in compartment
X, respectively
(X = A, B,
C, or D).
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Table 2.
Structural parameters of an ARD representing an element structure of
cerebral artery and arteriole network to control CBF compared with
related experimental data
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The diameter of each of the four classes of vessels was the same as the
experimental results of Kontos et al. (16). As assumed here, it is the
responses of cerebral arterial and arteriolar vessels in these four
compartments to changes in arterial blood pressure that regulate the
CBF (see below). These data of Kontos et al. were limited and did not
cover pressure below 40 mmHg. To predict the vessel diameters at <40
mmHg pressure, certain assumptions were necessary. According to the
observations of Kontos et al. and MacKenzie et al. (19), small cerebral
arteries and arterioles reach their maximum diameter at ~40 mmHg. We
used an empirical formula to fit the
diameter
![<IT>d</IT> = <IT>a</IT><SUB>0</SUB>(1 − <IT>e</IT><SUP>−<IT>a</IT><SUB>1</SUB>P</SUP>)[1 − <IT>a</IT><SUB>2</SUB><IT>e</IT><SUP>−0.5(P−<IT>a</IT><SUB>3</SUB>/<IT>a</IT><SUB>4</SUB>)<SUP>2</SUP></SUP>] + <IT>a</IT><SUB>5</SUB><IT>e</IT><SUP>−0.5(P−<IT>a</IT><SUB>6</SUB>/<IT>a</IT><SUB>7</SUB>)<SUP>2</SUP></SUP>](/content/vol274/issue3/fulltext/H1023/img003.gif)
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(3)
|
where
a0,
a1,
a2,
a3,
a4,
a5,
a6, and
a7 are parameters
fitted by using the data of Kontos et al.
From Eq. 1, the series resistance of
an ARD (treated as an electrical circuit) is
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(4)
|
where
i varies between 1 and 4 to account
for the four compartments of an ARD (Fig. 1). The CBF through one
ARD is
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(5)
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The
equivalent diameter,
de, of an ARD was
calculated by its resistance
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(6)
|
where
L = L1 + L2 + L3 + L4.
LLA, ULA, and the Two Slopes of the Autoregulation Curve
For comparison of our model with the experimental autoregulation
curves, data were taken from the original studies (if the data were
given in numeric form) or reconstructed by us from the experimental
curves (if the data were given in graphical form). These data included
the LLA, the ULA, and two slopes of the CBF-pressure curve in the
ranges below LLA (slopelower)
and beyond ULA (slopeupper). For
reconstructed data, the following rules were applied:
1) the data were taken from
graphically represented curves for normal individuals (normotensive,
normocapnia, and without vasodilator or vasoconstrictor) with the sole
exception being the human data reported by Strandgaard et al. (37) (see
DISCUSSION);
2) the data were mean values if more
than one curve was reported; 3) the
LLA was estimated at the point where CBF decreased by 10% of baseline,
and ULA was estimated where CBF increased by 10% of baseline;
4)
slopelower was estimated as the
average value over the pressure range between the point where CBF
decreased by 50% from baseline and where pressure was equal to LLA;
5)
slopeupper was estimated at the
pressure where the autoregulation curve turns to a straight line right
to ULA.
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RESULTS |
Literature Review
The reviewed articles are summarized in Table 1. For each study, two
limits of autoregulation and two slopes of CBF-pressure curves in the
range of pressure below LLA
(slopelower) and beyond ULA
(slopeupper) were measured and
listed in Table 1. The average calculated data are given in Table 4 for
animals, humans, and animals and humans combined. The average values of
LLA and ULA and slopelower and
slopeupper for humans are 80 and
161 mmHg and 1.5 and 3.7% change in CBF per mmHg, respectively. The
data for animals are almost identical. These data are compared with
those calculated for the reviewed autoregulation curves and our model in Table 4 (see below).
Synthesis of Reviewed Autoregulation Curves
Our calculations revealed some limitations and disadvantages of three
previously used descriptions of autoregulation curves.
Type 1.
As shown in Fig. 2, the slopes,
slopelower and
slopeupper, predicted by this type
of autoregulation curve are 2 and 0.7% change in CBF per mmHg,
respectively. Slopelower is larger
and slopeupper is smaller than the
mean calculated from experimental data. Type 1 is inconsistent with most experimental data, which
demonstrate that the slope of the flow-pressure relationship in the
range of pressure below the LLA is not smaller than that above the ULA and that resistance does not remain constant when pressure increases above the ULA (6, 14, 37).
Type 2.
Although vascular diameter and resistance remain constant below the
LLA, the pressure increases above the ULA are associated with a
decrease in vascular resistance resulting from vasodilation (Fig. 3).
Compared with type 1, the curve
description of type 2 provides an
improved description of pressure-induced changes in upper-range CBF.
However, there are still discrepancies between this curve and
experimental observations. As observed in the cat (19), maximum levels
of vascular dilation and constriction do not necessarily occur at the
LLA and the ULA, respectively. Thus LLA (~65 mmHg) occurred at a
pressure significantly higher than that at which the vessels were
maximally dilated (~35 mmHg). The direct observation of vasodilation
in resistive arteries and arterioles at a pressure below the LLA has
also been described (16). Similarly, vasoconstriction continues as the
arterial pressure rises above the ULA. In general, because the vascular
resistance changes with the vascular diameter, the sections of the
autoregulation curve below the LLA and above the ULA cannot be
represented as simple straight lines.
The slopes, slopelower and
slopeupper, predicted by this type
of autoregulation curve are 2% change in CBF per mmHg.
Slopelower is larger and
slopeupper is smaller than the
mean experimental data. The prediction that
slopelower equals
slopeupper is not consistent with
experimental data.
Type 3.
A CBF-pressure curve was generated using an empirical third-order
polynomial, which is obtained by fitting to the data reported by
Dirnagl and Pulsinelli (6)
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(7)
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Another CBF-pressure curve was calculated with a third-order
polynomial by fitting to the data reported by Olsen et al.
(26)
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(8)
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Figure 4 shows the CBF calculated by using Eqs.
7 (dashed line), and 8 (solid line). From Fig. 4, one can see that Eqs.
7 and 8 cannot be used
to predict autoregulation at <30 mmHg pressure, inasmuch as they
predict that blood flow ceases at <30 mmHg pressure.
The slopes, slopelower and
slopeupper, predicted by this type
of autoregulation curve are 1.7 and 2.0% change in CBF per mmHg, respectively. Slopelower is
similar to the mean experimental data; slopeupper is smaller.
Compartmental model.
The results of the coefficients for Eq. 3 are listed in Table 3.
Figure 5 shows the diameter-pressure
relationships of the vessels in the four compartments compared with the
data of Kontos et al. (16). The fitted curves for vessel diameters
agree with the experimental data and provide reasonable predictive
values for pressures outside the range of the experimental data. The vessels of three smaller compartments (50, 150, and 200 µm) begin to
constrict if the pressure decreases below 40 mmHg, whereas this
threshold pressure is 70 mmHg for the largest vessels.
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Table 3.
Coefficients used to calculate diameters of vessels in the four
compartments of our compartmental ARD model
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Fig. 5.
Diameter regression results for vessels used in compartmental model
based on experimental data of Kontos et al. (16). Coefficients of
Eq. 3 were calculated and are listed
in Table 3. Data for <40 mmHg pressure were not presented by Kontos
et al.; however, regression results were consistent with experimental
data of MacKenzie et al. (19).
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The composite or equivalent resistance of an ARD was calculated using
Eq. 4, and a regression was carried
out for the calculated resistance, resulting in
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(9)
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Figure 6 shows the predicted curves of
the vascular resistance and the resultant equivalent diameter and CBF
of an ARD based on Eq. 9. The two
limits, LLA and ULA, were calculated from the CBF-pressure curve,
inasmuch as the LLA and ULA are defined as the pressure where the CBF
changes 10% from the baseline (the mean value of the plateau). When
the pressure decreases below the LLA, the equivalent diameter of an ARD
increases until it reaches the maximum at a lower pressure of ~60
mmHg. As the pressure approaches zero, the vascular resistance
approaches infinity, because the diameters of all vessels in the model
are approaching zero. Some experimental data are also plotted in Fig.
6. These individual data points were obtained from hand-fitted curves
of reported data, because the numerical values were not reported in the
original reports. Some deviations in resistance between the
experimental data and the regression curve occur at ~70 and >150
mmHg pressure. An attempt was made to reduce the deviation by using the
best-fitted curve. However, the best-fitted curve resulted in a
CBF-pressure curve that slightly underestimates the plateau at
100-150 mmHg pressure.
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Fig. 6.
Regression results of cerebrovascular resistance, blood flow, and
effective diameter of an ARD in compartmental model. When pressure
decreases below LLA, vessel continues to dilate until it finally
reaches maximum at a lower pressure of 40 mmHg for 3 small vessels
(diameter = 50, 150, and 200 µm) or 70 mmHg for large vessel
(diameter = 300 µm). Some experimental data are also plotted.
Resistance is calculated directly from experimental data of Kontos et
al. (16) (see METHODS,
Construction of Compartmental
Model). Experimental data of CBF are readings of 2 hand-fitted curves of data reported by MacKenzie et al. (19, 20).
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Our compartmental ARD model predicted LLA and ULA that were similar to
those averaged over all reviewed articles (Table
4). The predicted LLA is 69 mmHg, which is
between a most frequently cited value of 50 mmHg and an estimated
experimental value from the reviewed articles of 73 mmHg. The predicted
ULA is 153 mmHg, similar to a frequently cited value of 150 mmHg. It
is, however, slightly lower than 157 mmHg, a value estimated from the
reviewed experimental articles (Table 4). The mean values of
slopelower and
slopeupper predicted by our model
are 1.3 and 3.3% change in CBF per mmHg, respectively; these values
compare favorably with the experimental data obtained from the
literature review: 1.5 and 3.4% change in CBF per mmHg. Our model
produces a 13 and 3% difference in
slopelower and
slopeupper, respectively, from the
experimental data, being much smaller than those for the three types of
autoregulation curves reviewed here: 33 and 79% for
type 1, 33 and 41% for
type 2, and 13 and 41% for
type 3 (Table 4).
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Table 4.
Principal autoregulatory parameters predicted by our compartmental ARD
model compared with experimental data of reviewed literature
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DISCUSSION |
In this study the network of small arteries and arterioles was
simplified as an ARD that controls CBF. A compartmental ARD model was
constructed to simulate the autoregulation function of an ARD. The
structural parameters of the model and the response of the vessel
diameters to the changes in arterial pressure were based on reported
experimental data. The blood flow through the ARD was calculated by
Eqs. 4 and 5 and was represented as a function of
pressure (autoregulation curve). The estimated values of LLA, ULA, and
two slopes of the autoregulation curve of this model were consistent
with previously reported experimental data. A summary of the comparison
of our model with the previous curves is shown in Table
5.
Regarding the previously used curves, two simple types of
autoregulation curves, i.e., fixed maximal vasoreactivity
(type 1) and variable maximal
vasoreactivity (type 2), are
frequently used or implied by previous studies. However, the simplified
assumptions used to generate these curves and their limitations have
not been previously addressed. The common characteristic of these two
curves is that the CBF-pressure curve is generated with three straight lines. The experimental data, however, demonstrate that in ranges of
pressure below the LLA and above the ULA the diameter of small arteries
or arterioles changes with arterial blood pressure. Thus in these two
ranges the autoregulation curve cannot be represented by straight
lines, as types 1 and
2 would necessitate. Furthermore, although type 3 is based on
experimental data, it does not account for the behavior of different
vascular hierarchies.
In this study, absolute CBF values were not discussed, because CBF is
proportional to the weight of the tissue perfused. This model suggests
that, in constructing an autoregulatory curve with a morphology
consistent with observed data, vessels of different hierarchical
levels, including small arteries and arterioles, must be taken into
consideration. The traditionally used "diameter" of
autoregulatory vessels, as embedded in the terminology "maximal vasodilation" or "maximal vasoconstriction"(29), is simply a
weighted average for an effective "composite" ARD, which has no
clear biophysical or anatomic basis and is not supported by
experimental observations.
The discrepancies between the experimental data and our simulated curve
occur at ~70 and >150 mmHg pressure. Our model minimally underestimates the plateau at 100-150 mmHg pressure. This may be
due to the omission of some hierarchy of circulation, the effects of
which on autoregulation are not negligible. It may well be that the
circulatory level not accounted for includes intraparenchymal vessels
(38). Although the pial circulation is the most readily accessible for
study of vascular behavior, there may be underlying differences from
the intraparenchymal resistive bed, as suggested by experimental
observations (9, 38).
Our proposed ARD compartmental model might be termed an instrumentalist
approach to describing autoregulatory behavior. Such an instrumental
approach in the present application is merely a mathematical tool for
deducing one set of variables from another (30). We cannot claim that
the ARD compartmental model that we have described is based on the
essential mechanistic properties of the cerebral vascular network. The
ARD model is simply a means for predicting the behavior of an
autoregulating vasculature by knowledge of arterial perfusion pressure.
Our model (instrument) predicts observational or experimental data
better than previously proposed systems, i.e., curves for
types 1-3.
Besides the three types of autoregulation curves reviewed, several
mathematical models have been proposed to simulate CBF autoregulation
(8, 28, 39) or regulation in other vascular beds (4). In general, they
have been primarily focused on the time-dependent responses of the
cerebral vessels. Borgstrom et al. (4) used the forces exerted on the
vessel wall to determine the vessel radius and then used radius to
determine flow. The forces used by Borgstrom et al. included passive
pressure force, passive elastic force, passive wall viscosity, dynamic
myogenic force, and control contractile force. Ursino and Di Giammarco (39) and Giulioni and Ursino (8) used three feedback mechanisms (myogenic, sympathetic, and cholinergic) to control the muscle tension
of the vessel wall and used vessel tension (and pressure) to determine
the vessel diameter and then determine CBF. Panerai (28) used fast
Fourier transform methods to analyze the experimental data relating
pressure, CBF velocity, and resistance-area product to obtain a
spectrum of gain functions, which as an indicator of CBF was then used
to determine resistance-area product in terms of CBF velocity, and then
determined CBF. Our model directly applied experimental data to a new
compartmental ARD model to simulate the CBF-arterial pressure curve.
In comparison to these previously proposed models, our approach is
novel and offers some improvements over existing models for several
reasons. First, the experimental data used in this model are based on
direct observations of diameter responses of vessels to changes in
arterial pressure (16). These vessels were selected from a relatively
wide range of arterial and arteriolar sizes that are considered to be
important in CBF autoregulation. Pressure-induced changes in vessel
diameter were the controlling factor for CBF. Previously described
models (4, 8, 28, 39) used indirect factors such as myogenic force,
sympathetic feedback, or muscle tension of the vessel wall to determine
diameter, then used diameter to determine blood flow. In these models,
therefore, the assumptions of the relationship between indirect factors
and diameters will affect the predicted CBF. The use of previously reported experimental data allowed our model to avoid using indirect factors and thus reduced the error associated with the assumed relationship between indirect factors and vascular diameters. Second,
our compartmental ARD model is structurally different from the
compartmental models used in other autoregulation modeling studies. In
our model there are four compartments, each of which contains a number
of identical vessels. The number of vessels inside a compartment was
determined by Murray's law. In the other models the number of vessels
in each compartment was not discussed in detail. Therefore, a
compartment in our ARD model contains more information about the
structure of the vascular network. This information could be used to
investigate the hemodynamic properties of intracranial arterial vessels
in different levels of cerebral circulation. Our ARD approach may also
be used to improve our previously developed compartmental model of a
complete cerebral blood circulation (7). Third, one result of our ARD model is a simple formula that can easily predict a CBF response that
is in reasonable agreement with experimental mean values (Table 4).
There are several important limitations to our approach. First, our
model did not include mechanisms describing how pressure influences
diameter. Second, of the many factors that influence CBF, only the most
important one, arterial pressure, has been taken into account. Other
factors, such as ICP or the effect of arterial
PCO2, have not been included. Third,
in our model, time-averaged values were used for all parameters;
therefore, the dynamic aspects of autoregulation were not considered.
Although it is a "black-box" approach, the fit of observed data
from a wide range of human and animal preparations supports the
underlying mechanistic proposition that vascular hierarchies are of
critical importance in flow regulation. In our model, physiological,
biomechanical, and anatomic concepts, such as Murray's law,
Poiseuille's law, and the structural parameters of vessels (diameter
and length), are used.
Because capillaries and veins were not included in the present model,
ICP and cerebral venous pressure were set at zero. The effects of
changing ICP and cerebral venous pressure on CBF autoregulation have
been investigated experimentally (2, 32, 41) and theoretically (8).
Some groups (41) have reported that the cerebral vascular bed responds
to changes in the perfusion pressure gradient in a similar fashion,
whether these changes are obtained by decreasing mean arterial
pressure, increasing jugular venous pressure, or increasing
cerebrospinal fluid pressure. However, others (2) have reported that
the effect of changing ICP on inner radius may be quite different from
the effect of changing arterial blood pressure. Further work on the
current model needs to be done to take into consideration
pathophysiological changes in ICP and cerebral venous pressure.
The contribution of postarteriolar resistance in the capillary and
venous vascular beds was not included in this study. If we assume that
postarteriolar resistance is constant, our model will overestimate CBF
when the precapillary resistance is lower and underestimate CBF when
the precapillary resistance is higher. If we further assume that
postarteriolar resistance is ~20% of total cerebrovascular
resistance at 120 mmHg pressure, then CBF is overestimated by 10% at
minimal resistance (60 mmHg pressure) and underestimated by 6% at
maximal resistance (170 mmHg pressure).
Because the experimental data for vessel diameters at profound
hypotension (<40 mmHg) or severe hypertension (>200 mmHg) are not
available, the fitted parameters (Table 3) of Eq. 3 are not necessarily able to reproduce correct
diameters at these extremes of pressure. By use of reported data above
40 mmHg and with the assumption that CBF is zero at zero pressure and
that the resistive vessels reach their maximal dilation at 40 mmHg
pressure, the inner diameter-pressure curve was fixed at 0 and 40 mmHg
pressure. Therefore, the shape of this curve at <40 mmHg pressure
will not be significantly affected by changing the fitted parameters
(Table 3). However, at >200 mmHg pressure, inner diameter will be
affected by changing the parameters. Consequently, the morphology of
the autoregulation curve above 200 mmHg cannot be predicted by this model. Our model is constructed with experimental data that are limited
in pressure range and species (cat). Because of insufficient data,
examination of the variability of parameter estimates is not possible
here. Because Eq. 3 is empirical and
provides no information about the physiological nature of the vessels,
there is no indication of how the parameters may change as a
consequence of external stimuli. Parameter sensitivity was estimated
for Eq. 3 by increasing each of the
parameters listed in Table 3 by 10%, resulting in <5.4% change in
diameter, except for
a2 of the largest two vessels (compartments C and
D). A 10% change in
a2 induces changes in diameter of 15 and 20% for the vessels in
compartments C and
D, respectively. We do not wish to
attach physiological significance to each parameter in
Eq. 3; we merely state that it is an
empirical approach to better simulate autoregulation.
The true vascular network is a nonlinear, time-dependent system. For
simplicity in our model, we assumed that the vessels were linearly
combined in a compartment and that the compartments were linearly
combined in the ARD. In our model, time-averaged values were used for
all parameters.
Our compartmental ARD model provides a mathematical function to
describe a CBF-pressure autoregulation curve that is based on observed
data. This study supports the hypothesis that there are multiple sites
of autoregulation in animal and human cerebral circulation. In future
studies the model can be used to explore or predict responses to
experimental interventions that preferentially affect different
hierarchical levels of the cerebral circulation.
| |
ACKNOWLEDGEMENTS |
The authors thank Joyce Ouchi and Steven Marshall for assistance in
preparation of the manuscript. The authors gratefully acknowledge the
support and contributions of the other members of the Columbia
University Arteriovenous Malformation Study Project.
| |
FOOTNOTES |
This work was supported in part by National Institute of Neurological
Disorders and Stroke Grants RO1-NS-27713 and RO1-NS-34949 and in part
by a Clinical Scholar Grant from the International Anesthesia Research
Society.
Received 8 April 1997; accepted in final form 26 November 1997.
| |
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Abstract
Introduction
