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1 Faculty of Design, Engineering, and Production, Department of Medical Technology and Mechanics, Man Machine Systems and Control Group, Delft University of Technology, 2628 CD Delft; and 2 Department of Medical Physics, Cardiovascular Research Institute Amsterdam, Academic Medical Center, University of Amsterdam, 1105 AZ Amsterdam, The Netherlands
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Myogenic response, flow-dependent dilation, and direct metabolic control are important mechanisms controlling coronary flow. A model was developed to study how these control mechanisms interact at different locations in the arteriolar tree and to evaluate their contribution to autoregulatory and metabolic flow control. The model consists of 10 resistance compartments in series, each representing parallel vessel units, with their diameters determined by tone depending on either flow and pressure [flow-dependent tone reduction factor (TRFflow) × Tonemyo] or directly on metabolic factors (Tonemeta). The pressure-Tonemyo and flow-TRFflow relations depend on the vessel size obtained from interpolation of data on isolated vessels. Flow-dependent dilation diminishes autoregulatory properties compared with pressure-flow lines obtained from vessels solely influenced by Tonemyo. By applying Tonemeta to the four distal compartments, the autoregulatory properties are restored and tone is equally distributed over the compartments. Also, metabolic control and blockage of nitric oxide are simulated. We conclude that a balance is required between the flow-dependent properties upstream and the constrictive metabolic properties downstream. Myogenic response contributes significantly to flow regulation.
myogenic response; flow-dependent dilation; metabolic control; autoregulation; mathematical model
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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CORONARY FLOW CONTROL is the result of modulation of tone in resistance arteries. On the basis of measurements and theoretical predictions of pressure profiles, the resistance vasculature spans vessels with diameter up to 400 µm (3, 20, 43). Because of this substantial distribution of resistance, a coordinated action of control mechanisms at different domains in the arterial tree is required to adapt flow to metabolism. The control mechanisms proposed are 1) metabolic factors, which are believed to have an effect only on the small arterioles (2, 18, 19); 2) myogenic control, constricting the arteries and arterioles with rising pressure and vice versa (5, 6, 14, 15, 21-23, 30, 37); and 3) flow-induced dilation, dilating the vessels in response to an increase in shear stress, modulated by nitric oxide (NO) (18, 22, 24, 25, 40). These mechanisms have been shown to play a role in regulating tone both in isolated coronary arteries and arterioles and in vivo. Theories integrating these mechanisms are proposed in the literature (7, 8, 17, 28, 31); however, a quantitative analysis that bridges the knowledge from isolated vessel studies and whole organ studies is lacking. We developed a mathematical model to study the possible interactions of these control mechanisms during flow control.
In an earlier study, we (5) demonstrated that pressure-induced myogenic tone alone, taking into account the diameter dependence of myogenic strength, could predict well the course of coronary autoregulation curves. However, this model raises two concerns: 1) flow-dependent dilation will diminish autoregulation; and 2) tone (myogenic) in the smallest vessels is almost absent because of the low level of pressure. The current study has the hypothesis that flow-dependent dilation in upstream vessels is compensated by vasoconstriction of the metabolically controlled smaller vessels, resulting in 1) restoration of autoregulation properties to the levels found for the model with pressure-induced myogenic tone alone and 2) increase in tone in the smaller vessels, leading to the possibility of dilation by increasing metabolism and to a more equal distribution of tone over the resistance vessels.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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A theoretical model of flow control in the arterial tree was made based on a series arrangement of vessel units, each unit having either myogenic and flow-induced dilatory properties or metabolic properties. The parameters for these units are obtained from experimental data of Liao and Kuo (27) and Kuo et al. (25) on the myogenic response and flow-dependent dilation.
Compartmental Model of Arterial Tree
The model of the arterial tree was described previously (5). The first nine compartments describe the behavior of the coronary arterial tree covering diameters ranging from 10 to 500 µm. The resistance of the capillaries and the venules are assumed to be constant and are lumped in the tenth compartment.We assume that each compartment i (i = 1...9), represents Ni identical vessels in
parallel. According to Poiseuille, the resistance of the first nine
compartments (Ri) can then be described with
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(1) |
i is the viscosity of the
perfusate and is taken to be dependent on the diameter of the vessel as
determined in the rat mesentery vasculature by Pries et al.
(35), assuming a systemic hematocrit of 45%
![&eegr;<SUB>i</SUB>=<FENCE>1+[1.6×10<SUP>−3</SUP>(6<IT>e</IT><SUP>−0.085<IT>d<SUB>i</SUB></IT></SUP><IT>+</IT>3.2<IT>−</IT>2.44<IT>e</IT><SUP>−0.06<IT>d</IT><SUP>0.645</SUP><SUB><IT>i</IT></SUB></SUP>)<IT>−</IT>1]</FENCE>](/content/vol282/issue6/fulltext/H2224/img002.gif)
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(2) |
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The vessels in each compartment are characterized by their anatomic
diameter, danat,i. This diameter is
defined as the passive diameter at 100 mmHg. The choice for the
anatomic diameters for each compartment was justified in a previous
study (5), and these diameters are shown in Table
1.
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The number of parallel vessels per compartment
(Ni) follows from the condition that the total
flow in each compartment
(
iNi) should be equal. Flow
(i) in each vessel unit was assumed to
match the velocity (vi)-diameter data of passive
vessels measured by Stepp et al. (40) in the intact canine
epicardial circulation
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i
for the fully dilated tree at 90 mmHg, resulting in values for
Ni, shown in Table 1.
The resistance of the vessel compartments
(Ri) at a perfusion pressure of 90 mmHg at full
dilation can be calculated from the assumed pressure drop (
P = 60/9 mmHg) and the flow distribution over the tree:
Ri = P/(iNi). With Eq. 1, the values for the length of the vessel units
(Li) follow, which are also shown in Table 1.
The constant resistance in the tenth compartment equals 15.4 × 103 mmHg · ml
1 · s [=
30/(iNi)].
Control Mechanisms of Isolated Vessels
Experimental data defining control mechanisms. Liao and Kuo (27) and Kuo et al. (25) performed experiments in four different sizes of isolated porcine subepicardial microvessels with anatomic diameters of 65, 100, 165, and 255 µm. The vessels constricted to stepwise increase in luminal pressure (20, 40, 60, 80, 100, and 120 cmH2O), and the strongest myogenic response was found in arterioles of 100 µm. The vessels also demonstrated flow-dependent dilation. The strongest response to shear occurred in arterioles of 165 µm. Furthermore, because NO mediates the flow-dependent response, they investigated the dose-response curve for nitroprusside in these vessels. For the four different sizes of microvessels, identical dose-dependent dilation for this NO donor was found.
Description of vessel unit in model. A set of equations was derived that describes the data of Liao and Kuo (27) and Kuo et al. (25) and allows for interpolation and extrapolation of responses within and outside the range of tone and diameters studied by these authors. We developed these equations on the basis of a biophysical model of vessel wall behavior as described earlier by VanBavel and Mulvany (42).
The biophysical model for the isolated vessel is based on a mechanical equilibrium involving passive and active tensions in the arteriolar wall, luminal pressure, and vessel diameter (36, 42). The relations on which this equilibrium is based are shown in Fig. 1. The passive tension (Tpas) is the tension developed when the vessel wall is stretched while the smooth muscle is completely inactive. When the smooth muscle is active, an active component of tension (Tact) comes into play. For a particular diameter, when the vessel wall is in an equilibrium state the wall tension (Twall), defined as the sum of these tensions (Twall = Tact + Tpas), is assumed to fulfill the law of Laplace: Tlaplace = Twall = P × d/2. In this equation Tlaplace is the Laplace tension, which is the tangential force component acting on the wall per unit length; P is the average transmural pressure of the vessel; and d is the inner diameter of the vessel. The tensions are normalized to the Laplace tension of a passive vessel at 100 mmHg, in which the diameter by definition equals the anatomic diameter (danat). Therefore, normalized wall tension equals T
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i) or pressure and an intrinsic vascular
tone modulated by metabolism (Tonemeta); the output of the
model is the normalized diameter (d
i, Pi, and Tonemeta follow from the
behavior of the model tree as described in Algorithm for Network
Simulations.
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) indicating the imbalance in tensions in the vessel wall.
When TRelations within vessel units.
We first present equations describing T
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(3) |
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(4) |
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(5) |
![TRF<SUB>flow</SUB> = 1 − <FR><NU>[NO]<SUP>HS<SUB>TRF<SUB>flow</SUB></SUB></SUP></NU><DE>[NO]<SUP>HS<SUB>TRF<SUB>flow</SUB></SUB></SUP><SUB>50</SUB> + [NO]<SUP>HS<SUB>TRF<SUB>flow</SUB></SUB></SUP></DE></FR>](/content/vol282/issue6/fulltext/H2224/img017.gif)
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(6) |
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![[NO] = [NO]<SUB>max,<IT>i</IT></SUB> <FR><NU><IT>&tgr;</IT><SUP>HS<SUB>NO,<IT>i</IT></SUB></SUP></NU><DE><IT>&tgr;</IT><SUP>HS<SUB>NO,<IT>i</IT></SUB></SUP><SUB>50,<IT>i</IT></SUB><IT>+&tgr;</IT><SUP>HS<SUB>NO,<IT>i</IT></SUB></SUP></DE></FR>](/content/vol282/issue6/fulltext/H2224/img019.gif)
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(7) |
50,i is
the shear stress where 50% of [NO]max,i is
reached. In this way, we obtained for each diameter group a set of
parameters that can be used for prediction of shear stress-[NO] relations for vessel units with different anatomic diameters (for details see APPENDIX).
Algorithm for Network Simulations
The iterative procedure used for the network simulations can be found in Cornelissen et al. (5) and is described briefly below. For a certain initial resistance distribution (Ri = Rstart,i), the flow () and the
pressure distribution [average pressure (Pi),
inflow pressure (Pin,i), and outflow pressure (Pout,i) of the vessel units] are calculated.
With the model for the vessel unit (Fig. 2), di
is determined. Applying Poiseuille's law (Eq. 2), a new
resistance distribution (Ri) follows. With the
pressure drop over the compartments, new
i for each compartment can be calculated.
With the new resistance distribution, the flow () and the
pressure distribution (Pi, Pin,i, and Pout,i) are
calculated. When the relative error between
i and is <1010,
the iterative process stops.
The data of Kuo et al. (25) show that the effect of shear
stress on TRFflow is saturated above 4 dyn/cm2. The estimates for shear stresses in vivo
are in general >4 dyn/cm2, whereas
flow-dependent dilation is believed to be still involved in control of
blood flow. We considered the possibility that shear stress sensitivity
is different in vivo from that in vitro and accordingly studied the
effect of shifts in 50,i of Eq. 7.
Therefore, we introduce an attenuation factor A by which
50 is divided.
When metabolism-dependent tone was added to a number of the most distal compartments the procedure was as follows. The model solutions were obtained for pressure-induced myogenic tone alone, and the resulting flow was used as set point flow for the network. Flow-dependent dilation was made active, resulting in an increase in flow above the set point flow. Tone in the metabolic units was then increased to bring back flow to its set point level. The level of tone in the distal vessel units needed for this flow decrease is referred as metabolism-dependent tone (Tonemeta).
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Simulations were performed for perfusion pressures ranging from 5 to 140 mmHg and outflow pressure of 0 mmHg. The simulated pressure-flow
relations are shown in Fig.
5A. The simulated
pressure-flow relation for maximal dilation is slightly curved as a
result of the pressure dependence of resistances. When tone in the
vessel units is determined by pressure-induced myogenic tone alone,
autoregulation is revealed: flow rises less than proportionally with
pressure. Flow-dependent dilation reduces the autoregulation
characteristics of the pressure-flow line. The attenuation factor
(A) that modifies the shear stress-[NO] relation is set to
0.046.
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In Fig. 5B, the effect of the attenuation factor on normalized flow at perfusion pressure of 90 mmHg is demonstrated. When A = 1, the flow equals 74% of the flow at maximal dilation; when A < 0.2, the flow decreases rapidly until A approaches zero and the normalized flow converges to the flow in the case of pressure-induced myogenic tone alone. To make sure that the flow-dependent mechanism is effective, the attenuation factor was set for further analyses to 0.046.
The distributions of normalized diameter and tone at a perfusion
pressure of 90 mmHg are depicted in Fig.
6. With pressure-induced myogenic
activity alone, tone drops gradually from the larger resistance vessels
to the smaller vessels. Note that normalized diameters of the largest
and smallest vessels are close to one. Flow-dependent dilation changes
the tone and diameter distribution over the compartments, but the
resulting changes in pressure distribution introduce marginal changes
in pressure-induced myogenic tone distribution (Fig. 6C).
Flow-dependent dilation gives a plateau in the diameter-tone relation
and gives an increase in diameter, most emphatically apparent in the
larger resistance vessels (Fig. 6A).
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Figure 7 demonstrates the effect of
balancing flow-dependent dilation with metabolic constriction on the
distribution of diameter and tone. This effect is presented for a
perfusion pressure of 90 mmHg and an attenuation factor of 0.046. Restricting metabolic compensation of flow to the most distal
compartment requires a high tone (Tonemeta = 0.99).
Distributing the required resistance increase equally over more than
one compartment reduces the amount of tone per vessel unit (Fig. 7).
Having four compartments under the influence of metabolism-dependent
tone results in a more or less equal tone distribution over the
compartments (Fig. 7A).
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In Fig. 8, the distributions of tone at
perfusion pressures of 60, 90, and 120 mmHg are depicted. To bring flow
back to the autoregulation curve obtained with pressure-induced
myogenic tone alone, a lower level of metabolism-dependent tone is
required at lower perfusion pressure. Most of the changes in tone of
the larger vessels are due to pressure-induced myogenic tone
variations. The changes in TRFflow oppose the desired
effect; however, the changes are small, which is to be expected because
flow is kept in a narrow range.
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Full dilation of the smaller vessels is simulated by reducing the tone
of the metabolic vessels to zero. Such local dilation has been observed
to result from adenosine infusion and metabolic dilation induced by NO
synthesis inhibition of larger resistance vessels (18).
The effect of metabolic dilation for these two conditions is
demonstrated in Fig. 9. Distal dilation
without inhibition of NO synthesis increases flow from 37% to 80% of
maximal dilated values. A substantial part of the decrease in total
resistance is due to the dilation of the proximal vessel units (14%).
Tone in these vessels decreases (Fig. 9A), and it is clear
that both the pressure-induced myogenic mechanism (Fig. 9B)
and the flow-dependent mechanism (Fig. 9C) enhance the
effect of distal dilation on reduction of overall coronary resistance.
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Inhibition of NO synthesis in our model is simulated by setting the TRFflow of the proximal vessels to 1. Flow increases only from 37% to 47% of maximal dilated values. The behavior of the vessels in the proximal compartments depends on location in the circuit. As indicated in Fig. 9A, the vessels in the third compartment decrease diameter with increasing tone, whereas in the fifth compartment both diameter and tone decrease. The pressure-induced myogenic tone (Fig. 9B) in compartments 1-5 is always reduced by dilation of compartments 6-9. Thus constriction of the larger vessels by inhibition of NO synthesis is attenuated by the myogenic mechanism, indicating that the upstream constriction is not only compensated by the metabolic mechanism but the myogenic mechanism also contributes to this compensating effect.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Pressure-induced myogenic tone alone results in autoregulatory properties of the coronary model, independent of factors related to metabolic control, i.e., the flow is rather independent of arterial pressure. When the flow rate through this myogenically controlled tree matches flow demand, a stable perfusion system is obtained. However, flow-dependent dilation induces a flow rate higher than needed. Therefore, an increase in tone in the smaller downstream vessels should compensate for this. When vessels with diameters ranging from ~10 to ~40 µm are assumed to have a higher intrinsic tone than found in isolated vessels, this tone can then be modulated by metabolism and tone is more or less equally distributed over the arterial tree. Moreover, the level of tone in all compartments is such that a sufficient range of tone in these compartments is available to adjust flow to the large variations in tissue oxygen demand. To distinguish the responses in the smallest resistance vessels from those in the larger ones we introduced the notion of "metabolism-dependent tone" to indicate that this tone is modulated predominantly by metabolism.
Previous Model Studies
Liao and Kuo (27) used a network model similar to ours to investigate the interaction of shear stress, pressure, and adenosine, an endogenous vasodilator. They used a four-compartment model with vessel units based on the same data as used in our study.However, an essential difference is that their model is diameter based and ours is wall tension based, which seems to better fit a biophysical description of vessel wall mechanics. The effect of shear stress-dependent dilation on the adenosine-flow dose-response relation was investigated, a problem different from the one addressed here.
Granger (11) used a three-compartment model. However, each compartment was characterized by a single control mechanism, which is different from the present model, in which the strength of myogenic tone and flow-dependent dilation vary gradually in a realistic way over the different compartments. Ursino et al. (41) developed a theoretical network model based on wall tension with five branched elements to analyze the functional role of dynamic vasomotion in blood flow control, but they did not study interactions between control mechanisms.
Evaluation of Model Structure and Parameter Choice
The model consists of a series of resistances, neglecting the stochastic nature of branching within the coronary tree. Obviously, the responses of individual vessel units depend on the assumption of the resistance and velocity distributions over the vessel units. The resistance distribution under dilated conditions was based on the epicardial pressure diameter measurements of Chilian et al. (1) and was justified in our earlier study on myogenic tone (5). The velocity distribution over the different units was based on the average data of Stepp et al. (40). Both experiments demonstrated a considerable intervessel variability. We have not addressed this variability because it would require a stochastic branching model that would have obscured the interactions between control mechanisms. This interaction was the main purpose of this study. The limitation of the model structure is an important reason for some quantitative differences between predictions and experimental data, although the characteristics of the model and the experimental responses tally well.In the vascular wall of veins and venules smooth muscle cells are observed, as well as diameter changes in response to sympathetic activity and certain agonists. However, the contribution to resistance of this part of the circulation under physiological conditions is small compared with the contribution to resistance of the vessels studied here (resistance vessels ranging from ~20 to 500 µm). Therefore, the assumption of a constant resistance contribution of capillaries and veins during flow control under physiological conditions seems quite realistic.
Important to the model is the law of Laplace. We used the law of
Laplace for thin-walled vessels, whereas a more general formula is
T = 0.5[P1d 
P2(d + 2t)] (34),
where P1 and P2 are the pressure inside and
outside the vessel, respectively, d is the inner diameter, and t is the wall thickness. The deviation between the thin
and thick wall equations increases with the wall thickness-to-diameter ratio but is diminished when the transmural pressure
(P1 P2) increases. This deviation is
not the same for all vessel diameters, and for a single vessel it
varies with tone. All these effects are neglected in the model. The
application of the thin wall formula simplifies the mathematics of the
model because it results in a linear tension-diameter relation for a
given pressure. The simplification seems justified in the context of
other uncertainties, especially with respect to the
diameter-Tmax, pressure-Tonemyo, and
shear-TRFflow relations in the model.
The relations chosen to determine Tmax and Tonemyo are not uniquely defined by the data of Liao and Kuo (27). For example, applying a different diameter-Tmax relation, where Tmaxtop,i is twice the original value and dm is 90% of the original value, results in less steep pressure-Tonemyo,i relations. However, the resulting autoregulation curve for myogenic tone alone is hardly different from the original one and, furthermore, the number of compartments under influence of metabolic control required to have an approximately equal distribution of tone is still four. The only difference is that the value of tone is smaller under these circumstances than when the original diameter-Tmax and pressure-Tonemyo relations are used.
Flow-dependent dilation is mediated by shear stress-induced production of NO. We have modeled this in three steps to reconcile the observations that the shear stress-TRFflow relation is dependent on anatomic diameter but the [NO]-TRFflow relation is not. Both relations were measured by Kuo et al. (25). Our model fits their relations by having components providing the diameter-dependent relation between shear stress and [NO] and the relation between [NO] and TRFflow. The data of Kuo et al. (25) suggest that for high shear stresses, [NO] saturates such that vasodilatation is submaximal. The level of [NO] would therefore depend on vessel size. The reason for submaximal vasodilatation is unknown.
The attenuation factor (A) in the model modifies the relation between shear stress and [NO] and by definition equals unity for the experiments of Kuo et al. (25). This factor deviates from unity to compensate for scavengers and other factors that may affect the shear stress-TRFflow relations. This factor only affects the effective working range of the [NO]-shear stress relation and not the level of [NO] at infinitely high shear stress. As shown in Fig. 5B, a large A results in a large flow, and thus, in our concept, a large counteracting metabolic resistance is needed. However, because the shear stresses in the vessel units of the network are all >7 dyn/cm2, the flow-dependent mechanism would hardly be effective: possible changes in shear stress do not result in changes in [NO] and thus do not result in changes in TRFflow. When A is too low, the counteracting metabolic resistance is low and metabolic control in the sense of metabolic dilatation is limited. With, for instance, A = 0.01, it is sufficient to set only the last compartment under the influence of metabolic control. Tonemeta would then be only 0.47.
There may be several reasons why the in vitro vessels behave differently in response to shear stress compared with the in vivo vessels. An important reason is obviously the fact that in vitro the red blood cells are missing that are metabolizing NO at a high rate. To study this would require a dedicated model in itself, to compensate for all the factors involved in controlling it, including the endothelial glycocalyx (4). Moreover, we left out other mechanisms that influence local vascular tone and concentrated on the interaction of two mechanisms well defined in in vitro studies. Hence, it is assumed that the attenuation factor, scaling the data from isolated single vessels to the whole heart, takes all these unknown factors into account.
Application of Model to Interpretation of Distributed Response in Coronary Circulation
Distributed vascular diameter response during metabolic regulation. Several elegant experiments (18, 40) have attempted to study the distributed response of flow-dependent dilation and metabolic influence over the vessels of different diameters in the coronary tree. However, the pressure-induced myogenic tone is always present as an additional factor of tone generation, and it is not always clear how this is affected by the interventions.
In experiments of Jones et al. (18) on subepicardial microcirculation in the beating dog heart it was demonstrated that NO blockage by NG-nitro-L-arginine methyl ester resulted in vasoconstriction of resistance vessels >100 µm, but in dilation of vessels smaller than this threshold diameter [in another study of Jones et al. (16) NO synthesis inhibitor was administered intravenously, and these data are not considered here]. This dilation was explained as a metabolic dilation and/or myogenic mechanism for compensation of the upstream constriction. Because adenosine could not further dilate the vessels smaller than 100 µm it was assumed that a metabolic stimulus for dilation was already maximal. Further evidence for a 100-µm threshold for feedback of metabolic vasodilatation was obtained from the pacing experiments reported in the same study. Adenosine administration alone also dilated only the vessels with diameter <100 µm. In our study we also found a threshold for metabolic dominance in the smaller resistance vessels. It is difficult to give an exact threshold. We did not go beyond 40-µm vessels under metabolic influence. As is clear from Fig. 7, we would be able to further equalize tone over the compartments by including more compartments under metabolic influence. However, we are then confronted with the problem introduced by considering only nine discrete compartments, and defining a threshold for metabolic control becomes arbitrary. On the other hand, most likely, the segments now assumed to be affected by metabolism alone will also have a shear stress-related dilatory factor. Without knowing the mediator responsible for metabolic vasomotor control, the combination of these mechanisms is difficult to model, which is why we used a strict threshold. Besides these limitations, the model behavior clearly agrees with metabolic compensation by distal resistance vessels for proximal flow-dependent dilatation, and it demonstrates that pressure-induced myogenic tone in the proximal vessels remains important for coronary flow control. In our model the effect of full dilation of the smaller vessels was simulated by reducing the tone of the metabolic vessels to zero. In Fig. 10A, the percent change in diameter of the vessel units is compared with the diameter change after administration of adenosine as observed by Jones et al. (18). The model predictions deviate from the experimental obtained data in two ways. First, for the larger vessels, Jones et al. observed small constriction of the vessels, which we could not predict. Second, Fig. 10A also shows that the threshold for metabolic dominance should shift to somewhat larger vessels. In our model the threshold can be shifted either by changing the number of compartments assumed to be under metabolic control or by assuming a different anatomic diameter distribution.
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Distributed shear stress and velocity response during metabolic
regulation.
In a study of Stepp et al. (40) microvascular diameters
and microsphere velocities were measured. Measurements in arterioles (30-160 µm) and small arteries (160-450 µm) were obtained
under basal conditions and after administration of adenosine. Wall
shear stress was calculated using the formula wall = 8v/d, where wall is the wall
shear stress, is blood viscosity, d is vascular diameter, and v is the mean velocity in the vessel
cross-sectional area, which was assumed to equal the microsphere
velocity. The data of these experiments are presented in Fig.
11, A and B.
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Predictions of flow and flow reserve by model. The model predicts a flow reserve a little over a factor of 2, whereas in humans and dogs this can easily be a factor of 4, although in goats it is less; a factor of ~3. The only conclusion can be that tone in our reference network is too low to provide sufficient vasoconstriction to allow the flow reserve to be a factor of 4. It should be noted, however, that we found an agreement between experimental and predicted diameter changes, which suggests that the distribution in tone is quite realistic. Therefore, both tone in the proximal vessels as well as tone in the distal metabolic vessels are somewhat too low. Tone in the proximal vessels was based on in vitro studies, and the conclusion must then be that in these studies the level of intrinsic tone was too low as well. Such differences between in vivo and in vitro observations may well be possible by changing vascular smooth muscle preconstrictor factors between dissection and mounting in the pressure myograph. For example, endothelin concentrations (26) might have been altered. The required increase in tone does not need to be very large. The law of Poiseuille dictates an inverse fourth-power relation between resistance and diameter. Hence, a 10% reduction in diameter of all segments would result in an increase of 40% of resistance and almost double the flow reserve in the model. Hence, the usefulness of the model is not in predicting quantitatively changes in flow but in the distribution of responses of control mechanisms in terms of vessel diameters and vascular tone.
An additional factor that must be considered when comparing flow predictions with real physiological responses is the beating of the heart. This makes the coronary circulation a very dynamic one, with pulsating flow therefore pulsating shear stress, pulsating pressure, and an additional resistance component related to contraction (39). All these factors have an effect on tone (10, 38). These factors are in addition to the effects that heart performance has on metabolism and, consequently, flow control (13). Moreover, perfusion by itself has an effect on heart performance as it changes diastolic time fraction (29). Although we recognize the importance of all these factors for the integral system of flow control, it does not alter our main conclusion on interactions between the three mechanisms studied, because it is assumed that all interventions considered here are performed at a constant heart rate. It should be noted that arteriolar diameter varies during the heartbeat, being smaller in systole than diastole (12). The experimental observations used in this study are diastolic diameters, assuming that these diameters are most relevant for myocardial perfusion. In conclusion, control of coronary blood flow requires a balance between flow-dependent properties of the larger coronary microvessels upstream and constriction of the small vessels downstream, e.g., basal metabolism-dependent tone. This study underlines the relevance of isolated-vessel studies to understanding of coronary flow control. Furthermore, the myogenic control mechanism plays a significant role both in autoregulatory flow control and in metabolic flow control.| |
APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The control behavior of isolated vessels as described by Liao and Kuo (27) and Kuo et al. (25) is implemented in the model of the vessel unit depicted in Fig. 2. The switch in Fig. 2 points from Tonemyo × TRFflow to Tone, i.e., tone of the vessel unit is determined by myogenic and flow-dependent properties. Here we give empirical relations of the model and estimate their variables based on the data experimentally obtained by Liao and Kuo (27) and Kuo et al (25). This allows for interpolation and extrapolation of these data, giving the necessary relations for each vessel segment used in the model. We first estimated parameters of our model equations without flow-dependent tone (TRFflow = 1).
Parameter Estimation of Pressure-Myogenic Tone Relation and Diameter-Maximal Tension Relation
The fitting procedure was done in two steps. First, rough fits through the experimental data of the four vessel groups of Liao and Kuo (Ref. 27; danat,i = 255, 165, 100, and 65 µm) resulted in estimates of parameters in Eqs. 4 and 5. At this stage, only Tmaxtop,i, a variable in Eq. 4, is dependent on the anatomic diameter, and the relation was assumed to be log normal
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(A1) |
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The fits to the experimental pressure-diameter data of the four
vessel groups were not satisfactory. The pressure-diameter data of Liao
and Kuo (27) was transformed to pressure-tone data with
Eqs. 3, 4, and A1. For the four vessel
groups, different relations were found and Eq. 5 was fitted
to the pressure-tone data of each vessel group (Fig. 3B). As
shown in Fig. 12, middle, the parameters found for Eq. 5 were related to danat,i. The
following equations were fitted to the
danat,i-HSmyo,i, danat,i-y0myo,i,
and
danat,i-P50,i relations
|
(A2) |
|
(A3) |
|
(A4) |
Parameter Estimation for Shear Stress-[NO] Relation
The shear stress-[NO] relation as obtained for the four vessel groups of Liao and Kuo (27) are clearly different (Fig. 4C). The parameters of Eq. 7, found by fitting this equation to the transformed data of Liao and Kuo (27), are related to their anatomic diameter and are shown in the bottom panels in Fig. 12. The following equations were fitted to the danat,i-HSNO,i, danat,i-50,i,
and
danat,i-NOmax,i relations
![HS<SUB>NO,<IT>i</IT></SUB><IT>=</IT>HS<SUB>[NO]<SUB>max</SUB></SUB> <FR><NU><IT>d</IT><SUP>HS<SUB>HS<SUB>NO</SUB></SUB></SUP><SUB>anat,<IT>i</IT></SUB></NU><DE><IT>d</IT><SUP>HS<SUB>HS<SUB>NO</SUB></SUB></SUP><SUB>50</SUB><IT>+d</IT><SUP>HS<SUB>HS<SUB>NO</SUB></SUB></SUP><SUB>anat,<IT>i</IT></SUB></DE></FR>](/content/vol282/issue6/fulltext/H2224/img024.gif)
|
(A5) |
|
(A6) |
![<SUP>10</SUP>log([NO]<SUB>max,<IT>i</IT></SUB>)<IT>=O</IT><SUB>[NO]<SUB>max</SUB></SUB><IT>+a</IT><SUB>[NO]<SUB>max</SUB></SUB><IT>e</IT><SUP><IT>−</IT>0.5<FENCE><FR><NU><IT>d</IT><SUB>anat,<IT>i</IT></SUB><IT>−</IT>x0<SUB>[NO]<SUB>max</SUB></SUB></NU><DE><IT>b</IT><SUB>[NO]<SUB>max</SUB></SUB></DE></FR></FENCE><SUP>2</SUP></SUP>](/content/vol282/issue6/fulltext/H2224/img026.gif)
|
(A7) |
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ACKNOWLEDGEMENTS |
|---|
The authors thank Joep Lagro for the preliminary studies and discussions that contributed to this paper.
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FOOTNOTES |
|---|
This work was supported by The Netherlands Heart Foundation Grant 95.020 (to J. Dankelman).
Address for reprint requests and other correspondence: J. A. E. Spaan, Dept. of Medical Physics, Cardiovascular Research Institute Amsterdam, Academic Medical Center, Univ. of Amsterdam, PO Box 22700, 1100 DE Amsterdam, The Netherlands (E-mail: j.a.spaan{at}amc.uva.nl).
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.
First published January 31, 2002;10.1152/ajpheart.00491.2001
Received 5 June 2001; accepted in final form 29 January 2002.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
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· (d · danat,i)3].
Second, the effective concentration of nitric oxide (NO) in the vessel
([NO]) follows from the shear stress-[NO] relation, which depends
on the attenuation factor (A) and
danat,i. Third, TRFflow
follows from the [NO]-TRFflow relation.
Tonemeta is only applied in the distal compartments in the
network, and its value is determined by assuming a certain flow and
having Tonemyo and TRFflow in the other
proximal compartments intact.
