On the dynamics of the coronary circulation

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ON THE DYNAMICS OF TMÜ

COEONAEY C1ECULATIOK

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<q> «Uu O H

ON THE DYNAMICS OF THE

CORONARY CIRCULATION

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CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Dankelman, Jennigje

On the dynamics of the coronary circulation / Jennigje Dankelman. Delft : Faculty of Mechanical Engineering and Marine Engineering, Delft University of Technology. - lil.

Thesis Delft. - Uith ref. - Uith summary in Dutch. ISBN 90-370-0023-1

SISO 642.3 UDC 513.8:611.13(043.3)

Subject headings: coronary regulation / coronary mechanics.

© J. Dankelman, Delft, 1989

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ON THE DYNAMICS OF THE

CORONARY CIRCULATION

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, prof. drs P.A. Schenck, in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College

van Dekanen op 15 juni te 14.00 uur

door

Jennigje Dankelman,

geboren te Ommen, Doctorandus in de Wiskunde.

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Dit proefschrift is goedgekeurd door de promotoren:

Prof. Dr. Ir. J.A.E. Spaan, Prof. Dr..Ir. H.G. Stassen.

Financiai support by the Netherlands Heart Foundation for the publication of this thesis is gratefully acknowledged.

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Chapter 1

INTRODUCTION

1.1 GENERAL INTRODUCTION

The heart is the pump that supplies all organs with blood, and hence must work continuously to sustain life. It is aided in this by the coronary circulation system, which ensures that the heart muscle, or myocardium, is perfused with blood. The vascular bed distributes the blood flow and, therefore, functions as a network to supply nutrients (e.g. oxygen) and to remove waste products (e.g. adenosine).

The heart contains four chambers (Fig. 1.1A). During ventricular contraction (systole), blood inside the left ventricle is squeezed into the aorta, while blood inside the right ventricle is squeezed into the pulmonary artery. During relaxation (diastole) the right and left ventricles are refilled with blood from the right and left atria, respectively. The atria contract slightly earlier than the ventricles.

The two main feeding arteries for the coronary circulation originate from the aorta just above the aortic valve. These arterial vessels lie on the epicardium of the heart and branch into smaller arteries penetratlng the myocardium (Fig. 1.1B). These arteries branch into arterioles which in turn supply the capillary bed, where material exchange with the surrounding tissue takes place. The capillaries drain into the venules which join into veins. Finally, the blood is drained into the atria. Apart from this main pathway, some blood flows directly into the ventricles through the thebesian channels. This thesis is mainly concerned with the flow through the left main coronary artery which will, therefore, be referred to as the coronary blood flow throughout this work.

The sound coronary circulation system is a prerequisite for the heart to work properly. The amount of blood flowing through the myocardium depends on (1) mechanical factors and (2) the activation of smooth muscle tone in the arterioles (regulatlon). Some phenomena related to mechanical factors and smooth muscle activation are described in the sections below.

Very of ten, the information required to identify the mechanisms determining the coronary blood flow is not available in the literature. The problems arise for a variety of reasons, e.g. experimental conditions are different, ways of

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Fig. 1.1 Anatomy of the heart.

Panel A shows a schematic cross-section of the heart. For the left part of the heart, blood from the pulmonary veins (PVs) flows into the left atrium (LA). During atrial contraction it flows into the left ventricle (LV) via the mitral valve (MV). When the left ventricle contracts, the blood is squeezed, via the aortic valve (AV) into the aorta (AO). For the right part of the heart a similar circuit is passed through. From the vena cava (VC) to the right atrium (RA) and via the tricuspid valve (TV) into the right ventricle (RV). After ventricular contraction it flows through the pulmonary valve (PV) into the pulmonary trunk (PT).

Panel B shows the position of the epicardial coronary arteries on the epicardium of the heart. RC: right coronary artery, LM: left main coronary artery, LC: circumflex artery, LAD: left anterior descending branch.

analyzing data are different, interaction of different effects, the models used are too simple etc. The complexity of the material is illustrated by a block diagram of the coronary circulation (Fig. 1.2). In this figure, distinction has been made between the following factors: the structure of the coronary vascular system, mechanical factors and the activation of smooth muscle tone. There are many studies directed to identifying the subsystems by means of experiments. However, in the experimental situation not all input and output variables are measurable. In Fig. 1.2 the inputs u(t) were split up into measurable inputs (e.g. heart rate) , u ^ t ) , and non-measurable ones (e.g. inputs from the nervous system), u,(t) . Similarly, the output variables were split up into measurable, y.(t), and non-measurable ones, y2(t).

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i -*■ INPUT MATRIX 1 INPUT MATRIX 2

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CORONARY ViECHANICAL DYNAMICS CORONARY CIRCULATION STRUCTURE SYSTEM SMOOTH ^USCLE T O N ! DYNAMICS CIRCULATION SYSTEM y(t)

~^-OUTPUT MATRIX 1 OUTPUT MATRIX 2 j y , ( t ) 1 y2«

Fig. 1.2 Schematic representation of the coronary circulation. The coronary circulation is split up into three parts. The structure of the coronary circulation is influenced by mechanical factors and by change in smooth muscle activation. u.,(t) and y ^ t ) represents the measurable input and output variables, u2(t) and y2(t) represents the non-measurable ones, respectively.

enhanced if quantities such as pressures in the myocardium and flow in the capillaries could be measured. Also, an accurate description of the structure of the vascular network would be helpful in predicting input-output relations. However, lack of information on all these points has resulted in the development of various hypotheses for mechanical interaction between heart contraction and coronary flow on the one hand and the control of flow on the other hand. Some of these hypotheses are provided in Section 1.3. The difficulty in discriminating between them is illustrated by an example in Section 1.4.

This thesis describes the results of an experimental study coupled to mathematical model analysis. The dynamic behavior of the coronary system was chosen as topic since dynamic analysis may discriminate between hypotheses where static analysis fails. The remaining sections of this chapter are devoted to outlining the phenomena and theoretical models relevant to the present study.

1.2 SOME EXPERIMENTAL OBSERVATIONS CLARIFYING THE MECHANISMS UNDER STUDY

This section specifies the topics of study described in general terms above. First, some examples ascribed to mechanical factors are presented. Next, experimental results will be presented illustrating the effect of change in smooth muscle tone (regulation).

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1.2.1 Mechanical interactions

The coronary arterial blood flow is pulsatile in the beating heart. This is illustrated in Fig. 1.3 by the coronary flow signal measured at constant pressure perfusion of the left raain coronary artery of an anesthetized goat. During systole the arterial blood flow is significantly lower than during

1 0 0 LEFT VENTRICULAR P R E S S U R E 5 0 ( m m Hg) C O R O N A R Y ARTERIAL F L O W (ml/s) TIME (sec)

Fig. 1.3 Simultaneously measured left ventricular pressure and arterial flow at constant perfusion pressure. During systole arterial flow is low, while at diastole the flow is high.

diastole. The pulsatility of the arterial flow signal is a general observation (Porter 1898, Sabiston and Gregg 1957, Wiggers 1954, Spaan et al. 1981a,1981b) and may result in retrograde flow at low perfusion pressures (Panerai et al. 1979, Chilian and Marcus 1982, Spaan et al. 1981b). The coronary venous flow, however, is higher during systole than during diastole (Anrep 1927, Johnson and Wiggers 1937, Wiggers 1954, Stein et al. 1969). The opposite variation of coronary venous and arterial flow induced by heart contraction indicates that flow variations are not simply due to resistance variations, but that intramyocardial blood-volume varies periodically. One cannot conclude from the decreased systolic arterial flow that contraction impedes coronary flow. The same reasoning applied to the coronary venous flow would lead to the conclusion that cardiac contraction augments flow. In fact, the classical discussion at the end of the 19th century went along those lines (Porter 1898).

That contraction impedes net coronary blood flow Is illustrated by the observation that contraction of the heart not only results in a pulsatile flow,

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Fig. 1.4 Coronary blood flow plotted against perfusion pressure for the beating state (right line) and the non-beating state (left line). The data are from one dog and are typical of the data from all other dogs. The lines were drawn by eye. Reproduced from Downey and Kirk (1975) by permission of the Am. Heart Ass. Ine.

O 50 100 150 200 PERFUSION P R E S S U R E (mmHg)

but decreases the time averaged flow when driving pressure is maintained. This is nicely illustrated by the experiments of Downey and Kirk (1975) reproduced in Fig. 1.4. The pressure-flow relationships were obtained in the maximally dilated bed when the heart was beating and during cardiac arrest. At equal perfusion pressures the coronary blood flow, averaged per beat, is lower in the beating than in the non-beating heart.

The effect of heart contraction on the coronary vessels is also illustrated by the change in coronary blood volume when the heart stops beating (long diastole). Vergroesen et al. (1987b) measured the change in coronary volume in the autoregulated and vasodilated bed. After the onset of a long diastole the decrease in myocardial compression caused the volume to increase, which is the result of a transient increase in coronary inflow and decrease in coronary outflow. 67% of the volume was achieved within 1.6 seconds with autoregulation intact and 1.0 second after vasodilation. These times are longer than the duration of a normal diastole, indicating that at the end of a normal diastole a steady state for blood volume has not yet been reached. Since volume variations are related to flow by capacitive effects, and to resistance, one must be cautious in assuming steady coronary arterial and venous flow in diastole of the beating heart.

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During systole, left ventricular pressure increases. In many studies it is assumed that this affects coronary circulation. Recently, however, experiments were performed by Kraras et al. (1988) in the isolated cat heart perfused at constant pressure perfusion. Coronary flow during isovolumic beats were compared with flow during isobaric beats. During isovolumic conditions the systolic inflow hardly depended on systolic left ventricular pressure and was similar to those obtained during isobaric beats. Enhancing the cardiac contractility, however, did result in a reduction of systolic flow. These measurements at least indicate that left ventricular pressure is not the sole determinant impeding coronary flow. Krams (1988) suggested that the phasic flow is related to time varying properties of the muscle fibers. However, recent measurements in our laboratory illustrate that, under different experimental conditions, coronary flow is influenced by left ventricular pressure.

That the coronary circulation does not behave as a pure resistance is illustrated by the pressure-flow relation measured during long diastoles. Instantaneous diastolic pressure-flow relationships have been presented by Bellamy 1980, Dole et al. 1982a, Eng et al. 1981,1982, Aversano 1984 and Canty et al. 1987. These relations were obtained by measuring flow during a continuous decrease of perfusion pressure during long diastoles. Bellamy (1978) was the first to analyze instantaneous diastolic coronary pressure-flow relationships. He found them to be linear but also to intercept the pressure axis (Pf=0) a t a value as high as 50 mm Hg in presence of coronary tone. This value for Pf_0 is

larger than venous outflow pressure, which seems an anomaly. During vasodilatation a higher slope and lower Pf.0 (Pf_0=20mm Hg) were measured. The diastolic pressure flow lines have long been interpreted as evidence for closure

Fig. 1.5 Diastolic coronary artery pressure-flow curves obtained with decreasing and with constant pressure during maximum coronary vasodilation. Initial coronary pressure was 125 mm Hg. The pressure-flow curve with decreasing pressure is indicated by open circles and the curve with constant pressure by closed circles. The pressure-flow curve obtained with decreasing pressure was characterized by a steeper slope and a higher zero flow pressure intercept. Reproduced with permission from Dole et al.

(1982b). 400 J

ff

3001 5 ?2 Coronary Pressure (mmHg)

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of the microvessels at high luminal pressure, due to smooth muscle tone and/or tissue pressure. The inverse of the slope would represent resistance of the vascular bed proximal of the site of vessel closure. Dole et al. (1982b) measured instantaneous and steady state pressure-flow curves and showed that the steady state curve had lower Pf=0 values and smaller slopes (Fig. 1.5). Especially at low perfusion pressures the difference between steady state and dynamic measurements becomes clear.

The measurements above give an illustration of the different mechanical effects. It is obvious that the heart contraction has an important influence on the coronary blood flow. The fact that during long diastole the flow is not linearly related to arterial-venous pressure difference indicates that the coronary system is a non-linear system.

1.2.2 Regulatlon

Contraction and dilation of smooth muscle cells in the arteriolar vessel wall change the arteriolar resistance and regulate coronary flow. The control of coronary flow demonstrates itself by two well known phenomena: autoregulatlon and metabolic adaptation. Some experimental results will be provided to illustrate the dynamic behavior of regulation. First, however, autoregulatlon and metabolic adaptation will be explained by referring to steady state observations.

Autoregulatlon and metabolic adaptation

Autoregulation is the intrinsic abillty of an organ to maintain a relatively constant blood supply following changes in perfusion pressure (Johnson 1964, Feigl 1983). This phenomenon is responsible for keeping the blood flow constant at constant metabolism despite changes in arterial pressure. Autoregulation of coronary flow has been observed in blood-perfused dog hearts in situ by a number of authors (Cross 1964, Mosher et al. 1964, Driscol et al. 1964a, Rouleau et al. 1979).

Metabolic adaptation is the phenomenon of the adjustment of blood flow to the level of the metabolism of the heart. The response to increased heart rate is an increase in metabolic activity leading to an increase in coronary blood flow and oxygen supply to the heart. Metabolic adaptation increases blood flow with increasing metabolic demand to supply more oxygen. This adaptation is illustrated by the shift of the autoregulation curve to a higher value of flow at a higher metabolism (Mosher et al. 1964, Drake-Holland et al. 1984, Dole et

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al. 1986).

The effect of autoregulation and metabolic adaptation on coronary flow is illustrated in Fig. 1.6. It depicts the coronary flow as a function of perfusion pressure in the vasodilated bed, and at two levels of metabolism when autoregulation is intact. At equal levels of perfusion pressure the coronary flow in the vasodilated bed is higher than in the autoregulated bed. In the flat

HIGH m V 0 2

LOW m V 0 2 '

0 5 0 1 0 0 1 5 0

PERF. PRESSURE ( m m Hg)

Fig. 1.6 Schematic representation of the pressure-flow relation in the vasodilated coronary bed and in the autoregulated bed. In the autoregulated bed this relation is given at two levels of metabolism.

region of the autoregulated pressure-flow curves the flow is relatively independent of perfusion pressure (Berne 1959, 1964, Mosher et al. 1964). Furthermore, the flow in the autoregulated bed is higher at higher oxygen consumption.

1.2.3 Measurements of dynamic behavior reported in literature

The change in smooth muscle activation in an intact preparation can not be measured directly. The coronary vascular resistance is, therefore, often used as an index for the level of smooth muscle tone. Measurements relevant tö

this thesis are described below and, in some cases, are illustrated by figures. One of the first papers describing a systematic investigation of the dynamic response of coronary resistance and venous oxygen content was presented by Mohrman et al. (1973). They measured the time course of vascular resistance and venous oxygen saturation following a brief tetanus in Isolated canine skeletal muscle during constant pressure perfusion, and during constant high and constant low flow perfusion, respectively. During pressure perfusion and

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high flow perfusion the resistance showed a monophasic decrease and changed faster than the change in end capillary oxygen tension. The end capillary oxygen tension was calculated from venous oxygen tension. The venous oxygen tension was, therefore, corrected for distribution of vascular transit delays between capillaries and measuring site. During low flow perfusion a biphasic resistance

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oo « 0 eo 4 0 2 0 0 I f 1 i i i i i 0 12 24 3C 48 60 72 SECONDS

Fig. 1.7 Average time courses of changes in coronary vascular resistance following increases and decreases in heart rate. Curves associated with heart rate increases are inverted. Reproduced with permission from Belloni and Sparks (1977).

response was shown as having the same time course as tissue oxygen consumption. From these observations Mohrman et al. (1973) concluded that oxygen tension could not be the only factor responsible for resistance change. In a later study Belloni and Sparks (1977) measured the response of coronary resistance and venous oxygen content change to a heart rate step (Fig. 1.7). They showed that coronary sinus 02 content changes preceded the adjusted time course of vascular resistance. This supports the hypothesis that coronary vascular resistance is regulated in part by factors closely linked to oxidative metabolism. The responses of resistance in both studies had a half time in the order of 12 seconds. These two papers show that the dynamic response to a change in metabolism depends on the perfusion condition and on the level of metabolism. The responses are slow compared to the responses caused by mechanical effects.

The two papers above presume that oxygen is a mediator for regulation. This is supported by the influence of an abrupt change in arterial oxygen content on coronary blood flow as measured by Duruble et al. (1985). A sudden decrease in oxygen content resulted in a fast initial increase in coronary flow foliowed by a slow increase. The response of flow, q(t), depehded on the level of arterial P02. Fitting the data with q(t)=A(l-e't/T)+Bt, with A and B

Increaaed Heart Rate Decreaeed Heart Rote

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constants, r a time constant, resulted in a value for r for the fast response of 9.2 and 15.6 seconds with high and low arterial P02, respectively. Hence, the measurements of Duruble et al. (1985) suggest that oxygen indeed plays an important role in regulation and that the rate of response depends on the level of arterial oxygen content.

The response of coronary flow to a change in oxygen consumption can also be measured by introducing a long diastole. A decreasing diastolic coronary flow following abrupt cessation of rapid ventricular pacing, was reported by different authors. Dole and Bishop (1982a) reported that blood flow remained constant throughout individual diastoles lasting for more that 5 seconds and was then foliowed by a decrease in flow. Vergroesen et al. (1987b), however, measured a steadily decreasing flow after the onset of a long diastole with a half time of 15 s. This steady decrease was also measured by others (Spaan and Laird 1981, Eng et al. 1982, Kloeke et al. 1981) but they found a half time of approximately 4 seconds. The reason for the difference in observations is not yet clear.

The dynamic response of regulation has not only been measured after a change in oxygen consumption or arterial oxygen content but also after a change

HR=120 » 100 m .£ HR=40 ï- E c ilOOp

rt-f-yf-f-^^

Fig. 1.8 Effects of lowering heart rate on the dynamic coronary flow response to a step decrease in coronary artery pressure from 120 to 80 mm Hg. The autoregulatory response was attenuated at a heart rate of 40 beats/min. Numbers above the flow recordings indicate steady-state flow values. Arrowheads indicate the point at which pressure was suddenly decreased. Reproduced from Dole and Nuno (1986) by permission of the Am. Heart Ass. Ine.

in perfusion pressure. The response to a pressure perfusion step was measured by Dole et al. (1986). Although this study was mainly directed at the steady state, Fig. 2 from this study (Fig. 1.8) demonstrates that regulation depends on the level of oxygen consumption. Coronary flow increased f aster at high heart rates than at low heart rates after an abrupt decrease in perfusion pressure.

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Sometimes, an overshoot was observed at high heart rates. Driscol et al. (1964) measured the vascular effects of changes in perfusion pressure under different perfusion conditions. They showed that in the non-ischemic heart, flow changes are slow after perfusion pressure is increased. The flow response to a pressure decrease was faster and showed oscillations. In the hypoxic heart the response in flow slowed after both an increase as well as a decrease in perfusion pressure; no oscillation was observed. Frora these measureraents it can be concluded that the dynamic response of flow to a perfusion pressure step is both related to the level of oxygen consumption and dependent on the direction of the intervention.

Not only blood pressure, but also the outside pressure was changed for dynamical measurements in sorae experiments, however, not in the coronary system.

TIME (,ec) ° I 1 1 1 1 1 1 1 1 1 1 — I 1 1 1 1 1 1 1 > ^ 0 30 60 90 0 30 60 90

Fig. 1.9 Microvascular resistance responses in skeletal rauscle in the cat evoked by increase and subsequent decrease in vascular transmural pressure by 30 mm Hg applied at 5 different rates. The original consecutive resistance recordings are reproduced on top of each other. Note the pronounced dynamic constrictor and dilator responses during the period of changing transmural pressure in relation to its rate of changing both for positive and negative values, and the comparatively small static constrictor response in the steady state period of constant increased transmural pressure. Reproduced with permission from Grande and Mellander (1978).

Grande and Mellander (1978) measured the responses of resistance after an increase and decrease in transmural pressure in the vascular bed of skeletal muscle (Fig. 1.9). These experiments are of importance as they are interpreted by the myogenic mechanism. Transmural pressure was calculated as the pressure inside the vessels minus the pressure outside the skeletal muscle. Only the

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outside pressure was changed in their experiments. The response depended on the speed of transmural perfusion pressure change, and on the absolute magnitude of this change. The response after an increase in transmural pressure exhibited a fast initial increase in resistance foliowed by a slow (half time + 15 seconds) decrease to almost the initial value. A decrease in transmural pressure yielded similar reversed responses. However, the overshoot was less pronounced and was foliowed by an extra oscillation. Comparing the experimental results of Dole and Nuno (1986), Fig. 1.8, with the results of Grande and Mellander (1978), Fig. 1.9, some differences can be identified. A decrease in perfusion pressure (Fig.1.8) results in a decrease of transmural pressure. The initial decrease in flow of Fig. 1.8 must be interpreted by an increase in resistance which then was foliowed by a decrease in resistance. In contrast, in the experiments of Grande and Mellander (1978) the decrease in transmural pressure (Fig. 1.9 right panel) initially resulted in a strong decrease in resistance foliowed by a strong increase, ending at almost its initial level. Thus the initial phases of the responses are completely opposite to one another. Furthermore, it takes more time for the response of Fig. 1.9 to reach a steady state than the one in Fig. 1.8.

In all measurements described above, the response of regulation was slow. Some authors, however, reported measurements from which they concluded that regulation occurs much faster. Schwarz et al. (1982b) measured the response of the coronary diastolic index (diastolic pressure/flow ratio) after a single extra ventricular activation. They reported that vasodilation occurred after one beat and then returned to the initial level with a half time of 4 seconds. In another study Schwarz et al. (1982a) showed that the hyperemic response of the coronary circulation to brief diastolic occlusion is also extremely rapid. The maximal diastolic flow occurred in the diastole immediately following. The discrepancy between this fast dilatory response and the slow ones will be further discussed in Chapter 3.

The interventions described in this section illustrate different manifestations of coronary regulation. It was shown that the rate of the response depends on the intervention. Furthermore, the rate is sensitive to the direction of the intervention and to factors such as level of metabolism and perfusion conditions. The differences between the experimental results are so diverse that it will be difficult to determine the underlying mechanisms for all measurements described above.

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1.3 MECHANISMS PROPOSED FOR MECHANICAL INTERACTION AND CONTROL.

The section above showed different characteristics of the dynamic behavior of the coronary systera. The underlying mechanisms are still unclear, but different hypotheses have been proposed. Differentiation between the mechanisms is hampered by the fact that regulation and mechanical factors both influence the coronary blood flow simultaneously.

The interaction between mechanical effects and regulation is illustrated in Fig. 1.2. In many studies, the coronary bed has been maximally vasodilated for the investigation of the mechanical interactions. However, extrapolation of these results to the coronary system with regulation intact should be done with care. In the situation where smooth muscle cells are relaxed the coronary flow is high and coronary resistance low. Since the coronary system is non-linear (e.g. coronary resistance is not constant) the parameters of the system will be state dependent and consequently alter with vasodilation. It is, therefore, questionable whether quantitative information obtained under conditions of vasodilation are relevant for the autoregulated bed. Furthermore, during autoregulation sub-endocardial and sub-epicardial flow are virtually equal. During vasodilation there is no such equality. Therefore, the distribution of mechanical factors during autoregulation will not be the same as during vasodilation.

In this thesis regulation is defined as the control of activation of smooth muscle in the arteriolar wall. A higher activation leads to higher smooth muscle tone tending to decrease the diameter of the arterioles. Unfortunately, this activation can not be measured directly. In the coronary bed a change in smooth muscle activation results in a change of the arteriolar resistance. Often, therefore, regulation is indicated by the change of coronary resistance calculated from arterial pressure and flow signals. This approach often neglects the mechanical factors which influence the arterial flow and pressure signals other than due to a resistance effect. This will be further emphasized below.

1.3.1 Mechanical factors

As the experimental observations above illustrate, the mechanical interaction between cardiac contraction causes the arterial diastolic-systolic swing, non-linear pressure-flow relations and volume change of the intramyocardial vessels after the onset of a long diastole. In order to explain these phenomena, various mechanisms have been postulated (Downey and Kirk 1975,

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Kloeke et al. 1985, Spaan et al. 1981a, Spaan 1985). The most important are schematically illustrated in Fig. 1.10 and will be discussed below.

In the waterfall model (Downey and Kirk 1975) it is assumed that during heart contraction the intramyocardial pressure increases, mainly at the subendocardial layer of the left ventricular wall. When the intramyocardial pressure becomes higher than venous pressure, the vessels at the venous side will collapse, making arterial blood flow independent of venous pressure,

Another mechanism is the intramyocardial pump model as suggested by Spaan et al. (1981a). It is based on the assumption that coronary vessels are distensible. Data of distensibility and compliance of vessels are given by

Rcor P i m pim=f(Plv) Ra Rsys = f(Plv) C = f(Plv)

Rv

Fig. 1.10 Schematic representation of different models proposed to explain the diastolic-systolic flow profiles. a: Waterfall model, b: Intramyocardial pump model, c: Extravascular resistance model, d: Elastance model.

Rcor: coronary resistance. Rfl, B^: arterial inlet and venous outlet resistance respectively. R : systolic resistance. Pjm: intramyocardial pressure, C: intramyocardial compliance. -f(P. ) : function of left ventricular pressure.

several authors (Smaje et al. 1980, Kajiya et al. 1986, Morgenstern et al. 1973, Scharf et al. 1973, Wiederhielm 1965, Mulvany 1977,1984). The intramyocardial pump model assumes that when the heart contracts the pressure in the tissue surrounding the vessel increases (Stein et al. 1980, Heineman et al. 1985) resulting in a decrease in transmural pressure (pressure inside minus pressure outside of the vessel). A heart beat will then result in blood being squeezed out of the coronary vessels to the arterial and venous side without causing the vessels to collapse. On the arterial side this results in a lowering

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of the arterial inflow and on the venous side in an increase in outflow. During diastole these vessels are refilled.

A third raechanism is the extravascular resistance mechanism (Berne and Rubio 1979, Wusten et al. 1977, Anrep et al. 1927, Snijder et al. 1975). It is assumed that coronary resistance increases during systole while during diastole, the flow is unaffected by contraction.

A fourth mechanism has been recently proposed by Krams et al. (1989a,b) and is based on the changing elasticity of the heart wall during contraction (Suga et al. 1973). This model assumes that a time varying elastance of the vascular bed causes the systolic-diastolic swing. During systole the compliance

of the vessels changes resulting in blood being squeezed out of the vessels. In reality, resistance and compliance are continuously distributed over the large and small vessels. Furthermore, characteristics such as distensibility and the branching structure are vessel type dependent. In order to explain the dynamics of the mechanical factors entirely, a model consisting of many compartments is required. Every sub-system must be fully specified. However, the information available on e.g. the distribution of resistance and compliance is too limited to validate such a model. The use of simple lumped models, therefore, is the only possibility in describing the different mechanisms determining the coronary system.

The first two mechanism described above have received the most attention in the literature (Kloeke et al. 1985, Spaan 1985). The essential difference between these two mechanisms is that the waterfall mechanism assumes short, and the pump mechanism long time constants related to the mechanical effects. Analysis of the flow responses on the arterial side seems to justify a short time constant favoring the waterfall hypothesis. However, flow responses at the venous side provide evidence for long time constants. Chilian and Marcus (1984) measured venous flow after cessation of coronary artery flow. They demonstrated that during vasodilation, 40% of the total volume of diastolic coronary venous outflow appeared after cessation of inflow. Venous outflow was observed after more than 5 seconds of arterial occlusion which supports the pump model (Spaan et al. 1981a).

It has been shown, however, that both models are too simple and that they are not able to explain all the mechanical interactions discussed above. Therefore, suggestions have been made to extend the models. The waterfall has been extended by the addition of a small compliance (Lee et al. 1984) in order to explain the diastolic-systolic flow difference (see also Sectlon 1.4). The

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linear pump model was not capable of describing the change in the mean flow and had to be altered as well. Bruinsma et al. (1985, 1988) developed a model following a model constructed by Arts (1978) based on distributed resistance and compliance which were transmural pressure dependent.

Intramyocardial pressure

In three of the four models presented in this section the intramyocardial pressure or tissue pressure plays an important role, which requires more extensive discussion. Many attempts were made to measure the intramyocardial pressure in different layers of the ventricular wall by using various methods, mostly by the insertion of small balloons or needies or other mechanical structures into the myocardium. A review of these measurements has been presented by Hoffman and Spaan (1989). All measurements were consistent in that the intramyocardial pressure at the endocardial layer is higher than at the epicardial layer. There was, however, disagreement on the magnitude of the pressure at the endocardium. Some authors measured endocardial intramyocardial pressures exceeding the left ventricular pressure. The probable reason for this disagreement was shown by Gregg et al. (1941). They noted that the greater the size of the foreign body introduced into the myocardium, the higher the pressure recorded. Inserting e.g. needies into the dense myocardium distorts the tissue and alters the local forces that determine tissue pressure. Recently, measurements of intramyocardial pressure were made by micropipettes with tips less than 15 firn in diameter (Heineman et al. 1985). Heineman et al. (1985)

showed that the intramyocardial pressure during systole was linearly related to the depth of the micropipette tip in the ventricular wall. The largest intramyocardial pressures measured at the endocardium were close to left ventricular pressure.

From these measurements it is concluded that the intramyocardial pressure plays an important role in the beating heart. The pressure is high in systole and probably similar to the systolic left ventricular pressure at the subendocardium. Furthermore, the pressure falls to near atmospheric pressure at the epicardium.

1.3.2 Regulation of coronary blood flow

The mechanisms underlylng the phenomena of autoregulation and metabolic adaptation are still unknown. However, different mechanisms for the control of local flow have been proposed in the literature (Berne 1964, Rubio and Berne

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1975, Belloni 1979, Feigl 1983). Two important suggested mechanisms are described below.

The metabolic hypothesis assumes that the level of a metabolic substrate or metabolite controls the degree of vascular smooth muscle contraction. Two different explanations of this ' hypothesis are: reduction of flow af ter a decrease in coronary artery pressure causes coronary vasodilation by 1) increasing production of metabolites or 2) decreasing myocardial substrate availability. A favorite metabolite was adenosine (Berne 1963, Berne and Rubio 1979, Feigl 1983, Wilcken 1983). However, experiments with infusion of adenosine deaminase (Dole et al. 1985, Hanley et al. 1986) could not demonstrate a sufficiënt effect on the autoregulatory response for adenosine to be a candidate for local control. Therefore, oxygen became more favorite (Guyton et al. 1964, Laird et al 1981a,b, Drake-Holland et al. 1984, Vergroesen et al. 1987a, Dole and Nuno 1986). Drake-Holland et al. (1984) developed a model based on the assumption that tissue oxygen tension controls coronary vascular resistance. The model could predict the steady state blood flow and arterial-venous oxygen content difference, measured over a wide range of perfusion pressure (which caused autoregulation) and heart rates (which caused metabolic adaptation). Vergroesen et al. (1987a) showed that in steady state, coronary blood flow depends on both oxygen consumption and arterial pressure in an independent way. They conclude that any factor regulating coronary . arterial flow would be influenced independently by both oxygen consumption and perfusion pressure.

The second mechanism is the myogenic mechanism. It is the intrinsic mechanism in vascular smooth muscle regulating resistance in response to a change in transmural pressure (Bayliss 1902, Folkow 1949, Johnson 1964, 1980, Borgstrom et al. 1984, Grande et al. 1977,1978, McHale et al. 1987). A decrease in coronary arterial pressure results in coronary vasodilation independent of changes in blood flow. The response was first described by Bayliss (1902). Borgstrom et al. (1982) formulated a mathematical description of the myogenic response capable of explaining the static and dynamic myogenic response to a change of vascular transmural pressure in the arterioles of skeletal muscle (Fig. 1.8).

It is difficult to explain all regulator responses by a single mechanism. E.g. a change in perfusion pressure is 1) the initiator for a myogenic response, and 2) the cause of changing coronary blood flow and thus changing the amount of a metabolite (e.g. oxygen or adenosine). Further, as shown in Section 1.2.3 large differences in experimental results have been obtained depending on the

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experimental conditions. It will be difficult to describe all the experimental results with the mechanisms proposed above.

1.4 THE DIFFICULTY IN DISCRIMINATING BEWEEN MECHANICAL MODELS

A major problem relates to the validation of the different mechanical models. Several experimental results can be explained by more than one model. This section will emphasize the similarity in predictive value of two different models, one based on the waterfall and the other on the intramyocardial compliance. The experimental observations by which the models will be compared are (1) the phasic arterial flow profile at constant pressure perfusion, and (2) the arterial flow signal to a step in perfusion pressure in a long diastole, and (3) the reduction of the magnitude of the pulse on the coronary arterial pressure at step wise lowering of flow perfusion.

Pim

Roep Rq

¥

Rv

H l '

oep~j~

_im

Pim

Fig. 1.11 Electrical analog of the extended waterfall model (upper panel) as proposed by Lee et al. (1984), and the intramyocardial pump model (lower panel) as proposed by Spaan et al. (1981a). The arterial epicardial compliance, ca

transmural pressure dependent.

was in this section assumed to be

The model based on the waterfall model is the model as proposed by Lee et al. (1984). In order to predict the transients in phasic flow Lee et al. (1984) extended the waterfall model with an intramyocardial compliance (Fig. 1.11, upper panel). Estimation of the compliance value was obtained from the flow signal following an abrupt change in perfusion pressure in a long diastole. The estimated compliance value was small (+ 0.007 ml/mm Hg/lOOg, with tone present).

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The experiment in which the magnitude of the arterial pressure pulse decreases at lower constant flow perfusion (Lee et al 1984, their Fig. 8) is, according to Lee et al. (1984), only compatible with a model incorporating vascular waterfalls in addition to a capacitance.

The second model is based on the linear pump model as proposed by Spaan et al. (1981a). It consists of an intramyocardial compliance (C=0.08 ml/mm Hg/lOOg) and an eplcardial compliance (00.005 ml/mm Hg/lOOg), Fig. 1.11 (lower panel). The model was capable of explaining the phasic arterial flow profile in the beating heart. The procedure for the estimation of the intramyocardial compliance value as foliowed by Lee et al (1984) was also carried out on this linear pump model. The compliance value obtained according to this procedure was almost equal to the epicardial compliance, and was in the same order as the compliance value found by Lee et al. (1984). In order to explain the experiment with lowering constant flow perfusion, the epicardial compliance was made transmural pressure dependent. Such a dependency was demonstrated by Canty et al. (1985). A hyperbolic function, Fig. 1.12, was used to calculate the

O O £ 0 . 0 2

E

\

E

0.01 LU ü

z

< _l Q. 2 O ü 1 3 0 P R E S S U R E ( m m H g )

Fig. 1.12 Transmural pressure versus arterial compliance as given by Canty et al. (1985) (+) and as used in our simulation (solid line).

epicardial compliance throughout the cardiac cycle. The epicardial compliance and the first resistance act as a low pass filter with a cut-off frequency dependent on the values of this resistance and epicardial compliance. Hence the

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damping effect of this filter will increase when the compliance value increases as is the case at lower arterial pressure values. Simulation results of the arterial pressure at lowering constant flow are provided in Fig. 1.13. It shows

Qa = ( m l / s / 1 0 0 g ) 1.6 1.2

'\WWWV

O.B 0.4

'\ZWWW

_0.2 5 - 1 — 10 l 15 — I — 20 2 5 TIME ( s e c )

Fig. 1.13 Simulated coronary phasic perfusion pressure as a result of decreasing mean perfusion pressure. For this simulation the epicardial compartment of the intramyocardial pump model is made pressure dependent.

that the magnitude of.the arterial pressure pulse is reduced at lower flow values as was the case with the experimental findings of Lee et al. (1984).

Simulation with the two compartmental pump model showed that the total intramyocardial compliance could not be measured from arterial flow signal with the method Lee et al. (1984) used. Furthermore, the simulations above illustrated that both models could explain the experimental findings described above. Neither models, therefore, can be validated by these experiments.

1.5 SCOPE OF THIS RESEARCH

This thesis deals with the modelling of the dynamics of the coronary circulation. Only dynamical effects which were measured in a cannulated preparation are investigated. Influences of pathological, nervous or hormonal source will not be considered here. Two dynamical aspects of the system are distinguished: 1) dynamics related to mechanical factors and 2) dynamics related

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to regulation of smooth muscle tone.

With respect to the mechanical factors, the distensibility of coronary vessels is an utmost important aspect of the mechanical determinants. Chapter 2 reports an attempt to model the coronary mechanlcs of the vasodilated bed incorporating distribution of transmural pressure dependent resistances and compliances. Especially, the sensitivity of coronary arterial flow to the different parameters of the model has been tested.

The dynamics of regulation has been studied by a heart rate step under different perfusion conditions. The data has been analyzed with a model described in Chapter 3. The response to a heart rate step clearly shows a mechanical effect, especially in the initial phase. These mechanical factors have been analyzed with mechanical models in Chapter 4. In Chapter 3 and 4 only the arterial response has been considered. The venous response has been analyzed in Chapter 5. The consequences of the various mechanical models for the venous response has also been given in that chapter. The model for regulation (Chapter 3) also predicts the response to a step in perfusion under different perfusion conditions. These predictions have been experimentally tested in Chapter 6 as another test for the dynamic model for regulation.

An eight layer compartimental model based on tissue oxygen pressure control of the arteriolar pressure-volume relation has been constructed. In this model the model given in Chapter 2 (mechanical) and an adapted control model

(regulatory) has been integrated. It has been extended to incorporate the behavior of the epicardial arterial and venous vessels as well. Simulation results of this overall model has been compared with experimentally obtained data and are discussed in Chapter 7. Furthermore, in this chapter some recommendations for further research have also been given. A summary of the present study is provided in Chapter 8.

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Chapter 2

SENSITIVITY OF CORONARY FLOW SIGNAL TO DISTRIBUTION OF VASCULAR DISTENSIBILITY ANALYZED BY A COMPARTMENTAL MODEL

2.1 INTRODUCTION

In Chapter 1 several mechanisms were hypotheslzed for as being responsible for the arterial pressure and flow signals. An important notion incorporated in most of the models discussed is the ability of vessels to change volume and hence also resistance. For the dynamic situation the rate of change of volume contributes transiently to the arterial and/or venous flow signal. Thé relation between these quantities is complex for several reasons. Important complications relate to 1) the magnitude of the compliance of the vessels and the forces required to deform these vessels, and 2) the distribution of the compliance and resistance over the vascular bed. In this chapter the importance of these two factors in determining the coronary arterial flow signal, with constant pressure perfusion of the coronary bed, will be assessed using the model proposed by Bruinsma et al. (1988). This model has been chosen because of its focus on the pressure dependency of compliance and resistance as a central issue. Moreover, it considers the distribution of these properties by compartmentalisation of the vascular bed from endocardium to epicardium and within each myocardial layer over three compartments: arteriolar, capillary and venular. Hence, this model enables us to assess the effect of changes in distribution of parameters over the compartments.

For the simulation of phasic arterial flow signals the model was altered to make it applicable to autoregulation. Furthermore, for the simulation of phasic arterial flow signals only one layer will be used, because as changes can then be explained on basis of changes in compliances and resistances and without being affected by changes in endocardial and epicardial flow. For the simulations of steady state pressure-flow relations the original eight layer model of Bruinsma et al. (1988) has been used.

In the discussion, an attempt will be made to generalize the conclusions of this specific model study.

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2.2 NON-LINEAR MODEL FOR THE VASODILATED BED

In the non-linear compartmental model as presented by Bruinsma et al. (1988) the vasculature is represented by eight layers frora endocardium to epicardium. Each layer consists of three compartraents, the arteriolar, capillary and venular compartment, in series. An electric analog of one compartraent is given in Fig. 2.IA. Each compartment consists of a transmural pressure dependent

ONE COMPARTMENT

B

2

r

contf

y 2' COC

Pim

com

0 5 0 1 0 0 1 5 0 Ptr ( m m H g )

Fig. 2.1 Behavior of a vessel compartment.

Panel A: Electric analog of a vessel compartment. Resistance and compliance depend on volume

(indicated by arrows), which in turn depends on transmural pressure.

Panel B: Transmural pressure-volume relations for the arteriolar (a) , capillary (c) and venular (v) compartment as used in the non-linear compartmental model of Bruinsma et al. (1988). The filled circles refer to the transmural pressure and the volume values of the three compartments

in the reference condition.

- a

compliance in between two transmural pressure dependent resistances. The blood volume depends on the transmural pressure according to a non-linear transmural pressure-volume relation. The transmural pressure-volume curves are assumed to be sigmoid, having a final volume at transmural pressure zero. In Fig. 2.1B the three curves are depicted for the arteriolar, capillary and venular compartment, indicated by a,c and v, respectively.

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For every compartment the resistance, rcom(t) , is defined according to the law of Poiseuille for tubes with constant length:

Kcom

r~ < * > ' v (t)2 ' ( 2 A )

where v (t) is the volume of the compartment and Kcom is a constant specific for each compartment. The transmural pressure, pt r com(t) , is the difference between the pressure in the vessel (intravascular), pc o m(t), and the pressure outside the vessel (extravascular), Pjm(t):

Ptr.eo.W " Pean^) " P f O O • <2'2>

The index "im" stands for intramyocardial and refers to the assumption that the extravascular pressure equals the intramyocardial tissue pressure. The compliance, c (t), is defined by:

d

W

fc

>

< w t > = - i r — ( ^ ) • <

2

-

3

>

^tr.conp '

The compliance is equal to the inverse of the slope of the transmural pressure -volume relation of the compartment. The distensibility of a compartment is then given by:

c (t) dv (t)

com* ' cour ' (2.4)

The volume of a compartment is related to the inflow, q,n(t) , and outflow, qout(t) , of the compartment according to:

— S " <!,-„(*>-<W(t). (2-5)

The intramyocardial pressure, pjm(t) , is assumed to decrease linearly from the left ventricular pressure at the endocardial layer to zero at the epicardial layer (Heineman et al. 1985). When tissue pressure rises in systole, blood is squeezed out of the compartment. The volume change of a compartment is related to the change in transmural pressure by the transmural pressure-volume relation (Fig. 2.1B). Due to alterations of the volume, the resistance is changed (Eq. (2.1)). For a more detailed justification and discussion on the transmural pressure-volume relation and initlal values used in the reference conditions see Bruinsma et al. (1988).

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\

For the model calculations below, the left ventricular systolic pressure is siraulated by a fourth-order polynomial resulting in a reasonable shape of a left ventricular pressure signal.

In determining the influences of the parameters of the different compartments on the phasic arterial flow signal, in Section 2.3 only one layer will be used. The changes in phasic arterial flow signals will then only be due to changes in compliances and resistances and not to differences in endocardial and epicardial flow. In order to predict the phasic arterial flow signal for the autoregulated bed, the arteriolar compartment is changed. The arteriolar resistance in the autoregulated bed is higher in comparison with the vasodilated bed. In the model used in this section, the arteriolar resistance in the reference conditlon has been increased by using a higher value for Kcom in Eq. (2.1) for the arteriolar compartment. By also using a higher perfusion pressure the phasic arterial flow signal for the autoregulated bed can be predicted, although, only at fixed vasomotor tone.

With the model formulated above, coronary arterial flow signals can be predicted (Bruinsma et al. 1988). These signals are obviously determined by the choice of the respective pressure-volume relations. Quantification of the effect of a change in pressure-volume relations on these signals is hampered by their complex shape. Moreover, it would be preferable to relate the effects on the coronary flow signals to a change in compliance and resistance rather than to alterations in the pressure-volume curve since the former quantities have direct hemodynamic effects, while the pressure-volume curve has only an indirect impact. Simplification of the analysis by linearization of the pressure-volume curves around their working point in the reference condition has been attempted. These linearized pressure-volume relations can be altered in two ways: 1) by rotating them around the reference point, or 2) shift them in parallel. This latter alteration affects the reference condition of volume. However, the reference distribution of resistance is not affected by this. The first alteration yields a change primarily in compliance, whereas the latter results primarily in a change in the sensitivity of resistance to volume change. This sensitivity of resistance on volume follows from Eq. (2.1) by differentiation yielding:

drcon(t) rccm

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Eq. (2.6) explains that when the reference volume is decreased but the reference resistance is kept constant, the sensitivity of resistance to volume is increased. The reference resistance can be kept constant by alterations of the constant Km_{com ^} in Eq. (2.1)

N '

2.3 PHASIC FLOW SIGNALS

The effect on the arterial flow signal of compliances and resistances which are dependent on transmural pressure is illustrated in Fig. 2.2. All three signals were calculated by the model described above with one layer. The signal in the top panel was obtained by keeping resistances and compliances at their reference values. Hence, the model was reduced to a linear model with

o>

° \ 1

E

2 - i

0 -

1

Constant compl. and resist.

A

C o n s t a n t compliances

1 2 TIME (sec)

Fig. 2.2 Instantaneous arterial inflow and resistive flow as predicted by the model in which the arterial compartment is changed in order to predict arterial flow signals in the autoregulated bed. In these simulations only one layer is used. After three heart beats a long diastole is slmulated.

Panel A: Simulations with constant compliances and resistances in all compartments. The resistive flow is thus constant. In this situation it takes a longer time before a steady state flow is reached.

Panel B: Simulation with constant compliances in the compartments. Panel C: Arterial and resistive flow as predicted by the non-linear model with transmural pressure dependent resistances and compliances.

constant resistances and compliances. The straight horizontal line represents the flow calculated from the ratio between total pressure difference over all three compartments and the sum of the compartmental resistances. This ratio will be referred to as resistive flow from now on. Since the resistances were kept constant, the resistive flow is independent of time in the top panel. The phasic flow signal has the shape to be expected from a linear RC-model. In diastole,

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flow decays to the resistive value which will be reached when the heart is arrested. For the normal beating heart, however, diastole is too short to reach this value and hence, end diastolic flow will stay above mean flow.

In the middle panel the flow signal depicted is obtained when, in the model, the resistances of the compartment are allowed to change according to volume (Eq. (2.1)) but compliance is kept constant. The diastolic transients in flow still exhibit the decay characteristics of an RC-circuit. However, the resistive flow component varies in time. The average resistive flow component for the beating heart is still close to the average of the total flow signal. In diastole, however, the resistive flow component increases because of the decreasing resistance. Hence, with the prolonged diastole, actual flow will decay to a value above the mean flow for the beating heart. Consequently, the time varying resistance suggests that end diastolic flow is close to the steady value in the arrested heart, while, however, the time varying processes are still far from steady state. The compliance on the one hand, and resistance change on the other both affect the arterial flow, but in opposite ways. This apparently lead to a steady state to be reached more rapidly.

The results obtained if the compliance is also allowed to change in the model are depicted in Fig. 2.2C. The f act that compliance is transmural pressure dependent hardly had any additional influence on the arterial flow signal. However, this conclusion may not be generalized but is parameter dependent, as will be discussed below.

2.3.1 Influence of the arteriolar compartment on the phasic arterial flow signal

The influence of the arteriolar compartment on the arterial flow profile is tested by using different linear pressure-volume curves for the arteriolar compartment. Different slopes of the linear pressure-volume curves lead to different arteriolar compliances. A shift of these curve by using different initial volumes results in a different sensitivity of resistance to a volume change. The various linearized transmural pressure-volume relations used are shown in Fig. 2.3 (left hand panel). -The compliance and initial volumes related to these curves are given in the right hand panel. The simulated flow profiles with these transmural volume curves are depicted in Fig. 2.4.

A comparison of the profiles in Fig. 2.4 demonstrates that when the compliance and initial volume are decreased together (from left upper panel to right lower panel) the flow profile is hardly affected. On the other hand when

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ARTERIOLAR

CQMPARTMENT

INITIAL VOLUME

( m l / 1 OOg) 2 - i O O

s

E ^^ Ld D _l O > 1

-o

o

\

O! I

0.01

8 z

< .

- 0.005

* I

O £ ° ^> 0.0025 I 50 Ptr (mm Hg) 1 0 0 1.8

&

-A — A -0.8 0.4 - O Q— - O —

B-Fig. 2.3 Linear transmural pressure-volume relations for the arteriolar compartment. The transmural pressure-volume relations in the left panel are obtained by using different slopes and different initial values in the reference condition. The different values related to the transmural pressure-volume curves are given in the right panel.

volume is decreased and compliance is increased large changes in diastolic flow profile were obtained (lower left panel to upper right panel). In the upper right panel the change in arteriolar resistance is more important than the compliance effect.

The findings in Fig. 2.4 can also be described in terms of arteriolar distensibility. The distensibility is the ratio between compliance and volume (Eq. (2.4)). In Fig. 2.4 this ratio shows virtually no changes from left upper panel to right lower panel. The changes in distensibility from left lower panel to right upper panel are large, leading to considerable changes in diastolic flow profiles.

2.3.2 Influence of the capillary and venular compartments on the arterial flow profile

The influence of the capillary and venular compartments on the phasic flow signal will be assessed in this section. The influences of the capillary and venular distensibility on the phasic arteriolar flow signal has been calculated

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o

° 2

\

^

O-Ï

2

w

o-5

2-O

e!

o-Va=1.8 m l / 1 0 0 g 0.8 m l / 1 OOg

RAAT

IAAT

TAAA

RAAT

hAAT

ihht

0 . 4 m l / 1 OOg Ca (ml/mmHg/100g)

?vuir

3 0 1 2 3 0

TIME (sec)

0.01 0.005 0.0025

Fig. 2.4 Instantaneous arterial inflow predicted by a model with different linear arteriolar transmural pressure-volume relations. Left.middle and right panels correspond with arteriolar initial volume of 1.8, 0.8 and 0.4 ml/lOOg, respectively; whereas upper, middle and lower panel correspond with an arteriolar compliance of 0.01, 0.005 and 0.0025 ml/mm Hg/lOOg. After three heart beats a long diastole was simulated.

by using different compliances and initial volumes for the capillary and venular compartment.

The capillary compliance has been changed from 0.002 to 1.0 ml/mm Hg/lOOg at a capillary volume of 4.0 ml/lOOg. The venular compliance has been modified from 0.01 to 2.0 ml/mm Hg/lOOg at a venular volume of 3.6 ml/lOOg. The capillary and the venular volume in the reference condition have been changed from 1.0 to 10 ml/lOOg; the capillary and venular compliance were 0.08 and 0.15 ml/mm Hg/lOOg, respectively.

Notwithstanding that these parameters varied over more than one order of magnitude, the changes above hardly influence the arterial diastolic flow profile. Larger compliance and lower volume values both result in a slight decrease in systolic flow, leading to a slight reduction of the mean flow. With larger compliances, it takes a little longer time to obtain steady inflow after the onset of a long diastole. Compared with the effects of the arteriolar compartment on the arterial flow profile, the influences of the capillary and venular compartments on the arterial flow profile are only very small.

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2.4 STEADY STATE PRESSURE-FLOW CURVES.

In this section the protocol of Hanley et al. (1984) and the protocol of Downey and Kirk (1975) will be used to analyze the influence of the different compartments on the steady state pressure-flow curves. Hanley et al. (1984) performed experiments in the non-beating heart, whereas Downey and Kirk (1975) compared results from beating and arrested hearts. Both protocols were performed during vasodilation. Therefore, the original eight layer model for the vasodilated bed of Section 2.2 has been used in this section. These protocols will be simulated to investigate the influence of the different compartments of the non-linear compartmental model on the pressure-flow relations. Moreover, the parameters can be optimized to fit the model better to the experimental results. Because of the large variation in volumes and transmural pressures needed to simulate these protocols, the sigmoid pressure-volume relations are used.

2.4.1 Protocol of Hanley et al. (1984)

The first measurement of pressure-flow relations that will be used as a test for the non-linear compartmental model is the one obtained by Hanley et al. (1984). These authors measured the pressure dependency of coronary resistance in the non-beating heart. The influence ofthree levels of arterial-venous pressure dlfferences on coronary blood flow was measured as a function of the perfusion pressure. The experimental results and the simulation results with the non-linear compartmental model (heavy lines) are given in Fig. 2.5. The experimental results are scaled in such a way that, at a perfusion pressure of 50 mm Hg, with a pressure difference of 20 mm Hg, the flow value coincides with the non-linear compartmental model results. The dependency of coronary blood flow on perfusion and venous pressure appeared to be higher than predicted by the model.

Simulations with different pressure-volume curves in the arteriolar, capillary and venular compartment show that the dependency of flow on perfusion pressure at fixed arterial-venous pressure differences in the non-beating heart mainly depends on the arteriolar compartment. Therefore, only the influence of the arteriolar compartment will be given.

A change in the distensibility of the arteriolar compartment changes the dependency of flow on the perfusion pressure at a constant arterial-venous

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- A P-30 mmHg i- A P = 20 mmHg |- A P- 10 mmHg J

100 (mmHg)

Fig. 2.5 Experimental results (symbols) and model simulations (lines) of diastolic coronary flow as a function of perfusion pressure, at fixed arterial-venous pressure difference, AP. The results of one experiment of Hanley et al. (1984), are represented by squares, triangles and filled circles for AP-30, 20 and 10 mm Hg, respectively. All experimental results are scaled in such a way that experimental results and model results coincide for Pp=50 mm Hg and AP=20 mm Hg. The solid lines represent the results of the non-linear compartmental model. The dotted lines are model simulations in which the dash-dotted curve of Fig. 2.6 is used as pressure-volume relation in the arteriolar compartment. The simulations are scaled in the same way as the experimental results.

pressure difference. As may expected, an increase in the distensibility resulted in a larger dependency of coronary blood flow on the level of perfusion and venous pressure. To illustrate this effect, the distensibility was increased by changing the initial arteriolar volume in the reference condition from 1.8 ml/lOOg to 1.0 ml/lOOg. The corresponding transmural pressure-volume curve is depicted by the dash-dotted curve in Fig. 2.6. The slmulation results obtained with this altered pressure-volume curve are given in Fig. 2.5. As is clear from Fig. 2.5, the results of Hanley et al. (1984) are reasonably described by the alternative curve. It should be noted, however, that the results achieved by Hanley et al (1984), depicted in Fig. 2.5, are not typical of all his data. It is the result with the most outspoken pressure dependency of his series.

O) O O co

Ë

u.

m

o

5 4 J

3

24

1

0 5 0

Perfusion pressure

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Fig. 2.6 Transmural pressure-volume relations. The solid lines are the sarae relation as depicted in Fig. 2.1B. The dash-dotted curve (a') is an alternative transmural pressure-volume relation for the arteriolar (see Fig. 2.5 and 2.7). The broken curve (v') is an alternative pressure-volume relation for the venular compartment.

The second protocol used to test the sensitivity of the arterial flow to the model parameters is presented by Downey and Kirk (1975) . They measured pressure-flow relations in the beating heart and at cardiac arrest. Furthermore, they showed a parallel shift in pressure-flow relation from the beating heart (low flow values) to the arrested heart (high flow values). Model simulation with the non-linear compartmental model could satisfactory explain the shift in transmural pressure-flow curve found by Downey and Kirk (Fig. 2.7). To allow the comparison of shapes of the theoretical and experimental flow curves, the flow data of Downey and kirk (1975) are scaled at a perfusion pressure of 130 mm Hg in the arrested heart.

The influence of the distensibility of the different compartments on the pressure-flow relations was simulated. The simulations show that again the capillary compartment has hardly any influence on the results. Increasing the distensibility of the arteriolar compartment by e.g. using the dash-dotted curve of Fig. 2.6 results in transmural pressure-flow curves, which at higher perfusion pressure, are not straight enough. The shift, however, was barely affected by the change in arteriolar distensibility. The distensibility of the venular compartment was also changed. As an example the dashed curve of Fig. 2.6 was used for the venular compartment. For the low transmural pressure values as will exist in the venules in the beating heart the dashed curve represents a stiffer compartment than the original curve. The curvature of the simulated pressure-flow relation was hardly changed. The shift in the pressure-flow relation from the beating heart to the arrested heart, however, did depend on

50 100 Ptr ( m m Hg)

150 200

On the dynamics of the coronary circulation

ON THE DYNAMICS OF TMÜ

COEONAEY C1ECULATIOK

<q> «Uu O H

ON THE DYNAMICS OF THE

CORONARY CIRCULATION

ON THE DYNAMICS OF THE

CORONARY CIRCULATION

CONTENTS

Chapter 1

r

u/O'

u

w

<

s

+ "(0

.

ff

HIGH m V 0 2

LOW m V 0 2 '

0 5 0 1 0 0 1 5 0

PERF. PRESSURE ( m m Hg)

#

>

2

rt-f-yf-f-^^

Rv

Pim

Pim

Roep Rq

¥

Rv

H l '

oep~j~

Pim

E

E

\

E

z

'\WWWV

'\ZWWW

Chapter 2

ONE COMPARTMENT

B

2

contf

Pim

com

- a

W

>

< w t > = - i r — ( ^ ) • <

-

>

— S " <!,-„(*>-<W(t). (2-5)

\

° \ 1

E

0 -

A

ARTERIOLAR

CQMPARTMENT

s

-o

o

\

0.01

8

z

< .

* I

&

o

° 2

\

^

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2

_s